void eraPvxpv(double a[2][3], double b[2][3], double axb[2][3])
/*
** - - - - - - - - -
** e r a P v x p v
** - - - - - - - - -
**
** Outer (=vector=cross) product of two pv-vectors.
**
** Given:
** a double[2][3] first pv-vector
** b double[2][3] second pv-vector
**
** Returned:
** axb double[2][3] a x b
**
** Notes:
**
** 1) If the position and velocity components of the two pv-vectors are
** ( ap, av ) and ( bp, bv ), the result, a x b, is the pair of
** vectors ( ap x bp, ap x bv + av x bp ). The two vectors are the
** cross-product of the two p-vectors and its derivative.
**
** 2) It is permissible to re-use the same array for any of the
** arguments.
**
** Called:
** eraCpv copy pv-vector
** eraPxp vector product of two p-vectors
** eraPpp p-vector plus p-vector
**
** Copyright (C) 2013-2016, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double wa[2][3], wb[2][3], axbd[3], adxb[3];
/* Make copies of the inputs. */
eraCpv(a, wa);
eraCpv(b, wb);
/* a x b = position part of result. */
eraPxp(wa[0], wb[0], axb[0]);
/* a x bdot + adot x b = velocity part of result. */
eraPxp(wa[0], wb[1], axbd);
eraPxp(wa[1], wb[0], adxb);
eraPpp(axbd, adxb, axb[1]);
return;
}
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;
}
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;
}
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 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 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. */
}
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];
}
void palDvxv ( double va[3], double vb[3], double vc[3] ) {
eraPxp( va, vb, vc );
}
