void eraRxpv(double r[3][3], double pv[2][3], double rpv[2][3])
/*
** - - - - - - - -
** 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
**
** Note:
** It is permissible for pv and rpv to be the same array.
**
** Called:
** eraRxp product of r-matrix and p-vector
**
** Copyright (C) 2013-2014, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
eraRxp(r, pv[0], rpv[0]);
eraRxp(r, pv[1], rpv[1]);
return;
}
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 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 );
}
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 eraTrxp(double r[3][3], double p[3], double trp[3])
/*
** - - - - - - - -
** 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 * 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
**
** Copyright (C) 2013-2016, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double tr[3][3];
/* Transpose of matrix r. */
eraTr(r, tr);
/* Matrix tr * vector p -> vector trp. */
eraRxp(tr, p, trp);
return;
}
void eraH2fk5(double rh, double dh,
double drh, double ddh, double pxh, double rvh,
double *r5, double *d5,
double *dr5, double *dd5, double *px5, double *rv5)
/*
** - - - - - - - - -
** 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).
**
** Copyright (C) 2013-2015, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
int i;
double pvh[2][3], r5h[3][3], s5h[3], sh[3], wxp[3], vv[3], pv5[2][3];
/* Hipparcos barycentric position/velocity pv-vector (normalized). */
eraStarpv(rh, dh, drh, ddh, pxh, rvh, pvh);
/* FK5 to Hipparcos orientation matrix and spin vector. */
eraFk5hip(r5h, s5h);
/* Make spin units per day instead of per year. */
for ( i = 0; i < 3; s5h[i++] /= 365.25 );
/* Orient the spin into the Hipparcos system. */
eraRxp(r5h, s5h, sh);
/* De-orient the Hipparcos position into the FK5 system. */
eraTrxp(r5h, pvh[0], pv5[0]);
/* Apply spin to the position giving an extra space motion component. */
eraPxp(pvh[0], sh, wxp);
/* Subtract this component from the Hipparcos space motion. */
eraPmp(pvh[1], wxp, vv);
/* De-orient the Hipparcos space motion into the FK5 system. */
eraTrxp(r5h, vv, pv5[1]);
/* FK5 pv-vector to spherical. */
eraPvstar(pv5, r5, d5, dr5, dd5, px5, rv5);
return;
}
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 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 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 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;
}
void palPertue( double date, double u[13], int *jstat ) {
/* Distance from EMB at which Earth and Moon are treated separately */
const double RNE=1.0;
/* Coincidence with major planet distance */
const double COINC=0.0001;
/* Coefficient relating timestep to perturbing force */
const double TSC=1e-4;
/* Minimum and maximum timestep (days) */
const double TSMIN = 0.01;
const double TSMAX = 10.0;
/* Age limit for major-planet state vector (days) */
const double AGEPMO=5.0;
/* Age limit for major-planet mean elements (days) */
const double AGEPEL=50.0;
/* Margin for error when deciding whether to renew the planetary data */
const double TINY=1e-6;
/* Age limit for the body's osculating elements (before rectification) */
const double AGEBEL=100.0;
/* Gaussian gravitational constant squared */
const double GCON2 = PAL__GCON * PAL__GCON;
/* The final epoch */
double TFINAL;
/* The body's current universal elements */
double UL[13];
/* Current reference epoch */
double T0;
/* Timespan from latest orbit rectification to final epoch (days) */
double TSPAN;
/* Time left to go before integration is complete */
double TLEFT;
/* Time direction flag: +1=forwards, -1=backwards */
double FB;
/* First-time flag */
int FIRST = 0;
/*
* The current perturbations
*/
/* Epoch (days relative to current reference epoch) */
double RTN;
/* Position (AU) */
double PERP[3];
/* Velocity (AU/d) */
double PERV[3];
/* Acceleration (AU/d/d) */
double PERA[3];
/* Length of current timestep (days), and half that */
double TS,HTS;
/* Epoch of middle of timestep */
double T;
/* Epoch of planetary mean elements */
double TPEL = 0.0;
/* Planet number (1=Mercury, 2=Venus, 3=EMB...8=Neptune) */
int NP;
/* Planetary universal orbital elements */
double UP[8][13];
/* Epoch of planetary state vectors */
double TPMO = 0.0;
/* State vectors for the major planets (AU,AU/s) */
double PVIN[8][6];
/* Earth velocity and position vectors (AU,AU/s) */
double VB[3],PB[3],VH[3],PE[3];
/* Moon geocentric state vector (AU,AU/s) and position part */
double PVM[6],PM[3];
/* Date to J2000 de-precession matrix */
double PMAT[3][3];
/*
* Correction terms for extrapolated major planet vectors
*/
/* Sun-to-planet distances squared multiplied by 3 */
double R2X3[8];
/* Sunward acceleration terms, G/2R^3 */
double GC[8];
/* Tangential-to-circular correction factor */
double FC;
/* Radial correction factor due to Sunwards acceleration */
double FG;
/* The body's unperturbed and perturbed state vectors (AU,AU/s) */
double PV0[6],PV[6];
/* The body's perturbed and unperturbed heliocentric distances (AU) cubed */
double R03,R3;
/* The perturbating accelerations, indirect and direct */
double FI[3],FD[3];
/* Sun-to-planet vector, and distance cubed */
double RHO[3],RHO3;
/* Body-to-planet vector, and distance cubed */
double DELTA[3],DELTA3;
/* Miscellaneous */
int I,J;
double R2,W,DT,DT2,R,FT;
int NE;
/* Planetary inverse masses, Mercury through Neptune then Earth and Moon */
const double AMAS[10] = {
6023600., 408523.5, 328900.5, 3098710.,
1047.355, 3498.5, 22869., 19314.,
332946.038, 27068709.
};
/* Preset the status to OK. */
*jstat = 0;
/* Copy the final epoch. */
TFINAL = date;
/* Copy the elements (which will be periodically updated). */
for (I=0; I<13; I++) {
UL[I] = u[I];
}
/* Initialize the working reference epoch. */
T0=UL[2];
/* Total timespan (days) and hence time left. */
TSPAN = TFINAL-T0;
TLEFT = TSPAN;
/* Warn if excessive. */
if (fabs(TSPAN) > 36525.0) *jstat=101;
/* Time direction: +1 for forwards, -1 for backwards. */
FB = COPYSIGN(1.0,TSPAN);
/* Initialize relative epoch for start of current timestep. */
RTN = 0.0;
/* Reset the perturbations (position, velocity, acceleration). */
for (I=0; I<3; I++) {
PERP[I] = 0.0;
PERV[I] = 0.0;
PERA[I] = 0.0;
}
/* Set "first iteration" flag. */
FIRST = 1;
/* Step through the time left. */
while (FB*TLEFT > 0.0) {
/* Magnitude of current acceleration due to planetary attractions. */
if (FIRST) {
TS = TSMIN;
} else {
R2 = 0.0;
for (I=0; I<3; I++) {
W = FD[I];
R2 = R2+W*W;
}
W = sqrt(R2);
/* Use the acceleration to decide how big a timestep can be tolerated. */
if (W != 0.0) {
TS = DMIN(TSMAX,DMAX(TSMIN,TSC/W));
} else {
TS = TSMAX;
}
}
TS = TS*FB;
/* Override if final epoch is imminent. */
TLEFT = TSPAN-RTN;
if (fabs(TS) > fabs(TLEFT)) TS=TLEFT;
/* Epoch of middle of timestep. */
HTS = TS/2.0;
T = T0+RTN+HTS;
/* Is it time to recompute the major-planet elements? */
if (FIRST || fabs(T-TPEL)-AGEPEL >= TINY) {
/* Yes: go forward in time by just under the maximum allowed. */
TPEL = T+FB*AGEPEL;
/* Compute the state vector for the new epoch. */
for (NP=1; NP<=8; NP++) {
palPlanet(TPEL,NP,PV,&J);
/* Warning if remote epoch, abort if error. */
if (J == 1) {
*jstat = 102;
} else if (J != 0) {
goto ABORT;
}
/* Transform the vector into universal elements. */
palPv2ue(PV,TPEL,0.0,&(UP[NP-1][0]),&J);
if (J != 0) goto ABORT;
}
}
/* Is it time to recompute the major-planet motions? */
if (FIRST || fabs(T-TPMO)-AGEPMO >= TINY) {
/* Yes: look ahead. */
TPMO = T+FB*AGEPMO;
/* Compute the motions of each planet (AU,AU/d). */
for (NP=1; NP<=8; NP++) {
/* The planet's position and velocity (AU,AU/s). */
palUe2pv(TPMO,&(UP[NP-1][0]),&(PVIN[NP-1][0]),&J);
if (J != 0) goto ABORT;
/* Scale velocity to AU/d. */
for (J=3; J<6; J++) {
PVIN[NP-1][J] = PVIN[NP-1][J]*PAL__SPD;
}
/* Precompute also the extrapolation correction terms. */
R2 = 0.0;
for (I=0; I<3; I++) {
W = PVIN[NP-1][I];
R2 = R2+W*W;
}
R2X3[NP-1] = R2*3.0;
GC[NP-1] = GCON2/(2.0*R2*sqrt(R2));
}
}
/* Reset the first-time flag. */
FIRST = 0;
/* Unperturbed motion of the body at middle of timestep (AU,AU/s). */
palUe2pv(T,UL,PV0,&J);
if (J != 0) goto ABORT;
/* Perturbed position of the body (AU) and heliocentric distance cubed. */
R2 = 0.0;
for (I=0; I<3; I++) {
W = PV0[I]+PERP[I]+(PERV[I]+PERA[I]*HTS/2.0)*HTS;
PV[I] = W;
R2 = R2+W*W;
}
R3 = R2*sqrt(R2);
/* The body's unperturbed heliocentric distance cubed. */
R2 = 0.0;
for (I=0; I<3; I++) {
W = PV0[I];
R2 = R2+W*W;
}
R03 = R2*sqrt(R2);
/* Compute indirect and initialize direct parts of the perturbation. */
for (I=0; I<3; I++) {
FI[I] = PV0[I]/R03-PV[I]/R3;
FD[I] = 0.0;
}
/* Ready to compute the direct planetary effects. */
/* Reset the "near-Earth" flag. */
NE = 0;
/* Interval from state-vector epoch to middle of current timestep. */
DT = T-TPMO;
DT2 = DT*DT;
/* Planet by planet, including separate Earth and Moon. */
for (NP=1; NP<10; NP++) {
/* Which perturbing body? */
if (NP <= 8) {
/* Planet: compute the extrapolation in longitude (squared). */
R2 = 0.0;
for (J=3; J<6; J++) {
W = PVIN[NP-1][J]*DT;
R2 = R2+W*W;
}
/* Hence the tangential-to-circular correction factor. */
FC = 1.0+R2/R2X3[NP-1];
/* The radial correction factor due to the inwards acceleration. */
FG = 1.0-GC[NP-1]*DT2;
/* Planet's position. */
for (I=0; I<3; I++) {
RHO[I] = FG*(PVIN[NP-1][I]+FC*PVIN[NP-1][I+3]*DT);
}
} else if (NE) {
/* Near-Earth and either Earth or Moon. */
if (NP == 9) {
/* Earth: position. */
palEpv(T,PE,VH,PB,VB);
for (I=0; I<3; I++) {
RHO[I] = PE[I];
}
} else {
/* Moon: position. */
palPrec(palEpj(T),2000.0,PMAT);
palDmoon(T,PVM);
eraRxp(PMAT,PVM,PM);
for (I=0; I<3; I++) {
RHO[I] = PM[I]+PE[I];
}
}
}
/* Proceed unless Earth or Moon and not the near-Earth case. */
if (NP <= 8 || NE) {
/* Heliocentric distance cubed. */
R2 = 0.0;
for (I=0; I<3; I++) {
W = RHO[I];
R2 = R2+W*W;
}
R = sqrt(R2);
RHO3 = R2*R;
/* Body-to-planet vector, and distance. */
R2 = 0.0;
for (I=0; I<3; I++) {
W = RHO[I]-PV[I];
DELTA[I] = W;
R2 = R2+W*W;
}
R = sqrt(R2);
/* If this is the EMB, set the near-Earth flag appropriately. */
if (NP == 3 && R < RNE) NE = 1;
/* Proceed unless EMB and this is the near-Earth case. */
if ( ! (NE && NP == 3) ) {
/* If too close, ignore this planet and set a warning. */
if (R < COINC) {
*jstat = NP;
} else {
/* Accumulate "direct" part of perturbation acceleration. */
DELTA3 = R2*R;
W = AMAS[NP-1];
for (I=0; I<3; I++) {
FD[I] = FD[I]+(DELTA[I]/DELTA3-RHO[I]/RHO3)/W;
}
}
}
}
}
/* Update the perturbations to the end of the timestep. */
RTN += TS;
for (I=0; I<3; I++) {
W = (FI[I]+FD[I])*GCON2;
FT = W*TS;
PERP[I] = PERP[I]+(PERV[I]+FT/2.0)*TS;
PERV[I] = PERV[I]+FT;
PERA[I] = W;
}
/* Time still to go. */
TLEFT = TSPAN-RTN;
/* Is it either time to rectify the orbit or the last time through? */
if (fabs(RTN) >= AGEBEL || FB*TLEFT <= 0.0) {
/* Yes: update to the end of the current timestep. */
T0 += RTN;
RTN = 0.0;
/* The body's unperturbed motion (AU,AU/s). */
palUe2pv(T0,UL,PV0,&J);
if (J != 0) goto ABORT;
/* Add and re-initialize the perturbations. */
for (I=0; I<3; I++) {
J = I+3;
PV[I] = PV0[I]+PERP[I];
PV[J] = PV0[J]+PERV[I]/PAL__SPD;
PERP[I] = 0.0;
PERV[I] = 0.0;
PERA[I] = FD[I]*GCON2;
}
/* Use the position and velocity to set up new universal elements. */
palPv2ue(PV,T0,0.0,UL,&J);
if (J != 0) goto ABORT;
/* Adjust the timespan and time left. */
TSPAN = TFINAL-T0;
TLEFT = TSPAN;
}
/* Next timestep. */
}
/* Return the updated universal-element set. */
for (I=0; I<13; I++) {
u[I] = UL[I];
}
/* Finished. */
return;
/* Miscellaneous numerical error. */
ABORT:
*jstat = -1;
return;
}
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);
}
void palDmxv ( double dm[3][3], double va[3], double vb[3] ) {
eraRxp( dm, va, vb );
}
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);
}
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. */
}
