double palRvlg( double r2000, double d2000 ){
/* Local Variables: */
double vb[ 3 ];
/*
*
* Solar velocity due to Galactic rotation and translation
*
* Speed = 300 km/s
*
* Apex = L2,B2 90deg, 0deg
* = RA,Dec 21 12 01.1 +48 19 47 J2000.0
*
* This is expressed in the form of a J2000.0 x,y,z vector:
*
* VA(1) = X = -SPEED*COS(RA)*COS(DEC)
* VA(2) = Y = -SPEED*SIN(RA)*COS(DEC)
* VA(3) = Z = -SPEED*SIN(DEC)
*/
double va[ 3 ] = { -148.23284, +133.44888, -224.09467 };
/* Convert given J2000 RA,Dec to x,y,z. */
eraS2c( r2000, d2000, vb );
/* Compute dot product with Solar motion vector. */
return eraPdp( va, vb );
}
double palRvlsrd( double r2000, double d2000 ){
/* Local Variables: */
double vb[ 3 ];
/*
* Peculiar solar motion from Delhaye 1965: in Galactic Cartesian
* coordinates (+9,+12,+7) km/s. This corresponds to about 16.6 km/s
* towards Galactic coordinates L2 = 53 deg, B2 = +25 deg, or RA,Dec
* 17 49 58.7 +28 07 04 J2000.
*
* The solar motion is expressed here in the form of a J2000.0
* equatorial Cartesian vector:
*
* VA(1) = X = -SPEED*COS(RA)*COS(DEC)
* VA(2) = Y = -SPEED*SIN(RA)*COS(DEC)
* VA(3) = Z = -SPEED*SIN(DEC)
*/
double va[ 3 ] = { +0.63823, +14.58542, -7.80116 };
/* Convert given J2000 RA,Dec to x,y,z. */
eraS2c( r2000, d2000, vb );
/* Compute dot product with Solar motion vector. */
return eraPdp( va, vb );
}
double palRvlsrk( double r2000, double d2000 ){
/* Local Variables: */
double vb[ 3 ];
/*
* Standard solar motion (from Methods of Experimental Physics, ed Meeks,
* vol 12, part C, sec 6.1.5.2, p281):
*
* 20 km/s towards RA 18h Dec +30d (1900).
*
* The solar motion is expressed here in the form of a J2000.0
* equatorial Cartesian vector:
*
* VA(1) = X = -SPEED*COS(RA)*COS(DEC)
* VA(2) = Y = -SPEED*SIN(RA)*COS(DEC)
* VA(3) = Z = -SPEED*SIN(DEC)
*/
double va[ 3 ] = { -0.29000, +17.31726, -10.00141 };
/* Convert given J2000 RA,Dec to x,y,z. */
eraS2c( r2000, d2000, vb );
/* Compute dot product with Solar motion vector. */
return eraPdp( va, vb );
}
void eraPvdpv(double a[2][3], double b[2][3], double adb[2])
/*
** - - - - - - - - -
** e r a P v d p v
** - - - - - - - - -
**
** Inner (=scalar=dot) product of two pv-vectors.
**
** Given:
** a double[2][3] first pv-vector
** b double[2][3] second pv-vector
**
** Returned:
** adb double[2] a . b (see note)
**
** Note:
**
** If the position and velocity components of the two pv-vectors are
** ( ap, av ) and ( bp, bv ), the result, a . b, is the pair of
** numbers ( ap . bp , ap . bv + av . bp ). The two numbers are the
** dot-product of the two p-vectors and its derivative.
**
** Called:
** eraPdp scalar product of two p-vectors
**
** Copyright (C) 2013-2016, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double adbd, addb;
/* a . b = constant part of result. */
adb[0] = eraPdp(a[0], b[0]);
/* a . bdot */
adbd = eraPdp(a[0], b[1]);
/* adot . b */
addb = eraPdp(a[1], b[0]);
/* Velocity part of result. */
adb[1] = adbd + addb;
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 );
}
double eraSepp(double a[3], double b[3])
/*
** - - - - - - - -
** e r a S e p p
** - - - - - - - -
**
** Angular separation between two p-vectors.
**
** Given:
** a double[3] first p-vector (not necessarily unit length)
** b double[3] second p-vector (not necessarily unit length)
**
** Returned (function value):
** double angular separation (radians, always positive)
**
** Notes:
**
** 1) If either vector is null, a zero result is returned.
**
** 2) The angular separation is most simply formulated in terms of
** scalar product. However, this gives poor accuracy for angles
** near zero and pi. The present algorithm uses both cross product
** and dot product, to deliver full accuracy whatever the size of
** the angle.
**
** Called:
** eraPxp vector product of two p-vectors
** eraPm modulus of p-vector
** eraPdp scalar product of two p-vectors
**
** Copyright (C) 2013-2016, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double axb[3], ss, cs, s;
/* Sine of angle between the vectors, multiplied by the two moduli. */
eraPxp(a, b, axb);
ss = eraPm(axb);
/* Cosine of the angle, multiplied by the two moduli. */
cs = eraPdp(a, b);
/* The angle. */
s = ((ss != 0.0) || (cs != 0.0)) ? atan2(ss, cs) : 0.0;
return s;
}
double eraPap(double a[3], double b[3])
/*
** - - - - - - -
** e r a P a p
** - - - - - - -
**
** Position-angle from two p-vectors.
**
** Given:
** a double[3] direction of reference point
** b double[3] direction of point whose PA is required
**
** Returned (function value):
** double position angle of b with respect to a (radians)
**
** Notes:
**
** 1) The result is the position angle, in radians, of direction b with
** respect to direction a. It is in the range -pi to +pi. The
** sense is such that if b is a small distance "north" of a the
** position angle is approximately zero, and if b is a small
** distance "east" of a the position angle is approximately +pi/2.
**
** 2) The vectors a and b need not be of unit length.
**
** 3) Zero is returned if the two directions are the same or if either
** vector is null.
**
** 4) If vector a is at a pole, the result is ill-defined.
**
** Called:
** eraPn decompose p-vector into modulus and direction
** eraPm modulus of p-vector
** eraPxp vector product of two p-vectors
** eraPmp p-vector minus p-vector
** eraPdp scalar product of two p-vectors
**
** Copyright (C) 2013-2015, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double am, au[3], bm, st, ct, xa, ya, za, eta[3], xi[3], a2b[3], pa;
/* Modulus and direction of the a vector. */
eraPn(a, &am, au);
/* Modulus of the b vector. */
bm = eraPm(b);
/* Deal with the case of a null vector. */
if ((am == 0.0) || (bm == 0.0)) {
st = 0.0;
ct = 1.0;
} else {
/* The "north" axis tangential from a (arbitrary length). */
xa = a[0];
ya = a[1];
za = a[2];
eta[0] = -xa * za;
eta[1] = -ya * za;
eta[2] = xa*xa + ya*ya;
/* The "east" axis tangential from a (same length). */
eraPxp(eta, au, xi);
/* The vector from a to b. */
eraPmp(b, a, a2b);
/* Resolve into components along the north and east axes. */
st = eraPdp(a2b, xi);
ct = eraPdp(a2b, eta);
/* Deal with degenerate cases. */
if ((st == 0.0) && (ct == 0.0)) ct = 1.0;
}
/* Position angle. */
pa = atan2(st, ct);
return pa;
}
void eraLd(double bm, double p[3], double q[3], double e[3],
double em, double dlim, double p1[3])
/*
** - - - - - -
** e r a L d
** - - - - - -
**
** Apply light deflection by a solar-system body, as part of
** transforming coordinate direction into natural direction.
**
** Given:
** bm double mass of the gravitating body (solar masses)
** p double[3] direction from observer to source (unit vector)
** q double[3] direction from body to source (unit vector)
** e double[3] direction from body to observer (unit vector)
** em double distance from body to observer (au)
** dlim double deflection limiter (Note 4)
**
** Returned:
** p1 double[3] observer to deflected source (unit vector)
**
** Notes:
**
** 1) The algorithm is based on Expr. (70) in Klioner (2003) and
** Expr. (7.63) in the Explanatory Supplement (Urban & Seidelmann
** 2013), with some rearrangement to minimize the effects of machine
** precision.
**
** 2) The mass parameter bm can, as required, be adjusted in order to
** allow for such effects as quadrupole field.
**
** 3) The barycentric position of the deflecting body should ideally
** correspond to the time of closest approach of the light ray to
** the body.
**
** 4) The deflection limiter parameter dlim is phi^2/2, where phi is
** the angular separation (in radians) between source and body at
** which limiting is applied. As phi shrinks below the chosen
** threshold, the deflection is artificially reduced, reaching zero
** for phi = 0.
**
** 5) The returned vector p1 is not normalized, but the consequential
** departure from unit magnitude is always negligible.
**
** 6) The arguments p and p1 can be the same array.
**
** 7) To accumulate total light deflection taking into account the
** contributions from several bodies, call the present function for
** each body in succession, in decreasing order of distance from the
** observer.
**
** 8) For efficiency, validation is omitted. The supplied vectors must
** be of unit magnitude, and the deflection limiter non-zero and
** positive.
**
** References:
**
** Urban, S. & Seidelmann, P. K. (eds), Explanatory Supplement to
** the Astronomical Almanac, 3rd ed., University Science Books
** (2013).
**
** Klioner, Sergei A., "A practical relativistic model for micro-
** arcsecond astrometry in space", Astr. J. 125, 1580-1597 (2003).
**
** Called:
** eraPdp scalar product of two p-vectors
** eraPxp vector product of two p-vectors
**
** Copyright (C) 2013-2014, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
int i;
double qpe[3], qdqpe, w, eq[3], peq[3];
/* q . (q + e). */
for (i = 0; i < 3; i++) {
qpe[i] = q[i] + e[i];
}
qdqpe = eraPdp(q, qpe);
/* 2 x G x bm / ( em x c^2 x ( q . (q + e) ) ). */
w = bm * ERFA_SRS / em / ERFA_GMAX(qdqpe,dlim);
/* p x (e x q). */
eraPxp(e, q, eq);
eraPxp(p, eq, peq);
/* Apply the deflection. */
for (i = 0; i < 3; i++) {
p1[i] = p[i] + w*peq[i];
}
/* Finished. */
}
int eraStarpv(double ra, double dec,
double pmr, double pmd, double px, double rv,
double pv[2][3])
/*
** - - - - - - - - - -
** e r a S t a r p v
** - - - - - - - - - -
**
** Convert star catalog coordinates to position+velocity vector.
**
** Given (Note 1):
** ra double right ascension (radians)
** dec double declination (radians)
** pmr double RA proper motion (radians/year)
** pmd double Dec proper motion (radians/year)
** px double parallax (arcseconds)
** rv double radial velocity (km/s, positive = receding)
**
** Returned (Note 2):
** pv double[2][3] pv-vector (AU, AU/day)
**
** Returned (function value):
** int status:
** 0 = no warnings
** 1 = distance overridden (Note 6)
** 2 = excessive speed (Note 7)
** 4 = solution didn't converge (Note 8)
** else = binary logical OR of the above
**
** Notes:
**
** 1) The star data accepted by this function are "observables" for an
** imaginary observer at the solar-system barycenter. Proper motion
** and radial velocity are, strictly, in terms of barycentric
** coordinate time, TCB. For most practical applications, it is
** permissible to neglect the distinction between TCB and ordinary
** "proper" time on Earth (TT/TAI). The result will, as a rule, be
** limited by the intrinsic accuracy of the proper-motion and
** radial-velocity data; moreover, the pv-vector is likely to be
** merely an intermediate result, so that a change of time unit
** would cancel out overall.
**
** In accordance with normal star-catalog conventions, the object's
** right ascension and declination are freed from the effects of
** secular aberration. The frame, which is aligned to the catalog
** equator and equinox, is Lorentzian and centered on the SSB.
**
** 2) The resulting position and velocity pv-vector is with respect to
** the same frame and, like the catalog coordinates, is freed from
** the effects of secular aberration. Should the "coordinate
** direction", where the object was located at the catalog epoch, be
** required, it may be obtained by calculating the magnitude of the
** position vector pv[0][0-2] dividing by the speed of light in
** AU/day to give the light-time, and then multiplying the space
** velocity pv[1][0-2] by this light-time and adding the result to
** pv[0][0-2].
**
** Summarizing, the pv-vector returned is for most stars almost
** identical to the result of applying the standard geometrical
** "space motion" transformation. The differences, which are the
** subject of the Stumpff paper referenced below, are:
**
** (i) In stars with significant radial velocity and proper motion,
** the constantly changing light-time distorts the apparent proper
** motion. Note that this is a classical, not a relativistic,
** effect.
**
** (ii) The transformation complies with special relativity.
**
** 3) Care is needed with units. The star coordinates are in radians
** and the proper motions in radians per Julian year, but the
** parallax is in arcseconds; the radial velocity is in km/s, but
** the pv-vector result is in AU and AU/day.
**
** 4) The RA proper motion is in terms of coordinate angle, not true
** angle. If the catalog uses arcseconds for both RA and Dec proper
** motions, the RA proper motion will need to be divided by cos(Dec)
** before use.
**
** 5) Straight-line motion at constant speed, in the inertial frame,
** is assumed.
**
** 6) An extremely small (or zero or negative) parallax is interpreted
** to mean that the object is on the "celestial sphere", the radius
** of which is an arbitrary (large) value (see the constant PXMIN).
** When the distance is overridden in this way, the status,
** initially zero, has 1 added to it.
**
** 7) If the space velocity is a significant fraction of c (see the
** constant VMAX), it is arbitrarily set to zero. When this action
** occurs, 2 is added to the status.
**
** 8) The relativistic adjustment involves an iterative calculation.
** If the process fails to converge within a set number (IMAX) of
** iterations, 4 is added to the status.
**
** 9) The inverse transformation is performed by the function
** eraPvstar.
**
** Called:
** eraS2pv spherical coordinates to pv-vector
** eraPm modulus of p-vector
** eraZp zero p-vector
** eraPn decompose p-vector into modulus and direction
** eraPdp scalar product of two p-vectors
** eraSxp multiply p-vector by scalar
** eraPmp p-vector minus p-vector
** eraPpp p-vector plus p-vector
**
** Reference:
**
** Stumpff, P., 1985, Astron.Astrophys. 144, 232-240.
**
** Copyright (C) 2013-2015, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
/* Smallest allowed parallax */
static const double PXMIN = 1e-7;
/* Largest allowed speed (fraction of c) */
static const double VMAX = 0.5;
/* Maximum number of iterations for relativistic solution */
static const int IMAX = 100;
int i, iwarn;
double w, r, rd, rad, decd, v, x[3], usr[3], ust[3],
vsr, vst, betst, betsr, bett, betr,
dd, ddel, ur[3], ut[3],
d = 0.0, del = 0.0, /* to prevent */
odd = 0.0, oddel = 0.0, /* compiler */
od = 0.0, odel = 0.0; /* warnings */
/* Distance (AU). */
if (px >= PXMIN) {
w = px;
iwarn = 0;
} else {
w = PXMIN;
iwarn = 1;
}
r = ERFA_DR2AS / w;
/* Radial velocity (AU/day). */
rd = ERFA_DAYSEC * rv * 1e3 / ERFA_DAU;
/* Proper motion (radian/day). */
rad = pmr / ERFA_DJY;
decd = pmd / ERFA_DJY;
/* To pv-vector (AU,AU/day). */
eraS2pv(ra, dec, r, rad, decd, rd, pv);
/* If excessive velocity, arbitrarily set it to zero. */
v = eraPm(pv[1]);
if (v / ERFA_DC > VMAX) {
eraZp(pv[1]);
iwarn += 2;
}
/* Isolate the radial component of the velocity (AU/day). */
eraPn(pv[0], &w, x);
vsr = eraPdp(x, pv[1]);
eraSxp(vsr, x, usr);
/* Isolate the transverse component of the velocity (AU/day). */
eraPmp(pv[1], usr, ust);
vst = eraPm(ust);
/* Special-relativity dimensionless parameters. */
betsr = vsr / ERFA_DC;
betst = vst / ERFA_DC;
/* Determine the inertial-to-observed relativistic correction terms. */
bett = betst;
betr = betsr;
for (i = 0; i < IMAX; i++) {
d = 1.0 + betr;
del = sqrt(1.0 - betr*betr - bett*bett) - 1.0;
betr = d * betsr + del;
bett = d * betst;
if (i > 0) {
dd = fabs(d - od);
ddel = fabs(del - odel);
if ((i > 1) && (dd >= odd) && (ddel >= oddel)) break;
odd = dd;
oddel = ddel;
}
od = d;
odel = del;
}
if (i >= IMAX) iwarn += 4;
/* Replace observed radial velocity with inertial value. */
w = (betsr != 0.0) ? d + del / betsr : 1.0;
eraSxp(w, usr, ur);
/* Replace observed tangential velocity with inertial value. */
eraSxp(d, ust, ut);
/* Combine the two to obtain the inertial space velocity. */
eraPpp(ur, ut, pv[1]);
/* Return the status. */
return iwarn;
}
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;
}
double palDvdv ( double va[3], double vb[3] ) {
return eraPdp( va, vb );
}
int eraStarpm(double ra1, double dec1,
double pmr1, double pmd1, double px1, double rv1,
double ep1a, double ep1b, double ep2a, double ep2b,
double *ra2, double *dec2,
double *pmr2, double *pmd2, double *px2, double *rv2)
/*
** - - - - - - - - - -
** e r a S t a r p m
** - - - - - - - - - -
**
** Star proper motion: update star catalog data for space motion.
**
** Given:
** ra1 double right ascension (radians), before
** dec1 double declination (radians), before
** pmr1 double RA proper motion (radians/year), before
** pmd1 double Dec proper motion (radians/year), before
** px1 double parallax (arcseconds), before
** rv1 double radial velocity (km/s, +ve = receding), before
** ep1a double "before" epoch, part A (Note 1)
** ep1b double "before" epoch, part B (Note 1)
** ep2a double "after" epoch, part A (Note 1)
** ep2b double "after" epoch, part B (Note 1)
**
** Returned:
** ra2 double right ascension (radians), after
** dec2 double declination (radians), after
** pmr2 double RA proper motion (radians/year), after
** pmd2 double Dec proper motion (radians/year), after
** px2 double parallax (arcseconds), after
** rv2 double radial velocity (km/s, +ve = receding), after
**
** Returned (function value):
** int status:
** -1 = system error (should not occur)
** 0 = no warnings or errors
** 1 = distance overridden (Note 6)
** 2 = excessive velocity (Note 7)
** 4 = solution didn't converge (Note 8)
** else = binary logical OR of the above warnings
**
** Notes:
**
** 1) The starting and ending TDB dates ep1a+ep1b and ep2a+ep2b are
** Julian Dates, apportioned in any convenient way between the two
** parts (A and B). For example, JD(TDB)=2450123.7 could be
** expressed in any of these ways, among others:
**
** epna epnb
**
** 2450123.7 0.0 (JD method)
** 2451545.0 -1421.3 (J2000 method)
** 2400000.5 50123.2 (MJD method)
** 2450123.5 0.2 (date & time method)
**
** The JD method is the most natural and convenient to use in
** cases where the loss of several decimal digits of resolution
** is acceptable. The J2000 method is best matched to the way
** the argument is handled internally and will deliver the
** optimum resolution. The MJD method and the date & time methods
** are both good compromises between resolution and convenience.
**
** 2) In accordance with normal star-catalog conventions, the object's
** right ascension and declination are freed from the effects of
** secular aberration. The frame, which is aligned to the catalog
** equator and equinox, is Lorentzian and centered on the SSB.
**
** The proper motions are the rate of change of the right ascension
** and declination at the catalog epoch and are in radians per TDB
** Julian year.
**
** The parallax and radial velocity are in the same frame.
**
** 3) Care is needed with units. The star coordinates are in radians
** and the proper motions in radians per Julian year, but the
** parallax is in arcseconds.
**
** 4) The RA proper motion is in terms of coordinate angle, not true
** angle. If the catalog uses arcseconds for both RA and Dec proper
** motions, the RA proper motion will need to be divided by cos(Dec)
** before use.
**
** 5) Straight-line motion at constant speed, in the inertial frame,
** is assumed.
**
** 6) An extremely small (or zero or negative) parallax is interpreted
** to mean that the object is on the "celestial sphere", the radius
** of which is an arbitrary (large) value (see the eraStarpv
** function for the value used). When the distance is overridden in
** this way, the status, initially zero, has 1 added to it.
**
** 7) If the space velocity is a significant fraction of c (see the
** constant VMAX in the function eraStarpv), it is arbitrarily set
** to zero. When this action occurs, 2 is added to the status.
**
** 8) The relativistic adjustment carried out in the eraStarpv function
** involves an iterative calculation. If the process fails to
** converge within a set number of iterations, 4 is added to the
** status.
**
** Called:
** eraStarpv star catalog data to space motion pv-vector
** eraPvu update a pv-vector
** eraPdp scalar product of two p-vectors
** eraPvstar space motion pv-vector to star catalog data
**
** Copyright (C) 2013, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double pv1[2][3], tl1, dt, pv[2][3], r2, rdv, v2, c2mv2, tl2,
pv2[2][3];
int j1, j2, j;
/* RA,Dec etc. at the "before" epoch to space motion pv-vector. */
j1 = eraStarpv(ra1, dec1, pmr1, pmd1, px1, rv1, pv1);
/* Light time when observed (days). */
tl1 = eraPm(pv1[0]) / DC;
/* Time interval, "before" to "after" (days). */
dt = (ep2a - ep1a) + (ep2b - ep1b);
/* Move star along track from the "before" observed position to the */
/* "after" geometric position. */
eraPvu(dt + tl1, pv1, pv);
/* From this geometric position, deduce the observed light time (days) */
/* at the "after" epoch (with theoretically unneccessary error check). */
r2 = eraPdp(pv[0], pv[0]);
rdv = eraPdp(pv[0], pv[1]);
v2 = eraPdp(pv[1], pv[1]);
c2mv2 = DC*DC - v2;
if (c2mv2 <= 0) return -1;
tl2 = (-rdv + sqrt(rdv*rdv + c2mv2*r2)) / c2mv2;
/* Move the position along track from the observed place at the */
/* "before" epoch to the observed place at the "after" epoch. */
eraPvu(dt + (tl1 - tl2), pv1, pv2);
/* Space motion pv-vector to RA,Dec etc. at the "after" epoch. */
j2 = eraPvstar(pv2, ra2, dec2, pmr2, pmd2, px2, rv2);
/* Final status. */
j = (j2 == 0) ? j1 : -1;
return j;
}
void eraPmpx(double rc, double dc, double pr, double pd,
double px, double rv, double pmt, double pob[3],
double pco[3])
/*
** - - - - - - - -
** e r a P m p x
** - - - - - - - -
**
** Proper motion and parallax.
**
** Given:
** rc,dc double ICRS RA,Dec at catalog epoch (radians)
** pr double RA proper motion (radians/year; Note 1)
** pd double Dec proper motion (radians/year)
** px double parallax (arcsec)
** rv double radial velocity (km/s, +ve if receding)
** pmt double proper motion time interval (SSB, Julian years)
** pob double[3] SSB to observer vector (au)
**
** Returned:
** pco double[3] coordinate direction (BCRS unit vector)
**
** Notes:
**
** 1) The proper motion in RA is dRA/dt rather than cos(Dec)*dRA/dt.
**
** 2) The proper motion time interval is for when the starlight
** reaches the solar system barycenter.
**
** 3) To avoid the need for iteration, the Roemer effect (i.e. the
** small annual modulation of the proper motion coming from the
** changing light time) is applied approximately, using the
** direction of the star at the catalog epoch.
**
** References:
**
** 1984 Astronomical Almanac, pp B39-B41.
**
** Urban, S. & Seidelmann, P. K. (eds), Explanatory Supplement to
** the Astronomical Almanac, 3rd ed., University Science Books
** (2013), Section 7.2.
**
** Called:
** eraPdp scalar product of two p-vectors
** eraPn decompose p-vector into modulus and direction
**
** Copyright (C) 2013-2017, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
/* Km/s to au/year */
const double VF = ERFA_DAYSEC*ERFA_DJM/ERFA_DAU;
/* Light time for 1 au, Julian years */
const double AULTY = ERFA_AULT/ERFA_DAYSEC/ERFA_DJY;
int i;
double sr, cr, sd, cd, x, y, z, p[3], dt, pxr, w, pdz, pm[3];
/* Spherical coordinates to unit vector (and useful functions). */
sr = sin(rc);
cr = cos(rc);
sd = sin(dc);
cd = cos(dc);
p[0] = x = cr*cd;
p[1] = y = sr*cd;
p[2] = z = sd;
/* Proper motion time interval (y) including Roemer effect. */
dt = pmt + eraPdp(p,pob)*AULTY;
/* Space motion (radians per year). */
pxr = px * ERFA_DAS2R;
w = VF * rv * pxr;
pdz = pd * z;
pm[0] = - pr*y - pdz*cr + w*x;
pm[1] = pr*x - pdz*sr + w*y;
pm[2] = pd*cd + w*z;
/* Coordinate direction of star (unit vector, BCRS). */
for (i = 0; i < 3; i++) {
p[i] += dt*pm[i] - pxr*pob[i];
}
eraPn(p, &w, pco);
/* Finished. */
}
