# Licensed under a 3-clause BSD style license - see LICENSE.rst # "core.py" is auto-generated by erfa_generator.py from the template # "core.py.templ". Do *not* edit "core.py" directly, instead edit # "core.py.templ" and run erfa_generator.py from the source directory to # update it. """ Python wrappers for the ufunc wrappers of the ERFA library. The key idea is that any function can be called with inputs that are arrays, and the ufuncs will automatically vectorize and call the ERFA functions for each item using broadcasting rules for numpy. So the return values are always numpy arrays of some sort. For ERFA functions that take/return vectors or matrices, the vector/matrix dimension(s) are always the *last* dimension(s). For example, if you want to give ten matrices (i.e., the ERFA input type is double[3][3]), you would pass in a (10, 3, 3) numpy array. If the output of the ERFA function is scalar, you'll get back a length-10 1D array. (Note that the ufuncs take this into account using structured dtypes.) Note that the ufunc part of these functions are implemented in a separate module (compiled as ``ufunc``), derived from the ``ufunc.c`` file. """ import warnings import numpy from . import ufunc __all__ = [ 'ErfaError', 'ErfaWarning', 'cal2jd', 'epb', 'epb2jd', 'epj', 'epj2jd', 'jd2cal', 'jdcalf', 'ab', 'apcg', 'apcg13', 'apci', 'apci13', 'apco', 'apco13', 'apcs', 'apcs13', 'aper', 'aper13', 'apio', 'apio13', 'atcc13', 'atccq', 'atci13', 'atciq', 'atciqn', 'atciqz', 'atco13', 'atic13', 'aticq', 'aticqn', 'atio13', 'atioq', 'atoc13', 'atoi13', 'atoiq', 'ld', 'ldn', 'ldsun', 'pmpx', 'pmsafe', 'pvtob', 'refco', 'epv00', 'moon98', 'plan94', 'fad03', 'fae03', 'faf03', 'faju03', 'fal03', 'falp03', 'fama03', 'fame03', 'fane03', 'faom03', 'fapa03', 'fasa03', 'faur03', 'fave03', 'bi00', 'bp00', 'bp06', 'bpn2xy', 'c2i00a', 'c2i00b', 'c2i06a', 'c2ibpn', 'c2ixy', 'c2ixys', 'c2t00a', 'c2t00b', 'c2t06a', 'c2tcio', 'c2teqx', 'c2tpe', 'c2txy', 'eo06a', 'eors', 'fw2m', 'fw2xy', 'ltp', 'ltpb', 'ltpecl', 'ltpequ', 'num00a', 'num00b', 'num06a', 'numat', 'nut00a', 'nut00b', 'nut06a', 'nut80', 'nutm80', 'obl06', 'obl80', 'p06e', 'pb06', 'pfw06', 'pmat00', 'pmat06', 'pmat76', 'pn00', 'pn00a', 'pn00b', 'pn06', 'pn06a', 'pnm00a', 'pnm00b', 'pnm06a', 'pnm80', 'pom00', 'pr00', 'prec76', 's00', 's00a', 's00b', 's06', 's06a', 'sp00', 'xy06', 'xys00a', 'xys00b', 'xys06a', 'ee00', 'ee00a', 'ee00b', 'ee06a', 'eect00', 'eqeq94', 'era00', 'gmst00', 'gmst06', 'gmst82', 'gst00a', 'gst00b', 'gst06', 'gst06a', 'gst94', 'pvstar', 'starpv', 'fk425', 'fk45z', 'fk524', 'fk52h', 'fk54z', 'fk5hip', 'fk5hz', 'h2fk5', 'hfk5z', 'starpm', 'eceq06', 'ecm06', 'eqec06', 'lteceq', 'ltecm', 'lteqec', 'g2icrs', 'icrs2g', 'eform', 'gc2gd', 'gc2gde', 'gd2gc', 'gd2gce', 'd2dtf', 'dat', 'dtdb', 'dtf2d', 'taitt', 'taiut1', 'taiutc', 'tcbtdb', 'tcgtt', 'tdbtcb', 'tdbtt', 'tttai', 'tttcg', 'tttdb', 'ttut1', 'ut1tai', 'ut1tt', 'ut1utc', 'utctai', 'utcut1', 'ae2hd', 'hd2ae', 'hd2pa', 'tpors', 'tporv', 'tpsts', 'tpstv', 'tpxes', 'tpxev', 'a2af', 'a2tf', 'af2a', 'anp', 'anpm', 'd2tf', 'tf2a', 'tf2d', 'rx', 'ry', 'rz', 'cp', 'cpv', 'cr', 'p2pv', 'pv2p', 'ir', 'zp', 'zpv', 'zr', 'rxr', 'tr', 'rxp', 'rxpv', 'trxp', 'trxpv', 'rm2v', 'rv2m', 'pap', 'pas', 'sepp', 'seps', 'c2s', 'p2s', 'pv2s', 's2c', 's2p', 's2pv', 'pdp', 'pm', 'pmp', 'pn', 'ppp', 'ppsp', 'pvdpv', 'pvm', 'pvmpv', 'pvppv', 'pvu', 'pvup', 'pvxpv', 'pxp', 's2xpv', 'sxp', 'sxpv', 'DPI', 'D2PI', 'DR2D', 'DD2R', 'DR2AS', 'DAS2R', 'DS2R', 'TURNAS', 'DMAS2R', 'DTY', 'DAYSEC', 'DJY', 'DJC', 'DJM', 'DJ00', 'DJM0', 'DJM00', 'DJM77', 'TTMTAI', 'DAU', 'CMPS', 'AULT', 'DC', 'ELG', 'ELB', 'TDB0', 'SRS', 'WGS84', 'GRS80', 'WGS72'] class ErfaError(ValueError): """ A class for errors triggered by ERFA functions (status codes < 0) Note: this class should *not* be referenced by fully-qualified name, because it may move to ERFA in a future version. In a future such move it will still be imported here as an alias, but the true namespace of the class may change. """ class ErfaWarning(UserWarning): """ A class for warnings triggered by ERFA functions (status codes > 0) Note: this class should *not* be referenced by fully-qualified name, because it may move to ERFA in a future version. In a future such move it will still be imported here as an alias, but the true namespace of the class may change. """ # <---------------------------------Error-handling----------------------------> STATUS_CODES = {} # populated below before each function that returns an int # This is a hard-coded list of status codes that need to be remapped, # such as to turn errors into warnings. STATUS_CODES_REMAP = { 'cal2jd': {-3: 3} } def check_errwarn(statcodes, func_name): if not numpy.any(statcodes): return # Remap any errors into warnings in the STATUS_CODES_REMAP dict. if func_name in STATUS_CODES_REMAP: for before, after in STATUS_CODES_REMAP[func_name].items(): statcodes[statcodes == before] = after STATUS_CODES[func_name][after] = STATUS_CODES[func_name][before] if numpy.any(statcodes < 0): # Errors present - only report the errors. if statcodes.shape: statcodes = statcodes[statcodes < 0] errcodes = numpy.unique(statcodes) errcounts = dict([(e, numpy.sum(statcodes == e)) for e in errcodes]) elsemsg = STATUS_CODES[func_name].get('else', None) if elsemsg is None: errmsgs = dict([(e, STATUS_CODES[func_name].get( e, 'Return code ' + str(e))) for e in errcodes]) else: errmsgs = dict([(e, STATUS_CODES[func_name].get( e, elsemsg)) for e in errcodes]) emsg = ', '.join(['{0} of "{1}"'.format(errcounts[e], errmsgs[e]) for e in errcodes]) raise ErfaError('ERFA function "{}" yielded {}' .format(func_name, emsg)) elif numpy.any(statcodes > 0): # Only warnings present. if statcodes.shape: statcodes = statcodes[statcodes > 0] warncodes = numpy.unique(statcodes) warncounts = dict([(w, numpy.sum(statcodes == w)) for w in warncodes]) elsemsg = STATUS_CODES[func_name].get('else', None) if elsemsg is None: warnmsgs = dict([(w, STATUS_CODES[func_name].get( w, 'Return code ' + str(w))) for w in warncodes]) else: warnmsgs = dict([(w, STATUS_CODES[func_name].get( w, elsemsg)) for w in warncodes]) wmsg = ', '.join(['{0} of "{1}"'.format(warncounts[w], warnmsgs[w]) for w in warncodes]) warnings.warn('ERFA function "{}" yielded {}'.format(func_name, wmsg), ErfaWarning) # <------------------------structured dtype conversion------------------------> dt_bytes1 = numpy.dtype('S1') dt_bytes12 = numpy.dtype('S12') # <--------------------------Actual ERFA-wrapping code------------------------> DPI = (3.141592653589793238462643) """Pi""" D2PI = (6.283185307179586476925287) """2Pi""" DR2D = (57.29577951308232087679815) """Radians to degrees""" DD2R = (1.745329251994329576923691e-2) """Degrees to radians""" DR2AS = (206264.8062470963551564734) """Radians to arcseconds""" DAS2R = (4.848136811095359935899141e-6) """Arcseconds to radians""" DS2R = (7.272205216643039903848712e-5) """Seconds of time to radians""" TURNAS = (1296000.0) """Arcseconds in a full circle""" DMAS2R = (DAS2R / 1e3) """Milliarcseconds to radians""" DTY = (365.242198781) """Length of tropical year B1900 (days)""" DAYSEC = (86400.0) """Seconds per day.""" DJY = (365.25) """Days per Julian year""" DJC = (36525.0) """Days per Julian century""" DJM = (365250.0) """Days per Julian millennium""" DJ00 = (2451545.0) """Reference epoch (J2000.0), Julian Date""" DJM0 = (2400000.5) """Julian Date of Modified Julian Date zero""" DJM00 = (51544.5) """Reference epoch (J2000.0), Modified Julian Date""" DJM77 = (43144.0) """1977 Jan 1.0 as MJD""" TTMTAI = (32.184) """TT minus TAI (s)""" DAU = (149597870.7e3) """Astronomical unit (m, IAU 2012)""" CMPS = 299792458.0 """Speed of light (m/s)""" AULT = (DAU/CMPS) """Light time for 1 au (s)""" DC = (DAYSEC/AULT) """Speed of light (au per day)""" ELG = (6.969290134e-10) """L_G = 1 - d(TT)/d(TCG)""" ELB = (1.550519768e-8) """L_B = 1 - d(TDB)/d(TCB), and TDB (s) at TAI 1977/1/1.0""" TDB0 = (-6.55e-5) """L_B = 1 - d(TDB)/d(TCB), and TDB (s) at TAI 1977/1/1.0""" SRS = 1.97412574336e-8 """Schwarzschild radius of the Sun (au) = 2 * 1.32712440041e20 / (2.99792458e8)^2 / 1.49597870700e11""" WGS84 = 1 """Reference ellipsoids""" GRS80 = 2 """Reference ellipsoids""" WGS72 = 3 """Reference ellipsoids""" def cal2jd(iy, im, id): """ Gregorian Calendar to Julian Date. Parameters ---------- iy : int array im : int array id : int array Returns ------- djm0 : double array djm : double array Notes ----- Wraps ERFA function ``eraCal2jd``. The ERFA documentation is:: - - - - - - - - - - e r a C a l 2 j d - - - - - - - - - - Gregorian Calendar to Julian Date. Given: iy,im,id int year, month, day in Gregorian calendar (Note 1) Returned: djm0 double MJD zero-point: always 2400000.5 djm double Modified Julian Date for 0 hrs Returned (function value): int status: 0 = OK -1 = bad year (Note 3: JD not computed) -2 = bad month (JD not computed) -3 = bad day (JD computed) Notes: 1) The algorithm used is valid from -4800 March 1, but this implementation rejects dates before -4799 January 1. 2) The Julian Date is returned in two pieces, in the usual ERFA manner, which is designed to preserve time resolution. The Julian Date is available as a single number by adding djm0 and djm. 3) In early eras the conversion is from the "Proleptic Gregorian Calendar"; no account is taken of the date(s) of adoption of the Gregorian Calendar, nor is the AD/BC numbering convention observed. Reference: Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), Section 12.92 (p604). This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ djm0, djm, c_retval = ufunc.cal2jd(iy, im, id) check_errwarn(c_retval, 'cal2jd') return djm0, djm STATUS_CODES['cal2jd'] = { 0: 'OK', -1: 'bad year (Note 3: JD not computed)', -2: 'bad month (JD not computed)', -3: 'bad day (JD computed)', } def epb(dj1, dj2): """ Julian Date to Besselian Epoch. Parameters ---------- dj1 : double array dj2 : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraEpb``. The ERFA documentation is:: - - - - - - - e r a E p b - - - - - - - Julian Date to Besselian Epoch. Given: dj1,dj2 double Julian Date (see note) Returned (function value): double Besselian Epoch. Note: The Julian Date is supplied in two pieces, in the usual ERFA manner, which is designed to preserve time resolution. The Julian Date is available as a single number by adding dj1 and dj2. The maximum resolution is achieved if dj1 is 2451545.0 (J2000.0). Reference: Lieske, J.H., 1979. Astron.Astrophys., 73, 282. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.epb(dj1, dj2) return c_retval def epb2jd(epb): """ Besselian Epoch to Julian Date. Parameters ---------- epb : double array Returns ------- djm0 : double array djm : double array Notes ----- Wraps ERFA function ``eraEpb2jd``. The ERFA documentation is:: - - - - - - - - - - e r a E p b 2 j d - - - - - - - - - - Besselian Epoch to Julian Date. Given: epb double Besselian Epoch (e.g. 1957.3) Returned: djm0 double MJD zero-point: always 2400000.5 djm double Modified Julian Date Note: The Julian Date is returned in two pieces, in the usual ERFA manner, which is designed to preserve time resolution. The Julian Date is available as a single number by adding djm0 and djm. Reference: Lieske, J.H., 1979, Astron.Astrophys. 73, 282. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ djm0, djm = ufunc.epb2jd(epb) return djm0, djm def epj(dj1, dj2): """ Julian Date to Julian Epoch. Parameters ---------- dj1 : double array dj2 : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraEpj``. The ERFA documentation is:: - - - - - - - e r a E p j - - - - - - - Julian Date to Julian Epoch. Given: dj1,dj2 double Julian Date (see note) Returned (function value): double Julian Epoch Note: The Julian Date is supplied in two pieces, in the usual ERFA manner, which is designed to preserve time resolution. The Julian Date is available as a single number by adding dj1 and dj2. The maximum resolution is achieved if dj1 is 2451545.0 (J2000.0). Reference: Lieske, J.H., 1979, Astron.Astrophys. 73, 282. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.epj(dj1, dj2) return c_retval def epj2jd(epj): """ Julian Epoch to Julian Date. Parameters ---------- epj : double array Returns ------- djm0 : double array djm : double array Notes ----- Wraps ERFA function ``eraEpj2jd``. The ERFA documentation is:: - - - - - - - - - - e r a E p j 2 j d - - - - - - - - - - Julian Epoch to Julian Date. Given: epj double Julian Epoch (e.g. 1996.8) Returned: djm0 double MJD zero-point: always 2400000.5 djm double Modified Julian Date Note: The Julian Date is returned in two pieces, in the usual ERFA manner, which is designed to preserve time resolution. The Julian Date is available as a single number by adding djm0 and djm. Reference: Lieske, J.H., 1979, Astron.Astrophys. 73, 282. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ djm0, djm = ufunc.epj2jd(epj) return djm0, djm def jd2cal(dj1, dj2): """ Julian Date to Gregorian year, month, day, and fraction of a day. Parameters ---------- dj1 : double array dj2 : double array Returns ------- iy : int array im : int array id : int array fd : double array Notes ----- Wraps ERFA function ``eraJd2cal``. The ERFA documentation is:: - - - - - - - - - - e r a J d 2 c a l - - - - - - - - - - Julian Date to Gregorian year, month, day, and fraction of a day. Given: dj1,dj2 double Julian Date (Notes 1, 2) Returned (arguments): iy int year im int month id int day fd double fraction of day Returned (function value): int status: 0 = OK -1 = unacceptable date (Note 1) Notes: 1) The earliest valid date is -68569.5 (-4900 March 1). The largest value accepted is 1e9. 2) The Julian Date is apportioned in any convenient way between the arguments dj1 and dj2. For example, JD=2450123.7 could be expressed in any of these ways, among others: dj1 dj2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) Separating integer and fraction uses the "compensated summation" algorithm of Kahan-Neumaier to preserve as much precision as possible irrespective of the jd1+jd2 apportionment. 3) In early eras the conversion is from the "proleptic Gregorian calendar"; no account is taken of the date(s) of adoption of the Gregorian calendar, nor is the AD/BC numbering convention observed. References: Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), Section 12.92 (p604). Klein, A., A Generalized Kahan-Babuska-Summation-Algorithm. Computing, 76, 279-293 (2006), Section 3. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ iy, im, id, fd, c_retval = ufunc.jd2cal(dj1, dj2) check_errwarn(c_retval, 'jd2cal') return iy, im, id, fd STATUS_CODES['jd2cal'] = { 0: 'OK', -1: 'unacceptable date (Note 1)', } def jdcalf(ndp, dj1, dj2): """ Julian Date to Gregorian Calendar, expressed in a form convenient for formatting messages: rounded to a specified precision. Parameters ---------- ndp : int array dj1 : double array dj2 : double array Returns ------- iymdf : int array Notes ----- Wraps ERFA function ``eraJdcalf``. The ERFA documentation is:: - - - - - - - - - - e r a J d c a l f - - - - - - - - - - Julian Date to Gregorian Calendar, expressed in a form convenient for formatting messages: rounded to a specified precision. Given: ndp int number of decimal places of days in fraction dj1,dj2 double dj1+dj2 = Julian Date (Note 1) Returned: iymdf int[4] year, month, day, fraction in Gregorian calendar Returned (function value): int status: -1 = date out of range 0 = OK +1 = NDP not 0-9 (interpreted as 0) Notes: 1) The Julian Date is apportioned in any convenient way between the arguments dj1 and dj2. For example, JD=2450123.7 could be expressed in any of these ways, among others: dj1 dj2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) 2) In early eras the conversion is from the "Proleptic Gregorian Calendar"; no account is taken of the date(s) of adoption of the Gregorian Calendar, nor is the AD/BC numbering convention observed. 3) See also the function eraJd2cal. 4) The number of decimal places ndp should be 4 or less if internal overflows are to be avoided on platforms which use 16-bit integers. Called: eraJd2cal JD to Gregorian calendar Reference: Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), Section 12.92 (p604). This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ iymdf, c_retval = ufunc.jdcalf(ndp, dj1, dj2) check_errwarn(c_retval, 'jdcalf') return iymdf STATUS_CODES['jdcalf'] = { -1: 'date out of range', 0: 'OK', 1: 'NDP not 0-9 (interpreted as 0)', } def ab(pnat, v, s, bm1): """ Apply aberration to transform natural direction into proper direction. Parameters ---------- pnat : double array v : double array s : double array bm1 : double array Returns ------- ppr : double array Notes ----- Wraps ERFA function ``eraAb``. The ERFA documentation is:: - - - - - - e r a A b - - - - - - Apply aberration to transform natural direction into proper direction. Given: pnat double[3] natural direction to the source (unit vector) v double[3] observer barycentric velocity in units of c s double distance between the Sun and the observer (au) bm1 double sqrt(1-|v|^2): reciprocal of Lorenz factor Returned: ppr double[3] proper direction to source (unit vector) Notes: 1) The algorithm is based on Expr. (7.40) in the Explanatory Supplement (Urban & Seidelmann 2013), but with the following changes: o Rigorous rather than approximate normalization is applied. o The gravitational potential term from Expr. (7) in Klioner (2003) is added, taking into account only the Sun's contribution. This has a maximum effect of about 0.4 microarcsecond. 2) In almost all cases, the maximum accuracy will be limited by the supplied velocity. For example, if the ERFA eraEpv00 function is used, errors of up to 5 microarcseconds could occur. References: Urban, S. & Seidelmann, P. K. (eds), Explanatory Supplement to the Astronomical Almanac, 3rd ed., University Science Books (2013). Klioner, Sergei A., "A practical relativistic model for micro- arcsecond astrometry in space", Astr. J. 125, 1580-1597 (2003). Called: eraPdp scalar product of two p-vectors This revision: 2021 February 24 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ ppr = ufunc.ab(pnat, v, s, bm1) return ppr def apcg(date1, date2, ebpv, ehp): """ For a geocentric observer, prepare star-independent astrometry parameters for transformations between ICRS and GCRS coordinates. Parameters ---------- date1 : double array date2 : double array ebpv : double array ehp : double array Returns ------- astrom : eraASTROM array Notes ----- Wraps ERFA function ``eraApcg``. The ERFA documentation is:: - - - - - - - - e r a A p c g - - - - - - - - For a geocentric observer, prepare star-independent astrometry parameters for transformations between ICRS and GCRS coordinates. The Earth ephemeris is supplied by the caller. The parameters produced by this function are required in the parallax, light deflection and aberration parts of the astrometric transformation chain. Given: date1 double TDB as a 2-part... date2 double ...Julian Date (Note 1) ebpv double[2][3] Earth barycentric pos/vel (au, au/day) ehp double[3] Earth heliocentric position (au) Returned: astrom eraASTROM star-independent astrometry parameters: pmt double PM time interval (SSB, Julian years) eb double[3] SSB to observer (vector, au) eh double[3] Sun to observer (unit vector) em double distance from Sun to observer (au) v double[3] barycentric observer velocity (vector, c) bm1 double sqrt(1-|v|^2): reciprocal of Lorenz factor bpn double[3][3] bias-precession-nutation matrix along double unchanged xpl double unchanged ypl double unchanged sphi double unchanged cphi double unchanged diurab double unchanged eral double unchanged refa double unchanged refb double unchanged Notes: 1) The TDB date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TDB)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. For most applications of this function the choice will not be at all critical. TT can be used instead of TDB without any significant impact on accuracy. 2) All the vectors are with respect to BCRS axes. 3) This is one of several functions that inserts into the astrom structure star-independent parameters needed for the chain of astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed. The various functions support different classes of observer and portions of the transformation chain: functions observer transformation eraApcg eraApcg13 geocentric ICRS <-> GCRS eraApci eraApci13 terrestrial ICRS <-> CIRS eraApco eraApco13 terrestrial ICRS <-> observed eraApcs eraApcs13 space ICRS <-> GCRS eraAper eraAper13 terrestrial update Earth rotation eraApio eraApio13 terrestrial CIRS <-> observed Those with names ending in "13" use contemporary ERFA models to compute the various ephemerides. The others accept ephemerides supplied by the caller. The transformation from ICRS to GCRS covers space motion, parallax, light deflection, and aberration. From GCRS to CIRS comprises frame bias and precession-nutation. From CIRS to observed takes account of Earth rotation, polar motion, diurnal aberration and parallax (unless subsumed into the ICRS <-> GCRS transformation), and atmospheric refraction. 4) The context structure astrom produced by this function is used by eraAtciq* and eraAticq*. Called: eraApcs astrometry parameters, ICRS-GCRS, space observer This revision: 2013 October 9 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ astrom = ufunc.apcg(date1, date2, ebpv, ehp) return astrom def apcg13(date1, date2): """ For a geocentric observer, prepare star-independent astrometry parameters for transformations between ICRS and GCRS coordinates. Parameters ---------- date1 : double array date2 : double array Returns ------- astrom : eraASTROM array Notes ----- Wraps ERFA function ``eraApcg13``. The ERFA documentation is:: - - - - - - - - - - e r a A p c g 1 3 - - - - - - - - - - For a geocentric observer, prepare star-independent astrometry parameters for transformations between ICRS and GCRS coordinates. The caller supplies the date, and ERFA models are used to predict the Earth ephemeris. The parameters produced by this function are required in the parallax, light deflection and aberration parts of the astrometric transformation chain. Given: date1 double TDB as a 2-part... date2 double ...Julian Date (Note 1) Returned: astrom eraASTROM* star-independent astrometry parameters: pmt double PM time interval (SSB, Julian years) eb double[3] SSB to observer (vector, au) eh double[3] Sun to observer (unit vector) em double distance from Sun to observer (au) v double[3] barycentric observer velocity (vector, c) bm1 double sqrt(1-|v|^2): reciprocal of Lorenz factor bpn double[3][3] bias-precession-nutation matrix along double unchanged xpl double unchanged ypl double unchanged sphi double unchanged cphi double unchanged diurab double unchanged eral double unchanged refa double unchanged refb double unchanged Notes: 1) The TDB date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TDB)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. For most applications of this function the choice will not be at all critical. TT can be used instead of TDB without any significant impact on accuracy. 2) All the vectors are with respect to BCRS axes. 3) In cases where the caller wishes to supply his own Earth ephemeris, the function eraApcg can be used instead of the present function. 4) This is one of several functions that inserts into the astrom structure star-independent parameters needed for the chain of astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed. The various functions support different classes of observer and portions of the transformation chain: functions observer transformation eraApcg eraApcg13 geocentric ICRS <-> GCRS eraApci eraApci13 terrestrial ICRS <-> CIRS eraApco eraApco13 terrestrial ICRS <-> observed eraApcs eraApcs13 space ICRS <-> GCRS eraAper eraAper13 terrestrial update Earth rotation eraApio eraApio13 terrestrial CIRS <-> observed Those with names ending in "13" use contemporary ERFA models to compute the various ephemerides. The others accept ephemerides supplied by the caller. The transformation from ICRS to GCRS covers space motion, parallax, light deflection, and aberration. From GCRS to CIRS comprises frame bias and precession-nutation. From CIRS to observed takes account of Earth rotation, polar motion, diurnal aberration and parallax (unless subsumed into the ICRS <-> GCRS transformation), and atmospheric refraction. 5) The context structure astrom produced by this function is used by eraAtciq* and eraAticq*. Called: eraEpv00 Earth position and velocity eraApcg astrometry parameters, ICRS-GCRS, geocenter This revision: 2013 October 9 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ astrom = ufunc.apcg13(date1, date2) return astrom def apci(date1, date2, ebpv, ehp, x, y, s): """ For a terrestrial observer, prepare star-independent astrometry parameters for transformations between ICRS and geocentric CIRS coordinates. Parameters ---------- date1 : double array date2 : double array ebpv : double array ehp : double array x : double array y : double array s : double array Returns ------- astrom : eraASTROM array Notes ----- Wraps ERFA function ``eraApci``. The ERFA documentation is:: - - - - - - - - e r a A p c i - - - - - - - - For a terrestrial observer, prepare star-independent astrometry parameters for transformations between ICRS and geocentric CIRS coordinates. The Earth ephemeris and CIP/CIO are supplied by the caller. The parameters produced by this function are required in the parallax, light deflection, aberration, and bias-precession-nutation parts of the astrometric transformation chain. Given: date1 double TDB as a 2-part... date2 double ...Julian Date (Note 1) ebpv double[2][3] Earth barycentric position/velocity (au, au/day) ehp double[3] Earth heliocentric position (au) x,y double CIP X,Y (components of unit vector) s double the CIO locator s (radians) Returned: astrom eraASTROM star-independent astrometry parameters: pmt double PM time interval (SSB, Julian years) eb double[3] SSB to observer (vector, au) eh double[3] Sun to observer (unit vector) em double distance from Sun to observer (au) v double[3] barycentric observer velocity (vector, c) bm1 double sqrt(1-|v|^2): reciprocal of Lorenz factor bpn double[3][3] bias-precession-nutation matrix along double unchanged xpl double unchanged ypl double unchanged sphi double unchanged cphi double unchanged diurab double unchanged eral double unchanged refa double unchanged refb double unchanged Notes: 1) The TDB date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TDB)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. For most applications of this function the choice will not be at all critical. TT can be used instead of TDB without any significant impact on accuracy. 2) All the vectors are with respect to BCRS axes. 3) In cases where the caller does not wish to provide the Earth ephemeris and CIP/CIO, the function eraApci13 can be used instead of the present function. This computes the required quantities using other ERFA functions. 4) This is one of several functions that inserts into the astrom structure star-independent parameters needed for the chain of astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed. The various functions support different classes of observer and portions of the transformation chain: functions observer transformation eraApcg eraApcg13 geocentric ICRS <-> GCRS eraApci eraApci13 terrestrial ICRS <-> CIRS eraApco eraApco13 terrestrial ICRS <-> observed eraApcs eraApcs13 space ICRS <-> GCRS eraAper eraAper13 terrestrial update Earth rotation eraApio eraApio13 terrestrial CIRS <-> observed Those with names ending in "13" use contemporary ERFA models to compute the various ephemerides. The others accept ephemerides supplied by the caller. The transformation from ICRS to GCRS covers space motion, parallax, light deflection, and aberration. From GCRS to CIRS comprises frame bias and precession-nutation. From CIRS to observed takes account of Earth rotation, polar motion, diurnal aberration and parallax (unless subsumed into the ICRS <-> GCRS transformation), and atmospheric refraction. 5) The context structure astrom produced by this function is used by eraAtciq* and eraAticq*. Called: eraApcg astrometry parameters, ICRS-GCRS, geocenter eraC2ixys celestial-to-intermediate matrix, given X,Y and s This revision: 2013 September 25 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ astrom = ufunc.apci(date1, date2, ebpv, ehp, x, y, s) return astrom def apci13(date1, date2): """ For a terrestrial observer, prepare star-independent astrometry parameters for transformations between ICRS and geocentric CIRS coordinates. Parameters ---------- date1 : double array date2 : double array Returns ------- astrom : eraASTROM array eo : double array Notes ----- Wraps ERFA function ``eraApci13``. The ERFA documentation is:: - - - - - - - - - - e r a A p c i 1 3 - - - - - - - - - - For a terrestrial observer, prepare star-independent astrometry parameters for transformations between ICRS and geocentric CIRS coordinates. The caller supplies the date, and ERFA models are used to predict the Earth ephemeris and CIP/CIO. The parameters produced by this function are required in the parallax, light deflection, aberration, and bias-precession-nutation parts of the astrometric transformation chain. Given: date1 double TDB as a 2-part... date2 double ...Julian Date (Note 1) Returned: astrom eraASTROM star-independent astrometry parameters: pmt double PM time interval (SSB, Julian years) eb double[3] SSB to observer (vector, au) eh double[3] Sun to observer (unit vector) em double distance from Sun to observer (au) v double[3] barycentric observer velocity (vector, c) bm1 double sqrt(1-|v|^2): reciprocal of Lorenz factor bpn double[3][3] bias-precession-nutation matrix along double unchanged xpl double unchanged ypl double unchanged sphi double unchanged cphi double unchanged diurab double unchanged eral double unchanged refa double unchanged refb double unchanged eo double equation of the origins (ERA-GST) Notes: 1) The TDB date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TDB)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. For most applications of this function the choice will not be at all critical. TT can be used instead of TDB without any significant impact on accuracy. 2) All the vectors are with respect to BCRS axes. 3) In cases where the caller wishes to supply his own Earth ephemeris and CIP/CIO, the function eraApci can be used instead of the present function. 4) This is one of several functions that inserts into the astrom structure star-independent parameters needed for the chain of astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed. The various functions support different classes of observer and portions of the transformation chain: functions observer transformation eraApcg eraApcg13 geocentric ICRS <-> GCRS eraApci eraApci13 terrestrial ICRS <-> CIRS eraApco eraApco13 terrestrial ICRS <-> observed eraApcs eraApcs13 space ICRS <-> GCRS eraAper eraAper13 terrestrial update Earth rotation eraApio eraApio13 terrestrial CIRS <-> observed Those with names ending in "13" use contemporary ERFA models to compute the various ephemerides. The others accept ephemerides supplied by the caller. The transformation from ICRS to GCRS covers space motion, parallax, light deflection, and aberration. From GCRS to CIRS comprises frame bias and precession-nutation. From CIRS to observed takes account of Earth rotation, polar motion, diurnal aberration and parallax (unless subsumed into the ICRS <-> GCRS transformation), and atmospheric refraction. 5) The context structure astrom produced by this function is used by eraAtciq* and eraAticq*. Called: eraEpv00 Earth position and velocity eraPnm06a classical NPB matrix, IAU 2006/2000A eraBpn2xy extract CIP X,Y coordinates from NPB matrix eraS06 the CIO locator s, given X,Y, IAU 2006 eraApci astrometry parameters, ICRS-CIRS eraEors equation of the origins, given NPB matrix and s This revision: 2013 October 9 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ astrom, eo = ufunc.apci13(date1, date2) return astrom, eo def apco(date1, date2, ebpv, ehp, x, y, s, theta, elong, phi, hm, xp, yp, sp, refa, refb): """ For a terrestrial observer, prepare star-independent astrometry parameters for transformations between ICRS and observed coordinates. Parameters ---------- date1 : double array date2 : double array ebpv : double array ehp : double array x : double array y : double array s : double array theta : double array elong : double array phi : double array hm : double array xp : double array yp : double array sp : double array refa : double array refb : double array Returns ------- astrom : eraASTROM array Notes ----- Wraps ERFA function ``eraApco``. The ERFA documentation is:: - - - - - - - - e r a A p c o - - - - - - - - For a terrestrial observer, prepare star-independent astrometry parameters for transformations between ICRS and observed coordinates. The caller supplies the Earth ephemeris, the Earth rotation information and the refraction constants as well as the site coordinates. Given: date1 double TDB as a 2-part... date2 double ...Julian Date (Note 1) ebpv double[2][3] Earth barycentric PV (au, au/day, Note 2) ehp double[3] Earth heliocentric P (au, Note 2) x,y double CIP X,Y (components of unit vector) s double the CIO locator s (radians) theta double Earth rotation angle (radians) elong double longitude (radians, east +ve, Note 3) phi double latitude (geodetic, radians, Note 3) hm double height above ellipsoid (m, geodetic, Note 3) xp,yp double polar motion coordinates (radians, Note 4) sp double the TIO locator s' (radians, Note 4) refa double refraction constant A (radians, Note 5) refb double refraction constant B (radians, Note 5) Returned: astrom eraASTROM star-independent astrometry parameters: pmt double PM time interval (SSB, Julian years) eb double[3] SSB to observer (vector, au) eh double[3] Sun to observer (unit vector) em double distance from Sun to observer (au) v double[3] barycentric observer velocity (vector, c) bm1 double sqrt(1-|v|^2): reciprocal of Lorenz factor bpn double[3][3] bias-precession-nutation matrix along double adjusted longitude (radians) xpl double polar motion xp wrt local meridian (radians) ypl double polar motion yp wrt local meridian (radians) sphi double sine of geodetic latitude cphi double cosine of geodetic latitude diurab double magnitude of diurnal aberration vector eral double "local" Earth rotation angle (radians) refa double refraction constant A (radians) refb double refraction constant B (radians) Notes: 1) The TDB date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TDB)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. For most applications of this function the choice will not be at all critical. TT can be used instead of TDB without any significant impact on accuracy. 2) The vectors eb, eh, and all the astrom vectors, are with respect to BCRS axes. 3) The geographical coordinates are with respect to the ERFA_WGS84 reference ellipsoid. TAKE CARE WITH THE LONGITUDE SIGN CONVENTION: the longitude required by the present function is right-handed, i.e. east-positive, in accordance with geographical convention. The adjusted longitude stored in the astrom array takes into account the TIO locator and polar motion. 4) xp and yp are the coordinates (in radians) of the Celestial Intermediate Pole with respect to the International Terrestrial Reference System (see IERS Conventions), measured along the meridians 0 and 90 deg west respectively. sp is the TIO locator s', in radians, which positions the Terrestrial Intermediate Origin on the equator. For many applications, xp, yp and (especially) sp can be set to zero. Internally, the polar motion is stored in a form rotated onto the local meridian. 5) The refraction constants refa and refb are for use in a dZ = A*tan(Z)+B*tan^3(Z) model, where Z is the observed (i.e. refracted) zenith distance and dZ is the amount of refraction. 6) It is advisable to take great care with units, as even unlikely values of the input parameters are accepted and processed in accordance with the models used. 7) In cases where the caller does not wish to provide the Earth Ephemeris, the Earth rotation information and refraction constants, the function eraApco13 can be used instead of the present function. This starts from UTC and weather readings etc. and computes suitable values using other ERFA functions. 8) This is one of several functions that inserts into the astrom structure star-independent parameters needed for the chain of astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed. The various functions support different classes of observer and portions of the transformation chain: functions observer transformation eraApcg eraApcg13 geocentric ICRS <-> GCRS eraApci eraApci13 terrestrial ICRS <-> CIRS eraApco eraApco13 terrestrial ICRS <-> observed eraApcs eraApcs13 space ICRS <-> GCRS eraAper eraAper13 terrestrial update Earth rotation eraApio eraApio13 terrestrial CIRS <-> observed Those with names ending in "13" use contemporary ERFA models to compute the various ephemerides. The others accept ephemerides supplied by the caller. The transformation from ICRS to GCRS covers space motion, parallax, light deflection, and aberration. From GCRS to CIRS comprises frame bias and precession-nutation. From CIRS to observed takes account of Earth rotation, polar motion, diurnal aberration and parallax (unless subsumed into the ICRS <-> GCRS transformation), and atmospheric refraction. 9) The context structure astrom produced by this function is used by eraAtioq, eraAtoiq, eraAtciq* and eraAticq*. Called: eraIr initialize r-matrix to identity eraRz rotate around Z-axis eraRy rotate around Y-axis eraRx rotate around X-axis eraAnpm normalize angle into range +/- pi eraC2ixys celestial-to-intermediate matrix, given X,Y and s eraPvtob position/velocity of terrestrial station eraTrxpv product of transpose of r-matrix and pv-vector eraApcs astrometry parameters, ICRS-GCRS, space observer eraCr copy r-matrix This revision: 2021 February 24 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ astrom = ufunc.apco( date1, date2, ebpv, ehp, x, y, s, theta, elong, phi, hm, xp, yp, sp, refa, refb) return astrom def apco13(utc1, utc2, dut1, elong, phi, hm, xp, yp, phpa, tc, rh, wl): """ For a terrestrial observer, prepare star-independent astrometry parameters for transformations between ICRS and observed coordinates. Parameters ---------- utc1 : double array utc2 : double array dut1 : double array elong : double array phi : double array hm : double array xp : double array yp : double array phpa : double array tc : double array rh : double array wl : double array Returns ------- astrom : eraASTROM array eo : double array Notes ----- Wraps ERFA function ``eraApco13``. The ERFA documentation is:: - - - - - - - - - - e r a A p c o 1 3 - - - - - - - - - - For a terrestrial observer, prepare star-independent astrometry parameters for transformations between ICRS and observed coordinates. The caller supplies UTC, site coordinates, ambient air conditions and observing wavelength, and ERFA models are used to obtain the Earth ephemeris, CIP/CIO and refraction constants. The parameters produced by this function are required in the parallax, light deflection, aberration, and bias-precession-nutation parts of the ICRS/CIRS transformations. Given: utc1 double UTC as a 2-part... utc2 double ...quasi Julian Date (Notes 1,2) dut1 double UT1-UTC (seconds, Note 3) elong double longitude (radians, east +ve, Note 4) phi double latitude (geodetic, radians, Note 4) hm double height above ellipsoid (m, geodetic, Notes 4,6) xp,yp double polar motion coordinates (radians, Note 5) phpa double pressure at the observer (hPa = mB, Note 6) tc double ambient temperature at the observer (deg C) rh double relative humidity at the observer (range 0-1) wl double wavelength (micrometers, Note 7) Returned: astrom eraASTROM* star-independent astrometry parameters: pmt double PM time interval (SSB, Julian years) eb double[3] SSB to observer (vector, au) eh double[3] Sun to observer (unit vector) em double distance from Sun to observer (au) v double[3] barycentric observer velocity (vector, c) bm1 double sqrt(1-|v|^2): reciprocal of Lorenz factor bpn double[3][3] bias-precession-nutation matrix along double longitude + s' (radians) xpl double polar motion xp wrt local meridian (radians) ypl double polar motion yp wrt local meridian (radians) sphi double sine of geodetic latitude cphi double cosine of geodetic latitude diurab double magnitude of diurnal aberration vector eral double "local" Earth rotation angle (radians) refa double refraction constant A (radians) refb double refraction constant B (radians) eo double equation of the origins (ERA-GST) Returned (function value): int status: +1 = dubious year (Note 2) 0 = OK -1 = unacceptable date Notes: 1) utc1+utc2 is quasi Julian Date (see Note 2), apportioned in any convenient way between the two arguments, for example where utc1 is the Julian Day Number and utc2 is the fraction of a day. However, JD cannot unambiguously represent UTC during a leap second unless special measures are taken. The convention in the present function is that the JD day represents UTC days whether the length is 86399, 86400 or 86401 SI seconds. Applications should use the function eraDtf2d to convert from calendar date and time of day into 2-part quasi Julian Date, as it implements the leap-second-ambiguity convention just described. 2) The warning status "dubious year" flags UTCs that predate the introduction of the time scale or that are too far in the future to be trusted. See eraDat for further details. 3) UT1-UTC is tabulated in IERS bulletins. It increases by exactly one second at the end of each positive UTC leap second, introduced in order to keep UT1-UTC within +/- 0.9s. n.b. This practice is under review, and in the future UT1-UTC may grow essentially without limit. 4) The geographical coordinates are with respect to the ERFA_WGS84 reference ellipsoid. TAKE CARE WITH THE LONGITUDE SIGN: the longitude required by the present function is east-positive (i.e. right-handed), in accordance with geographical convention. 5) The polar motion xp,yp can be obtained from IERS bulletins. The values are the coordinates (in radians) of the Celestial Intermediate Pole with respect to the International Terrestrial Reference System (see IERS Conventions 2003), measured along the meridians 0 and 90 deg west respectively. For many applications, xp and yp can be set to zero. Internally, the polar motion is stored in a form rotated onto the local meridian. 6) If hm, the height above the ellipsoid of the observing station in meters, is not known but phpa, the pressure in hPa (=mB), is available, an adequate estimate of hm can be obtained from the expression hm = -29.3 * tsl * log ( phpa / 1013.25 ); where tsl is the approximate sea-level air temperature in K (See Astrophysical Quantities, C.W.Allen, 3rd edition, section 52). Similarly, if the pressure phpa is not known, it can be estimated from the height of the observing station, hm, as follows: phpa = 1013.25 * exp ( -hm / ( 29.3 * tsl ) ); Note, however, that the refraction is nearly proportional to the pressure and that an accurate phpa value is important for precise work. 7) The argument wl specifies the observing wavelength in micrometers. The transition from optical to radio is assumed to occur at 100 micrometers (about 3000 GHz). 8) It is advisable to take great care with units, as even unlikely values of the input parameters are accepted and processed in accordance with the models used. 9) In cases where the caller wishes to supply his own Earth ephemeris, Earth rotation information and refraction constants, the function eraApco can be used instead of the present function. 10) This is one of several functions that inserts into the astrom structure star-independent parameters needed for the chain of astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed. The various functions support different classes of observer and portions of the transformation chain: functions observer transformation eraApcg eraApcg13 geocentric ICRS <-> GCRS eraApci eraApci13 terrestrial ICRS <-> CIRS eraApco eraApco13 terrestrial ICRS <-> observed eraApcs eraApcs13 space ICRS <-> GCRS eraAper eraAper13 terrestrial update Earth rotation eraApio eraApio13 terrestrial CIRS <-> observed Those with names ending in "13" use contemporary ERFA models to compute the various ephemerides. The others accept ephemerides supplied by the caller. The transformation from ICRS to GCRS covers space motion, parallax, light deflection, and aberration. From GCRS to CIRS comprises frame bias and precession-nutation. From CIRS to observed takes account of Earth rotation, polar motion, diurnal aberration and parallax (unless subsumed into the ICRS <-> GCRS transformation), and atmospheric refraction. 11) The context structure astrom produced by this function is used by eraAtioq, eraAtoiq, eraAtciq* and eraAticq*. Called: eraUtctai UTC to TAI eraTaitt TAI to TT eraUtcut1 UTC to UT1 eraEpv00 Earth position and velocity eraPnm06a classical NPB matrix, IAU 2006/2000A eraBpn2xy extract CIP X,Y coordinates from NPB matrix eraS06 the CIO locator s, given X,Y, IAU 2006 eraEra00 Earth rotation angle, IAU 2000 eraSp00 the TIO locator s', IERS 2000 eraRefco refraction constants for given ambient conditions eraApco astrometry parameters, ICRS-observed eraEors equation of the origins, given NPB matrix and s This revision: 2021 February 24 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ astrom, eo, c_retval = ufunc.apco13( utc1, utc2, dut1, elong, phi, hm, xp, yp, phpa, tc, rh, wl) check_errwarn(c_retval, 'apco13') return astrom, eo STATUS_CODES['apco13'] = { 1: 'dubious year (Note 2)', 0: 'OK', -1: 'unacceptable date', } def apcs(date1, date2, pv, ebpv, ehp): """ For an observer whose geocentric position and velocity are known, prepare star-independent astrometry parameters for transformations between ICRS and GCRS. Parameters ---------- date1 : double array date2 : double array pv : double array ebpv : double array ehp : double array Returns ------- astrom : eraASTROM array Notes ----- Wraps ERFA function ``eraApcs``. The ERFA documentation is:: - - - - - - - - e r a A p c s - - - - - - - - For an observer whose geocentric position and velocity are known, prepare star-independent astrometry parameters for transformations between ICRS and GCRS. The Earth ephemeris is supplied by the caller. The parameters produced by this function are required in the space motion, parallax, light deflection and aberration parts of the astrometric transformation chain. Given: date1 double TDB as a 2-part... date2 double ...Julian Date (Note 1) pv double[2][3] observer's geocentric pos/vel (m, m/s) ebpv double[2][3] Earth barycentric PV (au, au/day) ehp double[3] Earth heliocentric P (au) Returned: astrom eraASTROM star-independent astrometry parameters: pmt double PM time interval (SSB, Julian years) eb double[3] SSB to observer (vector, au) eh double[3] Sun to observer (unit vector) em double distance from Sun to observer (au) v double[3] barycentric observer velocity (vector, c) bm1 double sqrt(1-|v|^2): reciprocal of Lorenz factor bpn double[3][3] bias-precession-nutation matrix along double unchanged xpl double unchanged ypl double unchanged sphi double unchanged cphi double unchanged diurab double unchanged eral double unchanged refa double unchanged refb double unchanged Notes: 1) The TDB date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TDB)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. For most applications of this function the choice will not be at all critical. TT can be used instead of TDB without any significant impact on accuracy. 2) All the vectors are with respect to BCRS axes. 3) Providing separate arguments for (i) the observer's geocentric position and velocity and (ii) the Earth ephemeris is done for convenience in the geocentric, terrestrial and Earth orbit cases. For deep space applications it maybe more convenient to specify zero geocentric position and velocity and to supply the observer's position and velocity information directly instead of with respect to the Earth. However, note the different units: m and m/s for the geocentric vectors, au and au/day for the heliocentric and barycentric vectors. 4) In cases where the caller does not wish to provide the Earth ephemeris, the function eraApcs13 can be used instead of the present function. This computes the Earth ephemeris using the ERFA function eraEpv00. 5) This is one of several functions that inserts into the astrom structure star-independent parameters needed for the chain of astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed. The various functions support different classes of observer and portions of the transformation chain: functions observer transformation eraApcg eraApcg13 geocentric ICRS <-> GCRS eraApci eraApci13 terrestrial ICRS <-> CIRS eraApco eraApco13 terrestrial ICRS <-> observed eraApcs eraApcs13 space ICRS <-> GCRS eraAper eraAper13 terrestrial update Earth rotation eraApio eraApio13 terrestrial CIRS <-> observed Those with names ending in "13" use contemporary ERFA models to compute the various ephemerides. The others accept ephemerides supplied by the caller. The transformation from ICRS to GCRS covers space motion, parallax, light deflection, and aberration. From GCRS to CIRS comprises frame bias and precession-nutation. From CIRS to observed takes account of Earth rotation, polar motion, diurnal aberration and parallax (unless subsumed into the ICRS <-> GCRS transformation), and atmospheric refraction. 6) The context structure astrom produced by this function is used by eraAtciq* and eraAticq*. Called: eraCp copy p-vector eraPm modulus of p-vector eraPn decompose p-vector into modulus and direction eraIr initialize r-matrix to identity This revision: 2021 February 24 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ astrom = ufunc.apcs(date1, date2, pv, ebpv, ehp) return astrom def apcs13(date1, date2, pv): """ For an observer whose geocentric position and velocity are known, prepare star-independent astrometry parameters for transformations between ICRS and GCRS. Parameters ---------- date1 : double array date2 : double array pv : double array Returns ------- astrom : eraASTROM array Notes ----- Wraps ERFA function ``eraApcs13``. The ERFA documentation is:: - - - - - - - - - - e r a A p c s 1 3 - - - - - - - - - - For an observer whose geocentric position and velocity are known, prepare star-independent astrometry parameters for transformations between ICRS and GCRS. The Earth ephemeris is from ERFA models. The parameters produced by this function are required in the space motion, parallax, light deflection and aberration parts of the astrometric transformation chain. Given: date1 double TDB as a 2-part... date2 double ...Julian Date (Note 1) pv double[2][3] observer's geocentric pos/vel (Note 3) Returned: astrom eraASTROM star-independent astrometry parameters: pmt double PM time interval (SSB, Julian years) eb double[3] SSB to observer (vector, au) eh double[3] Sun to observer (unit vector) em double distance from Sun to observer (au) v double[3] barycentric observer velocity (vector, c) bm1 double sqrt(1-|v|^2): reciprocal of Lorenz factor bpn double[3][3] bias-precession-nutation matrix along double unchanged xpl double unchanged ypl double unchanged sphi double unchanged cphi double unchanged diurab double unchanged eral double unchanged refa double unchanged refb double unchanged Notes: 1) The TDB date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TDB)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. For most applications of this function the choice will not be at all critical. TT can be used instead of TDB without any significant impact on accuracy. 2) All the vectors are with respect to BCRS axes. 3) The observer's position and velocity pv are geocentric but with respect to BCRS axes, and in units of m and m/s. No assumptions are made about proximity to the Earth, and the function can be used for deep space applications as well as Earth orbit and terrestrial. 4) In cases where the caller wishes to supply his own Earth ephemeris, the function eraApcs can be used instead of the present function. 5) This is one of several functions that inserts into the astrom structure star-independent parameters needed for the chain of astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed. The various functions support different classes of observer and portions of the transformation chain: functions observer transformation eraApcg eraApcg13 geocentric ICRS <-> GCRS eraApci eraApci13 terrestrial ICRS <-> CIRS eraApco eraApco13 terrestrial ICRS <-> observed eraApcs eraApcs13 space ICRS <-> GCRS eraAper eraAper13 terrestrial update Earth rotation eraApio eraApio13 terrestrial CIRS <-> observed Those with names ending in "13" use contemporary ERFA models to compute the various ephemerides. The others accept ephemerides supplied by the caller. The transformation from ICRS to GCRS covers space motion, parallax, light deflection, and aberration. From GCRS to CIRS comprises frame bias and precession-nutation. From CIRS to observed takes account of Earth rotation, polar motion, diurnal aberration and parallax (unless subsumed into the ICRS <-> GCRS transformation), and atmospheric refraction. 6) The context structure astrom produced by this function is used by eraAtciq* and eraAticq*. Called: eraEpv00 Earth position and velocity eraApcs astrometry parameters, ICRS-GCRS, space observer This revision: 2013 October 9 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ astrom = ufunc.apcs13(date1, date2, pv) return astrom def aper(theta, astrom): """ In the star-independent astrometry parameters, update only the Earth rotation angle, supplied by the caller explicitly. Parameters ---------- theta : double array astrom : eraASTROM array Returns ------- astrom : eraASTROM array Notes ----- Wraps ERFA function ``eraAper``. Note that, unlike the erfa routine, the python wrapper does not change astrom in-place. The ERFA documentation is:: - - - - - - - - e r a A p e r - - - - - - - - In the star-independent astrometry parameters, update only the Earth rotation angle, supplied by the caller explicitly. Given: theta double Earth rotation angle (radians, Note 2) astrom eraASTROM star-independent astrometry parameters: pmt double not used eb double[3] not used eh double[3] not used em double not used v double[3] not used bm1 double not used bpn double[3][3] not used along double longitude + s' (radians) xpl double not used ypl double not used sphi double not used cphi double not used diurab double not used eral double not used refa double not used refb double not used Returned: astrom eraASTROM star-independent astrometry parameters: pmt double unchanged eb double[3] unchanged eh double[3] unchanged em double unchanged v double[3] unchanged bm1 double unchanged bpn double[3][3] unchanged along double unchanged xpl double unchanged ypl double unchanged sphi double unchanged cphi double unchanged diurab double unchanged eral double "local" Earth rotation angle (radians) refa double unchanged refb double unchanged Notes: 1) This function exists to enable sidereal-tracking applications to avoid wasteful recomputation of the bulk of the astrometry parameters: only the Earth rotation is updated. 2) For targets expressed as equinox based positions, such as classical geocentric apparent (RA,Dec), the supplied theta can be Greenwich apparent sidereal time rather than Earth rotation angle. 3) The function eraAper13 can be used instead of the present function, and starts from UT1 rather than ERA itself. 4) This is one of several functions that inserts into the astrom structure star-independent parameters needed for the chain of astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed. The various functions support different classes of observer and portions of the transformation chain: functions observer transformation eraApcg eraApcg13 geocentric ICRS <-> GCRS eraApci eraApci13 terrestrial ICRS <-> CIRS eraApco eraApco13 terrestrial ICRS <-> observed eraApcs eraApcs13 space ICRS <-> GCRS eraAper eraAper13 terrestrial update Earth rotation eraApio eraApio13 terrestrial CIRS <-> observed Those with names ending in "13" use contemporary ERFA models to compute the various ephemerides. The others accept ephemerides supplied by the caller. The transformation from ICRS to GCRS covers space motion, parallax, light deflection, and aberration. From GCRS to CIRS comprises frame bias and precession-nutation. From CIRS to observed takes account of Earth rotation, polar motion, diurnal aberration and parallax (unless subsumed into the ICRS <-> GCRS transformation), and atmospheric refraction. This revision: 2013 September 25 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ astrom = ufunc.aper(theta, astrom) return astrom def aper13(ut11, ut12, astrom): """ In the star-independent astrometry parameters, update only the Earth rotation angle. Parameters ---------- ut11 : double array ut12 : double array astrom : eraASTROM array Returns ------- astrom : eraASTROM array Notes ----- Wraps ERFA function ``eraAper13``. Note that, unlike the erfa routine, the python wrapper does not change astrom in-place. The ERFA documentation is:: - - - - - - - - - - e r a A p e r 1 3 - - - - - - - - - - In the star-independent astrometry parameters, update only the Earth rotation angle. The caller provides UT1, (n.b. not UTC). Given: ut11 double UT1 as a 2-part... ut12 double ...Julian Date (Note 1) astrom eraASTROM star-independent astrometry parameters: pmt double not used eb double[3] not used eh double[3] not used em double not used v double[3] not used bm1 double not used bpn double[3][3] not used along double longitude + s' (radians) xpl double not used ypl double not used sphi double not used cphi double not used diurab double not used eral double not used refa double not used refb double not used Returned: astrom eraASTROM star-independent astrometry parameters: pmt double unchanged eb double[3] unchanged eh double[3] unchanged em double unchanged v double[3] unchanged bm1 double unchanged bpn double[3][3] unchanged along double unchanged xpl double unchanged ypl double unchanged sphi double unchanged cphi double unchanged diurab double unchanged eral double "local" Earth rotation angle (radians) refa double unchanged refb double unchanged Notes: 1) The UT1 date (n.b. not UTC) ut11+ut12 is a Julian Date, apportioned in any convenient way between the arguments ut11 and ut12. For example, JD(UT1)=2450123.7 could be expressed in any of these ways, among others: ut11 ut12 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 and MJD methods are good compromises between resolution and convenience. The date & time method is best matched to the algorithm used: maximum precision is delivered when the ut11 argument is for 0hrs UT1 on the day in question and the ut12 argument lies in the range 0 to 1, or vice versa. 2) If the caller wishes to provide the Earth rotation angle itself, the function eraAper can be used instead. One use of this technique is to substitute Greenwich apparent sidereal time and thereby to support equinox based transformations directly. 3) This is one of several functions that inserts into the astrom structure star-independent parameters needed for the chain of astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed. The various functions support different classes of observer and portions of the transformation chain: functions observer transformation eraApcg eraApcg13 geocentric ICRS <-> GCRS eraApci eraApci13 terrestrial ICRS <-> CIRS eraApco eraApco13 terrestrial ICRS <-> observed eraApcs eraApcs13 space ICRS <-> GCRS eraAper eraAper13 terrestrial update Earth rotation eraApio eraApio13 terrestrial CIRS <-> observed Those with names ending in "13" use contemporary ERFA models to compute the various ephemerides. The others accept ephemerides supplied by the caller. The transformation from ICRS to GCRS covers space motion, parallax, light deflection, and aberration. From GCRS to CIRS comprises frame bias and precession-nutation. From CIRS to observed takes account of Earth rotation, polar motion, diurnal aberration and parallax (unless subsumed into the ICRS <-> GCRS transformation), and atmospheric refraction. Called: eraAper astrometry parameters: update ERA eraEra00 Earth rotation angle, IAU 2000 This revision: 2013 September 25 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ astrom = ufunc.aper13(ut11, ut12, astrom) return astrom def apio(sp, theta, elong, phi, hm, xp, yp, refa, refb): """ For a terrestrial observer, prepare star-independent astrometry parameters for transformations between CIRS and observed coordinates. Parameters ---------- sp : double array theta : double array elong : double array phi : double array hm : double array xp : double array yp : double array refa : double array refb : double array Returns ------- astrom : eraASTROM array Notes ----- Wraps ERFA function ``eraApio``. The ERFA documentation is:: - - - - - - - - e r a A p i o - - - - - - - - For a terrestrial observer, prepare star-independent astrometry parameters for transformations between CIRS and observed coordinates. The caller supplies the Earth orientation information and the refraction constants as well as the site coordinates. Given: sp double the TIO locator s' (radians, Note 1) theta double Earth rotation angle (radians) elong double longitude (radians, east +ve, Note 2) phi double geodetic latitude (radians, Note 2) hm double height above ellipsoid (m, geodetic Note 2) xp,yp double polar motion coordinates (radians, Note 3) refa double refraction constant A (radians, Note 4) refb double refraction constant B (radians, Note 4) Returned: astrom eraASTROM star-independent astrometry parameters: pmt double unchanged eb double[3] unchanged eh double[3] unchanged em double unchanged v double[3] unchanged bm1 double unchanged bpn double[3][3] unchanged along double adjusted longitude (radians) xpl double polar motion xp wrt local meridian (radians) ypl double polar motion yp wrt local meridian (radians) sphi double sine of geodetic latitude cphi double cosine of geodetic latitude diurab double magnitude of diurnal aberration vector eral double "local" Earth rotation angle (radians) refa double refraction constant A (radians) refb double refraction constant B (radians) Notes: 1) sp, the TIO locator s', is a tiny quantity needed only by the most precise applications. It can either be set to zero or predicted using the ERFA function eraSp00. 2) The geographical coordinates are with respect to the ERFA_WGS84 reference ellipsoid. TAKE CARE WITH THE LONGITUDE SIGN: the longitude required by the present function is east-positive (i.e. right-handed), in accordance with geographical convention. 3) The polar motion xp,yp can be obtained from IERS bulletins. The values are the coordinates (in radians) of the Celestial Intermediate Pole with respect to the International Terrestrial Reference System (see IERS Conventions 2003), measured along the meridians 0 and 90 deg west respectively. For many applications, xp and yp can be set to zero. Internally, the polar motion is stored in a form rotated onto the local meridian. 4) The refraction constants refa and refb are for use in a dZ = A*tan(Z)+B*tan^3(Z) model, where Z is the observed (i.e. refracted) zenith distance and dZ is the amount of refraction. 5) It is advisable to take great care with units, as even unlikely values of the input parameters are accepted and processed in accordance with the models used. 6) In cases where the caller does not wish to provide the Earth rotation information and refraction constants, the function eraApio13 can be used instead of the present function. This starts from UTC and weather readings etc. and computes suitable values using other ERFA functions. 7) This is one of several functions that inserts into the astrom structure star-independent parameters needed for the chain of astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed. The various functions support different classes of observer and portions of the transformation chain: functions observer transformation eraApcg eraApcg13 geocentric ICRS <-> GCRS eraApci eraApci13 terrestrial ICRS <-> CIRS eraApco eraApco13 terrestrial ICRS <-> observed eraApcs eraApcs13 space ICRS <-> GCRS eraAper eraAper13 terrestrial update Earth rotation eraApio eraApio13 terrestrial CIRS <-> observed Those with names ending in "13" use contemporary ERFA models to compute the various ephemerides. The others accept ephemerides supplied by the caller. The transformation from ICRS to GCRS covers space motion, parallax, light deflection, and aberration. From GCRS to CIRS comprises frame bias and precession-nutation. From CIRS to observed takes account of Earth rotation, polar motion, diurnal aberration and parallax (unless subsumed into the ICRS <-> GCRS transformation), and atmospheric refraction. 8) The context structure astrom produced by this function is used by eraAtioq and eraAtoiq. Called: eraIr initialize r-matrix to identity eraRz rotate around Z-axis eraRy rotate around Y-axis eraRx rotate around X-axis eraAnpm normalize angle into range +/- pi eraPvtob position/velocity of terrestrial station This revision: 2021 February 24 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ astrom = ufunc.apio(sp, theta, elong, phi, hm, xp, yp, refa, refb) return astrom def apio13(utc1, utc2, dut1, elong, phi, hm, xp, yp, phpa, tc, rh, wl): """ For a terrestrial observer, prepare star-independent astrometry parameters for transformations between CIRS and observed coordinates. Parameters ---------- utc1 : double array utc2 : double array dut1 : double array elong : double array phi : double array hm : double array xp : double array yp : double array phpa : double array tc : double array rh : double array wl : double array Returns ------- astrom : eraASTROM array Notes ----- Wraps ERFA function ``eraApio13``. The ERFA documentation is:: - - - - - - - - - - e r a A p i o 1 3 - - - - - - - - - - For a terrestrial observer, prepare star-independent astrometry parameters for transformations between CIRS and observed coordinates. The caller supplies UTC, site coordinates, ambient air conditions and observing wavelength. Given: utc1 double UTC as a 2-part... utc2 double ...quasi Julian Date (Notes 1,2) dut1 double UT1-UTC (seconds) elong double longitude (radians, east +ve, Note 3) phi double geodetic latitude (radians, Note 3) hm double height above ellipsoid (m, geodetic Notes 4,6) xp,yp double polar motion coordinates (radians, Note 5) phpa double pressure at the observer (hPa = mB, Note 6) tc double ambient temperature at the observer (deg C) rh double relative humidity at the observer (range 0-1) wl double wavelength (micrometers, Note 7) Returned: astrom eraASTROM star-independent astrometry parameters: pmt double unchanged eb double[3] unchanged eh double[3] unchanged em double unchanged v double[3] unchanged bm1 double unchanged bpn double[3][3] unchanged along double longitude + s' (radians) xpl double polar motion xp wrt local meridian (radians) ypl double polar motion yp wrt local meridian (radians) sphi double sine of geodetic latitude cphi double cosine of geodetic latitude diurab double magnitude of diurnal aberration vector eral double "local" Earth rotation angle (radians) refa double refraction constant A (radians) refb double refraction constant B (radians) Returned (function value): int status: +1 = dubious year (Note 2) 0 = OK -1 = unacceptable date Notes: 1) utc1+utc2 is quasi Julian Date (see Note 2), apportioned in any convenient way between the two arguments, for example where utc1 is the Julian Day Number and utc2 is the fraction of a day. However, JD cannot unambiguously represent UTC during a leap second unless special measures are taken. The convention in the present function is that the JD day represents UTC days whether the length is 86399, 86400 or 86401 SI seconds. Applications should use the function eraDtf2d to convert from calendar date and time of day into 2-part quasi Julian Date, as it implements the leap-second-ambiguity convention just described. 2) The warning status "dubious year" flags UTCs that predate the introduction of the time scale or that are too far in the future to be trusted. See eraDat for further details. 3) UT1-UTC is tabulated in IERS bulletins. It increases by exactly one second at the end of each positive UTC leap second, introduced in order to keep UT1-UTC within +/- 0.9s. n.b. This practice is under review, and in the future UT1-UTC may grow essentially without limit. 4) The geographical coordinates are with respect to the ERFA_WGS84 reference ellipsoid. TAKE CARE WITH THE LONGITUDE SIGN: the longitude required by the present function is east-positive (i.e. right-handed), in accordance with geographical convention. 5) The polar motion xp,yp can be obtained from IERS bulletins. The values are the coordinates (in radians) of the Celestial Intermediate Pole with respect to the International Terrestrial Reference System (see IERS Conventions 2003), measured along the meridians 0 and 90 deg west respectively. For many applications, xp and yp can be set to zero. Internally, the polar motion is stored in a form rotated onto the local meridian. 6) If hm, the height above the ellipsoid of the observing station in meters, is not known but phpa, the pressure in hPa (=mB), is available, an adequate estimate of hm can be obtained from the expression hm = -29.3 * tsl * log ( phpa / 1013.25 ); where tsl is the approximate sea-level air temperature in K (See Astrophysical Quantities, C.W.Allen, 3rd edition, section 52). Similarly, if the pressure phpa is not known, it can be estimated from the height of the observing station, hm, as follows: phpa = 1013.25 * exp ( -hm / ( 29.3 * tsl ) ); Note, however, that the refraction is nearly proportional to the pressure and that an accurate phpa value is important for precise work. 7) The argument wl specifies the observing wavelength in micrometers. The transition from optical to radio is assumed to occur at 100 micrometers (about 3000 GHz). 8) It is advisable to take great care with units, as even unlikely values of the input parameters are accepted and processed in accordance with the models used. 9) In cases where the caller wishes to supply his own Earth rotation information and refraction constants, the function eraApc can be used instead of the present function. 10) This is one of several functions that inserts into the astrom structure star-independent parameters needed for the chain of astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed. The various functions support different classes of observer and portions of the transformation chain: functions observer transformation eraApcg eraApcg13 geocentric ICRS <-> GCRS eraApci eraApci13 terrestrial ICRS <-> CIRS eraApco eraApco13 terrestrial ICRS <-> observed eraApcs eraApcs13 space ICRS <-> GCRS eraAper eraAper13 terrestrial update Earth rotation eraApio eraApio13 terrestrial CIRS <-> observed Those with names ending in "13" use contemporary ERFA models to compute the various ephemerides. The others accept ephemerides supplied by the caller. The transformation from ICRS to GCRS covers space motion, parallax, light deflection, and aberration. From GCRS to CIRS comprises frame bias and precession-nutation. From CIRS to observed takes account of Earth rotation, polar motion, diurnal aberration and parallax (unless subsumed into the ICRS <-> GCRS transformation), and atmospheric refraction. 11) The context structure astrom produced by this function is used by eraAtioq and eraAtoiq. Called: eraUtctai UTC to TAI eraTaitt TAI to TT eraUtcut1 UTC to UT1 eraSp00 the TIO locator s', IERS 2000 eraEra00 Earth rotation angle, IAU 2000 eraRefco refraction constants for given ambient conditions eraApio astrometry parameters, CIRS-observed This revision: 2021 February 24 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ astrom, c_retval = ufunc.apio13( utc1, utc2, dut1, elong, phi, hm, xp, yp, phpa, tc, rh, wl) check_errwarn(c_retval, 'apio13') return astrom STATUS_CODES['apio13'] = { 1: 'dubious year (Note 2)', 0: 'OK', -1: 'unacceptable date', } def atcc13(rc, dc, pr, pd, px, rv, date1, date2): """ Transform a star's ICRS catalog entry (epoch J2000.0) into ICRS astrometric place. Parameters ---------- rc : double array dc : double array pr : double array pd : double array px : double array rv : double array date1 : double array date2 : double array Returns ------- ra : double array da : double array Notes ----- Wraps ERFA function ``eraAtcc13``. The ERFA documentation is:: - - - - - - - - - - e r a A t c c 1 3 - - - - - - - - - - Transform a star's ICRS catalog entry (epoch J2000.0) into ICRS astrometric place. Given: rc double ICRS right ascension at J2000.0 (radians, Note 1) dc double ICRS declination at J2000.0 (radians, Note 1) pr double RA proper motion (radians/year, Note 2) pd double Dec proper motion (radians/year) px double parallax (arcsec) rv double radial velocity (km/s, +ve if receding) date1 double TDB as a 2-part... date2 double ...Julian Date (Note 3) Returned: ra,da double ICRS astrometric RA,Dec (radians) Notes: 1) Star data for an epoch other than J2000.0 (for example from the Hipparcos catalog, which has an epoch of J1991.25) will require a preliminary call to eraPmsafe before use. 2) The proper motion in RA is dRA/dt rather than cos(Dec)*dRA/dt. 3) The TDB date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TDB)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. For most applications of this function the choice will not be at all critical. TT can be used instead of TDB without any significant impact on accuracy. Called: eraApci13 astrometry parameters, ICRS-CIRS, 2013 eraAtccq quick catalog ICRS to astrometric This revision: 2021 April 18 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ ra, da = ufunc.atcc13(rc, dc, pr, pd, px, rv, date1, date2) return ra, da def atccq(rc, dc, pr, pd, px, rv, astrom): """ Quick transformation of a star's ICRS catalog entry (epoch J2000.0) into ICRS astrometric place, given precomputed star-independent astrometry parameters. Parameters ---------- rc : double array dc : double array pr : double array pd : double array px : double array rv : double array astrom : eraASTROM array Returns ------- ra : double array da : double array Notes ----- Wraps ERFA function ``eraAtccq``. The ERFA documentation is:: - - - - - - - - - e r a A t c c q - - - - - - - - - Quick transformation of a star's ICRS catalog entry (epoch J2000.0) into ICRS astrometric place, given precomputed star-independent astrometry parameters. Use of this function is appropriate when efficiency is important and where many star positions are to be transformed for one date. The star-independent parameters can be obtained by calling one of the functions eraApci[13], eraApcg[13], eraApco[13] or eraApcs[13]. If the parallax and proper motions are zero the transformation has no effect. Given: rc,dc double ICRS RA,Dec at J2000.0 (radians) pr double RA proper motion (radians/year, Note 3) pd double Dec proper motion (radians/year) px double parallax (arcsec) rv double radial velocity (km/s, +ve if receding) astrom eraASTROM* star-independent astrometry parameters: pmt double PM time interval (SSB, Julian years) eb double[3] SSB to observer (vector, au) eh double[3] Sun to observer (unit vector) em double distance from Sun to observer (au) v double[3] barycentric observer velocity (vector, c) bm1 double sqrt(1-|v|^2): reciprocal of Lorenz factor bpn double[3][3] bias-precession-nutation matrix along double longitude + s' (radians) xpl double polar motion xp wrt local meridian (radians) ypl double polar motion yp wrt local meridian (radians) sphi double sine of geodetic latitude cphi double cosine of geodetic latitude diurab double magnitude of diurnal aberration vector eral double "local" Earth rotation angle (radians) refa double refraction constant A (radians) refb double refraction constant B (radians) Returned: ra,da double ICRS astrometric RA,Dec (radians) Notes: 1) All the vectors are with respect to BCRS axes. 2) Star data for an epoch other than J2000.0 (for example from the Hipparcos catalog, which has an epoch of J1991.25) will require a preliminary call to eraPmsafe before use. 3) The proper motion in RA is dRA/dt rather than cos(Dec)*dRA/dt. Called: eraPmpx proper motion and parallax eraC2s p-vector to spherical eraAnp normalize angle into range 0 to 2pi This revision: 2021 April 18 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ ra, da = ufunc.atccq(rc, dc, pr, pd, px, rv, astrom) return ra, da def atci13(rc, dc, pr, pd, px, rv, date1, date2): """ Transform ICRS star data, epoch J2000.0, to CIRS. Parameters ---------- rc : double array dc : double array pr : double array pd : double array px : double array rv : double array date1 : double array date2 : double array Returns ------- ri : double array di : double array eo : double array Notes ----- Wraps ERFA function ``eraAtci13``. The ERFA documentation is:: - - - - - - - - - - e r a A t c i 1 3 - - - - - - - - - - Transform ICRS star data, epoch J2000.0, to CIRS. Given: rc double ICRS right ascension at J2000.0 (radians, Note 1) dc double ICRS declination at J2000.0 (radians, Note 1) pr double RA proper motion (radians/year, Note 2) pd double Dec proper motion (radians/year) px double parallax (arcsec) rv double radial velocity (km/s, +ve if receding) date1 double TDB as a 2-part... date2 double ...Julian Date (Note 3) Returned: ri,di double CIRS geocentric RA,Dec (radians) eo double equation of the origins (ERA-GST, Note 5) Notes: 1) Star data for an epoch other than J2000.0 (for example from the Hipparcos catalog, which has an epoch of J1991.25) will require a preliminary call to eraPmsafe before use. 2) The proper motion in RA is dRA/dt rather than cos(Dec)*dRA/dt. 3) The TDB date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TDB)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. For most applications of this function the choice will not be at all critical. TT can be used instead of TDB without any significant impact on accuracy. 4) The available accuracy is better than 1 milliarcsecond, limited mainly by the precession-nutation model that is used, namely IAU 2000A/2006. Very close to solar system bodies, additional errors of up to several milliarcseconds can occur because of unmodeled light deflection; however, the Sun's contribution is taken into account, to first order. The accuracy limitations of the ERFA function eraEpv00 (used to compute Earth position and velocity) can contribute aberration errors of up to 5 microarcseconds. Light deflection at the Sun's limb is uncertain at the 0.4 mas level. 5) Should the transformation to (equinox based) apparent place be required rather than (CIO based) intermediate place, subtract the equation of the origins from the returned right ascension: RA = RI - EO. (The eraAnp function can then be applied, as required, to keep the result in the conventional 0-2pi range.) Called: eraApci13 astrometry parameters, ICRS-CIRS, 2013 eraAtciq quick ICRS to CIRS This revision: 2021 April 3 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ ri, di, eo = ufunc.atci13(rc, dc, pr, pd, px, rv, date1, date2) return ri, di, eo def atciq(rc, dc, pr, pd, px, rv, astrom): """ Quick ICRS, epoch J2000.0, to CIRS transformation, given precomputed star-independent astrometry parameters. Parameters ---------- rc : double array dc : double array pr : double array pd : double array px : double array rv : double array astrom : eraASTROM array Returns ------- ri : double array di : double array Notes ----- Wraps ERFA function ``eraAtciq``. The ERFA documentation is:: - - - - - - - - - e r a A t c i q - - - - - - - - - Quick ICRS, epoch J2000.0, to CIRS transformation, given precomputed star-independent astrometry parameters. Use of this function is appropriate when efficiency is important and where many star positions are to be transformed for one date. The star-independent parameters can be obtained by calling one of the functions eraApci[13], eraApcg[13], eraApco[13] or eraApcs[13]. If the parallax and proper motions are zero the eraAtciqz function can be used instead. Given: rc,dc double ICRS RA,Dec at J2000.0 (radians, Note 1) pr double RA proper motion (radians/year, Note 2) pd double Dec proper motion (radians/year) px double parallax (arcsec) rv double radial velocity (km/s, +ve if receding) astrom eraASTROM* star-independent astrometry parameters: pmt double PM time interval (SSB, Julian years) eb double[3] SSB to observer (vector, au) eh double[3] Sun to observer (unit vector) em double distance from Sun to observer (au) v double[3] barycentric observer velocity (vector, c) bm1 double sqrt(1-|v|^2): reciprocal of Lorenz factor bpn double[3][3] bias-precession-nutation matrix along double longitude + s' (radians) xpl double polar motion xp wrt local meridian (radians) ypl double polar motion yp wrt local meridian (radians) sphi double sine of geodetic latitude cphi double cosine of geodetic latitude diurab double magnitude of diurnal aberration vector eral double "local" Earth rotation angle (radians) refa double refraction constant A (radians) refb double refraction constant B (radians) Returned: ri,di double CIRS RA,Dec (radians) Notes: 1) Star data for an epoch other than J2000.0 (for example from the Hipparcos catalog, which has an epoch of J1991.25) will require a preliminary call to eraPmsafe before use. 2) The proper motion in RA is dRA/dt rather than cos(Dec)*dRA/dt. Called: eraPmpx proper motion and parallax eraLdsun light deflection by the Sun eraAb stellar aberration eraRxp product of r-matrix and pv-vector eraC2s p-vector to spherical eraAnp normalize angle into range 0 to 2pi This revision: 2021 April 19 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ ri, di = ufunc.atciq(rc, dc, pr, pd, px, rv, astrom) return ri, di def atciqn(rc, dc, pr, pd, px, rv, astrom, b): """ Quick ICRS, epoch J2000.0, to CIRS transformation, given precomputed star-independent astrometry parameters plus a list of light- deflecting bodies. Parameters ---------- rc : double array dc : double array pr : double array pd : double array px : double array rv : double array astrom : eraASTROM array b : eraLDBODY array Returns ------- ri : double array di : double array Notes ----- Wraps ERFA function ``eraAtciqn``. The ERFA documentation is:: - - - - - - - - - - e r a A t c i q n - - - - - - - - - - Quick ICRS, epoch J2000.0, to CIRS transformation, given precomputed star-independent astrometry parameters plus a list of light- deflecting bodies. Use of this function is appropriate when efficiency is important and where many star positions are to be transformed for one date. The star-independent parameters can be obtained by calling one of the functions eraApci[13], eraApcg[13], eraApco[13] or eraApcs[13]. If the only light-deflecting body to be taken into account is the Sun, the eraAtciq function can be used instead. If in addition the parallax and proper motions are zero, the eraAtciqz function can be used. Given: rc,dc double ICRS RA,Dec at J2000.0 (radians) pr double RA proper motion (radians/year, Note 3) pd double Dec proper motion (radians/year) px double parallax (arcsec) rv double radial velocity (km/s, +ve if receding) astrom eraASTROM star-independent astrometry parameters: pmt double PM time interval (SSB, Julian years) eb double[3] SSB to observer (vector, au) eh double[3] Sun to observer (unit vector) em double distance from Sun to observer (au) v double[3] barycentric observer velocity (vector, c) bm1 double sqrt(1-|v|^2): reciprocal of Lorenz factor bpn double[3][3] bias-precession-nutation matrix along double longitude + s' (radians) xpl double polar motion xp wrt local meridian (radians) ypl double polar motion yp wrt local meridian (radians) sphi double sine of geodetic latitude cphi double cosine of geodetic latitude diurab double magnitude of diurnal aberration vector eral double "local" Earth rotation angle (radians) refa double refraction constant A (radians) refb double refraction constant B (radians) n int number of bodies (Note 3) b eraLDBODY[n] data for each of the n bodies (Notes 3,4): bm double mass of the body (solar masses, Note 5) dl double deflection limiter (Note 6) pv [2][3] barycentric PV of the body (au, au/day) Returned: ri,di double CIRS RA,Dec (radians) Notes: 1) Star data for an epoch other than J2000.0 (for example from the Hipparcos catalog, which has an epoch of J1991.25) will require a preliminary call to eraPmsafe before use. 2) The proper motion in RA is dRA/dt rather than cos(Dec)*dRA/dt. 3) The struct b contains n entries, one for each body to be considered. If n = 0, no gravitational light deflection will be applied, not even for the Sun. 4) The struct b should include an entry for the Sun as well as for any planet or other body to be taken into account. The entries should be in the order in which the light passes the body. 5) In the entry in the b struct for body i, the mass parameter b[i].bm can, as required, be adjusted in order to allow for such effects as quadrupole field. 6) The deflection limiter parameter b[i].dl is phi^2/2, where phi is the angular separation (in radians) between star and body at which limiting is applied. As phi shrinks below the chosen threshold, the deflection is artificially reduced, reaching zero for phi = 0. Example values suitable for a terrestrial observer, together with masses, are as follows: body i b[i].bm b[i].dl Sun 1.0 6e-6 Jupiter 0.00095435 3e-9 Saturn 0.00028574 3e-10 7) For efficiency, validation of the contents of the b array is omitted. The supplied masses must be greater than zero, the position and velocity vectors must be right, and the deflection limiter greater than zero. Called: eraPmpx proper motion and parallax eraLdn light deflection by n bodies eraAb stellar aberration eraRxp product of r-matrix and pv-vector eraC2s p-vector to spherical eraAnp normalize angle into range 0 to 2pi This revision: 2021 April 3 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ ri, di = ufunc.atciqn(rc, dc, pr, pd, px, rv, astrom, b) return ri, di def atciqz(rc, dc, astrom): """ Quick ICRS to CIRS transformation, given precomputed star- independent astrometry parameters, and assuming zero parallax and proper motion. Parameters ---------- rc : double array dc : double array astrom : eraASTROM array Returns ------- ri : double array di : double array Notes ----- Wraps ERFA function ``eraAtciqz``. The ERFA documentation is:: - - - - - - - - - - e r a A t c i q z - - - - - - - - - - Quick ICRS to CIRS transformation, given precomputed star- independent astrometry parameters, and assuming zero parallax and proper motion. Use of this function is appropriate when efficiency is important and where many star positions are to be transformed for one date. The star-independent parameters can be obtained by calling one of the functions eraApci[13], eraApcg[13], eraApco[13] or eraApcs[13]. The corresponding function for the case of non-zero parallax and proper motion is eraAtciq. Given: rc,dc double ICRS astrometric RA,Dec (radians) astrom eraASTROM* star-independent astrometry parameters: pmt double PM time interval (SSB, Julian years) eb double[3] SSB to observer (vector, au) eh double[3] Sun to observer (unit vector) em double distance from Sun to observer (au) v double[3] barycentric observer velocity (vector, c) bm1 double sqrt(1-|v|^2): reciprocal of Lorenz factor bpn double[3][3] bias-precession-nutation matrix along double longitude + s' (radians) xpl double polar motion xp wrt local meridian (radians) ypl double polar motion yp wrt local meridian (radians) sphi double sine of geodetic latitude cphi double cosine of geodetic latitude diurab double magnitude of diurnal aberration vector eral double "local" Earth rotation angle (radians) refa double refraction constant A (radians) refb double refraction constant B (radians) Returned: ri,di double CIRS RA,Dec (radians) Note: All the vectors are with respect to BCRS axes. References: Urban, S. & Seidelmann, P. K. (eds), Explanatory Supplement to the Astronomical Almanac, 3rd ed., University Science Books (2013). Klioner, Sergei A., "A practical relativistic model for micro- arcsecond astrometry in space", Astr. J. 125, 1580-1597 (2003). Called: eraS2c spherical coordinates to unit vector eraLdsun light deflection due to Sun eraAb stellar aberration eraRxp product of r-matrix and p-vector eraC2s p-vector to spherical eraAnp normalize angle into range +/- pi This revision: 2013 October 9 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ ri, di = ufunc.atciqz(rc, dc, astrom) return ri, di def atco13(rc, dc, pr, pd, px, rv, utc1, utc2, dut1, elong, phi, hm, xp, yp, phpa, tc, rh, wl): """ ICRS RA,Dec to observed place. The caller supplies UTC, site coordinates, ambient air conditions and observing wavelength. Parameters ---------- rc : double array dc : double array pr : double array pd : double array px : double array rv : double array utc1 : double array utc2 : double array dut1 : double array elong : double array phi : double array hm : double array xp : double array yp : double array phpa : double array tc : double array rh : double array wl : double array Returns ------- aob : double array zob : double array hob : double array dob : double array rob : double array eo : double array Notes ----- Wraps ERFA function ``eraAtco13``. The ERFA documentation is:: - - - - - - - - - - e r a A t c o 1 3 - - - - - - - - - - ICRS RA,Dec to observed place. The caller supplies UTC, site coordinates, ambient air conditions and observing wavelength. ERFA models are used for the Earth ephemeris, bias-precession- nutation, Earth orientation and refraction. Given: rc,dc double ICRS right ascension at J2000.0 (radians, Note 1) pr double RA proper motion (radians/year, Note 2) pd double Dec proper motion (radians/year) px double parallax (arcsec) rv double radial velocity (km/s, +ve if receding) utc1 double UTC as a 2-part... utc2 double ...quasi Julian Date (Notes 3-4) dut1 double UT1-UTC (seconds, Note 5) elong double longitude (radians, east +ve, Note 6) phi double latitude (geodetic, radians, Note 6) hm double height above ellipsoid (m, geodetic, Notes 6,8) xp,yp double polar motion coordinates (radians, Note 7) phpa double pressure at the observer (hPa = mB, Note 8) tc double ambient temperature at the observer (deg C) rh double relative humidity at the observer (range 0-1) wl double wavelength (micrometers, Note 9) Returned: aob double observed azimuth (radians: N=0,E=90) zob double observed zenith distance (radians) hob double observed hour angle (radians) dob double observed declination (radians) rob double observed right ascension (CIO-based, radians) eo double equation of the origins (ERA-GST) Returned (function value): int status: +1 = dubious year (Note 4) 0 = OK -1 = unacceptable date Notes: 1) Star data for an epoch other than J2000.0 (for example from the Hipparcos catalog, which has an epoch of J1991.25) will require a preliminary call to eraPmsafe before use. 2) The proper motion in RA is dRA/dt rather than cos(Dec)*dRA/dt. 3) utc1+utc2 is quasi Julian Date (see Note 2), apportioned in any convenient way between the two arguments, for example where utc1 is the Julian Day Number and utc2 is the fraction of a day. However, JD cannot unambiguously represent UTC during a leap second unless special measures are taken. The convention in the present function is that the JD day represents UTC days whether the length is 86399, 86400 or 86401 SI seconds. Applications should use the function eraDtf2d to convert from calendar date and time of day into 2-part quasi Julian Date, as it implements the leap-second-ambiguity convention just described. 4) The warning status "dubious year" flags UTCs that predate the introduction of the time scale or that are too far in the future to be trusted. See eraDat for further details. 5) UT1-UTC is tabulated in IERS bulletins. It increases by exactly one second at the end of each positive UTC leap second, introduced in order to keep UT1-UTC within +/- 0.9s. n.b. This practice is under review, and in the future UT1-UTC may grow essentially without limit. 6) The geographical coordinates are with respect to the ERFA_WGS84 reference ellipsoid. TAKE CARE WITH THE LONGITUDE SIGN: the longitude required by the present function is east-positive (i.e. right-handed), in accordance with geographical convention. 7) The polar motion xp,yp can be obtained from IERS bulletins. The values are the coordinates (in radians) of the Celestial Intermediate Pole with respect to the International Terrestrial Reference System (see IERS Conventions 2003), measured along the meridians 0 and 90 deg west respectively. For many applications, xp and yp can be set to zero. 8) If hm, the height above the ellipsoid of the observing station in meters, is not known but phpa, the pressure in hPa (=mB), is available, an adequate estimate of hm can be obtained from the expression hm = -29.3 * tsl * log ( phpa / 1013.25 ); where tsl is the approximate sea-level air temperature in K (See Astrophysical Quantities, C.W.Allen, 3rd edition, section 52). Similarly, if the pressure phpa is not known, it can be estimated from the height of the observing station, hm, as follows: phpa = 1013.25 * exp ( -hm / ( 29.3 * tsl ) ); Note, however, that the refraction is nearly proportional to the pressure and that an accurate phpa value is important for precise work. 9) The argument wl specifies the observing wavelength in micrometers. The transition from optical to radio is assumed to occur at 100 micrometers (about 3000 GHz). 10) The accuracy of the result is limited by the corrections for refraction, which use a simple A*tan(z) + B*tan^3(z) model. Providing the meteorological parameters are known accurately and there are no gross local effects, the predicted observed coordinates should be within 0.05 arcsec (optical) or 1 arcsec (radio) for a zenith distance of less than 70 degrees, better than 30 arcsec (optical or radio) at 85 degrees and better than 20 arcmin (optical) or 30 arcmin (radio) at the horizon. Without refraction, the complementary functions eraAtco13 and eraAtoc13 are self-consistent to better than 1 microarcsecond all over the celestial sphere. With refraction included, consistency falls off at high zenith distances, but is still better than 0.05 arcsec at 85 degrees. 11) "Observed" Az,ZD means the position that would be seen by a perfect geodetically aligned theodolite. (Zenith distance is used rather than altitude in order to reflect the fact that no allowance is made for depression of the horizon.) This is related to the observed HA,Dec via the standard rotation, using the geodetic latitude (corrected for polar motion), while the observed HA and RA are related simply through the Earth rotation angle and the site longitude. "Observed" RA,Dec or HA,Dec thus means the position that would be seen by a perfect equatorial with its polar axis aligned to the Earth's axis of rotation. 12) It is advisable to take great care with units, as even unlikely values of the input parameters are accepted and processed in accordance with the models used. Called: eraApco13 astrometry parameters, ICRS-observed, 2013 eraAtciq quick ICRS to CIRS eraAtioq quick CIRS to observed This revision: 2021 April 3 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ aob, zob, hob, dob, rob, eo, c_retval = ufunc.atco13( rc, dc, pr, pd, px, rv, utc1, utc2, dut1, elong, phi, hm, xp, yp, phpa, tc, rh, wl) check_errwarn(c_retval, 'atco13') return aob, zob, hob, dob, rob, eo STATUS_CODES['atco13'] = { 1: 'dubious year (Note 4)', 0: 'OK', -1: 'unacceptable date', } def atic13(ri, di, date1, date2): """ Transform star RA,Dec from geocentric CIRS to ICRS astrometric. Parameters ---------- ri : double array di : double array date1 : double array date2 : double array Returns ------- rc : double array dc : double array eo : double array Notes ----- Wraps ERFA function ``eraAtic13``. The ERFA documentation is:: - - - - - - - - - - e r a A t i c 1 3 - - - - - - - - - - Transform star RA,Dec from geocentric CIRS to ICRS astrometric. Given: ri,di double CIRS geocentric RA,Dec (radians) date1 double TDB as a 2-part... date2 double ...Julian Date (Note 1) Returned: rc,dc double ICRS astrometric RA,Dec (radians) eo double equation of the origins (ERA-GST, Note 4) Notes: 1) The TDB date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TDB)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. For most applications of this function the choice will not be at all critical. TT can be used instead of TDB without any significant impact on accuracy. 2) Iterative techniques are used for the aberration and light deflection corrections so that the functions eraAtic13 (or eraAticq) and eraAtci13 (or eraAtciq) are accurate inverses; even at the edge of the Sun's disk the discrepancy is only about 1 nanoarcsecond. 3) The available accuracy is better than 1 milliarcsecond, limited mainly by the precession-nutation model that is used, namely IAU 2000A/2006. Very close to solar system bodies, additional errors of up to several milliarcseconds can occur because of unmodeled light deflection; however, the Sun's contribution is taken into account, to first order. The accuracy limitations of the ERFA function eraEpv00 (used to compute Earth position and velocity) can contribute aberration errors of up to 5 microarcseconds. Light deflection at the Sun's limb is uncertain at the 0.4 mas level. 4) Should the transformation to (equinox based) J2000.0 mean place be required rather than (CIO based) ICRS coordinates, subtract the equation of the origins from the returned right ascension: RA = RI - EO. (The eraAnp function can then be applied, as required, to keep the result in the conventional 0-2pi range.) Called: eraApci13 astrometry parameters, ICRS-CIRS, 2013 eraAticq quick CIRS to ICRS astrometric This revision: 2013 October 9 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rc, dc, eo = ufunc.atic13(ri, di, date1, date2) return rc, dc, eo def aticq(ri, di, astrom): """ Quick CIRS RA,Dec to ICRS astrometric place, given the star- independent astrometry parameters. Parameters ---------- ri : double array di : double array astrom : eraASTROM array Returns ------- rc : double array dc : double array Notes ----- Wraps ERFA function ``eraAticq``. The ERFA documentation is:: - - - - - - - - - e r a A t i c q - - - - - - - - - Quick CIRS RA,Dec to ICRS astrometric place, given the star- independent astrometry parameters. Use of this function is appropriate when efficiency is important and where many star positions are all to be transformed for one date. The star-independent astrometry parameters can be obtained by calling one of the functions eraApci[13], eraApcg[13], eraApco[13] or eraApcs[13]. Given: ri,di double CIRS RA,Dec (radians) astrom eraASTROM* star-independent astrometry parameters: pmt double PM time interval (SSB, Julian years) eb double[3] SSB to observer (vector, au) eh double[3] Sun to observer (unit vector) em double distance from Sun to observer (au) v double[3] barycentric observer velocity (vector, c) bm1 double sqrt(1-|v|^2): reciprocal of Lorenz factor bpn double[3][3] bias-precession-nutation matrix along double longitude + s' (radians) xpl double polar motion xp wrt local meridian (radians) ypl double polar motion yp wrt local meridian (radians) sphi double sine of geodetic latitude cphi double cosine of geodetic latitude diurab double magnitude of diurnal aberration vector eral double "local" Earth rotation angle (radians) refa double refraction constant A (radians) refb double refraction constant B (radians) Returned: rc,dc double ICRS astrometric RA,Dec (radians) Notes: 1) Only the Sun is taken into account in the light deflection correction. 2) Iterative techniques are used for the aberration and light deflection corrections so that the functions eraAtic13 (or eraAticq) and eraAtci13 (or eraAtciq) are accurate inverses; even at the edge of the Sun's disk the discrepancy is only about 1 nanoarcsecond. Called: eraS2c spherical coordinates to unit vector eraTrxp product of transpose of r-matrix and p-vector eraZp zero p-vector eraAb stellar aberration eraLdsun light deflection by the Sun eraC2s p-vector to spherical eraAnp normalize angle into range +/- pi This revision: 2013 October 9 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rc, dc = ufunc.aticq(ri, di, astrom) return rc, dc def aticqn(ri, di, astrom, b): """ Quick CIRS to ICRS astrometric place transformation, given the star- independent astrometry parameters plus a list of light-deflecting bodies. Parameters ---------- ri : double array di : double array astrom : eraASTROM array b : eraLDBODY array Returns ------- rc : double array dc : double array Notes ----- Wraps ERFA function ``eraAticqn``. The ERFA documentation is:: - - - - - - - - - - e r a A t i c q n - - - - - - - - - - Quick CIRS to ICRS astrometric place transformation, given the star- independent astrometry parameters plus a list of light-deflecting bodies. Use of this function is appropriate when efficiency is important and where many star positions are all to be transformed for one date. The star-independent astrometry parameters can be obtained by calling one of the functions eraApci[13], eraApcg[13], eraApco[13] or eraApcs[13]. If the only light-deflecting body to be taken into account is the Sun, the eraAticq function can be used instead. Given: ri,di double CIRS RA,Dec (radians) astrom eraASTROM star-independent astrometry parameters: pmt double PM time interval (SSB, Julian years) eb double[3] SSB to observer (vector, au) eh double[3] Sun to observer (unit vector) em double distance from Sun to observer (au) v double[3] barycentric observer velocity (vector, c) bm1 double sqrt(1-|v|^2): reciprocal of Lorenz factor bpn double[3][3] bias-precession-nutation matrix along double longitude + s' (radians) xpl double polar motion xp wrt local meridian (radians) ypl double polar motion yp wrt local meridian (radians) sphi double sine of geodetic latitude cphi double cosine of geodetic latitude diurab double magnitude of diurnal aberration vector eral double "local" Earth rotation angle (radians) refa double refraction constant A (radians) refb double refraction constant B (radians) n int number of bodies (Note 3) b eraLDBODY[n] data for each of the n bodies (Notes 3,4): bm double mass of the body (solar masses, Note 5) dl double deflection limiter (Note 6) pv [2][3] barycentric PV of the body (au, au/day) Returned: rc,dc double ICRS astrometric RA,Dec (radians) Notes: 1) Iterative techniques are used for the aberration and light deflection corrections so that the functions eraAticqn and eraAtciqn are accurate inverses; even at the edge of the Sun's disk the discrepancy is only about 1 nanoarcsecond. 2) If the only light-deflecting body to be taken into account is the Sun, the eraAticq function can be used instead. 3) The struct b contains n entries, one for each body to be considered. If n = 0, no gravitational light deflection will be applied, not even for the Sun. 4) The struct b should include an entry for the Sun as well as for any planet or other body to be taken into account. The entries should be in the order in which the light passes the body. 5) In the entry in the b struct for body i, the mass parameter b[i].bm can, as required, be adjusted in order to allow for such effects as quadrupole field. 6) The deflection limiter parameter b[i].dl is phi^2/2, where phi is the angular separation (in radians) between star and body at which limiting is applied. As phi shrinks below the chosen threshold, the deflection is artificially reduced, reaching zero for phi = 0. Example values suitable for a terrestrial observer, together with masses, are as follows: body i b[i].bm b[i].dl Sun 1.0 6e-6 Jupiter 0.00095435 3e-9 Saturn 0.00028574 3e-10 7) For efficiency, validation of the contents of the b array is omitted. The supplied masses must be greater than zero, the position and velocity vectors must be right, and the deflection limiter greater than zero. Called: eraS2c spherical coordinates to unit vector eraTrxp product of transpose of r-matrix and p-vector eraZp zero p-vector eraAb stellar aberration eraLdn light deflection by n bodies eraC2s p-vector to spherical eraAnp normalize angle into range +/- pi This revision: 2021 January 6 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rc, dc = ufunc.aticqn(ri, di, astrom, b) return rc, dc def atio13(ri, di, utc1, utc2, dut1, elong, phi, hm, xp, yp, phpa, tc, rh, wl): """ CIRS RA,Dec to observed place. The caller supplies UTC, site coordinates, ambient air conditions and observing wavelength. Parameters ---------- ri : double array di : double array utc1 : double array utc2 : double array dut1 : double array elong : double array phi : double array hm : double array xp : double array yp : double array phpa : double array tc : double array rh : double array wl : double array Returns ------- aob : double array zob : double array hob : double array dob : double array rob : double array Notes ----- Wraps ERFA function ``eraAtio13``. The ERFA documentation is:: - - - - - - - - - - e r a A t i o 1 3 - - - - - - - - - - CIRS RA,Dec to observed place. The caller supplies UTC, site coordinates, ambient air conditions and observing wavelength. Given: ri double CIRS right ascension (CIO-based, radians) di double CIRS declination (radians) utc1 double UTC as a 2-part... utc2 double ...quasi Julian Date (Notes 1,2) dut1 double UT1-UTC (seconds, Note 3) elong double longitude (radians, east +ve, Note 4) phi double geodetic latitude (radians, Note 4) hm double height above ellipsoid (m, geodetic Notes 4,6) xp,yp double polar motion coordinates (radians, Note 5) phpa double pressure at the observer (hPa = mB, Note 6) tc double ambient temperature at the observer (deg C) rh double relative humidity at the observer (range 0-1) wl double wavelength (micrometers, Note 7) Returned: aob double observed azimuth (radians: N=0,E=90) zob double observed zenith distance (radians) hob double observed hour angle (radians) dob double observed declination (radians) rob double observed right ascension (CIO-based, radians) Returned (function value): int status: +1 = dubious year (Note 2) 0 = OK -1 = unacceptable date Notes: 1) utc1+utc2 is quasi Julian Date (see Note 2), apportioned in any convenient way between the two arguments, for example where utc1 is the Julian Day Number and utc2 is the fraction of a day. However, JD cannot unambiguously represent UTC during a leap second unless special measures are taken. The convention in the present function is that the JD day represents UTC days whether the length is 86399, 86400 or 86401 SI seconds. Applications should use the function eraDtf2d to convert from calendar date and time of day into 2-part quasi Julian Date, as it implements the leap-second-ambiguity convention just described. 2) The warning status "dubious year" flags UTCs that predate the introduction of the time scale or that are too far in the future to be trusted. See eraDat for further details. 3) UT1-UTC is tabulated in IERS bulletins. It increases by exactly one second at the end of each positive UTC leap second, introduced in order to keep UT1-UTC within +/- 0.9s. n.b. This practice is under review, and in the future UT1-UTC may grow essentially without limit. 4) The geographical coordinates are with respect to the ERFA_WGS84 reference ellipsoid. TAKE CARE WITH THE LONGITUDE SIGN: the longitude required by the present function is east-positive (i.e. right-handed), in accordance with geographical convention. 5) The polar motion xp,yp can be obtained from IERS bulletins. The values are the coordinates (in radians) of the Celestial Intermediate Pole with respect to the International Terrestrial Reference System (see IERS Conventions 2003), measured along the meridians 0 and 90 deg west respectively. For many applications, xp and yp can be set to zero. 6) If hm, the height above the ellipsoid of the observing station in meters, is not known but phpa, the pressure in hPa (=mB), is available, an adequate estimate of hm can be obtained from the expression hm = -29.3 * tsl * log ( phpa / 1013.25 ); where tsl is the approximate sea-level air temperature in K (See Astrophysical Quantities, C.W.Allen, 3rd edition, section 52). Similarly, if the pressure phpa is not known, it can be estimated from the height of the observing station, hm, as follows: phpa = 1013.25 * exp ( -hm / ( 29.3 * tsl ) ); Note, however, that the refraction is nearly proportional to the pressure and that an accurate phpa value is important for precise work. 7) The argument wl specifies the observing wavelength in micrometers. The transition from optical to radio is assumed to occur at 100 micrometers (about 3000 GHz). 8) "Observed" Az,ZD means the position that would be seen by a perfect geodetically aligned theodolite. (Zenith distance is used rather than altitude in order to reflect the fact that no allowance is made for depression of the horizon.) This is related to the observed HA,Dec via the standard rotation, using the geodetic latitude (corrected for polar motion), while the observed HA and RA are related simply through the Earth rotation angle and the site longitude. "Observed" RA,Dec or HA,Dec thus means the position that would be seen by a perfect equatorial with its polar axis aligned to the Earth's axis of rotation. 9) The accuracy of the result is limited by the corrections for refraction, which use a simple A*tan(z) + B*tan^3(z) model. Providing the meteorological parameters are known accurately and there are no gross local effects, the predicted astrometric coordinates should be within 0.05 arcsec (optical) or 1 arcsec (radio) for a zenith distance of less than 70 degrees, better than 30 arcsec (optical or radio) at 85 degrees and better than 20 arcmin (optical) or 30 arcmin (radio) at the horizon. 10) The complementary functions eraAtio13 and eraAtoi13 are self- consistent to better than 1 microarcsecond all over the celestial sphere. 11) It is advisable to take great care with units, as even unlikely values of the input parameters are accepted and processed in accordance with the models used. Called: eraApio13 astrometry parameters, CIRS-observed, 2013 eraAtioq quick CIRS to observed This revision: 2021 February 24 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ aob, zob, hob, dob, rob, c_retval = ufunc.atio13( ri, di, utc1, utc2, dut1, elong, phi, hm, xp, yp, phpa, tc, rh, wl) check_errwarn(c_retval, 'atio13') return aob, zob, hob, dob, rob STATUS_CODES['atio13'] = { 1: 'dubious year (Note 2)', 0: 'OK', -1: 'unacceptable date', } def atioq(ri, di, astrom): """ Quick CIRS to observed place transformation. Parameters ---------- ri : double array di : double array astrom : eraASTROM array Returns ------- aob : double array zob : double array hob : double array dob : double array rob : double array Notes ----- Wraps ERFA function ``eraAtioq``. The ERFA documentation is:: - - - - - - - - - e r a A t i o q - - - - - - - - - Quick CIRS to observed place transformation. Use of this function is appropriate when efficiency is important and where many star positions are all to be transformed for one date. The star-independent astrometry parameters can be obtained by calling eraApio[13] or eraApco[13]. Given: ri double CIRS right ascension di double CIRS declination astrom eraASTROM* star-independent astrometry parameters: pmt double PM time interval (SSB, Julian years) eb double[3] SSB to observer (vector, au) eh double[3] Sun to observer (unit vector) em double distance from Sun to observer (au) v double[3] barycentric observer velocity (vector, c) bm1 double sqrt(1-|v|^2): reciprocal of Lorenz factor bpn double[3][3] bias-precession-nutation matrix along double longitude + s' (radians) xpl double polar motion xp wrt local meridian (radians) ypl double polar motion yp wrt local meridian (radians) sphi double sine of geodetic latitude cphi double cosine of geodetic latitude diurab double magnitude of diurnal aberration vector eral double "local" Earth rotation angle (radians) refa double refraction constant A (radians) refb double refraction constant B (radians) Returned: aob double observed azimuth (radians: N=0,E=90) zob double observed zenith distance (radians) hob double observed hour angle (radians) dob double observed declination (radians) rob double observed right ascension (CIO-based, radians) Notes: 1) This function returns zenith distance rather than altitude in order to reflect the fact that no allowance is made for depression of the horizon. 2) The accuracy of the result is limited by the corrections for refraction, which use a simple A*tan(z) + B*tan^3(z) model. Providing the meteorological parameters are known accurately and there are no gross local effects, the predicted observed coordinates should be within 0.05 arcsec (optical) or 1 arcsec (radio) for a zenith distance of less than 70 degrees, better than 30 arcsec (optical or radio) at 85 degrees and better than 20 arcmin (optical) or 30 arcmin (radio) at the horizon. Without refraction, the complementary functions eraAtioq and eraAtoiq are self-consistent to better than 1 microarcsecond all over the celestial sphere. With refraction included, consistency falls off at high zenith distances, but is still better than 0.05 arcsec at 85 degrees. 3) It is advisable to take great care with units, as even unlikely values of the input parameters are accepted and processed in accordance with the models used. 4) The CIRS RA,Dec is obtained from a star catalog mean place by allowing for space motion, parallax, the Sun's gravitational lens effect, annual aberration and precession-nutation. For star positions in the ICRS, these effects can be applied by means of the eraAtci13 (etc.) functions. Starting from classical "mean place" systems, additional transformations will be needed first. 5) "Observed" Az,El means the position that would be seen by a perfect geodetically aligned theodolite. This is obtained from the CIRS RA,Dec by allowing for Earth orientation and diurnal aberration, rotating from equator to horizon coordinates, and then adjusting for refraction. The HA,Dec is obtained by rotating back into equatorial coordinates, and is the position that would be seen by a perfect equatorial with its polar axis aligned to the Earth's axis of rotation. Finally, the RA is obtained by subtracting the HA from the local ERA. 6) The star-independent CIRS-to-observed-place parameters in ASTROM may be computed with eraApio[13] or eraApco[13]. If nothing has changed significantly except the time, eraAper[13] may be used to perform the requisite adjustment to the astrom structure. Called: eraS2c spherical coordinates to unit vector eraC2s p-vector to spherical eraAnp normalize angle into range 0 to 2pi This revision: 2020 December 7 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ aob, zob, hob, dob, rob = ufunc.atioq(ri, di, astrom) return aob, zob, hob, dob, rob def atoc13(type, ob1, ob2, utc1, utc2, dut1, elong, phi, hm, xp, yp, phpa, tc, rh, wl): """ Observed place at a groundbased site to to ICRS astrometric RA,Dec. The caller supplies UTC, site coordinates, ambient air conditions and observing wavelength. Parameters ---------- type : const char array ob1 : double array ob2 : double array utc1 : double array utc2 : double array dut1 : double array elong : double array phi : double array hm : double array xp : double array yp : double array phpa : double array tc : double array rh : double array wl : double array Returns ------- rc : double array dc : double array Notes ----- Wraps ERFA function ``eraAtoc13``. The ERFA documentation is:: - - - - - - - - - - e r a A t o c 1 3 - - - - - - - - - - Observed place at a groundbased site to to ICRS astrometric RA,Dec. The caller supplies UTC, site coordinates, ambient air conditions and observing wavelength. Given: type char[] type of coordinates - "R", "H" or "A" (Notes 1,2) ob1 double observed Az, HA or RA (radians; Az is N=0,E=90) ob2 double observed ZD or Dec (radians) utc1 double UTC as a 2-part... utc2 double ...quasi Julian Date (Notes 3,4) dut1 double UT1-UTC (seconds, Note 5) elong double longitude (radians, east +ve, Note 6) phi double geodetic latitude (radians, Note 6) hm double height above ellipsoid (m, geodetic Notes 6,8) xp,yp double polar motion coordinates (radians, Note 7) phpa double pressure at the observer (hPa = mB, Note 8) tc double ambient temperature at the observer (deg C) rh double relative humidity at the observer (range 0-1) wl double wavelength (micrometers, Note 9) Returned: rc,dc double ICRS astrometric RA,Dec (radians) Returned (function value): int status: +1 = dubious year (Note 4) 0 = OK -1 = unacceptable date Notes: 1) "Observed" Az,ZD means the position that would be seen by a perfect geodetically aligned theodolite. (Zenith distance is used rather than altitude in order to reflect the fact that no allowance is made for depression of the horizon.) This is related to the observed HA,Dec via the standard rotation, using the geodetic latitude (corrected for polar motion), while the observed HA and RA are related simply through the Earth rotation angle and the site longitude. "Observed" RA,Dec or HA,Dec thus means the position that would be seen by a perfect equatorial with its polar axis aligned to the Earth's axis of rotation. 2) Only the first character of the type argument is significant. "R" or "r" indicates that ob1 and ob2 are the observed right ascension and declination; "H" or "h" indicates that they are hour angle (west +ve) and declination; anything else ("A" or "a" is recommended) indicates that ob1 and ob2 are azimuth (north zero, east 90 deg) and zenith distance. 3) utc1+utc2 is quasi Julian Date (see Note 2), apportioned in any convenient way between the two arguments, for example where utc1 is the Julian Day Number and utc2 is the fraction of a day. However, JD cannot unambiguously represent UTC during a leap second unless special measures are taken. The convention in the present function is that the JD day represents UTC days whether the length is 86399, 86400 or 86401 SI seconds. Applications should use the function eraDtf2d to convert from calendar date and time of day into 2-part quasi Julian Date, as it implements the leap-second-ambiguity convention just described. 4) The warning status "dubious year" flags UTCs that predate the introduction of the time scale or that are too far in the future to be trusted. See eraDat for further details. 5) UT1-UTC is tabulated in IERS bulletins. It increases by exactly one second at the end of each positive UTC leap second, introduced in order to keep UT1-UTC within +/- 0.9s. n.b. This practice is under review, and in the future UT1-UTC may grow essentially without limit. 6) The geographical coordinates are with respect to the ERFA_WGS84 reference ellipsoid. TAKE CARE WITH THE LONGITUDE SIGN: the longitude required by the present function is east-positive (i.e. right-handed), in accordance with geographical convention. 7) The polar motion xp,yp can be obtained from IERS bulletins. The values are the coordinates (in radians) of the Celestial Intermediate Pole with respect to the International Terrestrial Reference System (see IERS Conventions 2003), measured along the meridians 0 and 90 deg west respectively. For many applications, xp and yp can be set to zero. 8) If hm, the height above the ellipsoid of the observing station in meters, is not known but phpa, the pressure in hPa (=mB), is available, an adequate estimate of hm can be obtained from the expression hm = -29.3 * tsl * log ( phpa / 1013.25 ); where tsl is the approximate sea-level air temperature in K (See Astrophysical Quantities, C.W.Allen, 3rd edition, section 52). Similarly, if the pressure phpa is not known, it can be estimated from the height of the observing station, hm, as follows: phpa = 1013.25 * exp ( -hm / ( 29.3 * tsl ) ); Note, however, that the refraction is nearly proportional to the pressure and that an accurate phpa value is important for precise work. 9) The argument wl specifies the observing wavelength in micrometers. The transition from optical to radio is assumed to occur at 100 micrometers (about 3000 GHz). 10) The accuracy of the result is limited by the corrections for refraction, which use a simple A*tan(z) + B*tan^3(z) model. Providing the meteorological parameters are known accurately and there are no gross local effects, the predicted astrometric coordinates should be within 0.05 arcsec (optical) or 1 arcsec (radio) for a zenith distance of less than 70 degrees, better than 30 arcsec (optical or radio) at 85 degrees and better than 20 arcmin (optical) or 30 arcmin (radio) at the horizon. Without refraction, the complementary functions eraAtco13 and eraAtoc13 are self-consistent to better than 1 microarcsecond all over the celestial sphere. With refraction included, consistency falls off at high zenith distances, but is still better than 0.05 arcsec at 85 degrees. 11) It is advisable to take great care with units, as even unlikely values of the input parameters are accepted and processed in accordance with the models used. Called: eraApco13 astrometry parameters, ICRS-observed eraAtoiq quick observed to CIRS eraAticq quick CIRS to ICRS This revision: 2021 February 24 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rc, dc, c_retval = ufunc.atoc13( type, ob1, ob2, utc1, utc2, dut1, elong, phi, hm, xp, yp, phpa, tc, rh, wl) check_errwarn(c_retval, 'atoc13') return rc, dc STATUS_CODES['atoc13'] = { 1: 'dubious year (Note 4)', 0: 'OK', -1: 'unacceptable date', } def atoi13(type, ob1, ob2, utc1, utc2, dut1, elong, phi, hm, xp, yp, phpa, tc, rh, wl): """ Observed place to CIRS. The caller supplies UTC, site coordinates, ambient air conditions and observing wavelength. Parameters ---------- type : const char array ob1 : double array ob2 : double array utc1 : double array utc2 : double array dut1 : double array elong : double array phi : double array hm : double array xp : double array yp : double array phpa : double array tc : double array rh : double array wl : double array Returns ------- ri : double array di : double array Notes ----- Wraps ERFA function ``eraAtoi13``. The ERFA documentation is:: - - - - - - - - - - e r a A t o i 1 3 - - - - - - - - - - Observed place to CIRS. The caller supplies UTC, site coordinates, ambient air conditions and observing wavelength. Given: type char[] type of coordinates - "R", "H" or "A" (Notes 1,2) ob1 double observed Az, HA or RA (radians; Az is N=0,E=90) ob2 double observed ZD or Dec (radians) utc1 double UTC as a 2-part... utc2 double ...quasi Julian Date (Notes 3,4) dut1 double UT1-UTC (seconds, Note 5) elong double longitude (radians, east +ve, Note 6) phi double geodetic latitude (radians, Note 6) hm double height above the ellipsoid (meters, Notes 6,8) xp,yp double polar motion coordinates (radians, Note 7) phpa double pressure at the observer (hPa = mB, Note 8) tc double ambient temperature at the observer (deg C) rh double relative humidity at the observer (range 0-1) wl double wavelength (micrometers, Note 9) Returned: ri double CIRS right ascension (CIO-based, radians) di double CIRS declination (radians) Returned (function value): int status: +1 = dubious year (Note 2) 0 = OK -1 = unacceptable date Notes: 1) "Observed" Az,ZD means the position that would be seen by a perfect geodetically aligned theodolite. (Zenith distance is used rather than altitude in order to reflect the fact that no allowance is made for depression of the horizon.) This is related to the observed HA,Dec via the standard rotation, using the geodetic latitude (corrected for polar motion), while the observed HA and RA are related simply through the Earth rotation angle and the site longitude. "Observed" RA,Dec or HA,Dec thus means the position that would be seen by a perfect equatorial with its polar axis aligned to the Earth's axis of rotation. 2) Only the first character of the type argument is significant. "R" or "r" indicates that ob1 and ob2 are the observed right ascension and declination; "H" or "h" indicates that they are hour angle (west +ve) and declination; anything else ("A" or "a" is recommended) indicates that ob1 and ob2 are azimuth (north zero, east 90 deg) and zenith distance. 3) utc1+utc2 is quasi Julian Date (see Note 2), apportioned in any convenient way between the two arguments, for example where utc1 is the Julian Day Number and utc2 is the fraction of a day. However, JD cannot unambiguously represent UTC during a leap second unless special measures are taken. The convention in the present function is that the JD day represents UTC days whether the length is 86399, 86400 or 86401 SI seconds. Applications should use the function eraDtf2d to convert from calendar date and time of day into 2-part quasi Julian Date, as it implements the leap-second-ambiguity convention just described. 4) The warning status "dubious year" flags UTCs that predate the introduction of the time scale or that are too far in the future to be trusted. See eraDat for further details. 5) UT1-UTC is tabulated in IERS bulletins. It increases by exactly one second at the end of each positive UTC leap second, introduced in order to keep UT1-UTC within +/- 0.9s. n.b. This practice is under review, and in the future UT1-UTC may grow essentially without limit. 6) The geographical coordinates are with respect to the ERFA_WGS84 reference ellipsoid. TAKE CARE WITH THE LONGITUDE SIGN: the longitude required by the present function is east-positive (i.e. right-handed), in accordance with geographical convention. 7) The polar motion xp,yp can be obtained from IERS bulletins. The values are the coordinates (in radians) of the Celestial Intermediate Pole with respect to the International Terrestrial Reference System (see IERS Conventions 2003), measured along the meridians 0 and 90 deg west respectively. For many applications, xp and yp can be set to zero. 8) If hm, the height above the ellipsoid of the observing station in meters, is not known but phpa, the pressure in hPa (=mB), is available, an adequate estimate of hm can be obtained from the expression hm = -29.3 * tsl * log ( phpa / 1013.25 ); where tsl is the approximate sea-level air temperature in K (See Astrophysical Quantities, C.W.Allen, 3rd edition, section 52). Similarly, if the pressure phpa is not known, it can be estimated from the height of the observing station, hm, as follows: phpa = 1013.25 * exp ( -hm / ( 29.3 * tsl ) ); Note, however, that the refraction is nearly proportional to the pressure and that an accurate phpa value is important for precise work. 9) The argument wl specifies the observing wavelength in micrometers. The transition from optical to radio is assumed to occur at 100 micrometers (about 3000 GHz). 10) The accuracy of the result is limited by the corrections for refraction, which use a simple A*tan(z) + B*tan^3(z) model. Providing the meteorological parameters are known accurately and there are no gross local effects, the predicted astrometric coordinates should be within 0.05 arcsec (optical) or 1 arcsec (radio) for a zenith distance of less than 70 degrees, better than 30 arcsec (optical or radio) at 85 degrees and better than 20 arcmin (optical) or 30 arcmin (radio) at the horizon. Without refraction, the complementary functions eraAtio13 and eraAtoi13 are self-consistent to better than 1 microarcsecond all over the celestial sphere. With refraction included, consistency falls off at high zenith distances, but is still better than 0.05 arcsec at 85 degrees. 12) It is advisable to take great care with units, as even unlikely values of the input parameters are accepted and processed in accordance with the models used. Called: eraApio13 astrometry parameters, CIRS-observed, 2013 eraAtoiq quick observed to CIRS This revision: 2021 February 24 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ ri, di, c_retval = ufunc.atoi13( type, ob1, ob2, utc1, utc2, dut1, elong, phi, hm, xp, yp, phpa, tc, rh, wl) check_errwarn(c_retval, 'atoi13') return ri, di STATUS_CODES['atoi13'] = { 1: 'dubious year (Note 2)', 0: 'OK', -1: 'unacceptable date', } def atoiq(type, ob1, ob2, astrom): """ Quick observed place to CIRS, given the star-independent astrometry parameters. Parameters ---------- type : const char array ob1 : double array ob2 : double array astrom : eraASTROM array Returns ------- ri : double array di : double array Notes ----- Wraps ERFA function ``eraAtoiq``. The ERFA documentation is:: - - - - - - - - - e r a A t o i q - - - - - - - - - Quick observed place to CIRS, given the star-independent astrometry parameters. Use of this function is appropriate when efficiency is important and where many star positions are all to be transformed for one date. The star-independent astrometry parameters can be obtained by calling eraApio[13] or eraApco[13]. Given: type char[] type of coordinates: "R", "H" or "A" (Note 1) ob1 double observed Az, HA or RA (radians; Az is N=0,E=90) ob2 double observed ZD or Dec (radians) astrom eraASTROM* star-independent astrometry parameters: pmt double PM time interval (SSB, Julian years) eb double[3] SSB to observer (vector, au) eh double[3] Sun to observer (unit vector) em double distance from Sun to observer (au) v double[3] barycentric observer velocity (vector, c) bm1 double sqrt(1-|v|^2): reciprocal of Lorenz factor bpn double[3][3] bias-precession-nutation matrix along double longitude + s' (radians) xpl double polar motion xp wrt local meridian (radians) ypl double polar motion yp wrt local meridian (radians) sphi double sine of geodetic latitude cphi double cosine of geodetic latitude diurab double magnitude of diurnal aberration vector eral double "local" Earth rotation angle (radians) refa double refraction constant A (radians) refb double refraction constant B (radians) Returned: ri double CIRS right ascension (CIO-based, radians) di double CIRS declination (radians) Notes: 1) "Observed" Az,El means the position that would be seen by a perfect geodetically aligned theodolite. This is related to the observed HA,Dec via the standard rotation, using the geodetic latitude (corrected for polar motion), while the observed HA and RA are related simply through the Earth rotation angle and the site longitude. "Observed" RA,Dec or HA,Dec thus means the position that would be seen by a perfect equatorial with its polar axis aligned to the Earth's axis of rotation. By removing from the observed place the effects of atmospheric refraction and diurnal aberration, the CIRS RA,Dec is obtained. 2) Only the first character of the type argument is significant. "R" or "r" indicates that ob1 and ob2 are the observed right ascension and declination; "H" or "h" indicates that they are hour angle (west +ve) and declination; anything else ("A" or "a" is recommended) indicates that ob1 and ob2 are azimuth (north zero, east 90 deg) and zenith distance. (Zenith distance is used rather than altitude in order to reflect the fact that no allowance is made for depression of the horizon.) 3) The accuracy of the result is limited by the corrections for refraction, which use a simple A*tan(z) + B*tan^3(z) model. Providing the meteorological parameters are known accurately and there are no gross local effects, the predicted intermediate coordinates should be within 0.05 arcsec (optical) or 1 arcsec (radio) for a zenith distance of less than 70 degrees, better than 30 arcsec (optical or radio) at 85 degrees and better than 20 arcmin (optical) or 25 arcmin (radio) at the horizon. Without refraction, the complementary functions eraAtioq and eraAtoiq are self-consistent to better than 1 microarcsecond all over the celestial sphere. With refraction included, consistency falls off at high zenith distances, but is still better than 0.05 arcsec at 85 degrees. 4) It is advisable to take great care with units, as even unlikely values of the input parameters are accepted and processed in accordance with the models used. Called: eraS2c spherical coordinates to unit vector eraC2s p-vector to spherical eraAnp normalize angle into range 0 to 2pi This revision: 2020 December 7 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ ri, di = ufunc.atoiq(type, ob1, ob2, astrom) return ri, di def ld(bm, p, q, e, em, dlim): """ Apply light deflection by a solar-system body, as part of transforming coordinate direction into natural direction. Parameters ---------- bm : double array p : double array q : double array e : double array em : double array dlim : double array Returns ------- p1 : double array Notes ----- Wraps ERFA function ``eraLd``. The ERFA documentation is:: - - - - - - e r a L d - - - - - - Apply light deflection by a solar-system body, as part of transforming coordinate direction into natural direction. Given: bm double mass of the gravitating body (solar masses) p double[3] direction from observer to source (unit vector) q double[3] direction from body to source (unit vector) e double[3] direction from body to observer (unit vector) em double distance from body to observer (au) dlim double deflection limiter (Note 4) Returned: p1 double[3] observer to deflected source (unit vector) Notes: 1) The algorithm is based on Expr. (70) in Klioner (2003) and Expr. (7.63) in the Explanatory Supplement (Urban & Seidelmann 2013), with some rearrangement to minimize the effects of machine precision. 2) The mass parameter bm can, as required, be adjusted in order to allow for such effects as quadrupole field. 3) The barycentric position of the deflecting body should ideally correspond to the time of closest approach of the light ray to the body. 4) The deflection limiter parameter dlim is phi^2/2, where phi is the angular separation (in radians) between source and body at which limiting is applied. As phi shrinks below the chosen threshold, the deflection is artificially reduced, reaching zero for phi = 0. 5) The returned vector p1 is not normalized, but the consequential departure from unit magnitude is always negligible. 6) The arguments p and p1 can be the same array. 7) To accumulate total light deflection taking into account the contributions from several bodies, call the present function for each body in succession, in decreasing order of distance from the observer. 8) For efficiency, validation is omitted. The supplied vectors must be of unit magnitude, and the deflection limiter non-zero and positive. References: Urban, S. & Seidelmann, P. K. (eds), Explanatory Supplement to the Astronomical Almanac, 3rd ed., University Science Books (2013). Klioner, Sergei A., "A practical relativistic model for micro- arcsecond astrometry in space", Astr. J. 125, 1580-1597 (2003). Called: eraPdp scalar product of two p-vectors eraPxp vector product of two p-vectors This revision: 2021 February 24 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ p1 = ufunc.ld(bm, p, q, e, em, dlim) return p1 def ldn(b, ob, sc): """ For a star, apply light deflection by multiple solar-system bodies, as part of transforming coordinate direction into natural direction. Parameters ---------- b : eraLDBODY array ob : double array sc : double array Returns ------- sn : double array Notes ----- Wraps ERFA function ``eraLdn``. The ERFA documentation is:: - - - - - - - e r a L d n - - - - - - - For a star, apply light deflection by multiple solar-system bodies, as part of transforming coordinate direction into natural direction. Given: n int number of bodies (note 1) b eraLDBODY[n] data for each of the n bodies (Notes 1,2): bm double mass of the body (solar masses, Note 3) dl double deflection limiter (Note 4) pv [2][3] barycentric PV of the body (au, au/day) ob double[3] barycentric position of the observer (au) sc double[3] observer to star coord direction (unit vector) Returned: sn double[3] observer to deflected star (unit vector) 1) The array b contains n entries, one for each body to be considered. If n = 0, no gravitational light deflection will be applied, not even for the Sun. 2) The array b should include an entry for the Sun as well as for any planet or other body to be taken into account. The entries should be in the order in which the light passes the body. 3) In the entry in the b array for body i, the mass parameter b[i].bm can, as required, be adjusted in order to allow for such effects as quadrupole field. 4) The deflection limiter parameter b[i].dl is phi^2/2, where phi is the angular separation (in radians) between star and body at which limiting is applied. As phi shrinks below the chosen threshold, the deflection is artificially reduced, reaching zero for phi = 0. Example values suitable for a terrestrial observer, together with masses, are as follows: body i b[i].bm b[i].dl Sun 1.0 6e-6 Jupiter 0.00095435 3e-9 Saturn 0.00028574 3e-10 5) For cases where the starlight passes the body before reaching the observer, the body is placed back along its barycentric track by the light time from that point to the observer. For cases where the body is "behind" the observer no such shift is applied. If a different treatment is preferred, the user has the option of instead using the eraLd function. Similarly, eraLd can be used for cases where the source is nearby, not a star. 6) The returned vector sn is not normalized, but the consequential departure from unit magnitude is always negligible. 7) The arguments sc and sn can be the same array. 8) For efficiency, validation is omitted. The supplied masses must be greater than zero, the position and velocity vectors must be right, and the deflection limiter greater than zero. Reference: Urban, S. & Seidelmann, P. K. (eds), Explanatory Supplement to the Astronomical Almanac, 3rd ed., University Science Books (2013), Section 7.2.4. Called: eraCp copy p-vector eraPdp scalar product of two p-vectors eraPmp p-vector minus p-vector eraPpsp p-vector plus scaled p-vector eraPn decompose p-vector into modulus and direction eraLd light deflection by a solar-system body This revision: 2021 February 24 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ sn = ufunc.ldn(b, ob, sc) return sn def ldsun(p, e, em): """ Deflection of starlight by the Sun. Parameters ---------- p : double array e : double array em : double array Returns ------- p1 : double array Notes ----- Wraps ERFA function ``eraLdsun``. The ERFA documentation is:: - - - - - - - - - e r a L d s u n - - - - - - - - - Deflection of starlight by the Sun. Given: p double[3] direction from observer to star (unit vector) e double[3] direction from Sun to observer (unit vector) em double distance from Sun to observer (au) Returned: p1 double[3] observer to deflected star (unit vector) Notes: 1) The source is presumed to be sufficiently distant that its directions seen from the Sun and the observer are essentially the same. 2) The deflection is restrained when the angle between the star and the center of the Sun is less than a threshold value, falling to zero deflection for zero separation. The chosen threshold value is within the solar limb for all solar-system applications, and is about 5 arcminutes for the case of a terrestrial observer. 3) The arguments p and p1 can be the same array. Called: eraLd light deflection by a solar-system body This revision: 2016 June 16 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ p1 = ufunc.ldsun(p, e, em) return p1 def pmpx(rc, dc, pr, pd, px, rv, pmt, pob): """ Proper motion and parallax. Parameters ---------- rc : double array dc : double array pr : double array pd : double array px : double array rv : double array pmt : double array pob : double array Returns ------- pco : double array Notes ----- Wraps ERFA function ``eraPmpx``. The ERFA documentation is:: - - - - - - - - e r a P m p x - - - - - - - - Proper motion and parallax. Given: rc,dc double ICRS RA,Dec at catalog epoch (radians) pr double RA proper motion (radians/year, Note 1) pd double Dec proper motion (radians/year) px double parallax (arcsec) rv double radial velocity (km/s, +ve if receding) pmt double proper motion time interval (SSB, Julian years) pob double[3] SSB to observer vector (au) Returned: pco double[3] coordinate direction (BCRS unit vector) Notes: 1) The proper motion in RA is dRA/dt rather than cos(Dec)*dRA/dt. 2) The proper motion time interval is for when the starlight reaches the solar system barycenter. 3) To avoid the need for iteration, the Roemer effect (i.e. the small annual modulation of the proper motion coming from the changing light time) is applied approximately, using the direction of the star at the catalog epoch. References: 1984 Astronomical Almanac, pp B39-B41. Urban, S. & Seidelmann, P. K. (eds), Explanatory Supplement to the Astronomical Almanac, 3rd ed., University Science Books (2013), Section 7.2. Called: eraPdp scalar product of two p-vectors eraPn decompose p-vector into modulus and direction This revision: 2021 April 3 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ pco = ufunc.pmpx(rc, dc, pr, pd, px, rv, pmt, pob) return pco def pmsafe(ra1, dec1, pmr1, pmd1, px1, rv1, ep1a, ep1b, ep2a, ep2b): """ Star proper motion: update star catalog data for space motion, with special handling to handle the zero parallax case. Parameters ---------- ra1 : double array dec1 : double array pmr1 : double array pmd1 : double array px1 : double array rv1 : double array ep1a : double array ep1b : double array ep2a : double array ep2b : double array Returns ------- ra2 : double array dec2 : double array pmr2 : double array pmd2 : double array px2 : double array rv2 : double array Notes ----- Wraps ERFA function ``eraPmsafe``. The ERFA documentation is:: - - - - - - - - - - e r a P m s a f e - - - - - - - - - - Star proper motion: update star catalog data for space motion, with special handling to handle the zero parallax case. Given: ra1 double right ascension (radians), before dec1 double declination (radians), before pmr1 double RA proper motion (radians/year), before pmd1 double Dec proper motion (radians/year), before px1 double parallax (arcseconds), before rv1 double radial velocity (km/s, +ve = receding), before ep1a double "before" epoch, part A (Note 1) ep1b double "before" epoch, part B (Note 1) ep2a double "after" epoch, part A (Note 1) ep2b double "after" epoch, part B (Note 1) Returned: ra2 double right ascension (radians), after dec2 double declination (radians), after pmr2 double RA proper motion (radians/year), after pmd2 double Dec proper motion (radians/year), after px2 double parallax (arcseconds), after rv2 double radial velocity (km/s, +ve = receding), after Returned (function value): int status: -1 = system error (should not occur) 0 = no warnings or errors 1 = distance overridden (Note 6) 2 = excessive velocity (Note 7) 4 = solution didn't converge (Note 8) else = binary logical OR of the above warnings Notes: 1) The starting and ending TDB epochs ep1a+ep1b and ep2a+ep2b are Julian Dates, apportioned in any convenient way between the two parts (A and B). For example, JD(TDB)=2450123.7 could be expressed in any of these ways, among others: epNa epNb 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) In accordance with normal star-catalog conventions, the object's right ascension and declination are freed from the effects of secular aberration. The frame, which is aligned to the catalog equator and equinox, is Lorentzian and centered on the SSB. The proper motions are the rate of change of the right ascension and declination at the catalog epoch and are in radians per TDB Julian year. The parallax and radial velocity are in the same frame. 3) Care is needed with units. The star coordinates are in radians and the proper motions in radians per Julian year, but the parallax is in arcseconds. 4) The RA proper motion is in terms of coordinate angle, not true angle. If the catalog uses arcseconds for both RA and Dec proper motions, the RA proper motion will need to be divided by cos(Dec) before use. 5) Straight-line motion at constant speed, in the inertial frame, is assumed. 6) An extremely small (or zero or negative) parallax is overridden to ensure that the object is at a finite but very large distance, but not so large that the proper motion is equivalent to a large but safe speed (about 0.1c using the chosen constant). A warning status of 1 is added to the status if this action has been taken. 7) If the space velocity is a significant fraction of c (see the constant VMAX in the function eraStarpv), it is arbitrarily set to zero. When this action occurs, 2 is added to the status. 8) The relativistic adjustment carried out in the eraStarpv function involves an iterative calculation. If the process fails to converge within a set number of iterations, 4 is added to the status. Called: eraSeps angle between two points eraStarpm update star catalog data for space motion This revision: 2014 July 1 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ ra2, dec2, pmr2, pmd2, px2, rv2, c_retval = ufunc.pmsafe( ra1, dec1, pmr1, pmd1, px1, rv1, ep1a, ep1b, ep2a, ep2b) check_errwarn(c_retval, 'pmsafe') return ra2, dec2, pmr2, pmd2, px2, rv2 STATUS_CODES['pmsafe'] = { -1: 'system error (should not occur)', 0: 'no warnings or errors', 1: 'distance overridden (Note 6)', 2: 'excessive velocity (Note 7)', 4: "solution didn't converge (Note 8)", 'else': 'binary logical OR of the above warnings', } def pvtob(elong, phi, hm, xp, yp, sp, theta): """ Position and velocity of a terrestrial observing station. Parameters ---------- elong : double array phi : double array hm : double array xp : double array yp : double array sp : double array theta : double array Returns ------- pv : double array Notes ----- Wraps ERFA function ``eraPvtob``. The ERFA documentation is:: - - - - - - - - - e r a P v t o b - - - - - - - - - Position and velocity of a terrestrial observing station. Given: elong double longitude (radians, east +ve, Note 1) phi double latitude (geodetic, radians, Note 1) hm double height above ref. ellipsoid (geodetic, m) xp,yp double coordinates of the pole (radians, Note 2) sp double the TIO locator s' (radians, Note 2) theta double Earth rotation angle (radians, Note 3) Returned: pv double[2][3] position/velocity vector (m, m/s, CIRS) Notes: 1) The terrestrial coordinates are with respect to the ERFA_WGS84 reference ellipsoid. 2) xp and yp are the coordinates (in radians) of the Celestial Intermediate Pole with respect to the International Terrestrial Reference System (see IERS Conventions), measured along the meridians 0 and 90 deg west respectively. sp is the TIO locator s', in radians, which positions the Terrestrial Intermediate Origin on the equator. For many applications, xp, yp and (especially) sp can be set to zero. 3) If theta is Greenwich apparent sidereal time instead of Earth rotation angle, the result is with respect to the true equator and equinox of date, i.e. with the x-axis at the equinox rather than the celestial intermediate origin. 4) The velocity units are meters per UT1 second, not per SI second. This is unlikely to have any practical consequences in the modern era. 5) No validation is performed on the arguments. Error cases that could lead to arithmetic exceptions are trapped by the eraGd2gc function, and the result set to zeros. References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) Urban, S. & Seidelmann, P. K. (eds), Explanatory Supplement to the Astronomical Almanac, 3rd ed., University Science Books (2013), Section 7.4.3.3. Called: eraGd2gc geodetic to geocentric transformation eraPom00 polar motion matrix eraTrxp product of transpose of r-matrix and p-vector This revision: 2021 February 24 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ pv = ufunc.pvtob(elong, phi, hm, xp, yp, sp, theta) return pv def refco(phpa, tc, rh, wl): """ Determine the constants A and B in the atmospheric refraction model dZ = A tan Z + B tan^3 Z. Parameters ---------- phpa : double array tc : double array rh : double array wl : double array Returns ------- refa : double array refb : double array Notes ----- Wraps ERFA function ``eraRefco``. The ERFA documentation is:: - - - - - - - - - e r a R e f c o - - - - - - - - - Determine the constants A and B in the atmospheric refraction model dZ = A tan Z + B tan^3 Z. Z is the "observed" zenith distance (i.e. affected by refraction) and dZ is what to add to Z to give the "topocentric" (i.e. in vacuo) zenith distance. Given: phpa double pressure at the observer (hPa = millibar) tc double ambient temperature at the observer (deg C) rh double relative humidity at the observer (range 0-1) wl double wavelength (micrometers) Returned: refa double tan Z coefficient (radians) refb double tan^3 Z coefficient (radians) Notes: 1) The model balances speed and accuracy to give good results in applications where performance at low altitudes is not paramount. Performance is maintained across a range of conditions, and applies to both optical/IR and radio. 2) The model omits the effects of (i) height above sea level (apart from the reduced pressure itself), (ii) latitude (i.e. the flattening of the Earth), (iii) variations in tropospheric lapse rate and (iv) dispersive effects in the radio. The model was tested using the following range of conditions: lapse rates 0.0055, 0.0065, 0.0075 deg/meter latitudes 0, 25, 50, 75 degrees heights 0, 2500, 5000 meters ASL pressures mean for height -10% to +5% in steps of 5% temperatures -10 deg to +20 deg with respect to 280 deg at SL relative humidity 0, 0.5, 1 wavelengths 0.4, 0.6, ... 2 micron, + radio zenith distances 15, 45, 75 degrees The accuracy with respect to raytracing through a model atmosphere was as follows: worst RMS optical/IR 62 mas 8 mas radio 319 mas 49 mas For this particular set of conditions: lapse rate 0.0065 K/meter latitude 50 degrees sea level pressure 1005 mb temperature 280.15 K humidity 80% wavelength 5740 Angstroms the results were as follows: ZD raytrace eraRefco Saastamoinen 10 10.27 10.27 10.27 20 21.19 21.20 21.19 30 33.61 33.61 33.60 40 48.82 48.83 48.81 45 58.16 58.18 58.16 50 69.28 69.30 69.27 55 82.97 82.99 82.95 60 100.51 100.54 100.50 65 124.23 124.26 124.20 70 158.63 158.68 158.61 72 177.32 177.37 177.31 74 200.35 200.38 200.32 76 229.45 229.43 229.42 78 267.44 267.29 267.41 80 319.13 318.55 319.10 deg arcsec arcsec arcsec The values for Saastamoinen's formula (which includes terms up to tan^5) are taken from Hohenkerk and Sinclair (1985). 3) A wl value in the range 0-100 selects the optical/IR case and is wavelength in micrometers. Any value outside this range selects the radio case. 4) Outlandish input parameters are silently limited to mathematically safe values. Zero pressure is permissible, and causes zeroes to be returned. 5) The algorithm draws on several sources, as follows: a) The formula for the saturation vapour pressure of water as a function of temperature and temperature is taken from Equations (A4.5-A4.7) of Gill (1982). b) The formula for the water vapour pressure, given the saturation pressure and the relative humidity, is from Crane (1976), Equation (2.5.5). c) The refractivity of air is a function of temperature, total pressure, water-vapour pressure and, in the case of optical/IR, wavelength. The formulae for the two cases are developed from Hohenkerk & Sinclair (1985) and Rueger (2002). The IAG (1999) optical refractivity for dry air is used. d) The formula for beta, the ratio of the scale height of the atmosphere to the geocentric distance of the observer, is an adaption of Equation (9) from Stone (1996). The adaptations, arrived at empirically, consist of (i) a small adjustment to the coefficient and (ii) a humidity term for the radio case only. e) The formulae for the refraction constants as a function of n-1 and beta are from Green (1987), Equation (4.31). References: Crane, R.K., Meeks, M.L. (ed), "Refraction Effects in the Neutral Atmosphere", Methods of Experimental Physics: Astrophysics 12B, Academic Press, 1976. Gill, Adrian E., "Atmosphere-Ocean Dynamics", Academic Press, 1982. Green, R.M., "Spherical Astronomy", Cambridge University Press, 1987. Hohenkerk, C.Y., & Sinclair, A.T., NAO Technical Note No. 63, 1985. IAG Resolutions adopted at the XXIIth General Assembly in Birmingham, 1999, Resolution 3. Rueger, J.M., "Refractive Index Formulae for Electronic Distance Measurement with Radio and Millimetre Waves", in Unisurv Report S-68, School of Surveying and Spatial Information Systems, University of New South Wales, Sydney, Australia, 2002. Stone, Ronald C., P.A.S.P. 108, 1051-1058, 1996. This revision: 2021 February 24 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ refa, refb = ufunc.refco(phpa, tc, rh, wl) return refa, refb def epv00(date1, date2): """ Earth position and velocity, heliocentric and barycentric, with respect to the Barycentric Celestial Reference System. Parameters ---------- date1 : double array date2 : double array Returns ------- pvh : double array pvb : double array Notes ----- Wraps ERFA function ``eraEpv00``. The ERFA documentation is:: - - - - - - - - - e r a E p v 0 0 - - - - - - - - - Earth position and velocity, heliocentric and barycentric, with respect to the Barycentric Celestial Reference System. Given: date1,date2 double TDB date (Note 1) Returned: pvh double[2][3] heliocentric Earth position/velocity pvb double[2][3] barycentric Earth position/velocity Returned (function value): int status: 0 = OK +1 = warning: date outside the range 1900-2100 AD Notes: 1) The TDB date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TDB)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. However, the accuracy of the result is more likely to be limited by the algorithm itself than the way the date has been expressed. n.b. TT can be used instead of TDB in most applications. 2) On return, the arrays pvh and pvb contain the following: pvh[0][0] x } pvh[0][1] y } heliocentric position, au pvh[0][2] z } pvh[1][0] xdot } pvh[1][1] ydot } heliocentric velocity, au/d pvh[1][2] zdot } pvb[0][0] x } pvb[0][1] y } barycentric position, au pvb[0][2] z } pvb[1][0] xdot } pvb[1][1] ydot } barycentric velocity, au/d pvb[1][2] zdot } The vectors are with respect to the Barycentric Celestial Reference System. The time unit is one day in TDB. 3) The function is a SIMPLIFIED SOLUTION from the planetary theory VSOP2000 (X. Moisson, P. Bretagnon, 2001, Celes. Mechanics & Dyn. Astron., 80, 3/4, 205-213) and is an adaptation of original Fortran code supplied by P. Bretagnon (private comm., 2000). 4) Comparisons over the time span 1900-2100 with this simplified solution and the JPL DE405 ephemeris give the following results: RMS max Heliocentric: position error 3.7 11.2 km velocity error 1.4 5.0 mm/s Barycentric: position error 4.6 13.4 km velocity error 1.4 4.9 mm/s Comparisons with the JPL DE406 ephemeris show that by 1800 and 2200 the position errors are approximately double their 1900-2100 size. By 1500 and 2500 the deterioration is a factor of 10 and by 1000 and 3000 a factor of 60. The velocity accuracy falls off at about half that rate. 5) It is permissible to use the same array for pvh and pvb, which will receive the barycentric values. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ pvh, pvb, c_retval = ufunc.epv00(date1, date2) check_errwarn(c_retval, 'epv00') return pvh, pvb STATUS_CODES['epv00'] = { 0: 'OK', 1: 'warning: date outsidethe range 1900-2100 AD', } def moon98(date1, date2): """ Approximate geocentric position and velocity of the Moon. Parameters ---------- date1 : double array date2 : double array Returns ------- pv : double array Notes ----- Wraps ERFA function ``eraMoon98``. The ERFA documentation is:: - - - - - - - - - - e r a M o o n 9 8 - - - - - - - - - - Approximate geocentric position and velocity of the Moon. n.b. Not IAU-endorsed and without canonical status. Given: date1 double TT date part A (Notes 1,4) date2 double TT date part B (Notes 1,4) Returned: pv double[2][3] Moon p,v, GCRS (AU, AU/d, Note 5) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. The limited accuracy of the present algorithm is such that any of the methods is satisfactory. 2) This function is a full implementation of the algorithm published by Meeus (see reference) except that the light-time correction to the Moon's mean longitude has been omitted. 3) Comparisons with ELP/MPP02 over the interval 1950-2100 gave RMS errors of 2.9 arcsec in geocentric direction, 6.1 km in position and 36 mm/s in velocity. The worst case errors were 18.3 arcsec in geocentric direction, 31.7 km in position and 172 mm/s in velocity. 4) The original algorithm is expressed in terms of "dynamical time", which can either be TDB or TT without any significant change in accuracy. UT cannot be used without incurring significant errors (30 arcsec in the present era) due to the Moon's 0.5 arcsec/sec movement. 5) The result is with respect to the GCRS (the same as J2000.0 mean equator and equinox to within 23 mas). 6) Velocity is obtained by a complete analytical differentiation of the Meeus model. 7) The Meeus algorithm generates position and velocity in mean ecliptic coordinates of date, which the present function then rotates into GCRS. Because the ecliptic system is precessing, there is a coupling between this spin (about 1.4 degrees per century) and the Moon position that produces a small velocity contribution. In the present function this effect is neglected as it corresponds to a maximum difference of less than 3 mm/s and increases the RMS error by only 0.4%. References: Meeus, J., Astronomical Algorithms, 2nd edition, Willmann-Bell, 1998, p337. Simon, J.L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G. & Laskar, J., Astron.Astrophys., 1994, 282, 663 Defined in erfam.h: ERFA_DAU astronomical unit (m) ERFA_DJC days per Julian century ERFA_DJ00 reference epoch (J2000.0), Julian Date ERFA_DD2R degrees to radians Called: eraS2pv spherical coordinates to pv-vector eraPfw06 bias-precession F-W angles, IAU 2006 eraIr initialize r-matrix to identity eraRz rotate around Z-axis eraRx rotate around X-axis eraRxpv product of r-matrix and pv-vector This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ pv = ufunc.moon98(date1, date2) return pv def plan94(date1, date2, np): """ Approximate heliocentric position and velocity of a nominated major planet: Mercury, Venus, EMB, Mars, Jupiter, Saturn, Uranus or Neptune (but not the Earth itself). Parameters ---------- date1 : double array date2 : double array np : int array Returns ------- pv : double array Notes ----- Wraps ERFA function ``eraPlan94``. The ERFA documentation is:: - - - - - - - - - - e r a P l a n 9 4 - - - - - - - - - - Approximate heliocentric position and velocity of a nominated major planet: Mercury, Venus, EMB, Mars, Jupiter, Saturn, Uranus or Neptune (but not the Earth itself). n.b. Not IAU-endorsed and without canonical status. Given: date1 double TDB date part A (Note 1) date2 double TDB date part B (Note 1) np int planet (1=Mercury, 2=Venus, 3=EMB, 4=Mars, 5=Jupiter, 6=Saturn, 7=Uranus, 8=Neptune) Returned (argument): pv double[2][3] planet p,v (heliocentric, J2000.0, au,au/d) Returned (function value): int status: -1 = illegal NP (outside 1-8) 0 = OK +1 = warning: year outside 1000-3000 +2 = warning: failed to converge Notes: 1) The date date1+date2 is in the TDB time scale (in practice TT can be used) and is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TDB)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. The limited accuracy of the present algorithm is such that any of the methods is satisfactory. 2) If an np value outside the range 1-8 is supplied, an error status (function value -1) is returned and the pv vector set to zeroes. 3) For np=3 the result is for the Earth-Moon Barycenter. To obtain the heliocentric position and velocity of the Earth, use instead the ERFA function eraEpv00. 4) On successful return, the array pv contains the following: pv[0][0] x } pv[0][1] y } heliocentric position, au pv[0][2] z } pv[1][0] xdot } pv[1][1] ydot } heliocentric velocity, au/d pv[1][2] zdot } The reference frame is equatorial and is with respect to the mean equator and equinox of epoch J2000.0. 5) The algorithm is due to J.L. Simon, P. Bretagnon, J. Chapront, M. Chapront-Touze, G. Francou and J. Laskar (Bureau des Longitudes, Paris, France). From comparisons with JPL ephemeris DE102, they quote the following maximum errors over the interval 1800-2050: L (arcsec) B (arcsec) R (km) Mercury 4 1 300 Venus 5 1 800 EMB 6 1 1000 Mars 17 1 7700 Jupiter 71 5 76000 Saturn 81 13 267000 Uranus 86 7 712000 Neptune 11 1 253000 Over the interval 1000-3000, they report that the accuracy is no worse than 1.5 times that over 1800-2050. Outside 1000-3000 the accuracy declines. Comparisons of the present function with the JPL DE200 ephemeris give the following RMS errors over the interval 1960-2025: position (km) velocity (m/s) Mercury 334 0.437 Venus 1060 0.855 EMB 2010 0.815 Mars 7690 1.98 Jupiter 71700 7.70 Saturn 199000 19.4 Uranus 564000 16.4 Neptune 158000 14.4 Comparisons against DE200 over the interval 1800-2100 gave the following maximum absolute differences. (The results using DE406 were essentially the same.) L (arcsec) B (arcsec) R (km) Rdot (m/s) Mercury 7 1 500 0.7 Venus 7 1 1100 0.9 EMB 9 1 1300 1.0 Mars 26 1 9000 2.5 Jupiter 78 6 82000 8.2 Saturn 87 14 263000 24.6 Uranus 86 7 661000 27.4 Neptune 11 2 248000 21.4 6) The present ERFA re-implementation of the original Simon et al. Fortran code differs from the original in the following respects: C instead of Fortran. The date is supplied in two parts. The result is returned only in equatorial Cartesian form; the ecliptic longitude, latitude and radius vector are not returned. The result is in the J2000.0 equatorial frame, not ecliptic. More is done in-line: there are fewer calls to subroutines. Different error/warning status values are used. A different Kepler's-equation-solver is used (avoiding use of double precision complex). Polynomials in t are nested to minimize rounding errors. Explicit double constants are used to avoid mixed-mode expressions. None of the above changes affects the result significantly. 7) The returned status indicates the most serious condition encountered during execution of the function. Illegal np is considered the most serious, overriding failure to converge, which in turn takes precedence over the remote date warning. Called: eraAnpm normalize angle into range +/- pi Reference: Simon, J.L, Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., and Laskar, J., Astron.Astrophys., 282, 663 (1994). This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ pv, c_retval = ufunc.plan94(date1, date2, np) check_errwarn(c_retval, 'plan94') return pv STATUS_CODES['plan94'] = { -1: 'illegal NP (outside 1-8)', 0: 'OK', 1: 'warning: year outside 1000-3000', 2: 'warning: failed to converge', } def fad03(t): """ Fundamental argument, IERS Conventions (2003): mean elongation of the Moon from the Sun. Parameters ---------- t : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraFad03``. The ERFA documentation is:: - - - - - - - - - e r a F a d 0 3 - - - - - - - - - Fundamental argument, IERS Conventions (2003): mean elongation of the Moon from the Sun. Given: t double TDB, Julian centuries since J2000.0 (Note 1) Returned (function value): double D, radians (Note 2) Notes: 1) Though t is strictly TDB, it is usually more convenient to use TT, which makes no significant difference. 2) The expression used is as adopted in IERS Conventions (2003) and is from Simon et al. (1994). References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) Simon, J.-L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J. 1994, Astron.Astrophys. 282, 663-683 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.fad03(t) return c_retval def fae03(t): """ Fundamental argument, IERS Conventions (2003): mean longitude of Earth. Parameters ---------- t : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraFae03``. The ERFA documentation is:: - - - - - - - - - e r a F a e 0 3 - - - - - - - - - Fundamental argument, IERS Conventions (2003): mean longitude of Earth. Given: t double TDB, Julian centuries since J2000.0 (Note 1) Returned (function value): double mean longitude of Earth, radians (Note 2) Notes: 1) Though t is strictly TDB, it is usually more convenient to use TT, which makes no significant difference. 2) The expression used is as adopted in IERS Conventions (2003) and comes from Souchay et al. (1999) after Simon et al. (1994). References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) Simon, J.-L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J. 1994, Astron.Astrophys. 282, 663-683 Souchay, J., Loysel, B., Kinoshita, H., Folgueira, M. 1999, Astron.Astrophys.Supp.Ser. 135, 111 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.fae03(t) return c_retval def faf03(t): """ Fundamental argument, IERS Conventions (2003): mean longitude of the Moon minus mean longitude of the ascending node. Parameters ---------- t : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraFaf03``. The ERFA documentation is:: - - - - - - - - - e r a F a f 0 3 - - - - - - - - - Fundamental argument, IERS Conventions (2003): mean longitude of the Moon minus mean longitude of the ascending node. Given: t double TDB, Julian centuries since J2000.0 (Note 1) Returned (function value): double F, radians (Note 2) Notes: 1) Though t is strictly TDB, it is usually more convenient to use TT, which makes no significant difference. 2) The expression used is as adopted in IERS Conventions (2003) and is from Simon et al. (1994). References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) Simon, J.-L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J. 1994, Astron.Astrophys. 282, 663-683 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.faf03(t) return c_retval def faju03(t): """ Fundamental argument, IERS Conventions (2003): mean longitude of Jupiter. Parameters ---------- t : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraFaju03``. The ERFA documentation is:: - - - - - - - - - - e r a F a j u 0 3 - - - - - - - - - - Fundamental argument, IERS Conventions (2003): mean longitude of Jupiter. Given: t double TDB, Julian centuries since J2000.0 (Note 1) Returned (function value): double mean longitude of Jupiter, radians (Note 2) Notes: 1) Though t is strictly TDB, it is usually more convenient to use TT, which makes no significant difference. 2) The expression used is as adopted in IERS Conventions (2003) and comes from Souchay et al. (1999) after Simon et al. (1994). References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) Simon, J.-L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J. 1994, Astron.Astrophys. 282, 663-683 Souchay, J., Loysel, B., Kinoshita, H., Folgueira, M. 1999, Astron.Astrophys.Supp.Ser. 135, 111 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.faju03(t) return c_retval def fal03(t): """ Fundamental argument, IERS Conventions (2003): mean anomaly of the Moon. Parameters ---------- t : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraFal03``. The ERFA documentation is:: - - - - - - - - - e r a F a l 0 3 - - - - - - - - - Fundamental argument, IERS Conventions (2003): mean anomaly of the Moon. Given: t double TDB, Julian centuries since J2000.0 (Note 1) Returned (function value): double l, radians (Note 2) Notes: 1) Though t is strictly TDB, it is usually more convenient to use TT, which makes no significant difference. 2) The expression used is as adopted in IERS Conventions (2003) and is from Simon et al. (1994). References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) Simon, J.-L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J. 1994, Astron.Astrophys. 282, 663-683 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.fal03(t) return c_retval def falp03(t): """ Fundamental argument, IERS Conventions (2003): mean anomaly of the Sun. Parameters ---------- t : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraFalp03``. The ERFA documentation is:: - - - - - - - - - - e r a F a l p 0 3 - - - - - - - - - - Fundamental argument, IERS Conventions (2003): mean anomaly of the Sun. Given: t double TDB, Julian centuries since J2000.0 (Note 1) Returned (function value): double l', radians (Note 2) Notes: 1) Though t is strictly TDB, it is usually more convenient to use TT, which makes no significant difference. 2) The expression used is as adopted in IERS Conventions (2003) and is from Simon et al. (1994). References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) Simon, J.-L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J. 1994, Astron.Astrophys. 282, 663-683 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.falp03(t) return c_retval def fama03(t): """ Fundamental argument, IERS Conventions (2003): mean longitude of Mars. Parameters ---------- t : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraFama03``. The ERFA documentation is:: - - - - - - - - - - e r a F a m a 0 3 - - - - - - - - - - Fundamental argument, IERS Conventions (2003): mean longitude of Mars. Given: t double TDB, Julian centuries since J2000.0 (Note 1) Returned (function value): double mean longitude of Mars, radians (Note 2) Notes: 1) Though t is strictly TDB, it is usually more convenient to use TT, which makes no significant difference. 2) The expression used is as adopted in IERS Conventions (2003) and comes from Souchay et al. (1999) after Simon et al. (1994). References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) Simon, J.-L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J. 1994, Astron.Astrophys. 282, 663-683 Souchay, J., Loysel, B., Kinoshita, H., Folgueira, M. 1999, Astron.Astrophys.Supp.Ser. 135, 111 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.fama03(t) return c_retval def fame03(t): """ Fundamental argument, IERS Conventions (2003): mean longitude of Mercury. Parameters ---------- t : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraFame03``. The ERFA documentation is:: - - - - - - - - - - e r a F a m e 0 3 - - - - - - - - - - Fundamental argument, IERS Conventions (2003): mean longitude of Mercury. Given: t double TDB, Julian centuries since J2000.0 (Note 1) Returned (function value): double mean longitude of Mercury, radians (Note 2) Notes: 1) Though t is strictly TDB, it is usually more convenient to use TT, which makes no significant difference. 2) The expression used is as adopted in IERS Conventions (2003) and comes from Souchay et al. (1999) after Simon et al. (1994). References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) Simon, J.-L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J. 1994, Astron.Astrophys. 282, 663-683 Souchay, J., Loysel, B., Kinoshita, H., Folgueira, M. 1999, Astron.Astrophys.Supp.Ser. 135, 111 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.fame03(t) return c_retval def fane03(t): """ Fundamental argument, IERS Conventions (2003): mean longitude of Neptune. Parameters ---------- t : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraFane03``. The ERFA documentation is:: - - - - - - - - - - e r a F a n e 0 3 - - - - - - - - - - Fundamental argument, IERS Conventions (2003): mean longitude of Neptune. Given: t double TDB, Julian centuries since J2000.0 (Note 1) Returned (function value): double mean longitude of Neptune, radians (Note 2) Notes: 1) Though t is strictly TDB, it is usually more convenient to use TT, which makes no significant difference. 2) The expression used is as adopted in IERS Conventions (2003) and is adapted from Simon et al. (1994). References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) Simon, J.-L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J. 1994, Astron.Astrophys. 282, 663-683 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.fane03(t) return c_retval def faom03(t): """ Fundamental argument, IERS Conventions (2003): mean longitude of the Moon's ascending node. Parameters ---------- t : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraFaom03``. The ERFA documentation is:: - - - - - - - - - - e r a F a o m 0 3 - - - - - - - - - - Fundamental argument, IERS Conventions (2003): mean longitude of the Moon's ascending node. Given: t double TDB, Julian centuries since J2000.0 (Note 1) Returned (function value): double Omega, radians (Note 2) Notes: 1) Though t is strictly TDB, it is usually more convenient to use TT, which makes no significant difference. 2) The expression used is as adopted in IERS Conventions (2003) and is from Simon et al. (1994). References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) Simon, J.-L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J., 1994, Astron.Astrophys. 282, 663-683. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.faom03(t) return c_retval def fapa03(t): """ Fundamental argument, IERS Conventions (2003): general accumulated precession in longitude. Parameters ---------- t : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraFapa03``. The ERFA documentation is:: - - - - - - - - - - e r a F a p a 0 3 - - - - - - - - - - Fundamental argument, IERS Conventions (2003): general accumulated precession in longitude. Given: t double TDB, Julian centuries since J2000.0 (Note 1) Returned (function value): double general precession in longitude, radians (Note 2) Notes: 1) Though t is strictly TDB, it is usually more convenient to use TT, which makes no significant difference. 2) The expression used is as adopted in IERS Conventions (2003). It is taken from Kinoshita & Souchay (1990) and comes originally from Lieske et al. (1977). References: Kinoshita, H. and Souchay J. 1990, Celest.Mech. and Dyn.Astron. 48, 187 Lieske, J.H., Lederle, T., Fricke, W. & Morando, B. 1977, Astron.Astrophys. 58, 1-16 McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.fapa03(t) return c_retval def fasa03(t): """ Fundamental argument, IERS Conventions (2003): mean longitude of Saturn. Parameters ---------- t : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraFasa03``. The ERFA documentation is:: - - - - - - - - - - e r a F a s a 0 3 - - - - - - - - - - Fundamental argument, IERS Conventions (2003): mean longitude of Saturn. Given: t double TDB, Julian centuries since J2000.0 (Note 1) Returned (function value): double mean longitude of Saturn, radians (Note 2) Notes: 1) Though t is strictly TDB, it is usually more convenient to use TT, which makes no significant difference. 2) The expression used is as adopted in IERS Conventions (2003) and comes from Souchay et al. (1999) after Simon et al. (1994). References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) Simon, J.-L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J. 1994, Astron.Astrophys. 282, 663-683 Souchay, J., Loysel, B., Kinoshita, H., Folgueira, M. 1999, Astron.Astrophys.Supp.Ser. 135, 111 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.fasa03(t) return c_retval def faur03(t): """ Fundamental argument, IERS Conventions (2003): mean longitude of Uranus. Parameters ---------- t : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraFaur03``. The ERFA documentation is:: - - - - - - - - - - e r a F a u r 0 3 - - - - - - - - - - Fundamental argument, IERS Conventions (2003): mean longitude of Uranus. Given: t double TDB, Julian centuries since J2000.0 (Note 1) Returned (function value): double mean longitude of Uranus, radians (Note 2) Notes: 1) Though t is strictly TDB, it is usually more convenient to use TT, which makes no significant difference. 2) The expression used is as adopted in IERS Conventions (2003) and is adapted from Simon et al. (1994). References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) Simon, J.-L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J. 1994, Astron.Astrophys. 282, 663-683 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.faur03(t) return c_retval def fave03(t): """ Fundamental argument, IERS Conventions (2003): mean longitude of Venus. Parameters ---------- t : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraFave03``. The ERFA documentation is:: - - - - - - - - - - e r a F a v e 0 3 - - - - - - - - - - Fundamental argument, IERS Conventions (2003): mean longitude of Venus. Given: t double TDB, Julian centuries since J2000.0 (Note 1) Returned (function value): double mean longitude of Venus, radians (Note 2) Notes: 1) Though t is strictly TDB, it is usually more convenient to use TT, which makes no significant difference. 2) The expression used is as adopted in IERS Conventions (2003) and comes from Souchay et al. (1999) after Simon et al. (1994). References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) Simon, J.-L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J. 1994, Astron.Astrophys. 282, 663-683 Souchay, J., Loysel, B., Kinoshita, H., Folgueira, M. 1999, Astron.Astrophys.Supp.Ser. 135, 111 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.fave03(t) return c_retval def bi00(): """ Frame bias components of IAU 2000 precession-nutation models; part of the Mathews-Herring-Buffett (MHB2000) nutation series, with additions. Parameters ---------- Returns ------- dpsibi : double array depsbi : double array dra : double array Notes ----- Wraps ERFA function ``eraBi00``. The ERFA documentation is:: - - - - - - - - e r a B i 0 0 - - - - - - - - Frame bias components of IAU 2000 precession-nutation models; part of the Mathews-Herring-Buffett (MHB2000) nutation series, with additions. Returned: dpsibi,depsbi double longitude and obliquity corrections dra double the ICRS RA of the J2000.0 mean equinox Notes: 1) The frame bias corrections in longitude and obliquity (radians) are required in order to correct for the offset between the GCRS pole and the mean J2000.0 pole. They define, with respect to the GCRS frame, a J2000.0 mean pole that is consistent with the rest of the IAU 2000A precession-nutation model. 2) In addition to the displacement of the pole, the complete description of the frame bias requires also an offset in right ascension. This is not part of the IAU 2000A model, and is from Chapront et al. (2002). It is returned in radians. 3) This is a supplemented implementation of one aspect of the IAU 2000A nutation model, formally adopted by the IAU General Assembly in 2000, namely MHB2000 (Mathews et al. 2002). References: Chapront, J., Chapront-Touze, M. & Francou, G., Astron. Astrophys., 387, 700, 2002. Mathews, P.M., Herring, T.A., Buffet, B.A., "Modeling of nutation and precession: New nutation series for nonrigid Earth and insights into the Earth's interior", J.Geophys.Res., 107, B4, 2002. The MHB2000 code itself was obtained on 2002 September 9 from ftp://maia.usno.navy.mil/conv2000/chapter5/IAU2000A. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ dpsibi, depsbi, dra = ufunc.bi00() return dpsibi, depsbi, dra def bp00(date1, date2): """ Frame bias and precession, IAU 2000. Parameters ---------- date1 : double array date2 : double array Returns ------- rb : double array rp : double array rbp : double array Notes ----- Wraps ERFA function ``eraBp00``. The ERFA documentation is:: - - - - - - - - e r a B p 0 0 - - - - - - - - Frame bias and precession, IAU 2000. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: rb double[3][3] frame bias matrix (Note 2) rp double[3][3] precession matrix (Note 3) rbp double[3][3] bias-precession matrix (Note 4) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The matrix rb transforms vectors from GCRS to mean J2000.0 by applying frame bias. 3) The matrix rp transforms vectors from J2000.0 mean equator and equinox to mean equator and equinox of date by applying precession. 4) The matrix rbp transforms vectors from GCRS to mean equator and equinox of date by applying frame bias then precession. It is the product rp x rb. 5) It is permissible to re-use the same array in the returned arguments. The arrays are filled in the order given. Called: eraBi00 frame bias components, IAU 2000 eraPr00 IAU 2000 precession adjustments eraIr initialize r-matrix to identity eraRx rotate around X-axis eraRy rotate around Y-axis eraRz rotate around Z-axis eraCr copy r-matrix eraRxr product of two r-matrices Reference: "Expressions for the Celestial Intermediate Pole and Celestial Ephemeris Origin consistent with the IAU 2000A precession- nutation model", Astron.Astrophys. 400, 1145-1154 (2003) n.b. The celestial ephemeris origin (CEO) was renamed "celestial intermediate origin" (CIO) by IAU 2006 Resolution 2. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rb, rp, rbp = ufunc.bp00(date1, date2) return rb, rp, rbp def bp06(date1, date2): """ Frame bias and precession, IAU 2006. Parameters ---------- date1 : double array date2 : double array Returns ------- rb : double array rp : double array rbp : double array Notes ----- Wraps ERFA function ``eraBp06``. The ERFA documentation is:: - - - - - - - - e r a B p 0 6 - - - - - - - - Frame bias and precession, IAU 2006. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: rb double[3][3] frame bias matrix (Note 2) rp double[3][3] precession matrix (Note 3) rbp double[3][3] bias-precession matrix (Note 4) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The matrix rb transforms vectors from GCRS to mean J2000.0 by applying frame bias. 3) The matrix rp transforms vectors from mean J2000.0 to mean of date by applying precession. 4) The matrix rbp transforms vectors from GCRS to mean of date by applying frame bias then precession. It is the product rp x rb. 5) It is permissible to re-use the same array in the returned arguments. The arrays are filled in the order given. Called: eraPfw06 bias-precession F-W angles, IAU 2006 eraFw2m F-W angles to r-matrix eraPmat06 PB matrix, IAU 2006 eraTr transpose r-matrix eraRxr product of two r-matrices eraCr copy r-matrix References: Capitaine, N. & Wallace, P.T., 2006, Astron.Astrophys. 450, 855 Wallace, P.T. & Capitaine, N., 2006, Astron.Astrophys. 459, 981 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rb, rp, rbp = ufunc.bp06(date1, date2) return rb, rp, rbp def bpn2xy(rbpn): """ Extract from the bias-precession-nutation matrix the X,Y coordinates of the Celestial Intermediate Pole. Parameters ---------- rbpn : double array Returns ------- x : double array y : double array Notes ----- Wraps ERFA function ``eraBpn2xy``. The ERFA documentation is:: - - - - - - - - - - e r a B p n 2 x y - - - - - - - - - - Extract from the bias-precession-nutation matrix the X,Y coordinates of the Celestial Intermediate Pole. Given: rbpn double[3][3] celestial-to-true matrix (Note 1) Returned: x,y double Celestial Intermediate Pole (Note 2) Notes: 1) The matrix rbpn transforms vectors from GCRS to true equator (and CIO or equinox) of date, and therefore the Celestial Intermediate Pole unit vector is the bottom row of the matrix. 2) The arguments x,y are components of the Celestial Intermediate Pole unit vector in the Geocentric Celestial Reference System. Reference: "Expressions for the Celestial Intermediate Pole and Celestial Ephemeris Origin consistent with the IAU 2000A precession- nutation model", Astron.Astrophys. 400, 1145-1154 (2003) n.b. The celestial ephemeris origin (CEO) was renamed "celestial intermediate origin" (CIO) by IAU 2006 Resolution 2. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ x, y = ufunc.bpn2xy(rbpn) return x, y def c2i00a(date1, date2): """ Form the celestial-to-intermediate matrix for a given date using the IAU 2000A precession-nutation model. Parameters ---------- date1 : double array date2 : double array Returns ------- rc2i : double array Notes ----- Wraps ERFA function ``eraC2i00a``. The ERFA documentation is:: - - - - - - - - - - e r a C 2 i 0 0 a - - - - - - - - - - Form the celestial-to-intermediate matrix for a given date using the IAU 2000A precession-nutation model. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: rc2i double[3][3] celestial-to-intermediate matrix (Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The matrix rc2i is the first stage in the transformation from celestial to terrestrial coordinates: [TRS] = RPOM * R_3(ERA) * rc2i * [CRS] = rc2t * [CRS] where [CRS] is a vector in the Geocentric Celestial Reference System and [TRS] is a vector in the International Terrestrial Reference System (see IERS Conventions 2003), ERA is the Earth Rotation Angle and RPOM is the polar motion matrix. 3) A faster, but slightly less accurate, result (about 1 mas) can be obtained by using instead the eraC2i00b function. Called: eraPnm00a classical NPB matrix, IAU 2000A eraC2ibpn celestial-to-intermediate matrix, given NPB matrix References: "Expressions for the Celestial Intermediate Pole and Celestial Ephemeris Origin consistent with the IAU 2000A precession- nutation model", Astron.Astrophys. 400, 1145-1154 (2003) n.b. The celestial ephemeris origin (CEO) was renamed "celestial intermediate origin" (CIO) by IAU 2006 Resolution 2. McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rc2i = ufunc.c2i00a(date1, date2) return rc2i def c2i00b(date1, date2): """ Form the celestial-to-intermediate matrix for a given date using the IAU 2000B precession-nutation model. Parameters ---------- date1 : double array date2 : double array Returns ------- rc2i : double array Notes ----- Wraps ERFA function ``eraC2i00b``. The ERFA documentation is:: - - - - - - - - - - e r a C 2 i 0 0 b - - - - - - - - - - Form the celestial-to-intermediate matrix for a given date using the IAU 2000B precession-nutation model. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: rc2i double[3][3] celestial-to-intermediate matrix (Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The matrix rc2i is the first stage in the transformation from celestial to terrestrial coordinates: [TRS] = RPOM * R_3(ERA) * rc2i * [CRS] = rc2t * [CRS] where [CRS] is a vector in the Geocentric Celestial Reference System and [TRS] is a vector in the International Terrestrial Reference System (see IERS Conventions 2003), ERA is the Earth Rotation Angle and RPOM is the polar motion matrix. 3) The present function is faster, but slightly less accurate (about 1 mas), than the eraC2i00a function. Called: eraPnm00b classical NPB matrix, IAU 2000B eraC2ibpn celestial-to-intermediate matrix, given NPB matrix References: "Expressions for the Celestial Intermediate Pole and Celestial Ephemeris Origin consistent with the IAU 2000A precession- nutation model", Astron.Astrophys. 400, 1145-1154 (2003) n.b. The celestial ephemeris origin (CEO) was renamed "celestial intermediate origin" (CIO) by IAU 2006 Resolution 2. McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rc2i = ufunc.c2i00b(date1, date2) return rc2i def c2i06a(date1, date2): """ Form the celestial-to-intermediate matrix for a given date using the IAU 2006 precession and IAU 2000A nutation models. Parameters ---------- date1 : double array date2 : double array Returns ------- rc2i : double array Notes ----- Wraps ERFA function ``eraC2i06a``. The ERFA documentation is:: - - - - - - - - - - e r a C 2 i 0 6 a - - - - - - - - - - Form the celestial-to-intermediate matrix for a given date using the IAU 2006 precession and IAU 2000A nutation models. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: rc2i double[3][3] celestial-to-intermediate matrix (Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The matrix rc2i is the first stage in the transformation from celestial to terrestrial coordinates: [TRS] = RPOM * R_3(ERA) * rc2i * [CRS] = RC2T * [CRS] where [CRS] is a vector in the Geocentric Celestial Reference System and [TRS] is a vector in the International Terrestrial Reference System (see IERS Conventions 2003), ERA is the Earth Rotation Angle and RPOM is the polar motion matrix. Called: eraPnm06a classical NPB matrix, IAU 2006/2000A eraBpn2xy extract CIP X,Y coordinates from NPB matrix eraS06 the CIO locator s, given X,Y, IAU 2006 eraC2ixys celestial-to-intermediate matrix, given X,Y and s References: McCarthy, D. D., Petit, G. (eds.), 2004, IERS Conventions (2003), IERS Technical Note No. 32, BKG This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rc2i = ufunc.c2i06a(date1, date2) return rc2i def c2ibpn(date1, date2, rbpn): """ Form the celestial-to-intermediate matrix for a given date given the bias-precession-nutation matrix. Parameters ---------- date1 : double array date2 : double array rbpn : double array Returns ------- rc2i : double array Notes ----- Wraps ERFA function ``eraC2ibpn``. The ERFA documentation is:: - - - - - - - - - - e r a C 2 i b p n - - - - - - - - - - Form the celestial-to-intermediate matrix for a given date given the bias-precession-nutation matrix. IAU 2000. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) rbpn double[3][3] celestial-to-true matrix (Note 2) Returned: rc2i double[3][3] celestial-to-intermediate matrix (Note 3) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The matrix rbpn transforms vectors from GCRS to true equator (and CIO or equinox) of date. Only the CIP (bottom row) is used. 3) The matrix rc2i is the first stage in the transformation from celestial to terrestrial coordinates: [TRS] = RPOM * R_3(ERA) * rc2i * [CRS] = RC2T * [CRS] where [CRS] is a vector in the Geocentric Celestial Reference System and [TRS] is a vector in the International Terrestrial Reference System (see IERS Conventions 2003), ERA is the Earth Rotation Angle and RPOM is the polar motion matrix. 4) Although its name does not include "00", This function is in fact specific to the IAU 2000 models. Called: eraBpn2xy extract CIP X,Y coordinates from NPB matrix eraC2ixy celestial-to-intermediate matrix, given X,Y References: "Expressions for the Celestial Intermediate Pole and Celestial Ephemeris Origin consistent with the IAU 2000A precession- nutation model", Astron.Astrophys. 400, 1145-1154 (2003) n.b. The celestial ephemeris origin (CEO) was renamed "celestial intermediate origin" (CIO) by IAU 2006 Resolution 2. McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rc2i = ufunc.c2ibpn(date1, date2, rbpn) return rc2i def c2ixy(date1, date2, x, y): """ Form the celestial to intermediate-frame-of-date matrix for a given date when the CIP X,Y coordinates are known. Parameters ---------- date1 : double array date2 : double array x : double array y : double array Returns ------- rc2i : double array Notes ----- Wraps ERFA function ``eraC2ixy``. The ERFA documentation is:: - - - - - - - - - e r a C 2 i x y - - - - - - - - - Form the celestial to intermediate-frame-of-date matrix for a given date when the CIP X,Y coordinates are known. IAU 2000. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) x,y double Celestial Intermediate Pole (Note 2) Returned: rc2i double[3][3] celestial-to-intermediate matrix (Note 3) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The Celestial Intermediate Pole coordinates are the x,y components of the unit vector in the Geocentric Celestial Reference System. 3) The matrix rc2i is the first stage in the transformation from celestial to terrestrial coordinates: [TRS] = RPOM * R_3(ERA) * rc2i * [CRS] = RC2T * [CRS] where [CRS] is a vector in the Geocentric Celestial Reference System and [TRS] is a vector in the International Terrestrial Reference System (see IERS Conventions 2003), ERA is the Earth Rotation Angle and RPOM is the polar motion matrix. 4) Although its name does not include "00", This function is in fact specific to the IAU 2000 models. Called: eraC2ixys celestial-to-intermediate matrix, given X,Y and s eraS00 the CIO locator s, given X,Y, IAU 2000A Reference: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rc2i = ufunc.c2ixy(date1, date2, x, y) return rc2i def c2ixys(x, y, s): """ Form the celestial to intermediate-frame-of-date matrix given the CIP X,Y and the CIO locator s. Parameters ---------- x : double array y : double array s : double array Returns ------- rc2i : double array Notes ----- Wraps ERFA function ``eraC2ixys``. The ERFA documentation is:: - - - - - - - - - - e r a C 2 i x y s - - - - - - - - - - Form the celestial to intermediate-frame-of-date matrix given the CIP X,Y and the CIO locator s. Given: x,y double Celestial Intermediate Pole (Note 1) s double the CIO locator s (Note 2) Returned: rc2i double[3][3] celestial-to-intermediate matrix (Note 3) Notes: 1) The Celestial Intermediate Pole coordinates are the x,y components of the unit vector in the Geocentric Celestial Reference System. 2) The CIO locator s (in radians) positions the Celestial Intermediate Origin on the equator of the CIP. 3) The matrix rc2i is the first stage in the transformation from celestial to terrestrial coordinates: [TRS] = RPOM * R_3(ERA) * rc2i * [CRS] = RC2T * [CRS] where [CRS] is a vector in the Geocentric Celestial Reference System and [TRS] is a vector in the International Terrestrial Reference System (see IERS Conventions 2003), ERA is the Earth Rotation Angle and RPOM is the polar motion matrix. Called: eraIr initialize r-matrix to identity eraRz rotate around Z-axis eraRy rotate around Y-axis Reference: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rc2i = ufunc.c2ixys(x, y, s) return rc2i def c2t00a(tta, ttb, uta, utb, xp, yp): """ Form the celestial to terrestrial matrix given the date, the UT1 and the polar motion, using the IAU 2000A precession-nutation model. Parameters ---------- tta : double array ttb : double array uta : double array utb : double array xp : double array yp : double array Returns ------- rc2t : double array Notes ----- Wraps ERFA function ``eraC2t00a``. The ERFA documentation is:: - - - - - - - - - - e r a C 2 t 0 0 a - - - - - - - - - - Form the celestial to terrestrial matrix given the date, the UT1 and the polar motion, using the IAU 2000A precession-nutation model. Given: tta,ttb double TT as a 2-part Julian Date (Note 1) uta,utb double UT1 as a 2-part Julian Date (Note 1) xp,yp double CIP coordinates (radians, Note 2) Returned: rc2t double[3][3] celestial-to-terrestrial matrix (Note 3) Notes: 1) The TT and UT1 dates tta+ttb and uta+utb are Julian Dates, apportioned in any convenient way between the arguments uta and utb. For example, JD(UT1)=2450123.7 could be expressed in any of these ways, among others: uta utb 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 and MJD methods are good compromises between resolution and convenience. In the case of uta,utb, the date & time method is best matched to the Earth rotation angle algorithm used: maximum precision is delivered when the uta argument is for 0hrs UT1 on the day in question and the utb argument lies in the range 0 to 1, or vice versa. 2) The arguments xp and yp are the coordinates (in radians) of the Celestial Intermediate Pole with respect to the International Terrestrial Reference System (see IERS Conventions 2003), measured along the meridians 0 and 90 deg west respectively. 3) The matrix rc2t transforms from celestial to terrestrial coordinates: [TRS] = RPOM * R_3(ERA) * RC2I * [CRS] = rc2t * [CRS] where [CRS] is a vector in the Geocentric Celestial Reference System and [TRS] is a vector in the International Terrestrial Reference System (see IERS Conventions 2003), RC2I is the celestial-to-intermediate matrix, ERA is the Earth rotation angle and RPOM is the polar motion matrix. 4) A faster, but slightly less accurate, result (about 1 mas) can be obtained by using instead the eraC2t00b function. Called: eraC2i00a celestial-to-intermediate matrix, IAU 2000A eraEra00 Earth rotation angle, IAU 2000 eraSp00 the TIO locator s', IERS 2000 eraPom00 polar motion matrix eraC2tcio form CIO-based celestial-to-terrestrial matrix Reference: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rc2t = ufunc.c2t00a(tta, ttb, uta, utb, xp, yp) return rc2t def c2t00b(tta, ttb, uta, utb, xp, yp): """ Form the celestial to terrestrial matrix given the date, the UT1 and the polar motion, using the IAU 2000B precession-nutation model. Parameters ---------- tta : double array ttb : double array uta : double array utb : double array xp : double array yp : double array Returns ------- rc2t : double array Notes ----- Wraps ERFA function ``eraC2t00b``. The ERFA documentation is:: - - - - - - - - - - e r a C 2 t 0 0 b - - - - - - - - - - Form the celestial to terrestrial matrix given the date, the UT1 and the polar motion, using the IAU 2000B precession-nutation model. Given: tta,ttb double TT as a 2-part Julian Date (Note 1) uta,utb double UT1 as a 2-part Julian Date (Note 1) xp,yp double coordinates of the pole (radians, Note 2) Returned: rc2t double[3][3] celestial-to-terrestrial matrix (Note 3) Notes: 1) The TT and UT1 dates tta+ttb and uta+utb are Julian Dates, apportioned in any convenient way between the arguments uta and utb. For example, JD(UT1)=2450123.7 could be expressed in any of these ways, among others: uta utb 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 and MJD methods are good compromises between resolution and convenience. In the case of uta,utb, the date & time method is best matched to the Earth rotation angle algorithm used: maximum precision is delivered when the uta argument is for 0hrs UT1 on the day in question and the utb argument lies in the range 0 to 1, or vice versa. 2) The arguments xp and yp are the coordinates (in radians) of the Celestial Intermediate Pole with respect to the International Terrestrial Reference System (see IERS Conventions 2003), measured along the meridians 0 and 90 deg west respectively. 3) The matrix rc2t transforms from celestial to terrestrial coordinates: [TRS] = RPOM * R_3(ERA) * RC2I * [CRS] = rc2t * [CRS] where [CRS] is a vector in the Geocentric Celestial Reference System and [TRS] is a vector in the International Terrestrial Reference System (see IERS Conventions 2003), RC2I is the celestial-to-intermediate matrix, ERA is the Earth rotation angle and RPOM is the polar motion matrix. 4) The present function is faster, but slightly less accurate (about 1 mas), than the eraC2t00a function. Called: eraC2i00b celestial-to-intermediate matrix, IAU 2000B eraEra00 Earth rotation angle, IAU 2000 eraPom00 polar motion matrix eraC2tcio form CIO-based celestial-to-terrestrial matrix Reference: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rc2t = ufunc.c2t00b(tta, ttb, uta, utb, xp, yp) return rc2t def c2t06a(tta, ttb, uta, utb, xp, yp): """ Form the celestial to terrestrial matrix given the date, the UT1 and the polar motion, using the IAU 2006/2000A precession-nutation model. Parameters ---------- tta : double array ttb : double array uta : double array utb : double array xp : double array yp : double array Returns ------- rc2t : double array Notes ----- Wraps ERFA function ``eraC2t06a``. The ERFA documentation is:: - - - - - - - - - - e r a C 2 t 0 6 a - - - - - - - - - - Form the celestial to terrestrial matrix given the date, the UT1 and the polar motion, using the IAU 2006/2000A precession-nutation model. Given: tta,ttb double TT as a 2-part Julian Date (Note 1) uta,utb double UT1 as a 2-part Julian Date (Note 1) xp,yp double coordinates of the pole (radians, Note 2) Returned: rc2t double[3][3] celestial-to-terrestrial matrix (Note 3) Notes: 1) The TT and UT1 dates tta+ttb and uta+utb are Julian Dates, apportioned in any convenient way between the arguments uta and utb. For example, JD(UT1)=2450123.7 could be expressed in any of these ways, among others: uta utb 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 and MJD methods are good compromises between resolution and convenience. In the case of uta,utb, the date & time method is best matched to the Earth rotation angle algorithm used: maximum precision is delivered when the uta argument is for 0hrs UT1 on the day in question and the utb argument lies in the range 0 to 1, or vice versa. 2) The arguments xp and yp are the coordinates (in radians) of the Celestial Intermediate Pole with respect to the International Terrestrial Reference System (see IERS Conventions 2003), measured along the meridians 0 and 90 deg west respectively. 3) The matrix rc2t transforms from celestial to terrestrial coordinates: [TRS] = RPOM * R_3(ERA) * RC2I * [CRS] = rc2t * [CRS] where [CRS] is a vector in the Geocentric Celestial Reference System and [TRS] is a vector in the International Terrestrial Reference System (see IERS Conventions 2003), RC2I is the celestial-to-intermediate matrix, ERA is the Earth rotation angle and RPOM is the polar motion matrix. Called: eraC2i06a celestial-to-intermediate matrix, IAU 2006/2000A eraEra00 Earth rotation angle, IAU 2000 eraSp00 the TIO locator s', IERS 2000 eraPom00 polar motion matrix eraC2tcio form CIO-based celestial-to-terrestrial matrix Reference: McCarthy, D. D., Petit, G. (eds.), 2004, IERS Conventions (2003), IERS Technical Note No. 32, BKG This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rc2t = ufunc.c2t06a(tta, ttb, uta, utb, xp, yp) return rc2t def c2tcio(rc2i, era, rpom): """ Assemble the celestial to terrestrial matrix from CIO-based components (the celestial-to-intermediate matrix, the Earth Rotation Angle and the polar motion matrix). Parameters ---------- rc2i : double array era : double array rpom : double array Returns ------- rc2t : double array Notes ----- Wraps ERFA function ``eraC2tcio``. The ERFA documentation is:: - - - - - - - - - - e r a C 2 t c i o - - - - - - - - - - Assemble the celestial to terrestrial matrix from CIO-based components (the celestial-to-intermediate matrix, the Earth Rotation Angle and the polar motion matrix). Given: rc2i double[3][3] celestial-to-intermediate matrix era double Earth rotation angle (radians) rpom double[3][3] polar-motion matrix Returned: rc2t double[3][3] celestial-to-terrestrial matrix Notes: 1) This function constructs the rotation matrix that transforms vectors in the celestial system into vectors in the terrestrial system. It does so starting from precomputed components, namely the matrix which rotates from celestial coordinates to the intermediate frame, the Earth rotation angle and the polar motion matrix. One use of the present function is when generating a series of celestial-to-terrestrial matrices where only the Earth Rotation Angle changes, avoiding the considerable overhead of recomputing the precession-nutation more often than necessary to achieve given accuracy objectives. 2) The relationship between the arguments is as follows: [TRS] = RPOM * R_3(ERA) * rc2i * [CRS] = rc2t * [CRS] where [CRS] is a vector in the Geocentric Celestial Reference System and [TRS] is a vector in the International Terrestrial Reference System (see IERS Conventions 2003). Called: eraCr copy r-matrix eraRz rotate around Z-axis eraRxr product of two r-matrices Reference: McCarthy, D. D., Petit, G. (eds.), 2004, IERS Conventions (2003), IERS Technical Note No. 32, BKG This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rc2t = ufunc.c2tcio(rc2i, era, rpom) return rc2t def c2teqx(rbpn, gst, rpom): """ Assemble the celestial to terrestrial matrix from equinox-based components (the celestial-to-true matrix, the Greenwich Apparent Sidereal Time and the polar motion matrix). Parameters ---------- rbpn : double array gst : double array rpom : double array Returns ------- rc2t : double array Notes ----- Wraps ERFA function ``eraC2teqx``. The ERFA documentation is:: - - - - - - - - - - e r a C 2 t e q x - - - - - - - - - - Assemble the celestial to terrestrial matrix from equinox-based components (the celestial-to-true matrix, the Greenwich Apparent Sidereal Time and the polar motion matrix). Given: rbpn double[3][3] celestial-to-true matrix gst double Greenwich (apparent) Sidereal Time (radians) rpom double[3][3] polar-motion matrix Returned: rc2t double[3][3] celestial-to-terrestrial matrix (Note 2) Notes: 1) This function constructs the rotation matrix that transforms vectors in the celestial system into vectors in the terrestrial system. It does so starting from precomputed components, namely the matrix which rotates from celestial coordinates to the true equator and equinox of date, the Greenwich Apparent Sidereal Time and the polar motion matrix. One use of the present function is when generating a series of celestial-to-terrestrial matrices where only the Sidereal Time changes, avoiding the considerable overhead of recomputing the precession-nutation more often than necessary to achieve given accuracy objectives. 2) The relationship between the arguments is as follows: [TRS] = rpom * R_3(gst) * rbpn * [CRS] = rc2t * [CRS] where [CRS] is a vector in the Geocentric Celestial Reference System and [TRS] is a vector in the International Terrestrial Reference System (see IERS Conventions 2003). Called: eraCr copy r-matrix eraRz rotate around Z-axis eraRxr product of two r-matrices Reference: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rc2t = ufunc.c2teqx(rbpn, gst, rpom) return rc2t def c2tpe(tta, ttb, uta, utb, dpsi, deps, xp, yp): """ Form the celestial to terrestrial matrix given the date, the UT1, the nutation and the polar motion. Parameters ---------- tta : double array ttb : double array uta : double array utb : double array dpsi : double array deps : double array xp : double array yp : double array Returns ------- rc2t : double array Notes ----- Wraps ERFA function ``eraC2tpe``. The ERFA documentation is:: - - - - - - - - - e r a C 2 t p e - - - - - - - - - Form the celestial to terrestrial matrix given the date, the UT1, the nutation and the polar motion. IAU 2000. Given: tta,ttb double TT as a 2-part Julian Date (Note 1) uta,utb double UT1 as a 2-part Julian Date (Note 1) dpsi,deps double nutation (Note 2) xp,yp double coordinates of the pole (radians, Note 3) Returned: rc2t double[3][3] celestial-to-terrestrial matrix (Note 4) Notes: 1) The TT and UT1 dates tta+ttb and uta+utb are Julian Dates, apportioned in any convenient way between the arguments uta and utb. For example, JD(UT1)=2450123.7 could be expressed in any of these ways, among others: uta utb 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 and MJD methods are good compromises between resolution and convenience. In the case of uta,utb, the date & time method is best matched to the Earth rotation angle algorithm used: maximum precision is delivered when the uta argument is for 0hrs UT1 on the day in question and the utb argument lies in the range 0 to 1, or vice versa. 2) The caller is responsible for providing the nutation components; they are in longitude and obliquity, in radians and are with respect to the equinox and ecliptic of date. For high-accuracy applications, free core nutation should be included as well as any other relevant corrections to the position of the CIP. 3) The arguments xp and yp are the coordinates (in radians) of the Celestial Intermediate Pole with respect to the International Terrestrial Reference System (see IERS Conventions 2003), measured along the meridians 0 and 90 deg west respectively. 4) The matrix rc2t transforms from celestial to terrestrial coordinates: [TRS] = RPOM * R_3(GST) * RBPN * [CRS] = rc2t * [CRS] where [CRS] is a vector in the Geocentric Celestial Reference System and [TRS] is a vector in the International Terrestrial Reference System (see IERS Conventions 2003), RBPN is the bias-precession-nutation matrix, GST is the Greenwich (apparent) Sidereal Time and RPOM is the polar motion matrix. 5) Although its name does not include "00", This function is in fact specific to the IAU 2000 models. Called: eraPn00 bias/precession/nutation results, IAU 2000 eraGmst00 Greenwich mean sidereal time, IAU 2000 eraSp00 the TIO locator s', IERS 2000 eraEe00 equation of the equinoxes, IAU 2000 eraPom00 polar motion matrix eraC2teqx form equinox-based celestial-to-terrestrial matrix Reference: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rc2t = ufunc.c2tpe(tta, ttb, uta, utb, dpsi, deps, xp, yp) return rc2t def c2txy(tta, ttb, uta, utb, x, y, xp, yp): """ Form the celestial to terrestrial matrix given the date, the UT1, the CIP coordinates and the polar motion. Parameters ---------- tta : double array ttb : double array uta : double array utb : double array x : double array y : double array xp : double array yp : double array Returns ------- rc2t : double array Notes ----- Wraps ERFA function ``eraC2txy``. The ERFA documentation is:: - - - - - - - - - e r a C 2 t x y - - - - - - - - - Form the celestial to terrestrial matrix given the date, the UT1, the CIP coordinates and the polar motion. IAU 2000. Given: tta,ttb double TT as a 2-part Julian Date (Note 1) uta,utb double UT1 as a 2-part Julian Date (Note 1) x,y double Celestial Intermediate Pole (Note 2) xp,yp double coordinates of the pole (radians, Note 3) Returned: rc2t double[3][3] celestial-to-terrestrial matrix (Note 4) Notes: 1) The TT and UT1 dates tta+ttb and uta+utb are Julian Dates, apportioned in any convenient way between the arguments uta and utb. For example, JD(UT1)=2450123.7 could be expressed in any o these ways, among others: uta utb 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 and MJD methods are good compromises between resolution and convenience. In the case of uta,utb, the date & time method is best matched to the Earth rotation angle algorithm used: maximum precision is delivered when the uta argument is for 0hrs UT1 on the day in question and the utb argument lies in the range 0 to 1, or vice versa. 2) The Celestial Intermediate Pole coordinates are the x,y components of the unit vector in the Geocentric Celestial Reference System. 3) The arguments xp and yp are the coordinates (in radians) of the Celestial Intermediate Pole with respect to the International Terrestrial Reference System (see IERS Conventions 2003), measured along the meridians 0 and 90 deg west respectively. 4) The matrix rc2t transforms from celestial to terrestrial coordinates: [TRS] = RPOM * R_3(ERA) * RC2I * [CRS] = rc2t * [CRS] where [CRS] is a vector in the Geocentric Celestial Reference System and [TRS] is a vector in the International Terrestrial Reference System (see IERS Conventions 2003), ERA is the Earth Rotation Angle and RPOM is the polar motion matrix. 5) Although its name does not include "00", This function is in fact specific to the IAU 2000 models. Called: eraC2ixy celestial-to-intermediate matrix, given X,Y eraEra00 Earth rotation angle, IAU 2000 eraSp00 the TIO locator s', IERS 2000 eraPom00 polar motion matrix eraC2tcio form CIO-based celestial-to-terrestrial matrix Reference: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rc2t = ufunc.c2txy(tta, ttb, uta, utb, x, y, xp, yp) return rc2t def eo06a(date1, date2): """ Equation of the origins, IAU 2006 precession and IAU 2000A nutation. Parameters ---------- date1 : double array date2 : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraEo06a``. The ERFA documentation is:: - - - - - - - - - e r a E o 0 6 a - - - - - - - - - Equation of the origins, IAU 2006 precession and IAU 2000A nutation. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned (function value): double the equation of the origins in radians Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The equation of the origins is the distance between the true equinox and the celestial intermediate origin and, equivalently, the difference between Earth rotation angle and Greenwich apparent sidereal time (ERA-GST). It comprises the precession (since J2000.0) in right ascension plus the equation of the equinoxes (including the small correction terms). Called: eraPnm06a classical NPB matrix, IAU 2006/2000A eraBpn2xy extract CIP X,Y coordinates from NPB matrix eraS06 the CIO locator s, given X,Y, IAU 2006 eraEors equation of the origins, given NPB matrix and s References: Capitaine, N. & Wallace, P.T., 2006, Astron.Astrophys. 450, 855 Wallace, P.T. & Capitaine, N., 2006, Astron.Astrophys. 459, 981 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.eo06a(date1, date2) return c_retval def eors(rnpb, s): """ Equation of the origins, given the classical NPB matrix and the quantity s. Parameters ---------- rnpb : double array s : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraEors``. The ERFA documentation is:: - - - - - - - - e r a E o r s - - - - - - - - Equation of the origins, given the classical NPB matrix and the quantity s. Given: rnpb double[3][3] classical nutation x precession x bias matrix s double the quantity s (the CIO locator) in radians Returned (function value): double the equation of the origins in radians Notes: 1) The equation of the origins is the distance between the true equinox and the celestial intermediate origin and, equivalently, the difference between Earth rotation angle and Greenwich apparent sidereal time (ERA-GST). It comprises the precession (since J2000.0) in right ascension plus the equation of the equinoxes (including the small correction terms). 2) The algorithm is from Wallace & Capitaine (2006). References: Capitaine, N. & Wallace, P.T., 2006, Astron.Astrophys. 450, 855 Wallace, P. & Capitaine, N., 2006, Astron.Astrophys. 459, 981 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.eors(rnpb, s) return c_retval def fw2m(gamb, phib, psi, eps): """ Form rotation matrix given the Fukushima-Williams angles. Parameters ---------- gamb : double array phib : double array psi : double array eps : double array Returns ------- r : double array Notes ----- Wraps ERFA function ``eraFw2m``. The ERFA documentation is:: - - - - - - - - e r a F w 2 m - - - - - - - - Form rotation matrix given the Fukushima-Williams angles. Given: gamb double F-W angle gamma_bar (radians) phib double F-W angle phi_bar (radians) psi double F-W angle psi (radians) eps double F-W angle epsilon (radians) Returned: r double[3][3] rotation matrix Notes: 1) Naming the following points: e = J2000.0 ecliptic pole, p = GCRS pole, E = ecliptic pole of date, and P = CIP, the four Fukushima-Williams angles are as follows: gamb = gamma = epE phib = phi = pE psi = psi = pEP eps = epsilon = EP 2) The matrix representing the combined effects of frame bias, precession and nutation is: NxPxB = R_1(-eps).R_3(-psi).R_1(phib).R_3(gamb) 3) The present function can construct three different matrices, depending on which angles are supplied as the arguments gamb, phib, psi and eps: o To obtain the nutation x precession x frame bias matrix, first generate the four precession angles known conventionally as gamma_bar, phi_bar, psi_bar and epsilon_A, then generate the nutation components Dpsi and Depsilon and add them to psi_bar and epsilon_A, and finally call the present function using those four angles as arguments. o To obtain the precession x frame bias matrix, generate the four precession angles and call the present function. o To obtain the frame bias matrix, generate the four precession angles for date J2000.0 and call the present function. The nutation-only and precession-only matrices can if necessary be obtained by combining these three appropriately. Called: eraIr initialize r-matrix to identity eraRz rotate around Z-axis eraRx rotate around X-axis References: Capitaine, N. & Wallace, P.T., 2006, Astron.Astrophys. 450, 855 Hilton, J. et al., 2006, Celest.Mech.Dyn.Astron. 94, 351 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ r = ufunc.fw2m(gamb, phib, psi, eps) return r def fw2xy(gamb, phib, psi, eps): """ CIP X,Y given Fukushima-Williams bias-precession-nutation angles. Parameters ---------- gamb : double array phib : double array psi : double array eps : double array Returns ------- x : double array y : double array Notes ----- Wraps ERFA function ``eraFw2xy``. The ERFA documentation is:: - - - - - - - - - e r a F w 2 x y - - - - - - - - - CIP X,Y given Fukushima-Williams bias-precession-nutation angles. Given: gamb double F-W angle gamma_bar (radians) phib double F-W angle phi_bar (radians) psi double F-W angle psi (radians) eps double F-W angle epsilon (radians) Returned: x,y double CIP unit vector X,Y Notes: 1) Naming the following points: e = J2000.0 ecliptic pole, p = GCRS pole E = ecliptic pole of date, and P = CIP, the four Fukushima-Williams angles are as follows: gamb = gamma = epE phib = phi = pE psi = psi = pEP eps = epsilon = EP 2) The matrix representing the combined effects of frame bias, precession and nutation is: NxPxB = R_1(-epsA).R_3(-psi).R_1(phib).R_3(gamb) The returned values x,y are elements [2][0] and [2][1] of the matrix. Near J2000.0, they are essentially angles in radians. Called: eraFw2m F-W angles to r-matrix eraBpn2xy extract CIP X,Y coordinates from NPB matrix Reference: Hilton, J. et al., 2006, Celest.Mech.Dyn.Astron. 94, 351 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ x, y = ufunc.fw2xy(gamb, phib, psi, eps) return x, y def ltp(epj): """ Long-term precession matrix. Parameters ---------- epj : double array Returns ------- rp : double array Notes ----- Wraps ERFA function ``eraLtp``. The ERFA documentation is:: - - - - - - - e r a L t p - - - - - - - Long-term precession matrix. Given: epj double Julian epoch (TT) Returned: rp double[3][3] precession matrix, J2000.0 to date Notes: 1) The matrix is in the sense P_date = rp x P_J2000, where P_J2000 is a vector with respect to the J2000.0 mean equator and equinox and P_date is the same vector with respect to the equator and equinox of epoch epj. 2) The Vondrak et al. (2011, 2012) 400 millennia precession model agrees with the IAU 2006 precession at J2000.0 and stays within 100 microarcseconds during the 20th and 21st centuries. It is accurate to a few arcseconds throughout the historical period, worsening to a few tenths of a degree at the end of the +/- 200,000 year time span. Called: eraLtpequ equator pole, long term eraLtpecl ecliptic pole, long term eraPxp vector product eraPn normalize vector References: Vondrak, J., Capitaine, N. and Wallace, P., 2011, New precession expressions, valid for long time intervals, Astron.Astrophys. 534, A22 Vondrak, J., Capitaine, N. and Wallace, P., 2012, New precession expressions, valid for long time intervals (Corrigendum), Astron.Astrophys. 541, C1 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rp = ufunc.ltp(epj) return rp def ltpb(epj): """ Long-term precession matrix, including ICRS frame bias. Parameters ---------- epj : double array Returns ------- rpb : double array Notes ----- Wraps ERFA function ``eraLtpb``. The ERFA documentation is:: - - - - - - - - e r a L t p b - - - - - - - - Long-term precession matrix, including ICRS frame bias. Given: epj double Julian epoch (TT) Returned: rpb double[3][3] precession-bias matrix, J2000.0 to date Notes: 1) The matrix is in the sense P_date = rpb x P_ICRS, where P_ICRS is a vector in the Geocentric Celestial Reference System, and P_date is the vector with respect to the Celestial Intermediate Reference System at that date but with nutation neglected. 2) A first order frame bias formulation is used, of sub- microarcsecond accuracy compared with a full 3D rotation. 3) The Vondrak et al. (2011, 2012) 400 millennia precession model agrees with the IAU 2006 precession at J2000.0 and stays within 100 microarcseconds during the 20th and 21st centuries. It is accurate to a few arcseconds throughout the historical period, worsening to a few tenths of a degree at the end of the +/- 200,000 year time span. References: Vondrak, J., Capitaine, N. and Wallace, P., 2011, New precession expressions, valid for long time intervals, Astron.Astrophys. 534, A22 Vondrak, J., Capitaine, N. and Wallace, P., 2012, New precession expressions, valid for long time intervals (Corrigendum), Astron.Astrophys. 541, C1 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rpb = ufunc.ltpb(epj) return rpb def ltpecl(epj): """ Long-term precession of the ecliptic. Parameters ---------- epj : double array Returns ------- vec : double array Notes ----- Wraps ERFA function ``eraLtpecl``. The ERFA documentation is:: - - - - - - - - - - e r a L t p e c l - - - - - - - - - - Long-term precession of the ecliptic. Given: epj double Julian epoch (TT) Returned: vec double[3] ecliptic pole unit vector Notes: 1) The returned vector is with respect to the J2000.0 mean equator and equinox. 2) The Vondrak et al. (2011, 2012) 400 millennia precession model agrees with the IAU 2006 precession at J2000.0 and stays within 100 microarcseconds during the 20th and 21st centuries. It is accurate to a few arcseconds throughout the historical period, worsening to a few tenths of a degree at the end of the +/- 200,000 year time span. References: Vondrak, J., Capitaine, N. and Wallace, P., 2011, New precession expressions, valid for long time intervals, Astron.Astrophys. 534, A22 Vondrak, J., Capitaine, N. and Wallace, P., 2012, New precession expressions, valid for long time intervals (Corrigendum), Astron.Astrophys. 541, C1 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ vec = ufunc.ltpecl(epj) return vec def ltpequ(epj): """ Long-term precession of the equator. Parameters ---------- epj : double array Returns ------- veq : double array Notes ----- Wraps ERFA function ``eraLtpequ``. The ERFA documentation is:: - - - - - - - - - - e r a L t p e q u - - - - - - - - - - Long-term precession of the equator. Given: epj double Julian epoch (TT) Returned: veq double[3] equator pole unit vector Notes: 1) The returned vector is with respect to the J2000.0 mean equator and equinox. 2) The Vondrak et al. (2011, 2012) 400 millennia precession model agrees with the IAU 2006 precession at J2000.0 and stays within 100 microarcseconds during the 20th and 21st centuries. It is accurate to a few arcseconds throughout the historical period, worsening to a few tenths of a degree at the end of the +/- 200,000 year time span. References: Vondrak, J., Capitaine, N. and Wallace, P., 2011, New precession expressions, valid for long time intervals, Astron.Astrophys. 534, A22 Vondrak, J., Capitaine, N. and Wallace, P., 2012, New precession expressions, valid for long time intervals (Corrigendum), Astron.Astrophys. 541, C1 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ veq = ufunc.ltpequ(epj) return veq def num00a(date1, date2): """ Form the matrix of nutation for a given date, IAU 2000A model. Parameters ---------- date1 : double array date2 : double array Returns ------- rmatn : double array Notes ----- Wraps ERFA function ``eraNum00a``. The ERFA documentation is:: - - - - - - - - - - e r a N u m 0 0 a - - - - - - - - - - Form the matrix of nutation for a given date, IAU 2000A model. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: rmatn double[3][3] nutation matrix Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The matrix operates in the sense V(true) = rmatn * V(mean), where the p-vector V(true) is with respect to the true equatorial triad of date and the p-vector V(mean) is with respect to the mean equatorial triad of date. 3) A faster, but slightly less accurate, result (about 1 mas) can be obtained by using instead the eraNum00b function. Called: eraPn00a bias/precession/nutation, IAU 2000A Reference: Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), Section 3.222-3 (p114). This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rmatn = ufunc.num00a(date1, date2) return rmatn def num00b(date1, date2): """ Form the matrix of nutation for a given date, IAU 2000B model. Parameters ---------- date1 : double array date2 : double array Returns ------- rmatn : double array Notes ----- Wraps ERFA function ``eraNum00b``. The ERFA documentation is:: - - - - - - - - - - e r a N u m 0 0 b - - - - - - - - - - Form the matrix of nutation for a given date, IAU 2000B model. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: rmatn double[3][3] nutation matrix Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The matrix operates in the sense V(true) = rmatn * V(mean), where the p-vector V(true) is with respect to the true equatorial triad of date and the p-vector V(mean) is with respect to the mean equatorial triad of date. 3) The present function is faster, but slightly less accurate (about 1 mas), than the eraNum00a function. Called: eraPn00b bias/precession/nutation, IAU 2000B Reference: Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), Section 3.222-3 (p114). This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rmatn = ufunc.num00b(date1, date2) return rmatn def num06a(date1, date2): """ Form the matrix of nutation for a given date, IAU 2006/2000A model. Parameters ---------- date1 : double array date2 : double array Returns ------- rmatn : double array Notes ----- Wraps ERFA function ``eraNum06a``. The ERFA documentation is:: - - - - - - - - - - e r a N u m 0 6 a - - - - - - - - - - Form the matrix of nutation for a given date, IAU 2006/2000A model. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: rmatn double[3][3] nutation matrix Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The matrix operates in the sense V(true) = rmatn * V(mean), where the p-vector V(true) is with respect to the true equatorial triad of date and the p-vector V(mean) is with respect to the mean equatorial triad of date. Called: eraObl06 mean obliquity, IAU 2006 eraNut06a nutation, IAU 2006/2000A eraNumat form nutation matrix Reference: Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), Section 3.222-3 (p114). This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rmatn = ufunc.num06a(date1, date2) return rmatn def numat(epsa, dpsi, deps): """ Form the matrix of nutation. Parameters ---------- epsa : double array dpsi : double array deps : double array Returns ------- rmatn : double array Notes ----- Wraps ERFA function ``eraNumat``. The ERFA documentation is:: - - - - - - - - - e r a N u m a t - - - - - - - - - Form the matrix of nutation. Given: epsa double mean obliquity of date (Note 1) dpsi,deps double nutation (Note 2) Returned: rmatn double[3][3] nutation matrix (Note 3) Notes: 1) The supplied mean obliquity epsa, must be consistent with the precession-nutation models from which dpsi and deps were obtained. 2) The caller is responsible for providing the nutation components; they are in longitude and obliquity, in radians and are with respect to the equinox and ecliptic of date. 3) The matrix operates in the sense V(true) = rmatn * V(mean), where the p-vector V(true) is with respect to the true equatorial triad of date and the p-vector V(mean) is with respect to the mean equatorial triad of date. Called: eraIr initialize r-matrix to identity eraRx rotate around X-axis eraRz rotate around Z-axis Reference: Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), Section 3.222-3 (p114). This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rmatn = ufunc.numat(epsa, dpsi, deps) return rmatn def nut00a(date1, date2): """ Nutation, IAU 2000A model (MHB2000 luni-solar and planetary nutation with free core nutation omitted). Parameters ---------- date1 : double array date2 : double array Returns ------- dpsi : double array deps : double array Notes ----- Wraps ERFA function ``eraNut00a``. The ERFA documentation is:: - - - - - - - - - - e r a N u t 0 0 a - - - - - - - - - - Nutation, IAU 2000A model (MHB2000 luni-solar and planetary nutation with free core nutation omitted). Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: dpsi,deps double nutation, luni-solar + planetary (Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The nutation components in longitude and obliquity are in radians and with respect to the equinox and ecliptic of date. The obliquity at J2000.0 is assumed to be the Lieske et al. (1977) value of 84381.448 arcsec. Both the luni-solar and planetary nutations are included. The latter are due to direct planetary nutations and the perturbations of the lunar and terrestrial orbits. 3) The function computes the MHB2000 nutation series with the associated corrections for planetary nutations. It is an implementation of the nutation part of the IAU 2000A precession- nutation model, formally adopted by the IAU General Assembly in 2000, namely MHB2000 (Mathews et al. 2002), but with the free core nutation (FCN - see Note 4) omitted. 4) The full MHB2000 model also contains contributions to the nutations in longitude and obliquity due to the free-excitation of the free-core-nutation during the period 1979-2000. These FCN terms, which are time-dependent and unpredictable, are NOT included in the present function and, if required, must be independently computed. With the FCN corrections included, the present function delivers a pole which is at current epochs accurate to a few hundred microarcseconds. The omission of FCN introduces further errors of about that size. 5) The present function provides classical nutation. The MHB2000 algorithm, from which it is adapted, deals also with (i) the offsets between the GCRS and mean poles and (ii) the adjustments in longitude and obliquity due to the changed precession rates. These additional functions, namely frame bias and precession adjustments, are supported by the ERFA functions eraBi00 and eraPr00. 6) The MHB2000 algorithm also provides "total" nutations, comprising the arithmetic sum of the frame bias, precession adjustments, luni-solar nutation and planetary nutation. These total nutations can be used in combination with an existing IAU 1976 precession implementation, such as eraPmat76, to deliver GCRS- to-true predictions of sub-mas accuracy at current dates. However, there are three shortcomings in the MHB2000 model that must be taken into account if more accurate or definitive results are required (see Wallace 2002): (i) The MHB2000 total nutations are simply arithmetic sums, yet in reality the various components are successive Euler rotations. This slight lack of rigor leads to cross terms that exceed 1 mas after a century. The rigorous procedure is to form the GCRS-to-true rotation matrix by applying the bias, precession and nutation in that order. (ii) Although the precession adjustments are stated to be with respect to Lieske et al. (1977), the MHB2000 model does not specify which set of Euler angles are to be used and how the adjustments are to be applied. The most literal and straightforward procedure is to adopt the 4-rotation epsilon_0, psi_A, omega_A, xi_A option, and to add DPSIPR to psi_A and DEPSPR to both omega_A and eps_A. (iii) The MHB2000 model predates the determination by Chapront et al. (2002) of a 14.6 mas displacement between the J2000.0 mean equinox and the origin of the ICRS frame. It should, however, be noted that neglecting this displacement when calculating star coordinates does not lead to a 14.6 mas change in right ascension, only a small second- order distortion in the pattern of the precession-nutation effect. For these reasons, the ERFA functions do not generate the "total nutations" directly, though they can of course easily be generated by calling eraBi00, eraPr00 and the present function and adding the results. 7) The MHB2000 model contains 41 instances where the same frequency appears multiple times, of which 38 are duplicates and three are triplicates. To keep the present code close to the original MHB algorithm, this small inefficiency has not been corrected. Called: eraFal03 mean anomaly of the Moon eraFaf03 mean argument of the latitude of the Moon eraFaom03 mean longitude of the Moon's ascending node eraFame03 mean longitude of Mercury eraFave03 mean longitude of Venus eraFae03 mean longitude of Earth eraFama03 mean longitude of Mars eraFaju03 mean longitude of Jupiter eraFasa03 mean longitude of Saturn eraFaur03 mean longitude of Uranus eraFapa03 general accumulated precession in longitude References: Chapront, J., Chapront-Touze, M. & Francou, G. 2002, Astron.Astrophys. 387, 700 Lieske, J.H., Lederle, T., Fricke, W. & Morando, B. 1977, Astron.Astrophys. 58, 1-16 Mathews, P.M., Herring, T.A., Buffet, B.A. 2002, J.Geophys.Res. 107, B4. The MHB_2000 code itself was obtained on 9th September 2002 from ftp//maia.usno.navy.mil/conv2000/chapter5/IAU2000A. Simon, J.-L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J. 1994, Astron.Astrophys. 282, 663-683 Souchay, J., Loysel, B., Kinoshita, H., Folgueira, M. 1999, Astron.Astrophys.Supp.Ser. 135, 111 Wallace, P.T., "Software for Implementing the IAU 2000 Resolutions", in IERS Workshop 5.1 (2002) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ dpsi, deps = ufunc.nut00a(date1, date2) return dpsi, deps def nut00b(date1, date2): """ Nutation, IAU 2000B model. Parameters ---------- date1 : double array date2 : double array Returns ------- dpsi : double array deps : double array Notes ----- Wraps ERFA function ``eraNut00b``. The ERFA documentation is:: - - - - - - - - - - e r a N u t 0 0 b - - - - - - - - - - Nutation, IAU 2000B model. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: dpsi,deps double nutation, luni-solar + planetary (Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The nutation components in longitude and obliquity are in radians and with respect to the equinox and ecliptic of date. The obliquity at J2000.0 is assumed to be the Lieske et al. (1977) value of 84381.448 arcsec. (The errors that result from using this function with the IAU 2006 value of 84381.406 arcsec can be neglected.) The nutation model consists only of luni-solar terms, but includes also a fixed offset which compensates for certain long- period planetary terms (Note 7). 3) This function is an implementation of the IAU 2000B abridged nutation model formally adopted by the IAU General Assembly in 2000. The function computes the MHB_2000_SHORT luni-solar nutation series (Luzum 2001), but without the associated corrections for the precession rate adjustments and the offset between the GCRS and J2000.0 mean poles. 4) The full IAU 2000A (MHB2000) nutation model contains nearly 1400 terms. The IAU 2000B model (McCarthy & Luzum 2003) contains only 77 terms, plus additional simplifications, yet still delivers results of 1 mas accuracy at present epochs. This combination of accuracy and size makes the IAU 2000B abridged nutation model suitable for most practical applications. The function delivers a pole accurate to 1 mas from 1900 to 2100 (usually better than 1 mas, very occasionally just outside 1 mas). The full IAU 2000A model, which is implemented in the function eraNut00a (q.v.), delivers considerably greater accuracy at current dates; however, to realize this improved accuracy, corrections for the essentially unpredictable free-core-nutation (FCN) must also be included. 5) The present function provides classical nutation. The MHB_2000_SHORT algorithm, from which it is adapted, deals also with (i) the offsets between the GCRS and mean poles and (ii) the adjustments in longitude and obliquity due to the changed precession rates. These additional functions, namely frame bias and precession adjustments, are supported by the ERFA functions eraBi00 and eraPr00. 6) The MHB_2000_SHORT algorithm also provides "total" nutations, comprising the arithmetic sum of the frame bias, precession adjustments, and nutation (luni-solar + planetary). These total nutations can be used in combination with an existing IAU 1976 precession implementation, such as eraPmat76, to deliver GCRS- to-true predictions of mas accuracy at current epochs. However, for symmetry with the eraNut00a function (q.v. for the reasons), the ERFA functions do not generate the "total nutations" directly. Should they be required, they could of course easily be generated by calling eraBi00, eraPr00 and the present function and adding the results. 7) The IAU 2000B model includes "planetary bias" terms that are fixed in size but compensate for long-period nutations. The amplitudes quoted in McCarthy & Luzum (2003), namely Dpsi = -1.5835 mas and Depsilon = +1.6339 mas, are optimized for the "total nutations" method described in Note 6. The Luzum (2001) values used in this ERFA implementation, namely -0.135 mas and +0.388 mas, are optimized for the "rigorous" method, where frame bias, precession and nutation are applied separately and in that order. During the interval 1995-2050, the ERFA implementation delivers a maximum error of 1.001 mas (not including FCN). References: Lieske, J.H., Lederle, T., Fricke, W., Morando, B., "Expressions for the precession quantities based upon the IAU /1976/ system of astronomical constants", Astron.Astrophys. 58, 1-2, 1-16. (1977) Luzum, B., private communication, 2001 (Fortran code MHB_2000_SHORT) McCarthy, D.D. & Luzum, B.J., "An abridged model of the precession-nutation of the celestial pole", Cel.Mech.Dyn.Astron. 85, 37-49 (2003) Simon, J.-L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J., Astron.Astrophys. 282, 663-683 (1994) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ dpsi, deps = ufunc.nut00b(date1, date2) return dpsi, deps def nut06a(date1, date2): """ IAU 2000A nutation with adjustments to match the IAU 2006 precession. Parameters ---------- date1 : double array date2 : double array Returns ------- dpsi : double array deps : double array Notes ----- Wraps ERFA function ``eraNut06a``. The ERFA documentation is:: - - - - - - - - - - e r a N u t 0 6 a - - - - - - - - - - IAU 2000A nutation with adjustments to match the IAU 2006 precession. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: dpsi,deps double nutation, luni-solar + planetary (Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The nutation components in longitude and obliquity are in radians and with respect to the mean equinox and ecliptic of date, IAU 2006 precession model (Hilton et al. 2006, Capitaine et al. 2005). 3) The function first computes the IAU 2000A nutation, then applies adjustments for (i) the consequences of the change in obliquity from the IAU 1980 ecliptic to the IAU 2006 ecliptic and (ii) the secular variation in the Earth's dynamical form factor J2. 4) The present function provides classical nutation, complementing the IAU 2000 frame bias and IAU 2006 precession. It delivers a pole which is at current epochs accurate to a few tens of microarcseconds, apart from the free core nutation. Called: eraNut00a nutation, IAU 2000A References: Chapront, J., Chapront-Touze, M. & Francou, G. 2002, Astron.Astrophys. 387, 700 Lieske, J.H., Lederle, T., Fricke, W. & Morando, B. 1977, Astron.Astrophys. 58, 1-16 Mathews, P.M., Herring, T.A., Buffet, B.A. 2002, J.Geophys.Res. 107, B4. The MHB_2000 code itself was obtained on 9th September 2002 from ftp//maia.usno.navy.mil/conv2000/chapter5/IAU2000A. Simon, J.-L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., Laskar, J. 1994, Astron.Astrophys. 282, 663-683 Souchay, J., Loysel, B., Kinoshita, H., Folgueira, M. 1999, Astron.Astrophys.Supp.Ser. 135, 111 Wallace, P.T., "Software for Implementing the IAU 2000 Resolutions", in IERS Workshop 5.1 (2002) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ dpsi, deps = ufunc.nut06a(date1, date2) return dpsi, deps def nut80(date1, date2): """ Nutation, IAU 1980 model. Parameters ---------- date1 : double array date2 : double array Returns ------- dpsi : double array deps : double array Notes ----- Wraps ERFA function ``eraNut80``. The ERFA documentation is:: - - - - - - - - - e r a N u t 8 0 - - - - - - - - - Nutation, IAU 1980 model. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: dpsi double nutation in longitude (radians) deps double nutation in obliquity (radians) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The nutation components are with respect to the ecliptic of date. Called: eraAnpm normalize angle into range +/- pi Reference: Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), Section 3.222 (p111). This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ dpsi, deps = ufunc.nut80(date1, date2) return dpsi, deps def nutm80(date1, date2): """ Form the matrix of nutation for a given date, IAU 1980 model. Parameters ---------- date1 : double array date2 : double array Returns ------- rmatn : double array Notes ----- Wraps ERFA function ``eraNutm80``. The ERFA documentation is:: - - - - - - - - - - e r a N u t m 8 0 - - - - - - - - - - Form the matrix of nutation for a given date, IAU 1980 model. Given: date1,date2 double TDB date (Note 1) Returned: rmatn double[3][3] nutation matrix Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The matrix operates in the sense V(true) = rmatn * V(mean), where the p-vector V(true) is with respect to the true equatorial triad of date and the p-vector V(mean) is with respect to the mean equatorial triad of date. Called: eraNut80 nutation, IAU 1980 eraObl80 mean obliquity, IAU 1980 eraNumat form nutation matrix This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rmatn = ufunc.nutm80(date1, date2) return rmatn def obl06(date1, date2): """ Mean obliquity of the ecliptic, IAU 2006 precession model. Parameters ---------- date1 : double array date2 : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraObl06``. The ERFA documentation is:: - - - - - - - - - e r a O b l 0 6 - - - - - - - - - Mean obliquity of the ecliptic, IAU 2006 precession model. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned (function value): double obliquity of the ecliptic (radians, Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The result is the angle between the ecliptic and mean equator of date date1+date2. Reference: Hilton, J. et al., 2006, Celest.Mech.Dyn.Astron. 94, 351 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.obl06(date1, date2) return c_retval def obl80(date1, date2): """ Mean obliquity of the ecliptic, IAU 1980 model. Parameters ---------- date1 : double array date2 : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraObl80``. The ERFA documentation is:: - - - - - - - - - e r a O b l 8 0 - - - - - - - - - Mean obliquity of the ecliptic, IAU 1980 model. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned (function value): double obliquity of the ecliptic (radians, Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The result is the angle between the ecliptic and mean equator of date date1+date2. Reference: Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), Expression 3.222-1 (p114). This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.obl80(date1, date2) return c_retval def p06e(date1, date2): """ Precession angles, IAU 2006, equinox based. Parameters ---------- date1 : double array date2 : double array Returns ------- eps0 : double array psia : double array oma : double array bpa : double array bqa : double array pia : double array bpia : double array epsa : double array chia : double array za : double array zetaa : double array thetaa : double array pa : double array gam : double array phi : double array psi : double array Notes ----- Wraps ERFA function ``eraP06e``. The ERFA documentation is:: - - - - - - - - e r a P 0 6 e - - - - - - - - Precession angles, IAU 2006, equinox based. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned (see Note 2): eps0 double epsilon_0 psia double psi_A oma double omega_A bpa double P_A bqa double Q_A pia double pi_A bpia double Pi_A epsa double obliquity epsilon_A chia double chi_A za double z_A zetaa double zeta_A thetaa double theta_A pa double p_A gam double F-W angle gamma_J2000 phi double F-W angle phi_J2000 psi double F-W angle psi_J2000 Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) This function returns the set of equinox based angles for the Capitaine et al. "P03" precession theory, adopted by the IAU in 2006. The angles are set out in Table 1 of Hilton et al. (2006): eps0 epsilon_0 obliquity at J2000.0 psia psi_A luni-solar precession oma omega_A inclination of equator wrt J2000.0 ecliptic bpa P_A ecliptic pole x, J2000.0 ecliptic triad bqa Q_A ecliptic pole -y, J2000.0 ecliptic triad pia pi_A angle between moving and J2000.0 ecliptics bpia Pi_A longitude of ascending node of the ecliptic epsa epsilon_A obliquity of the ecliptic chia chi_A planetary precession za z_A equatorial precession: -3rd 323 Euler angle zetaa zeta_A equatorial precession: -1st 323 Euler angle thetaa theta_A equatorial precession: 2nd 323 Euler angle pa p_A general precession (n.b. see below) gam gamma_J2000 J2000.0 RA difference of ecliptic poles phi phi_J2000 J2000.0 codeclination of ecliptic pole psi psi_J2000 longitude difference of equator poles, J2000.0 The returned values are all radians. Note that the t^5 coefficient in the series for p_A from Capitaine et al. (2003) is incorrectly signed in Hilton et al. (2006). 3) Hilton et al. (2006) Table 1 also contains angles that depend on models distinct from the P03 precession theory itself, namely the IAU 2000A frame bias and nutation. The quoted polynomials are used in other ERFA functions: . eraXy06 contains the polynomial parts of the X and Y series. . eraS06 contains the polynomial part of the s+XY/2 series. . eraPfw06 implements the series for the Fukushima-Williams angles that are with respect to the GCRS pole (i.e. the variants that include frame bias). 4) The IAU resolution stipulated that the choice of parameterization was left to the user, and so an IAU compliant precession implementation can be constructed using various combinations of the angles returned by the present function. 5) The parameterization used by ERFA is the version of the Fukushima- Williams angles that refers directly to the GCRS pole. These angles may be calculated by calling the function eraPfw06. ERFA also supports the direct computation of the CIP GCRS X,Y by series, available by calling eraXy06. 6) The agreement between the different parameterizations is at the 1 microarcsecond level in the present era. 7) When constructing a precession formulation that refers to the GCRS pole rather than the dynamical pole, it may (depending on the choice of angles) be necessary to introduce the frame bias explicitly. 8) It is permissible to re-use the same variable in the returned arguments. The quantities are stored in the stated order. References: Capitaine, N., Wallace, P.T. & Chapront, J., 2003, Astron.Astrophys., 412, 567 Hilton, J. et al., 2006, Celest.Mech.Dyn.Astron. 94, 351 Called: eraObl06 mean obliquity, IAU 2006 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ (eps0, psia, oma, bpa, bqa, pia, bpia, epsa, chia, za, zetaa, thetaa, pa, gam, phi, psi) = ufunc.p06e(date1, date2) return eps0, psia, oma, bpa, bqa, pia, bpia, epsa, chia, za, zetaa, thetaa, pa, gam, phi, psi def pb06(date1, date2): """ This function forms three Euler angles which implement general precession from epoch J2000.0, using the IAU 2006 model. Parameters ---------- date1 : double array date2 : double array Returns ------- bzeta : double array bz : double array btheta : double array Notes ----- Wraps ERFA function ``eraPb06``. The ERFA documentation is:: - - - - - - - - e r a P b 0 6 - - - - - - - - This function forms three Euler angles which implement general precession from epoch J2000.0, using the IAU 2006 model. Frame bias (the offset between ICRS and mean J2000.0) is included. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: bzeta double 1st rotation: radians cw around z bz double 3rd rotation: radians cw around z btheta double 2nd rotation: radians ccw around y Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The traditional accumulated precession angles zeta_A, z_A, theta_A cannot be obtained in the usual way, namely through polynomial expressions, because of the frame bias. The latter means that two of the angles undergo rapid changes near this date. They are instead the results of decomposing the precession-bias matrix obtained by using the Fukushima-Williams method, which does not suffer from the problem. The decomposition returns values which can be used in the conventional formulation and which include frame bias. 3) The three angles are returned in the conventional order, which is not the same as the order of the corresponding Euler rotations. The precession-bias matrix is R_3(-z) x R_2(+theta) x R_3(-zeta). 4) Should zeta_A, z_A, theta_A angles be required that do not contain frame bias, they are available by calling the ERFA function eraP06e. Called: eraPmat06 PB matrix, IAU 2006 eraRz rotate around Z-axis This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ bzeta, bz, btheta = ufunc.pb06(date1, date2) return bzeta, bz, btheta def pfw06(date1, date2): """ Precession angles, IAU 2006 (Fukushima-Williams 4-angle formulation). Parameters ---------- date1 : double array date2 : double array Returns ------- gamb : double array phib : double array psib : double array epsa : double array Notes ----- Wraps ERFA function ``eraPfw06``. The ERFA documentation is:: - - - - - - - - - e r a P f w 0 6 - - - - - - - - - Precession angles, IAU 2006 (Fukushima-Williams 4-angle formulation). Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: gamb double F-W angle gamma_bar (radians) phib double F-W angle phi_bar (radians) psib double F-W angle psi_bar (radians) epsa double F-W angle epsilon_A (radians) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) Naming the following points: e = J2000.0 ecliptic pole, p = GCRS pole, E = mean ecliptic pole of date, and P = mean pole of date, the four Fukushima-Williams angles are as follows: gamb = gamma_bar = epE phib = phi_bar = pE psib = psi_bar = pEP epsa = epsilon_A = EP 3) The matrix representing the combined effects of frame bias and precession is: PxB = R_1(-epsa).R_3(-psib).R_1(phib).R_3(gamb) 4) The matrix representing the combined effects of frame bias, precession and nutation is simply: NxPxB = R_1(-epsa-dE).R_3(-psib-dP).R_1(phib).R_3(gamb) where dP and dE are the nutation components with respect to the ecliptic of date. Reference: Hilton, J. et al., 2006, Celest.Mech.Dyn.Astron. 94, 351 Called: eraObl06 mean obliquity, IAU 2006 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ gamb, phib, psib, epsa = ufunc.pfw06(date1, date2) return gamb, phib, psib, epsa def pmat00(date1, date2): """ Precession matrix (including frame bias) from GCRS to a specified date, IAU 2000 model. Parameters ---------- date1 : double array date2 : double array Returns ------- rbp : double array Notes ----- Wraps ERFA function ``eraPmat00``. The ERFA documentation is:: - - - - - - - - - - e r a P m a t 0 0 - - - - - - - - - - Precession matrix (including frame bias) from GCRS to a specified date, IAU 2000 model. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: rbp double[3][3] bias-precession matrix (Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The matrix operates in the sense V(date) = rbp * V(GCRS), where the p-vector V(GCRS) is with respect to the Geocentric Celestial Reference System (IAU, 2000) and the p-vector V(date) is with respect to the mean equatorial triad of the given date. Called: eraBp00 frame bias and precession matrices, IAU 2000 Reference: IAU: Trans. International Astronomical Union, Vol. XXIVB; Proc. 24th General Assembly, Manchester, UK. Resolutions B1.3, B1.6. (2000) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rbp = ufunc.pmat00(date1, date2) return rbp def pmat06(date1, date2): """ Precession matrix (including frame bias) from GCRS to a specified date, IAU 2006 model. Parameters ---------- date1 : double array date2 : double array Returns ------- rbp : double array Notes ----- Wraps ERFA function ``eraPmat06``. The ERFA documentation is:: - - - - - - - - - - e r a P m a t 0 6 - - - - - - - - - - Precession matrix (including frame bias) from GCRS to a specified date, IAU 2006 model. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: rbp double[3][3] bias-precession matrix (Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The matrix operates in the sense V(date) = rbp * V(GCRS), where the p-vector V(GCRS) is with respect to the Geocentric Celestial Reference System (IAU, 2000) and the p-vector V(date) is with respect to the mean equatorial triad of the given date. Called: eraPfw06 bias-precession F-W angles, IAU 2006 eraFw2m F-W angles to r-matrix References: Capitaine, N. & Wallace, P.T., 2006, Astron.Astrophys. 450, 855 IAU: Trans. International Astronomical Union, Vol. XXIVB; Proc. 24th General Assembly, Manchester, UK. Resolutions B1.3, B1.6. (2000) Wallace, P.T. & Capitaine, N., 2006, Astron.Astrophys. 459, 981 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rbp = ufunc.pmat06(date1, date2) return rbp def pmat76(date1, date2): """ Precession matrix from J2000.0 to a specified date, IAU 1976 model. Parameters ---------- date1 : double array date2 : double array Returns ------- rmatp : double array Notes ----- Wraps ERFA function ``eraPmat76``. The ERFA documentation is:: - - - - - - - - - - e r a P m a t 7 6 - - - - - - - - - - Precession matrix from J2000.0 to a specified date, IAU 1976 model. Given: date1,date2 double ending date, TT (Note 1) Returned: rmatp double[3][3] precession matrix, J2000.0 -> date1+date2 Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The matrix operates in the sense V(date) = RMATP * V(J2000), where the p-vector V(J2000) is with respect to the mean equatorial triad of epoch J2000.0 and the p-vector V(date) is with respect to the mean equatorial triad of the given date. 3) Though the matrix method itself is rigorous, the precession angles are expressed through canonical polynomials which are valid only for a limited time span. In addition, the IAU 1976 precession rate is known to be imperfect. The absolute accuracy of the present formulation is better than 0.1 arcsec from 1960AD to 2040AD, better than 1 arcsec from 1640AD to 2360AD, and remains below 3 arcsec for the whole of the period 500BC to 3000AD. The errors exceed 10 arcsec outside the range 1200BC to 3900AD, exceed 100 arcsec outside 4200BC to 5600AD and exceed 1000 arcsec outside 6800BC to 8200AD. Called: eraPrec76 accumulated precession angles, IAU 1976 eraIr initialize r-matrix to identity eraRz rotate around Z-axis eraRy rotate around Y-axis eraCr copy r-matrix References: Lieske, J.H., 1979, Astron.Astrophys. 73, 282. equations (6) & (7), p283. Kaplan,G.H., 1981. USNO circular no. 163, pA2. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rmatp = ufunc.pmat76(date1, date2) return rmatp def pn00(date1, date2, dpsi, deps): """ Precession-nutation, IAU 2000 model: a multi-purpose function, supporting classical (equinox-based) use directly and CIO-based use indirectly. Parameters ---------- date1 : double array date2 : double array dpsi : double array deps : double array Returns ------- epsa : double array rb : double array rp : double array rbp : double array rn : double array rbpn : double array Notes ----- Wraps ERFA function ``eraPn00``. The ERFA documentation is:: - - - - - - - - e r a P n 0 0 - - - - - - - - Precession-nutation, IAU 2000 model: a multi-purpose function, supporting classical (equinox-based) use directly and CIO-based use indirectly. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) dpsi,deps double nutation (Note 2) Returned: epsa double mean obliquity (Note 3) rb double[3][3] frame bias matrix (Note 4) rp double[3][3] precession matrix (Note 5) rbp double[3][3] bias-precession matrix (Note 6) rn double[3][3] nutation matrix (Note 7) rbpn double[3][3] GCRS-to-true matrix (Note 8) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The caller is responsible for providing the nutation components; they are in longitude and obliquity, in radians and are with respect to the equinox and ecliptic of date. For high-accuracy applications, free core nutation should be included as well as any other relevant corrections to the position of the CIP. 3) The returned mean obliquity is consistent with the IAU 2000 precession-nutation models. 4) The matrix rb transforms vectors from GCRS to J2000.0 mean equator and equinox by applying frame bias. 5) The matrix rp transforms vectors from J2000.0 mean equator and equinox to mean equator and equinox of date by applying precession. 6) The matrix rbp transforms vectors from GCRS to mean equator and equinox of date by applying frame bias then precession. It is the product rp x rb. 7) The matrix rn transforms vectors from mean equator and equinox of date to true equator and equinox of date by applying the nutation (luni-solar + planetary). 8) The matrix rbpn transforms vectors from GCRS to true equator and equinox of date. It is the product rn x rbp, applying frame bias, precession and nutation in that order. 9) It is permissible to re-use the same array in the returned arguments. The arrays are filled in the order given. Called: eraPr00 IAU 2000 precession adjustments eraObl80 mean obliquity, IAU 1980 eraBp00 frame bias and precession matrices, IAU 2000 eraCr copy r-matrix eraNumat form nutation matrix eraRxr product of two r-matrices Reference: Capitaine, N., Chapront, J., Lambert, S. and Wallace, P., "Expressions for the Celestial Intermediate Pole and Celestial Ephemeris Origin consistent with the IAU 2000A precession- nutation model", Astron.Astrophys. 400, 1145-1154 (2003) n.b. The celestial ephemeris origin (CEO) was renamed "celestial intermediate origin" (CIO) by IAU 2006 Resolution 2. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ epsa, rb, rp, rbp, rn, rbpn = ufunc.pn00(date1, date2, dpsi, deps) return epsa, rb, rp, rbp, rn, rbpn def pn00a(date1, date2): """ Precession-nutation, IAU 2000A model: a multi-purpose function, supporting classical (equinox-based) use directly and CIO-based use indirectly. Parameters ---------- date1 : double array date2 : double array Returns ------- dpsi : double array deps : double array epsa : double array rb : double array rp : double array rbp : double array rn : double array rbpn : double array Notes ----- Wraps ERFA function ``eraPn00a``. The ERFA documentation is:: - - - - - - - - - e r a P n 0 0 a - - - - - - - - - Precession-nutation, IAU 2000A model: a multi-purpose function, supporting classical (equinox-based) use directly and CIO-based use indirectly. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: dpsi,deps double nutation (Note 2) epsa double mean obliquity (Note 3) rb double[3][3] frame bias matrix (Note 4) rp double[3][3] precession matrix (Note 5) rbp double[3][3] bias-precession matrix (Note 6) rn double[3][3] nutation matrix (Note 7) rbpn double[3][3] GCRS-to-true matrix (Notes 8,9) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The nutation components (luni-solar + planetary, IAU 2000A) in longitude and obliquity are in radians and with respect to the equinox and ecliptic of date. Free core nutation is omitted; for the utmost accuracy, use the eraPn00 function, where the nutation components are caller-specified. For faster but slightly less accurate results, use the eraPn00b function. 3) The mean obliquity is consistent with the IAU 2000 precession. 4) The matrix rb transforms vectors from GCRS to J2000.0 mean equator and equinox by applying frame bias. 5) The matrix rp transforms vectors from J2000.0 mean equator and equinox to mean equator and equinox of date by applying precession. 6) The matrix rbp transforms vectors from GCRS to mean equator and equinox of date by applying frame bias then precession. It is the product rp x rb. 7) The matrix rn transforms vectors from mean equator and equinox of date to true equator and equinox of date by applying the nutation (luni-solar + planetary). 8) The matrix rbpn transforms vectors from GCRS to true equator and equinox of date. It is the product rn x rbp, applying frame bias, precession and nutation in that order. 9) The X,Y,Z coordinates of the IAU 2000A Celestial Intermediate Pole are elements (3,1-3) of the GCRS-to-true matrix, i.e. rbpn[2][0-2]. 10) It is permissible to re-use the same array in the returned arguments. The arrays are filled in the stated order. Called: eraNut00a nutation, IAU 2000A eraPn00 bias/precession/nutation results, IAU 2000 Reference: Capitaine, N., Chapront, J., Lambert, S. and Wallace, P., "Expressions for the Celestial Intermediate Pole and Celestial Ephemeris Origin consistent with the IAU 2000A precession- nutation model", Astron.Astrophys. 400, 1145-1154 (2003) n.b. The celestial ephemeris origin (CEO) was renamed "celestial intermediate origin" (CIO) by IAU 2006 Resolution 2. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ dpsi, deps, epsa, rb, rp, rbp, rn, rbpn = ufunc.pn00a(date1, date2) return dpsi, deps, epsa, rb, rp, rbp, rn, rbpn def pn00b(date1, date2): """ Precession-nutation, IAU 2000B model: a multi-purpose function, supporting classical (equinox-based) use directly and CIO-based use indirectly. Parameters ---------- date1 : double array date2 : double array Returns ------- dpsi : double array deps : double array epsa : double array rb : double array rp : double array rbp : double array rn : double array rbpn : double array Notes ----- Wraps ERFA function ``eraPn00b``. The ERFA documentation is:: - - - - - - - - - e r a P n 0 0 b - - - - - - - - - Precession-nutation, IAU 2000B model: a multi-purpose function, supporting classical (equinox-based) use directly and CIO-based use indirectly. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: dpsi,deps double nutation (Note 2) epsa double mean obliquity (Note 3) rb double[3][3] frame bias matrix (Note 4) rp double[3][3] precession matrix (Note 5) rbp double[3][3] bias-precession matrix (Note 6) rn double[3][3] nutation matrix (Note 7) rbpn double[3][3] GCRS-to-true matrix (Notes 8,9) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The nutation components (luni-solar + planetary, IAU 2000B) in longitude and obliquity are in radians and with respect to the equinox and ecliptic of date. For more accurate results, but at the cost of increased computation, use the eraPn00a function. For the utmost accuracy, use the eraPn00 function, where the nutation components are caller-specified. 3) The mean obliquity is consistent with the IAU 2000 precession. 4) The matrix rb transforms vectors from GCRS to J2000.0 mean equator and equinox by applying frame bias. 5) The matrix rp transforms vectors from J2000.0 mean equator and equinox to mean equator and equinox of date by applying precession. 6) The matrix rbp transforms vectors from GCRS to mean equator and equinox of date by applying frame bias then precession. It is the product rp x rb. 7) The matrix rn transforms vectors from mean equator and equinox of date to true equator and equinox of date by applying the nutation (luni-solar + planetary). 8) The matrix rbpn transforms vectors from GCRS to true equator and equinox of date. It is the product rn x rbp, applying frame bias, precession and nutation in that order. 9) The X,Y,Z coordinates of the IAU 2000B Celestial Intermediate Pole are elements (3,1-3) of the GCRS-to-true matrix, i.e. rbpn[2][0-2]. 10) It is permissible to re-use the same array in the returned arguments. The arrays are filled in the stated order. Called: eraNut00b nutation, IAU 2000B eraPn00 bias/precession/nutation results, IAU 2000 Reference: Capitaine, N., Chapront, J., Lambert, S. and Wallace, P., "Expressions for the Celestial Intermediate Pole and Celestial Ephemeris Origin consistent with the IAU 2000A precession- nutation model", Astron.Astrophys. 400, 1145-1154 (2003). n.b. The celestial ephemeris origin (CEO) was renamed "celestial intermediate origin" (CIO) by IAU 2006 Resolution 2. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ dpsi, deps, epsa, rb, rp, rbp, rn, rbpn = ufunc.pn00b(date1, date2) return dpsi, deps, epsa, rb, rp, rbp, rn, rbpn def pn06(date1, date2, dpsi, deps): """ Precession-nutation, IAU 2006 model: a multi-purpose function, supporting classical (equinox-based) use directly and CIO-based use indirectly. Parameters ---------- date1 : double array date2 : double array dpsi : double array deps : double array Returns ------- epsa : double array rb : double array rp : double array rbp : double array rn : double array rbpn : double array Notes ----- Wraps ERFA function ``eraPn06``. The ERFA documentation is:: - - - - - - - - e r a P n 0 6 - - - - - - - - Precession-nutation, IAU 2006 model: a multi-purpose function, supporting classical (equinox-based) use directly and CIO-based use indirectly. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) dpsi,deps double nutation (Note 2) Returned: epsa double mean obliquity (Note 3) rb double[3][3] frame bias matrix (Note 4) rp double[3][3] precession matrix (Note 5) rbp double[3][3] bias-precession matrix (Note 6) rn double[3][3] nutation matrix (Note 7) rbpn double[3][3] GCRS-to-true matrix (Notes 8,9) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The caller is responsible for providing the nutation components; they are in longitude and obliquity, in radians and are with respect to the equinox and ecliptic of date. For high-accuracy applications, free core nutation should be included as well as any other relevant corrections to the position of the CIP. 3) The returned mean obliquity is consistent with the IAU 2006 precession. 4) The matrix rb transforms vectors from GCRS to J2000.0 mean equator and equinox by applying frame bias. 5) The matrix rp transforms vectors from J2000.0 mean equator and equinox to mean equator and equinox of date by applying precession. 6) The matrix rbp transforms vectors from GCRS to mean equator and equinox of date by applying frame bias then precession. It is the product rp x rb. 7) The matrix rn transforms vectors from mean equator and equinox of date to true equator and equinox of date by applying the nutation (luni-solar + planetary). 8) The matrix rbpn transforms vectors from GCRS to true equator and equinox of date. It is the product rn x rbp, applying frame bias, precession and nutation in that order. 9) The X,Y,Z coordinates of the Celestial Intermediate Pole are elements (3,1-3) of the GCRS-to-true matrix, i.e. rbpn[2][0-2]. 10) It is permissible to re-use the same array in the returned arguments. The arrays are filled in the stated order. Called: eraPfw06 bias-precession F-W angles, IAU 2006 eraFw2m F-W angles to r-matrix eraCr copy r-matrix eraTr transpose r-matrix eraRxr product of two r-matrices References: Capitaine, N. & Wallace, P.T., 2006, Astron.Astrophys. 450, 855 Wallace, P.T. & Capitaine, N., 2006, Astron.Astrophys. 459, 981 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ epsa, rb, rp, rbp, rn, rbpn = ufunc.pn06(date1, date2, dpsi, deps) return epsa, rb, rp, rbp, rn, rbpn def pn06a(date1, date2): """ Precession-nutation, IAU 2006/2000A models: a multi-purpose function, supporting classical (equinox-based) use directly and CIO-based use indirectly. Parameters ---------- date1 : double array date2 : double array Returns ------- dpsi : double array deps : double array epsa : double array rb : double array rp : double array rbp : double array rn : double array rbpn : double array Notes ----- Wraps ERFA function ``eraPn06a``. The ERFA documentation is:: - - - - - - - - - e r a P n 0 6 a - - - - - - - - - Precession-nutation, IAU 2006/2000A models: a multi-purpose function, supporting classical (equinox-based) use directly and CIO-based use indirectly. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: dpsi,deps double nutation (Note 2) epsa double mean obliquity (Note 3) rb double[3][3] frame bias matrix (Note 4) rp double[3][3] precession matrix (Note 5) rbp double[3][3] bias-precession matrix (Note 6) rn double[3][3] nutation matrix (Note 7) rbpn double[3][3] GCRS-to-true matrix (Notes 8,9) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The nutation components (luni-solar + planetary, IAU 2000A) in longitude and obliquity are in radians and with respect to the equinox and ecliptic of date. Free core nutation is omitted; for the utmost accuracy, use the eraPn06 function, where the nutation components are caller-specified. 3) The mean obliquity is consistent with the IAU 2006 precession. 4) The matrix rb transforms vectors from GCRS to mean J2000.0 by applying frame bias. 5) The matrix rp transforms vectors from mean J2000.0 to mean of date by applying precession. 6) The matrix rbp transforms vectors from GCRS to mean of date by applying frame bias then precession. It is the product rp x rb. 7) The matrix rn transforms vectors from mean of date to true of date by applying the nutation (luni-solar + planetary). 8) The matrix rbpn transforms vectors from GCRS to true of date (CIP/equinox). It is the product rn x rbp, applying frame bias, precession and nutation in that order. 9) The X,Y,Z coordinates of the IAU 2006/2000A Celestial Intermediate Pole are elements (3,1-3) of the GCRS-to-true matrix, i.e. rbpn[2][0-2]. 10) It is permissible to re-use the same array in the returned arguments. The arrays are filled in the stated order. Called: eraNut06a nutation, IAU 2006/2000A eraPn06 bias/precession/nutation results, IAU 2006 Reference: Capitaine, N. & Wallace, P.T., 2006, Astron.Astrophys. 450, 855 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ dpsi, deps, epsa, rb, rp, rbp, rn, rbpn = ufunc.pn06a(date1, date2) return dpsi, deps, epsa, rb, rp, rbp, rn, rbpn def pnm00a(date1, date2): """ Form the matrix of precession-nutation for a given date (including frame bias), equinox based, IAU 2000A model. Parameters ---------- date1 : double array date2 : double array Returns ------- rbpn : double array Notes ----- Wraps ERFA function ``eraPnm00a``. The ERFA documentation is:: - - - - - - - - - - e r a P n m 0 0 a - - - - - - - - - - Form the matrix of precession-nutation for a given date (including frame bias), equinox based, IAU 2000A model. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: rbpn double[3][3] bias-precession-nutation matrix (Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The matrix operates in the sense V(date) = rbpn * V(GCRS), where the p-vector V(date) is with respect to the true equatorial triad of date date1+date2 and the p-vector V(GCRS) is with respect to the Geocentric Celestial Reference System (IAU, 2000). 3) A faster, but slightly less accurate, result (about 1 mas) can be obtained by using instead the eraPnm00b function. Called: eraPn00a bias/precession/nutation, IAU 2000A Reference: IAU: Trans. International Astronomical Union, Vol. XXIVB; Proc. 24th General Assembly, Manchester, UK. Resolutions B1.3, B1.6. (2000) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rbpn = ufunc.pnm00a(date1, date2) return rbpn def pnm00b(date1, date2): """ Form the matrix of precession-nutation for a given date (including frame bias), equinox-based, IAU 2000B model. Parameters ---------- date1 : double array date2 : double array Returns ------- rbpn : double array Notes ----- Wraps ERFA function ``eraPnm00b``. The ERFA documentation is:: - - - - - - - - - - e r a P n m 0 0 b - - - - - - - - - - Form the matrix of precession-nutation for a given date (including frame bias), equinox-based, IAU 2000B model. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: rbpn double[3][3] bias-precession-nutation matrix (Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The matrix operates in the sense V(date) = rbpn * V(GCRS), where the p-vector V(date) is with respect to the true equatorial triad of date date1+date2 and the p-vector V(GCRS) is with respect to the Geocentric Celestial Reference System (IAU, 2000). 3) The present function is faster, but slightly less accurate (about 1 mas), than the eraPnm00a function. Called: eraPn00b bias/precession/nutation, IAU 2000B Reference: IAU: Trans. International Astronomical Union, Vol. XXIVB; Proc. 24th General Assembly, Manchester, UK. Resolutions B1.3, B1.6. (2000) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rbpn = ufunc.pnm00b(date1, date2) return rbpn def pnm06a(date1, date2): """ Form the matrix of precession-nutation for a given date (including frame bias), equinox based, IAU 2006 precession and IAU 2000A nutation models. Parameters ---------- date1 : double array date2 : double array Returns ------- rbpn : double array Notes ----- Wraps ERFA function ``eraPnm06a``. The ERFA documentation is:: - - - - - - - - - - e r a P n m 0 6 a - - - - - - - - - - Form the matrix of precession-nutation for a given date (including frame bias), equinox based, IAU 2006 precession and IAU 2000A nutation models. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: rbpn double[3][3] bias-precession-nutation matrix (Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The matrix operates in the sense V(date) = rbpn * V(GCRS), where the p-vector V(date) is with respect to the true equatorial triad of date date1+date2 and the p-vector V(GCRS) is with respect to the Geocentric Celestial Reference System (IAU, 2000). Called: eraPfw06 bias-precession F-W angles, IAU 2006 eraNut06a nutation, IAU 2006/2000A eraFw2m F-W angles to r-matrix Reference: Capitaine, N. & Wallace, P.T., 2006, Astron.Astrophys. 450, 855. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rbpn = ufunc.pnm06a(date1, date2) return rbpn def pnm80(date1, date2): """ Form the matrix of precession/nutation for a given date, IAU 1976 precession model, IAU 1980 nutation model. Parameters ---------- date1 : double array date2 : double array Returns ------- rmatpn : double array Notes ----- Wraps ERFA function ``eraPnm80``. The ERFA documentation is:: - - - - - - - - - e r a P n m 8 0 - - - - - - - - - Form the matrix of precession/nutation for a given date, IAU 1976 precession model, IAU 1980 nutation model. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: rmatpn double[3][3] combined precession/nutation matrix Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The matrix operates in the sense V(date) = rmatpn * V(J2000), where the p-vector V(date) is with respect to the true equatorial triad of date date1+date2 and the p-vector V(J2000) is with respect to the mean equatorial triad of epoch J2000.0. Called: eraPmat76 precession matrix, IAU 1976 eraNutm80 nutation matrix, IAU 1980 eraRxr product of two r-matrices Reference: Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), Section 3.3 (p145). This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rmatpn = ufunc.pnm80(date1, date2) return rmatpn def pom00(xp, yp, sp): """ Form the matrix of polar motion for a given date, IAU 2000. Parameters ---------- xp : double array yp : double array sp : double array Returns ------- rpom : double array Notes ----- Wraps ERFA function ``eraPom00``. The ERFA documentation is:: - - - - - - - - - - e r a P o m 0 0 - - - - - - - - - - Form the matrix of polar motion for a given date, IAU 2000. Given: xp,yp double coordinates of the pole (radians, Note 1) sp double the TIO locator s' (radians, Note 2) Returned: rpom double[3][3] polar-motion matrix (Note 3) Notes: 1) The arguments xp and yp are the coordinates (in radians) of the Celestial Intermediate Pole with respect to the International Terrestrial Reference System (see IERS Conventions 2003), measured along the meridians 0 and 90 deg west respectively. 2) The argument sp is the TIO locator s', in radians, which positions the Terrestrial Intermediate Origin on the equator. It is obtained from polar motion observations by numerical integration, and so is in essence unpredictable. However, it is dominated by a secular drift of about 47 microarcseconds per century, and so can be taken into account by using s' = -47*t, where t is centuries since J2000.0. The function eraSp00 implements this approximation. 3) The matrix operates in the sense V(TRS) = rpom * V(CIP), meaning that it is the final rotation when computing the pointing direction to a celestial source. Called: eraIr initialize r-matrix to identity eraRz rotate around Z-axis eraRy rotate around Y-axis eraRx rotate around X-axis Reference: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rpom = ufunc.pom00(xp, yp, sp) return rpom def pr00(date1, date2): """ Precession-rate part of the IAU 2000 precession-nutation models (part of MHB2000). Parameters ---------- date1 : double array date2 : double array Returns ------- dpsipr : double array depspr : double array Notes ----- Wraps ERFA function ``eraPr00``. The ERFA documentation is:: - - - - - - - - e r a P r 0 0 - - - - - - - - Precession-rate part of the IAU 2000 precession-nutation models (part of MHB2000). Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: dpsipr,depspr double precession corrections (Notes 2,3) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The precession adjustments are expressed as "nutation components", corrections in longitude and obliquity with respect to the J2000.0 equinox and ecliptic. 3) Although the precession adjustments are stated to be with respect to Lieske et al. (1977), the MHB2000 model does not specify which set of Euler angles are to be used and how the adjustments are to be applied. The most literal and straightforward procedure is to adopt the 4-rotation epsilon_0, psi_A, omega_A, xi_A option, and to add dpsipr to psi_A and depspr to both omega_A and eps_A. 4) This is an implementation of one aspect of the IAU 2000A nutation model, formally adopted by the IAU General Assembly in 2000, namely MHB2000 (Mathews et al. 2002). References: Lieske, J.H., Lederle, T., Fricke, W. & Morando, B., "Expressions for the precession quantities based upon the IAU (1976) System of Astronomical Constants", Astron.Astrophys., 58, 1-16 (1977) Mathews, P.M., Herring, T.A., Buffet, B.A., "Modeling of nutation and precession New nutation series for nonrigid Earth and insights into the Earth's interior", J.Geophys.Res., 107, B4, 2002. The MHB2000 code itself was obtained on 9th September 2002 from ftp://maia.usno.navy.mil/conv2000/chapter5/IAU2000A. Wallace, P.T., "Software for Implementing the IAU 2000 Resolutions", in IERS Workshop 5.1 (2002). This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ dpsipr, depspr = ufunc.pr00(date1, date2) return dpsipr, depspr def prec76(date01, date02, date11, date12): """ IAU 1976 precession model. Parameters ---------- date01 : double array date02 : double array date11 : double array date12 : double array Returns ------- zeta : double array z : double array theta : double array Notes ----- Wraps ERFA function ``eraPrec76``. The ERFA documentation is:: - - - - - - - - - - e r a P r e c 7 6 - - - - - - - - - - IAU 1976 precession model. This function forms the three Euler angles which implement general precession between two dates, using the IAU 1976 model (as for the FK5 catalog). Given: date01,date02 double TDB starting date (Note 1) date11,date12 double TDB ending date (Note 1) Returned: zeta double 1st rotation: radians cw around z z double 3rd rotation: radians cw around z theta double 2nd rotation: radians ccw around y Notes: 1) The dates date01+date02 and date11+date12 are Julian Dates, apportioned in any convenient way between the arguments daten1 and daten2. For example, JD(TDB)=2450123.7 could be expressed in any of these ways, among others: daten1 daten2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. The two dates may be expressed using different methods, but at the risk of losing some resolution. 2) The accumulated precession angles zeta, z, theta are expressed through canonical polynomials which are valid only for a limited time span. In addition, the IAU 1976 precession rate is known to be imperfect. The absolute accuracy of the present formulation is better than 0.1 arcsec from 1960AD to 2040AD, better than 1 arcsec from 1640AD to 2360AD, and remains below 3 arcsec for the whole of the period 500BC to 3000AD. The errors exceed 10 arcsec outside the range 1200BC to 3900AD, exceed 100 arcsec outside 4200BC to 5600AD and exceed 1000 arcsec outside 6800BC to 8200AD. 3) The three angles are returned in the conventional order, which is not the same as the order of the corresponding Euler rotations. The precession matrix is R_3(-z) x R_2(+theta) x R_3(-zeta). Reference: Lieske, J.H., 1979, Astron.Astrophys. 73, 282, equations (6) & (7), p283. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ zeta, z, theta = ufunc.prec76(date01, date02, date11, date12) return zeta, z, theta def s00(date1, date2, x, y): """ The CIO locator s, positioning the Celestial Intermediate Origin on the equator of the Celestial Intermediate Pole, given the CIP's X,Y coordinates. Parameters ---------- date1 : double array date2 : double array x : double array y : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraS00``. The ERFA documentation is:: - - - - - - - e r a S 0 0 - - - - - - - The CIO locator s, positioning the Celestial Intermediate Origin on the equator of the Celestial Intermediate Pole, given the CIP's X,Y coordinates. Compatible with IAU 2000A precession-nutation. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) x,y double CIP coordinates (Note 3) Returned (function value): double the CIO locator s in radians (Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The CIO locator s is the difference between the right ascensions of the same point in two systems: the two systems are the GCRS and the CIP,CIO, and the point is the ascending node of the CIP equator. The quantity s remains below 0.1 arcsecond throughout 1900-2100. 3) The series used to compute s is in fact for s+XY/2, where X and Y are the x and y components of the CIP unit vector; this series is more compact than a direct series for s would be. This function requires X,Y to be supplied by the caller, who is responsible for providing values that are consistent with the supplied date. 4) The model is consistent with the IAU 2000A precession-nutation. Called: eraFal03 mean anomaly of the Moon eraFalp03 mean anomaly of the Sun eraFaf03 mean argument of the latitude of the Moon eraFad03 mean elongation of the Moon from the Sun eraFaom03 mean longitude of the Moon's ascending node eraFave03 mean longitude of Venus eraFae03 mean longitude of Earth eraFapa03 general accumulated precession in longitude References: Capitaine, N., Chapront, J., Lambert, S. and Wallace, P., "Expressions for the Celestial Intermediate Pole and Celestial Ephemeris Origin consistent with the IAU 2000A precession- nutation model", Astron.Astrophys. 400, 1145-1154 (2003) n.b. The celestial ephemeris origin (CEO) was renamed "celestial intermediate origin" (CIO) by IAU 2006 Resolution 2. McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.s00(date1, date2, x, y) return c_retval def s00a(date1, date2): """ The CIO locator s, positioning the Celestial Intermediate Origin on the equator of the Celestial Intermediate Pole, using the IAU 2000A precession-nutation model. Parameters ---------- date1 : double array date2 : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraS00a``. The ERFA documentation is:: - - - - - - - - e r a S 0 0 a - - - - - - - - The CIO locator s, positioning the Celestial Intermediate Origin on the equator of the Celestial Intermediate Pole, using the IAU 2000A precession-nutation model. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned (function value): double the CIO locator s in radians (Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The CIO locator s is the difference between the right ascensions of the same point in two systems. The two systems are the GCRS and the CIP,CIO, and the point is the ascending node of the CIP equator. The CIO locator s remains a small fraction of 1 arcsecond throughout 1900-2100. 3) The series used to compute s is in fact for s+XY/2, where X and Y are the x and y components of the CIP unit vector; this series is more compact than a direct series for s would be. The present function uses the full IAU 2000A nutation model when predicting the CIP position. Faster results, with no significant loss of accuracy, can be obtained via the function eraS00b, which uses instead the IAU 2000B truncated model. Called: eraPnm00a classical NPB matrix, IAU 2000A eraBnp2xy extract CIP X,Y from the BPN matrix eraS00 the CIO locator s, given X,Y, IAU 2000A References: Capitaine, N., Chapront, J., Lambert, S. and Wallace, P., "Expressions for the Celestial Intermediate Pole and Celestial Ephemeris Origin consistent with the IAU 2000A precession- nutation model", Astron.Astrophys. 400, 1145-1154 (2003) n.b. The celestial ephemeris origin (CEO) was renamed "celestial intermediate origin" (CIO) by IAU 2006 Resolution 2. McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.s00a(date1, date2) return c_retval def s00b(date1, date2): """ The CIO locator s, positioning the Celestial Intermediate Origin on the equator of the Celestial Intermediate Pole, using the IAU 2000B precession-nutation model. Parameters ---------- date1 : double array date2 : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraS00b``. The ERFA documentation is:: - - - - - - - - e r a S 0 0 b - - - - - - - - The CIO locator s, positioning the Celestial Intermediate Origin on the equator of the Celestial Intermediate Pole, using the IAU 2000B precession-nutation model. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned (function value): double the CIO locator s in radians (Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The CIO locator s is the difference between the right ascensions of the same point in two systems. The two systems are the GCRS and the CIP,CIO, and the point is the ascending node of the CIP equator. The CIO locator s remains a small fraction of 1 arcsecond throughout 1900-2100. 3) The series used to compute s is in fact for s+XY/2, where X and Y are the x and y components of the CIP unit vector; this series is more compact than a direct series for s would be. The present function uses the IAU 2000B truncated nutation model when predicting the CIP position. The function eraS00a uses instead the full IAU 2000A model, but with no significant increase in accuracy and at some cost in speed. Called: eraPnm00b classical NPB matrix, IAU 2000B eraBnp2xy extract CIP X,Y from the BPN matrix eraS00 the CIO locator s, given X,Y, IAU 2000A References: Capitaine, N., Chapront, J., Lambert, S. and Wallace, P., "Expressions for the Celestial Intermediate Pole and Celestial Ephemeris Origin consistent with the IAU 2000A precession- nutation model", Astron.Astrophys. 400, 1145-1154 (2003) n.b. The celestial ephemeris origin (CEO) was renamed "celestial intermediate origin" (CIO) by IAU 2006 Resolution 2. McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.s00b(date1, date2) return c_retval def s06(date1, date2, x, y): """ The CIO locator s, positioning the Celestial Intermediate Origin on the equator of the Celestial Intermediate Pole, given the CIP's X,Y coordinates. Parameters ---------- date1 : double array date2 : double array x : double array y : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraS06``. The ERFA documentation is:: - - - - - - - e r a S 0 6 - - - - - - - The CIO locator s, positioning the Celestial Intermediate Origin on the equator of the Celestial Intermediate Pole, given the CIP's X,Y coordinates. Compatible with IAU 2006/2000A precession-nutation. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) x,y double CIP coordinates (Note 3) Returned (function value): double the CIO locator s in radians (Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The CIO locator s is the difference between the right ascensions of the same point in two systems: the two systems are the GCRS and the CIP,CIO, and the point is the ascending node of the CIP equator. The quantity s remains below 0.1 arcsecond throughout 1900-2100. 3) The series used to compute s is in fact for s+XY/2, where X and Y are the x and y components of the CIP unit vector; this series is more compact than a direct series for s would be. This function requires X,Y to be supplied by the caller, who is responsible for providing values that are consistent with the supplied date. 4) The model is consistent with the "P03" precession (Capitaine et al. 2003), adopted by IAU 2006 Resolution 1, 2006, and the IAU 2000A nutation (with P03 adjustments). Called: eraFal03 mean anomaly of the Moon eraFalp03 mean anomaly of the Sun eraFaf03 mean argument of the latitude of the Moon eraFad03 mean elongation of the Moon from the Sun eraFaom03 mean longitude of the Moon's ascending node eraFave03 mean longitude of Venus eraFae03 mean longitude of Earth eraFapa03 general accumulated precession in longitude References: Capitaine, N., Wallace, P.T. & Chapront, J., 2003, Astron. Astrophys. 432, 355 McCarthy, D.D., Petit, G. (eds.) 2004, IERS Conventions (2003), IERS Technical Note No. 32, BKG This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.s06(date1, date2, x, y) return c_retval def s06a(date1, date2): """ The CIO locator s, positioning the Celestial Intermediate Origin on the equator of the Celestial Intermediate Pole, using the IAU 2006 precession and IAU 2000A nutation models. Parameters ---------- date1 : double array date2 : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraS06a``. The ERFA documentation is:: - - - - - - - - e r a S 0 6 a - - - - - - - - The CIO locator s, positioning the Celestial Intermediate Origin on the equator of the Celestial Intermediate Pole, using the IAU 2006 precession and IAU 2000A nutation models. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned (function value): double the CIO locator s in radians (Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The CIO locator s is the difference between the right ascensions of the same point in two systems. The two systems are the GCRS and the CIP,CIO, and the point is the ascending node of the CIP equator. The CIO locator s remains a small fraction of 1 arcsecond throughout 1900-2100. 3) The series used to compute s is in fact for s+XY/2, where X and Y are the x and y components of the CIP unit vector; this series is more compact than a direct series for s would be. The present function uses the full IAU 2000A nutation model when predicting the CIP position. Called: eraPnm06a classical NPB matrix, IAU 2006/2000A eraBpn2xy extract CIP X,Y coordinates from NPB matrix eraS06 the CIO locator s, given X,Y, IAU 2006 References: Capitaine, N., Chapront, J., Lambert, S. and Wallace, P., "Expressions for the Celestial Intermediate Pole and Celestial Ephemeris Origin consistent with the IAU 2000A precession- nutation model", Astron.Astrophys. 400, 1145-1154 (2003) n.b. The celestial ephemeris origin (CEO) was renamed "celestial intermediate origin" (CIO) by IAU 2006 Resolution 2. Capitaine, N. & Wallace, P.T., 2006, Astron.Astrophys. 450, 855 McCarthy, D. D., Petit, G. (eds.), 2004, IERS Conventions (2003), IERS Technical Note No. 32, BKG Wallace, P.T. & Capitaine, N., 2006, Astron.Astrophys. 459, 981 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.s06a(date1, date2) return c_retval def sp00(date1, date2): """ The TIO locator s', positioning the Terrestrial Intermediate Origin on the equator of the Celestial Intermediate Pole. Parameters ---------- date1 : double array date2 : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraSp00``. The ERFA documentation is:: - - - - - - - - e r a S p 0 0 - - - - - - - - The TIO locator s', positioning the Terrestrial Intermediate Origin on the equator of the Celestial Intermediate Pole. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned (function value): double the TIO locator s' in radians (Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The TIO locator s' is obtained from polar motion observations by numerical integration, and so is in essence unpredictable. However, it is dominated by a secular drift of about 47 microarcseconds per century, which is the approximation evaluated by the present function. Reference: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.sp00(date1, date2) return c_retval def xy06(date1, date2): """ X,Y coordinates of celestial intermediate pole from series based on IAU 2006 precession and IAU 2000A nutation. Parameters ---------- date1 : double array date2 : double array Returns ------- x : double array y : double array Notes ----- Wraps ERFA function ``eraXy06``. The ERFA documentation is:: - - - - - - - - e r a X y 0 6 - - - - - - - - X,Y coordinates of celestial intermediate pole from series based on IAU 2006 precession and IAU 2000A nutation. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: x,y double CIP X,Y coordinates (Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The X,Y coordinates are those of the unit vector towards the celestial intermediate pole. They represent the combined effects of frame bias, precession and nutation. 3) The fundamental arguments used are as adopted in IERS Conventions (2003) and are from Simon et al. (1994) and Souchay et al. (1999). 4) This is an alternative to the angles-based method, via the ERFA function eraFw2xy and as used in eraXys06a for example. The two methods agree at the 1 microarcsecond level (at present), a negligible amount compared with the intrinsic accuracy of the models. However, it would be unwise to mix the two methods (angles-based and series-based) in a single application. Called: eraFal03 mean anomaly of the Moon eraFalp03 mean anomaly of the Sun eraFaf03 mean argument of the latitude of the Moon eraFad03 mean elongation of the Moon from the Sun eraFaom03 mean longitude of the Moon's ascending node eraFame03 mean longitude of Mercury eraFave03 mean longitude of Venus eraFae03 mean longitude of Earth eraFama03 mean longitude of Mars eraFaju03 mean longitude of Jupiter eraFasa03 mean longitude of Saturn eraFaur03 mean longitude of Uranus eraFane03 mean longitude of Neptune eraFapa03 general accumulated precession in longitude References: Capitaine, N., Wallace, P.T. & Chapront, J., 2003, Astron.Astrophys., 412, 567 Capitaine, N. & Wallace, P.T., 2006, Astron.Astrophys. 450, 855 McCarthy, D. D., Petit, G. (eds.), 2004, IERS Conventions (2003), IERS Technical Note No. 32, BKG Simon, J.L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G. & Laskar, J., Astron.Astrophys., 1994, 282, 663 Souchay, J., Loysel, B., Kinoshita, H., Folgueira, M., 1999, Astron.Astrophys.Supp.Ser. 135, 111 Wallace, P.T. & Capitaine, N., 2006, Astron.Astrophys. 459, 981 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ x, y = ufunc.xy06(date1, date2) return x, y def xys00a(date1, date2): """ For a given TT date, compute the X,Y coordinates of the Celestial Intermediate Pole and the CIO locator s, using the IAU 2000A precession-nutation model. Parameters ---------- date1 : double array date2 : double array Returns ------- x : double array y : double array s : double array Notes ----- Wraps ERFA function ``eraXys00a``. The ERFA documentation is:: - - - - - - - - - - e r a X y s 0 0 a - - - - - - - - - - For a given TT date, compute the X,Y coordinates of the Celestial Intermediate Pole and the CIO locator s, using the IAU 2000A precession-nutation model. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: x,y double Celestial Intermediate Pole (Note 2) s double the CIO locator s (Note 3) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The Celestial Intermediate Pole coordinates are the x,y components of the unit vector in the Geocentric Celestial Reference System. 3) The CIO locator s (in radians) positions the Celestial Intermediate Origin on the equator of the CIP. 4) A faster, but slightly less accurate result (about 1 mas for X,Y), can be obtained by using instead the eraXys00b function. Called: eraPnm00a classical NPB matrix, IAU 2000A eraBpn2xy extract CIP X,Y coordinates from NPB matrix eraS00 the CIO locator s, given X,Y, IAU 2000A Reference: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ x, y, s = ufunc.xys00a(date1, date2) return x, y, s def xys00b(date1, date2): """ For a given TT date, compute the X,Y coordinates of the Celestial Intermediate Pole and the CIO locator s, using the IAU 2000B precession-nutation model. Parameters ---------- date1 : double array date2 : double array Returns ------- x : double array y : double array s : double array Notes ----- Wraps ERFA function ``eraXys00b``. The ERFA documentation is:: - - - - - - - - - - e r a X y s 0 0 b - - - - - - - - - - For a given TT date, compute the X,Y coordinates of the Celestial Intermediate Pole and the CIO locator s, using the IAU 2000B precession-nutation model. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: x,y double Celestial Intermediate Pole (Note 2) s double the CIO locator s (Note 3) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The Celestial Intermediate Pole coordinates are the x,y components of the unit vector in the Geocentric Celestial Reference System. 3) The CIO locator s (in radians) positions the Celestial Intermediate Origin on the equator of the CIP. 4) The present function is faster, but slightly less accurate (about 1 mas in X,Y), than the eraXys00a function. Called: eraPnm00b classical NPB matrix, IAU 2000B eraBpn2xy extract CIP X,Y coordinates from NPB matrix eraS00 the CIO locator s, given X,Y, IAU 2000A Reference: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ x, y, s = ufunc.xys00b(date1, date2) return x, y, s def xys06a(date1, date2): """ For a given TT date, compute the X,Y coordinates of the Celestial Intermediate Pole and the CIO locator s, using the IAU 2006 precession and IAU 2000A nutation models. Parameters ---------- date1 : double array date2 : double array Returns ------- x : double array y : double array s : double array Notes ----- Wraps ERFA function ``eraXys06a``. The ERFA documentation is:: - - - - - - - - - - e r a X y s 0 6 a - - - - - - - - - - For a given TT date, compute the X,Y coordinates of the Celestial Intermediate Pole and the CIO locator s, using the IAU 2006 precession and IAU 2000A nutation models. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned: x,y double Celestial Intermediate Pole (Note 2) s double the CIO locator s (Note 3) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The Celestial Intermediate Pole coordinates are the x,y components of the unit vector in the Geocentric Celestial Reference System. 3) The CIO locator s (in radians) positions the Celestial Intermediate Origin on the equator of the CIP. 4) Series-based solutions for generating X and Y are also available: see Capitaine & Wallace (2006) and eraXy06. Called: eraPnm06a classical NPB matrix, IAU 2006/2000A eraBpn2xy extract CIP X,Y coordinates from NPB matrix eraS06 the CIO locator s, given X,Y, IAU 2006 References: Capitaine, N. & Wallace, P.T., 2006, Astron.Astrophys. 450, 855 Wallace, P.T. & Capitaine, N., 2006, Astron.Astrophys. 459, 981 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ x, y, s = ufunc.xys06a(date1, date2) return x, y, s def ee00(date1, date2, epsa, dpsi): """ The equation of the equinoxes, compatible with IAU 2000 resolutions, given the nutation in longitude and the mean obliquity. Parameters ---------- date1 : double array date2 : double array epsa : double array dpsi : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraEe00``. The ERFA documentation is:: - - - - - - - - e r a E e 0 0 - - - - - - - - The equation of the equinoxes, compatible with IAU 2000 resolutions, given the nutation in longitude and the mean obliquity. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) epsa double mean obliquity (Note 2) dpsi double nutation in longitude (Note 3) Returned (function value): double equation of the equinoxes (Note 4) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The obliquity, in radians, is mean of date. 3) The result, which is in radians, operates in the following sense: Greenwich apparent ST = GMST + equation of the equinoxes 4) The result is compatible with the IAU 2000 resolutions. For further details, see IERS Conventions 2003 and Capitaine et al. (2002). Called: eraEect00 equation of the equinoxes complementary terms References: Capitaine, N., Wallace, P.T. and McCarthy, D.D., "Expressions to implement the IAU 2000 definition of UT1", Astronomy & Astrophysics, 406, 1135-1149 (2003) McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.ee00(date1, date2, epsa, dpsi) return c_retval def ee00a(date1, date2): """ Equation of the equinoxes, compatible with IAU 2000 resolutions. Parameters ---------- date1 : double array date2 : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraEe00a``. The ERFA documentation is:: - - - - - - - - - e r a E e 0 0 a - - - - - - - - - Equation of the equinoxes, compatible with IAU 2000 resolutions. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned (function value): double equation of the equinoxes (Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The result, which is in radians, operates in the following sense: Greenwich apparent ST = GMST + equation of the equinoxes 3) The result is compatible with the IAU 2000 resolutions. For further details, see IERS Conventions 2003 and Capitaine et al. (2002). Called: eraPr00 IAU 2000 precession adjustments eraObl80 mean obliquity, IAU 1980 eraNut00a nutation, IAU 2000A eraEe00 equation of the equinoxes, IAU 2000 References: Capitaine, N., Wallace, P.T. and McCarthy, D.D., "Expressions to implement the IAU 2000 definition of UT1", Astronomy & Astrophysics, 406, 1135-1149 (2003). McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004). This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.ee00a(date1, date2) return c_retval def ee00b(date1, date2): """ Equation of the equinoxes, compatible with IAU 2000 resolutions but using the truncated nutation model IAU 2000B. Parameters ---------- date1 : double array date2 : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraEe00b``. The ERFA documentation is:: - - - - - - - - - e r a E e 0 0 b - - - - - - - - - Equation of the equinoxes, compatible with IAU 2000 resolutions but using the truncated nutation model IAU 2000B. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned (function value): double equation of the equinoxes (Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The result, which is in radians, operates in the following sense: Greenwich apparent ST = GMST + equation of the equinoxes 3) The result is compatible with the IAU 2000 resolutions except that accuracy has been compromised (1 mas) for the sake of speed. For further details, see McCarthy & Luzum (2003), IERS Conventions 2003 and Capitaine et al. (2003). Called: eraPr00 IAU 2000 precession adjustments eraObl80 mean obliquity, IAU 1980 eraNut00b nutation, IAU 2000B eraEe00 equation of the equinoxes, IAU 2000 References: Capitaine, N., Wallace, P.T. and McCarthy, D.D., "Expressions to implement the IAU 2000 definition of UT1", Astronomy & Astrophysics, 406, 1135-1149 (2003) McCarthy, D.D. & Luzum, B.J., "An abridged model of the precession-nutation of the celestial pole", Celestial Mechanics & Dynamical Astronomy, 85, 37-49 (2003) McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.ee00b(date1, date2) return c_retval def ee06a(date1, date2): """ Equation of the equinoxes, compatible with IAU 2000 resolutions and IAU 2006/2000A precession-nutation. Parameters ---------- date1 : double array date2 : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraEe06a``. The ERFA documentation is:: - - - - - - - - - e r a E e 0 6 a - - - - - - - - - Equation of the equinoxes, compatible with IAU 2000 resolutions and IAU 2006/2000A precession-nutation. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned (function value): double equation of the equinoxes (Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The result, which is in radians, operates in the following sense: Greenwich apparent ST = GMST + equation of the equinoxes Called: eraAnpm normalize angle into range +/- pi eraGst06a Greenwich apparent sidereal time, IAU 2006/2000A eraGmst06 Greenwich mean sidereal time, IAU 2006 Reference: McCarthy, D. D., Petit, G. (eds.), 2004, IERS Conventions (2003), IERS Technical Note No. 32, BKG This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.ee06a(date1, date2) return c_retval def eect00(date1, date2): """ Equation of the equinoxes complementary terms, consistent with IAU 2000 resolutions. Parameters ---------- date1 : double array date2 : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraEect00``. The ERFA documentation is:: - - - - - - - - - - e r a E e c t 0 0 - - - - - - - - - - Equation of the equinoxes complementary terms, consistent with IAU 2000 resolutions. Given: date1,date2 double TT as a 2-part Julian Date (Note 1) Returned (function value): double complementary terms (Note 2) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The "complementary terms" are part of the equation of the equinoxes (EE), classically the difference between apparent and mean Sidereal Time: GAST = GMST + EE with: EE = dpsi * cos(eps) where dpsi is the nutation in longitude and eps is the obliquity of date. However, if the rotation of the Earth were constant in an inertial frame the classical formulation would lead to apparent irregularities in the UT1 timescale traceable to side- effects of precession-nutation. In order to eliminate these effects from UT1, "complementary terms" were introduced in 1994 (IAU, 1994) and took effect from 1997 (Capitaine and Gontier, 1993): GAST = GMST + CT + EE By convention, the complementary terms are included as part of the equation of the equinoxes rather than as part of the mean Sidereal Time. This slightly compromises the "geometrical" interpretation of mean sidereal time but is otherwise inconsequential. The present function computes CT in the above expression, compatible with IAU 2000 resolutions (Capitaine et al., 2002, and IERS Conventions 2003). Called: eraFal03 mean anomaly of the Moon eraFalp03 mean anomaly of the Sun eraFaf03 mean argument of the latitude of the Moon eraFad03 mean elongation of the Moon from the Sun eraFaom03 mean longitude of the Moon's ascending node eraFave03 mean longitude of Venus eraFae03 mean longitude of Earth eraFapa03 general accumulated precession in longitude References: Capitaine, N. & Gontier, A.-M., Astron.Astrophys., 275, 645-650 (1993) Capitaine, N., Wallace, P.T. and McCarthy, D.D., "Expressions to implement the IAU 2000 definition of UT1", Astron.Astrophys., 406, 1135-1149 (2003) IAU Resolution C7, Recommendation 3 (1994) McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.eect00(date1, date2) return c_retval def eqeq94(date1, date2): """ Equation of the equinoxes, IAU 1994 model. Parameters ---------- date1 : double array date2 : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraEqeq94``. The ERFA documentation is:: - - - - - - - - - - e r a E q e q 9 4 - - - - - - - - - - Equation of the equinoxes, IAU 1994 model. Given: date1,date2 double TDB date (Note 1) Returned (function value): double equation of the equinoxes (Note 2) Notes: 1) The date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The result, which is in radians, operates in the following sense: Greenwich apparent ST = GMST + equation of the equinoxes Called: eraAnpm normalize angle into range +/- pi eraNut80 nutation, IAU 1980 eraObl80 mean obliquity, IAU 1980 References: IAU Resolution C7, Recommendation 3 (1994). Capitaine, N. & Gontier, A.-M., 1993, Astron.Astrophys., 275, 645-650. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.eqeq94(date1, date2) return c_retval def era00(dj1, dj2): """ Earth rotation angle (IAU 2000 model). Parameters ---------- dj1 : double array dj2 : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraEra00``. The ERFA documentation is:: - - - - - - - - - e r a E r a 0 0 - - - - - - - - - Earth rotation angle (IAU 2000 model). Given: dj1,dj2 double UT1 as a 2-part Julian Date (see note) Returned (function value): double Earth rotation angle (radians), range 0-2pi Notes: 1) The UT1 date dj1+dj2 is a Julian Date, apportioned in any convenient way between the arguments dj1 and dj2. For example, JD(UT1)=2450123.7 could be expressed in any of these ways, among others: dj1 dj2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 and MJD methods are good compromises between resolution and convenience. The date & time method is best matched to the algorithm used: maximum precision is delivered when the dj1 argument is for 0hrs UT1 on the day in question and the dj2 argument lies in the range 0 to 1, or vice versa. 2) The algorithm is adapted from Expression 22 of Capitaine et al. 2000. The time argument has been expressed in days directly, and, to retain precision, integer contributions have been eliminated. The same formulation is given in IERS Conventions (2003), Chap. 5, Eq. 14. Called: eraAnp normalize angle into range 0 to 2pi References: Capitaine N., Guinot B. and McCarthy D.D, 2000, Astron. Astrophys., 355, 398-405. McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.era00(dj1, dj2) return c_retval def gmst00(uta, utb, tta, ttb): """ Greenwich mean sidereal time (model consistent with IAU 2000 resolutions). Parameters ---------- uta : double array utb : double array tta : double array ttb : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraGmst00``. The ERFA documentation is:: - - - - - - - - - - e r a G m s t 0 0 - - - - - - - - - - Greenwich mean sidereal time (model consistent with IAU 2000 resolutions). Given: uta,utb double UT1 as a 2-part Julian Date (Notes 1,2) tta,ttb double TT as a 2-part Julian Date (Notes 1,2) Returned (function value): double Greenwich mean sidereal time (radians) Notes: 1) The UT1 and TT dates uta+utb and tta+ttb respectively, are both Julian Dates, apportioned in any convenient way between the argument pairs. For example, JD(UT1)=2450123.7 could be expressed in any of these ways, among others: Part A Part B 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable (in the case of UT; the TT is not at all critical in this respect). The J2000 and MJD methods are good compromises between resolution and convenience. For UT, the date & time method is best matched to the algorithm that is used by the Earth Rotation Angle function, called internally: maximum precision is delivered when the uta argument is for 0hrs UT1 on the day in question and the utb argument lies in the range 0 to 1, or vice versa. 2) Both UT1 and TT are required, UT1 to predict the Earth rotation and TT to predict the effects of precession. If UT1 is used for both purposes, errors of order 100 microarcseconds result. 3) This GMST is compatible with the IAU 2000 resolutions and must be used only in conjunction with other IAU 2000 compatible components such as precession-nutation and equation of the equinoxes. 4) The result is returned in the range 0 to 2pi. 5) The algorithm is from Capitaine et al. (2003) and IERS Conventions 2003. Called: eraEra00 Earth rotation angle, IAU 2000 eraAnp normalize angle into range 0 to 2pi References: Capitaine, N., Wallace, P.T. and McCarthy, D.D., "Expressions to implement the IAU 2000 definition of UT1", Astronomy & Astrophysics, 406, 1135-1149 (2003) McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.gmst00(uta, utb, tta, ttb) return c_retval def gmst06(uta, utb, tta, ttb): """ Greenwich mean sidereal time (consistent with IAU 2006 precession). Parameters ---------- uta : double array utb : double array tta : double array ttb : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraGmst06``. The ERFA documentation is:: - - - - - - - - - - e r a G m s t 0 6 - - - - - - - - - - Greenwich mean sidereal time (consistent with IAU 2006 precession). Given: uta,utb double UT1 as a 2-part Julian Date (Notes 1,2) tta,ttb double TT as a 2-part Julian Date (Notes 1,2) Returned (function value): double Greenwich mean sidereal time (radians) Notes: 1) The UT1 and TT dates uta+utb and tta+ttb respectively, are both Julian Dates, apportioned in any convenient way between the argument pairs. For example, JD=2450123.7 could be expressed in any of these ways, among others: Part A Part B 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable (in the case of UT; the TT is not at all critical in this respect). The J2000 and MJD methods are good compromises between resolution and convenience. For UT, the date & time method is best matched to the algorithm that is used by the Earth rotation angle function, called internally: maximum precision is delivered when the uta argument is for 0hrs UT1 on the day in question and the utb argument lies in the range 0 to 1, or vice versa. 2) Both UT1 and TT are required, UT1 to predict the Earth rotation and TT to predict the effects of precession. If UT1 is used for both purposes, errors of order 100 microarcseconds result. 3) This GMST is compatible with the IAU 2006 precession and must not be used with other precession models. 4) The result is returned in the range 0 to 2pi. Called: eraEra00 Earth rotation angle, IAU 2000 eraAnp normalize angle into range 0 to 2pi Reference: Capitaine, N., Wallace, P.T. & Chapront, J., 2005, Astron.Astrophys. 432, 355 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.gmst06(uta, utb, tta, ttb) return c_retval def gmst82(dj1, dj2): """ Universal Time to Greenwich mean sidereal time (IAU 1982 model). Parameters ---------- dj1 : double array dj2 : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraGmst82``. The ERFA documentation is:: - - - - - - - - - - e r a G m s t 8 2 - - - - - - - - - - Universal Time to Greenwich mean sidereal time (IAU 1982 model). Given: dj1,dj2 double UT1 Julian Date (see note) Returned (function value): double Greenwich mean sidereal time (radians) Notes: 1) The UT1 date dj1+dj2 is a Julian Date, apportioned in any convenient way between the arguments dj1 and dj2. For example, JD(UT1)=2450123.7 could be expressed in any of these ways, among others: dj1 dj2 2450123.7 0 (JD method) 2451545 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 and MJD methods are good compromises between resolution and convenience. The date & time method is best matched to the algorithm used: maximum accuracy (or, at least, minimum noise) is delivered when the dj1 argument is for 0hrs UT1 on the day in question and the dj2 argument lies in the range 0 to 1, or vice versa. 2) The algorithm is based on the IAU 1982 expression. This is always described as giving the GMST at 0 hours UT1. In fact, it gives the difference between the GMST and the UT, the steady 4-minutes-per-day drawing-ahead of ST with respect to UT. When whole days are ignored, the expression happens to equal the GMST at 0 hours UT1 each day. 3) In this function, the entire UT1 (the sum of the two arguments dj1 and dj2) is used directly as the argument for the standard formula, the constant term of which is adjusted by 12 hours to take account of the noon phasing of Julian Date. The UT1 is then added, but omitting whole days to conserve accuracy. Called: eraAnp normalize angle into range 0 to 2pi References: Transactions of the International Astronomical Union, XVIII B, 67 (1983). Aoki et al., Astron.Astrophys., 105, 359-361 (1982). This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.gmst82(dj1, dj2) return c_retval def gst00a(uta, utb, tta, ttb): """ Greenwich apparent sidereal time (consistent with IAU 2000 resolutions). Parameters ---------- uta : double array utb : double array tta : double array ttb : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraGst00a``. The ERFA documentation is:: - - - - - - - - - - e r a G s t 0 0 a - - - - - - - - - - Greenwich apparent sidereal time (consistent with IAU 2000 resolutions). Given: uta,utb double UT1 as a 2-part Julian Date (Notes 1,2) tta,ttb double TT as a 2-part Julian Date (Notes 1,2) Returned (function value): double Greenwich apparent sidereal time (radians) Notes: 1) The UT1 and TT dates uta+utb and tta+ttb respectively, are both Julian Dates, apportioned in any convenient way between the argument pairs. For example, JD(UT1)=2450123.7 could be expressed in any of these ways, among others: uta utb 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable (in the case of UT; the TT is not at all critical in this respect). The J2000 and MJD methods are good compromises between resolution and convenience. For UT, the date & time method is best matched to the algorithm that is used by the Earth Rotation Angle function, called internally: maximum precision is delivered when the uta argument is for 0hrs UT1 on the day in question and the utb argument lies in the range 0 to 1, or vice versa. 2) Both UT1 and TT are required, UT1 to predict the Earth rotation and TT to predict the effects of precession-nutation. If UT1 is used for both purposes, errors of order 100 microarcseconds result. 3) This GAST is compatible with the IAU 2000 resolutions and must be used only in conjunction with other IAU 2000 compatible components such as precession-nutation. 4) The result is returned in the range 0 to 2pi. 5) The algorithm is from Capitaine et al. (2003) and IERS Conventions 2003. Called: eraGmst00 Greenwich mean sidereal time, IAU 2000 eraEe00a equation of the equinoxes, IAU 2000A eraAnp normalize angle into range 0 to 2pi References: Capitaine, N., Wallace, P.T. and McCarthy, D.D., "Expressions to implement the IAU 2000 definition of UT1", Astronomy & Astrophysics, 406, 1135-1149 (2003) McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.gst00a(uta, utb, tta, ttb) return c_retval def gst00b(uta, utb): """ Greenwich apparent sidereal time (consistent with IAU 2000 resolutions but using the truncated nutation model IAU 2000B). Parameters ---------- uta : double array utb : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraGst00b``. The ERFA documentation is:: - - - - - - - - - - e r a G s t 0 0 b - - - - - - - - - - Greenwich apparent sidereal time (consistent with IAU 2000 resolutions but using the truncated nutation model IAU 2000B). Given: uta,utb double UT1 as a 2-part Julian Date (Notes 1,2) Returned (function value): double Greenwich apparent sidereal time (radians) Notes: 1) The UT1 date uta+utb is a Julian Date, apportioned in any convenient way between the argument pair. For example, JD(UT1)=2450123.7 could be expressed in any of these ways, among others: uta utb 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 and MJD methods are good compromises between resolution and convenience. For UT, the date & time method is best matched to the algorithm that is used by the Earth Rotation Angle function, called internally: maximum precision is delivered when the uta argument is for 0hrs UT1 on the day in question and the utb argument lies in the range 0 to 1, or vice versa. 2) The result is compatible with the IAU 2000 resolutions, except that accuracy has been compromised for the sake of speed and convenience in two respects: . UT is used instead of TDB (or TT) to compute the precession component of GMST and the equation of the equinoxes. This results in errors of order 0.1 mas at present. . The IAU 2000B abridged nutation model (McCarthy & Luzum, 2003) is used, introducing errors of up to 1 mas. 3) This GAST is compatible with the IAU 2000 resolutions and must be used only in conjunction with other IAU 2000 compatible components such as precession-nutation. 4) The result is returned in the range 0 to 2pi. 5) The algorithm is from Capitaine et al. (2003) and IERS Conventions 2003. Called: eraGmst00 Greenwich mean sidereal time, IAU 2000 eraEe00b equation of the equinoxes, IAU 2000B eraAnp normalize angle into range 0 to 2pi References: Capitaine, N., Wallace, P.T. and McCarthy, D.D., "Expressions to implement the IAU 2000 definition of UT1", Astronomy & Astrophysics, 406, 1135-1149 (2003) McCarthy, D.D. & Luzum, B.J., "An abridged model of the precession-nutation of the celestial pole", Celestial Mechanics & Dynamical Astronomy, 85, 37-49 (2003) McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.gst00b(uta, utb) return c_retval def gst06(uta, utb, tta, ttb, rnpb): """ Greenwich apparent sidereal time, IAU 2006, given the NPB matrix. Parameters ---------- uta : double array utb : double array tta : double array ttb : double array rnpb : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraGst06``. The ERFA documentation is:: - - - - - - - - - e r a G s t 0 6 - - - - - - - - - Greenwich apparent sidereal time, IAU 2006, given the NPB matrix. Given: uta,utb double UT1 as a 2-part Julian Date (Notes 1,2) tta,ttb double TT as a 2-part Julian Date (Notes 1,2) rnpb double[3][3] nutation x precession x bias matrix Returned (function value): double Greenwich apparent sidereal time (radians) Notes: 1) The UT1 and TT dates uta+utb and tta+ttb respectively, are both Julian Dates, apportioned in any convenient way between the argument pairs. For example, JD(UT1)=2450123.7 could be expressed in any of these ways, among others: uta utb 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable (in the case of UT; the TT is not at all critical in this respect). The J2000 and MJD methods are good compromises between resolution and convenience. For UT, the date & time method is best matched to the algorithm that is used by the Earth rotation angle function, called internally: maximum precision is delivered when the uta argument is for 0hrs UT1 on the day in question and the utb argument lies in the range 0 to 1, or vice versa. 2) Both UT1 and TT are required, UT1 to predict the Earth rotation and TT to predict the effects of precession-nutation. If UT1 is used for both purposes, errors of order 100 microarcseconds result. 3) Although the function uses the IAU 2006 series for s+XY/2, it is otherwise independent of the precession-nutation model and can in practice be used with any equinox-based NPB matrix. 4) The result is returned in the range 0 to 2pi. Called: eraBpn2xy extract CIP X,Y coordinates from NPB matrix eraS06 the CIO locator s, given X,Y, IAU 2006 eraAnp normalize angle into range 0 to 2pi eraEra00 Earth rotation angle, IAU 2000 eraEors equation of the origins, given NPB matrix and s Reference: Wallace, P.T. & Capitaine, N., 2006, Astron.Astrophys. 459, 981 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.gst06(uta, utb, tta, ttb, rnpb) return c_retval def gst06a(uta, utb, tta, ttb): """ Greenwich apparent sidereal time (consistent with IAU 2000 and 2006 resolutions). Parameters ---------- uta : double array utb : double array tta : double array ttb : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraGst06a``. The ERFA documentation is:: - - - - - - - - - - e r a G s t 0 6 a - - - - - - - - - - Greenwich apparent sidereal time (consistent with IAU 2000 and 2006 resolutions). Given: uta,utb double UT1 as a 2-part Julian Date (Notes 1,2) tta,ttb double TT as a 2-part Julian Date (Notes 1,2) Returned (function value): double Greenwich apparent sidereal time (radians) Notes: 1) The UT1 and TT dates uta+utb and tta+ttb respectively, are both Julian Dates, apportioned in any convenient way between the argument pairs. For example, JD(UT1)=2450123.7 could be expressed in any of these ways, among others: uta utb 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable (in the case of UT; the TT is not at all critical in this respect). The J2000 and MJD methods are good compromises between resolution and convenience. For UT, the date & time method is best matched to the algorithm that is used by the Earth rotation angle function, called internally: maximum precision is delivered when the uta argument is for 0hrs UT1 on the day in question and the utb argument lies in the range 0 to 1, or vice versa. 2) Both UT1 and TT are required, UT1 to predict the Earth rotation and TT to predict the effects of precession-nutation. If UT1 is used for both purposes, errors of order 100 microarcseconds result. 3) This GAST is compatible with the IAU 2000/2006 resolutions and must be used only in conjunction with IAU 2006 precession and IAU 2000A nutation. 4) The result is returned in the range 0 to 2pi. Called: eraPnm06a classical NPB matrix, IAU 2006/2000A eraGst06 Greenwich apparent ST, IAU 2006, given NPB matrix Reference: Wallace, P.T. & Capitaine, N., 2006, Astron.Astrophys. 459, 981 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.gst06a(uta, utb, tta, ttb) return c_retval def gst94(uta, utb): """ Greenwich apparent sidereal time (consistent with IAU 1982/94 resolutions). Parameters ---------- uta : double array utb : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraGst94``. The ERFA documentation is:: - - - - - - - - - e r a G s t 9 4 - - - - - - - - - Greenwich apparent sidereal time (consistent with IAU 1982/94 resolutions). Given: uta,utb double UT1 as a 2-part Julian Date (Notes 1,2) Returned (function value): double Greenwich apparent sidereal time (radians) Notes: 1) The UT1 date uta+utb is a Julian Date, apportioned in any convenient way between the argument pair. For example, JD(UT1)=2450123.7 could be expressed in any of these ways, among others: uta utb 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 and MJD methods are good compromises between resolution and convenience. For UT, the date & time method is best matched to the algorithm that is used by the Earth Rotation Angle function, called internally: maximum precision is delivered when the uta argument is for 0hrs UT1 on the day in question and the utb argument lies in the range 0 to 1, or vice versa. 2) The result is compatible with the IAU 1982 and 1994 resolutions, except that accuracy has been compromised for the sake of convenience in that UT is used instead of TDB (or TT) to compute the equation of the equinoxes. 3) This GAST must be used only in conjunction with contemporaneous IAU standards such as 1976 precession, 1980 obliquity and 1982 nutation. It is not compatible with the IAU 2000 resolutions. 4) The result is returned in the range 0 to 2pi. Called: eraGmst82 Greenwich mean sidereal time, IAU 1982 eraEqeq94 equation of the equinoxes, IAU 1994 eraAnp normalize angle into range 0 to 2pi References: Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992) IAU Resolution C7, Recommendation 3 (1994) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.gst94(uta, utb) return c_retval def pvstar(pv): """ Convert star position+velocity vector to catalog coordinates. Parameters ---------- pv : double array Returns ------- ra : double array dec : double array pmr : double array pmd : double array px : double array rv : double array Notes ----- Wraps ERFA function ``eraPvstar``. The ERFA documentation is:: - - - - - - - - - - e r a P v s t a r - - - - - - - - - - Convert star position+velocity vector to catalog coordinates. Given (Note 1): pv double[2][3] pv-vector (au, au/day) Returned (Note 2): ra double right ascension (radians) dec double declination (radians) pmr double RA proper motion (radians/year) pmd double Dec proper motion (radians/year) px double parallax (arcsec) rv double radial velocity (km/s, positive = receding) Returned (function value): int status: 0 = OK -1 = superluminal speed (Note 5) -2 = null position vector Notes: 1) The specified pv-vector is the coordinate direction (and its rate of change) for the date at which the light leaving the star reached the solar-system barycenter. 2) The star data returned by this function are "observables" for an imaginary observer at the solar-system barycenter. Proper motion and radial velocity are, strictly, in terms of barycentric coordinate time, TCB. For most practical applications, it is permissible to neglect the distinction between TCB and ordinary "proper" time on Earth (TT/TAI). The result will, as a rule, be limited by the intrinsic accuracy of the proper-motion and radial-velocity data; moreover, the supplied pv-vector is likely to be merely an intermediate result (for example generated by the function eraStarpv), so that a change of time unit will cancel out overall. In accordance with normal star-catalog conventions, the object's right ascension and declination are freed from the effects of secular aberration. The frame, which is aligned to the catalog equator and equinox, is Lorentzian and centered on the SSB. Summarizing, the specified pv-vector is for most stars almost identical to the result of applying the standard geometrical "space motion" transformation to the catalog data. The differences, which are the subject of the Stumpff paper cited below, are: (i) In stars with significant radial velocity and proper motion, the constantly changing light-time distorts the apparent proper motion. Note that this is a classical, not a relativistic, effect. (ii) The transformation complies with special relativity. 3) Care is needed with units. The star coordinates are in radians and the proper motions in radians per Julian year, but the parallax is in arcseconds; the radial velocity is in km/s, but the pv-vector result is in au and au/day. 4) The proper motions are the rate of change of the right ascension and declination at the catalog epoch and are in radians per Julian year. The RA proper motion is in terms of coordinate angle, not true angle, and will thus be numerically larger at high declinations. 5) Straight-line motion at constant speed in the inertial frame is assumed. If the speed is greater than or equal to the speed of light, the function aborts with an error status. 6) The inverse transformation is performed by the function eraStarpv. Called: eraPn decompose p-vector into modulus and direction eraPdp scalar product of two p-vectors eraSxp multiply p-vector by scalar eraPmp p-vector minus p-vector eraPm modulus of p-vector eraPpp p-vector plus p-vector eraPv2s pv-vector to spherical eraAnp normalize angle into range 0 to 2pi Reference: Stumpff, P., 1985, Astron.Astrophys. 144, 232-240. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ ra, dec, pmr, pmd, px, rv, c_retval = ufunc.pvstar(pv) check_errwarn(c_retval, 'pvstar') return ra, dec, pmr, pmd, px, rv STATUS_CODES['pvstar'] = { 0: 'OK', -1: 'superluminal speed (Note 5)', -2: 'null position vector', } def starpv(ra, dec, pmr, pmd, px, rv): """ Convert star catalog coordinates to position+velocity vector. Parameters ---------- ra : double array dec : double array pmr : double array pmd : double array px : double array rv : double array Returns ------- pv : double array Notes ----- Wraps ERFA function ``eraStarpv``. The ERFA documentation is:: - - - - - - - - - - e r a S t a r p v - - - - - - - - - - Convert star catalog coordinates to position+velocity vector. Given (Note 1): ra double right ascension (radians) dec double declination (radians) pmr double RA proper motion (radians/year) pmd double Dec proper motion (radians/year) px double parallax (arcseconds) rv double radial velocity (km/s, positive = receding) Returned (Note 2): pv double[2][3] pv-vector (au, au/day) Returned (function value): int status: 0 = no warnings 1 = distance overridden (Note 6) 2 = excessive speed (Note 7) 4 = solution didn't converge (Note 8) else = binary logical OR of the above Notes: 1) The star data accepted by this function are "observables" for an imaginary observer at the solar-system barycenter. Proper motion and radial velocity are, strictly, in terms of barycentric coordinate time, TCB. For most practical applications, it is permissible to neglect the distinction between TCB and ordinary "proper" time on Earth (TT/TAI). The result will, as a rule, be limited by the intrinsic accuracy of the proper-motion and radial-velocity data; moreover, the pv-vector is likely to be merely an intermediate result, so that a change of time unit would cancel out overall. In accordance with normal star-catalog conventions, the object's right ascension and declination are freed from the effects of secular aberration. The frame, which is aligned to the catalog equator and equinox, is Lorentzian and centered on the SSB. 2) The resulting position and velocity pv-vector is with respect to the same frame and, like the catalog coordinates, is freed from the effects of secular aberration. Should the "coordinate direction", where the object was located at the catalog epoch, be required, it may be obtained by calculating the magnitude of the position vector pv[0][0-2] dividing by the speed of light in au/day to give the light-time, and then multiplying the space velocity pv[1][0-2] by this light-time and adding the result to pv[0][0-2]. Summarizing, the pv-vector returned is for most stars almost identical to the result of applying the standard geometrical "space motion" transformation. The differences, which are the subject of the Stumpff paper referenced below, are: (i) In stars with significant radial velocity and proper motion, the constantly changing light-time distorts the apparent proper motion. Note that this is a classical, not a relativistic, effect. (ii) The transformation complies with special relativity. 3) Care is needed with units. The star coordinates are in radians and the proper motions in radians per Julian year, but the parallax is in arcseconds; the radial velocity is in km/s, but the pv-vector result is in au and au/day. 4) The RA proper motion is in terms of coordinate angle, not true angle. If the catalog uses arcseconds for both RA and Dec proper motions, the RA proper motion will need to be divided by cos(Dec) before use. 5) Straight-line motion at constant speed, in the inertial frame, is assumed. 6) An extremely small (or zero or negative) parallax is interpreted to mean that the object is on the "celestial sphere", the radius of which is an arbitrary (large) value (see the constant PXMIN). When the distance is overridden in this way, the status, initially zero, has 1 added to it. 7) If the space velocity is a significant fraction of c (see the constant VMAX), it is arbitrarily set to zero. When this action occurs, 2 is added to the status. 8) The relativistic adjustment involves an iterative calculation. If the process fails to converge within a set number (IMAX) of iterations, 4 is added to the status. 9) The inverse transformation is performed by the function eraPvstar. Called: eraS2pv spherical coordinates to pv-vector eraPm modulus of p-vector eraZp zero p-vector eraPn decompose p-vector into modulus and direction eraPdp scalar product of two p-vectors eraSxp multiply p-vector by scalar eraPmp p-vector minus p-vector eraPpp p-vector plus p-vector Reference: Stumpff, P., 1985, Astron.Astrophys. 144, 232-240. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ pv, c_retval = ufunc.starpv(ra, dec, pmr, pmd, px, rv) check_errwarn(c_retval, 'starpv') return pv STATUS_CODES['starpv'] = { 0: 'no warnings', 1: 'distance overridden (Note 6)', 2: 'excessive speed (Note 7)', 4: "solution didn't converge (Note 8)", 'else': 'binary logical OR of the above', } def fk425(r1950, d1950, dr1950, dd1950, p1950, v1950): """ Convert B1950.0 FK4 star catalog data to J2000.0 FK5. Parameters ---------- r1950 : double array d1950 : double array dr1950 : double array dd1950 : double array p1950 : double array v1950 : double array Returns ------- r2000 : double array d2000 : double array dr2000 : double array dd2000 : double array p2000 : double array v2000 : double array Notes ----- Wraps ERFA function ``eraFk425``. The ERFA documentation is:: - - - - - - - - - e r a F k 4 2 5 - - - - - - - - - Convert B1950.0 FK4 star catalog data to J2000.0 FK5. This function converts a star's catalog data from the old FK4 (Bessel-Newcomb) system to the later IAU 1976 FK5 (Fricke) system. Given: (all B1950.0, FK4) r1950,d1950 double B1950.0 RA,Dec (rad) dr1950,dd1950 double B1950.0 proper motions (rad/trop.yr) p1950 double parallax (arcsec) v1950 double radial velocity (km/s, +ve = moving away) Returned: (all J2000.0, FK5) r2000,d2000 double J2000.0 RA,Dec (rad) dr2000,dd2000 double J2000.0 proper motions (rad/Jul.yr) p2000 double parallax (arcsec) v2000 double radial velocity (km/s, +ve = moving away) Notes: 1) The proper motions in RA are dRA/dt rather than cos(Dec)*dRA/dt, and are per year rather than per century. 2) The conversion is somewhat complicated, for several reasons: . Change of standard epoch from B1950.0 to J2000.0. . An intermediate transition date of 1984 January 1.0 TT. . A change of precession model. . Change of time unit for proper motion (tropical to Julian). . FK4 positions include the E-terms of aberration, to simplify the hand computation of annual aberration. FK5 positions assume a rigorous aberration computation based on the Earth's barycentric velocity. . The E-terms also affect proper motions, and in particular cause objects at large distances to exhibit fictitious proper motions. The algorithm is based on Smith et al. (1989) and Yallop et al. (1989), which presented a matrix method due to Standish (1982) as developed by Aoki et al. (1983), using Kinoshita's development of Andoyer's post-Newcomb precession. The numerical constants from Seidelmann (1992) are used canonically. 3) Conversion from B1950.0 FK4 to J2000.0 FK5 only is provided for. Conversions for different epochs and equinoxes would require additional treatment for precession, proper motion and E-terms. 4) In the FK4 catalog the proper motions of stars within 10 degrees of the poles do not embody differential E-terms effects and should, strictly speaking, be handled in a different manner from stars outside these regions. However, given the general lack of homogeneity of the star data available for routine astrometry, the difficulties of handling positions that may have been determined from astrometric fields spanning the polar and non- polar regions, the likelihood that the differential E-terms effect was not taken into account when allowing for proper motion in past astrometry, and the undesirability of a discontinuity in the algorithm, the decision has been made in this ERFA algorithm to include the effects of differential E-terms on the proper motions for all stars, whether polar or not. At epoch J2000.0, and measuring "on the sky" rather than in terms of RA change, the errors resulting from this simplification are less than 1 milliarcsecond in position and 1 milliarcsecond per century in proper motion. Called: eraAnp normalize angle into range 0 to 2pi eraPv2s pv-vector to spherical coordinates eraPdp scalar product of two p-vectors eraPvmpv pv-vector minus pv_vector eraPvppv pv-vector plus pv_vector eraS2pv spherical coordinates to pv-vector eraSxp multiply p-vector by scalar References: Aoki, S. et al., 1983, "Conversion matrix of epoch B1950.0 FK4-based positions of stars to epoch J2000.0 positions in accordance with the new IAU resolutions". Astron.Astrophys. 128, 263-267. Seidelmann, P.K. (ed), 1992, "Explanatory Supplement to the Astronomical Almanac", ISBN 0-935702-68-7. Smith, C.A. et al., 1989, "The transformation of astrometric catalog systems to the equinox J2000.0". Astron.J. 97, 265. Standish, E.M., 1982, "Conversion of positions and proper motions from B1950.0 to the IAU system at J2000.0". Astron.Astrophys., 115, 1, 20-22. Yallop, B.D. et al., 1989, "Transformation of mean star places from FK4 B1950.0 to FK5 J2000.0 using matrices in 6-space". Astron.J. 97, 274. This revision: 2021 February 24 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ r2000, d2000, dr2000, dd2000, p2000, v2000 = ufunc.fk425( r1950, d1950, dr1950, dd1950, p1950, v1950) return r2000, d2000, dr2000, dd2000, p2000, v2000 def fk45z(r1950, d1950, bepoch): """ Convert a B1950.0 FK4 star position to J2000.0 FK5, assuming zero proper motion in the FK5 system. Parameters ---------- r1950 : double array d1950 : double array bepoch : double array Returns ------- r2000 : double array d2000 : double array Notes ----- Wraps ERFA function ``eraFk45z``. The ERFA documentation is:: - - - - - - - - - e r a F k 4 5 z - - - - - - - - - Convert a B1950.0 FK4 star position to J2000.0 FK5, assuming zero proper motion in the FK5 system. This function converts a star's catalog data from the old FK4 (Bessel-Newcomb) system to the later IAU 1976 FK5 (Fricke) system, in such a way that the FK5 proper motion is zero. Because such a star has, in general, a non-zero proper motion in the FK4 system, the function requires the epoch at which the position in the FK4 system was determined. Given: r1950,d1950 double B1950.0 FK4 RA,Dec at epoch (rad) bepoch double Besselian epoch (e.g. 1979.3) Returned: r2000,d2000 double J2000.0 FK5 RA,Dec (rad) Notes: 1) The epoch bepoch is strictly speaking Besselian, but if a Julian epoch is supplied the result will be affected only to a negligible extent. 2) The method is from Appendix 2 of Aoki et al. (1983), but using the constants of Seidelmann (1992). See the function eraFk425 for a general introduction to the FK4 to FK5 conversion. 3) Conversion from equinox B1950.0 FK4 to equinox J2000.0 FK5 only is provided for. Conversions for different starting and/or ending epochs would require additional treatment for precession, proper motion and E-terms. 4) In the FK4 catalog the proper motions of stars within 10 degrees of the poles do not embody differential E-terms effects and should, strictly speaking, be handled in a different manner from stars outside these regions. However, given the general lack of homogeneity of the star data available for routine astrometry, the difficulties of handling positions that may have been determined from astrometric fields spanning the polar and non- polar regions, the likelihood that the differential E-terms effect was not taken into account when allowing for proper motion in past astrometry, and the undesirability of a discontinuity in the algorithm, the decision has been made in this ERFA algorithm to include the effects of differential E-terms on the proper motions for all stars, whether polar or not. At epoch 2000.0, and measuring "on the sky" rather than in terms of RA change, the errors resulting from this simplification are less than 1 milliarcsecond in position and 1 milliarcsecond per century in proper motion. References: Aoki, S. et al., 1983, "Conversion matrix of epoch B1950.0 FK4-based positions of stars to epoch J2000.0 positions in accordance with the new IAU resolutions". Astron.Astrophys. 128, 263-267. Seidelmann, P.K. (ed), 1992, "Explanatory Supplement to the Astronomical Almanac", ISBN 0-935702-68-7. Called: eraAnp normalize angle into range 0 to 2pi eraC2s p-vector to spherical eraEpb2jd Besselian epoch to Julian date eraEpj Julian date to Julian epoch eraPdp scalar product of two p-vectors eraPmp p-vector minus p-vector eraPpsp p-vector plus scaled p-vector eraPvu update a pv-vector eraS2c spherical to p-vector This revision: 2021 February 24 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ r2000, d2000 = ufunc.fk45z(r1950, d1950, bepoch) return r2000, d2000 def fk524(r2000, d2000, dr2000, dd2000, p2000, v2000): """ Convert J2000.0 FK5 star catalog data to B1950.0 FK4. Parameters ---------- r2000 : double array d2000 : double array dr2000 : double array dd2000 : double array p2000 : double array v2000 : double array Returns ------- r1950 : double array d1950 : double array dr1950 : double array dd1950 : double array p1950 : double array v1950 : double array Notes ----- Wraps ERFA function ``eraFk524``. The ERFA documentation is:: - - - - - - - - - e r a F k 5 2 4 - - - - - - - - - Convert J2000.0 FK5 star catalog data to B1950.0 FK4. Given: (all J2000.0, FK5) r2000,d2000 double J2000.0 RA,Dec (rad) dr2000,dd2000 double J2000.0 proper motions (rad/Jul.yr) p2000 double parallax (arcsec) v2000 double radial velocity (km/s, +ve = moving away) Returned: (all B1950.0, FK4) r1950,d1950 double B1950.0 RA,Dec (rad) dr1950,dd1950 double B1950.0 proper motions (rad/trop.yr) p1950 double parallax (arcsec) v1950 double radial velocity (km/s, +ve = moving away) Notes: 1) The proper motions in RA are dRA/dt rather than cos(Dec)*dRA/dt, and are per year rather than per century. 2) The conversion is somewhat complicated, for several reasons: . Change of standard epoch from J2000.0 to B1950.0. . An intermediate transition date of 1984 January 1.0 TT. . A change of precession model. . Change of time unit for proper motion (Julian to tropical). . FK4 positions include the E-terms of aberration, to simplify the hand computation of annual aberration. FK5 positions assume a rigorous aberration computation based on the Earth's barycentric velocity. . The E-terms also affect proper motions, and in particular cause objects at large distances to exhibit fictitious proper motions. The algorithm is based on Smith et al. (1989) and Yallop et al. (1989), which presented a matrix method due to Standish (1982) as developed by Aoki et al. (1983), using Kinoshita's development of Andoyer's post-Newcomb precession. The numerical constants from Seidelmann (1992) are used canonically. 4) In the FK4 catalog the proper motions of stars within 10 degrees of the poles do not embody differential E-terms effects and should, strictly speaking, be handled in a different manner from stars outside these regions. However, given the general lack of homogeneity of the star data available for routine astrometry, the difficulties of handling positions that may have been determined from astrometric fields spanning the polar and non- polar regions, the likelihood that the differential E-terms effect was not taken into account when allowing for proper motion in past astrometry, and the undesirability of a discontinuity in the algorithm, the decision has been made in this ERFA algorithm to include the effects of differential E-terms on the proper motions for all stars, whether polar or not. At epoch J2000.0, and measuring "on the sky" rather than in terms of RA change, the errors resulting from this simplification are less than 1 milliarcsecond in position and 1 milliarcsecond per century in proper motion. Called: eraAnp normalize angle into range 0 to 2pi eraPdp scalar product of two p-vectors eraPm modulus of p-vector eraPmp p-vector minus p-vector eraPpp p-vector pluus p-vector eraPv2s pv-vector to spherical coordinates eraS2pv spherical coordinates to pv-vector eraSxp multiply p-vector by scalar References: Aoki, S. et al., 1983, "Conversion matrix of epoch B1950.0 FK4-based positions of stars to epoch J2000.0 positions in accordance with the new IAU resolutions". Astron.Astrophys. 128, 263-267. Seidelmann, P.K. (ed), 1992, "Explanatory Supplement to the Astronomical Almanac", ISBN 0-935702-68-7. Smith, C.A. et al., 1989, "The transformation of astrometric catalog systems to the equinox J2000.0". Astron.J. 97, 265. Standish, E.M., 1982, "Conversion of positions and proper motions from B1950.0 to the IAU system at J2000.0". Astron.Astrophys., 115, 1, 20-22. Yallop, B.D. et al., 1989, "Transformation of mean star places from FK4 B1950.0 to FK5 J2000.0 using matrices in 6-space". Astron.J. 97, 274. This revision: 2021 February 24 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ r1950, d1950, dr1950, dd1950, p1950, v1950 = ufunc.fk524( r2000, d2000, dr2000, dd2000, p2000, v2000) return r1950, d1950, dr1950, dd1950, p1950, v1950 def fk52h(r5, d5, dr5, dd5, px5, rv5): """ Transform FK5 (J2000.0) star data into the Hipparcos system. Parameters ---------- r5 : double array d5 : double array dr5 : double array dd5 : double array px5 : double array rv5 : double array Returns ------- rh : double array dh : double array drh : double array ddh : double array pxh : double array rvh : double array Notes ----- Wraps ERFA function ``eraFk52h``. The ERFA documentation is:: - - - - - - - - - e r a F k 5 2 h - - - - - - - - - Transform FK5 (J2000.0) star data into the Hipparcos system. Given (all FK5, equinox J2000.0, epoch J2000.0): r5 double RA (radians) d5 double Dec (radians) dr5 double proper motion in RA (dRA/dt, rad/Jyear) dd5 double proper motion in Dec (dDec/dt, rad/Jyear) px5 double parallax (arcsec) rv5 double radial velocity (km/s, positive = receding) Returned (all Hipparcos, epoch J2000.0): rh double RA (radians) dh double Dec (radians) drh double proper motion in RA (dRA/dt, rad/Jyear) ddh double proper motion in Dec (dDec/dt, rad/Jyear) pxh double parallax (arcsec) rvh double radial velocity (km/s, positive = receding) Notes: 1) This function transforms FK5 star positions and proper motions into the system of the Hipparcos catalog. 2) The proper motions in RA are dRA/dt rather than cos(Dec)*dRA/dt, and are per year rather than per century. 3) The FK5 to Hipparcos transformation is modeled as a pure rotation and spin; zonal errors in the FK5 catalog are not taken into account. 4) See also eraH2fk5, eraFk5hz, eraHfk5z. Called: eraStarpv star catalog data to space motion pv-vector eraFk5hip FK5 to Hipparcos rotation and spin eraRxp product of r-matrix and p-vector eraPxp vector product of two p-vectors eraPpp p-vector plus p-vector eraPvstar space motion pv-vector to star catalog data Reference: F.Mignard & M.Froeschle, Astron.Astrophys., 354, 732-739 (2000). This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rh, dh, drh, ddh, pxh, rvh = ufunc.fk52h(r5, d5, dr5, dd5, px5, rv5) return rh, dh, drh, ddh, pxh, rvh def fk54z(r2000, d2000, bepoch): """ Convert a J2000.0 FK5 star position to B1950.0 FK4, assuming zero proper motion in FK5 and parallax. Parameters ---------- r2000 : double array d2000 : double array bepoch : double array Returns ------- r1950 : double array d1950 : double array dr1950 : double array dd1950 : double array Notes ----- Wraps ERFA function ``eraFk54z``. The ERFA documentation is:: - - - - - - - - - e r a F k 5 4 z - - - - - - - - - Convert a J2000.0 FK5 star position to B1950.0 FK4, assuming zero proper motion in FK5 and parallax. Given: r2000,d2000 double J2000.0 FK5 RA,Dec (rad) bepoch double Besselian epoch (e.g. 1950.0) Returned: r1950,d1950 double B1950.0 FK4 RA,Dec (rad) at epoch BEPOCH dr1950,dd1950 double B1950.0 FK4 proper motions (rad/trop.yr) Notes: 1) In contrast to the eraFk524 function, here the FK5 proper motions, the parallax and the radial velocity are presumed zero. 2) This function converts a star position from the IAU 1976 FK5 (Fricke) system to the former FK4 (Bessel-Newcomb) system, for cases such as distant radio sources where it is presumed there is zero parallax and no proper motion. Because of the E-terms of aberration, such objects have (in general) non-zero proper motion in FK4, and the present function returns those fictitious proper motions. 3) Conversion from B1950.0 FK4 to J2000.0 FK5 only is provided for. Conversions involving other equinoxes would require additional treatment for precession. 4) The position returned by this function is in the B1950.0 FK4 reference system but at Besselian epoch BEPOCH. For comparison with catalogs the BEPOCH argument will frequently be 1950.0. (In this context the distinction between Besselian and Julian epoch is insignificant.) 5) The RA component of the returned (fictitious) proper motion is dRA/dt rather than cos(Dec)*dRA/dt. Called: eraAnp normalize angle into range 0 to 2pi eraC2s p-vector to spherical eraFk524 FK4 to FK5 eraS2c spherical to p-vector This revision: 2020 November 19 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ r1950, d1950, dr1950, dd1950 = ufunc.fk54z(r2000, d2000, bepoch) return r1950, d1950, dr1950, dd1950 def fk5hip(): """ FK5 to Hipparcos rotation and spin. Parameters ---------- Returns ------- r5h : double array s5h : double array Notes ----- Wraps ERFA function ``eraFk5hip``. The ERFA documentation is:: - - - - - - - - - - e r a F k 5 h i p - - - - - - - - - - FK5 to Hipparcos rotation and spin. Returned: r5h double[3][3] r-matrix: FK5 rotation wrt Hipparcos (Note 2) s5h double[3] r-vector: FK5 spin wrt Hipparcos (Note 3) Notes: 1) This function models the FK5 to Hipparcos transformation as a pure rotation and spin; zonal errors in the FK5 catalogue are not taken into account. 2) The r-matrix r5h operates in the sense: P_Hipparcos = r5h x P_FK5 where P_FK5 is a p-vector in the FK5 frame, and P_Hipparcos is the equivalent Hipparcos p-vector. 3) The r-vector s5h represents the time derivative of the FK5 to Hipparcos rotation. The units are radians per year (Julian, TDB). Called: eraRv2m r-vector to r-matrix Reference: F.Mignard & M.Froeschle, Astron.Astrophys., 354, 732-739 (2000). This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ r5h, s5h = ufunc.fk5hip() return r5h, s5h def fk5hz(r5, d5, date1, date2): """ Transform an FK5 (J2000.0) star position into the system of the Hipparcos catalogue, assuming zero Hipparcos proper motion. Parameters ---------- r5 : double array d5 : double array date1 : double array date2 : double array Returns ------- rh : double array dh : double array Notes ----- Wraps ERFA function ``eraFk5hz``. The ERFA documentation is:: - - - - - - - - - e r a F k 5 h z - - - - - - - - - Transform an FK5 (J2000.0) star position into the system of the Hipparcos catalogue, assuming zero Hipparcos proper motion. Given: r5 double FK5 RA (radians), equinox J2000.0, at date d5 double FK5 Dec (radians), equinox J2000.0, at date date1,date2 double TDB date (Notes 1,2) Returned: rh double Hipparcos RA (radians) dh double Hipparcos Dec (radians) Notes: 1) This function converts a star position from the FK5 system to the Hipparcos system, in such a way that the Hipparcos proper motion is zero. Because such a star has, in general, a non-zero proper motion in the FK5 system, the function requires the date at which the position in the FK5 system was determined. 2) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 3) The FK5 to Hipparcos transformation is modeled as a pure rotation and spin; zonal errors in the FK5 catalogue are not taken into account. 4) The position returned by this function is in the Hipparcos reference system but at date date1+date2. 5) See also eraFk52h, eraH2fk5, eraHfk5z. Called: eraS2c spherical coordinates to unit vector eraFk5hip FK5 to Hipparcos rotation and spin eraSxp multiply p-vector by scalar eraRv2m r-vector to r-matrix eraTrxp product of transpose of r-matrix and p-vector eraPxp vector product of two p-vectors eraC2s p-vector to spherical eraAnp normalize angle into range 0 to 2pi Reference: F.Mignard & M.Froeschle, 2000, Astron.Astrophys. 354, 732-739. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rh, dh = ufunc.fk5hz(r5, d5, date1, date2) return rh, dh def h2fk5(rh, dh, drh, ddh, pxh, rvh): """ Transform Hipparcos star data into the FK5 (J2000.0) system. Parameters ---------- rh : double array dh : double array drh : double array ddh : double array pxh : double array rvh : double array Returns ------- r5 : double array d5 : double array dr5 : double array dd5 : double array px5 : double array rv5 : double array Notes ----- Wraps ERFA function ``eraH2fk5``. The ERFA documentation is:: - - - - - - - - - e r a H 2 f k 5 - - - - - - - - - Transform Hipparcos star data into the FK5 (J2000.0) system. Given (all Hipparcos, epoch J2000.0): rh double RA (radians) dh double Dec (radians) drh double proper motion in RA (dRA/dt, rad/Jyear) ddh double proper motion in Dec (dDec/dt, rad/Jyear) pxh double parallax (arcsec) rvh double radial velocity (km/s, positive = receding) Returned (all FK5, equinox J2000.0, epoch J2000.0): r5 double RA (radians) d5 double Dec (radians) dr5 double proper motion in RA (dRA/dt, rad/Jyear) dd5 double proper motion in Dec (dDec/dt, rad/Jyear) px5 double parallax (arcsec) rv5 double radial velocity (km/s, positive = receding) Notes: 1) This function transforms Hipparcos star positions and proper motions into FK5 J2000.0. 2) The proper motions in RA are dRA/dt rather than cos(Dec)*dRA/dt, and are per year rather than per century. 3) The FK5 to Hipparcos transformation is modeled as a pure rotation and spin; zonal errors in the FK5 catalog are not taken into account. 4) See also eraFk52h, eraFk5hz, eraHfk5z. Called: eraStarpv star catalog data to space motion pv-vector eraFk5hip FK5 to Hipparcos rotation and spin eraRv2m r-vector to r-matrix eraRxp product of r-matrix and p-vector eraTrxp product of transpose of r-matrix and p-vector eraPxp vector product of two p-vectors eraPmp p-vector minus p-vector eraPvstar space motion pv-vector to star catalog data Reference: F.Mignard & M.Froeschle, Astron.Astrophys., 354, 732-739 (2000). This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ r5, d5, dr5, dd5, px5, rv5 = ufunc.h2fk5(rh, dh, drh, ddh, pxh, rvh) return r5, d5, dr5, dd5, px5, rv5 def hfk5z(rh, dh, date1, date2): """ Transform a Hipparcos star position into FK5 J2000.0, assuming zero Hipparcos proper motion. Parameters ---------- rh : double array dh : double array date1 : double array date2 : double array Returns ------- r5 : double array d5 : double array dr5 : double array dd5 : double array Notes ----- Wraps ERFA function ``eraHfk5z``. The ERFA documentation is:: - - - - - - - - - e r a H f k 5 z - - - - - - - - - Transform a Hipparcos star position into FK5 J2000.0, assuming zero Hipparcos proper motion. Given: rh double Hipparcos RA (radians) dh double Hipparcos Dec (radians) date1,date2 double TDB date (Note 1) Returned (all FK5, equinox J2000.0, date date1+date2): r5 double RA (radians) d5 double Dec (radians) dr5 double FK5 RA proper motion (rad/year, Note 4) dd5 double Dec proper motion (rad/year, Note 4) Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) The proper motion in RA is dRA/dt rather than cos(Dec)*dRA/dt. 3) The FK5 to Hipparcos transformation is modeled as a pure rotation and spin; zonal errors in the FK5 catalogue are not taken into account. 4) It was the intention that Hipparcos should be a close approximation to an inertial frame, so that distant objects have zero proper motion; such objects have (in general) non-zero proper motion in FK5, and this function returns those fictitious proper motions. 5) The position returned by this function is in the FK5 J2000.0 reference system but at date date1+date2. 6) See also eraFk52h, eraH2fk5, eraFk5zhz. Called: eraS2c spherical coordinates to unit vector eraFk5hip FK5 to Hipparcos rotation and spin eraRxp product of r-matrix and p-vector eraSxp multiply p-vector by scalar eraRxr product of two r-matrices eraTrxp product of transpose of r-matrix and p-vector eraPxp vector product of two p-vectors eraPv2s pv-vector to spherical eraAnp normalize angle into range 0 to 2pi Reference: F.Mignard & M.Froeschle, 2000, Astron.Astrophys. 354, 732-739. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ r5, d5, dr5, dd5 = ufunc.hfk5z(rh, dh, date1, date2) return r5, d5, dr5, dd5 def starpm(ra1, dec1, pmr1, pmd1, px1, rv1, ep1a, ep1b, ep2a, ep2b): """ Star proper motion: update star catalog data for space motion. Parameters ---------- ra1 : double array dec1 : double array pmr1 : double array pmd1 : double array px1 : double array rv1 : double array ep1a : double array ep1b : double array ep2a : double array ep2b : double array Returns ------- ra2 : double array dec2 : double array pmr2 : double array pmd2 : double array px2 : double array rv2 : double array Notes ----- Wraps ERFA function ``eraStarpm``. The ERFA documentation is:: - - - - - - - - - - e r a S t a r p m - - - - - - - - - - Star proper motion: update star catalog data for space motion. Given: ra1 double right ascension (radians), before dec1 double declination (radians), before pmr1 double RA proper motion (radians/year), before pmd1 double Dec proper motion (radians/year), before px1 double parallax (arcseconds), before rv1 double radial velocity (km/s, +ve = receding), before ep1a double "before" epoch, part A (Note 1) ep1b double "before" epoch, part B (Note 1) ep2a double "after" epoch, part A (Note 1) ep2b double "after" epoch, part B (Note 1) Returned: ra2 double right ascension (radians), after dec2 double declination (radians), after pmr2 double RA proper motion (radians/year), after pmd2 double Dec proper motion (radians/year), after px2 double parallax (arcseconds), after rv2 double radial velocity (km/s, +ve = receding), after Returned (function value): int status: -1 = system error (should not occur) 0 = no warnings or errors 1 = distance overridden (Note 6) 2 = excessive velocity (Note 7) 4 = solution didn't converge (Note 8) else = binary logical OR of the above warnings Notes: 1) The starting and ending TDB dates ep1a+ep1b and ep2a+ep2b are Julian Dates, apportioned in any convenient way between the two parts (A and B). For example, JD(TDB)=2450123.7 could be expressed in any of these ways, among others: epna epnb 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) In accordance with normal star-catalog conventions, the object's right ascension and declination are freed from the effects of secular aberration. The frame, which is aligned to the catalog equator and equinox, is Lorentzian and centered on the SSB. The proper motions are the rate of change of the right ascension and declination at the catalog epoch and are in radians per TDB Julian year. The parallax and radial velocity are in the same frame. 3) Care is needed with units. The star coordinates are in radians and the proper motions in radians per Julian year, but the parallax is in arcseconds. 4) The RA proper motion is in terms of coordinate angle, not true angle. If the catalog uses arcseconds for both RA and Dec proper motions, the RA proper motion will need to be divided by cos(Dec) before use. 5) Straight-line motion at constant speed, in the inertial frame, is assumed. 6) An extremely small (or zero or negative) parallax is interpreted to mean that the object is on the "celestial sphere", the radius of which is an arbitrary (large) value (see the eraStarpv function for the value used). When the distance is overridden in this way, the status, initially zero, has 1 added to it. 7) If the space velocity is a significant fraction of c (see the constant VMAX in the function eraStarpv), it is arbitrarily set to zero. When this action occurs, 2 is added to the status. 8) The relativistic adjustment carried out in the eraStarpv function involves an iterative calculation. If the process fails to converge within a set number of iterations, 4 is added to the status. Called: eraStarpv star catalog data to space motion pv-vector eraPvu update a pv-vector eraPdp scalar product of two p-vectors eraPvstar space motion pv-vector to star catalog data This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ ra2, dec2, pmr2, pmd2, px2, rv2, c_retval = ufunc.starpm( ra1, dec1, pmr1, pmd1, px1, rv1, ep1a, ep1b, ep2a, ep2b) check_errwarn(c_retval, 'starpm') return ra2, dec2, pmr2, pmd2, px2, rv2 STATUS_CODES['starpm'] = { -1: 'system error (should not occur)', 0: 'no warnings or errors', 1: 'distance overridden (Note 6)', 2: 'excessive velocity (Note 7)', 4: "solution didn't converge (Note 8)", 'else': 'binary logical OR of the above warnings', } def eceq06(date1, date2, dl, db): """ Transformation from ecliptic coordinates (mean equinox and ecliptic of date) to ICRS RA,Dec, using the IAU 2006 precession model. Parameters ---------- date1 : double array date2 : double array dl : double array db : double array Returns ------- dr : double array dd : double array Notes ----- Wraps ERFA function ``eraEceq06``. The ERFA documentation is:: - - - - - - - - - - e r a E c e q 0 6 - - - - - - - - - - Transformation from ecliptic coordinates (mean equinox and ecliptic of date) to ICRS RA,Dec, using the IAU 2006 precession model. Given: date1,date2 double TT as a 2-part Julian date (Note 1) dl,db double ecliptic longitude and latitude (radians) Returned: dr,dd double ICRS right ascension and declination (radians) 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) No assumptions are made about whether the coordinates represent starlight and embody astrometric effects such as parallax or aberration. 3) The transformation is approximately that from ecliptic longitude and latitude (mean equinox and ecliptic of date) to mean J2000.0 right ascension and declination, with only frame bias (always less than 25 mas) to disturb this classical picture. Called: eraS2c spherical coordinates to unit vector eraEcm06 J2000.0 to ecliptic rotation matrix, IAU 2006 eraTrxp product of transpose of r-matrix and p-vector eraC2s unit vector to spherical coordinates eraAnp normalize angle into range 0 to 2pi eraAnpm normalize angle into range +/- pi This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ dr, dd = ufunc.eceq06(date1, date2, dl, db) return dr, dd def ecm06(date1, date2): """ ICRS equatorial to ecliptic rotation matrix, IAU 2006. Parameters ---------- date1 : double array date2 : double array Returns ------- rm : double array Notes ----- Wraps ERFA function ``eraEcm06``. The ERFA documentation is:: - - - - - - - - - e r a E c m 0 6 - - - - - - - - - ICRS equatorial to ecliptic rotation matrix, IAU 2006. Given: date1,date2 double TT as a 2-part Julian date (Note 1) Returned: rm double[3][3] ICRS to ecliptic rotation matrix Notes: 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 1) The matrix is in the sense E_ep = rm x P_ICRS, where P_ICRS is a vector with respect to ICRS right ascension and declination axes and E_ep is the same vector with respect to the (inertial) ecliptic and equinox of date. 2) P_ICRS is a free vector, merely a direction, typically of unit magnitude, and not bound to any particular spatial origin, such as the Earth, Sun or SSB. No assumptions are made about whether it represents starlight and embodies astrometric effects such as parallax or aberration. The transformation is approximately that between mean J2000.0 right ascension and declination and ecliptic longitude and latitude, with only frame bias (always less than 25 mas) to disturb this classical picture. Called: eraObl06 mean obliquity, IAU 2006 eraPmat06 PB matrix, IAU 2006 eraIr initialize r-matrix to identity eraRx rotate around X-axis eraRxr product of two r-matrices This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rm = ufunc.ecm06(date1, date2) return rm def eqec06(date1, date2, dr, dd): """ Transformation from ICRS equatorial coordinates to ecliptic coordinates (mean equinox and ecliptic of date) using IAU 2006 precession model. Parameters ---------- date1 : double array date2 : double array dr : double array dd : double array Returns ------- dl : double array db : double array Notes ----- Wraps ERFA function ``eraEqec06``. The ERFA documentation is:: - - - - - - - - - - e r a E q e c 0 6 - - - - - - - - - - Transformation from ICRS equatorial coordinates to ecliptic coordinates (mean equinox and ecliptic of date) using IAU 2006 precession model. Given: date1,date2 double TT as a 2-part Julian date (Note 1) dr,dd double ICRS right ascension and declination (radians) Returned: dl,db double ecliptic longitude and latitude (radians) 1) The TT date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. 2) No assumptions are made about whether the coordinates represent starlight and embody astrometric effects such as parallax or aberration. 3) The transformation is approximately that from mean J2000.0 right ascension and declination to ecliptic longitude and latitude (mean equinox and ecliptic of date), with only frame bias (always less than 25 mas) to disturb this classical picture. Called: eraS2c spherical coordinates to unit vector eraEcm06 J2000.0 to ecliptic rotation matrix, IAU 2006 eraRxp product of r-matrix and p-vector eraC2s unit vector to spherical coordinates eraAnp normalize angle into range 0 to 2pi eraAnpm normalize angle into range +/- pi This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ dl, db = ufunc.eqec06(date1, date2, dr, dd) return dl, db def lteceq(epj, dl, db): """ Transformation from ecliptic coordinates (mean equinox and ecliptic of date) to ICRS RA,Dec, using a long-term precession model. Parameters ---------- epj : double array dl : double array db : double array Returns ------- dr : double array dd : double array Notes ----- Wraps ERFA function ``eraLteceq``. The ERFA documentation is:: - - - - - - - - - - e r a L t e c e q - - - - - - - - - - Transformation from ecliptic coordinates (mean equinox and ecliptic of date) to ICRS RA,Dec, using a long-term precession model. Given: epj double Julian epoch (TT) dl,db double ecliptic longitude and latitude (radians) Returned: dr,dd double ICRS right ascension and declination (radians) 1) No assumptions are made about whether the coordinates represent starlight and embody astrometric effects such as parallax or aberration. 2) The transformation is approximately that from ecliptic longitude and latitude (mean equinox and ecliptic of date) to mean J2000.0 right ascension and declination, with only frame bias (always less than 25 mas) to disturb this classical picture. 3) The Vondrak et al. (2011, 2012) 400 millennia precession model agrees with the IAU 2006 precession at J2000.0 and stays within 100 microarcseconds during the 20th and 21st centuries. It is accurate to a few arcseconds throughout the historical period, worsening to a few tenths of a degree at the end of the +/- 200,000 year time span. Called: eraS2c spherical coordinates to unit vector eraLtecm J2000.0 to ecliptic rotation matrix, long term eraTrxp product of transpose of r-matrix and p-vector eraC2s unit vector to spherical coordinates eraAnp normalize angle into range 0 to 2pi eraAnpm normalize angle into range +/- pi References: Vondrak, J., Capitaine, N. and Wallace, P., 2011, New precession expressions, valid for long time intervals, Astron.Astrophys. 534, A22 Vondrak, J., Capitaine, N. and Wallace, P., 2012, New precession expressions, valid for long time intervals (Corrigendum), Astron.Astrophys. 541, C1 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ dr, dd = ufunc.lteceq(epj, dl, db) return dr, dd def ltecm(epj): """ ICRS equatorial to ecliptic rotation matrix, long-term. Parameters ---------- epj : double array Returns ------- rm : double array Notes ----- Wraps ERFA function ``eraLtecm``. The ERFA documentation is:: - - - - - - - - - e r a L t e c m - - - - - - - - - ICRS equatorial to ecliptic rotation matrix, long-term. Given: epj double Julian epoch (TT) Returned: rm double[3][3] ICRS to ecliptic rotation matrix Notes: 1) The matrix is in the sense E_ep = rm x P_ICRS, where P_ICRS is a vector with respect to ICRS right ascension and declination axes and E_ep is the same vector with respect to the (inertial) ecliptic and equinox of epoch epj. 2) P_ICRS is a free vector, merely a direction, typically of unit magnitude, and not bound to any particular spatial origin, such as the Earth, Sun or SSB. No assumptions are made about whether it represents starlight and embodies astrometric effects such as parallax or aberration. The transformation is approximately that between mean J2000.0 right ascension and declination and ecliptic longitude and latitude, with only frame bias (always less than 25 mas) to disturb this classical picture. 3) The Vondrak et al. (2011, 2012) 400 millennia precession model agrees with the IAU 2006 precession at J2000.0 and stays within 100 microarcseconds during the 20th and 21st centuries. It is accurate to a few arcseconds throughout the historical period, worsening to a few tenths of a degree at the end of the +/- 200,000 year time span. Called: eraLtpequ equator pole, long term eraLtpecl ecliptic pole, long term eraPxp vector product eraPn normalize vector References: Vondrak, J., Capitaine, N. and Wallace, P., 2011, New precession expressions, valid for long time intervals, Astron.Astrophys. 534, A22 Vondrak, J., Capitaine, N. and Wallace, P., 2012, New precession expressions, valid for long time intervals (Corrigendum), Astron.Astrophys. 541, C1 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rm = ufunc.ltecm(epj) return rm def lteqec(epj, dr, dd): """ Transformation from ICRS equatorial coordinates to ecliptic coordinates (mean equinox and ecliptic of date) using a long-term precession model. Parameters ---------- epj : double array dr : double array dd : double array Returns ------- dl : double array db : double array Notes ----- Wraps ERFA function ``eraLteqec``. The ERFA documentation is:: - - - - - - - - - - e r a L t e q e c - - - - - - - - - - Transformation from ICRS equatorial coordinates to ecliptic coordinates (mean equinox and ecliptic of date) using a long-term precession model. Given: epj double Julian epoch (TT) dr,dd double ICRS right ascension and declination (radians) Returned: dl,db double ecliptic longitude and latitude (radians) 1) No assumptions are made about whether the coordinates represent starlight and embody astrometric effects such as parallax or aberration. 2) The transformation is approximately that from mean J2000.0 right ascension and declination to ecliptic longitude and latitude (mean equinox and ecliptic of date), with only frame bias (always less than 25 mas) to disturb this classical picture. 3) The Vondrak et al. (2011, 2012) 400 millennia precession model agrees with the IAU 2006 precession at J2000.0 and stays within 100 microarcseconds during the 20th and 21st centuries. It is accurate to a few arcseconds throughout the historical period, worsening to a few tenths of a degree at the end of the +/- 200,000 year time span. Called: eraS2c spherical coordinates to unit vector eraLtecm J2000.0 to ecliptic rotation matrix, long term eraRxp product of r-matrix and p-vector eraC2s unit vector to spherical coordinates eraAnp normalize angle into range 0 to 2pi eraAnpm normalize angle into range +/- pi References: Vondrak, J., Capitaine, N. and Wallace, P., 2011, New precession expressions, valid for long time intervals, Astron.Astrophys. 534, A22 Vondrak, J., Capitaine, N. and Wallace, P., 2012, New precession expressions, valid for long time intervals (Corrigendum), Astron.Astrophys. 541, C1 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ dl, db = ufunc.lteqec(epj, dr, dd) return dl, db def g2icrs(dl, db): """ Transformation from Galactic Coordinates to ICRS. Parameters ---------- dl : double array db : double array Returns ------- dr : double array dd : double array Notes ----- Wraps ERFA function ``eraG2icrs``. The ERFA documentation is:: - - - - - - - - - - e r a G 2 i c r s - - - - - - - - - - Transformation from Galactic Coordinates to ICRS. Given: dl double galactic longitude (radians) db double galactic latitude (radians) Returned: dr double ICRS right ascension (radians) dd double ICRS declination (radians) Notes: 1) The IAU 1958 system of Galactic coordinates was defined with respect to the now obsolete reference system FK4 B1950.0. When interpreting the system in a modern context, several factors have to be taken into account: . The inclusion in FK4 positions of the E-terms of aberration. . The distortion of the FK4 proper motion system by differential Galactic rotation. . The use of the B1950.0 equinox rather than the now-standard J2000.0. . The frame bias between ICRS and the J2000.0 mean place system. The Hipparcos Catalogue (Perryman & ESA 1997) provides a rotation matrix that transforms directly between ICRS and Galactic coordinates with the above factors taken into account. The matrix is derived from three angles, namely the ICRS coordinates of the Galactic pole and the longitude of the ascending node of the galactic equator on the ICRS equator. They are given in degrees to five decimal places and for canonical purposes are regarded as exact. In the Hipparcos Catalogue the matrix elements are given to 10 decimal places (about 20 microarcsec). In the present ERFA function the matrix elements have been recomputed from the canonical three angles and are given to 30 decimal places. 2) The inverse transformation is performed by the function eraIcrs2g. Called: eraAnp normalize angle into range 0 to 2pi eraAnpm normalize angle into range +/- pi eraS2c spherical coordinates to unit vector eraTrxp product of transpose of r-matrix and p-vector eraC2s p-vector to spherical Reference: Perryman M.A.C. & ESA, 1997, ESA SP-1200, The Hipparcos and Tycho catalogues. Astrometric and photometric star catalogues derived from the ESA Hipparcos Space Astrometry Mission. ESA Publications Division, Noordwijk, Netherlands. This revision: 2021 January 25 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ dr, dd = ufunc.g2icrs(dl, db) return dr, dd def icrs2g(dr, dd): """ Transformation from ICRS to Galactic Coordinates. Parameters ---------- dr : double array dd : double array Returns ------- dl : double array db : double array Notes ----- Wraps ERFA function ``eraIcrs2g``. The ERFA documentation is:: - - - - - - - - - - e r a I c r s 2 g - - - - - - - - - - Transformation from ICRS to Galactic Coordinates. Given: dr double ICRS right ascension (radians) dd double ICRS declination (radians) Returned: dl double galactic longitude (radians) db double galactic latitude (radians) Notes: 1) The IAU 1958 system of Galactic coordinates was defined with respect to the now obsolete reference system FK4 B1950.0. When interpreting the system in a modern context, several factors have to be taken into account: . The inclusion in FK4 positions of the E-terms of aberration. . The distortion of the FK4 proper motion system by differential Galactic rotation. . The use of the B1950.0 equinox rather than the now-standard J2000.0. . The frame bias between ICRS and the J2000.0 mean place system. The Hipparcos Catalogue (Perryman & ESA 1997) provides a rotation matrix that transforms directly between ICRS and Galactic coordinates with the above factors taken into account. The matrix is derived from three angles, namely the ICRS coordinates of the Galactic pole and the longitude of the ascending node of the galactic equator on the ICRS equator. They are given in degrees to five decimal places and for canonical purposes are regarded as exact. In the Hipparcos Catalogue the matrix elements are given to 10 decimal places (about 20 microarcsec). In the present ERFA function the matrix elements have been recomputed from the canonical three angles and are given to 30 decimal places. 2) The inverse transformation is performed by the function eraG2icrs. Called: eraAnp normalize angle into range 0 to 2pi eraAnpm normalize angle into range +/- pi eraS2c spherical coordinates to unit vector eraRxp product of r-matrix and p-vector eraC2s p-vector to spherical Reference: Perryman M.A.C. & ESA, 1997, ESA SP-1200, The Hipparcos and Tycho catalogues. Astrometric and photometric star catalogues derived from the ESA Hipparcos Space Astrometry Mission. ESA Publications Division, Noordwijk, Netherlands. This revision: 2021 January 25 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ dl, db = ufunc.icrs2g(dr, dd) return dl, db def eform(n): """ Earth reference ellipsoids. Parameters ---------- n : int array Returns ------- a : double array f : double array Notes ----- Wraps ERFA function ``eraEform``. The ERFA documentation is:: - - - - - - - - - e r a E f o r m - - - - - - - - - Earth reference ellipsoids. Given: n int ellipsoid identifier (Note 1) Returned: a double equatorial radius (meters, Note 2) f double flattening (Note 2) Returned (function value): int status: 0 = OK -1 = illegal identifier (Note 3) Notes: 1) The identifier n is a number that specifies the choice of reference ellipsoid. The following are supported: n ellipsoid 1 ERFA_WGS84 2 ERFA_GRS80 3 ERFA_WGS72 The n value has no significance outside the ERFA software. For convenience, symbols ERFA_WGS84 etc. are defined in erfam.h. 2) The ellipsoid parameters are returned in the form of equatorial radius in meters (a) and flattening (f). The latter is a number around 0.00335, i.e. around 1/298. 3) For the case where an unsupported n value is supplied, zero a and f are returned, as well as error status. References: Department of Defense World Geodetic System 1984, National Imagery and Mapping Agency Technical Report 8350.2, Third Edition, p3-2. Moritz, H., Bull. Geodesique 66-2, 187 (1992). The Department of Defense World Geodetic System 1972, World Geodetic System Committee, May 1974. Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), p220. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ a, f, c_retval = ufunc.eform(n) check_errwarn(c_retval, 'eform') return a, f STATUS_CODES['eform'] = { 0: 'OK', -1: 'illegal identifier (Note 3)', } def gc2gd(n, xyz): """ Transform geocentric coordinates to geodetic using the specified reference ellipsoid. Parameters ---------- n : int array xyz : double array Returns ------- elong : double array phi : double array height : double array Notes ----- Wraps ERFA function ``eraGc2gd``. The ERFA documentation is:: - - - - - - - - - e r a G c 2 g d - - - - - - - - - Transform geocentric coordinates to geodetic using the specified reference ellipsoid. Given: n int ellipsoid identifier (Note 1) xyz double[3] geocentric vector (Note 2) Returned: elong double longitude (radians, east +ve, Note 3) phi double latitude (geodetic, radians, Note 3) height double height above ellipsoid (geodetic, Notes 2,3) Returned (function value): int status: 0 = OK -1 = illegal identifier (Note 3) -2 = internal error (Note 3) Notes: 1) The identifier n is a number that specifies the choice of reference ellipsoid. The following are supported: n ellipsoid 1 ERFA_WGS84 2 ERFA_GRS80 3 ERFA_WGS72 The n value has no significance outside the ERFA software. For convenience, symbols ERFA_WGS84 etc. are defined in erfam.h. 2) The geocentric vector (xyz, given) and height (height, returned) are in meters. 3) An error status -1 means that the identifier n is illegal. An error status -2 is theoretically impossible. In all error cases, all three results are set to -1e9. 4) The inverse transformation is performed in the function eraGd2gc. Called: eraEform Earth reference ellipsoids eraGc2gde geocentric to geodetic transformation, general This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ elong, phi, height, c_retval = ufunc.gc2gd(n, xyz) check_errwarn(c_retval, 'gc2gd') return elong, phi, height STATUS_CODES['gc2gd'] = { 0: 'OK', -1: 'illegal identifier (Note 3)', -2: 'internal error (Note 3)', } def gc2gde(a, f, xyz): """ Transform geocentric coordinates to geodetic for a reference ellipsoid of specified form. Parameters ---------- a : double array f : double array xyz : double array Returns ------- elong : double array phi : double array height : double array Notes ----- Wraps ERFA function ``eraGc2gde``. The ERFA documentation is:: - - - - - - - - - - e r a G c 2 g d e - - - - - - - - - - Transform geocentric coordinates to geodetic for a reference ellipsoid of specified form. Given: a double equatorial radius (Notes 2,4) f double flattening (Note 3) xyz double[3] geocentric vector (Note 4) Returned: elong double longitude (radians, east +ve) phi double latitude (geodetic, radians) height double height above ellipsoid (geodetic, Note 4) Returned (function value): int status: 0 = OK -1 = illegal f -2 = illegal a Notes: 1) This function is based on the GCONV2H Fortran subroutine by Toshio Fukushima (see reference). 2) The equatorial radius, a, can be in any units, but meters is the conventional choice. 3) The flattening, f, is (for the Earth) a value around 0.00335, i.e. around 1/298. 4) The equatorial radius, a, and the geocentric vector, xyz, must be given in the same units, and determine the units of the returned height, height. 5) If an error occurs (status < 0), elong, phi and height are unchanged. 6) The inverse transformation is performed in the function eraGd2gce. 7) The transformation for a standard ellipsoid (such as ERFA_WGS84) can more conveniently be performed by calling eraGc2gd, which uses a numerical code to identify the required A and F values. Reference: Fukushima, T., "Transformation from Cartesian to geodetic coordinates accelerated by Halley's method", J.Geodesy (2006) 79: 689-693 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ elong, phi, height, c_retval = ufunc.gc2gde(a, f, xyz) check_errwarn(c_retval, 'gc2gde') return elong, phi, height STATUS_CODES['gc2gde'] = { 0: 'OK', -1: 'illegal f', -2: 'illegal a', } def gd2gc(n, elong, phi, height): """ Transform geodetic coordinates to geocentric using the specified reference ellipsoid. Parameters ---------- n : int array elong : double array phi : double array height : double array Returns ------- xyz : double array Notes ----- Wraps ERFA function ``eraGd2gc``. The ERFA documentation is:: - - - - - - - - - e r a G d 2 g c - - - - - - - - - Transform geodetic coordinates to geocentric using the specified reference ellipsoid. Given: n int ellipsoid identifier (Note 1) elong double longitude (radians, east +ve) phi double latitude (geodetic, radians, Note 3) height double height above ellipsoid (geodetic, Notes 2,3) Returned: xyz double[3] geocentric vector (Note 2) Returned (function value): int status: 0 = OK -1 = illegal identifier (Note 3) -2 = illegal case (Note 3) Notes: 1) The identifier n is a number that specifies the choice of reference ellipsoid. The following are supported: n ellipsoid 1 ERFA_WGS84 2 ERFA_GRS80 3 ERFA_WGS72 The n value has no significance outside the ERFA software. For convenience, symbols ERFA_WGS84 etc. are defined in erfam.h. 2) The height (height, given) and the geocentric vector (xyz, returned) are in meters. 3) No validation is performed on the arguments elong, phi and height. An error status -1 means that the identifier n is illegal. An error status -2 protects against cases that would lead to arithmetic exceptions. In all error cases, xyz is set to zeros. 4) The inverse transformation is performed in the function eraGc2gd. Called: eraEform Earth reference ellipsoids eraGd2gce geodetic to geocentric transformation, general eraZp zero p-vector This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ xyz, c_retval = ufunc.gd2gc(n, elong, phi, height) check_errwarn(c_retval, 'gd2gc') return xyz STATUS_CODES['gd2gc'] = { 0: 'OK', -1: 'illegal identifier (Note 3)', -2: 'illegal case (Note 3)', } def gd2gce(a, f, elong, phi, height): """ Transform geodetic coordinates to geocentric for a reference ellipsoid of specified form. Parameters ---------- a : double array f : double array elong : double array phi : double array height : double array Returns ------- xyz : double array Notes ----- Wraps ERFA function ``eraGd2gce``. The ERFA documentation is:: - - - - - - - - - - e r a G d 2 g c e - - - - - - - - - - Transform geodetic coordinates to geocentric for a reference ellipsoid of specified form. Given: a double equatorial radius (Notes 1,4) f double flattening (Notes 2,4) elong double longitude (radians, east +ve) phi double latitude (geodetic, radians, Note 4) height double height above ellipsoid (geodetic, Notes 3,4) Returned: xyz double[3] geocentric vector (Note 3) Returned (function value): int status: 0 = OK -1 = illegal case (Note 4) Notes: 1) The equatorial radius, a, can be in any units, but meters is the conventional choice. 2) The flattening, f, is (for the Earth) a value around 0.00335, i.e. around 1/298. 3) The equatorial radius, a, and the height, height, must be given in the same units, and determine the units of the returned geocentric vector, xyz. 4) No validation is performed on individual arguments. The error status -1 protects against (unrealistic) cases that would lead to arithmetic exceptions. If an error occurs, xyz is unchanged. 5) The inverse transformation is performed in the function eraGc2gde. 6) The transformation for a standard ellipsoid (such as ERFA_WGS84) can more conveniently be performed by calling eraGd2gc, which uses a numerical code to identify the required a and f values. References: Green, R.M., Spherical Astronomy, Cambridge University Press, (1985) Section 4.5, p96. Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992), Section 4.22, p202. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ xyz, c_retval = ufunc.gd2gce(a, f, elong, phi, height) check_errwarn(c_retval, 'gd2gce') return xyz STATUS_CODES['gd2gce'] = { 0: 'OK', -1: 'illegal case (Note 4)Notes:', } def d2dtf(scale, ndp, d1, d2): """ Format for output a 2-part Julian Date (or in the case of UTC a quasi-JD form that includes special provision for leap seconds). Parameters ---------- scale : const char array ndp : int array d1 : double array d2 : double array Returns ------- iy : int array im : int array id : int array ihmsf : int array Notes ----- Wraps ERFA function ``eraD2dtf``. The ERFA documentation is:: - - - - - - - - - e r a D 2 d t f - - - - - - - - - Format for output a 2-part Julian Date (or in the case of UTC a quasi-JD form that includes special provision for leap seconds). Given: scale char[] time scale ID (Note 1) ndp int resolution (Note 2) d1,d2 double time as a 2-part Julian Date (Notes 3,4) Returned: iy,im,id int year, month, day in Gregorian calendar (Note 5) ihmsf int[4] hours, minutes, seconds, fraction (Note 1) Returned (function value): int status: +1 = dubious year (Note 5) 0 = OK -1 = unacceptable date (Note 6) Notes: 1) scale identifies the time scale. Only the value "UTC" (in upper case) is significant, and enables handling of leap seconds (see Note 4). 2) ndp is the number of decimal places in the seconds field, and can have negative as well as positive values, such as: ndp resolution -4 1 00 00 -3 0 10 00 -2 0 01 00 -1 0 00 10 0 0 00 01 1 0 00 00.1 2 0 00 00.01 3 0 00 00.001 The limits are platform dependent, but a safe range is -5 to +9. 3) d1+d2 is Julian Date, apportioned in any convenient way between the two arguments, for example where d1 is the Julian Day Number and d2 is the fraction of a day. In the case of UTC, where the use of JD is problematical, special conventions apply: see the next note. 4) JD cannot unambiguously represent UTC during a leap second unless special measures are taken. The ERFA internal convention is that the quasi-JD day represents UTC days whether the length is 86399, 86400 or 86401 SI seconds. In the 1960-1972 era there were smaller jumps (in either direction) each time the linear UTC(TAI) expression was changed, and these "mini-leaps" are also included in the ERFA convention. 5) The warning status "dubious year" flags UTCs that predate the introduction of the time scale or that are too far in the future to be trusted. See eraDat for further details. 6) For calendar conventions and limitations, see eraCal2jd. Called: eraJd2cal JD to Gregorian calendar eraD2tf decompose days to hms eraDat delta(AT) = TAI-UTC This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ iy, im, id, ihmsf, c_retval = ufunc.d2dtf(scale, ndp, d1, d2) check_errwarn(c_retval, 'd2dtf') return iy, im, id, ihmsf STATUS_CODES['d2dtf'] = { 1: 'dubious year (Note 5)', 0: 'OK', -1: 'unacceptable date (Note 6)', } def dat(iy, im, id, fd): """ For a given UTC date, calculate Delta(AT) = TAI-UTC. Parameters ---------- iy : int array im : int array id : int array fd : double array Returns ------- deltat : double array Notes ----- Wraps ERFA function ``eraDat``. The ERFA documentation is:: - - - - - - - e r a D a t - - - - - - - For a given UTC date, calculate Delta(AT) = TAI-UTC. :------------------------------------------: : : : IMPORTANT : : : : A new version of this function must be : : produced whenever a new leap second is : : announced. There are four items to : : change on each such occasion: : : : : 1) A new line must be added to the set : : of statements that initialize the : : array "changes". : : : : 2) The constant IYV must be set to the : : current year. : : : : 3) The "Latest leap second" comment : : below must be set to the new leap : : second date. : : : : 4) The "This revision" comment, later, : : must be set to the current date. : : : : Change (2) must also be carried out : : whenever the function is re-issued, : : even if no leap seconds have been : : added. : : : : Latest leap second: 2016 December 31 : : : :__________________________________________: Given: iy int UTC: year (Notes 1 and 2) im int month (Note 2) id int day (Notes 2 and 3) fd double fraction of day (Note 4) Returned: deltat double TAI minus UTC, seconds Returned (function value): int status (Note 5): 1 = dubious year (Note 1) 0 = OK -1 = bad year -2 = bad month -3 = bad day (Note 3) -4 = bad fraction (Note 4) -5 = internal error (Note 5) Notes: 1) UTC began at 1960 January 1.0 (JD 2436934.5) and it is improper to call the function with an earlier date. If this is attempted, zero is returned together with a warning status. Because leap seconds cannot, in principle, be predicted in advance, a reliable check for dates beyond the valid range is impossible. To guard against gross errors, a year five or more after the release year of the present function (see the constant IYV) is considered dubious. In this case a warning status is returned but the result is computed in the normal way. For both too-early and too-late years, the warning status is +1. This is distinct from the error status -1, which signifies a year so early that JD could not be computed. 2) If the specified date is for a day which ends with a leap second, the TAI-UTC value returned is for the period leading up to the leap second. If the date is for a day which begins as a leap second ends, the TAI-UTC returned is for the period following the leap second. 3) The day number must be in the normal calendar range, for example 1 through 30 for April. The "almanac" convention of allowing such dates as January 0 and December 32 is not supported in this function, in order to avoid confusion near leap seconds. 4) The fraction of day is used only for dates before the introduction of leap seconds, the first of which occurred at the end of 1971. It is tested for validity (0 to 1 is the valid range) even if not used; if invalid, zero is used and status -4 is returned. For many applications, setting fd to zero is acceptable; the resulting error is always less than 3 ms (and occurs only pre-1972). 5) The status value returned in the case where there are multiple errors refers to the first error detected. For example, if the month and day are 13 and 32 respectively, status -2 (bad month) will be returned. The "internal error" status refers to a case that is impossible but causes some compilers to issue a warning. 6) In cases where a valid result is not available, zero is returned. References: 1) For dates from 1961 January 1 onwards, the expressions from the file ftp://maia.usno.navy.mil/ser7/tai-utc.dat are used. 2) The 5ms timestep at 1961 January 1 is taken from 2.58.1 (p87) of the 1992 Explanatory Supplement. Called: eraCal2jd Gregorian calendar to JD This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ deltat, c_retval = ufunc.dat(iy, im, id, fd) check_errwarn(c_retval, 'dat') return deltat STATUS_CODES['dat'] = { 1: 'dubious year (Note 1)', 0: 'OK', -1: 'bad year', -2: 'bad month', -3: 'bad day (Note 3)', -4: 'bad fraction (Note 4)', -5: 'internal error (Note 5)', } def dtdb(date1, date2, ut, elong, u, v): """ An approximation to TDB-TT, the difference between barycentric dynamical time and terrestrial time, for an observer on the Earth. Parameters ---------- date1 : double array date2 : double array ut : double array elong : double array u : double array v : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraDtdb``. The ERFA documentation is:: - - - - - - - - e r a D t d b - - - - - - - - An approximation to TDB-TT, the difference between barycentric dynamical time and terrestrial time, for an observer on the Earth. The different time scales - proper, coordinate and realized - are related to each other: TAI <- physically realized : offset <- observed (nominally +32.184s) : TT <- terrestrial time : rate adjustment (L_G) <- definition of TT : TCG <- time scale for GCRS : "periodic" terms <- eraDtdb is an implementation : rate adjustment (L_C) <- function of solar-system ephemeris : TCB <- time scale for BCRS : rate adjustment (-L_B) <- definition of TDB : TDB <- TCB scaled to track TT : "periodic" terms <- -eraDtdb is an approximation : TT <- terrestrial time Adopted values for the various constants can be found in the IERS Conventions (McCarthy & Petit 2003). Given: date1,date2 double date, TDB (Notes 1-3) ut double universal time (UT1, fraction of one day) elong double longitude (east positive, radians) u double distance from Earth spin axis (km) v double distance north of equatorial plane (km) Returned (function value): double TDB-TT (seconds) Notes: 1) The date date1+date2 is a Julian Date, apportioned in any convenient way between the two arguments. For example, JD(TT)=2450123.7 could be expressed in any of these ways, among others: date1 date2 2450123.7 0.0 (JD method) 2451545.0 -1421.3 (J2000 method) 2400000.5 50123.2 (MJD method) 2450123.5 0.2 (date & time method) The JD method is the most natural and convenient to use in cases where the loss of several decimal digits of resolution is acceptable. The J2000 method is best matched to the way the argument is handled internally and will deliver the optimum resolution. The MJD method and the date & time methods are both good compromises between resolution and convenience. Although the date is, formally, barycentric dynamical time (TDB), the terrestrial dynamical time (TT) can be used with no practical effect on the accuracy of the prediction. 2) TT can be regarded as a coordinate time that is realized as an offset of 32.184s from International Atomic Time, TAI. TT is a specific linear transformation of geocentric coordinate time TCG, which is the time scale for the Geocentric Celestial Reference System, GCRS. 3) TDB is a coordinate time, and is a specific linear transformation of barycentric coordinate time TCB, which is the time scale for the Barycentric Celestial Reference System, BCRS. 4) The difference TCG-TCB depends on the masses and positions of the bodies of the solar system and the velocity of the Earth. It is dominated by a rate difference, the residual being of a periodic character. The latter, which is modeled by the present function, comprises a main (annual) sinusoidal term of amplitude approximately 0.00166 seconds, plus planetary terms up to about 20 microseconds, and lunar and diurnal terms up to 2 microseconds. These effects come from the changing transverse Doppler effect and gravitational red-shift as the observer (on the Earth's surface) experiences variations in speed (with respect to the BCRS) and gravitational potential. 5) TDB can be regarded as the same as TCB but with a rate adjustment to keep it close to TT, which is convenient for many applications. The history of successive attempts to define TDB is set out in Resolution 3 adopted by the IAU General Assembly in 2006, which defines a fixed TDB(TCB) transformation that is consistent with contemporary solar-system ephemerides. Future ephemerides will imply slightly changed transformations between TCG and TCB, which could introduce a linear drift between TDB and TT; however, any such drift is unlikely to exceed 1 nanosecond per century. 6) The geocentric TDB-TT model used in the present function is that of Fairhead & Bretagnon (1990), in its full form. It was originally supplied by Fairhead (private communications with P.T.Wallace, 1990) as a Fortran subroutine. The present C function contains an adaptation of the Fairhead code. The numerical results are essentially unaffected by the changes, the differences with respect to the Fairhead & Bretagnon original being at the 1e-20 s level. The topocentric part of the model is from Moyer (1981) and Murray (1983), with fundamental arguments adapted from Simon et al. 1994. It is an approximation to the expression ( v / c ) . ( r / c ), where v is the barycentric velocity of the Earth, r is the geocentric position of the observer and c is the speed of light. By supplying zeroes for u and v, the topocentric part of the model can be nullified, and the function will return the Fairhead & Bretagnon result alone. 7) During the interval 1950-2050, the absolute accuracy is better than +/- 3 nanoseconds relative to time ephemerides obtained by direct numerical integrations based on the JPL DE405 solar system ephemeris. 8) It must be stressed that the present function is merely a model, and that numerical integration of solar-system ephemerides is the definitive method for predicting the relationship between TCG and TCB and hence between TT and TDB. References: Fairhead, L., & Bretagnon, P., Astron.Astrophys., 229, 240-247 (1990). IAU 2006 Resolution 3. McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) Moyer, T.D., Cel.Mech., 23, 33 (1981). Murray, C.A., Vectorial Astrometry, Adam Hilger (1983). Seidelmann, P.K. et al., Explanatory Supplement to the Astronomical Almanac, Chapter 2, University Science Books (1992). Simon, J.L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G. & Laskar, J., Astron.Astrophys., 282, 663-683 (1994). This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.dtdb(date1, date2, ut, elong, u, v) return c_retval def dtf2d(scale, iy, im, id, ihr, imn, sec): """ Encode date and time fields into 2-part Julian Date (or in the case of UTC a quasi-JD form that includes special provision for leap seconds). Parameters ---------- scale : const char array iy : int array im : int array id : int array ihr : int array imn : int array sec : double array Returns ------- d1 : double array d2 : double array Notes ----- Wraps ERFA function ``eraDtf2d``. The ERFA documentation is:: - - - - - - - - - e r a D t f 2 d - - - - - - - - - Encode date and time fields into 2-part Julian Date (or in the case of UTC a quasi-JD form that includes special provision for leap seconds). Given: scale char[] time scale ID (Note 1) iy,im,id int year, month, day in Gregorian calendar (Note 2) ihr,imn int hour, minute sec double seconds Returned: d1,d2 double 2-part Julian Date (Notes 3,4) Returned (function value): int status: +3 = both of next two +2 = time is after end of day (Note 5) +1 = dubious year (Note 6) 0 = OK -1 = bad year -2 = bad month -3 = bad day -4 = bad hour -5 = bad minute -6 = bad second (<0) Notes: 1) scale identifies the time scale. Only the value "UTC" (in upper case) is significant, and enables handling of leap seconds (see Note 4). 2) For calendar conventions and limitations, see eraCal2jd. 3) The sum of the results, d1+d2, is Julian Date, where normally d1 is the Julian Day Number and d2 is the fraction of a day. In the case of UTC, where the use of JD is problematical, special conventions apply: see the next note. 4) JD cannot unambiguously represent UTC during a leap second unless special measures are taken. The ERFA internal convention is that the quasi-JD day represents UTC days whether the length is 86399, 86400 or 86401 SI seconds. In the 1960-1972 era there were smaller jumps (in either direction) each time the linear UTC(TAI) expression was changed, and these "mini-leaps" are also included in the ERFA convention. 5) The warning status "time is after end of day" usually means that the sec argument is greater than 60.0. However, in a day ending in a leap second the limit changes to 61.0 (or 59.0 in the case of a negative leap second). 6) The warning status "dubious year" flags UTCs that predate the introduction of the time scale or that are too far in the future to be trusted. See eraDat for further details. 7) Only in the case of continuous and regular time scales (TAI, TT, TCG, TCB and TDB) is the result d1+d2 a Julian Date, strictly speaking. In the other cases (UT1 and UTC) the result must be used with circumspection; in particular the difference between two such results cannot be interpreted as a precise time interval. Called: eraCal2jd Gregorian calendar to JD eraDat delta(AT) = TAI-UTC eraJd2cal JD to Gregorian calendar This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ d1, d2, c_retval = ufunc.dtf2d(scale, iy, im, id, ihr, imn, sec) check_errwarn(c_retval, 'dtf2d') return d1, d2 STATUS_CODES['dtf2d'] = { 3: 'both of next two', 2: 'time is after end of day (Note 5)', 1: 'dubious year (Note 6)', 0: 'OK', -1: 'bad year', -2: 'bad month', -3: 'bad day', -4: 'bad hour', -5: 'bad minute', -6: 'bad second (<0)', } def taitt(tai1, tai2): """ Time scale transformation: International Atomic Time, TAI, to Terrestrial Time, TT. Parameters ---------- tai1 : double array tai2 : double array Returns ------- tt1 : double array tt2 : double array Notes ----- Wraps ERFA function ``eraTaitt``. The ERFA documentation is:: - - - - - - - - - e r a T a i t t - - - - - - - - - Time scale transformation: International Atomic Time, TAI, to Terrestrial Time, TT. Given: tai1,tai2 double TAI as a 2-part Julian Date Returned: tt1,tt2 double TT as a 2-part Julian Date Returned (function value): int status: 0 = OK Note: tai1+tai2 is Julian Date, apportioned in any convenient way between the two arguments, for example where tai1 is the Julian Day Number and tai2 is the fraction of a day. The returned tt1,tt2 follow suit. References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ tt1, tt2, c_retval = ufunc.taitt(tai1, tai2) check_errwarn(c_retval, 'taitt') return tt1, tt2 STATUS_CODES['taitt'] = { 0: 'OK', } def taiut1(tai1, tai2, dta): """ Time scale transformation: International Atomic Time, TAI, to Universal Time, UT1. Parameters ---------- tai1 : double array tai2 : double array dta : double array Returns ------- ut11 : double array ut12 : double array Notes ----- Wraps ERFA function ``eraTaiut1``. The ERFA documentation is:: - - - - - - - - - - e r a T a i u t 1 - - - - - - - - - - Time scale transformation: International Atomic Time, TAI, to Universal Time, UT1. Given: tai1,tai2 double TAI as a 2-part Julian Date dta double UT1-TAI in seconds Returned: ut11,ut12 double UT1 as a 2-part Julian Date Returned (function value): int status: 0 = OK Notes: 1) tai1+tai2 is Julian Date, apportioned in any convenient way between the two arguments, for example where tai1 is the Julian Day Number and tai2 is the fraction of a day. The returned UT11,UT12 follow suit. 2) The argument dta, i.e. UT1-TAI, is an observed quantity, and is available from IERS tabulations. Reference: Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ ut11, ut12, c_retval = ufunc.taiut1(tai1, tai2, dta) check_errwarn(c_retval, 'taiut1') return ut11, ut12 STATUS_CODES['taiut1'] = { 0: 'OK', } def taiutc(tai1, tai2): """ Time scale transformation: International Atomic Time, TAI, to Coordinated Universal Time, UTC. Parameters ---------- tai1 : double array tai2 : double array Returns ------- utc1 : double array utc2 : double array Notes ----- Wraps ERFA function ``eraTaiutc``. The ERFA documentation is:: - - - - - - - - - - e r a T a i u t c - - - - - - - - - - Time scale transformation: International Atomic Time, TAI, to Coordinated Universal Time, UTC. Given: tai1,tai2 double TAI as a 2-part Julian Date (Note 1) Returned: utc1,utc2 double UTC as a 2-part quasi Julian Date (Notes 1-3) Returned (function value): int status: +1 = dubious year (Note 4) 0 = OK -1 = unacceptable date Notes: 1) tai1+tai2 is Julian Date, apportioned in any convenient way between the two arguments, for example where tai1 is the Julian Day Number and tai2 is the fraction of a day. The returned utc1 and utc2 form an analogous pair, except that a special convention is used, to deal with the problem of leap seconds - see the next note. 2) JD cannot unambiguously represent UTC during a leap second unless special measures are taken. The convention in the present function is that the JD day represents UTC days whether the length is 86399, 86400 or 86401 SI seconds. In the 1960-1972 era there were smaller jumps (in either direction) each time the linear UTC(TAI) expression was changed, and these "mini-leaps" are also included in the ERFA convention. 3) The function eraD2dtf can be used to transform the UTC quasi-JD into calendar date and clock time, including UTC leap second handling. 4) The warning status "dubious year" flags UTCs that predate the introduction of the time scale or that are too far in the future to be trusted. See eraDat for further details. Called: eraUtctai UTC to TAI References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ utc1, utc2, c_retval = ufunc.taiutc(tai1, tai2) check_errwarn(c_retval, 'taiutc') return utc1, utc2 STATUS_CODES['taiutc'] = { 1: 'dubious year (Note 4)', 0: 'OK', -1: 'unacceptable date', } def tcbtdb(tcb1, tcb2): """ Time scale transformation: Barycentric Coordinate Time, TCB, to Barycentric Dynamical Time, TDB. Parameters ---------- tcb1 : double array tcb2 : double array Returns ------- tdb1 : double array tdb2 : double array Notes ----- Wraps ERFA function ``eraTcbtdb``. The ERFA documentation is:: - - - - - - - - - - e r a T c b t d b - - - - - - - - - - Time scale transformation: Barycentric Coordinate Time, TCB, to Barycentric Dynamical Time, TDB. Given: tcb1,tcb2 double TCB as a 2-part Julian Date Returned: tdb1,tdb2 double TDB as a 2-part Julian Date Returned (function value): int status: 0 = OK Notes: 1) tcb1+tcb2 is Julian Date, apportioned in any convenient way between the two arguments, for example where tcb1 is the Julian Day Number and tcb2 is the fraction of a day. The returned tdb1,tdb2 follow suit. 2) The 2006 IAU General Assembly introduced a conventional linear transformation between TDB and TCB. This transformation compensates for the drift between TCB and terrestrial time TT, and keeps TDB approximately centered on TT. Because the relationship between TT and TCB depends on the adopted solar system ephemeris, the degree of alignment between TDB and TT over long intervals will vary according to which ephemeris is used. Former definitions of TDB attempted to avoid this problem by stipulating that TDB and TT should differ only by periodic effects. This is a good description of the nature of the relationship but eluded precise mathematical formulation. The conventional linear relationship adopted in 2006 sidestepped these difficulties whilst delivering a TDB that in practice was consistent with values before that date. 3) TDB is essentially the same as Teph, the time argument for the JPL solar system ephemerides. Reference: IAU 2006 Resolution B3 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ tdb1, tdb2, c_retval = ufunc.tcbtdb(tcb1, tcb2) check_errwarn(c_retval, 'tcbtdb') return tdb1, tdb2 STATUS_CODES['tcbtdb'] = { 0: 'OK', } def tcgtt(tcg1, tcg2): """ Time scale transformation: Geocentric Coordinate Time, TCG, to Terrestrial Time, TT. Parameters ---------- tcg1 : double array tcg2 : double array Returns ------- tt1 : double array tt2 : double array Notes ----- Wraps ERFA function ``eraTcgtt``. The ERFA documentation is:: - - - - - - - - - e r a T c g t t - - - - - - - - - Time scale transformation: Geocentric Coordinate Time, TCG, to Terrestrial Time, TT. Given: tcg1,tcg2 double TCG as a 2-part Julian Date Returned: tt1,tt2 double TT as a 2-part Julian Date Returned (function value): int status: 0 = OK Note: tcg1+tcg2 is Julian Date, apportioned in any convenient way between the two arguments, for example where tcg1 is the Julian Day Number and tcg22 is the fraction of a day. The returned tt1,tt2 follow suit. References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) IAU 2000 Resolution B1.9 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ tt1, tt2, c_retval = ufunc.tcgtt(tcg1, tcg2) check_errwarn(c_retval, 'tcgtt') return tt1, tt2 STATUS_CODES['tcgtt'] = { 0: 'OK', } def tdbtcb(tdb1, tdb2): """ Time scale transformation: Barycentric Dynamical Time, TDB, to Barycentric Coordinate Time, TCB. Parameters ---------- tdb1 : double array tdb2 : double array Returns ------- tcb1 : double array tcb2 : double array Notes ----- Wraps ERFA function ``eraTdbtcb``. The ERFA documentation is:: - - - - - - - - - - e r a T d b t c b - - - - - - - - - - Time scale transformation: Barycentric Dynamical Time, TDB, to Barycentric Coordinate Time, TCB. Given: tdb1,tdb2 double TDB as a 2-part Julian Date Returned: tcb1,tcb2 double TCB as a 2-part Julian Date Returned (function value): int status: 0 = OK Notes: 1) tdb1+tdb2 is Julian Date, apportioned in any convenient way between the two arguments, for example where tdb1 is the Julian Day Number and tdb2 is the fraction of a day. The returned tcb1,tcb2 follow suit. 2) The 2006 IAU General Assembly introduced a conventional linear transformation between TDB and TCB. This transformation compensates for the drift between TCB and terrestrial time TT, and keeps TDB approximately centered on TT. Because the relationship between TT and TCB depends on the adopted solar system ephemeris, the degree of alignment between TDB and TT over long intervals will vary according to which ephemeris is used. Former definitions of TDB attempted to avoid this problem by stipulating that TDB and TT should differ only by periodic effects. This is a good description of the nature of the relationship but eluded precise mathematical formulation. The conventional linear relationship adopted in 2006 sidestepped these difficulties whilst delivering a TDB that in practice was consistent with values before that date. 3) TDB is essentially the same as Teph, the time argument for the JPL solar system ephemerides. Reference: IAU 2006 Resolution B3 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ tcb1, tcb2, c_retval = ufunc.tdbtcb(tdb1, tdb2) check_errwarn(c_retval, 'tdbtcb') return tcb1, tcb2 STATUS_CODES['tdbtcb'] = { 0: 'OK', } def tdbtt(tdb1, tdb2, dtr): """ Time scale transformation: Barycentric Dynamical Time, TDB, to Terrestrial Time, TT. Parameters ---------- tdb1 : double array tdb2 : double array dtr : double array Returns ------- tt1 : double array tt2 : double array Notes ----- Wraps ERFA function ``eraTdbtt``. The ERFA documentation is:: - - - - - - - - - e r a T d b t t - - - - - - - - - Time scale transformation: Barycentric Dynamical Time, TDB, to Terrestrial Time, TT. Given: tdb1,tdb2 double TDB as a 2-part Julian Date dtr double TDB-TT in seconds Returned: tt1,tt2 double TT as a 2-part Julian Date Returned (function value): int status: 0 = OK Notes: 1) tdb1+tdb2 is Julian Date, apportioned in any convenient way between the two arguments, for example where tdb1 is the Julian Day Number and tdb2 is the fraction of a day. The returned tt1,tt2 follow suit. 2) The argument dtr represents the quasi-periodic component of the GR transformation between TT and TCB. It is dependent upon the adopted solar-system ephemeris, and can be obtained by numerical integration, by interrogating a precomputed time ephemeris or by evaluating a model such as that implemented in the ERFA function eraDtdb. The quantity is dominated by an annual term of 1.7 ms amplitude. 3) TDB is essentially the same as Teph, the time argument for the JPL solar system ephemerides. References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) IAU 2006 Resolution 3 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ tt1, tt2, c_retval = ufunc.tdbtt(tdb1, tdb2, dtr) check_errwarn(c_retval, 'tdbtt') return tt1, tt2 STATUS_CODES['tdbtt'] = { 0: 'OK', } def tttai(tt1, tt2): """ Time scale transformation: Terrestrial Time, TT, to International Atomic Time, TAI. Parameters ---------- tt1 : double array tt2 : double array Returns ------- tai1 : double array tai2 : double array Notes ----- Wraps ERFA function ``eraTttai``. The ERFA documentation is:: - - - - - - - - - e r a T t t a i - - - - - - - - - Time scale transformation: Terrestrial Time, TT, to International Atomic Time, TAI. Given: tt1,tt2 double TT as a 2-part Julian Date Returned: tai1,tai2 double TAI as a 2-part Julian Date Returned (function value): int status: 0 = OK Note: tt1+tt2 is Julian Date, apportioned in any convenient way between the two arguments, for example where tt1 is the Julian Day Number and tt2 is the fraction of a day. The returned tai1,tai2 follow suit. References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ tai1, tai2, c_retval = ufunc.tttai(tt1, tt2) check_errwarn(c_retval, 'tttai') return tai1, tai2 STATUS_CODES['tttai'] = { 0: 'OK', } def tttcg(tt1, tt2): """ Time scale transformation: Terrestrial Time, TT, to Geocentric Coordinate Time, TCG. Parameters ---------- tt1 : double array tt2 : double array Returns ------- tcg1 : double array tcg2 : double array Notes ----- Wraps ERFA function ``eraTttcg``. The ERFA documentation is:: - - - - - - - - - e r a T t t c g - - - - - - - - - Time scale transformation: Terrestrial Time, TT, to Geocentric Coordinate Time, TCG. Given: tt1,tt2 double TT as a 2-part Julian Date Returned: tcg1,tcg2 double TCG as a 2-part Julian Date Returned (function value): int status: 0 = OK Note: tt1+tt2 is Julian Date, apportioned in any convenient way between the two arguments, for example where tt1 is the Julian Day Number and tt2 is the fraction of a day. The returned tcg1,tcg2 follow suit. References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) IAU 2000 Resolution B1.9 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ tcg1, tcg2, c_retval = ufunc.tttcg(tt1, tt2) check_errwarn(c_retval, 'tttcg') return tcg1, tcg2 STATUS_CODES['tttcg'] = { 0: 'OK', } def tttdb(tt1, tt2, dtr): """ Time scale transformation: Terrestrial Time, TT, to Barycentric Dynamical Time, TDB. Parameters ---------- tt1 : double array tt2 : double array dtr : double array Returns ------- tdb1 : double array tdb2 : double array Notes ----- Wraps ERFA function ``eraTttdb``. The ERFA documentation is:: - - - - - - - - - e r a T t t d b - - - - - - - - - Time scale transformation: Terrestrial Time, TT, to Barycentric Dynamical Time, TDB. Given: tt1,tt2 double TT as a 2-part Julian Date dtr double TDB-TT in seconds Returned: tdb1,tdb2 double TDB as a 2-part Julian Date Returned (function value): int status: 0 = OK Notes: 1) tt1+tt2 is Julian Date, apportioned in any convenient way between the two arguments, for example where tt1 is the Julian Day Number and tt2 is the fraction of a day. The returned tdb1,tdb2 follow suit. 2) The argument dtr represents the quasi-periodic component of the GR transformation between TT and TCB. It is dependent upon the adopted solar-system ephemeris, and can be obtained by numerical integration, by interrogating a precomputed time ephemeris or by evaluating a model such as that implemented in the ERFA function eraDtdb. The quantity is dominated by an annual term of 1.7 ms amplitude. 3) TDB is essentially the same as Teph, the time argument for the JPL solar system ephemerides. References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) IAU 2006 Resolution 3 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ tdb1, tdb2, c_retval = ufunc.tttdb(tt1, tt2, dtr) check_errwarn(c_retval, 'tttdb') return tdb1, tdb2 STATUS_CODES['tttdb'] = { 0: 'OK', } def ttut1(tt1, tt2, dt): """ Time scale transformation: Terrestrial Time, TT, to Universal Time, UT1. Parameters ---------- tt1 : double array tt2 : double array dt : double array Returns ------- ut11 : double array ut12 : double array Notes ----- Wraps ERFA function ``eraTtut1``. The ERFA documentation is:: - - - - - - - - - e r a T t u t 1 - - - - - - - - - Time scale transformation: Terrestrial Time, TT, to Universal Time, UT1. Given: tt1,tt2 double TT as a 2-part Julian Date dt double TT-UT1 in seconds Returned: ut11,ut12 double UT1 as a 2-part Julian Date Returned (function value): int status: 0 = OK Notes: 1) tt1+tt2 is Julian Date, apportioned in any convenient way between the two arguments, for example where tt1 is the Julian Day Number and tt2 is the fraction of a day. The returned ut11,ut12 follow suit. 2) The argument dt is classical Delta T. Reference: Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ ut11, ut12, c_retval = ufunc.ttut1(tt1, tt2, dt) check_errwarn(c_retval, 'ttut1') return ut11, ut12 STATUS_CODES['ttut1'] = { 0: 'OK', } def ut1tai(ut11, ut12, dta): """ Time scale transformation: Universal Time, UT1, to International Atomic Time, TAI. Parameters ---------- ut11 : double array ut12 : double array dta : double array Returns ------- tai1 : double array tai2 : double array Notes ----- Wraps ERFA function ``eraUt1tai``. The ERFA documentation is:: - - - - - - - - - - e r a U t 1 t a i - - - - - - - - - - Time scale transformation: Universal Time, UT1, to International Atomic Time, TAI. Given: ut11,ut12 double UT1 as a 2-part Julian Date dta double UT1-TAI in seconds Returned: tai1,tai2 double TAI as a 2-part Julian Date Returned (function value): int status: 0 = OK Notes: 1) ut11+ut12 is Julian Date, apportioned in any convenient way between the two arguments, for example where ut11 is the Julian Day Number and ut12 is the fraction of a day. The returned tai1,tai2 follow suit. 2) The argument dta, i.e. UT1-TAI, is an observed quantity, and is available from IERS tabulations. Reference: Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ tai1, tai2, c_retval = ufunc.ut1tai(ut11, ut12, dta) check_errwarn(c_retval, 'ut1tai') return tai1, tai2 STATUS_CODES['ut1tai'] = { 0: 'OK', } def ut1tt(ut11, ut12, dt): """ Time scale transformation: Universal Time, UT1, to Terrestrial Time, TT. Parameters ---------- ut11 : double array ut12 : double array dt : double array Returns ------- tt1 : double array tt2 : double array Notes ----- Wraps ERFA function ``eraUt1tt``. The ERFA documentation is:: - - - - - - - - - e r a U t 1 t t - - - - - - - - - Time scale transformation: Universal Time, UT1, to Terrestrial Time, TT. Given: ut11,ut12 double UT1 as a 2-part Julian Date dt double TT-UT1 in seconds Returned: tt1,tt2 double TT as a 2-part Julian Date Returned (function value): int status: 0 = OK Notes: 1) ut11+ut12 is Julian Date, apportioned in any convenient way between the two arguments, for example where ut11 is the Julian Day Number and ut12 is the fraction of a day. The returned tt1,tt2 follow suit. 2) The argument dt is classical Delta T. Reference: Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ tt1, tt2, c_retval = ufunc.ut1tt(ut11, ut12, dt) check_errwarn(c_retval, 'ut1tt') return tt1, tt2 STATUS_CODES['ut1tt'] = { 0: 'OK', } def ut1utc(ut11, ut12, dut1): """ Time scale transformation: Universal Time, UT1, to Coordinated Universal Time, UTC. Parameters ---------- ut11 : double array ut12 : double array dut1 : double array Returns ------- utc1 : double array utc2 : double array Notes ----- Wraps ERFA function ``eraUt1utc``. The ERFA documentation is:: - - - - - - - - - - e r a U t 1 u t c - - - - - - - - - - Time scale transformation: Universal Time, UT1, to Coordinated Universal Time, UTC. Given: ut11,ut12 double UT1 as a 2-part Julian Date (Note 1) dut1 double Delta UT1: UT1-UTC in seconds (Note 2) Returned: utc1,utc2 double UTC as a 2-part quasi Julian Date (Notes 3,4) Returned (function value): int status: +1 = dubious year (Note 5) 0 = OK -1 = unacceptable date Notes: 1) ut11+ut12 is Julian Date, apportioned in any convenient way between the two arguments, for example where ut11 is the Julian Day Number and ut12 is the fraction of a day. The returned utc1 and utc2 form an analogous pair, except that a special convention is used, to deal with the problem of leap seconds - see Note 3. 2) Delta UT1 can be obtained from tabulations provided by the International Earth Rotation and Reference Systems Service. The value changes abruptly by 1s at a leap second; however, close to a leap second the algorithm used here is tolerant of the "wrong" choice of value being made. 3) JD cannot unambiguously represent UTC during a leap second unless special measures are taken. The convention in the present function is that the returned quasi-JD UTC1+UTC2 represents UTC days whether the length is 86399, 86400 or 86401 SI seconds. 4) The function eraD2dtf can be used to transform the UTC quasi-JD into calendar date and clock time, including UTC leap second handling. 5) The warning status "dubious year" flags UTCs that predate the introduction of the time scale or that are too far in the future to be trusted. See eraDat for further details. Called: eraJd2cal JD to Gregorian calendar eraDat delta(AT) = TAI-UTC eraCal2jd Gregorian calendar to JD References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ utc1, utc2, c_retval = ufunc.ut1utc(ut11, ut12, dut1) check_errwarn(c_retval, 'ut1utc') return utc1, utc2 STATUS_CODES['ut1utc'] = { 1: 'dubious year (Note 5)', 0: 'OK', -1: 'unacceptable date', } def utctai(utc1, utc2): """ Time scale transformation: Coordinated Universal Time, UTC, to International Atomic Time, TAI. Parameters ---------- utc1 : double array utc2 : double array Returns ------- tai1 : double array tai2 : double array Notes ----- Wraps ERFA function ``eraUtctai``. The ERFA documentation is:: - - - - - - - - - - e r a U t c t a i - - - - - - - - - - Time scale transformation: Coordinated Universal Time, UTC, to International Atomic Time, TAI. Given: utc1,utc2 double UTC as a 2-part quasi Julian Date (Notes 1-4) Returned: tai1,tai2 double TAI as a 2-part Julian Date (Note 5) Returned (function value): int status: +1 = dubious year (Note 3) 0 = OK -1 = unacceptable date Notes: 1) utc1+utc2 is quasi Julian Date (see Note 2), apportioned in any convenient way between the two arguments, for example where utc1 is the Julian Day Number and utc2 is the fraction of a day. 2) JD cannot unambiguously represent UTC during a leap second unless special measures are taken. The convention in the present function is that the JD day represents UTC days whether the length is 86399, 86400 or 86401 SI seconds. In the 1960-1972 era there were smaller jumps (in either direction) each time the linear UTC(TAI) expression was changed, and these "mini-leaps" are also included in the ERFA convention. 3) The warning status "dubious year" flags UTCs that predate the introduction of the time scale or that are too far in the future to be trusted. See eraDat for further details. 4) The function eraDtf2d converts from calendar date and time of day into 2-part Julian Date, and in the case of UTC implements the leap-second-ambiguity convention described above. 5) The returned TAI1,TAI2 are such that their sum is the TAI Julian Date. Called: eraJd2cal JD to Gregorian calendar eraDat delta(AT) = TAI-UTC eraCal2jd Gregorian calendar to JD References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ tai1, tai2, c_retval = ufunc.utctai(utc1, utc2) check_errwarn(c_retval, 'utctai') return tai1, tai2 STATUS_CODES['utctai'] = { 1: 'dubious year (Note 3)', 0: 'OK', -1: 'unacceptable date', } def utcut1(utc1, utc2, dut1): """ Time scale transformation: Coordinated Universal Time, UTC, to Universal Time, UT1. Parameters ---------- utc1 : double array utc2 : double array dut1 : double array Returns ------- ut11 : double array ut12 : double array Notes ----- Wraps ERFA function ``eraUtcut1``. The ERFA documentation is:: - - - - - - - - - - e r a U t c u t 1 - - - - - - - - - - Time scale transformation: Coordinated Universal Time, UTC, to Universal Time, UT1. Given: utc1,utc2 double UTC as a 2-part quasi Julian Date (Notes 1-4) dut1 double Delta UT1 = UT1-UTC in seconds (Note 5) Returned: ut11,ut12 double UT1 as a 2-part Julian Date (Note 6) Returned (function value): int status: +1 = dubious year (Note 3) 0 = OK -1 = unacceptable date Notes: 1) utc1+utc2 is quasi Julian Date (see Note 2), apportioned in any convenient way between the two arguments, for example where utc1 is the Julian Day Number and utc2 is the fraction of a day. 2) JD cannot unambiguously represent UTC during a leap second unless special measures are taken. The convention in the present function is that the JD day represents UTC days whether the length is 86399, 86400 or 86401 SI seconds. 3) The warning status "dubious year" flags UTCs that predate the introduction of the time scale or that are too far in the future to be trusted. See eraDat for further details. 4) The function eraDtf2d converts from calendar date and time of day into 2-part Julian Date, and in the case of UTC implements the leap-second-ambiguity convention described above. 5) Delta UT1 can be obtained from tabulations provided by the International Earth Rotation and Reference Systems Service. It is the caller's responsibility to supply a dut1 argument containing the UT1-UTC value that matches the given UTC. 6) The returned ut11,ut12 are such that their sum is the UT1 Julian Date. References: McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003), IERS Technical Note No. 32, BKG (2004) Explanatory Supplement to the Astronomical Almanac, P. Kenneth Seidelmann (ed), University Science Books (1992) Called: eraJd2cal JD to Gregorian calendar eraDat delta(AT) = TAI-UTC eraUtctai UTC to TAI eraTaiut1 TAI to UT1 This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ ut11, ut12, c_retval = ufunc.utcut1(utc1, utc2, dut1) check_errwarn(c_retval, 'utcut1') return ut11, ut12 STATUS_CODES['utcut1'] = { 1: 'dubious year (Note 3)', 0: 'OK', -1: 'unacceptable date', } def ae2hd(az, el, phi): """ Horizon to equatorial coordinates: transform azimuth and altitude to hour angle and declination. Parameters ---------- az : double array el : double array phi : double array Returns ------- ha : double array dec : double array Notes ----- Wraps ERFA function ``eraAe2hd``. The ERFA documentation is:: - - - - - - - - - e r a A e 2 h d - - - - - - - - - Horizon to equatorial coordinates: transform azimuth and altitude to hour angle and declination. Given: az double azimuth el double altitude (informally, elevation) phi double site latitude Returned: ha double hour angle (local) dec double declination Notes: 1) All the arguments are angles in radians. 2) The sign convention for azimuth is north zero, east +pi/2. 3) HA is returned in the range +/-pi. Declination is returned in the range +/-pi/2. 4) The latitude phi is pi/2 minus the angle between the Earth's rotation axis and the adopted zenith. In many applications it will be sufficient to use the published geodetic latitude of the site. In very precise (sub-arcsecond) applications, phi can be corrected for polar motion. 5) The azimuth az must be with respect to the rotational north pole, as opposed to the ITRS pole, and an azimuth with respect to north on a map of the Earth's surface will need to be adjusted for polar motion if sub-arcsecond accuracy is required. 6) Should the user wish to work with respect to the astronomical zenith rather than the geodetic zenith, phi will need to be adjusted for deflection of the vertical (often tens of arcseconds), and the zero point of ha will also be affected. 7) The transformation is the same as Ve = Ry(phi-pi/2)*Rz(pi)*Vh, where Ve and Vh are lefthanded unit vectors in the (ha,dec) and (az,el) systems respectively and Rz and Ry are rotations about first the z-axis and then the y-axis. (n.b. Rz(pi) simply reverses the signs of the x and y components.) For efficiency, the algorithm is written out rather than calling other utility functions. For applications that require even greater efficiency, additional savings are possible if constant terms such as functions of latitude are computed once and for all. 8) Again for efficiency, no range checking of arguments is carried out. Last revision: 2017 September 12 ERFA release 2021-05-12 Copyright (C) 2021 IAU ERFA Board. See notes at end. """ ha, dec = ufunc.ae2hd(az, el, phi) return ha, dec def hd2ae(ha, dec, phi): """ Equatorial to horizon coordinates: transform hour angle and declination to azimuth and altitude. Parameters ---------- ha : double array dec : double array phi : double array Returns ------- az : double array el : double array Notes ----- Wraps ERFA function ``eraHd2ae``. The ERFA documentation is:: - - - - - - - - - e r a H d 2 a e - - - - - - - - - Equatorial to horizon coordinates: transform hour angle and declination to azimuth and altitude. Given: ha double hour angle (local) dec double declination phi double site latitude Returned: *az double azimuth *el double altitude (informally, elevation) Notes: 1) All the arguments are angles in radians. 2) Azimuth is returned in the range 0-2pi; north is zero, and east is +pi/2. Altitude is returned in the range +/- pi/2. 3) The latitude phi is pi/2 minus the angle between the Earth's rotation axis and the adopted zenith. In many applications it will be sufficient to use the published geodetic latitude of the site. In very precise (sub-arcsecond) applications, phi can be corrected for polar motion. 4) The returned azimuth az is with respect to the rotational north pole, as opposed to the ITRS pole, and for sub-arcsecond accuracy will need to be adjusted for polar motion if it is to be with respect to north on a map of the Earth's surface. 5) Should the user wish to work with respect to the astronomical zenith rather than the geodetic zenith, phi will need to be adjusted for deflection of the vertical (often tens of arcseconds), and the zero point of the hour angle ha will also be affected. 6) The transformation is the same as Vh = Rz(pi)*Ry(pi/2-phi)*Ve, where Vh and Ve are lefthanded unit vectors in the (az,el) and (ha,dec) systems respectively and Ry and Rz are rotations about first the y-axis and then the z-axis. (n.b. Rz(pi) simply reverses the signs of the x and y components.) For efficiency, the algorithm is written out rather than calling other utility functions. For applications that require even greater efficiency, additional savings are possible if constant terms such as functions of latitude are computed once and for all. 7) Again for efficiency, no range checking of arguments is carried out. Last revision: 2021 February 24 ERFA release 2021-05-12 Copyright (C) 2021 IAU ERFA Board. See notes at end. """ az, el = ufunc.hd2ae(ha, dec, phi) return az, el def hd2pa(ha, dec, phi): """ Parallactic angle for a given hour angle and declination. Parameters ---------- ha : double array dec : double array phi : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraHd2pa``. The ERFA documentation is:: - - - - - - - - - e r a H d 2 p a - - - - - - - - - Parallactic angle for a given hour angle and declination. Given: ha double hour angle dec double declination phi double site latitude Returned (function value): double parallactic angle Notes: 1) All the arguments are angles in radians. 2) The parallactic angle at a point in the sky is the position angle of the vertical, i.e. the angle between the directions to the north celestial pole and to the zenith respectively. 3) The result is returned in the range -pi to +pi. 4) At the pole itself a zero result is returned. 5) The latitude phi is pi/2 minus the angle between the Earth's rotation axis and the adopted zenith. In many applications it will be sufficient to use the published geodetic latitude of the site. In very precise (sub-arcsecond) applications, phi can be corrected for polar motion. 6) Should the user wish to work with respect to the astronomical zenith rather than the geodetic zenith, phi will need to be adjusted for deflection of the vertical (often tens of arcseconds), and the zero point of the hour angle ha will also be affected. Reference: Smart, W.M., "Spherical Astronomy", Cambridge University Press, 6th edition (Green, 1977), p49. Last revision: 2017 September 12 ERFA release 2021-05-12 Copyright (C) 2021 IAU ERFA Board. See notes at end. """ c_retval = ufunc.hd2pa(ha, dec, phi) return c_retval def tpors(xi, eta, a, b): """ In the tangent plane projection, given the rectangular coordinates of a star and its spherical coordinates, determine the spherical coordinates of the tangent point. Parameters ---------- xi : double array eta : double array a : double array b : double array Returns ------- a01 : double array b01 : double array a02 : double array b02 : double array Notes ----- Wraps ERFA function ``eraTpors``. The ERFA documentation is:: - - - - - - - - - e r a T p o r s - - - - - - - - - In the tangent plane projection, given the rectangular coordinates of a star and its spherical coordinates, determine the spherical coordinates of the tangent point. Given: xi,eta double rectangular coordinates of star image (Note 2) a,b double star's spherical coordinates (Note 3) Returned: *a01,*b01 double tangent point's spherical coordinates, Soln. 1 *a02,*b02 double tangent point's spherical coordinates, Soln. 2 Returned (function value): int number of solutions: 0 = no solutions returned (Note 5) 1 = only the first solution is useful (Note 6) 2 = both solutions are useful (Note 6) Notes: 1) The tangent plane projection is also called the "gnomonic projection" and the "central projection". 2) The eta axis points due north in the adopted coordinate system. If the spherical coordinates are observed (RA,Dec), the tangent plane coordinates (xi,eta) are conventionally called the "standard coordinates". If the spherical coordinates are with respect to a right-handed triad, (xi,eta) are also right-handed. The units of (xi,eta) are, effectively, radians at the tangent point. 3) All angular arguments are in radians. 4) The angles a01 and a02 are returned in the range 0-2pi. The angles b01 and b02 are returned in the range +/-pi, but in the usual, non-pole-crossing, case, the range is +/-pi/2. 5) Cases where there is no solution can arise only near the poles. For example, it is clearly impossible for a star at the pole itself to have a non-zero xi value, and hence it is meaningless to ask where the tangent point would have to be to bring about this combination of xi and dec. 6) Also near the poles, cases can arise where there are two useful solutions. The return value indicates whether the second of the two solutions returned is useful; 1 indicates only one useful solution, the usual case. 7) The basis of the algorithm is to solve the spherical triangle PSC, where P is the north celestial pole, S is the star and C is the tangent point. The spherical coordinates of the tangent point are [a0,b0]; writing rho^2 = (xi^2+eta^2) and r^2 = (1+rho^2), side c is then (pi/2-b), side p is sqrt(xi^2+eta^2) and side s (to be found) is (pi/2-b0). Angle C is given by sin(C) = xi/rho and cos(C) = eta/rho. Angle P (to be found) is the longitude difference between star and tangent point (a-a0). 8) This function is a member of the following set: spherical vector solve for eraTpxes eraTpxev xi,eta eraTpsts eraTpstv star > eraTpors < eraTporv origin Called: eraAnp normalize angle into range 0 to 2pi References: Calabretta M.R. & Greisen, E.W., 2002, "Representations of celestial coordinates in FITS", Astron.Astrophys. 395, 1077 Green, R.M., "Spherical Astronomy", Cambridge University Press, 1987, Chapter 13. This revision: 2018 January 2 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ a01, b01, a02, b02, c_retval = ufunc.tpors(xi, eta, a, b) check_errwarn(c_retval, 'tpors') return a01, b01, a02, b02 def tporv(xi, eta, v): """ In the tangent plane projection, given the rectangular coordinates of a star and its direction cosines, determine the direction cosines of the tangent point. Parameters ---------- xi : double array eta : double array v : double array Returns ------- v01 : double array v02 : double array Notes ----- Wraps ERFA function ``eraTporv``. The ERFA documentation is:: - - - - - - - - - e r a T p o r v - - - - - - - - - In the tangent plane projection, given the rectangular coordinates of a star and its direction cosines, determine the direction cosines of the tangent point. Given: xi,eta double rectangular coordinates of star image (Note 2) v double[3] star's direction cosines (Note 3) Returned: v01 double[3] tangent point's direction cosines, Solution 1 v02 double[3] tangent point's direction cosines, Solution 2 Returned (function value): int number of solutions: 0 = no solutions returned (Note 4) 1 = only the first solution is useful (Note 5) 2 = both solutions are useful (Note 5) Notes: 1) The tangent plane projection is also called the "gnomonic projection" and the "central projection". 2) The eta axis points due north in the adopted coordinate system. If the direction cosines represent observed (RA,Dec), the tangent plane coordinates (xi,eta) are conventionally called the "standard coordinates". If the direction cosines are with respect to a right-handed triad, (xi,eta) are also right-handed. The units of (xi,eta) are, effectively, radians at the tangent point. 3) The vector v must be of unit length or the result will be wrong. 4) Cases where there is no solution can arise only near the poles. For example, it is clearly impossible for a star at the pole itself to have a non-zero xi value, and hence it is meaningless to ask where the tangent point would have to be. 5) Also near the poles, cases can arise where there are two useful solutions. The return value indicates whether the second of the two solutions returned is useful; 1 indicates only one useful solution, the usual case. 6) The basis of the algorithm is to solve the spherical triangle PSC, where P is the north celestial pole, S is the star and C is the tangent point. Calling the celestial spherical coordinates of the star and tangent point (a,b) and (a0,b0) respectively, and writing rho^2 = (xi^2+eta^2) and r^2 = (1+rho^2), and transforming the vector v into (a,b) in the normal way, side c is then (pi/2-b), side p is sqrt(xi^2+eta^2) and side s (to be found) is (pi/2-b0), while angle C is given by sin(C) = xi/rho and cos(C) = eta/rho; angle P (to be found) is (a-a0). After solving the spherical triangle, the result (a0,b0) can be expressed in vector form as v0. 7) This function is a member of the following set: spherical vector solve for eraTpxes eraTpxev xi,eta eraTpsts eraTpstv star eraTpors > eraTporv < origin References: Calabretta M.R. & Greisen, E.W., 2002, "Representations of celestial coordinates in FITS", Astron.Astrophys. 395, 1077 Green, R.M., "Spherical Astronomy", Cambridge University Press, 1987, Chapter 13. This revision: 2018 January 2 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ v01, v02, c_retval = ufunc.tporv(xi, eta, v) check_errwarn(c_retval, 'tporv') return v01, v02 def tpsts(xi, eta, a0, b0): """ In the tangent plane projection, given the star's rectangular coordinates and the spherical coordinates of the tangent point, solve for the spherical coordinates of the star. Parameters ---------- xi : double array eta : double array a0 : double array b0 : double array Returns ------- a : double array b : double array Notes ----- Wraps ERFA function ``eraTpsts``. The ERFA documentation is:: - - - - - - - - - e r a T p s t s - - - - - - - - - In the tangent plane projection, given the star's rectangular coordinates and the spherical coordinates of the tangent point, solve for the spherical coordinates of the star. Given: xi,eta double rectangular coordinates of star image (Note 2) a0,b0 double tangent point's spherical coordinates Returned: *a,*b double star's spherical coordinates 1) The tangent plane projection is also called the "gnomonic projection" and the "central projection". 2) The eta axis points due north in the adopted coordinate system. If the spherical coordinates are observed (RA,Dec), the tangent plane coordinates (xi,eta) are conventionally called the "standard coordinates". If the spherical coordinates are with respect to a right-handed triad, (xi,eta) are also right-handed. The units of (xi,eta) are, effectively, radians at the tangent point. 3) All angular arguments are in radians. 4) This function is a member of the following set: spherical vector solve for eraTpxes eraTpxev xi,eta > eraTpsts < eraTpstv star eraTpors eraTporv origin Called: eraAnp normalize angle into range 0 to 2pi References: Calabretta M.R. & Greisen, E.W., 2002, "Representations of celestial coordinates in FITS", Astron.Astrophys. 395, 1077 Green, R.M., "Spherical Astronomy", Cambridge University Press, 1987, Chapter 13. This revision: 2018 January 2 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ a, b = ufunc.tpsts(xi, eta, a0, b0) return a, b def tpstv(xi, eta, v0): """ In the tangent plane projection, given the star's rectangular coordinates and the direction cosines of the tangent point, solve for the direction cosines of the star. Parameters ---------- xi : double array eta : double array v0 : double array Returns ------- v : double array Notes ----- Wraps ERFA function ``eraTpstv``. The ERFA documentation is:: - - - - - - - - - e r a T p s t v - - - - - - - - - In the tangent plane projection, given the star's rectangular coordinates and the direction cosines of the tangent point, solve for the direction cosines of the star. Given: xi,eta double rectangular coordinates of star image (Note 2) v0 double[3] tangent point's direction cosines Returned: v double[3] star's direction cosines 1) The tangent plane projection is also called the "gnomonic projection" and the "central projection". 2) The eta axis points due north in the adopted coordinate system. If the direction cosines represent observed (RA,Dec), the tangent plane coordinates (xi,eta) are conventionally called the "standard coordinates". If the direction cosines are with respect to a right-handed triad, (xi,eta) are also right-handed. The units of (xi,eta) are, effectively, radians at the tangent point. 3) The method used is to complete the star vector in the (xi,eta) based triad and normalize it, then rotate the triad to put the tangent point at the pole with the x-axis aligned to zero longitude. Writing (a0,b0) for the celestial spherical coordinates of the tangent point, the sequence of rotations is (b-pi/2) around the x-axis followed by (-a-pi/2) around the z-axis. 4) If vector v0 is not of unit length, the returned vector v will be wrong. 5) If vector v0 points at a pole, the returned vector v will be based on the arbitrary assumption that the longitude coordinate of the tangent point is zero. 6) This function is a member of the following set: spherical vector solve for eraTpxes eraTpxev xi,eta eraTpsts > eraTpstv < star eraTpors eraTporv origin References: Calabretta M.R. & Greisen, E.W., 2002, "Representations of celestial coordinates in FITS", Astron.Astrophys. 395, 1077 Green, R.M., "Spherical Astronomy", Cambridge University Press, 1987, Chapter 13. This revision: 2018 January 2 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ v = ufunc.tpstv(xi, eta, v0) return v def tpxes(a, b, a0, b0): """ In the tangent plane projection, given celestial spherical coordinates for a star and the tangent point, solve for the star's rectangular coordinates in the tangent plane. Parameters ---------- a : double array b : double array a0 : double array b0 : double array Returns ------- xi : double array eta : double array Notes ----- Wraps ERFA function ``eraTpxes``. The ERFA documentation is:: - - - - - - - - - e r a T p x e s - - - - - - - - - In the tangent plane projection, given celestial spherical coordinates for a star and the tangent point, solve for the star's rectangular coordinates in the tangent plane. Given: a,b double star's spherical coordinates a0,b0 double tangent point's spherical coordinates Returned: *xi,*eta double rectangular coordinates of star image (Note 2) Returned (function value): int status: 0 = OK 1 = star too far from axis 2 = antistar on tangent plane 3 = antistar too far from axis Notes: 1) The tangent plane projection is also called the "gnomonic projection" and the "central projection". 2) The eta axis points due north in the adopted coordinate system. If the spherical coordinates are observed (RA,Dec), the tangent plane coordinates (xi,eta) are conventionally called the "standard coordinates". For right-handed spherical coordinates, (xi,eta) are also right-handed. The units of (xi,eta) are, effectively, radians at the tangent point. 3) All angular arguments are in radians. 4) This function is a member of the following set: spherical vector solve for > eraTpxes < eraTpxev xi,eta eraTpsts eraTpstv star eraTpors eraTporv origin References: Calabretta M.R. & Greisen, E.W., 2002, "Representations of celestial coordinates in FITS", Astron.Astrophys. 395, 1077 Green, R.M., "Spherical Astronomy", Cambridge University Press, 1987, Chapter 13. This revision: 2018 January 2 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ xi, eta, c_retval = ufunc.tpxes(a, b, a0, b0) check_errwarn(c_retval, 'tpxes') return xi, eta STATUS_CODES['tpxes'] = { 0: 'OK', 1: 'star too far from axis', 2: 'antistar on tangent plane', 3: 'antistar too far from axis', } def tpxev(v, v0): """ In the tangent plane projection, given celestial direction cosines for a star and the tangent point, solve for the star's rectangular coordinates in the tangent plane. Parameters ---------- v : double array v0 : double array Returns ------- xi : double array eta : double array Notes ----- Wraps ERFA function ``eraTpxev``. The ERFA documentation is:: - - - - - - - - - e r a T p x e v - - - - - - - - - In the tangent plane projection, given celestial direction cosines for a star and the tangent point, solve for the star's rectangular coordinates in the tangent plane. Given: v double[3] direction cosines of star (Note 4) v0 double[3] direction cosines of tangent point (Note 4) Returned: *xi,*eta double tangent plane coordinates of star Returned (function value): int status: 0 = OK 1 = star too far from axis 2 = antistar on tangent plane 3 = antistar too far from axis Notes: 1) The tangent plane projection is also called the "gnomonic projection" and the "central projection". 2) The eta axis points due north in the adopted coordinate system. If the direction cosines represent observed (RA,Dec), the tangent plane coordinates (xi,eta) are conventionally called the "standard coordinates". If the direction cosines are with respect to a right-handed triad, (xi,eta) are also right-handed. The units of (xi,eta) are, effectively, radians at the tangent point. 3) The method used is to extend the star vector to the tangent plane and then rotate the triad so that (x,y) becomes (xi,eta). Writing (a,b) for the celestial spherical coordinates of the star, the sequence of rotations is (a+pi/2) around the z-axis followed by (pi/2-b) around the x-axis. 4) If vector v0 is not of unit length, or if vector v is of zero length, the results will be wrong. 5) If v0 points at a pole, the returned (xi,eta) will be based on the arbitrary assumption that the longitude coordinate of the tangent point is zero. 6) This function is a member of the following set: spherical vector solve for eraTpxes > eraTpxev < xi,eta eraTpsts eraTpstv star eraTpors eraTporv origin References: Calabretta M.R. & Greisen, E.W., 2002, "Representations of celestial coordinates in FITS", Astron.Astrophys. 395, 1077 Green, R.M., "Spherical Astronomy", Cambridge University Press, 1987, Chapter 13. This revision: 2018 January 2 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ xi, eta, c_retval = ufunc.tpxev(v, v0) check_errwarn(c_retval, 'tpxev') return xi, eta STATUS_CODES['tpxev'] = { 0: 'OK', 1: 'star too far from axis', 2: 'antistar on tangent plane', 3: 'antistar too far from axis', } def a2af(ndp, angle): """ Decompose radians into degrees, arcminutes, arcseconds, fraction. Parameters ---------- ndp : int array angle : double array Returns ------- sign : char array idmsf : int array Notes ----- Wraps ERFA function ``eraA2af``. The ERFA documentation is:: - - - - - - - - e r a A 2 a f - - - - - - - - Decompose radians into degrees, arcminutes, arcseconds, fraction. Given: ndp int resolution (Note 1) angle double angle in radians Returned: sign char '+' or '-' idmsf int[4] degrees, arcminutes, arcseconds, fraction Notes: 1) The argument ndp is interpreted as follows: ndp resolution : ...0000 00 00 -7 1000 00 00 -6 100 00 00 -5 10 00 00 -4 1 00 00 -3 0 10 00 -2 0 01 00 -1 0 00 10 0 0 00 01 1 0 00 00.1 2 0 00 00.01 3 0 00 00.001 : 0 00 00.000... 2) The largest positive useful value for ndp is determined by the size of angle, the format of doubles on the target platform, and the risk of overflowing idmsf[3]. On a typical platform, for angle up to 2pi, the available floating-point precision might correspond to ndp=12. However, the practical limit is typically ndp=9, set by the capacity of a 32-bit int, or ndp=4 if int is only 16 bits. 3) The absolute value of angle may exceed 2pi. In cases where it does not, it is up to the caller to test for and handle the case where angle is very nearly 2pi and rounds up to 360 degrees, by testing for idmsf[0]=360 and setting idmsf[0-3] to zero. Called: eraD2tf decompose days to hms This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ sign, idmsf = ufunc.a2af(ndp, angle) sign = sign.view(dt_bytes1) return sign, idmsf def a2tf(ndp, angle): """ Decompose radians into hours, minutes, seconds, fraction. Parameters ---------- ndp : int array angle : double array Returns ------- sign : char array ihmsf : int array Notes ----- Wraps ERFA function ``eraA2tf``. The ERFA documentation is:: - - - - - - - - e r a A 2 t f - - - - - - - - Decompose radians into hours, minutes, seconds, fraction. Given: ndp int resolution (Note 1) angle double angle in radians Returned: sign char '+' or '-' ihmsf int[4] hours, minutes, seconds, fraction Notes: 1) The argument ndp is interpreted as follows: ndp resolution : ...0000 00 00 -7 1000 00 00 -6 100 00 00 -5 10 00 00 -4 1 00 00 -3 0 10 00 -2 0 01 00 -1 0 00 10 0 0 00 01 1 0 00 00.1 2 0 00 00.01 3 0 00 00.001 : 0 00 00.000... 2) The largest positive useful value for ndp is determined by the size of angle, the format of doubles on the target platform, and the risk of overflowing ihmsf[3]. On a typical platform, for angle up to 2pi, the available floating-point precision might correspond to ndp=12. However, the practical limit is typically ndp=9, set by the capacity of a 32-bit int, or ndp=4 if int is only 16 bits. 3) The absolute value of angle may exceed 2pi. In cases where it does not, it is up to the caller to test for and handle the case where angle is very nearly 2pi and rounds up to 24 hours, by testing for ihmsf[0]=24 and setting ihmsf[0-3] to zero. Called: eraD2tf decompose days to hms This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ sign, ihmsf = ufunc.a2tf(ndp, angle) sign = sign.view(dt_bytes1) return sign, ihmsf def af2a(s, ideg, iamin, asec): """ Convert degrees, arcminutes, arcseconds to radians. Parameters ---------- s : char array ideg : int array iamin : int array asec : double array Returns ------- rad : double array Notes ----- Wraps ERFA function ``eraAf2a``. The ERFA documentation is:: - - - - - - - - e r a A f 2 a - - - - - - - - Convert degrees, arcminutes, arcseconds to radians. Given: s char sign: '-' = negative, otherwise positive ideg int degrees iamin int arcminutes asec double arcseconds Returned: rad double angle in radians Returned (function value): int status: 0 = OK 1 = ideg outside range 0-359 2 = iamin outside range 0-59 3 = asec outside range 0-59.999... Notes: 1) The result is computed even if any of the range checks fail. 2) Negative ideg, iamin and/or asec produce a warning status, but the absolute value is used in the conversion. 3) If there are multiple errors, the status value reflects only the first, the smallest taking precedence. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rad, c_retval = ufunc.af2a(s, ideg, iamin, asec) check_errwarn(c_retval, 'af2a') return rad STATUS_CODES['af2a'] = { 0: 'OK', 1: 'ideg outside range 0-359', 2: 'iamin outside range 0-59', 3: 'asec outside range 0-59.999...', } def anp(a): """ Normalize angle into the range 0 <= a < 2pi. Parameters ---------- a : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraAnp``. The ERFA documentation is:: - - - - - - - e r a A n p - - - - - - - Normalize angle into the range 0 <= a < 2pi. Given: a double angle (radians) Returned (function value): double angle in range 0-2pi This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.anp(a) return c_retval def anpm(a): """ Normalize angle into the range -pi <= a < +pi. Parameters ---------- a : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraAnpm``. The ERFA documentation is:: - - - - - - - - e r a A n p m - - - - - - - - Normalize angle into the range -pi <= a < +pi. Given: a double angle (radians) Returned (function value): double angle in range +/-pi This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.anpm(a) return c_retval def d2tf(ndp, days): """ Decompose days to hours, minutes, seconds, fraction. Parameters ---------- ndp : int array days : double array Returns ------- sign : char array ihmsf : int array Notes ----- Wraps ERFA function ``eraD2tf``. The ERFA documentation is:: - - - - - - - - e r a D 2 t f - - - - - - - - Decompose days to hours, minutes, seconds, fraction. Given: ndp int resolution (Note 1) days double interval in days Returned: sign char '+' or '-' ihmsf int[4] hours, minutes, seconds, fraction Notes: 1) The argument ndp is interpreted as follows: ndp resolution : ...0000 00 00 -7 1000 00 00 -6 100 00 00 -5 10 00 00 -4 1 00 00 -3 0 10 00 -2 0 01 00 -1 0 00 10 0 0 00 01 1 0 00 00.1 2 0 00 00.01 3 0 00 00.001 : 0 00 00.000... 2) The largest positive useful value for ndp is determined by the size of days, the format of double on the target platform, and the risk of overflowing ihmsf[3]. On a typical platform, for days up to 1.0, the available floating-point precision might correspond to ndp=12. However, the practical limit is typically ndp=9, set by the capacity of a 32-bit int, or ndp=4 if int is only 16 bits. 3) The absolute value of days may exceed 1.0. In cases where it does not, it is up to the caller to test for and handle the case where days is very nearly 1.0 and rounds up to 24 hours, by testing for ihmsf[0]=24 and setting ihmsf[0-3] to zero. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ sign, ihmsf = ufunc.d2tf(ndp, days) sign = sign.view(dt_bytes1) return sign, ihmsf def tf2a(s, ihour, imin, sec): """ Convert hours, minutes, seconds to radians. Parameters ---------- s : char array ihour : int array imin : int array sec : double array Returns ------- rad : double array Notes ----- Wraps ERFA function ``eraTf2a``. The ERFA documentation is:: - - - - - - - - e r a T f 2 a - - - - - - - - Convert hours, minutes, seconds to radians. Given: s char sign: '-' = negative, otherwise positive ihour int hours imin int minutes sec double seconds Returned: rad double angle in radians Returned (function value): int status: 0 = OK 1 = ihour outside range 0-23 2 = imin outside range 0-59 3 = sec outside range 0-59.999... Notes: 1) The result is computed even if any of the range checks fail. 2) Negative ihour, imin and/or sec produce a warning status, but the absolute value is used in the conversion. 3) If there are multiple errors, the status value reflects only the first, the smallest taking precedence. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rad, c_retval = ufunc.tf2a(s, ihour, imin, sec) check_errwarn(c_retval, 'tf2a') return rad STATUS_CODES['tf2a'] = { 0: 'OK', 1: 'ihour outside range 0-23', 2: 'imin outside range 0-59', 3: 'sec outside range 0-59.999...', } def tf2d(s, ihour, imin, sec): """ Convert hours, minutes, seconds to days. Parameters ---------- s : char array ihour : int array imin : int array sec : double array Returns ------- days : double array Notes ----- Wraps ERFA function ``eraTf2d``. The ERFA documentation is:: - - - - - - - - e r a T f 2 d - - - - - - - - Convert hours, minutes, seconds to days. Given: s char sign: '-' = negative, otherwise positive ihour int hours imin int minutes sec double seconds Returned: days double interval in days Returned (function value): int status: 0 = OK 1 = ihour outside range 0-23 2 = imin outside range 0-59 3 = sec outside range 0-59.999... Notes: 1) The result is computed even if any of the range checks fail. 2) Negative ihour, imin and/or sec produce a warning status, but the absolute value is used in the conversion. 3) If there are multiple errors, the status value reflects only the first, the smallest taking precedence. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ days, c_retval = ufunc.tf2d(s, ihour, imin, sec) check_errwarn(c_retval, 'tf2d') return days STATUS_CODES['tf2d'] = { 0: 'OK', 1: 'ihour outside range 0-23', 2: 'imin outside range 0-59', 3: 'sec outside range 0-59.999...', } def rx(phi, r): """ Rotate an r-matrix about the x-axis. Parameters ---------- phi : double array r : double array Returns ------- r : double array Notes ----- Wraps ERFA function ``eraRx``. Note that, unlike the erfa routine, the python wrapper does not change r in-place. The ERFA documentation is:: - - - - - - e r a R x - - - - - - Rotate an r-matrix about the x-axis. Given: phi double angle (radians) Given and returned: r double[3][3] r-matrix, rotated Notes: 1) Calling this function with positive phi incorporates in the supplied r-matrix r an additional rotation, about the x-axis, anticlockwise as seen looking towards the origin from positive x. 2) The additional rotation can be represented by this matrix: ( 1 0 0 ) ( ) ( 0 + cos(phi) + sin(phi) ) ( ) ( 0 - sin(phi) + cos(phi) ) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ r = ufunc.rx(phi, r) return r def ry(theta, r): """ Rotate an r-matrix about the y-axis. Parameters ---------- theta : double array r : double array Returns ------- r : double array Notes ----- Wraps ERFA function ``eraRy``. Note that, unlike the erfa routine, the python wrapper does not change r in-place. The ERFA documentation is:: - - - - - - e r a R y - - - - - - Rotate an r-matrix about the y-axis. Given: theta double angle (radians) Given and returned: r double[3][3] r-matrix, rotated Notes: 1) Calling this function with positive theta incorporates in the supplied r-matrix r an additional rotation, about the y-axis, anticlockwise as seen looking towards the origin from positive y. 2) The additional rotation can be represented by this matrix: ( + cos(theta) 0 - sin(theta) ) ( ) ( 0 1 0 ) ( ) ( + sin(theta) 0 + cos(theta) ) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ r = ufunc.ry(theta, r) return r def rz(psi, r): """ Rotate an r-matrix about the z-axis. Parameters ---------- psi : double array r : double array Returns ------- r : double array Notes ----- Wraps ERFA function ``eraRz``. Note that, unlike the erfa routine, the python wrapper does not change r in-place. The ERFA documentation is:: - - - - - - e r a R z - - - - - - Rotate an r-matrix about the z-axis. Given: psi double angle (radians) Given and returned: r double[3][3] r-matrix, rotated Notes: 1) Calling this function with positive psi incorporates in the supplied r-matrix r an additional rotation, about the z-axis, anticlockwise as seen looking towards the origin from positive z. 2) The additional rotation can be represented by this matrix: ( + cos(psi) + sin(psi) 0 ) ( ) ( - sin(psi) + cos(psi) 0 ) ( ) ( 0 0 1 ) This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ r = ufunc.rz(psi, r) return r def cp(p): """ Copy a p-vector. Parameters ---------- p : double array Returns ------- c : double array Notes ----- Wraps ERFA function ``eraCp``. The ERFA documentation is:: - - - - - - e r a C p - - - - - - Copy a p-vector. Given: p double[3] p-vector to be copied Returned: c double[3] copy This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c = ufunc.cp(p) return c def cpv(pv): """ Copy a position/velocity vector. Parameters ---------- pv : double array Returns ------- c : double array Notes ----- Wraps ERFA function ``eraCpv``. The ERFA documentation is:: - - - - - - - e r a C p v - - - - - - - Copy a position/velocity vector. Given: pv double[2][3] position/velocity vector to be copied Returned: c double[2][3] copy Called: eraCp copy p-vector This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c = ufunc.cpv(pv) return c def cr(r): """ Copy an r-matrix. Parameters ---------- r : double array Returns ------- c : double array Notes ----- Wraps ERFA function ``eraCr``. The ERFA documentation is:: - - - - - - e r a C r - - - - - - Copy an r-matrix. Given: r double[3][3] r-matrix to be copied Returned: c double[3][3] copy Called: eraCp copy p-vector This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c = ufunc.cr(r) return c def p2pv(p): """ Extend a p-vector to a pv-vector by appending a zero velocity. Parameters ---------- p : double array Returns ------- pv : double array Notes ----- Wraps ERFA function ``eraP2pv``. The ERFA documentation is:: - - - - - - - - e r a P 2 p v - - - - - - - - Extend a p-vector to a pv-vector by appending a zero velocity. Given: p double[3] p-vector Returned: pv double[2][3] pv-vector Called: eraCp copy p-vector eraZp zero p-vector This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ pv = ufunc.p2pv(p) return pv def pv2p(pv): """ Discard velocity component of a pv-vector. Parameters ---------- pv : double array Returns ------- p : double array Notes ----- Wraps ERFA function ``eraPv2p``. The ERFA documentation is:: - - - - - - - - e r a P v 2 p - - - - - - - - Discard velocity component of a pv-vector. Given: pv double[2][3] pv-vector Returned: p double[3] p-vector Called: eraCp copy p-vector This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ p = ufunc.pv2p(pv) return p def ir(): """ Initialize an r-matrix to the identity matrix. Parameters ---------- Returns ------- r : double array Notes ----- Wraps ERFA function ``eraIr``. The ERFA documentation is:: - - - - - - e r a I r - - - - - - Initialize an r-matrix to the identity matrix. Returned: r double[3][3] r-matrix This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ r = ufunc.ir() return r def zp(): """ Zero a p-vector. Parameters ---------- Returns ------- p : double array Notes ----- Wraps ERFA function ``eraZp``. The ERFA documentation is:: - - - - - - e r a Z p - - - - - - Zero a p-vector. Returned: p double[3] zero p-vector This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ p = ufunc.zp() return p def zpv(): """ Zero a pv-vector. Parameters ---------- Returns ------- pv : double array Notes ----- Wraps ERFA function ``eraZpv``. The ERFA documentation is:: - - - - - - - e r a Z p v - - - - - - - Zero a pv-vector. Returned: pv double[2][3] zero pv-vector Called: eraZp zero p-vector This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ pv = ufunc.zpv() return pv def zr(): """ Initialize an r-matrix to the null matrix. Parameters ---------- Returns ------- r : double array Notes ----- Wraps ERFA function ``eraZr``. The ERFA documentation is:: - - - - - - e r a Z r - - - - - - Initialize an r-matrix to the null matrix. Returned: r double[3][3] r-matrix This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ r = ufunc.zr() return r def rxr(a, b): """ Multiply two r-matrices. Parameters ---------- a : double array b : double array Returns ------- atb : double array Notes ----- Wraps ERFA function ``eraRxr``. The ERFA documentation is:: - - - - - - - e r a R x r - - - - - - - Multiply two r-matrices. Given: a double[3][3] first r-matrix b double[3][3] second r-matrix Returned: atb double[3][3] a * b Note: It is permissible to re-use the same array for any of the arguments. Called: eraCr copy r-matrix This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ atb = ufunc.rxr(a, b) return atb def tr(r): """ Transpose an r-matrix. Parameters ---------- r : double array Returns ------- rt : double array Notes ----- Wraps ERFA function ``eraTr``. The ERFA documentation is:: - - - - - - e r a T r - - - - - - Transpose an r-matrix. Given: r double[3][3] r-matrix Returned: rt double[3][3] transpose Note: It is permissible for r and rt to be the same array. Called: eraCr copy r-matrix This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rt = ufunc.tr(r) return rt def rxp(r, p): """ Multiply a p-vector by an r-matrix. Parameters ---------- r : double array p : double array Returns ------- rp : double array Notes ----- Wraps ERFA function ``eraRxp``. The ERFA documentation is:: - - - - - - - e r a R x p - - - - - - - Multiply a p-vector by an r-matrix. Given: r double[3][3] r-matrix p double[3] p-vector Returned: rp double[3] r * p Note: It is permissible for p and rp to be the same array. Called: eraCp copy p-vector This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rp = ufunc.rxp(r, p) return rp def rxpv(r, pv): """ Multiply a pv-vector by an r-matrix. Parameters ---------- r : double array pv : double array Returns ------- rpv : double array Notes ----- Wraps ERFA function ``eraRxpv``. The ERFA documentation is:: - - - - - - - - e r a R x p v - - - - - - - - Multiply a pv-vector by an r-matrix. Given: r double[3][3] r-matrix pv double[2][3] pv-vector Returned: rpv double[2][3] r * pv Notes: 1) The algorithm is for the simple case where the r-matrix r is not a function of time. The case where r is a function of time leads to an additional velocity component equal to the product of the derivative of r and the position vector. 2) It is permissible for pv and rpv to be the same array. Called: eraRxp product of r-matrix and p-vector This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ rpv = ufunc.rxpv(r, pv) return rpv def trxp(r, p): """ Multiply a p-vector by the transpose of an r-matrix. Parameters ---------- r : double array p : double array Returns ------- trp : double array Notes ----- Wraps ERFA function ``eraTrxp``. The ERFA documentation is:: - - - - - - - - e r a T r x p - - - - - - - - Multiply a p-vector by the transpose of an r-matrix. Given: r double[3][3] r-matrix p double[3] p-vector Returned: trp double[3] r^T * p Note: It is permissible for p and trp to be the same array. Called: eraTr transpose r-matrix eraRxp product of r-matrix and p-vector This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ trp = ufunc.trxp(r, p) return trp def trxpv(r, pv): """ Multiply a pv-vector by the transpose of an r-matrix. Parameters ---------- r : double array pv : double array Returns ------- trpv : double array Notes ----- Wraps ERFA function ``eraTrxpv``. The ERFA documentation is:: - - - - - - - - - e r a T r x p v - - - - - - - - - Multiply a pv-vector by the transpose of an r-matrix. Given: r double[3][3] r-matrix pv double[2][3] pv-vector Returned: trpv double[2][3] r^T * pv Notes: 1) The algorithm is for the simple case where the r-matrix r is not a function of time. The case where r is a function of time leads to an additional velocity component equal to the product of the derivative of the transpose of r and the position vector. 2) It is permissible for pv and rpv to be the same array. Called: eraTr transpose r-matrix eraRxpv product of r-matrix and pv-vector This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ trpv = ufunc.trxpv(r, pv) return trpv def rm2v(r): """ Express an r-matrix as an r-vector. Parameters ---------- r : double array Returns ------- w : double array Notes ----- Wraps ERFA function ``eraRm2v``. The ERFA documentation is:: - - - - - - - - e r a R m 2 v - - - - - - - - Express an r-matrix as an r-vector. Given: r double[3][3] rotation matrix Returned: w double[3] rotation vector (Note 1) Notes: 1) A rotation matrix describes a rotation through some angle about some arbitrary axis called the Euler axis. The "rotation vector" returned by this function has the same direction as the Euler axis, and its magnitude is the angle in radians. (The magnitude and direction can be separated by means of the function eraPn.) 2) If r is null, so is the result. If r is not a rotation matrix the result is undefined; r must be proper (i.e. have a positive determinant) and real orthogonal (inverse = transpose). 3) The reference frame rotates clockwise as seen looking along the rotation vector from the origin. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ w = ufunc.rm2v(r) return w def rv2m(w): """ Form the r-matrix corresponding to a given r-vector. Parameters ---------- w : double array Returns ------- r : double array Notes ----- Wraps ERFA function ``eraRv2m``. The ERFA documentation is:: - - - - - - - - e r a R v 2 m - - - - - - - - Form the r-matrix corresponding to a given r-vector. Given: w double[3] rotation vector (Note 1) Returned: r double[3][3] rotation matrix Notes: 1) A rotation matrix describes a rotation through some angle about some arbitrary axis called the Euler axis. The "rotation vector" supplied to This function has the same direction as the Euler axis, and its magnitude is the angle in radians. 2) If w is null, the identity matrix is returned. 3) The reference frame rotates clockwise as seen looking along the rotation vector from the origin. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ r = ufunc.rv2m(w) return r def pap(a, b): """ Position-angle from two p-vectors. Parameters ---------- a : double array b : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraPap``. The ERFA documentation is:: - - - - - - - e r a P a p - - - - - - - Position-angle from two p-vectors. Given: a double[3] direction of reference point b double[3] direction of point whose PA is required Returned (function value): double position angle of b with respect to a (radians) Notes: 1) The result is the position angle, in radians, of direction b with respect to direction a. It is in the range -pi to +pi. The sense is such that if b is a small distance "north" of a the position angle is approximately zero, and if b is a small distance "east" of a the position angle is approximately +pi/2. 2) The vectors a and b need not be of unit length. 3) Zero is returned if the two directions are the same or if either vector is null. 4) If vector a is at a pole, the result is ill-defined. Called: eraPn decompose p-vector into modulus and direction eraPm modulus of p-vector eraPxp vector product of two p-vectors eraPmp p-vector minus p-vector eraPdp scalar product of two p-vectors This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.pap(a, b) return c_retval def pas(al, ap, bl, bp): """ Position-angle from spherical coordinates. Parameters ---------- al : double array ap : double array bl : double array bp : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraPas``. The ERFA documentation is:: - - - - - - - e r a P a s - - - - - - - Position-angle from spherical coordinates. Given: al double longitude of point A (e.g. RA) in radians ap double latitude of point A (e.g. Dec) in radians bl double longitude of point B bp double latitude of point B Returned (function value): double position angle of B with respect to A Notes: 1) The result is the bearing (position angle), in radians, of point B with respect to point A. It is in the range -pi to +pi. The sense is such that if B is a small distance "east" of point A, the bearing is approximately +pi/2. 2) Zero is returned if the two points are coincident. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.pas(al, ap, bl, bp) return c_retval def sepp(a, b): """ Angular separation between two p-vectors. Parameters ---------- a : double array b : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraSepp``. The ERFA documentation is:: - - - - - - - - e r a S e p p - - - - - - - - Angular separation between two p-vectors. Given: a double[3] first p-vector (not necessarily unit length) b double[3] second p-vector (not necessarily unit length) Returned (function value): double angular separation (radians, always positive) Notes: 1) If either vector is null, a zero result is returned. 2) The angular separation is most simply formulated in terms of scalar product. However, this gives poor accuracy for angles near zero and pi. The present algorithm uses both cross product and dot product, to deliver full accuracy whatever the size of the angle. Called: eraPxp vector product of two p-vectors eraPm modulus of p-vector eraPdp scalar product of two p-vectors This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.sepp(a, b) return c_retval def seps(al, ap, bl, bp): """ Angular separation between two sets of spherical coordinates. Parameters ---------- al : double array ap : double array bl : double array bp : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraSeps``. The ERFA documentation is:: - - - - - - - - e r a S e p s - - - - - - - - Angular separation between two sets of spherical coordinates. Given: al double first longitude (radians) ap double first latitude (radians) bl double second longitude (radians) bp double second latitude (radians) Returned (function value): double angular separation (radians) Called: eraS2c spherical coordinates to unit vector eraSepp angular separation between two p-vectors This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.seps(al, ap, bl, bp) return c_retval def c2s(p): """ P-vector to spherical coordinates. Parameters ---------- p : double array Returns ------- theta : double array phi : double array Notes ----- Wraps ERFA function ``eraC2s``. The ERFA documentation is:: - - - - - - - e r a C 2 s - - - - - - - P-vector to spherical coordinates. Given: p double[3] p-vector Returned: theta double longitude angle (radians) phi double latitude angle (radians) Notes: 1) The vector p can have any magnitude; only its direction is used. 2) If p is null, zero theta and phi are returned. 3) At either pole, zero theta is returned. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ theta, phi = ufunc.c2s(p) return theta, phi def p2s(p): """ P-vector to spherical polar coordinates. Parameters ---------- p : double array Returns ------- theta : double array phi : double array r : double array Notes ----- Wraps ERFA function ``eraP2s``. The ERFA documentation is:: - - - - - - - e r a P 2 s - - - - - - - P-vector to spherical polar coordinates. Given: p double[3] p-vector Returned: theta double longitude angle (radians) phi double latitude angle (radians) r double radial distance Notes: 1) If P is null, zero theta, phi and r are returned. 2) At either pole, zero theta is returned. Called: eraC2s p-vector to spherical eraPm modulus of p-vector This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ theta, phi, r = ufunc.p2s(p) return theta, phi, r def pv2s(pv): """ Convert position/velocity from Cartesian to spherical coordinates. Parameters ---------- pv : double array Returns ------- theta : double array phi : double array r : double array td : double array pd : double array rd : double array Notes ----- Wraps ERFA function ``eraPv2s``. The ERFA documentation is:: - - - - - - - - e r a P v 2 s - - - - - - - - Convert position/velocity from Cartesian to spherical coordinates. Given: pv double[2][3] pv-vector Returned: theta double longitude angle (radians) phi double latitude angle (radians) r double radial distance td double rate of change of theta pd double rate of change of phi rd double rate of change of r Notes: 1) If the position part of pv is null, theta, phi, td and pd are indeterminate. This is handled by extrapolating the position through unit time by using the velocity part of pv. This moves the origin without changing the direction of the velocity component. If the position and velocity components of pv are both null, zeroes are returned for all six results. 2) If the position is a pole, theta, td and pd are indeterminate. In such cases zeroes are returned for all three. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ theta, phi, r, td, pd, rd = ufunc.pv2s(pv) return theta, phi, r, td, pd, rd def s2c(theta, phi): """ Convert spherical coordinates to Cartesian. Parameters ---------- theta : double array phi : double array Returns ------- c : double array Notes ----- Wraps ERFA function ``eraS2c``. The ERFA documentation is:: - - - - - - - e r a S 2 c - - - - - - - Convert spherical coordinates to Cartesian. Given: theta double longitude angle (radians) phi double latitude angle (radians) Returned: c double[3] direction cosines This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c = ufunc.s2c(theta, phi) return c def s2p(theta, phi, r): """ Convert spherical polar coordinates to p-vector. Parameters ---------- theta : double array phi : double array r : double array Returns ------- p : double array Notes ----- Wraps ERFA function ``eraS2p``. The ERFA documentation is:: - - - - - - - e r a S 2 p - - - - - - - Convert spherical polar coordinates to p-vector. Given: theta double longitude angle (radians) phi double latitude angle (radians) r double radial distance Returned: p double[3] Cartesian coordinates Called: eraS2c spherical coordinates to unit vector eraSxp multiply p-vector by scalar This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ p = ufunc.s2p(theta, phi, r) return p def s2pv(theta, phi, r, td, pd, rd): """ Convert position/velocity from spherical to Cartesian coordinates. Parameters ---------- theta : double array phi : double array r : double array td : double array pd : double array rd : double array Returns ------- pv : double array Notes ----- Wraps ERFA function ``eraS2pv``. The ERFA documentation is:: - - - - - - - - e r a S 2 p v - - - - - - - - Convert position/velocity from spherical to Cartesian coordinates. Given: theta double longitude angle (radians) phi double latitude angle (radians) r double radial distance td double rate of change of theta pd double rate of change of phi rd double rate of change of r Returned: pv double[2][3] pv-vector This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ pv = ufunc.s2pv(theta, phi, r, td, pd, rd) return pv def pdp(a, b): """ p-vector inner (=scalar=dot) product. Parameters ---------- a : double array b : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraPdp``. The ERFA documentation is:: - - - - - - - e r a P d p - - - - - - - p-vector inner (=scalar=dot) product. Given: a double[3] first p-vector b double[3] second p-vector Returned (function value): double a . b This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.pdp(a, b) return c_retval def pm(p): """ Modulus of p-vector. Parameters ---------- p : double array Returns ------- c_retval : double array Notes ----- Wraps ERFA function ``eraPm``. The ERFA documentation is:: - - - - - - e r a P m - - - - - - Modulus of p-vector. Given: p double[3] p-vector Returned (function value): double modulus This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ c_retval = ufunc.pm(p) return c_retval def pmp(a, b): """ P-vector subtraction. Parameters ---------- a : double array b : double array Returns ------- amb : double array Notes ----- Wraps ERFA function ``eraPmp``. The ERFA documentation is:: - - - - - - - e r a P m p - - - - - - - P-vector subtraction. Given: a double[3] first p-vector b double[3] second p-vector Returned: amb double[3] a - b Note: It is permissible to re-use the same array for any of the arguments. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ amb = ufunc.pmp(a, b) return amb def pn(p): """ Convert a p-vector into modulus and unit vector. Parameters ---------- p : double array Returns ------- r : double array u : double array Notes ----- Wraps ERFA function ``eraPn``. The ERFA documentation is:: - - - - - - e r a P n - - - - - - Convert a p-vector into modulus and unit vector. Given: p double[3] p-vector Returned: r double modulus u double[3] unit vector Notes: 1) If p is null, the result is null. Otherwise the result is a unit vector. 2) It is permissible to re-use the same array for any of the arguments. Called: eraPm modulus of p-vector eraZp zero p-vector eraSxp multiply p-vector by scalar This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ r, u = ufunc.pn(p) return r, u def ppp(a, b): """ P-vector addition. Parameters ---------- a : double array b : double array Returns ------- apb : double array Notes ----- Wraps ERFA function ``eraPpp``. The ERFA documentation is:: - - - - - - - e r a P p p - - - - - - - P-vector addition. Given: a double[3] first p-vector b double[3] second p-vector Returned: apb double[3] a + b Note: It is permissible to re-use the same array for any of the arguments. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ apb = ufunc.ppp(a, b) return apb def ppsp(a, s, b): """ P-vector plus scaled p-vector. Parameters ---------- a : double array s : double array b : double array Returns ------- apsb : double array Notes ----- Wraps ERFA function ``eraPpsp``. The ERFA documentation is:: - - - - - - - - e r a P p s p - - - - - - - - P-vector plus scaled p-vector. Given: a double[3] first p-vector s double scalar (multiplier for b) b double[3] second p-vector Returned: apsb double[3] a + s*b Note: It is permissible for any of a, b and apsb to be the same array. Called: eraSxp multiply p-vector by scalar eraPpp p-vector plus p-vector This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ apsb = ufunc.ppsp(a, s, b) return apsb def pvdpv(a, b): """ Inner (=scalar=dot) product of two pv-vectors. Parameters ---------- a : double array b : double array Returns ------- adb : double array Notes ----- Wraps ERFA function ``eraPvdpv``. The ERFA documentation is:: - - - - - - - - - e r a P v d p v - - - - - - - - - Inner (=scalar=dot) product of two pv-vectors. Given: a double[2][3] first pv-vector b double[2][3] second pv-vector Returned: adb double[2] a . b (see note) Note: If the position and velocity components of the two pv-vectors are ( ap, av ) and ( bp, bv ), the result, a . b, is the pair of numbers ( ap . bp , ap . bv + av . bp ). The two numbers are the dot-product of the two p-vectors and its derivative. Called: eraPdp scalar product of two p-vectors This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ adb = ufunc.pvdpv(a, b) return adb def pvm(pv): """ Modulus of pv-vector. Parameters ---------- pv : double array Returns ------- r : double array s : double array Notes ----- Wraps ERFA function ``eraPvm``. The ERFA documentation is:: - - - - - - - e r a P v m - - - - - - - Modulus of pv-vector. Given: pv double[2][3] pv-vector Returned: r double modulus of position component s double modulus of velocity component Called: eraPm modulus of p-vector This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ r, s = ufunc.pvm(pv) return r, s def pvmpv(a, b): """ Subtract one pv-vector from another. Parameters ---------- a : double array b : double array Returns ------- amb : double array Notes ----- Wraps ERFA function ``eraPvmpv``. The ERFA documentation is:: - - - - - - - - - e r a P v m p v - - - - - - - - - Subtract one pv-vector from another. Given: a double[2][3] first pv-vector b double[2][3] second pv-vector Returned: amb double[2][3] a - b Note: It is permissible to re-use the same array for any of the arguments. Called: eraPmp p-vector minus p-vector This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ amb = ufunc.pvmpv(a, b) return amb def pvppv(a, b): """ Add one pv-vector to another. Parameters ---------- a : double array b : double array Returns ------- apb : double array Notes ----- Wraps ERFA function ``eraPvppv``. The ERFA documentation is:: - - - - - - - - - e r a P v p p v - - - - - - - - - Add one pv-vector to another. Given: a double[2][3] first pv-vector b double[2][3] second pv-vector Returned: apb double[2][3] a + b Note: It is permissible to re-use the same array for any of the arguments. Called: eraPpp p-vector plus p-vector This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ apb = ufunc.pvppv(a, b) return apb def pvu(dt, pv): """ Update a pv-vector. Parameters ---------- dt : double array pv : double array Returns ------- upv : double array Notes ----- Wraps ERFA function ``eraPvu``. The ERFA documentation is:: - - - - - - - e r a P v u - - - - - - - Update a pv-vector. Given: dt double time interval pv double[2][3] pv-vector Returned: upv double[2][3] p updated, v unchanged Notes: 1) "Update" means "refer the position component of the vector to a new date dt time units from the existing date". 2) The time units of dt must match those of the velocity. 3) It is permissible for pv and upv to be the same array. Called: eraPpsp p-vector plus scaled p-vector eraCp copy p-vector This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ upv = ufunc.pvu(dt, pv) return upv def pvup(dt, pv): """ Update a pv-vector, discarding the velocity component. Parameters ---------- dt : double array pv : double array Returns ------- p : double array Notes ----- Wraps ERFA function ``eraPvup``. The ERFA documentation is:: - - - - - - - - e r a P v u p - - - - - - - - Update a pv-vector, discarding the velocity component. Given: dt double time interval pv double[2][3] pv-vector Returned: p double[3] p-vector Notes: 1) "Update" means "refer the position component of the vector to a new date dt time units from the existing date". 2) The time units of dt must match those of the velocity. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ p = ufunc.pvup(dt, pv) return p def pvxpv(a, b): """ Outer (=vector=cross) product of two pv-vectors. Parameters ---------- a : double array b : double array Returns ------- axb : double array Notes ----- Wraps ERFA function ``eraPvxpv``. The ERFA documentation is:: - - - - - - - - - e r a P v x p v - - - - - - - - - Outer (=vector=cross) product of two pv-vectors. Given: a double[2][3] first pv-vector b double[2][3] second pv-vector Returned: axb double[2][3] a x b Notes: 1) If the position and velocity components of the two pv-vectors are ( ap, av ) and ( bp, bv ), the result, a x b, is the pair of vectors ( ap x bp, ap x bv + av x bp ). The two vectors are the cross-product of the two p-vectors and its derivative. 2) It is permissible to re-use the same array for any of the arguments. Called: eraCpv copy pv-vector eraPxp vector product of two p-vectors eraPpp p-vector plus p-vector This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ axb = ufunc.pvxpv(a, b) return axb def pxp(a, b): """ p-vector outer (=vector=cross) product. Parameters ---------- a : double array b : double array Returns ------- axb : double array Notes ----- Wraps ERFA function ``eraPxp``. The ERFA documentation is:: - - - - - - - e r a P x p - - - - - - - p-vector outer (=vector=cross) product. Given: a double[3] first p-vector b double[3] second p-vector Returned: axb double[3] a x b Note: It is permissible to re-use the same array for any of the arguments. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ axb = ufunc.pxp(a, b) return axb def s2xpv(s1, s2, pv): """ Multiply a pv-vector by two scalars. Parameters ---------- s1 : double array s2 : double array pv : double array Returns ------- spv : double array Notes ----- Wraps ERFA function ``eraS2xpv``. The ERFA documentation is:: - - - - - - - - - e r a S 2 x p v - - - - - - - - - Multiply a pv-vector by two scalars. Given: s1 double scalar to multiply position component by s2 double scalar to multiply velocity component by pv double[2][3] pv-vector Returned: spv double[2][3] pv-vector: p scaled by s1, v scaled by s2 Note: It is permissible for pv and spv to be the same array. Called: eraSxp multiply p-vector by scalar This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ spv = ufunc.s2xpv(s1, s2, pv) return spv def sxp(s, p): """ Multiply a p-vector by a scalar. Parameters ---------- s : double array p : double array Returns ------- sp : double array Notes ----- Wraps ERFA function ``eraSxp``. The ERFA documentation is:: - - - - - - - e r a S x p - - - - - - - Multiply a p-vector by a scalar. Given: s double scalar p double[3] p-vector Returned: sp double[3] s * p Note: It is permissible for p and sp to be the same array. This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ sp = ufunc.sxp(s, p) return sp def sxpv(s, pv): """ Multiply a pv-vector by a scalar. Parameters ---------- s : double array pv : double array Returns ------- spv : double array Notes ----- Wraps ERFA function ``eraSxpv``. The ERFA documentation is:: - - - - - - - - e r a S x p v - - - - - - - - Multiply a pv-vector by a scalar. Given: s double scalar pv double[2][3] pv-vector Returned: spv double[2][3] s * pv Note: It is permissible for pv and spv to be the same array. Called: eraS2xpv multiply pv-vector by two scalars This revision: 2021 May 11 Copyright (C) 2013-2021, NumFOCUS Foundation. Derived, with permission, from the SOFA library. See notes at end of file. """ spv = ufunc.sxpv(s, pv) return spv