void palGalsup ( double dl, double db, double *dsl, double *dsb ) {
double v1[3];
double v2[3];
/*
* System of supergalactic coordinates:
*
* SGL SGB L2 B2 (deg)
* - +90 47.37 +6.32
* 0 0 - 0
*
* Galactic to supergalactic rotation matrix:
*/
double rmat[3][3] = {
{ -0.735742574804,+0.677261296414,+0.000000000000 },
{ -0.074553778365,-0.080991471307,+0.993922590400 },
{ +0.673145302109,+0.731271165817,+0.110081262225 }
};
/* Spherical to Cartesian */
eraS2c( dl, db, v1 );
/* Galactic to Supergalactic */
eraRxp( rmat, v1, v2 );
/* Cartesian to spherical */
eraC2s( v2, dsl, dsb );
/* Express in conventional ranges */
*dsl = eraAnp( *dsl );
*dsb = eraAnpm( *dsb );
}
void palGe50 ( double dl, double db, double * dr, double * dd ) {
/*
* L2,B2 system of galactic coordinates
*
* P = 192.25 RA of galactic north pole (mean B1950.0)
* Q = 62.6 inclination of galactic to mean B1950.0 equator
* R = 33 longitude of ascending node
*
* P,Q,R are degrees
*
*
* Equatorial to galactic rotation matrix
*
* The Euler angles are P, Q, 90-R, about the z then y then
* z axes.
*
* +CP.CQ.SR-SP.CR +SP.CQ.SR+CP.CR -SQ.SR
*
* -CP.CQ.CR-SP.SR -SP.CQ.CR+CP.SR +SQ.CR
*
* +CP.SQ +SP.SQ +CQ
*
*/
double rmat[3][3] = {
{ -0.066988739415,-0.872755765852,-0.483538914632 },
{ +0.492728466075,-0.450346958020,+0.744584633283 },
{ -0.867600811151,-0.188374601723,+0.460199784784 }
};
double v1[3], v2[3], r, d, re, de;
/* Spherical to cartesian */
eraS2c( dl, db, v1 );
/* Rotate to mean B1950.0 */
eraTrxp( rmat, v1, v2 );
/* Cartesian to spherical */
eraC2s( v2, &r, &d );
/* Introduce E-terms */
palAddet( r, d, 1950.0, &re, &de );
/* Express in conventional ranges */
*dr = eraAnp( re );
*dd = eraAnpm( de );
}
void
palDtp2s ( double xi, double eta, double raz, double decz,
double *ra, double *dec ) {
double cdecz;
double denom;
double sdecz;
double d;
sdecz = sin(decz);
cdecz = cos(decz);
denom = cdecz - eta * sdecz;
d = atan2(xi, denom) + raz;
*ra = eraAnp(d);
*dec = atan2(sdecz + eta * cdecz, sqrt(xi * xi + denom * denom));
return;
}
void palMapqkz ( double rm, double dm, double amprms[21], double *ra,
double *da ){
/* Local Variables: */
int i;
double ab1, abv[3], p[3], w, p1dv, p2[3], p3[3];
double gr2e, pde, pdep1, ehn[3], p1[3];
/* Unpack scalar and vector parameters. */
ab1 = amprms[11];
gr2e = amprms[7];
for( i = 0; i < 3; i++ ) {
abv[i] = amprms[i+8];
ehn[i] = amprms[i+4];
}
/* Spherical to x,y,z. */
eraS2c( rm, dm, p );
/* Light deflection (restrained within the Sun's disc) */
pde = eraPdp( p, ehn );
pdep1 = pde + 1.0;
w = gr2e / ( pdep1 > 1.0e-5 ? pdep1 : 1.0e-5 );
for( i = 0; i < 3; i++) {
p1[i] = p[i] + w * ( ehn[i] - pde * p[i] );
}
/* Aberration. */
p1dv = eraPdp( p1, abv );
w = 1.0 + p1dv / ( ab1 + 1.0 );
for( i = 0; i < 3; i++ ) {
p2[i] = ( ( ab1 * p1[i] ) + ( w * abv[i] ) );
}
/* Precession and nutation. */
eraRxp( (double(*)[3]) &rms[12], p2, p3 );
/* Geocentric apparent RA,dec. */
eraC2s( p3, ra, da );
*ra = eraAnp( *ra );
}
void palEqecl ( double dr, double dd, double date, double *dl, double *db ) {
double v1[3], v2[3];
double rmat[3][3];
/* Spherical to Cartesian */
eraS2c( dr, dd, v1 );
/* Mean J2000 to mean of date */
palPrec( 2000.0, palEpj(date), rmat );
eraRxp( rmat, v1, v2 );
/* Equatorial to ecliptic */
palEcmat( date, rmat );
eraRxp( rmat, v2, v1 );
/* Cartesian to spherical */
eraC2s( v1, dl, db );
/* Express in conventional range */
*dl = eraAnp( *dl );
*db = palDrange( *db );
}
double eraGmst06(double uta, double utb, double tta, double ttb)
/*
** - - - - - - - - - -
** 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
**
** Copyright (C) 2013, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double t, gmst;
/* TT Julian centuries since J2000.0. */
t = ((tta - DJ00) + ttb) / DJC;
/* Greenwich mean sidereal time, IAU 2006. */
gmst = eraAnp(eraEra00(uta, utb) +
( 0.014506 +
( 4612.156534 +
( 1.3915817 +
( -0.00000044 +
( -0.000029956 +
( -0.0000000368 )
* t) * t) * t) * t) * t) * DAS2R);
return gmst;
}
double eraGmst00(double uta, double utb, double tta, double ttb)
/*
** - - - - - - - - - -
** 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=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)
**
** Copyright (C) 2013, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double t, gmst;
/* TT Julian centuries since J2000.0. */
t = ((tta - ERFA_DJ00) + ttb) / ERFA_DJC;
/* Greenwich Mean Sidereal Time, IAU 2000. */
gmst = eraAnp(eraEra00(uta, utb) +
( 0.014506 +
( 4612.15739966 +
( 1.39667721 +
( -0.00009344 +
( 0.00001882 )
* t) * t) * t) * t) * ERFA_DAS2R);
return gmst;
}
void eraAtciq(double rc, double dc,
double pr, double pd, double px, double rv,
eraASTROM *astrom, double *ri, double *di)
/*
** - - - - - - - - -
** 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)
** 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:
** ri,di double CIRS 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
** 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
**
** Copyright (C) 2013-2016, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double pco[3], pnat[3], ppr[3], pi[3], w;
/* Proper motion and parallax, giving BCRS coordinate direction. */
eraPmpx(rc, dc, pr, pd, px, rv, astrom->pmt, astrom->eb, pco);
/* Light deflection by the Sun, giving BCRS natural direction. */
eraLdsun(pco, astrom->eh, astrom->em, pnat);
/* Aberration, giving GCRS proper direction. */
eraAb(pnat, astrom->v, astrom->em, astrom->bm1, ppr);
/* Bias-precession-nutation, giving CIRS proper direction. */
eraRxp(astrom->bpn, ppr, pi);
/* CIRS RA,Dec. */
eraC2s(pi, &w, di);
*ri = eraAnp(w);
/* Finished. */
}
void eraAtciqz(double rc, double dc, eraASTROM *astrom,
double *ri, double *di)
/*
** - - - - - - - - - -
** 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
**
** Copyright (C) 2013-2015, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double pco[3], pnat[3], ppr[3], pi[3], w;
/* BCRS coordinate direction (unit vector). */
eraS2c(rc, dc, pco);
/* Light deflection by the Sun, giving BCRS natural direction. */
eraLdsun(pco, astrom->eh, astrom->em, pnat);
/* Aberration, giving GCRS proper direction. */
eraAb(pnat, astrom->v, astrom->em, astrom->bm1, ppr);
/* Bias-precession-nutation, giving CIRS proper direction. */
eraRxp(astrom->bpn, ppr, pi);
/* CIRS RA,Dec. */
eraC2s(pi, &w, di);
*ri = eraAnp(w);
/* Finished. */
}
void palFk45z( double r1950, double d1950, double bepoch, double *r2000,
double *d2000 ){
/* Local Variables: */
double w;
int i;
int j;
double r0[3], a1[3], v1[3], v2[6]; /* Position and position+velocity vectors */
/* CANONICAL CONSTANTS (see references) */
/* Vector A. */
double a[3] = { -1.62557E-6, -0.31919E-6, -0.13843E-6 };
/* Vectors Adot. */
double ad[3] = { 1.245E-3, -1.580E-3, -0.659E-3 };
/* Matrix M (only half of which is needed here). */
double em[6][3] = { {0.9999256782, -0.0111820611, -0.0048579477},
{0.0111820610, 0.9999374784, -0.0000271765},
{0.0048579479, -0.0000271474, 0.9999881997},
{-0.000551, -0.238565, 0.435739},
{0.238514, -0.002667, -0.008541},
{-0.435623, 0.012254, 0.002117} };
/* Spherical to Cartesian. */
eraS2c( r1950, d1950, r0 );
/* Adjust vector A to give zero proper motion in FK5. */
w = ( bepoch - 1950.0 )/PAL__PMF;
for( i = 0; i < 3; i++ ) {
a1[ i ] = a[ i ] + w*ad[ i ];
}
/* Remove e-terms. */
w = r0[ 0 ]*a1[ 0 ] + r0[ 1 ]*a1[ 1 ] + r0[ 2 ]*a1[ 2 ];
for( i = 0; i < 3; i++ ) {
v1[ i ] = r0[ i ] - a1[ i ] + w*r0[ i ];
}
/* Convert position vector to Fricke system. */
for( i = 0; i < 6; i++ ) {
w = 0.0;
for( j = 0; j < 3; j++ ) {
w += em[ i ][ j ]*v1[ j ];
}
v2[ i ] = w;
}
/* Allow for fictitious proper motion in FK4. */
w = ( palEpj( palEpb2d( bepoch ) ) - 2000.0 )/PAL__PMF;
for( i = 0; i < 3; i++ ) {
v2[ i ] += w*v2[ i + 3 ];
}
/* Revert to spherical coordinates. */
eraC2s( v2, &w, d2000 );
*r2000 = eraAnp( w );
}
double eraGst94(double uta, double utb)
/*
** - - - - - - - - -
** 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=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)
**
** Copyright (C) 2013, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double gmst82, eqeq94, gst;
gmst82 = eraGmst82(uta, utb);
eqeq94 = eraEqeq94(uta, utb);
gst = eraAnp(gmst82 + eqeq94);
return gst;
}
double eraGmst82(double dj1, double dj2)
/*
** - - - - - - - - - -
** 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
**
** 2450123.7D0 0D0 (JD method)
** 2451545D0 -1421.3D0 (J2000 method)
** 2400000.5D0 50123.2D0 (MJD method)
** 2450123.5D0 0.2D0 (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).
**
** Copyright (C) 2013, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
/* Coefficients of IAU 1982 GMST-UT1 model */
double A = 24110.54841 - DAYSEC / 2.0;
double B = 8640184.812866;
double C = 0.093104;
double D = -6.2e-6;
/* Note: the first constant, A, has to be adjusted by 12 hours */
/* because the UT1 is supplied as a Julian date, which begins */
/* at noon. */
double d1, d2, t, f, gmst;
/* Julian centuries since fundamental epoch. */
if (dj1 < dj2) {
d1 = dj1;
d2 = dj2;
} else {
d1 = dj2;
d2 = dj1;
}
t = (d1 + (d2 - DJ00)) / DJC;
/* Fractional part of JD(UT1), in seconds. */
f = DAYSEC * (fmod(d1, 1.0) + fmod(d2, 1.0));
/* GMST at this UT1. */
gmst = eraAnp(DS2R * ((A + (B + (C + D * t) * t) * t) + f));
return gmst;
}
void eraHfk5z(double rh, double dh, double date1, double date2,
double *r5, double *d5, double *dr5, double *dd5)
/*
** - - - - - - - - -
** 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.
**
** Copyright (C) 2013, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double t, ph[3], r5h[3][3], s5h[3], sh[3], vst[3],
rst[3][3], r5ht[3][3], pv5e[2][3], vv[3],
w, r, v;
/* Time interval from fundamental epoch J2000.0 to given date (JY). */
t = ((date1 - DJ00) + date2) / DJY;
/* Hipparcos barycentric position vector (normalized). */
eraS2c(rh, dh, ph);
/* FK5 to Hipparcos orientation matrix and spin vector. */
eraFk5hip(r5h, s5h);
/* Rotate the spin into the Hipparcos system. */
eraRxp(r5h, s5h, sh);
/* Accumulated Hipparcos wrt FK5 spin over that interval. */
eraSxp(t, s5h, vst);
/* Express the accumulated spin as a rotation matrix. */
eraRv2m(vst, rst);
/* Rotation matrix: accumulated spin, then FK5 to Hipparcos. */
eraRxr(r5h, rst, r5ht);
/* De-orient & de-spin the Hipparcos position into FK5 J2000.0. */
eraTrxp(r5ht, ph, pv5e[0]);
/* Apply spin to the position giving a space motion. */
eraPxp(sh, ph, vv);
/* De-orient & de-spin the Hipparcos space motion into FK5 J2000.0. */
eraTrxp(r5ht, vv, pv5e[1]);
/* FK5 position/velocity pv-vector to spherical. */
eraPv2s(pv5e, &w, d5, &r, dr5, dd5, &v);
*r5 = eraAnp(w);
return;
}
void eraFk5hz(double r5, double d5, double date1, double date2,
double *rh, double *dh)
/*
** - - - - - - - - -
** 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.
**
** Copyright (C) 2013-2015, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double t, p5e[3], r5h[3][3], s5h[3], vst[3], rst[3][3], p5[3],
ph[3], w;
/* Interval from given date to fundamental epoch J2000.0 (JY). */
t = - ((date1 - ERFA_DJ00) + date2) / ERFA_DJY;
/* FK5 barycentric position vector. */
eraS2c(r5, d5, p5e);
/* FK5 to Hipparcos orientation matrix and spin vector. */
eraFk5hip(r5h, s5h);
/* Accumulated Hipparcos wrt FK5 spin over that interval. */
eraSxp(t, s5h, vst);
/* Express the accumulated spin as a rotation matrix. */
eraRv2m(vst, rst);
/* Derotate the vector's FK5 axes back to date. */
eraTrxp(rst, p5e, p5);
/* Rotate the vector into the Hipparcos system. */
eraRxp(r5h, p5, ph);
/* Hipparcos vector to spherical. */
eraC2s(ph, &w, dh);
*rh = eraAnp(w);
return;
}
int eraPvstar(double pv[2][3], double *ra, double *dec,
double *pmr, double *pmd, double *px, double *rv)
/*
** - - - - - - - - - -
** 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.
**
** Copyright (C) 2013-2016, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double r, x[3], vr, ur[3], vt, ut[3], bett, betr, d, w, del,
usr[3], ust[3], a, rad, decd, rd;
/* Isolate the radial component of the velocity (AU/day, inertial). */
eraPn(pv[0], &r, x);
vr = eraPdp(x, pv[1]);
eraSxp(vr, x, ur);
/* Isolate the transverse component of the velocity (AU/day, inertial). */
eraPmp(pv[1], ur, ut);
vt = eraPm(ut);
/* Special-relativity dimensionless parameters. */
bett = vt / ERFA_DC;
betr = vr / ERFA_DC;
/* The inertial-to-observed correction terms. */
d = 1.0 + betr;
w = 1.0 - betr*betr - bett*bett;
if (d == 0.0 || w < 0) return -1;
del = sqrt(w) - 1.0;
/* Apply relativistic correction factor to radial velocity component. */
w = (betr != 0) ? (betr - del) / (betr * d) : 1.0;
eraSxp(w, ur, usr);
/* Apply relativistic correction factor to tangential velocity */
/* component. */
eraSxp(1.0/d, ut, ust);
/* Combine the two to obtain the observed velocity vector (AU/day). */
eraPpp(usr, ust, pv[1]);
/* Cartesian to spherical. */
eraPv2s(pv, &a, dec, &r, &rad, &decd, &rd);
if (r == 0.0) return -2;
/* Return RA in range 0 to 2pi. */
*ra = eraAnp(a);
/* Return proper motions in radians per year. */
*pmr = rad * ERFA_DJY;
*pmd = decd * ERFA_DJY;
/* Return parallax in arcsec. */
*px = ERFA_DR2AS / r;
/* Return radial velocity in km/s. */
*rv = 1e-3 * rd * ERFA_DAU / ERFA_DAYSEC;
/* OK status. */
return 0;
}
void eraAtoiq(const char *type,
double ob1, double ob2, eraASTROM *astrom,
double *ri, double *di)
/*
** - - - - - - - - -
** 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 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.
**
** 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
**
** Copyright (C) 2013-2015, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
int c;
double c1, c2, sphi, cphi, ce, xaeo, yaeo, zaeo, v[3],
xmhdo, ymhdo, zmhdo, az, sz, zdo, refa, refb, tz, dref,
zdt, xaet, yaet, zaet, xmhda, ymhda, zmhda,
f, xhd, yhd, zhd, xpl, ypl, w, hma;
/* Coordinate type. */
c = (int) type[0];
/* Coordinates. */
c1 = ob1;
c2 = ob2;
/* Sin, cos of latitude. */
sphi = astrom->sphi;
cphi = astrom->cphi;
/* Standardize coordinate type. */
if ( c == 'r' || c == 'R' ) {
c = 'R';
} else if ( c == 'h' || c == 'H' ) {
c = 'H';
} else {
c = 'A';
}
/* If Az,ZD, convert to Cartesian (S=0,E=90). */
if ( c == 'A' ) {
ce = sin(c2);
xaeo = - cos(c1) * ce;
yaeo = sin(c1) * ce;
zaeo = cos(c2);
} else {
/* If RA,Dec, convert to HA,Dec. */
if ( c == 'R' ) c1 = astrom->eral - c1;
/* To Cartesian -HA,Dec. */
eraS2c ( -c1, c2, v );
xmhdo = v[0];
ymhdo = v[1];
zmhdo = v[2];
/* To Cartesian Az,El (S=0,E=90). */
xaeo = sphi*xmhdo - cphi*zmhdo;
yaeo = ymhdo;
zaeo = cphi*xmhdo + sphi*zmhdo;
}
/* Azimuth (S=0,E=90). */
az = ( xaeo != 0.0 || yaeo != 0.0 ) ? atan2(yaeo,xaeo) : 0.0;
/* Sine of observed ZD, and observed ZD. */
sz = sqrt ( xaeo*xaeo + yaeo*yaeo );
zdo = atan2 ( sz, zaeo );
/*
** Refraction
** ----------
*/
/* Fast algorithm using two constant model. */
refa = astrom->refa;
refb = astrom->refb;
tz = sz / zaeo;
dref = ( refa + refb*tz*tz ) * tz;
zdt = zdo + dref;
/* To Cartesian Az,ZD. */
ce = sin(zdt);
xaet = cos(az) * ce;
yaet = sin(az) * ce;
zaet = cos(zdt);
/* Cartesian Az,ZD to Cartesian -HA,Dec. */
xmhda = sphi*xaet + cphi*zaet;
ymhda = yaet;
zmhda = - cphi*xaet + sphi*zaet;
/* Diurnal aberration. */
f = ( 1.0 + astrom->diurab*ymhda );
xhd = f * xmhda;
yhd = f * ( ymhda - astrom->diurab );
zhd = f * zmhda;
/* Polar motion. */
xpl = astrom->xpl;
ypl = astrom->ypl;
w = xpl*xhd - ypl*yhd + zhd;
v[0] = xhd - xpl*w;
v[1] = yhd + ypl*w;
v[2] = w - ( xpl*xpl + ypl*ypl ) * zhd;
/* To spherical -HA,Dec. */
eraC2s(v, &hma, di);
/* Right ascension. */
*ri = eraAnp(astrom->eral + hma);
/* Finished. */
}
void eraEqec06(double date1, double date2, double dr, double dd,
double *dl, double *db)
/*
** - - - - - - - - - -
** 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
**
** Copyright (C) 2013-2016, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double rm[3][3], v1[3], v2[3], a, b;
/* Spherical to Cartesian. */
eraS2c(dr, dd, v1);
/* Rotation matrix, ICRS equatorial to ecliptic. */
eraEcm06(date1, date2, rm);
/* The transformation from ICRS to ecliptic. */
eraRxp(rm, v1, v2);
/* Cartesian to spherical. */
eraC2s(v2, &a, &b);
/* Express in conventional ranges. */
*dl = eraAnp(a);
*db = eraAnpm(b);
}
double palDranrm ( double angle ) {
return eraAnp( angle );
}
double eraEra00(double dj1, double dj2)
/*
** - - - - - - - - -
** 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)
**
** Copyright (C) 2013-2014, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double d1, d2, t, f, theta;
/* Days since fundamental epoch. */
if (dj1 < dj2) {
d1 = dj1;
d2 = dj2;
} else {
d1 = dj2;
d2 = dj1;
}
t = d1 + (d2- ERFA_DJ00);
/* Fractional part of T (days). */
f = fmod(d1, 1.0) + fmod(d2, 1.0);
/* Earth rotation angle at this UT1. */
theta = eraAnp(ERFA_D2PI * (f + 0.7790572732640
+ 0.00273781191135448 * t));
return theta;
}
double eraGst00b(double uta, double utb)
/*
** - - - - - - - - - -
** 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=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, 2001)
** 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)
**
** Copyright (C) 2013, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double gmst00, ee00b, gst;
gmst00 = eraGmst00(uta, utb, uta, utb);
ee00b = eraEe00b(uta, utb);
gst = eraAnp(gmst00 + ee00b);
return gst;
}
void eraIcrs2g ( double dr, double dd, double *dl, double *db )
/*
** - - - - - - - - - -
** 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.
**
** Copyright (C) 2013-2016, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double v1[3], v2[3];
/*
** L2,B2 system of galactic coordinates in the form presented in the
** Hipparcos Catalogue. In degrees:
**
** P = 192.85948 right ascension of the Galactic north pole in ICRS
** Q = 27.12825 declination of the Galactic north pole in ICRS
** R = 32.93192 longitude of the ascending node of the Galactic
** plane on the ICRS equator
**
** ICRS to galactic rotation matrix, obtained by computing
** R_3(-R) R_1(pi/2-Q) R_3(pi/2+P) to the full precision shown:
*/
double r[3][3] = { { -0.054875560416215368492398900454,
-0.873437090234885048760383168409,
-0.483835015548713226831774175116 },
{ +0.494109427875583673525222371358,
-0.444829629960011178146614061616,
+0.746982244497218890527388004556 },
{ -0.867666149019004701181616534570,
-0.198076373431201528180486091412,
+0.455983776175066922272100478348 } };
/* Spherical to Cartesian. */
eraS2c(dr, dd, v1);
/* ICRS to Galactic. */
eraRxp(r, v1, v2);
/* Cartesian to spherical. */
eraC2s(v2, dl, db);
/* Express in conventional ranges. */
*dl = eraAnp(*dl);
*db = eraAnpm(*db);
/* Finished. */
}
void eraAtioq(double ri, double di, eraASTROM *astrom,
double *aob, double *zob,
double *hob, double *dob, double *rob)
/*
** - - - - - - - - -
** 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
**
** Copyright (C) 2013-2015, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
/* Minimum cos(alt) and sin(alt) for refraction purposes */
const double CELMIN = 1e-6;
const double SELMIN = 0.05;
double v[3], x, y, z, xhd, yhd, zhd, f, xhdt, yhdt, zhdt,
xaet, yaet, zaet, azobs, r, tz, w, del, cosdel,
xaeo, yaeo, zaeo, zdobs, hmobs, dcobs, raobs;
/*--------------------------------------------------------------------*/
/* CIRS RA,Dec to Cartesian -HA,Dec. */
eraS2c(ri-astrom->eral, di, v);
x = v[0];
y = v[1];
z = v[2];
/* Polar motion. */
xhd = x + astrom->xpl*z;
yhd = y - astrom->ypl*z;
zhd = z - astrom->xpl*x + astrom->ypl*y;
/* Diurnal aberration. */
f = ( 1.0 - astrom->diurab*yhd );
xhdt = f * xhd;
yhdt = f * ( yhd + astrom->diurab );
zhdt = f * zhd;
/* Cartesian -HA,Dec to Cartesian Az,El (S=0,E=90). */
xaet = astrom->sphi*xhdt - astrom->cphi*zhdt;
yaet = yhdt;
zaet = astrom->cphi*xhdt + astrom->sphi*zhdt;
/* Azimuth (N=0,E=90). */
azobs = ( xaet != 0.0 || yaet != 0.0 ) ? atan2(yaet,-xaet) : 0.0;
/* ---------- */
/* Refraction */
/* ---------- */
/* Cosine and sine of altitude, with precautions. */
r = sqrt(xaet*xaet + yaet*yaet);
r = r > CELMIN ? r : CELMIN;
z = zaet > SELMIN ? zaet : SELMIN;
/* A*tan(z)+B*tan^3(z) model, with Newton-Raphson correction. */
tz = r/z;
w = astrom->refb*tz*tz;
del = ( astrom->refa + w ) * tz /
( 1.0 + ( astrom->refa + 3.0*w ) / ( z*z ) );
/* Apply the change, giving observed vector. */
cosdel = 1.0 - del*del/2.0;
f = cosdel - del*z/r;
xaeo = xaet*f;
yaeo = yaet*f;
zaeo = cosdel*zaet + del*r;
/* Observed ZD. */
zdobs = atan2(sqrt(xaeo*xaeo+yaeo*yaeo), zaeo);
/* Az/El vector to HA,Dec vector (both right-handed). */
v[0] = astrom->sphi*xaeo + astrom->cphi*zaeo;
v[1] = yaeo;
v[2] = - astrom->cphi*xaeo + astrom->sphi*zaeo;
/* To spherical -HA,Dec. */
eraC2s ( v, &hmobs, &dcobs );
/* Right ascension (with respect to CIO). */
raobs = astrom->eral + hmobs;
/* Return the results. */
*aob = eraAnp(azobs);
*zob = zdobs;
*hob = -hmobs;
*dob = dcobs;
*rob = eraAnp(raobs);
/* Finished. */
}
void eraAticq(double ri, double di, eraASTROM *astrom,
double *rc, double *dc)
/*
** - - - - - - - - -
** 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
**
** Copyright (C) 2013-2015, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
int j, i;
double pi[3], ppr[3], pnat[3], pco[3], w, d[3], before[3], r2, r,
after[3];
/* CIRS RA,Dec to Cartesian. */
eraS2c(ri, di, pi);
/* Bias-precession-nutation, giving GCRS proper direction. */
eraTrxp(astrom->bpn, pi, ppr);
/* Aberration, giving GCRS natural direction. */
eraZp(d);
for (j = 0; j < 2; j++) {
r2 = 0.0;
for (i = 0; i < 3; i++) {
w = ppr[i] - d[i];
before[i] = w;
r2 += w*w;
}
r = sqrt(r2);
for (i = 0; i < 3; i++) {
before[i] /= r;
}
eraAb(before, astrom->v, astrom->em, astrom->bm1, after);
r2 = 0.0;
for (i = 0; i < 3; i++) {
d[i] = after[i] - before[i];
w = ppr[i] - d[i];
pnat[i] = w;
r2 += w*w;
}
r = sqrt(r2);
for (i = 0; i < 3; i++) {
pnat[i] /= r;
}
}
/* Light deflection by the Sun, giving BCRS coordinate direction. */
eraZp(d);
for (j = 0; j < 5; j++) {
r2 = 0.0;
for (i = 0; i < 3; i++) {
w = pnat[i] - d[i];
before[i] = w;
r2 += w*w;
}
r = sqrt(r2);
for (i = 0; i < 3; i++) {
before[i] /= r;
}
eraLdsun(before, astrom->eh, astrom->em, after);
r2 = 0.0;
for (i = 0; i < 3; i++) {
d[i] = after[i] - before[i];
w = pnat[i] - d[i];
pco[i] = w;
r2 += w*w;
}
r = sqrt(r2);
for (i = 0; i < 3; i++) {
pco[i] /= r;
}
}
/* ICRS astrometric RA,Dec. */
eraC2s(pco, &w, dc);
*rc = eraAnp(w);
/* Finished. */
}
void eraLteqec(double epj, double dr, double dd, double *dl, double *db)
/*
** - - - - - - - - - -
** 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
**
** Copyright (C) 2013-2016, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double rm[3][3], v1[3], v2[3], a, b;
/* Spherical to Cartesian. */
eraS2c(dr, dd, v1);
/* Rotation matrix, ICRS equatorial to ecliptic. */
eraLtecm(epj, rm);
/* The transformation from ICRS to ecliptic. */
eraRxp(rm, v1, v2);
/* Cartesian to spherical. */
eraC2s(v2, &a, &b);
/* Express in conventional ranges. */
*dl = eraAnp(a);
*db = eraAnpm(b);
}
