void palMappa( double eq, double date, double amprms[21] ){
/* Local constants */
/* Gravitational radius of the Sun x 2 (2*mu/c**2, AU) */
const double GR2 = 2.0 * 9.87063e-9;
/* Local Variables; */
int i;
double ebd[ 3 ], ehd[ 3 ], eh[ 3 ], e, vn[ 3 ], vm;
/* Initialise so that unsused values are returned holding zero */
memset( amprms, 0, 21*sizeof( *amprms ) );
/* Time interval for proper motion correction. */
amprms[ 0 ] = eraEpj( PAL__MJD0, date ) - eq;
/* Get Earth barycentric and heliocentric position and velocity. */
palEvp( date, eq, ebd, &rms[ 1 ], ehd, eh );
/* Heliocentric direction of Earth (normalized) and modulus. */
eraPn( eh, &e, &rms[ 4 ] );
/* Light deflection parameter */
amprms[7] = GR2 / e;
/* Aberration parameters. */
for( i = 0; i < 3; i++ ) {
amprms[ i + 8 ] = ebd[ i ]*PAL__CR;
}
eraPn( &rms[8], &vm, vn );
amprms[ 11 ] = sqrt( 1.0 - vm*vm );
/* NPB matrix. */
palPrenut( eq, date, (double(*)[ 3 ]) &rms[ 12 ] );
}
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;
}
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;
}
void eraLtecm(double epj, double rm[3][3])
/*
** - - - - - - - - -
** e r a L t e c m
** - - - - - - - - -
**
** ICRS equatorial to ecliptic rotation matrix, long-term.
**
** Given:
** epj double Julian epoch (TT)
**
** Returned:
** rm double[3][3] ICRS to ecliptic rotation matrix
**
** Notes:
**
** 1) The matrix is in the sense
**
** E_ep = rm x P_ICRS,
**
** where P_ICRS is a vector with respect to ICRS right ascension
** and declination axes and E_ep is the same vector with respect to
** the (inertial) ecliptic and equinox of epoch epj.
**
** 2) P_ICRS is a free vector, merely a direction, typically of unit
** magnitude, and not bound to any particular spatial origin, such
** as the Earth, Sun or SSB. No assumptions are made about whether
** it represents starlight and embodies astrometric effects such as
** parallax or aberration. The transformation is approximately that
** between mean J2000.0 right ascension and declination and ecliptic
** longitude and latitude, with only frame bias (always less than
** 25 mas) to disturb this classical picture.
**
** 3) The Vondrak et al. (2011, 2012) 400 millennia precession model
** agrees with the IAU 2006 precession at J2000.0 and stays within
** 100 microarcseconds during the 20th and 21st centuries. It is
** accurate to a few arcseconds throughout the historical period,
** worsening to a few tenths of a degree at the end of the
** +/- 200,000 year time span.
**
** Called:
** eraLtpequ equator pole, long term
** eraLtpecl ecliptic pole, long term
** eraPxp vector product
** eraPn normalize vector
**
** References:
**
** Vondrak, J., Capitaine, N. and Wallace, P., 2011, New precession
** expressions, valid for long time intervals, Astron.Astrophys. 534,
** A22
**
** Vondrak, J., Capitaine, N. and Wallace, P., 2012, New precession
** expressions, valid for long time intervals (Corrigendum),
** Astron.Astrophys. 541, C1
**
** Copyright (C) 2013-2017, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
/* Frame bias (IERS Conventions 2010, Eqs. 5.21 and 5.33) */
const double dx = -0.016617 * ERFA_DAS2R,
de = -0.0068192 * ERFA_DAS2R,
dr = -0.0146 * ERFA_DAS2R;
double p[3], z[3], w[3], s, x[3], y[3];
/* Equator pole. */
eraLtpequ(epj, p);
/* Ecliptic pole (bottom row of equatorial to ecliptic matrix). */
eraLtpecl(epj, z);
/* Equinox (top row of matrix). */
eraPxp(p, z, w);
eraPn(w, &s, x);
/* Middle row of matrix. */
eraPxp(z, x, y);
/* Combine with frame bias. */
rm[0][0] = x[0] - x[1]*dr + x[2]*dx;
rm[0][1] = x[0]*dr + x[1] + x[2]*de;
rm[0][2] = - x[0]*dx - x[1]*de + x[2];
rm[1][0] = y[0] - y[1]*dr + y[2]*dx;
rm[1][1] = y[0]*dr + y[1] + y[2]*de;
rm[1][2] = - y[0]*dx - y[1]*de + y[2];
rm[2][0] = z[0] - z[1]*dr + z[2]*dx;
rm[2][1] = z[0]*dr + z[1] + z[2]*de;
rm[2][2] = - z[0]*dx - z[1]*de + z[2];
}
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;
}
/* Note that the arguments are flipped */
void palDvn ( double v[3], double uv[3], double *vm ) {
eraPn( v, vm, uv );
}
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. */
}
