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sjy01-image-proc/vendor/github.com/hebl/gofa/astrometry.go
nuknal 1161e8d054 fixed dependencies
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// Copyright 2022 HE Boliang
// All rights reserved.
package gofa
// SOFA Astrometry Tools
/*
Ab Apply stellar aberration
Apply aberration to transform natural direction into proper
direction.
Given:
pnat [3]float64 natural direction to the source (unit vector)
v [3]float64 observer barycentric velocity in units of c
s float64 distance between the Sun and the observer (au)
bm1 float64 sqrt(1-|v|^2): reciprocal of Lorenz factor
Returned:
ppr [3]float64 proper direction to source (unit vector)
Notes:
1. The algorithm is based on Expr. (7.40) in the Explanatory
Supplement (Urban & Seidelmann 2013), but with the following
changes:
o Rigorous rather than approximate normalization is applied.
o The gravitational potential term from Expr. (7) in
Klioner (2003) is added, taking into account only the Sun's
contribution. This has a maximum effect of about
0.4 microarcsecond.
2. In almost all cases, the maximum accuracy will be limited by the
supplied velocity. For example, if the SOFA Epv00 function is
used, errors of up to 5 microarcseconds could occur.
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:
Pdp scalar product of two p-vectors
*/
func Ab(pnat, v [3]float64, s, bm1 float64, ppr *[3]float64) {
var i int
var pdv, w1, w2, r2, w, r float64
var p [3]float64
pdv = Pdp(pnat, v)
w1 = 1.0 + pdv/(1.0+bm1)
w2 = SRS / s
r2 = 0.0
for i = 0; i < 3; i++ {
w = pnat[i]*bm1 + w1*v[i] + w2*(v[i]-pdv*pnat[i])
p[i] = w
r2 = r2 + w*w
}
r = sqrt(r2)
for i = 0; i < 3; i++ {
ppr[i] = p[i] / r
}
}
/*
Apcg Prepare for ICRS <-> GCRS, geocentric, special
For a geocentric observer, prepare star-independent astrometry
parameters for transformations between ICRS and GCRS coordinates.
The Earth ephemeris is supplied by the caller.
The parameters produced by this function are required in the
parallax, light deflection and aberration parts of the astrometric
transformation chain.
Given:
date1 float64 TDB as a 2-part...
date2 float64 ...Julian Date (Note 1)
ebpv [2][3]float64 Earth barycentric pos/vel (au, au/day)
ehp [3]float64 Earth heliocentric position (au)
Returned:
astrom ASTROM star-independent astrometry parameters:
pmt float64 PM time interval (SSB, Julian years)
eb [3]float64 SSB to observer (vector, au)
eh [3]float64 Sun to observer (unit vector)
em float64 distance from Sun to observer (au)
v [3]float64 barycentric observer velocity (vector, c)
bm1 float64 sqrt(1-|v|^2): reciprocal of Lorenz factor
bpn [3][3]float64 bias-precession-nutation matrix
along float64 unchanged
xpl float64 unchanged
ypl float64 unchanged
sphi float64 unchanged
cphi float64 unchanged
diurab float64 unchanged
eral float64 unchanged
refa float64 unchanged
refb float64 unchanged
Notes:
1. The TDB date date1+date2 is a Julian Date, apportioned in any
convenient way between the two arguments. For example,
JD(TDB)=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. For most
applications of this function the choice will not be at all
critical.
TT can be used instead of TDB without any significant impact on
accuracy.
2. All the vectors are with respect to BCRS axes.
3. This is one of several functions that inserts into the astrom
structure star-independent parameters needed for the chain of
astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed.
The various functions support different classes of observer and
portions of the transformation chain:
functions observer transformation
Apcg Apcg13 geocentric ICRS <-> GCRS
Apci Apci13 terrestrial ICRS <-> CIRS
Apco Apco13 terrestrial ICRS <-> observed
Apcs Apcs13 space ICRS <-> GCRS
Aper Aper13 terrestrial update Earth rotation
Apio Apio13 terrestrial CIRS <-> observed
Those with names ending in "13" use contemporary SOFA models to
compute the various ephemerides. The others accept ephemerides
supplied by the caller.
The transformation from ICRS to GCRS covers space motion,
parallax, light deflection, and aberration. From GCRS to CIRS
comprises frame bias and precession-nutation. From CIRS to
observed takes account of Earth rotation, polar motion, diurnal
aberration and parallax (unless subsumed into the ICRS <-> GCRS
transformation), and atmospheric refraction.
4. The context structure astrom produced by this function is used by
Atciq* and Aticq*.
Called:
Apcs astrometry parameters, ICRS-GCRS, space observer
*/
func Apcg(date1, date2 float64, ebpv [2][3]float64, ehp [3]float64, astrom *ASTROM) {
/* Geocentric observer */
pv := [2][3]float64{
{0.0, 0.0, 0.0},
{0.0, 0.0, 0.0},
}
/* Compute the star-independent astrometry parameters. */
Apcs(date1, date2, pv, ebpv, ehp, astrom)
}
/*
Apcg13 Prepare for ICRS <-> GCRS, geocentric
For a geocentric observer, prepare star-independent astrometry
parameters for transformations between ICRS and GCRS coordinates.
The caller supplies the date, and SOFA models are used to predict
the Earth ephemeris.
The parameters produced by this function are required in the
parallax, light deflection and aberration parts of the astrometric
transformation chain.
Given:
date1 float64 TDB as a 2-part...
date2 float64 ...Julian Date (Note 1)
Returned:
astrom ASTROM star-independent astrometry parameters:
pmt float64 PM time interval (SSB, Julian years)
eb [3]float64 SSB to observer (vector, au)
eh [3]float64 Sun to observer (unit vector)
em float64 distance from Sun to observer (au)
v [3]float64 barycentric observer velocity (vector, c)
bm1 float64 sqrt(1-|v|^2): reciprocal of Lorenz factor
bpn [3][3]float64 bias-precession-nutation matrix
along float64 unchanged
xpl float64 unchanged
ypl float64 unchanged
sphi float64 unchanged
cphi float64 unchanged
diurab float64 unchanged
eral float64 unchanged
refa float64 unchanged
refb float64 unchanged
Notes:
1. The TDB date date1+date2 is a Julian Date, apportioned in any
convenient way between the two arguments. For example,
JD(TDB)=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. For most
applications of this function the choice will not be at all
critical.
TT can be used instead of TDB without any significant impact on
accuracy.
2. All the vectors are with respect to BCRS axes.
3. In cases where the caller wishes to supply his own Earth
ephemeris, the function Apcg can be used instead of the present
function.
4. This is one of several functions that inserts into the astrom
structure star-independent parameters needed for the chain of
astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed.
The various functions support different classes of observer and
portions of the transformation chain:
functions observer transformation
Apcg Apcg13 geocentric ICRS <-> GCRS
Apci Apci13 terrestrial ICRS <-> CIRS
Apco Apco13 terrestrial ICRS <-> observed
Apcs Apcs13 space ICRS <-> GCRS
Aper Aper13 terrestrial update Earth rotation
Apio Apio13 terrestrial CIRS <-> observed
Those with names ending in "13" use contemporary SOFA models to
compute the various ephemerides. The others accept ephemerides
supplied by the caller.
The transformation from ICRS to GCRS covers space motion,
parallax, light deflection, and aberration. From GCRS to CIRS
comprises frame bias and precession-nutation. From CIRS to
observed takes account of Earth rotation, polar motion, diurnal
aberration and parallax (unless subsumed into the ICRS <-> GCRS
transformation), and atmospheric refraction.
5. The context structure astrom produced by this function is used by
Atciq* and Aticq*.
Called:
Epv00 Earth position and velocity
Apcg astrometry parameters, ICRS-GCRS, geocenter
*/
func Apcg13(date1, date2 float64, astrom *ASTROM) {
var ehpv, ebpv [2][3]float64
/* Earth barycentric & heliocentric position/velocity (au, au/d). */
Epv00(date1, date2, &ehpv, &ebpv)
/* Compute the star-independent astrometry parameters. */
Apcg(date1, date2, ebpv, ehpv[0], astrom)
}
/*
Apci Prepare for ICRS <-> CIRS, terrestrial, special
For a terrestrial observer, prepare star-independent astrometry
parameters for transformations between ICRS and geocentric CIRS
coordinates. The Earth ephemeris and CIP/CIO are supplied by the
caller.
The parameters produced by this function are required in the
parallax, light deflection, aberration, and bias-precession-nutation
parts of the astrometric transformation chain.
Given:
date1 float64 TDB as a 2-part...
date2 float64 ...Julian Date (Note 1)
ebpv [2][3]float64 Earth barycentric position/velocity (au, au/day)
ehp [3]float64 Earth heliocentric position (au)
x,y float64 CIP X,Y (components of unit vector)
s float64 the CIO locator s (radians)
Returned:
astrom ASTROM star-independent astrometry parameters:
pmt float64 PM time interval (SSB, Julian years)
eb [3]float64 SSB to observer (vector, au)
eh [3]float64 Sun to observer (unit vector)
em float64 distance from Sun to observer (au)
v [3]float64 barycentric observer velocity (vector, c)
bm1 float64 sqrt(1-|v|^2): reciprocal of Lorenz factor
bpn [3][3]float64 bias-precession-nutation matrix
along float64 unchanged
xpl float64 unchanged
ypl float64 unchanged
sphi float64 unchanged
cphi float64 unchanged
diurab float64 unchanged
eral float64 unchanged
refa float64 unchanged
refb float64 unchanged
Notes:
1. The TDB date date1+date2 is a Julian Date, apportioned in any
convenient way between the two arguments. For example,
JD(TDB)=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. For most
applications of this function the choice will not be at all
critical.
TT can be used instead of TDB without any significant impact on
accuracy.
2. All the vectors are with respect to BCRS axes.
3. In cases where the caller does not wish to provide the Earth
ephemeris and CIP/CIO, the function Apci13 can be used instead
of the present function. This computes the required quantities
using other SOFA functions.
4. This is one of several functions that inserts into the astrom
structure star-independent parameters needed for the chain of
astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed.
The various functions support different classes of observer and
portions of the transformation chain:
functions observer transformation
Apcg Apcg13 geocentric ICRS <-> GCRS
Apci Apci13 terrestrial ICRS <-> CIRS
Apco Apco13 terrestrial ICRS <-> observed
Apcs Apcs13 space ICRS <-> GCRS
Aper Aper13 terrestrial update Earth rotation
Apio Apio13 terrestrial CIRS <-> observed
Those with names ending in "13" use contemporary SOFA models to
compute the various ephemerides. The others accept ephemerides
supplied by the caller.
The transformation from ICRS to GCRS covers space motion,
parallax, light deflection, and aberration. From GCRS to CIRS
comprises frame bias and precession-nutation. From CIRS to
observed takes account of Earth rotation, polar motion, diurnal
aberration and parallax (unless subsumed into the ICRS <-> GCRS
transformation), and atmospheric refraction.
5. The context structure astrom produced by this function is used by
Atciq* and Aticq*.
Called:
Apcg astrometry parameters, ICRS-GCRS, geocenter
C2ixys celestial-to-intermediate matrix, given X,Y and s
*/
func Apci(date1, date2 float64, ebpv [2][3]float64, ehp [3]float64, x, y, s float64, astrom *ASTROM) {
/* Star-independent astrometry parameters for geocenter. */
Apcg(date1, date2, ebpv, ehp, astrom)
/* CIO based BPN matrix. */
C2ixys(x, y, s, &astrom.Bpn)
}
/*
Apci13 Prepare for ICRS <-> CIRS, terrestrial
For a terrestrial observer, prepare star-independent astrometry
parameters for transformations between ICRS and geocentric CIRS
coordinates. The caller supplies the date, and SOFA models are used
to predict the Earth ephemeris and CIP/CIO.
The parameters produced by this function are required in the
parallax, light deflection, aberration, and bias-precession-nutation
parts of the astrometric transformation chain.
Given:
date1 float64 TDB as a 2-part...
date2 float64 ...Julian Date (Note 1)
Returned:
astrom ASTROM star-independent astrometry parameters:
pmt float64 PM time interval (SSB, Julian years)
eb [3]float64 SSB to observer (vector, au)
eh [3]float64 Sun to observer (unit vector)
em float64 distance from Sun to observer (au)
v [3]float64 barycentric observer velocity (vector, c)
bm1 float64 sqrt(1-|v|^2): reciprocal of Lorenz factor
bpn [3][3]float64 bias-precession-nutation matrix
along float64 unchanged
xpl float64 unchanged
ypl float64 unchanged
sphi float64 unchanged
cphi float64 unchanged
diurab float64 unchanged
eral float64 unchanged
refa float64 unchanged
refb float64 unchanged
eo float64 equation of the origins (ERA-GST)
Notes:
1. The TDB date date1+date2 is a Julian Date, apportioned in any
convenient way between the two arguments. For example,
JD(TDB)=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. For most
applications of this function the choice will not be at all
critical.
TT can be used instead of TDB without any significant impact on
accuracy.
2. All the vectors are with respect to BCRS axes.
3. In cases where the caller wishes to supply his own Earth
ephemeris and CIP/CIO, the function Apci can be used instead
of the present function.
4. This is one of several functions that inserts into the astrom
structure star-independent parameters needed for the chain of
astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed.
The various functions support different classes of observer and
portions of the transformation chain:
functions observer transformation
Apcg Apcg13 geocentric ICRS <-> GCRS
Apci Apci13 terrestrial ICRS <-> CIRS
Apco Apco13 terrestrial ICRS <-> observed
Apcs Apcs13 space ICRS <-> GCRS
Aper Aper13 terrestrial update Earth rotation
Apio Apio13 terrestrial CIRS <-> observed
Those with names ending in "13" use contemporary SOFA models to
compute the various ephemerides. The others accept ephemerides
supplied by the caller.
The transformation from ICRS to GCRS covers space motion,
parallax, light deflection, and aberration. From GCRS to CIRS
comprises frame bias and precession-nutation. From CIRS to
observed takes account of Earth rotation, polar motion, diurnal
aberration and parallax (unless subsumed into the ICRS <-> GCRS
transformation), and atmospheric refraction.
5. The context structure astrom produced by this function is used by
Atciq* and Aticq*.
Called:
Epv00 Earth position and velocity
Pnm06a classical NPB matrix, IAU 2006/2000A
Bpn2xy extract CIP X,Y coordinates from NPB matrix
S06 the CIO locator s, given X,Y, IAU 2006
Apci astrometry parameters, ICRS-CIRS
Eors equation of the origins, given NPB matrix and s
*/
func Apci13(date1, date2 float64, astrom *ASTROM, eo *float64) {
var ehpv, ebpv [2][3]float64
var r [3][3]float64
var x, y, s float64
/* Earth barycentric & heliocentric position/velocity (au, au/d). */
Epv00(date1, date2, &ehpv, &ebpv)
/* Form the equinox based BPN matrix, IAU 2006/2000A. */
Pnm06a(date1, date2, &r)
/* Extract CIP X,Y. */
Bpn2xy(r, &x, &y)
/* Obtain CIO locator s. */
s = S06(date1, date2, x, y)
/* Compute the star-independent astrometry parameters. */
Apci(date1, date2, ebpv, ehpv[0], x, y, s, astrom)
/* Equation of the origins. */
*eo = Eors(r, s)
}
/*
Apco Prepare for ICRS <-> observed, terrestrial, special
For a terrestrial observer, prepare star-independent astrometry
parameters for transformations between ICRS and observed
coordinates. The caller supplies the Earth ephemeris, the Earth
rotation information and the refraction constants as well as the
site coordinates.
Given:
date1 float64 TDB as a 2-part...
date2 float64 ...Julian Date (Note 1)
ebpv [2][3]float64 Earth barycentric PV (au, au/day, Note 2)
ehp [3]float64 Earth heliocentric P (au, Note 2)
x,y float64 CIP X,Y (components of unit vector)
s float64 the CIO locator s (radians)
theta float64 Earth rotation angle (radians)
elong float64 longitude (radians, east +ve, Note 3)
phi float64 latitude (geodetic, radians, Note 3)
hm float64 height above ellipsoid (m, geodetic, Note 3)
xp,yp float64 polar motion coordinates (radians, Note 4)
sp float64 the TIO locator s' (radians, Note 4)
refa float64 refraction constant A (radians, Note 5)
refb float64 refraction constant B (radians, Note 5)
Returned:
astrom ASTROM star-independent astrometry parameters:
pmt float64 PM time interval (SSB, Julian years)
eb [3]float64 SSB to observer (vector, au)
eh [3]float64 Sun to observer (unit vector)
em float64 distance from Sun to observer (au)
v [3]float64 barycentric observer velocity (vector, c)
bm1 float64 sqrt(1-|v|^2): reciprocal of Lorenz factor
bpn [3][3]float64 bias-precession-nutation matrix
along float64 adjusted longitude (radians)
xpl float64 polar motion xp wrt local meridian (radians)
ypl float64 polar motion yp wrt local meridian (radians)
sphi float64 sine of geodetic latitude
cphi float64 cosine of geodetic latitude
diurab float64 magnitude of diurnal aberration vector
eral float64 "local" Earth rotation angle (radians)
refa float64 refraction constant A (radians)
refb float64 refraction constant B (radians)
Notes:
1. The TDB date date1+date2 is a Julian Date, apportioned in any
convenient way between the two arguments. For example,
JD(TDB)=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. For most
applications of this function the choice will not be at all
critical.
TT can be used instead of TDB without any significant impact on
accuracy.
2. The vectors eb, eh, and all the astrom vectors, are with respect
to BCRS axes.
3. The geographical coordinates are with respect to the WGS84
reference ellipsoid. TAKE CARE WITH THE LONGITUDE SIGN
CONVENTION: the longitude required by the present function is
right-handed, i.e. east-positive, in accordance with geographical
convention.
The adjusted longitude stored in the astrom array takes into
account the TIO locator and polar motion.
4. xp and yp are the coordinates (in radians) of the Celestial
Intermediate Pole with respect to the International Terrestrial
Reference System (see IERS Conventions), measured along the
meridians 0 and 90 deg west respectively. sp is the TIO locator
s', in radians, which positions the Terrestrial Intermediate
Origin on the equator. For many applications, xp, yp and
(especially) sp can be set to zero.
Internally, the polar motion is stored in a form rotated onto the
local meridian.
5. The refraction constants refa and refb are for use in a
dZ = A*tan(Z)+B*tan^3(Z) model, where Z is the observed
(i.e. refracted) zenith distance and dZ is the amount of
refraction.
6. It is advisable to take great care with units, as even unlikely
values of the input parameters are accepted and processed in
accordance with the models used.
7. In cases where the caller does not wish to provide the Earth
Ephemeris, the Earth rotation information and refraction
constants, the function Apco13 can be used instead of the
present function. This starts from UTC and weather readings etc.
and computes suitable values using other SOFA functions.
8. This is one of several functions that inserts into the astrom
structure star-independent parameters needed for the chain of
astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed.
The various functions support different classes of observer and
portions of the transformation chain:
functions observer transformation
Apcg Apcg13 geocentric ICRS <-> GCRS
Apci Apci13 terrestrial ICRS <-> CIRS
Apco Apco13 terrestrial ICRS <-> observed
Apcs Apcs13 space ICRS <-> GCRS
Aper Aper13 terrestrial update Earth rotation
Apio Apio13 terrestrial CIRS <-> observed
Those with names ending in "13" use contemporary SOFA models to
compute the various ephemerides. The others accept ephemerides
supplied by the caller.
The transformation from ICRS to GCRS covers space motion,
parallax, light deflection, and aberration. From GCRS to CIRS
comprises frame bias and precession-nutation. From CIRS to
observed takes account of Earth rotation, polar motion, diurnal
aberration and parallax (unless subsumed into the ICRS <-> GCRS
transformation), and atmospheric refraction.
9. The context structure astrom produced by this function is used by
Atioq, Atoiq, Atciq* and Aticq*.
Called:
Ir initialize r-matrix to identity
Rz rotate around Z-axis
Ry rotate around Y-axis
Rx rotate around X-axis
Anpm normalize angle into range +/- pi
C2ixys celestial-to-intermediate matrix, given X,Y and s
Pvtob position/velocity of terrestrial station
Trxpv product of transpose of r-matrix and pv-vector
Apcs astrometry parameters, ICRS-GCRS, space observer
Cr copy r-matrix
*/
func Apco(date1, date2 float64, ebpv [2][3]float64, ehp [3]float64, x, y, s float64, theta,
elong, phi float64, hm, xp, yp, sp float64, refa, refb float64, astrom *ASTROM) {
var r [3][3]float64
var a, b, eral, c float64
var pvc, pv [2][3]float64
/* Form the rotation matrix, CIRS to apparent [HA,Dec]. */
Ir(&r)
Rz(theta+sp, &r)
Ry(-xp, &r)
Rx(-yp, &r)
Rz(elong, &r)
/* Solve for local Earth rotation angle. */
a = r[0][0]
b = r[0][1]
// eral = ( a != 0.0 || b != 0.0 ) ? atan2(b, a) : 0.0;
if a != 0.0 || b != 0.0 {
eral = atan2(b, a)
} else {
eral = 0.0
}
astrom.Eral = eral
/* Solve for polar motion [X,Y] with respect to local meridian. */
a = r[0][0]
c = r[0][2]
astrom.Xpl = atan2(c, sqrt(a*a+b*b))
a = r[1][2]
b = r[2][2]
// astrom.Ypl = ( a != 0.0 || b != 0.0 ) ? -atan2(a, b) : 0.0;
if a != 0.0 || b != 0.0 {
astrom.Ypl = -atan2(a, b)
} else {
astrom.Ypl = 0.0
}
/* Adjusted longitude. */
astrom.Along = Anpm(eral - theta)
/* Functions of latitude. */
astrom.Sphi = sin(phi)
astrom.Cphi = cos(phi)
/* Refraction constants. */
astrom.Refa = refa
astrom.Refb = refb
/* Disable the (redundant) diurnal aberration step. */
astrom.Diurab = 0.0
/* CIO based BPN matrix. */
C2ixys(x, y, s, &r)
/* Observer's geocentric position and velocity (m, m/s, CIRS). */
Pvtob(elong, phi, hm, xp, yp, sp, theta, &pvc)
/* Rotate into GCRS. */
Trxpv(r, pvc, &pv)
/* ICRS <-> GCRS parameters. */
Apcs(date1, date2, pv, ebpv, ehp, astrom)
/* Store the CIO based BPN matrix. */
Cr(r, &astrom.Bpn)
}
/*
Apco13 Prepare for ICRS <-> observed, terrestrial
For a terrestrial observer, prepare star-independent astrometry
parameters for transformations between ICRS and observed
coordinates. The caller supplies UTC, site coordinates, ambient air
conditions and observing wavelength, and SOFA models are used to
obtain the Earth ephemeris, CIP/CIO and refraction constants.
The parameters produced by this function are required in the
parallax, light deflection, aberration, and bias-precession-nutation
parts of the ICRS/CIRS transformations.
Given:
utc1 float64 UTC as a 2-part...
utc2 float64 ...quasi Julian Date (Notes 1,2)
dut1 float64 UT1-UTC (seconds, Note 3)
elong float64 longitude (radians, east +ve, Note 4)
phi float64 latitude (geodetic, radians, Note 4)
hm float64 height above ellipsoid (m, geodetic, Notes 4,6)
xp,yp float64 polar motion coordinates (radians, Note 5)
phpa float64 pressure at the observer (hPa = mB, Note 6)
tc float64 ambient temperature at the observer (deg C)
rh float64 relative humidity at the observer (range 0-1)
wl float64 wavelength (micrometers, Note 7)
Returned:
astrom ASTROM star-independent astrometry parameters:
pmt float64 PM time interval (SSB, Julian years)
eb [3]float64 SSB to observer (vector, au)
eh [3]float64 Sun to observer (unit vector)
em float64 distance from Sun to observer (au)
v [3]float64 barycentric observer velocity (vector, c)
bm1 float64 sqrt(1-|v|^2): reciprocal of Lorenz factor
bpn [3][3]float64 bias-precession-nutation matrix
along float64 adjusted longitude (radians)
xpl float64 polar motion xp wrt local meridian (radians)
ypl float64 polar motion yp wrt local meridian (radians)
sphi float64 sine of geodetic latitude
cphi float64 cosine of geodetic latitude
diurab float64 magnitude of diurnal aberration vector
eral float64 "local" Earth rotation angle (radians)
refa float64 refraction constant A (radians)
refb float64 refraction constant B (radians)
eo float64 equation of the origins (ERA-GST)
Returned (function value):
int status: +1 = dubious year (Note 2)
0 = OK
-1 = unacceptable date
Notes:
1. utc1+utc2 is quasi Julian Date (see Note 2), apportioned in any
convenient way between the two arguments, for example where utc1
is the Julian Day Number and utc2 is the fraction of a day.
However, JD cannot unambiguously represent UTC during a leap
second unless special measures are taken. The convention in the
present function is that the JD day represents UTC days whether
the length is 86399, 86400 or 86401 SI seconds.
Applications should use the function Dtf2d to convert from
calendar date and time of day into 2-part quasi Julian Date, as
it implements the leap-second-ambiguity convention just
described.
2. The warning status "dubious year" flags UTCs that predate the
introduction of the time scale or that are too far in the
future to be trusted. See Dat for further details.
3. UT1-UTC is tabulated in IERS bulletins. It increases by exactly
one second at the end of each positive UTC leap second,
introduced in order to keep UT1-UTC within +/- 0.9s. n.b. This
practice is under review, and in the future UT1-UTC may grow
essentially without limit.
4. The geographical coordinates are with respect to the WGS84
reference ellipsoid. TAKE CARE WITH THE LONGITUDE SIGN: the
longitude required by the present function is east-positive
(i.e. right-handed), in accordance with geographical convention.
5. The polar motion xp,yp can be obtained from IERS bulletins. The
values are the coordinates (in radians) of the Celestial
Intermediate Pole with respect to the International Terrestrial
Reference System (see IERS Conventions 2003), measured along the
meridians 0 and 90 deg west respectively. For many
applications, xp and yp can be set to zero.
Internally, the polar motion is stored in a form rotated onto
the local meridian.
6. If hm, the height above the ellipsoid of the observing station
in meters, is not known but phpa, the pressure in hPa (=mB), is
available, an adequate estimate of hm can be obtained from the
expression
hm = -29.3 * tsl * log ( phpa / 1013.25 );
where tsl is the approximate sea-level air temperature in K
(See Astrophysical Quantities, C.W.Allen, 3rd edition, section
52). Similarly, if the pressure phpa is not known, it can be
estimated from the height of the observing station, hm, as
follows:
phpa = 1013.25 * exp ( -hm / ( 29.3 * tsl ) );
Note, however, that the refraction is nearly proportional to
the pressure and that an accurate phpa value is important for
precise work.
7. The argument wl specifies the observing wavelength in
micrometers. The transition from optical to radio is assumed to
occur at 100 micrometers (about 3000 GHz).
8. It is advisable to take great care with units, as even unlikely
values of the input parameters are accepted and processed in
accordance with the models used.
9. In cases where the caller wishes to supply his own Earth
ephemeris, Earth rotation information and refraction constants,
the function Apco can be used instead of the present function.
10)This is one of several functions that inserts into the astrom
structure star-independent parameters needed for the chain of
astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed.
The various functions support different classes of observer and
portions of the transformation chain:
functions observer transformation
Apcg Apcg13 geocentric ICRS <-> GCRS
Apci Apci13 terrestrial ICRS <-> CIRS
Apco Apco13 terrestrial ICRS <-> observed
Apcs Apcs13 space ICRS <-> GCRS
Aper Aper13 terrestrial update Earth rotation
Apio Apio13 terrestrial CIRS <-> observed
Those with names ending in "13" use contemporary SOFA models to
compute the various ephemerides. The others accept ephemerides
supplied by the caller.
The transformation from ICRS to GCRS covers space motion,
parallax, light deflection, and aberration. From GCRS to CIRS
comprises frame bias and precession-nutation. From CIRS to
observed takes account of Earth rotation, polar motion, diurnal
aberration and parallax (unless subsumed into the ICRS <-> GCRS
transformation), and atmospheric refraction.
11)The context structure astrom produced by this function is used
by Atioq, Atoiq, Atciq* and Aticq*.
Called:
Utctai UTC to TAI
Taitt TAI to TT
Utcut1 UTC to UT1
Epv00 Earth position and velocity
Pnm06a classical NPB matrix, IAU 2006/2000A
Bpn2xy extract CIP X,Y coordinates from NPB matrix
S06 the CIO locator s, given X,Y, IAU 2006
Era00 Earth rotation angle, IAU 2000
Sp00 the TIO locator s', IERS 2000
Refco refraction constants for given ambient conditions
Apco astrometry parameters, ICRS-observed
Eors equation of the origins, given NPB matrix and s
*/
func Apco13(utc1, utc2, dut1 float64, elong, phi, hm, xp, yp float64, phpa, tc, rh, wl float64, astrom *ASTROM, eo *float64) int {
var j int
var tai1, tai2, tt1, tt2, ut11, ut12 float64
var ehpv, ebpv [2][3]float64
var r [3][3]float64
var x, y, s, theta, sp, refa, refb float64
/* UTC to other time scales. */
j = Utctai(utc1, utc2, &tai1, &tai2)
if j < 0 {
return -1
}
Taitt(tai1, tai2, &tt1, &tt2)
j = Utcut1(utc1, utc2, dut1, &ut11, &ut12)
if j < 0 {
return -1
}
/* Earth barycentric & heliocentric position/velocity (au, au/d). */
Epv00(tt1, tt2, &ehpv, &ebpv)
/* Form the equinox based BPN matrix, IAU 2006/2000A. */
Pnm06a(tt1, tt2, &r)
/* Extract CIP X,Y. */
Bpn2xy(r, &x, &y)
/* Obtain CIO locator s. */
s = S06(tt1, tt2, x, y)
/* Earth rotation angle. */
theta = Era00(ut11, ut12)
/* TIO locator s'. */
sp = Sp00(tt1, tt2)
/* Refraction constants A and B. */
Refco(phpa, tc, rh, wl, &refa, &refb)
/* Compute the star-independent astrometry parameters. */
Apco(tt1, tt2, ebpv, ehpv[0], x, y, s, theta,
elong, phi, hm, xp, yp, sp, refa, refb, astrom)
/* Equation of the origins. */
*eo = Eors(r, s)
/* Return any warning status. */
return j
}
/*
Apcs Prepare for ICRS <-> CIRS, space, special
For an observer whose geocentric position and velocity are known,
prepare star-independent astrometry parameters for transformations
between ICRS and GCRS. The Earth ephemeris is supplied by the
caller.
The parameters produced by this function are required in the space
motion, parallax, light deflection and aberration parts of the
astrometric transformation chain.
Given:
date1 float64 TDB as a 2-part...
date2 float64 ...Julian Date (Note 1)
pv [2][3]float64 observer's geocentric pos/vel (m, m/s)
ebpv [2][3]float64 Earth barycentric PV (au, au/day)
ehp [3]float64 Earth heliocentric P (au)
Returned:
astrom ASTROM star-independent astrometry parameters:
pmt float64 PM time interval (SSB, Julian years)
eb [3]float64 SSB to observer (vector, au)
eh [3]float64 Sun to observer (unit vector)
em float64 distance from Sun to observer (au)
v [3]float64 barycentric observer velocity (vector, c)
bm1 float64 sqrt(1-|v|^2): reciprocal of Lorenz factor
bpn [3][3]float64 bias-precession-nutation matrix
along float64 unchanged
xpl float64 unchanged
ypl float64 unchanged
sphi float64 unchanged
cphi float64 unchanged
diurab float64 unchanged
eral float64 unchanged
refa float64 unchanged
refb float64 unchanged
Notes:
1. The TDB date date1+date2 is a Julian Date, apportioned in any
convenient way between the two arguments. For example,
JD(TDB)=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. For most
applications of this function the choice will not be at all
critical.
TT can be used instead of TDB without any significant impact on
accuracy.
2. All the vectors are with respect to BCRS axes.
3. Providing separate arguments for (i) the observer's geocentric
position and velocity and (ii) the Earth ephemeris is done for
convenience in the geocentric, terrestrial and Earth orbit cases.
For deep space applications it maybe more convenient to specify
zero geocentric position and velocity and to supply the
observer's position and velocity information directly instead of
with respect to the Earth. However, note the different units:
m and m/s for the geocentric vectors, au and au/day for the
heliocentric and barycentric vectors.
4. In cases where the caller does not wish to provide the Earth
ephemeris, the function Apcs13 can be used instead of the
present function. This computes the Earth ephemeris using the
SOFA function Epv00.
5. This is one of several functions that inserts into the astrom
structure star-independent parameters needed for the chain of
astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed.
The various functions support different classes of observer and
portions of the transformation chain:
functions observer transformation
Apcg Apcg13 geocentric ICRS <-> GCRS
Apci Apci13 terrestrial ICRS <-> CIRS
Apco Apco13 terrestrial ICRS <-> observed
Apcs Apcs13 space ICRS <-> GCRS
Aper Aper13 terrestrial update Earth rotation
Apio Apio13 terrestrial CIRS <-> observed
Those with names ending in "13" use contemporary SOFA models to
compute the various ephemerides. The others accept ephemerides
supplied by the caller.
The transformation from ICRS to GCRS covers space motion,
parallax, light deflection, and aberration. From GCRS to CIRS
comprises frame bias and precession-nutation. From CIRS to
observed takes account of Earth rotation, polar motion, diurnal
aberration and parallax (unless subsumed into the ICRS <-> GCRS
transformation), and atmospheric refraction.
6. The context structure astrom produced by this function is used by
Atciq* and Aticq*.
Called:
Cp copy p-vector
Pm modulus of p-vector
Pn decompose p-vector into modulus and direction
Ir initialize r-matrix to identity
*/
func Apcs(date1, date2 float64, pv, ebpv [2][3]float64, ehp [3]float64, astrom *ASTROM) {
/* au/d to m/s */
const AUDMS = DAU / DAYSEC
/* Light time for 1 au (day) */
const CR = AULT / DAYSEC
var i int
var dp, dv float64
var pb, vb, ph [3]float64
var v2, w float64
/* Time since reference epoch, years (for proper motion calculation). */
astrom.Pmt = ((date1 - DJ00) + date2) / DJY
/* Adjust Earth ephemeris to observer. */
for i = 0; i < 3; i++ {
dp = pv[0][i] / DAU
dv = pv[1][i] / AUDMS
pb[i] = ebpv[0][i] + dp
vb[i] = ebpv[1][i] + dv
ph[i] = ehp[i] + dp
}
/* Barycentric position of observer (au). */
Cp(pb, &astrom.Eb)
/* Heliocentric direction and distance (unit vector and au). */
Pn(ph, &astrom.Em, &astrom.Eh)
/* Barycentric vel. in units of c, and reciprocal of Lorenz factor. */
v2 = 0.0
for i = 0; i < 3; i++ {
w = vb[i] * CR
astrom.V[i] = w
v2 += w * w
}
astrom.Bm1 = sqrt(1.0 - v2)
/* Reset the NPB matrix. */
Ir(&astrom.Bpn)
}
/*
Apcs13 Prepare for ICRS <-> CIRS, space
For an observer whose geocentric position and velocity are known,
prepare star-independent astrometry parameters for transformations
between ICRS and GCRS. The Earth ephemeris is from SOFA models.
The parameters produced by this function are required in the space
motion, parallax, light deflection and aberration parts of the
astrometric transformation chain.
Given:
date1 float64 TDB as a 2-part...
date2 float64 ...Julian Date (Note 1)
pv [2][3]float64 observer's geocentric pos/vel (Note 3)
Returned:
astrom ASTROM star-independent astrometry parameters:
pmt float64 PM time interval (SSB, Julian years)
eb [3]float64 SSB to observer (vector, au)
eh [3]float64 Sun to observer (unit vector)
em float64 distance from Sun to observer (au)
v [3]float64 barycentric observer velocity (vector, c)
bm1 float64 sqrt(1-|v|^2): reciprocal of Lorenz factor
bpn [3][3]float64 bias-precession-nutation matrix
along float64 unchanged
xpl float64 unchanged
ypl float64 unchanged
sphi float64 unchanged
cphi float64 unchanged
diurab float64 unchanged
eral float64 unchanged
refa float64 unchanged
refb float64 unchanged
Notes:
1. The TDB date date1+date2 is a Julian Date, apportioned in any
convenient way between the two arguments. For example,
JD(TDB)=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. For most
applications of this function the choice will not be at all
critical.
TT can be used instead of TDB without any significant impact on
accuracy.
2. All the vectors are with respect to BCRS axes.
3. The observer's position and velocity pv are geocentric but with
respect to BCRS axes, and in units of m and m/s. No assumptions
are made about proximity to the Earth, and the function can be
used for deep space applications as well as Earth orbit and
terrestrial.
4. In cases where the caller wishes to supply his own Earth
ephemeris, the function Apcs can be used instead of the present
function.
5. This is one of several functions that inserts into the astrom
structure star-independent parameters needed for the chain of
astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed.
The various functions support different classes of observer and
portions of the transformation chain:
functions observer transformation
Apcg Apcg13 geocentric ICRS <-> GCRS
Apci Apci13 terrestrial ICRS <-> CIRS
Apco Apco13 terrestrial ICRS <-> observed
Apcs Apcs13 space ICRS <-> GCRS
Aper Aper13 terrestrial update Earth rotation
Apio Apio13 terrestrial CIRS <-> observed
Those with names ending in "13" use contemporary SOFA models to
compute the various ephemerides. The others accept ephemerides
supplied by the caller.
The transformation from ICRS to GCRS covers space motion,
parallax, light deflection, and aberration. From GCRS to CIRS
comprises frame bias and precession-nutation. From CIRS to
observed takes account of Earth rotation, polar motion, diurnal
aberration and parallax (unless subsumed into the ICRS <-> GCRS
transformation), and atmospheric refraction.
6. The context structure astrom produced by this function is used by
Atciq* and Aticq*.
Called:
Epv00 Earth position and velocity
Apcs astrometry parameters, ICRS-GCRS, space observer
*/
func Apcs13(date1, date2 float64, pv [2][3]float64, astrom *ASTROM) {
var ehpv, ebpv [2][3]float64
/* Earth barycentric & heliocentric position/velocity (au, au/d). */
Epv00(date1, date2, &ehpv, &ebpv)
/* Compute the star-independent astrometry parameters. */
Apcs(date1, date2, pv, ebpv, ehpv[0], astrom)
}
/*
Aper Insert ERA into context
In the star-independent astrometry parameters, update only the
Earth rotation angle, supplied by the caller explicitly.
Given:
theta float64 Earth rotation angle (radians, Note 2)
astrom ASTROM star-independent astrometry parameters:
pmt float64 not used
eb [3]float64 not used
eh [3]float64 not used
em float64 not used
v [3]float64 not used
bm1 float64 not used
bpn [3][3]float64 not used
along float64 longitude + s' (radians)
xpl float64 not used
ypl float64 not used
sphi float64 not used
cphi float64 not used
diurab float64 not used
eral float64 not used
refa float64 not used
refb float64 not used
Returned:
astrom ASTROM star-independent astrometry parameters:
pmt float64 unchanged
eb [3]float64 unchanged
eh [3]float64 unchanged
em float64 unchanged
v [3]float64 unchanged
bm1 float64 unchanged
bpn [3][3]float64 unchanged
along float64 unchanged
xpl float64 unchanged
ypl float64 unchanged
sphi float64 unchanged
cphi float64 unchanged
diurab float64 unchanged
eral float64 "local" Earth rotation angle (radians)
refa float64 unchanged
refb float64 unchanged
Notes:
1. This function exists to enable sidereal-tracking applications to
avoid wasteful recomputation of the bulk of the astrometry
parameters: only the Earth rotation is updated.
2. For targets expressed as equinox based positions, such as
classical geocentric apparent (RA,Dec), the supplied theta can be
Greenwich apparent sidereal time rather than Earth rotation
angle.
3. The function Aper13 can be used instead of the present
function, and starts from UT1 rather than ERA itself.
4. This is one of several functions that inserts into the astrom
structure star-independent parameters needed for the chain of
astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed.
The various functions support different classes of observer and
portions of the transformation chain:
functions observer transformation
Apcg Apcg13 geocentric ICRS <-> GCRS
Apci Apci13 terrestrial ICRS <-> CIRS
Apco Apco13 terrestrial ICRS <-> observed
Apcs Apcs13 space ICRS <-> GCRS
Aper Aper13 terrestrial update Earth rotation
Apio Apio13 terrestrial CIRS <-> observed
Those with names ending in "13" use contemporary SOFA models to
compute the various ephemerides. The others accept ephemerides
supplied by the caller.
The transformation from ICRS to GCRS covers space motion,
parallax, light deflection, and aberration. From GCRS to CIRS
comprises frame bias and precession-nutation. From CIRS to
observed takes account of Earth rotation, polar motion, diurnal
aberration and parallax (unless subsumed into the ICRS <-> GCRS
transformation), and atmospheric refraction.
*/
func Aper(theta float64, astrom *ASTROM) {
astrom.Eral = theta + astrom.Along
}
/*
Aper13 Update context for Earth rotation
In the star-independent astrometry parameters, update only the
Earth rotation angle. The caller provides UT1, (n.b. not UTC).
Given:
ut11 float64 UT1 as a 2-part...
ut12 float64 ...Julian Date (Note 1)
astrom ASTROM star-independent astrometry parameters:
pmt float64 not used
eb [3]float64 not used
eh [3]float64 not used
em float64 not used
v [3]float64 not used
bm1 float64 not used
bpn [3][3]float64 not used
along float64 longitude + s' (radians)
xpl float64 not used
ypl float64 not used
sphi float64 not used
cphi float64 not used
diurab float64 not used
eral float64 not used
refa float64 not used
refb float64 not used
Returned:
astrom ASTROM star-independent astrometry parameters:
pmt float64 unchanged
eb [3]float64 unchanged
eh [3]float64 unchanged
em float64 unchanged
v [3]float64 unchanged
bm1 float64 unchanged
bpn [3][3]float64 unchanged
along float64 unchanged
xpl float64 unchanged
ypl float64 unchanged
sphi float64 unchanged
cphi float64 unchanged
diurab float64 unchanged
eral float64 "local" Earth rotation angle (radians)
refa float64 unchanged
refb float64 unchanged
Notes:
1. The UT1 date (n.b. not UTC) ut11+ut12 is a Julian Date,
apportioned in any convenient way between the arguments ut11 and
ut12. For example, JD(UT1)=2450123.7 could be expressed in any
of these ways, among others:
ut11 ut12
2450123.7 0.0 (JD method)
2451545.0 -1421.3 (J2000 method)
2400000.5 50123.2 (MJD method)
2450123.5 0.2 (date & time method)
The JD method is the most natural and convenient to use in cases
where the loss of several decimal digits of resolution is
acceptable. The J2000 and MJD methods are good compromises
between resolution and convenience. The date & time method is
best matched to the algorithm used: maximum precision is
delivered when the ut11 argument is for 0hrs UT1 on the day in
question and the ut12 argument lies in the range 0 to 1, or vice
versa.
2. If the caller wishes to provide the Earth rotation angle itself,
the function Aper can be used instead. One use of this
technique is to substitute Greenwich apparent sidereal time and
thereby to support equinox based transformations directly.
3. This is one of several functions that inserts into the astrom
structure star-independent parameters needed for the chain of
astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed.
The various functions support different classes of observer and
portions of the transformation chain:
functions observer transformation
Apcg Apcg13 geocentric ICRS <-> GCRS
Apci Apci13 terrestrial ICRS <-> CIRS
Apco Apco13 terrestrial ICRS <-> observed
Apcs Apcs13 space ICRS <-> GCRS
Aper Aper13 terrestrial update Earth rotation
Apio Apio13 terrestrial CIRS <-> observed
Those with names ending in "13" use contemporary SOFA models to
compute the various ephemerides. The others accept ephemerides
supplied by the caller.
The transformation from ICRS to GCRS covers space motion,
parallax, light deflection, and aberration. From GCRS to CIRS
comprises frame bias and precession-nutation. From CIRS to
observed takes account of Earth rotation, polar motion, diurnal
aberration and parallax (unless subsumed into the ICRS <-> GCRS
transformation), and atmospheric refraction.
Called:
Aper astrometry parameters: update ERA
Era00 Earth rotation angle, IAU 2000
*/
func Aper13(ut11, ut12 float64, astrom *ASTROM) {
Aper(Era00(ut11, ut12), astrom)
}
/*
Apio Prepare for CIRS <-> observed, terrestrial, special
For a terrestrial observer, prepare star-independent astrometry
parameters for transformations between CIRS and observed
coordinates. The caller supplies the Earth orientation information
and the refraction constants as well as the site coordinates.
Given:
sp float64 the TIO locator s' (radians, Note 1)
theta float64 Earth rotation angle (radians)
elong float64 longitude (radians, east +ve, Note 2)
phi float64 geodetic latitude (radians, Note 2)
hm float64 height above ellipsoid (m, geodetic Note 2)
xp,yp float64 polar motion coordinates (radians, Note 3)
refa float64 refraction constant A (radians, Note 4)
refb float64 refraction constant B (radians, Note 4)
Returned:
astrom ASTROM star-independent astrometry parameters:
pmt float64 unchanged
eb [3]float64 unchanged
eh [3]float64 unchanged
em float64 unchanged
v [3]float64 unchanged
bm1 float64 unchanged
bpn [3][3]float64 unchanged
along float64 adjusted longitude (radians)
xpl float64 polar motion xp wrt local meridian (radians)
ypl float64 polar motion yp wrt local meridian (radians)
sphi float64 sine of geodetic latitude
cphi float64 cosine of geodetic latitude
diurab float64 magnitude of diurnal aberration vector
eral float64 "local" Earth rotation angle (radians)
refa float64 refraction constant A (radians)
refb float64 refraction constant B (radians)
Notes:
1. sp, the TIO locator s', is a tiny quantity needed only by the
most precise applications. It can either be set to zero or
predicted using the SOFA function Sp00.
2. The geographical coordinates are with respect to the WGS84
reference ellipsoid. TAKE CARE WITH THE LONGITUDE SIGN: the
longitude required by the present function is east-positive
(i.e. right-handed), in accordance with geographical convention.
3. The polar motion xp,yp can be obtained from IERS bulletins. The
values are the coordinates (in radians) of the Celestial
Intermediate Pole with respect to the International Terrestrial
Reference System (see IERS Conventions 2003), measured along the
meridians 0 and 90 deg west respectively. For many applications,
xp and yp can be set to zero.
Internally, the polar motion is stored in a form rotated onto the
local meridian.
4. The refraction constants refa and refb are for use in a
dZ = A*tan(Z)+B*tan^3(Z) model, where Z is the observed
(i.e. refracted) zenith distance and dZ is the amount of
refraction.
5. It is advisable to take great care with units, as even unlikely
values of the input parameters are accepted and processed in
accordance with the models used.
6. In cases where the caller does not wish to provide the Earth
rotation information and refraction constants, the function
Apio13 can be used instead of the present function. This
starts from UTC and weather readings etc. and computes suitable
values using other SOFA functions.
7. This is one of several functions that inserts into the astrom
structure star-independent parameters needed for the chain of
astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed.
The various functions support different classes of observer and
portions of the transformation chain:
functions observer transformation
Apcg Apcg13 geocentric ICRS <-> GCRS
Apci Apci13 terrestrial ICRS <-> CIRS
Apco Apco13 terrestrial ICRS <-> observed
Apcs Apcs13 space ICRS <-> GCRS
Aper Aper13 terrestrial update Earth rotation
Apio Apio13 terrestrial CIRS <-> observed
Those with names ending in "13" use contemporary SOFA models to
compute the various ephemerides. The others accept ephemerides
supplied by the caller.
The transformation from ICRS to GCRS covers space motion,
parallax, light deflection, and aberration. From GCRS to CIRS
comprises frame bias and precession-nutation. From CIRS to
observed takes account of Earth rotation, polar motion, diurnal
aberration and parallax (unless subsumed into the ICRS <-> GCRS
transformation), and atmospheric refraction.
8. The context structure astrom produced by this function is used by
Atioq and Atoiq.
Called:
Ir initialize r-matrix to identity
Rz rotate around Z-axis
Ry rotate around Y-axis
Rx rotate around X-axis
Anpm normalize angle into range +/- pi
Pvtob position/velocity of terrestrial station
*/
func Apio(sp, theta float64, elong, phi, hm, xp, yp float64, refa, refb float64, astrom *ASTROM) {
var r [3][3]float64
var a, b, eral, c float64
var pv [2][3]float64
/* Form the rotation matrix, CIRS to apparent [HA,Dec]. */
Ir(&r)
Rz(theta+sp, &r)
Ry(-xp, &r)
Rx(-yp, &r)
Rz(elong, &r)
/* Solve for local Earth rotation angle. */
a = r[0][0]
b = r[0][1]
// eral = ( a != 0.0 || b != 0.0 ) ? atan2(b, a) : 0.0;
if a != 0.0 || b != 0.0 {
eral = atan2(b, a)
} else {
eral = 0.0
}
astrom.Eral = eral
/* Solve for polar motion [X,Y] with respect to local meridian. */
a = r[0][0]
c = r[0][2]
astrom.Xpl = atan2(c, sqrt(a*a+b*b))
a = r[1][2]
b = r[2][2]
// astrom.Ypl = ( a != 0.0 || b != 0.0 ) ? -atan2(a, b) : 0.0;
if a != 0.0 || b != 0.0 {
astrom.Ypl = -atan2(a, b)
} else {
astrom.Ypl = 0.0
}
/* Adjusted longitude. */
astrom.Along = Anpm(eral - theta)
/* Functions of latitude. */
astrom.Sphi = sin(phi)
astrom.Cphi = cos(phi)
/* Observer's geocentric position and velocity (m, m/s, CIRS). */
Pvtob(elong, phi, hm, xp, yp, sp, theta, &pv)
/* Magnitude of diurnal aberration vector. */
astrom.Diurab = sqrt(pv[1][0]*pv[1][0]+pv[1][1]*pv[1][1]) / CMPS
/* Refraction constants. */
astrom.Refa = refa
astrom.Refb = refb
}
/*
Apio13 Prepare for CIRS <-> observed, terrestrial
For a terrestrial observer, prepare star-independent astrometry
parameters for transformations between CIRS and observed
coordinates. The caller supplies UTC, site coordinates, ambient air
conditions and observing wavelength.
Given:
utc1 float64 UTC as a 2-part...
utc2 float64 ...quasi Julian Date (Notes 1,2)
dut1 float64 UT1-UTC (seconds)
elong float64 longitude (radians, east +ve, Note 3)
phi float64 geodetic latitude (radians, Note 3)
hm float64 height above ellipsoid (m, geodetic Notes 4,6)
xp,yp float64 polar motion coordinates (radians, Note 5)
phpa float64 pressure at the observer (hPa = mB, Note 6)
tc float64 ambient temperature at the observer (deg C)
rh float64 relative humidity at the observer (range 0-1)
wl float64 wavelength (micrometers, Note 7)
Returned:
astrom ASTROM star-independent astrometry parameters:
pmt float64 unchanged
eb [3]float64 unchanged
eh [3]float64 unchanged
em float64 unchanged
v [3]float64 unchanged
bm1 float64 unchanged
bpn [3][3]float64 unchanged
along float64 longitude + s' (radians)
xpl float64 polar motion xp wrt local meridian (radians)
ypl float64 polar motion yp wrt local meridian (radians)
sphi float64 sine of geodetic latitude
cphi float64 cosine of geodetic latitude
diurab float64 magnitude of diurnal aberration vector
eral float64 "local" Earth rotation angle (radians)
refa float64 refraction constant A (radians)
refb float64 refraction constant B (radians)
Returned (function value):
int status: +1 = dubious year (Note 2)
0 = OK
-1 = unacceptable date
Notes:
1. utc1+utc2 is quasi Julian Date (see Note 2), apportioned in any
convenient way between the two arguments, for example where utc1
is the Julian Day Number and utc2 is the fraction of a day.
However, JD cannot unambiguously represent UTC during a leap
second unless special measures are taken. The convention in the
present function is that the JD day represents UTC days whether
the length is 86399, 86400 or 86401 SI seconds.
Applications should use the function Dtf2d to convert from
calendar date and time of day into 2-part quasi Julian Date, as
it implements the leap-second-ambiguity convention just
described.
2. The warning status "dubious year" flags UTCs that predate the
introduction of the time scale or that are too far in the future
to be trusted. See Dat for further details.
3. UT1-UTC is tabulated in IERS bulletins. It increases by exactly
one second at the end of each positive UTC leap second,
introduced in order to keep UT1-UTC within +/- 0.9s. n.b. This
practice is under review, and in the future UT1-UTC may grow
essentially without limit.
4. The geographical coordinates are with respect to the WGS84
reference ellipsoid. TAKE CARE WITH THE LONGITUDE SIGN: the
longitude required by the present function is east-positive
(i.e. right-handed), in accordance with geographical convention.
5. The polar motion xp,yp can be obtained from IERS bulletins. The
values are the coordinates (in radians) of the Celestial
Intermediate Pole with respect to the International Terrestrial
Reference System (see IERS Conventions 2003), measured along the
meridians 0 and 90 deg west respectively. For many applications,
xp and yp can be set to zero.
Internally, the polar motion is stored in a form rotated onto
the local meridian.
6. If hm, the height above the ellipsoid of the observing station
in meters, is not known but phpa, the pressure in hPa (=mB), is
available, an adequate estimate of hm can be obtained from the
expression
hm = -29.3 * tsl * log ( phpa / 1013.25 );
where tsl is the approximate sea-level air temperature in K
(See Astrophysical Quantities, C.W.Allen, 3rd edition, section
52). Similarly, if the pressure phpa is not known, it can be
estimated from the height of the observing station, hm, as
follows:
phpa = 1013.25 * exp ( -hm / ( 29.3 * tsl ) );
Note, however, that the refraction is nearly proportional to the
pressure and that an accurate phpa value is important for
precise work.
7. The argument wl specifies the observing wavelength in
micrometers. The transition from optical to radio is assumed to
occur at 100 micrometers (about 3000 GHz).
8. It is advisable to take great care with units, as even unlikely
values of the input parameters are accepted and processed in
accordance with the models used.
9. In cases where the caller wishes to supply his own Earth
rotation information and refraction constants, the function
Apc can be used instead of the present function.
10)This is one of several functions that inserts into the astrom
structure star-independent parameters needed for the chain of
astrometric transformations ICRS <-> GCRS <-> CIRS <-> observed.
The various functions support different classes of observer and
portions of the transformation chain:
functions observer transformation
Apcg Apcg13 geocentric ICRS <-> GCRS
Apci Apci13 terrestrial ICRS <-> CIRS
Apco Apco13 terrestrial ICRS <-> observed
Apcs Apcs13 space ICRS <-> GCRS
Aper Aper13 terrestrial update Earth rotation
Apio Apio13 terrestrial CIRS <-> observed
Those with names ending in "13" use contemporary SOFA models to
compute the various ephemerides. The others accept ephemerides
supplied by the caller.
The transformation from ICRS to GCRS covers space motion,
parallax, light deflection, and aberration. From GCRS to CIRS
comprises frame bias and precession-nutation. From CIRS to
observed takes account of Earth rotation, polar motion, diurnal
aberration and parallax (unless subsumed into the ICRS <-> GCRS
transformation), and atmospheric refraction.
11)The context structure astrom produced by this function is used
by Atioq and Atoiq.
Called:
Utctai UTC to TAI
Taitt TAI to TT
Utcut1 UTC to UT1
Sp00 the TIO locator s', IERS 2000
Era00 Earth rotation angle, IAU 2000
Refco refraction constants for given ambient conditions
Apio astrometry parameters, CIRS-observed
*/
func Apio13(utc1, utc2, dut1 float64, elong, phi, hm, xp, yp float64, phpa, tc, rh, wl float64, astrom *ASTROM) int {
var j int
var tai1, tai2, tt1, tt2, ut11, ut12, sp, theta, refa, refb float64
/* UTC to other time scales. */
j = Utctai(utc1, utc2, &tai1, &tai2)
if j < 0 {
return -1
}
Taitt(tai1, tai2, &tt1, &tt2)
j = Utcut1(utc1, utc2, dut1, &ut11, &ut12)
if j < 0 {
return -1
}
/* TIO locator s'. */
sp = Sp00(tt1, tt2)
/* Earth rotation angle. */
theta = Era00(ut11, ut12)
/* Refraction constants A and B. */
Refco(phpa, tc, rh, wl, &refa, &refb)
/* CIRS <-> observed astrometry parameters. */
Apio(sp, theta, elong, phi, hm, xp, yp, refa, refb, astrom)
/* Return any warning status. */
return j
}
/*
Atcc13 J2000.0 catalog ICRS -> astrometric ICRS
Transform a star's ICRS catalog entry (epoch J2000.0) into ICRS
astrometric place.
Given:
rc float64 ICRS right ascension at J2000.0 (radians, Note 1)
dc float64 ICRS declination at J2000.0 (radians, Note 1)
pr float64 RA proper motion (radians/year, Note 2)
pd float64 Dec proper motion (radians/year)
px float64 parallax (arcsec)
rv float64 radial velocity (km/s, +ve if receding)
date1 float64 TDB as a 2-part...
date2 float64 ...Julian Date (Note 3)
Returned:
ra,da float64 ICRS astrometric RA,Dec (radians)
Notes:
1. 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 Pmsafe before use.
2. The proper motion in RA is dRA/dt rather than cos(Dec)*dRA/dt.
3. The TDB date date1+date2 is a Julian Date, apportioned in any
convenient way between the two arguments. For example,
JD(TDB)=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. For most
applications of this function the choice will not be at all
critical.
TT can be used instead of TDB without any significant impact on
accuracy.
Called:
Apci13 astrometry parameters, ICRS-CIRS, 2013
Atccq quick catalog ICRS to astrometric
*/
func Atcc13(rc, dc float64, pr, pd, px, rv float64, date1, date2 float64, ra, da *float64) {
/* Star-independent astrometry parameters */
var astrom ASTROM
var w float64
/* The transformation parameters. */
Apci13(date1, date2, &astrom, &w)
/* Catalog ICRS (epoch J2000.0) to astrometric. */
Atccq(rc, dc, pr, pd, px, rv, &astrom, ra, da)
}
/*
Atccq Astrometric ICRS -> J2000.0 catalog ICRS
Quick transformation of a star's ICRS catalog entry (epoch J2000.0)
into ICRS astrometric place, 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 Apci[13], Apcg[13], Apco[13] or Apcs[13].
If the parallax and proper motions are zero the transformation has
no effect.
Given:
rc,dc float64 ICRS RA,Dec at J2000.0 (radians)
pr float64 RA proper motion (radians/year, Note 3)
pd float64 Dec proper motion (radians/year)
px float64 parallax (arcsec)
rv float64 radial velocity (km/s, +ve if receding)
astrom ASTROM star-independent astrometry parameters:
pmt float64 PM time interval (SSB, Julian years)
eb [3]float64 SSB to observer (vector, au)
eh [3]float64 Sun to observer (unit vector)
em float64 distance from Sun to observer (au)
v [3]float64 barycentric observer velocity (vector, c)
bm1 float64 sqrt(1-|v|^2): reciprocal of Lorenz factor
bpn [3][3]float64 bias-precession-nutation matrix
along float64 longitude + s' (radians)
xpl float64 polar motion xp wrt local meridian (radians)
ypl float64 polar motion yp wrt local meridian (radians)
sphi float64 sine of geodetic latitude
cphi float64 cosine of geodetic latitude
diurab float64 magnitude of diurnal aberration vector
eral float64 "local" Earth rotation angle (radians)
refa float64 refraction constant A (radians)
refb float64 refraction constant B (radians)
Returned:
ra,da float64 ICRS astrometric 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 Pmsafe before use.
3. The proper motion in RA is dRA/dt rather than cos(Dec)*dRA/dt.
Called:
Pmpx proper motion and parallax
C2s p-vector to spherical
Anp normalize angle into range 0 to 2pi
*/
func Atccq(rc, dc float64, pr, pd, px, rv float64, astrom *ASTROM, ra, da *float64) {
var p [3]float64
var w float64
/* Proper motion and parallax, giving BCRS coordinate direction. */
Pmpx(rc, dc, pr, pd, px, rv, astrom.Pmt, astrom.Eb, &p)
/* ICRS astrometric RA,Dec. */
C2s(p, &w, da)
*ra = Anp(w)
}
/*
Atci13 Catalog -> CIRS
Transform ICRS star data, epoch J2000.0, to CIRS.
Given:
rc float64 ICRS right ascension at J2000.0 (radians, Note 1)
dc float64 ICRS declination at J2000.0 (radians, Note 1)
pr float64 RA proper motion (radians/year, Note 2)
pd float64 Dec proper motion (radians/year)
px float64 parallax (arcsec)
rv float64 radial velocity (km/s, +ve if receding)
date1 float64 TDB as a 2-part...
date2 float64 ...Julian Date (Note 3)
Returned:
ri,di float64 CIRS geocentric RA,Dec (radians)
eo float64 equation of the origins (ERA-GST, Note 5)
Notes:
1. 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 Pmsafe before use.
2. The proper motion in RA is dRA/dt rather than cos(Dec)*dRA/dt.
3. The TDB date date1+date2 is a Julian Date, apportioned in any
convenient way between the two arguments. For example,
JD(TDB)=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. For most
applications of this function the choice will not be at all
critical.
TT can be used instead of TDB without any significant impact on
accuracy.
4. The available accuracy is better than 1 milliarcsecond, limited
mainly by the precession-nutation model that is used, namely
2000A/2006. Very close to solar system bodies, additional
errors of up to several milliarcseconds can occur because of
unmodeled light deflection; however, the Sun's contribution is
taken into account, to first order. The accuracy limitations of
the SOFA function Epv00 (used to compute Earth position and
velocity) can contribute aberration errors of up to
5 microarcseconds. Light deflection at the Sun's limb is
uncertain at the 0.4 mas level.
5. Should the transformation to (equinox based) apparent place be
required rather than (CIO based) intermediate place, subtract the
equation of the origins from the returned right ascension:
RA = RI - EO. (The Anp function can then be applied, as
required, to keep the result in the conventional 0-2pi range.)
Called:
Apci13 astrometry parameters, ICRS-CIRS, 2013
Atciq quick ICRS to CIRS
*/
func Atci13(rc, dc float64, pr, pd, px, rv float64, date1, date2 float64, ri, di, eo *float64) {
/* Star-independent astrometry parameters */
var astrom ASTROM
/* The transformation parameters. */
Apci13(date1, date2, &astrom, eo)
/* ICRS (epoch J2000.0) to CIRS. */
Atciq(rc, dc, pr, pd, px, rv, &astrom, ri, di)
}
/*
Atciq Quick ICRS -> CIRS
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 Apci[13], Apcg[13], Apco[13] or Apcs[13].
If the parallax and proper motions are zero the Atciqz function
can be used instead.
Given:
rc,dc float64 ICRS RA,Dec at J2000.0 (radians)
pr float64 RA proper motion (radians/year, Note 3)
pd float64 Dec proper motion (radians/year)
px float64 parallax (arcsec)
rv float64 radial velocity (km/s, +ve if receding)
astrom ASTROM star-independent astrometry parameters:
pmt float64 PM time interval (SSB, Julian years)
eb [3]float64 SSB to observer (vector, au)
eh [3]float64 Sun to observer (unit vector)
em float64 distance from Sun to observer (au)
v [3]float64 barycentric observer velocity (vector, c)
bm1 float64 sqrt(1-|v|^2): reciprocal of Lorenz factor
bpn [3][3]float64 bias-precession-nutation matrix
along float64 longitude + s' (radians)
xpl float64 polar motion xp wrt local meridian (radians)
ypl float64 polar motion yp wrt local meridian (radians)
sphi float64 sine of geodetic latitude
cphi float64 cosine of geodetic latitude
diurab float64 magnitude of diurnal aberration vector
eral float64 "local" Earth rotation angle (radians)
refa float64 refraction constant A (radians)
refb float64 refraction constant B (radians)
Returned:
ri,di float64 CIRS RA,Dec (radians)
Notes:
1. 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 Pmsafe before use.
2. The proper motion in RA is dRA/dt rather than cos(Dec)*dRA/dt.
Called:
Pmpx proper motion and parallax
Ldsun light deflection by the Sun
Ab stellar aberration
Rxp product of r-matrix and pv-vector
C2s p-vector to spherical
Anp normalize angle into range 0 to 2pi
*/
func Atciq(rc, dc float64, pr, pd, px, rv float64, astrom *ASTROM, ri, di *float64) {
var pco, pnat, ppr, pi [3]float64
var w float64
/* Proper motion and parallax, giving BCRS coordinate direction. */
Pmpx(rc, dc, pr, pd, px, rv, astrom.Pmt, astrom.Eb, &pco)
/* Light deflection by the Sun, giving BCRS natural direction. */
Ldsun(pco, astrom.Eh, astrom.Em, &pnat)
/* Aberration, giving GCRS proper direction. */
Ab(pnat, astrom.V, astrom.Em, astrom.Bm1, &ppr)
/* Bias-precession-nutation, giving CIRS proper direction. */
Rxp(astrom.Bpn, ppr, &pi)
/* CIRS RA,Dec. */
C2s(pi, &w, di)
*ri = Anp(w)
}
/*
Atciqn Quick ICRS -> CIRS, multiple deflections
Quick ICRS, epoch J2000.0, to CIRS transformation, given precomputed
star-independent astrometry parameters plus a list of light-
deflecting bodies.
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 Apci[13], Apcg[13], Apco[13] or Apcs[13].
If the only light-deflecting body to be taken into account is the
Sun, the Atciq function can be used instead. If in addition the
parallax and proper motions are zero, the Atciqz function can be
used.
Given:
rc,dc float64 ICRS RA,Dec at J2000.0 (radians)
pr float64 RA proper motion (radians/year, Note 3)
pd float64 Dec proper motion (radians/year)
px float64 parallax (arcsec)
rv float64 radial velocity (km/s, +ve if receding)
astrom ASTROM star-independent astrometry parameters:
pmt float64 PM time interval (SSB, Julian years)
eb [3]float64 SSB to observer (vector, au)
eh [3]float64 Sun to observer (unit vector)
em float64 distance from Sun to observer (au)
v [3]float64 barycentric observer velocity (vector, c)
bm1 float64 sqrt(1-|v|^2): reciprocal of Lorenz factor
bpn [3][3]float64 bias-precession-nutation matrix
along float64 longitude + s' (radians)
xpl float64 polar motion xp wrt local meridian (radians)
ypl float64 polar motion yp wrt local meridian (radians)
sphi float64 sine of geodetic latitude
cphi float64 cosine of geodetic latitude
diurab float64 magnitude of diurnal aberration vector
eral float64 "local" Earth rotation angle (radians)
refa float64 refraction constant A (radians)
refb float64 refraction constant B (radians)
n int number of bodies (Note 3)
b LDBODY[n] data for each of the n bodies (Notes 3,4):
bm float64 mass of the body (solar masses, Note 5)
dl float64 deflection limiter (Note 6)
pv [2][3]float64 barycentric PV of the body (au, au/day)
Returned:
ri,di float64 CIRS RA,Dec (radians)
Notes:
1. 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 Pmsafe before use.
2. The proper motion in RA is dRA/dt rather than cos(Dec)*dRA/dt.
3. The struct b contains n entries, one for each body to be
considered. If n = 0, no gravitational light deflection will be
applied, not even for the Sun.
4. The struct b should include an entry for the Sun as well as for
any planet or other body to be taken into account. The entries
should be in the order in which the light passes the body.
5. In the entry in the b struct for body i, the mass parameter
b[i].bm can, as required, be adjusted in order to allow for such
effects as quadrupole field.
6. The deflection limiter parameter b[i].dl is phi^2/2, where phi is
the angular separation (in radians) between star and body at
which limiting is applied. As phi shrinks below the chosen
threshold, the deflection is artificially reduced, reaching zero
for phi = 0. Example values suitable for a terrestrial
observer, together with masses, are as follows:
body i b[i].bm b[i].dl
Sun 1.0 6e-6
Jupiter 0.00095435 3e-9
Saturn 0.00028574 3e-10
7. For efficiency, validation of the contents of the b array is
omitted. The supplied masses must be greater than zero, the
position and velocity vectors must be right, and the deflection
limiter greater than zero.
Called:
Pmpx proper motion and parallax
Ldn light deflection by n bodies
Ab stellar aberration
Rxp product of r-matrix and pv-vector
C2s p-vector to spherical
Anp normalize angle into range 0 to 2pi
*/
func Atciqn(rc, dc float64, pr, pd, px, rv float64, astrom *ASTROM, n int, b []LDBODY, ri, di *float64) {
var pco, pnat, ppr, pi [3]float64
var w float64
/* Proper motion and parallax, giving BCRS coordinate direction. */
Pmpx(rc, dc, pr, pd, px, rv, astrom.Pmt, astrom.Eb, &pco)
/* Light deflection, giving BCRS natural direction. */
Ldn(n, b, astrom.Eb, pco, &pnat)
/* Aberration, giving GCRS proper direction. */
Ab(pnat, astrom.V, astrom.Em, astrom.Bm1, &ppr)
/* Bias-precession-nutation, giving CIRS proper direction. */
Rxp(astrom.Bpn, ppr, &pi)
/* CIRS RA,Dec. */
C2s(pi, &w, di)
*ri = Anp(w)
}
/*
Atciqz Quick astrometric ICRS -> CIRS
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 Apci[13], Apcg[13], Apco[13] or Apcs[13].
The corresponding function for the case of non-zero parallax and
proper motion is Atciq.
Given:
rc,dc float64 ICRS astrometric RA,Dec (radians)
astrom ASTROM star-independent astrometry parameters:
pmt float64 PM time interval (SSB, Julian years)
eb [3]float64 SSB to observer (vector, au)
eh [3]float64 Sun to observer (unit vector)
em float64 distance from Sun to observer (au)
v [3]float64 barycentric observer velocity (vector, c)
bm1 float64 sqrt(1-|v|^2): reciprocal of Lorenz factor
bpn [3][3]float64 bias-precession-nutation matrix
along float64 longitude + s' (radians)
xpl float64 polar motion xp wrt local meridian (radians)
ypl float64 polar motion yp wrt local meridian (radians)
sphi float64 sine of geodetic latitude
cphi float64 cosine of geodetic latitude
diurab float64 magnitude of diurnal aberration vector
eral float64 "local" Earth rotation angle (radians)
refa float64 refraction constant A (radians)
refb float64 refraction constant B (radians)
Returned:
ri,di float64 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:
S2c spherical coordinates to unit vector
Ldsun light deflection due to Sun
Ab stellar aberration
Rxp product of r-matrix and p-vector
C2s p-vector to spherical
Anp normalize angle into range +/- pi
*/
func Atciqz(rc, dc float64, astrom *ASTROM, ri, di *float64) {
var pco, pnat, ppr, pi [3]float64
var w float64
/* BCRS coordinate direction (unit vector). */
S2c(rc, dc, &pco)
/* Light deflection by the Sun, giving BCRS natural direction. */
Ldsun(pco, astrom.Eh, astrom.Em, &pnat)
/* Aberration, giving GCRS proper direction. */
Ab(pnat, astrom.V, astrom.Em, astrom.Bm1, &ppr)
/* Bias-precession-nutation, giving CIRS proper direction. */
Rxp(astrom.Bpn, ppr, &pi)
/* CIRS RA,Dec. */
C2s(pi, &w, di)
*ri = Anp(w)
}
/*
Atco13 ICRS -> observed
ICRS RA,Dec to observed place. The caller supplies UTC, site
coordinates, ambient air conditions and observing wavelength.
SOFA models are used for the Earth ephemeris, bias-precession-
nutation, Earth orientation and refraction.
Given:
rc,dc float64 ICRS right ascension at J2000.0 (radians, Note 1)
pr float64 RA proper motion (radians/year, Note 2)
pd float64 Dec proper motion (radians/year)
px float64 parallax (arcsec)
rv float64 radial velocity (km/s, +ve if receding)
utc1 float64 UTC as a 2-part...
utc2 float64 ...quasi Julian Date (Notes 3-4)
dut1 float64 UT1-UTC (seconds, Note 5)
elong float64 longitude (radians, east +ve, Note 6)
phi float64 latitude (geodetic, radians, Note 6)
hm float64 height above ellipsoid (m, geodetic, Notes 6,8)
xp,yp float64 polar motion coordinates (radians, Note 7)
phpa float64 pressure at the observer (hPa = mB, Note 8)
tc float64 ambient temperature at the observer (deg C)
rh float64 relative humidity at the observer (range 0-1)
wl float64 wavelength (micrometers, Note 9)
Returned:
aob float64 observed azimuth (radians: N=0,E=90)
zob float64 observed zenith distance (radians)
hob float64 observed hour angle (radians)
dob float64 observed declination (radians)
rob float64 observed right ascension (CIO-based, radians)
eo float64 equation of the origins (ERA-GST)
Returned (function value):
int status: +1 = dubious year (Note 4)
0 = OK
-1 = unacceptable date
Notes:
1. 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 Pmsafe before use.
2. The proper motion in RA is dRA/dt rather than cos(Dec)*dRA/dt.
3. utc1+utc2 is quasi Julian Date (see Note 2), apportioned in any
convenient way between the two arguments, for example where utc1
is the Julian Day Number and utc2 is the fraction of a day.
However, JD cannot unambiguously represent UTC during a leap
second unless special measures are taken. The convention in the
present function is that the JD day represents UTC days whether
the length is 86399, 86400 or 86401 SI seconds.
Applications should use the function Dtf2d to convert from
calendar date and time of day into 2-part quasi Julian Date, as
it implements the leap-second-ambiguity convention just
described.
4. The warning status "dubious year" flags UTCs that predate the
introduction of the time scale or that are too far in the
future to be trusted. See Dat for further details.
5. UT1-UTC is tabulated in IERS bulletins. It increases by exactly
one second at the end of each positive UTC leap second,
introduced in order to keep UT1-UTC within +/- 0.9s. n.b. This
practice is under review, and in the future UT1-UTC may grow
essentially without limit.
6. The geographical coordinates are with respect to the WGS84
reference ellipsoid. TAKE CARE WITH THE LONGITUDE SIGN: the
longitude required by the present function is east-positive
(i.e. right-handed), in accordance with geographical convention.
7. The polar motion xp,yp can be obtained from IERS bulletins. The
values are the coordinates (in radians) of the Celestial
Intermediate Pole with respect to the International Terrestrial
Reference System (see IERS Conventions 2003), measured along the
meridians 0 and 90 deg west respectively. For many
applications, xp and yp can be set to zero.
8. If hm, the height above the ellipsoid of the observing station
in meters, is not known but phpa, the pressure in hPa (=mB),
is available, an adequate estimate of hm can be obtained from
the expression
hm = -29.3 * tsl * log ( phpa / 1013.25 );
where tsl is the approximate sea-level air temperature in K
(See Astrophysical Quantities, C.W.Allen, 3rd edition, section
52). Similarly, if the pressure phpa is not known, it can be
estimated from the height of the observing station, hm, as
follows:
phpa = 1013.25 * exp ( -hm / ( 29.3 * tsl ) );
Note, however, that the refraction is nearly proportional to
the pressure and that an accurate phpa value is important for
precise work.
9. The argument wl specifies the observing wavelength in
micrometers. The transition from optical to radio is assumed to
occur at 100 micrometers (about 3000 GHz).
10)The accuracy of the result is limited by the corrections for
refraction, which use a simple A*tan(z) + B*tan^3(z) model.
Providing the meteorological parameters are known accurately and
there are no gross local effects, the predicted observed
coordinates should be within 0.05 arcsec (optical) or 1 arcsec
(radio) for a zenith distance of less than 70 degrees, better
than 30 arcsec (optical or radio) at 85 degrees and better
than 20 arcmin (optical) or 30 arcmin (radio) at the horizon.
Without refraction, the complementary functions Atco13 and
Atoc13 are self-consistent to better than 1 microarcsecond
all over the celestial sphere. With refraction included,
consistency falls off at high zenith distances, but is still
better than 0.05 arcsec at 85 degrees.
11)"Observed" Az,ZD means the position that would be seen by a
perfect geodetically aligned theodolite. (Zenith distance is
used rather than altitude in order to reflect the fact that no
allowance is made for depression of the horizon.) This is
related to the observed HA,Dec via the standard rotation, using
the geodetic latitude (corrected for polar motion), while the
observed HA and RA are related simply through the Earth rotation
angle and the site longitude. "Observed" RA,Dec or HA,Dec thus
means the position that would be seen by a perfect equatorial
with its polar axis aligned to the Earth's axis of rotation.
12)It is advisable to take great care with units, as even unlikely
values of the input parameters are accepted and processed in
accordance with the models used.
Called:
Apco13 astrometry parameters, ICRS-observed, 2013
Atciq quick ICRS to CIRS
Atioq quick CIRS to observed
*/
func Atco13(rc, dc float64, pr, pd, px, rv float64, utc1, utc2, dut1 float64,
elong, phi, hm, xp, yp float64, phpa, tc, rh, wl float64,
aob, zob, hob *float64, dob, rob, eo *float64) int {
var j int
var astrom ASTROM
var ri, di float64
/* Star-independent astrometry parameters. */
j = Apco13(utc1, utc2, dut1, elong, phi, hm, xp, yp,
phpa, tc, rh, wl, &astrom, eo)
/* Abort if bad UTC. */
if j < 0 {
return j
}
/* Transform ICRS to CIRS. */
Atciq(rc, dc, pr, pd, px, rv, &astrom, &ri, &di)
/* Transform CIRS to observed. */
Atioq(ri, di, &astrom, aob, zob, hob, dob, rob)
/* Return OK/warning status. */
return j
}
/*
Atic13 CIRS -> ICRS
Transform star RA,Dec from geocentric CIRS to ICRS astrometric.
Given:
ri,di float64 CIRS geocentric RA,Dec (radians)
date1 float64 TDB as a 2-part...
date2 float64 ...Julian Date (Note 1)
Returned:
rc,dc float64 ICRS astrometric RA,Dec (radians)
eo float64 equation of the origins (ERA-GST, Note 4)
Notes:
1. The TDB date date1+date2 is a Julian Date, apportioned in any
convenient way between the two arguments. For example,
JD(TDB)=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. For most
applications of this function the choice will not be at all
critical.
TT can be used instead of TDB without any significant impact on
accuracy.
2. Iterative techniques are used for the aberration and light
deflection corrections so that the functions Atic13 (or
Aticq) and Atci13 (or Atciq) are accurate inverses;
even at the edge of the Sun's disk the discrepancy is only about
1 nanoarcsecond.
3. The available accuracy is better than 1 milliarcsecond, limited
mainly by the precession-nutation model that is used, namely
2000A/2006. Very close to solar system bodies, additional
errors of up to several milliarcseconds can occur because of
unmodeled light deflection; however, the Sun's contribution is
taken into account, to first order. The accuracy limitations of
the SOFA function Epv00 (used to compute Earth position and
velocity) can contribute aberration errors of up to
5 microarcseconds. Light deflection at the Sun's limb is
uncertain at the 0.4 mas level.
4. Should the transformation to (equinox based) J2000.0 mean place
be required rather than (CIO based) ICRS coordinates, subtract the
equation of the origins from the returned right ascension:
RA = RI - EO. (The Anp function can then be applied, as
required, to keep the result in the conventional 0-2pi range.)
Called:
Apci13 astrometry parameters, ICRS-CIRS, 2013
Aticq quick CIRS to ICRS astrometric
*/
func Atic13(ri, di float64, date1, date2 float64, rc, dc, eo *float64) {
/* Star-independent astrometry parameters */
var astrom ASTROM
/* Star-independent astrometry parameters. */
Apci13(date1, date2, &astrom, eo)
/* CIRS to ICRS astrometric. */
Aticq(ri, di, &astrom, rc, dc)
}
/*
Aticq Quick CIRS -> ICRS
Quick CIRS RA,Dec to ICRS astrometric place, given the star-
independent astrometry parameters.
Use of this function is appropriate when efficiency is important and
where many star positions are all to be transformed for one date.
The star-independent astrometry parameters can be obtained by
calling one of the functions Apci[13], Apcg[13], Apco[13]
or Apcs[13].
Given:
ri,di float64 CIRS RA,Dec (radians)
astrom ASTROM star-independent astrometry parameters:
pmt float64 PM time interval (SSB, Julian years)
eb [3]float64 SSB to observer (vector, au)
eh [3]float64 Sun to observer (unit vector)
em float64 distance from Sun to observer (au)
v [3]float64 barycentric observer velocity (vector, c)
bm1 float64 sqrt(1-|v|^2): reciprocal of Lorenz factor
bpn [3][3]float64 bias-precession-nutation matrix
along float64 longitude + s' (radians)
xpl float64 polar motion xp wrt local meridian (radians)
ypl float64 polar motion yp wrt local meridian (radians)
sphi float64 sine of geodetic latitude
cphi float64 cosine of geodetic latitude
diurab float64 magnitude of diurnal aberration vector
eral float64 "local" Earth rotation angle (radians)
refa float64 refraction constant A (radians)
refb float64 refraction constant B (radians)
Returned:
rc,dc float64 ICRS astrometric RA,Dec (radians)
Notes:
1. Only the Sun is taken into account in the light deflection
correction.
2. Iterative techniques are used for the aberration and light
deflection corrections so that the functions Atic13 (or
Aticq) and Atci13 (or Atciq) are accurate inverses;
even at the edge of the Sun's disk the discrepancy is only about
1 nanoarcsecond.
Called:
S2c spherical coordinates to unit vector
Trxp product of transpose of r-matrix and p-vector
Zp zero p-vector
Ab stellar aberration
Ldsun light deflection by the Sun
C2s p-vector to spherical
Anp normalize angle into range +/- pi
*/
func Aticq(ri, di float64, astrom *ASTROM, rc, dc *float64) {
var j, i int
var pi, ppr, pnat, pco, d, before, after [3]float64
var w, r2, r float64
/* CIRS RA,Dec to Cartesian. */
S2c(ri, di, &pi)
/* Bias-precession-nutation, giving GCRS proper direction. */
Trxp(astrom.Bpn, pi, &ppr)
/* Aberration, giving GCRS natural direction. */
Zp(&d)
for j = 0; j < 2; j++ {
r2 = 0.0
for i = 0; i < 3; i++ {
w = ppr[i] - d[i]
before[i] = w
r2 += w * w
}
r = sqrt(r2)
for i = 0; i < 3; i++ {
before[i] /= r
}
Ab(before, astrom.V, astrom.Em, astrom.Bm1, &after)
r2 = 0.0
for i = 0; i < 3; i++ {
d[i] = after[i] - before[i]
w = ppr[i] - d[i]
pnat[i] = w
r2 += w * w
}
r = sqrt(r2)
for i = 0; i < 3; i++ {
pnat[i] /= r
}
}
/* Light deflection by the Sun, giving BCRS coordinate direction. */
Zp(&d)
for j = 0; j < 5; j++ {
r2 = 0.0
for i = 0; i < 3; i++ {
w = pnat[i] - d[i]
before[i] = w
r2 += w * w
}
r = sqrt(r2)
for i = 0; i < 3; i++ {
before[i] /= r
}
Ldsun(before, astrom.Eh, astrom.Em, &after)
r2 = 0.0
for i = 0; i < 3; i++ {
d[i] = after[i] - before[i]
w = pnat[i] - d[i]
pco[i] = w
r2 += w * w
}
r = sqrt(r2)
for i = 0; i < 3; i++ {
pco[i] /= r
}
}
/* ICRS astrometric RA,Dec. */
C2s(pco, &w, dc)
*rc = Anp(w)
}
/*
Aticqn Quick CIRS -> ICRS, multiple deflections
Quick CIRS to ICRS astrometric place transformation, given the star-
independent astrometry parameters plus a list of light-deflecting
bodies.
Use of this function is appropriate when efficiency is important and
where many star positions are all to be transformed for one date.
The star-independent astrometry parameters can be obtained by
calling one of the functions Apci[13], Apcg[13], Apco[13]
or Apcs[13].
If the only light-deflecting body to be taken into account is the
Sun, the Aticq function can be used instead.
Given:
ri,di double CIRS RA,Dec (radians)
astrom ASTROM star-independent astrometry parameters:
pmt float64 PM time interval (SSB, Julian years)
eb [3]float64 SSB to observer (vector, au)
eh [3]float64 Sun to observer (unit vector)
em float64 distance from Sun to observer (au)
v [3]float64 barycentric observer velocity (vector, c)
bm1 float64 sqrt(1-|v|^2): reciprocal of Lorenz factor
bpn [3][3]float64 bias-precession-nutation matrix
along float64 longitude + s' (radians)
xpl float64 polar motion xp wrt local meridian (radians)
ypl float64 polar motion yp wrt local meridian (radians)
sphi float64 sine of geodetic latitude
cphi float64 cosine of geodetic latitude
diurab float64 magnitude of diurnal aberration vector
eral float64 "local" Earth rotation angle (radians)
refa float64 refraction constant A (radians)
refb float64 refraction constant B (radians)
n int number of bodies (Note 3)
b LDBODY[n] data for each of the n bodies (Notes 3,4):
bm float64 mass of the body (solar masses, Note 5)
dl float64 deflection limiter (Note 6)
pv [2][3]float64 barycentric PV of the body (au, au/day)
Returned:
rc,dc float64 ICRS astrometric RA,Dec (radians)
Notes:
1. Iterative techniques are used for the aberration and light
deflection corrections so that the functions Aticqn and
Atciqn are accurate inverses; even at the edge of the Sun's
disk the discrepancy is only about 1 nanoarcsecond.
2. If the only light-deflecting body to be taken into account is the
Sun, the Aticq function can be used instead.
3. The struct b contains n entries, one for each body to be
considered. If n = 0, no gravitational light deflection will be
applied, not even for the Sun.
4. The struct b should include an entry for the Sun as well as for
any planet or other body to be taken into account. The entries
should be in the order in which the light passes the body.
5. In the entry in the b struct for body i, the mass parameter
b[i].bm can, as required, be adjusted in order to allow for such
effects as quadrupole field.
6. The deflection limiter parameter b[i].dl is phi^2/2, where phi is
the angular separation (in radians) between star and body at
which limiting is applied. As phi shrinks below the chosen
threshold, the deflection is artificially reduced, reaching zero
for phi = 0. Example values suitable for a terrestrial
observer, together with masses, are as follows:
body i b[i].bm b[i].dl
Sun 1.0 6e-6
Jupiter 0.00095435 3e-9
Saturn 0.00028574 3e-10
7. For efficiency, validation of the contents of the b array is
omitted. The supplied masses must be greater than zero, the
position and velocity vectors must be right, and the deflection
limiter greater than zero.
Called:
S2c spherical coordinates to unit vector
Trxp product of transpose of r-matrix and p-vector
Zp zero p-vector
Ab stellar aberration
Ldn light deflection by n bodies
C2s p-vector to spherical
Anp normalize angle into range +/- pi
*/
func Aticqn(ri, di float64, astrom *ASTROM, n int, b []LDBODY, rc, dc *float64) {
var j, i int
var pi, ppr, pnat, pco, d, before, after [3]float64
var w, r2, r float64
/* CIRS RA,Dec to Cartesian. */
S2c(ri, di, &pi)
/* Bias-precession-nutation, giving GCRS proper direction. */
Trxp(astrom.Bpn, pi, &ppr)
/* Aberration, giving GCRS natural direction. */
Zp(&d)
for j = 0; j < 2; j++ {
r2 = 0.0
for i = 0; i < 3; i++ {
w = ppr[i] - d[i]
before[i] = w
r2 += w * w
}
r = sqrt(r2)
for i = 0; i < 3; i++ {
before[i] /= r
}
Ab(before, astrom.V, astrom.Em, astrom.Bm1, &after)
r2 = 0.0
for i = 0; i < 3; i++ {
d[i] = after[i] - before[i]
w = ppr[i] - d[i]
pnat[i] = w
r2 += w * w
}
r = sqrt(r2)
for i = 0; i < 3; i++ {
pnat[i] /= r
}
}
/* Light deflection, giving BCRS coordinate direction. */
Zp(&d)
for j = 0; j < 5; j++ {
r2 = 0.0
for i = 0; i < 3; i++ {
w = pnat[i] - d[i]
before[i] = w
r2 += w * w
}
r = sqrt(r2)
for i = 0; i < 3; i++ {
before[i] /= r
}
Ldn(n, b, astrom.Eb, before, &after)
r2 = 0.0
for i = 0; i < 3; i++ {
d[i] = after[i] - before[i]
w = pnat[i] - d[i]
pco[i] = w
r2 += w * w
}
r = sqrt(r2)
for i = 0; i < 3; i++ {
pco[i] /= r
}
}
/* ICRS astrometric RA,Dec. */
C2s(pco, &w, dc)
*rc = Anp(w)
}
/*
Atio13 CIRS -> observed
CIRS RA,Dec to observed place. The caller supplies UTC, site
coordinates, ambient air conditions and observing wavelength.
Given:
ri float64 CIRS right ascension (CIO-based, radians)
di float64 CIRS declination (radians)
utc1 float64 UTC as a 2-part...
utc2 float64 ...quasi Julian Date (Notes 1,2)
dut1 float64 UT1-UTC (seconds, Note 3)
elong float64 longitude (radians, east +ve, Note 4)
phi float64 geodetic latitude (radians, Note 4)
hm float64 height above ellipsoid (m, geodetic Notes 4,6)
xp,yp float64 polar motion coordinates (radians, Note 5)
phpa float64 pressure at the observer (hPa = mB, Note 6)
tc float64 ambient temperature at the observer (deg C)
rh float64 relative humidity at the observer (range 0-1)
wl float64 wavelength (micrometers, Note 7)
Returned:
aob float64 observed azimuth (radians: N=0,E=90)
zob float64 observed zenith distance (radians)
hob float64 observed hour angle (radians)
dob float64 observed declination (radians)
rob float64 observed right ascension (CIO-based, radians)
Returned (function value):
int status: +1 = dubious year (Note 2)
0 = OK
-1 = unacceptable date
Notes:
1. utc1+utc2 is quasi Julian Date (see Note 2), apportioned in any
convenient way between the two arguments, for example where utc1
is the Julian Day Number and utc2 is the fraction of a day.
However, JD cannot unambiguously represent UTC during a leap
second unless special measures are taken. The convention in the
present function is that the JD day represents UTC days whether
the length is 86399, 86400 or 86401 SI seconds.
Applications should use the function Dtf2d to convert from
calendar date and time of day into 2-part quasi Julian Date, as
it implements the leap-second-ambiguity convention just
described.
2. The warning status "dubious year" flags UTCs that predate the
introduction of the time scale or that are too far in the
future to be trusted. See Dat for further details.
3. UT1-UTC is tabulated in IERS bulletins. It increases by exactly
one second at the end of each positive UTC leap second,
introduced in order to keep UT1-UTC within +/- 0.9s. n.b. This
practice is under review, and in the future UT1-UTC may grow
essentially without limit.
4. The geographical coordinates are with respect to the WGS84
reference ellipsoid. TAKE CARE WITH THE LONGITUDE SIGN: the
longitude required by the present function is east-positive
(i.e. right-handed), in accordance with geographical convention.
5. The polar motion xp,yp can be obtained from IERS bulletins. The
values are the coordinates (in radians) of the Celestial
Intermediate Pole with respect to the International Terrestrial
Reference System (see IERS Conventions 2003), measured along the
meridians 0 and 90 deg west respectively. For many
applications, xp and yp can be set to zero.
6. If hm, the height above the ellipsoid of the observing station
in meters, is not known but phpa, the pressure in hPa (=mB), is
available, an adequate estimate of hm can be obtained from the
expression
hm = -29.3 * tsl * log ( phpa / 1013.25 );
where tsl is the approximate sea-level air temperature in K
(See Astrophysical Quantities, C.W.Allen, 3rd edition, section
52). Similarly, if the pressure phpa is not known, it can be
estimated from the height of the observing station, hm, as
follows:
phpa = 1013.25 * exp ( -hm / ( 29.3 * tsl ) );
Note, however, that the refraction is nearly proportional to
the pressure and that an accurate phpa value is important for
precise work.
7. The argument wl specifies the observing wavelength in
micrometers. The transition from optical to radio is assumed to
occur at 100 micrometers (about 3000 GHz).
8. "Observed" Az,ZD means the position that would be seen by a
perfect geodetically aligned theodolite. (Zenith distance is
used rather than altitude in order to reflect the fact that no
allowance is made for depression of the horizon.) This is
related to the observed HA,Dec via the standard rotation, using
the geodetic latitude (corrected for polar motion), while the
observed HA and RA are related simply through the Earth rotation
angle and the site longitude. "Observed" RA,Dec or HA,Dec thus
means the position that would be seen by a perfect equatorial
with its polar axis aligned to the Earth's axis of rotation.
9. The accuracy of the result is limited by the corrections for
refraction, which use a simple A*tan(z) + B*tan^3(z) model.
Providing the meteorological parameters are known accurately and
there are no gross local effects, the predicted astrometric
coordinates should be within 0.05 arcsec (optical) or 1 arcsec
(radio) for a zenith distance of less than 70 degrees, better
than 30 arcsec (optical or radio) at 85 degrees and better
than 20 arcmin (optical) or 30 arcmin (radio) at the horizon.
10)The complementary functions Atio13 and Atoi13 are self-
consistent to better than 1 microarcsecond all over the
celestial sphere.
11)It is advisable to take great care with units, as even unlikely
values of the input parameters are accepted and processed in
accordance with the models used.
Called:
Apio13 astrometry parameters, CIRS-observed, 2013
Atioq quick CIRS to observed
*/
func Atio13(ri, di float64, utc1, utc2, dut1 float64, elong, phi, hm, xp, yp float64, phpa, tc, rh, wl float64,
aob, zob, hob *float64, dob, rob *float64) int {
var j int
var astrom ASTROM
/* Star-independent astrometry parameters for CIRS->observed. */
j = Apio13(utc1, utc2, dut1, elong, phi, hm, xp, yp,
phpa, tc, rh, wl, &astrom)
/* Abort if bad UTC. */
if j < 0 {
return j
}
/* Transform CIRS to observed. */
Atioq(ri, di, &astrom, aob, zob, hob, dob, rob)
/* Return OK/warning status. */
return j
}
/*
Atioq Quick CIRS -> observed
Quick CIRS to observed place transformation.
Use of this function is appropriate when efficiency is important and
where many star positions are all to be transformed for one date.
The star-independent astrometry parameters can be obtained by
calling Apio[13] or Apco[13].
Given:
ri float64 CIRS right ascension
di float64 CIRS declination
astrom ASTROM star-independent astrometry parameters:
pmt float64 PM time interval (SSB, Julian years)
eb [3]float64 SSB to observer (vector, au)
eh [3]float64 Sun to observer (unit vector)
em float64 distance from Sun to observer (au)
v [3]float64 barycentric observer velocity (vector, c)
bm1 float64 sqrt(1-|v|^2): reciprocal of Lorenz factor
bpn [3][3]float64 bias-precession-nutation matrix
along float64 longitude + s' (radians)
xpl float64 polar motion xp wrt local meridian (radians)
ypl float64 polar motion yp wrt local meridian (radians)
sphi float64 sine of geodetic latitude
cphi float64 cosine of geodetic latitude
diurab float64 magnitude of diurnal aberration vector
eral float64 "local" Earth rotation angle (radians)
refa float64 refraction constant A (radians)
refb float64 refraction constant B (radians)
Returned:
aob float64 observed azimuth (radians: N=0,E=90)
zob float64 observed zenith distance (radians)
hob float64 observed hour angle (radians)
dob float64 observed declination (radians)
rob float64 observed right ascension (CIO-based, radians)
Notes:
1. This function returns zenith distance rather than altitude in
order to reflect the fact that no allowance is made for
depression of the horizon.
2. The accuracy of the result is limited by the corrections for
refraction, which use a simple A*tan(z) + B*tan^3(z) model.
Providing the meteorological parameters are known accurately and
there are no gross local effects, the predicted observed
coordinates should be within 0.05 arcsec (optical) or 1 arcsec
(radio) for a zenith distance of less than 70 degrees, better
than 30 arcsec (optical or radio) at 85 degrees and better
than 20 arcmin (optical) or 30 arcmin (radio) at the horizon.
Without refraction, the complementary functions Atioq and
Atoiq are self-consistent to better than 1 microarcsecond all
over the celestial sphere. With refraction included, consistency
falls off at high zenith distances, but is still better than
0.05 arcsec at 85 degrees.
3. It is advisable to take great care with units, as even unlikely
values of the input parameters are accepted and processed in
accordance with the models used.
4. The CIRS RA,Dec is obtained from a star catalog mean place by
allowing for space motion, parallax, the Sun's gravitational lens
effect, annual aberration and precession-nutation. For star
positions in the ICRS, these effects can be applied by means of
the Atci13 (etc.) functions. Starting from classical "mean
place" systems, additional transformations will be needed first.
5. "Observed" Az,El means the position that would be seen by a
perfect geodetically aligned theodolite. This is obtained from
the CIRS RA,Dec by allowing for Earth orientation and diurnal
aberration, rotating from equator to horizon coordinates, and
then adjusting for refraction. The HA,Dec is obtained by
rotating back into equatorial coordinates, and is the position
that would be seen by a perfect equatorial with its polar axis
aligned to the Earth's axis of rotation. Finally, the RA is
obtained by subtracting the HA from the local ERA.
6. The star-independent CIRS-to-observed-place parameters in ASTROM
may be computed with Apio[13] or Apco[13]. If nothing has
changed significantly except the time, Aper[13] may be used to
perform the requisite adjustment to the astrom structure.
Called:
S2c spherical coordinates to unit vector
C2s p-vector to spherical
Anp normalize angle into range 0 to 2pi
*/
func Atioq(ri, di float64, astrom *ASTROM, aob, zob *float64, hob, dob, rob *float64) {
/* Minimum cos(alt) and sin(alt) for refraction purposes */
const CELMIN = 1e-6
const SELMIN = 0.05
var v [3]float64
var x, y, z, sx, cx, sy, cy, xhd, yhd, zhd, f,
xhdt, yhdt, zhdt, xaet, yaet, zaet, azobs, r, tz, w, del,
cosdel, xaeo, yaeo, zaeo, zdobs, hmobs, dcobs, raobs float64
/* CIRS RA,Dec to Cartesian -HA,Dec. */
S2c(ri-astrom.Eral, di, &v)
x = v[0]
y = v[1]
z = v[2]
/* Polar motion. */
sx = sin(astrom.Xpl)
cx = cos(astrom.Xpl)
sy = sin(astrom.Ypl)
cy = cos(astrom.Ypl)
xhd = cx*x + sx*z
yhd = sx*sy*x + cy*y - cx*sy*z
zhd = -sx*cy*x + sy*y + cx*cy*z
/* Diurnal aberration. */
f = (1.0 - astrom.Diurab*yhd)
xhdt = f * xhd
yhdt = f * (yhd + astrom.Diurab)
zhdt = f * zhd
/* Cartesian -HA,Dec to Cartesian Az,El (S=0,E=90). */
xaet = astrom.Sphi*xhdt - astrom.Cphi*zhdt
yaet = yhdt
zaet = astrom.Cphi*xhdt + astrom.Sphi*zhdt
/* Azimuth (N=0,E=90). */
// azobs = ( xaet != 0.0 || yaet != 0.0 ) ? atan2(yaet,-xaet) : 0.0;
if xaet != 0.0 || yaet != 0.0 {
azobs = atan2(yaet, -xaet)
} else {
azobs = 0.0
}
/* ---------- */
/* Refraction */
/* ---------- */
/* Cosine and sine of altitude, with precautions. */
r = sqrt(xaet*xaet + yaet*yaet)
// r = r > CELMIN ? r : CELMIN;
if r <= CELMIN {
r = CELMIN
}
// z = zaet > SELMIN ? zaet : SELMIN;
if zaet > SELMIN {
z = zaet
} else {
z = SELMIN
}
/* A*tan(z)+B*tan^3(z) model, with Newton-Raphson correction. */
tz = r / z
w = astrom.Refb * tz * tz
del = (astrom.Refa + w) * tz /
(1.0 + (astrom.Refa+3.0*w)/(z*z))
/* Apply the change, giving observed vector. */
cosdel = 1.0 - del*del/2.0
f = cosdel - del*z/r
xaeo = xaet * f
yaeo = yaet * f
zaeo = cosdel*zaet + del*r
/* Observed ZD. */
zdobs = atan2(sqrt(xaeo*xaeo+yaeo*yaeo), zaeo)
/* Az/El vector to HA,Dec vector (both right-handed). */
v[0] = astrom.Sphi*xaeo + astrom.Cphi*zaeo
v[1] = yaeo
v[2] = -astrom.Cphi*xaeo + astrom.Sphi*zaeo
/* To spherical -HA,Dec. */
C2s(v, &hmobs, &dcobs)
/* Right ascension (with respect to CIO). */
raobs = astrom.Eral + hmobs
/* Return the results. */
*aob = Anp(azobs)
*zob = zdobs
*hob = -hmobs
*dob = dcobs
*rob = Anp(raobs)
}
/*
Atoc13 Observed -> astrometric ICRS
Observed place at a groundbased site to to ICRS astrometric RA,Dec.
The caller supplies UTC, site coordinates, ambient air conditions
and observing wavelength.
Given:
typ string type of coordinates - "R", "H" or "A" (Notes 1,2)
ob1 float64 observed Az, HA or RA (radians; Az is N=0,E=90)
ob2 float64 observed ZD or Dec (radians)
utc1 float64 UTC as a 2-part...
utc2 float64 ...quasi Julian Date (Notes 3,4)
dut1 float64 UT1-UTC (seconds, Note 5)
elong float64 longitude (radians, east +ve, Note 6)
phi float64 geodetic latitude (radians, Note 6)
hm float64 height above ellipsoid (m, geodetic Notes 6,8)
xp,yp float64 polar motion coordinates (radians, Note 7)
phpa float64 pressure at the observer (hPa = mB, Note 8)
tc float64 ambient temperature at the observer (deg C)
rh float64 relative humidity at the observer (range 0-1)
wl float64 wavelength (micrometers, Note 9)
Returned:
rc,dc float64 ICRS astrometric RA,Dec (radians)
Returned (function value):
int status: +1 = dubious year (Note 4)
0 = OK
-1 = unacceptable date
Notes:
1. "Observed" Az,ZD means the position that would be seen by a
perfect geodetically aligned theodolite. (Zenith distance is
used rather than altitude in order to reflect the fact that no
allowance is made for depression of the horizon.) This is
related to the observed HA,Dec via the standard rotation, using
the geodetic latitude (corrected for polar motion), while the
observed HA and RA are related simply through the Earth rotation
angle and the site longitude. "Observed" RA,Dec or HA,Dec thus
means the position that would be seen by a perfect equatorial
with its polar axis aligned to the Earth's axis of rotation.
2. Only the first character of the type argument is significant.
"R" or "r" indicates that ob1 and ob2 are the observed right
ascension and declination; "H" or "h" indicates that they are
hour angle (west +ve) and declination; anything else ("A" or
"a" is recommended) indicates that ob1 and ob2 are azimuth
(north zero, east 90 deg) and zenith distance.
3. utc1+utc2 is quasi Julian Date (see Note 2), apportioned in any
convenient way between the two arguments, for example where utc1
is the Julian Day Number and utc2 is the fraction of a day.
However, JD cannot unambiguously represent UTC during a leap
second unless special measures are taken. The convention in the
present function is that the JD day represents UTC days whether
the length is 86399, 86400 or 86401 SI seconds.
Applications should use the function Dtf2d to convert from
calendar date and time of day into 2-part quasi Julian Date, as
it implements the leap-second-ambiguity convention just
described.
4. The warning status "dubious year" flags UTCs that predate the
introduction of the time scale or that are too far in the
future to be trusted. See Dat for further details.
5. UT1-UTC is tabulated in IERS bulletins. It increases by exactly
one second at the end of each positive UTC leap second,
introduced in order to keep UT1-UTC within +/- 0.9s. n.b. This
practice is under review, and in the future UT1-UTC may grow
essentially without limit.
6. The geographical coordinates are with respect to the WGS84
reference ellipsoid. TAKE CARE WITH THE LONGITUDE SIGN: the
longitude required by the present function is east-positive
(i.e. right-handed), in accordance with geographical convention.
7. The polar motion xp,yp can be obtained from IERS bulletins. The
values are the coordinates (in radians) of the Celestial
Intermediate Pole with respect to the International Terrestrial
Reference System (see IERS Conventions 2003), measured along the
meridians 0 and 90 deg west respectively. For many
applications, xp and yp can be set to zero.
8. If hm, the height above the ellipsoid of the observing station
in meters, is not known but phpa, the pressure in hPa (=mB), is
available, an adequate estimate of hm can be obtained from the
expression
hm = -29.3 * tsl * log ( phpa / 1013.25 );
where tsl is the approximate sea-level air temperature in K
(See Astrophysical Quantities, C.W.Allen, 3rd edition, section
52). Similarly, if the pressure phpa is not known, it can be
estimated from the height of the observing station, hm, as
follows:
phpa = 1013.25 * exp ( -hm / ( 29.3 * tsl ) );
Note, however, that the refraction is nearly proportional to
the pressure and that an accurate phpa value is important for
precise work.
9. The argument wl specifies the observing wavelength in
micrometers. The transition from optical to radio is assumed to
occur at 100 micrometers (about 3000 GHz).
10)The accuracy of the result is limited by the corrections for
refraction, which use a simple A*tan(z) + B*tan^3(z) model.
Providing the meteorological parameters are known accurately and
there are no gross local effects, the predicted astrometric
coordinates should be within 0.05 arcsec (optical) or 1 arcsec
(radio) for a zenith distance of less than 70 degrees, better
than 30 arcsec (optical or radio) at 85 degrees and better
than 20 arcmin (optical) or 30 arcmin (radio) at the horizon.
Without refraction, the complementary functions Atco13 and
Atoc13 are self-consistent to better than 1 microarcsecond
all over the celestial sphere. With refraction included,
consistency falls off at high zenith distances, but is still
better than 0.05 arcsec at 85 degrees.
11)It is advisable to take great care with units, as even unlikely
values of the input parameters are accepted and processed in
accordance with the models used.
Called:
Apco13 astrometry parameters, ICRS-observed
Atoiq quick observed to CIRS
Aticq quick CIRS to ICRS
*/
func Atoc13(typ string, ob1, ob2 float64, utc1, utc2, dut1 float64,
elong, phi, hm, xp, yp float64, phpa, tc, rh, wl float64,
rc, dc *float64) int {
var j int
var astrom ASTROM
var eo, ri, di float64
/* Star-independent astrometry parameters. */
j = Apco13(utc1, utc2, dut1, elong, phi, hm, xp, yp,
phpa, tc, rh, wl, &astrom, &eo)
/* Abort if bad UTC. */
if j < 0 {
return j
}
/* Transform observed to CIRS. */
Atoiq(typ, ob1, ob2, &astrom, &ri, &di)
/* Transform CIRS to ICRS. */
Aticq(ri, di, &astrom, rc, dc)
/* Return OK/warning status. */
return j
}
/*
Atoi13 Observed -> CIRS
Observed place to CIRS. The caller supplies UTC, site coordinates,
ambient air conditions and observing wavelength.
Given:
typ string type of coordinates - "R", "H" or "A" (Notes 1,2)
ob1 float64 observed Az, HA or RA (radians; Az is N=0,E=90)
ob2 float64 observed ZD or Dec (radians)
utc1 float64 UTC as a 2-part...
utc2 float64 ...quasi Julian Date (Notes 3,4)
dut1 float64 UT1-UTC (seconds, Note 5)
elong float64 longitude (radians, east +ve, Note 6)
phi float64 geodetic latitude (radians, Note 6)
hm float64 height above the ellipsoid (meters, Notes 6,8)
xp,yp float64 polar motion coordinates (radians, Note 7)
phpa float64 pressure at the observer (hPa = mB, Note 8)
tc float64 ambient temperature at the observer (deg C)
rh float64 relative humidity at the observer (range 0-1)
wl float64 wavelength (micrometers, Note 9)
Returned:
ri float64 CIRS right ascension (CIO-based, radians)
di float64 CIRS declination (radians)
Returned (function value):
int status: +1 = dubious year (Note 2)
0 = OK
-1 = unacceptable date
Notes:
1. "Observed" Az,ZD means the position that would be seen by a
perfect geodetically aligned theodolite. (Zenith distance is
used rather than altitude in order to reflect the fact that no
allowance is made for depression of the horizon.) This is
related to the observed HA,Dec via the standard rotation, using
the geodetic latitude (corrected for polar motion), while the
observed HA and RA are related simply through the Earth rotation
angle and the site longitude. "Observed" RA,Dec or HA,Dec thus
means the position that would be seen by a perfect equatorial
with its polar axis aligned to the Earth's axis of rotation.
2. Only the first character of the type argument is significant.
"R" or "r" indicates that ob1 and ob2 are the observed right
ascension and declination; "H" or "h" indicates that they are
hour angle (west +ve) and declination; anything else ("A" or
"a" is recommended) indicates that ob1 and ob2 are azimuth
(north zero, east 90 deg) and zenith distance.
3. utc1+utc2 is quasi Julian Date (see Note 2), apportioned in any
convenient way between the two arguments, for example where utc1
is the Julian Day Number and utc2 is the fraction of a day.
However, JD cannot unambiguously represent UTC during a leap
second unless special measures are taken. The convention in the
present function is that the JD day represents UTC days whether
the length is 86399, 86400 or 86401 SI seconds.
Applications should use the function Dtf2d to convert from
calendar date and time of day into 2-part quasi Julian Date, as
it implements the leap-second-ambiguity convention just
described.
4. The warning status "dubious year" flags UTCs that predate the
introduction of the time scale or that are too far in the
future to be trusted. See Dat for further details.
5. UT1-UTC is tabulated in IERS bulletins. It increases by exactly
one second at the end of each positive UTC leap second,
introduced in order to keep UT1-UTC within +/- 0.9s. n.b. This
practice is under review, and in the future UT1-UTC may grow
essentially without limit.
6. The geographical coordinates are with respect to the WGS84
reference ellipsoid. TAKE CARE WITH THE LONGITUDE SIGN: the
longitude required by the present function is east-positive
(i.e. right-handed), in accordance with geographical convention.
7. The polar motion xp,yp can be obtained from IERS bulletins. The
values are the coordinates (in radians) of the Celestial
Intermediate Pole with respect to the International Terrestrial
Reference System (see IERS Conventions 2003), measured along the
meridians 0 and 90 deg west respectively. For many
applications, xp and yp can be set to zero.
8. If hm, the height above the ellipsoid of the observing station
in meters, is not known but phpa, the pressure in hPa (=mB), is
available, an adequate estimate of hm can be obtained from the
expression
hm = -29.3 * tsl * log ( phpa / 1013.25 );
where tsl is the approximate sea-level air temperature in K
(See Astrophysical Quantities, C.W.Allen, 3rd edition, section
52). Similarly, if the pressure phpa is not known, it can be
estimated from the height of the observing station, hm, as
follows:
phpa = 1013.25 * exp ( -hm / ( 29.3 * tsl ) );
Note, however, that the refraction is nearly proportional to
the pressure and that an accurate phpa value is important for
precise work.
9. The argument wl specifies the observing wavelength in
micrometers. The transition from optical to radio is assumed to
occur at 100 micrometers (about 3000 GHz).
10)The accuracy of the result is limited by the corrections for
refraction, which use a simple A*tan(z) + B*tan^3(z) model.
Providing the meteorological parameters are known accurately and
there are no gross local effects, the predicted astrometric
coordinates should be within 0.05 arcsec (optical) or 1 arcsec
(radio) for a zenith distance of less than 70 degrees, better
than 30 arcsec (optical or radio) at 85 degrees and better
than 20 arcmin (optical) or 30 arcmin (radio) at the horizon.
Without refraction, the complementary functions Atio13 and
Atoi13 are self-consistent to better than 1 microarcsecond
all over the celestial sphere. With refraction included,
consistency falls off at high zenith distances, but is still
better than 0.05 arcsec at 85 degrees.
12)It is advisable to take great care with units, as even unlikely
values of the input parameters are accepted and processed in
accordance with the models used.
Called:
Apio13 astrometry parameters, CIRS-observed, 2013
Atoiq quick observed to CIRS
*/
func Atoi13(typ string, ob1, ob2 float64, utc1, utc2, dut1 float64,
elong, phi, hm, xp, yp float64, phpa, tc, rh, wl float64,
ri, di *float64) int {
var j int
var astrom ASTROM
/* Star-independent astrometry parameters for CIRS->observed. */
j = Apio13(utc1, utc2, dut1, elong, phi, hm, xp, yp,
phpa, tc, rh, wl, &astrom)
/* Abort if bad UTC. */
if j < 0 {
return j
}
/* Transform observed to CIRS. */
Atoiq(typ, ob1, ob2, &astrom, ri, di)
/* Return OK/warning status. */
return j
}
/*
Atoiq Quick observed -> CIRS
Quick observed place to CIRS, given the star-independent astrometry
parameters.
Use of this function is appropriate when efficiency is important and
where many star positions are all to be transformed for one date.
The star-independent astrometry parameters can be obtained by
calling Apio[13] or Apco[13].
Given:
typ string type of coordinates: "R", "H" or "A" (Note 1)
ob1 float64 observed Az, HA or RA (radians; Az is N=0,E=90)
ob2 float64 observed ZD or Dec (radians)
astrom ASTROM star-independent astrometry parameters:
pmt float64 PM time interval (SSB, Julian years)
eb [3]float64 SSB to observer (vector, au)
eh [3]float64 Sun to observer (unit vector)
em float64 distance from Sun to observer (au)
v [3]float64 barycentric observer velocity (vector, c)
bm1 float64 sqrt(1-|v|^2): reciprocal of Lorenz factor
bpn [3][3]float64 bias-precession-nutation matrix
along float64 longitude + s' (radians)
xpl float64 polar motion xp wrt local meridian (radians)
ypl float64 polar motion yp wrt local meridian (radians)
sphi float64 sine of geodetic latitude
cphi float64 cosine of geodetic latitude
diurab float64 magnitude of diurnal aberration vector
eral float64 "local" Earth rotation angle (radians)
refa float64 refraction constant A (radians)
refb float64 refraction constant B (radians)
Returned:
ri float64 CIRS right ascension (CIO-based, radians)
di float64 CIRS declination (radians)
Notes:
1. "Observed" Az,ZD means the position that would be seen by a
perfect geodetically aligned theodolite. This is related to
the observed HA,Dec via the standard rotation, using the geodetic
latitude (corrected for polar motion), while the observed HA and
RA are related simply through the Earth rotation angle and the
site longitude. "Observed" RA,Dec or HA,Dec thus means the
position that would be seen by a perfect equatorial with its
polar axis aligned to the Earth's axis of rotation. By removing
from the observed place the effects of atmospheric refraction and
diurnal aberration, the CIRS RA,Dec is obtained.
2. Only the first character of the type argument is significant.
"R" or "r" indicates that ob1 and ob2 are the observed right
ascension and declination; "H" or "h" indicates that they are
hour angle (west +ve) and declination; anything else ("A" or
"a" is recommended) indicates that ob1 and ob2 are azimuth (north
zero, east 90 deg) and zenith distance. (Zenith distance is used
rather than altitude in order to reflect the fact that no
allowance is made for depression of the horizon.)
3. The accuracy of the result is limited by the corrections for
refraction, which use a simple A*tan(z) + B*tan^3(z) model.
Providing the meteorological parameters are known accurately and
there are no gross local effects, the predicted intermediate
coordinates should be within 0.05 arcsec (optical) or 1 arcsec
(radio) for a zenith distance of less than 70 degrees, better
than 30 arcsec (optical or radio) at 85 degrees and better than
20 arcmin (optical) or 25 arcmin (radio) at the horizon.
Without refraction, the complementary functions Atioq and
Atoiq are self-consistent to better than 1 microarcsecond all
over the celestial sphere. With refraction included, consistency
falls off at high zenith distances, but is still better than
0.05 arcsec at 85 degrees.
4. It is advisable to take great care with units, as even unlikely
values of the input parameters are accepted and processed in
accordance with the models used.
Called:
S2c spherical coordinates to unit vector
C2s p-vector to spherical
Anp normalize angle into range 0 to 2pi
*/
func Atoiq(typ string, ob1, ob2 float64, astrom *ASTROM, ri, di *float64) {
/* Minimum sin(alt) for refraction purposes */
const SELMIN = 0.05
var c byte
var c1, c2, sphi, cphi, ce, xaeo, yaeo, zaeo,
xmhdo, ymhdo, zmhdo, az, sz, zdo, refa, refb, tz, dref,
zdt, xaet, yaet, zaet, xmhda, ymhda, zmhda,
f, xhd, yhd, zhd, sx, cx, sy, cy, hma float64
var v [3]float64
/* Coordinate type. */
c = typ[0]
/* Coordinates. */
c1 = ob1
c2 = ob2
/* Sin, cos of latitude. */
sphi = astrom.Sphi
cphi = astrom.Cphi
/* Standardize coordinate type. */
if c == 'r' || c == 'R' {
c = 'R'
} else if c == 'h' || c == 'H' {
c = 'H'
} else {
c = 'A'
}
/* If Az,ZD, convert to Cartesian (S=0,E=90). */
if c == 'A' {
ce = sin(c2)
xaeo = -cos(c1) * ce
yaeo = sin(c1) * ce
zaeo = cos(c2)
} else {
/* If RA,Dec, convert to HA,Dec. */
if c == 'R' {
c1 = astrom.Eral - c1
}
/* To Cartesian -HA,Dec. */
S2c(-c1, c2, &v)
xmhdo = v[0]
ymhdo = v[1]
zmhdo = v[2]
/* To Cartesian Az,El (S=0,E=90). */
xaeo = sphi*xmhdo - cphi*zmhdo
yaeo = ymhdo
zaeo = cphi*xmhdo + sphi*zmhdo
}
/* Azimuth (S=0,E=90). */
// az = ( xaeo != 0.0 || yaeo != 0.0 ) ? atan2(yaeo,xaeo) : 0.0;
if xaeo != 0.0 || yaeo != 0.0 {
az = atan2(yaeo, xaeo)
} else {
az = 0.0
}
/* Sine of observed ZD, and observed ZD. */
sz = sqrt(xaeo*xaeo + yaeo*yaeo)
zdo = atan2(sz, zaeo)
/*
Refraction
----------
*/
/* Fast algorithm using two constant model. */
refa = astrom.Refa
refb = astrom.Refb
// tz = sz / ( zaeo > SELMIN ? zaeo : SELMIN );
if zaeo > SELMIN {
tz = sz / zaeo
} else {
tz = sz / SELMIN
}
dref = (refa + refb*tz*tz) * tz
zdt = zdo + dref
/* To Cartesian Az,ZD. */
ce = sin(zdt)
xaet = cos(az) * ce
yaet = sin(az) * ce
zaet = cos(zdt)
/* Cartesian Az,ZD to Cartesian -HA,Dec. */
xmhda = sphi*xaet + cphi*zaet
ymhda = yaet
zmhda = -cphi*xaet + sphi*zaet
/* Diurnal aberration. */
f = (1.0 + astrom.Diurab*ymhda)
xhd = f * xmhda
yhd = f * (ymhda - astrom.Diurab)
zhd = f * zmhda
/* Polar motion. */
sx = sin(astrom.Xpl)
cx = cos(astrom.Xpl)
sy = sin(astrom.Ypl)
cy = cos(astrom.Ypl)
v[0] = cx*xhd + sx*sy*yhd - sx*cy*zhd
v[1] = cy*yhd + sy*zhd
v[2] = sx*xhd - cx*sy*yhd + cx*cy*zhd
/* To spherical -HA,Dec. */
C2s(v, &hma, di)
/* Right ascension. */
*ri = Anp(astrom.Eral + hma)
}
/*
Ld Light deflection by a single solar-system body
Apply light deflection by a solar-system body, as part of
transforming coordinate direction into natural direction.
Given:
bm float64 mass of the gravitating body (solar masses)
p [3]float64 direction from observer to source (unit vector)
q [3]float64 direction from body to source (unit vector)
e [3]float64 direction from body to observer (unit vector)
em float64 distance from body to observer (au)
dlim float64 deflection limiter (Note 4)
Returned:
p1 [3]float64 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:
Pdp scalar product of two p-vectors
Pxp vector product of two p-vectors
*/
func Ld(bm float64, p, q, e [3]float64, em, dlim float64, p1 *[3]float64) {
var i int
var qpe, eq, peq [3]float64
var qdqpe, w float64
/* q . (q + e). */
for i = 0; i < 3; i++ {
qpe[i] = q[i] + e[i]
}
qdqpe = Pdp(q, qpe)
/* 2 x G x bm / ( em x c^2 x ( q . (q + e) ) ). */
w = bm * SRS / em / gmax(qdqpe, dlim)
/* p x (e x q). */
Pxp(e, q, &eq)
Pxp(p, eq, &peq)
/* Apply the deflection. */
for i = 0; i < 3; i++ {
p1[i] = p[i] + w*peq[i]
}
}
/*
Ldn Light deflection by multiple solar-system bodies
For a star, apply light deflection by multiple solar-system bodies,
as part of transforming coordinate direction into natural direction.
Given:
n int number of bodies (note 1)
b LDBODY[n] data for each of the n bodies (Notes 1,2):
bm float64 mass of the body (solar masses, Note 3)
dl float64 deflection limiter (Note 4)
pv [2][3]float64 barycentric PV of the body (au, au/day)
ob [3]float64 barycentric position of the observer (au)
sc [3]float64 observer to star coord direction (unit vector)
Returned:
sn [3]float64 observer to deflected star (unit vector)
Notes:
1. The array b contains n entries, one for each body to be
considered. If n = 0, no gravitational light deflection will be
applied, not even for the Sun.
2. The array b should include an entry for the Sun as well as for
any planet or other body to be taken into account. The entries
should be in the order in which the light passes the body.
3. In the entry in the b array for body i, the mass parameter
b[i].bm can, as required, be adjusted in order to allow for such
effects as quadrupole field.
4. The deflection limiter parameter b[i].dl is phi^2/2, where phi is
the angular separation (in radians) between star and body at
which limiting is applied. As phi shrinks below the chosen
threshold, the deflection is artificially reduced, reaching zero
for phi = 0. Example values suitable for a terrestrial
observer, together with masses, are as follows:
body i b[i].bm b[i].dl
Sun 1.0 6e-6
Jupiter 0.00095435 3e-9
Saturn 0.00028574 3e-10
5. For cases where the starlight passes the body before reaching the
observer, the body is placed back along its barycentric track by
the light time from that point to the observer. For cases where
the body is "behind" the observer no such shift is applied. If
a different treatment is preferred, the user has the option of
instead using the Ld function. Similarly, Ld can be used
for cases where the source is nearby, not a star.
6. The returned vector sn is not normalized, but the consequential
departure from unit magnitude is always negligible.
7. The arguments sc and sn can be the same array.
8. For efficiency, validation is omitted. The supplied masses must
be greater than zero, the position and velocity vectors must be
right, and the deflection limiter greater than zero.
Reference:
Urban, S. & Seidelmann, P. K. (eds), Explanatory Supplement to
the Astronomical Almanac, 3rd ed., University Science Books
(2013), Section 7.2.4.
Called:
Cp copy p-vector
Pdp scalar product of two p-vectors
Pmp p-vector minus p-vector
Ppsp p-vector plus scaled p-vector
Pn decompose p-vector into modulus and direction
Ld light deflection by a solar-system body
*/
func Ldn(n int, b []LDBODY, ob, sc [3]float64, sn *[3]float64) {
/* Light time for 1 au (days) */
const CR = AULT / DAYSEC
var i int
var v, ev, e [3]float64
var dt, em float64
/* Star direction prior to deflection. */
Cp(sc, sn)
/* Body by body. */
for i = 0; i < n; i++ {
/* Body to observer vector at epoch of observation (au). */
Pmp(ob, b[i].Pv[0], &v)
/* Minus the time since the light passed the body (days). */
dt = Pdp(*sn, v) * CR
/* Neutralize if the star is "behind" the observer. */
dt = gmin(dt, 0.0)
/* Backtrack the body to the time the light was passing the body. */
Ppsp(v, -dt, b[i].Pv[1], &ev)
/* Body to observer vector as magnitude and direction. */
Pn(ev, &em, &e)
/* Apply light deflection for this body. */
Ld(b[i].Bm, *sn, *sn, e, em, b[i].Dl, sn)
/* Next body. */
}
}
/*
Ldsun Light deflection by the Sun
Deflection of starlight by the Sun.
Given:
p [3]float64 direction from observer to star (unit vector)
e [3]float64 direction from Sun to observer (unit vector)
em float64 distance from Sun to observer (au)
Returned:
p1 [3]float64 observer to deflected star (unit vector)
Notes:
1. The source is presumed to be sufficiently distant that its
directions seen from the Sun and the observer are essentially
the same.
2. The deflection is restrained when the angle between the star and
the center of the Sun is less than a threshold value, falling to
zero deflection for zero separation. The chosen threshold value
is within the solar limb for all solar-system applications, and
is about 5 arcminutes for the case of a terrestrial observer.
3. The arguments p and p1 can be the same array.
Called:
Ld light deflection by a solar-system body
*/
func Ldsun(p, e [3]float64, em float64, p1 *[3]float64) {
var em2, dlim float64
/* Deflection limiter (smaller for distant observers). */
em2 = em * em
if em2 < 1.0 {
em2 = 1.0
}
// dlim = 1e-6 / (em2 > 1.0 ? em2 : 1.0);
if em2 > 1.0 {
dlim = 1e-6 / em2
} else {
dlim = 1e-6
}
/* Apply the deflection. */
Ld(1.0, p, p, e, em, dlim, p1)
}
/*
Pmpx Apply proper motion and parallax
Given:
rc,dc float64 ICRS RA,Dec at catalog epoch (radians)
pr float64 RA proper motion (radians/year, Note 1)
pd float64 Dec proper motion (radians/year)
px float64 parallax (arcsec)
rv float64 radial velocity (km/s, +ve if receding)
pmt float64 proper motion time interval (SSB, Julian years)
pob [3]float64 SSB to observer vector (au)
Returned:
pco [3]float64 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:
Pdp scalar product of two p-vectors
Pn decompose p-vector into modulus and direction
*/
func Pmpx(rc, dc float64, pr, pd, px, rv, pmt float64, pob [3]float64, pco *[3]float64) {
/* Km/s to au/year */
const VF = DAYSEC * DJM / DAU
/* Light time for 1 au, Julian years */
const AULTY = AULT / DAYSEC / DJY
var i int
var sr, cr, sd, cd, x, y, z, dt, pxr, w, pdz float64
var p, pm [3]float64
/* Spherical coordinates to unit vector (and useful functions). */
sr = sin(rc)
cr = cos(rc)
sd = sin(dc)
cd = cos(dc)
x = cr * cd
p[0] = x
y = sr * cd
p[1] = y
z = sd
p[2] = z
/* Proper motion time interval (y) including Roemer effect. */
dt = pmt + Pdp(p, pob)*AULTY
/* Space motion (radians per year). */
pxr = px * 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]
}
Pn(p, &w, pco)
}
/*
Pmsafe Apply proper motion, with zero-parallax precautions
Star proper motion: update star catalog data for space motion, with
special handling to handle the zero parallax case.
Given:
ra1 float64 right ascension (radians), before
dec1 float64 declination (radians), before
pmr1 float64 RA proper motion (radians/year), before
pmd1 float64 Dec proper motion (radians/year), before
px1 float64 parallax (arcseconds), before
rv1 float64 radial velocity (km/s, +ve = receding), before
ep1a float64 "before" epoch, part A (Note 1)
ep1b float64 "before" epoch, part B (Note 1)
ep2a float64 "after" epoch, part A (Note 1)
ep2b float64 "after" epoch, part B (Note 1)
Returned:
ra2 float64 right ascension (radians), after
dec2 float64 declination (radians), after
pmr2 float64 RA proper motion (radians/year), after
pmd2 float64 Dec proper motion (radians/year), after
px2 float64 parallax (arcseconds), after
rv2 float64 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 epochs 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 overridden
to ensure that the object is at a finite but very large distance,
but not so large that the proper motion is equivalent to a large
but safe speed (about 0.1c using the chosen constant). A warning
status of 1 is added to the status if this action has been taken.
7. If the space velocity is a significant fraction of c (see the
constant VMAX in the function Starpv), it is arbitrarily set
to zero. When this action occurs, 2 is added to the status.
8. The relativistic adjustment carried out in the Starpv function
involves an iterative calculation. If the process fails to
converge within a set number of iterations, 4 is added to the
status.
Called:
Seps angle between two points
Starpm update star catalog data for space motion
*/
func Pmsafe(ra1, dec1, pmr1, pmd1, px1, rv1 float64,
ep1a, ep1b, ep2a, ep2b float64,
ra2, dec2, pmr2, pmd2, px2, rv2 *float64) int {
/* Minimum allowed parallax (arcsec) */
const PXMIN = 5e-7
/* Factor giving maximum allowed transverse speed of about 1% c */
const F = 326.0
var jpx, j int
var pm, px1a float64
/* Proper motion in one year (radians). */
pm = Seps(ra1, dec1, ra1+pmr1, dec1+pmd1)
/* Override the parallax to reduce the chances of a warning status. */
jpx = 0
px1a = px1
pm *= F
if px1a < pm {
jpx = 1
px1a = pm
}
if px1a < PXMIN {
jpx = 1
px1a = PXMIN
}
/* Carry out the transformation using the modified parallax. */
j = Starpm(ra1, dec1, pmr1, pmd1, px1a, rv1,
ep1a, ep1b, ep2a, ep2b,
ra2, dec2, pmr2, pmd2, px2, rv2)
/* Revise and return the status. */
if (j % 2) != 1 {
j += jpx
}
return j
}
/*
Pvstar Star position+velocity vector to catalog coordinates
Convert star position+velocity vector to catalog coordinates.
Given (Note 1):
pv [2][3]float64 pv-vector (au, au/day)
Returned (Note 2):
ra float64 right ascension (radians)
dec float64 declination (radians)
pmr float64 RA proper motion (radians/year)
pmd float64 Dec proper motion (radians/year)
px float64 parallax (arcsec)
rv float64 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 Starpv), 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 Starpv.
Called:
Pn decompose p-vector into modulus and direction
Pdp scalar product of two p-vectors
Sxp multiply p-vector by scalar
Pmp p-vector minus p-vector
Pm modulus of p-vector
Ppp p-vector plus p-vector
Pv2s pv-vector to spherical
Anp normalize angle into range 0 to 2pi
Reference:
Stumpff, P., 1985, Astron.Astrophys. 144, 232-240.
*/
func Pvstar(pv [2][3]float64, ra, dec, pmr, pmd, px, rv *float64) int {
var r, vr, vt, bett, betr, d, w, del, a, rad, decd, rd float64
var pu, ur, ut, usr, ust [3]float64
/* Isolate the radial component of the velocity (au/day, inertial). */
Pn(pv[0], &r, &pu)
vr = Pdp(pu, pv[1])
Sxp(vr, pu, &ur)
/* Isolate the transverse component of the velocity (au/day, inertial). */
Pmp(pv[1], ur, &ut)
vt = Pm(ut)
/* Special-relativity dimensionless parameters. */
bett = vt / DC
betr = vr / DC
/* The observed-to-inertial correction terms. */
d = 1.0 + betr
w = betr*betr + bett*bett
if d == 0.0 || w > 1.0 {
return -1
}
del = -w / (sqrt(1.0-w) + 1.0)
/* Scale inertial tangential velocity vector into observed (au/d). */
Sxp(1.0/d, ut, &ust)
/* Compute observed radial velocity vector (au/d). */
Sxp(DC*(betr-del)/d, pu, &usr)
/* Combine the two to obtain the observed velocity vector (au/day). */
Ppp(usr, ust, &pv[1])
/* Cartesian to spherical. */
Pv2s(pv, &a, dec, &r, &rad, &decd, &rd)
if r == 0.0 {
return -2
}
/* Return RA in range 0 to 2pi. */
*ra = Anp(a)
/* Return proper motions in radians per year. */
*pmr = rad * DJY
*pmd = decd * DJY
/* Return parallax in arcsec. */
*px = DR2AS / r
/* Return radial velocity in km/s. */
*rv = 1e-3 * rd * DAU / DAYSEC
/* Success. */
return 0
}
/*
Pvtob Observatory position and velocity
Position and velocity of a terrestrial observing station.
Given:
elong float64 longitude (radians, east +ve, Note 1)
phi float64 latitude (geodetic, radians, Note 1)
hm float64 height above ref. ellipsoid (geodetic, m)
xp,yp float64 coordinates of the pole (radians, Note 2)
sp float64 the TIO locator s' (radians, Note 2)
theta float64 Earth rotation angle (radians, Note 3)
Returned:
pv [2][3]float64 position/velocity vector (m, m/s, CIRS)
Notes:
1. The terrestrial coordinates are with respect to the WGS84
reference ellipsoid.
2. xp and yp are the coordinates (in radians) of the Celestial
Intermediate Pole with respect to the International Terrestrial
Reference System (see IERS Conventions), measured along the
meridians 0 and 90 deg west respectively. sp is the TIO locator
s', in radians, which positions the Terrestrial Intermediate
Origin on the equator. For many applications, xp, yp and
(especially) sp can be set to zero.
3. If theta is Greenwich apparent sidereal time instead of Earth
rotation angle, the result is with respect to the true equator
and equinox of date, i.e. with the x-axis at the equinox rather
than the celestial intermediate origin.
4. The velocity units are meters per UT1 second, not per SI second.
This is unlikely to have any practical consequences in the modern
era.
5. No validation is performed on the arguments. Error cases that
could lead to arithmetic exceptions are trapped by the Gd2gc
function, and the result set to zeros.
References:
McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003),
IERS Technical Note No. 32, BKG (2004)
Urban, S. & Seidelmann, P. K. (eds), Explanatory Supplement to
the Astronomical Almanac, 3rd ed., University Science Books
(2013), Section 7.4.3.3.
Called:
Gd2gc geodetic to geocentric transformation
Pom00 polar motion matrix
Trxp product of transpose of r-matrix and p-vector
*/
func Pvtob(elong, phi, hm float64, xp, yp, sp, theta float64, pv *[2][3]float64) {
/* Earth rotation rate in radians per UT1 second */
const OM = 1.00273781191135448 * D2PI / DAYSEC
var xyz, xyzm [3]float64
var rpm [3][3]float64
var x, y, z, s, c float64
/* Geodetic to geocentric transformation (WGS84). */
Gd2gc(1, elong, phi, hm, &xyzm)
/* Polar motion and TIO position. */
Pom00(xp, yp, sp, &rpm)
Trxp(rpm, xyzm, &xyz)
x = xyz[0]
y = xyz[1]
z = xyz[2]
/* Functions of ERA. */
s = sin(theta)
c = cos(theta)
/* Position. */
pv[0][0] = c*x - s*y
pv[0][1] = s*x + c*y
pv[0][2] = z
/* Velocity. */
pv[1][0] = OM * (-s*x - c*y)
pv[1][1] = OM * (c*x - s*y)
pv[1][2] = 0.0
}
/*
Refco Refraction constants
Determine the constants A and B in the atmospheric refraction model
dZ = A tan Z + B tan^3 Z.
Z is the "observed" zenith distance (i.e. affected by refraction)
and dZ is what to add to Z to give the "topocentric" (i.e. in vacuo)
zenith distance.
Given:
phpa float64 pressure at the observer (hPa = millibar)
tc float64 ambient temperature at the observer (deg C)
rh float64 relative humidity at the observer (range 0-1)
wl float64 wavelength (micrometers)
Returned:
refa float64 tan Z coefficient (radians)
refb float64 tan^3 Z coefficient (radians)
Notes:
1. The model balances speed and accuracy to give good results in
applications where performance at low altitudes is not paramount.
Performance is maintained across a range of conditions, and
applies to both optical/IR and radio.
2. The model omits the effects of (i) height above sea level (apart
from the reduced pressure itself), (ii) latitude (i.e. the
flattening of the Earth), (iii) variations in tropospheric lapse
rate and (iv) dispersive effects in the radio.
The model was tested using the following range of conditions:
lapse rates 0.0055, 0.0065, 0.0075 deg/meter
latitudes 0, 25, 50, 75 degrees
heights 0, 2500, 5000 meters ASL
pressures mean for height -10% to +5% in steps of 5%
temperatures -10 deg to +20 deg with respect to 280 deg at SL
relative humidity 0, 0.5, 1
wavelengths 0.4, 0.6, ... 2 micron, + radio
zenith distances 15, 45, 75 degrees
The accuracy with respect to raytracing through a model
atmosphere was as follows:
worst RMS
optical/IR 62 mas 8 mas
radio 319 mas 49 mas
For this particular set of conditions:
lapse rate 0.0065 K/meter
latitude 50 degrees
sea level
pressure 1005 mb
temperature 280.15 K
humidity 80%
wavelength 5740 Angstroms
the results were as follows:
ZD raytrace Refco Saastamoinen
10 10.27 10.27 10.27
20 21.19 21.20 21.19
30 33.61 33.61 33.60
40 48.82 48.83 48.81
45 58.16 58.18 58.16
50 69.28 69.30 69.27
55 82.97 82.99 82.95
60 100.51 100.54 100.50
65 124.23 124.26 124.20
70 158.63 158.68 158.61
72 177.32 177.37 177.31
74 200.35 200.38 200.32
76 229.45 229.43 229.42
78 267.44 267.29 267.41
80 319.13 318.55 319.10
deg arcsec arcsec arcsec
The values for Saastamoinen's formula (which includes terms
up to tan^5) are taken from Hohenkerk and Sinclair (1985).
3. A wl value in the range 0-100 selects the optical/IR case and is
wavelength in micrometers. Any value outside this range selects
the radio case.
4. Outlandish input parameters are silently limited to
mathematically safe values. Zero pressure is permissible, and
causes zeroes to be returned.
5. The algorithm draws on several sources, as follows:
a) The formula for the saturation vapour pressure of water as
a function of temperature and temperature is taken from
Equations (A4.5-A4.7) of Gill (1982).
b) The formula for the water vapour pressure, given the
saturation pressure and the relative humidity, is from
Crane (1976), Equation (2.5.5).
c) The refractivity of air is a function of temperature,
total pressure, water-vapour pressure and, in the case
of optical/IR, wavelength. The formulae for the two cases are
developed from Hohenkerk & Sinclair (1985) and Rueger (2002).
The IAG (1999) optical refractivity for dry air is used.
d) The formula for beta, the ratio of the scale height of the
atmosphere to the geocentric distance of the observer, is
an adaption of Equation (9) from Stone (1996). The
adaptations, arrived at empirically, consist of (i) a small
adjustment to the coefficient and (ii) a humidity term for the
radio case only.
e) The formulae for the refraction constants as a function of
n-1 and beta are from Green (1987), Equation (4.31).
References:
Crane, R.K., Meeks, M.L. (ed), "Refraction Effects in the Neutral
Atmosphere", Methods of Experimental Physics: Astrophysics 12B,
Academic Press, 1976.
Gill, Adrian E., "Atmosphere-Ocean Dynamics", Academic Press,
1982.
Green, R.M., "Spherical Astronomy", Cambridge University Press,
1987.
Hohenkerk, C.Y., & Sinclair, A.T., NAO Technical Note No. 63,
1985.
IAG Resolutions adopted at the XXIIth General Assembly in
Birmingham, 1999, Resolution 3.
Rueger, J.M., "Refractive Index Formulae for Electronic Distance
Measurement with Radio and Millimetre Waves", in Unisurv Report
S-68, School of Surveying and Spatial Information Systems,
University of New South Wales, Sydney, Australia, 2002.
Stone, Ronald C., P.A.S.P. 108, 1051-1058, 1996.
*/
func Refco(phpa, tc, rh, wl float64, refa, refb *float64) {
var optic bool
var p, t, r, w, ps, pw, tk, wlsq, gamma, beta float64
/* Decide whether optical/IR or radio case: switch at 100 microns. */
optic = (wl <= 100.0)
/* Restrict parameters to safe values. */
t = gmax(tc, -150.0)
t = gmin(t, 200.0)
p = gmax(phpa, 0.0)
p = gmin(p, 10000.0)
r = gmax(rh, 0.0)
r = gmin(r, 1.0)
w = gmax(wl, 0.1)
w = gmin(w, 1e6)
/* Water vapour pressure at the observer. */
if p > 0.0 {
ps = pow(10.0, (0.7859+0.03477*t)/
(1.0+0.00412*t)) *
(1.0 + p*(4.5e-6+6e-10*t*t))
pw = r * ps / (1.0 - (1.0-r)*ps/p)
} else {
pw = 0.0
}
/* Refractive index minus 1 at the observer. */
tk = t + 273.15
if optic {
wlsq = w * w
gamma = ((77.53484e-6+
(4.39108e-7+3.666e-9/wlsq)/wlsq)*p -
11.2684e-6*pw) / tk
} else {
gamma = (77.6890e-6*p - (6.3938e-6-0.375463/tk)*pw) / tk
}
/* Formula for beta from Stone, with empirical adjustments. */
beta = 4.4474e-6 * tk
if !optic {
beta -= 0.0074 * pw * beta
}
/* Refraction constants from Green. */
*refa = gamma * (1.0 - beta)
*refb = -gamma * (beta - gamma/2.0)
}
/*
Starpm Proper motion between two epochs
Star proper motion: update star catalog data for space motion.
Given:
ra1 float64 right ascension (radians), before
dec1 float64 declination (radians), before
pmr1 float64 RA proper motion (radians/year), before
pmd1 float64 Dec proper motion (radians/year), before
px1 float64 parallax (arcseconds), before
rv1 float64 radial velocity (km/s, +ve = receding), before
ep1a float64 "before" epoch, part A (Note 1)
ep1b float64 "before" epoch, part B (Note 1)
ep2a float64 "after" epoch, part A (Note 1)
ep2b float64 "after" epoch, part B (Note 1)
Returned:
ra2 float64 right ascension (radians), after
dec2 float64 declination (radians), after
pmr2 float64 RA proper motion (radians/year), after
pmd2 float64 Dec proper motion (radians/year), after
px2 float64 parallax (arcseconds), after
rv2 float64 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 Starpv
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 Starpv), it is arbitrarily set
to zero. When this action occurs, 2 is added to the status.
8. The relativistic adjustment carried out in the Starpv function
involves an iterative calculation. If the process fails to
converge within a set number of iterations, 4 is added to the
status.
Called:
Starpv star catalog data to space motion pv-vector
Pvu update a pv-vector
Pdp scalar product of two p-vectors
Pvstar space motion pv-vector to star catalog data
*/
func Starpm(ra1, dec1, pmr1, pmd1, px1, rv1 float64, ep1a, ep1b, ep2a, ep2b float64,
ra2, dec2, pmr2, pmd2, px2, rv2 *float64) int {
var pv1, pv, pv2 [2][3]float64
var tl1, dt, r2, rdv, v2, c2mv2, tl2 float64
var j1, j2 int
/* RA,Dec etc. at the "before" epoch to space motion pv-vector. */
j1 = Starpv(ra1, dec1, pmr1, pmd1, px1, rv1, &pv1)
/* Light time when observed (days). */
tl1 = Pm(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. */
Pvu(dt+tl1, pv1, &pv)
/* From this geometric position, deduce the observed light time (days) */
/* at the "after" epoch (with theoretically unneccessary error check). */
r2 = Pdp(pv[0], pv[0])
rdv = Pdp(pv[0], pv[1])
v2 = Pdp(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. */
Pvu(dt+(tl1-tl2), pv1, &pv2)
/* Space motion pv-vector to RA,Dec etc. at the "after" epoch. */
j2 = Pvstar(pv2, ra2, dec2, pmr2, pmd2, px2, rv2)
/* Final status. */
if j2 == 0 {
return j1
}
return -1
}
/*
Starpv Star catalog coordinates to position+velocity vector
Convert star catalog coordinates to position+velocity vector.
Given (Note 1):
ra float64 right ascension (radians)
dec float64 declination (radians)
pmr float64 RA proper motion (radians/year)
pmd float64 Dec proper motion (radians/year)
px float64 parallax (arcseconds)
rv float64 radial velocity (km/s, positive = receding)
Returned (Note 2):
pv [2][3]float64 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
Pvstar.
Called:
S2pv spherical coordinates to pv-vector
Pm modulus of p-vector
Zp zero p-vector
Pn decompose p-vector into modulus and direction
Pdp scalar product of two p-vectors
Sxp multiply p-vector by scalar
Pmp p-vector minus p-vector
Ppp p-vector plus p-vector
*/
func Starpv(ra, dec, pmr, pmd, px, rv float64, pv *[2][3]float64) int {
/* Smallest allowed parallax */
const PXMIN = 1e-7
/* Largest allowed speed (fraction of c) */
const VMAX = 0.5
/* Maximum number of iterations for relativistic solution */
const IMAX = 100
var i, iwarn int
var w, r, rd, rad, decd, v,
vsr, vst, betst, betsr, bett, betr,
dd, ddel,
d, del,
odd, oddel,
od, odel float64
d, del = 0.0, 0.0 /* to prevent */
odd, oddel = 0.0, 0.0 /* compiler */
od, odel = 0.0, 0.0 /* warnings */
var pu, usr, ust, ur, ut [3]float64
/* Distance (au). */
if px >= PXMIN {
w = px
iwarn = 0
} else {
w = PXMIN
iwarn = 1
}
r = DR2AS / w
/* Radial speed (au/day). */
rd = DAYSEC * rv * 1e3 / DAU
/* Proper motion (radian/day). */
rad = pmr / DJY
decd = pmd / DJY
/* To pv-vector (au,au/day). */
S2pv(ra, dec, r, rad, decd, rd, pv)
/* If excessive velocity, arbitrarily set it to zero. */
v = Pm(pv[1])
if (v / DC) > VMAX {
Zp(&pv[1])
iwarn += 2
}
/* Isolate the radial component of the velocity (au/day). */
Pn(pv[0], &w, &pu)
vsr = Pdp(pu, pv[1])
Sxp(vsr, pu, &usr)
/* Isolate the transverse component of the velocity (au/day). */
Pmp(pv[1], usr, &ust)
vst = Pm(ust)
/* Special-relativity dimensionless parameters. */
betsr = vsr / DC
betst = vst / DC
/* Determine the observed-to-inertial relativistic correction terms. */
bett = betst
betr = betsr
for i = 0; i < IMAX; i++ {
d = 1.0 + betr
w = betr*betr + bett*bett
del = -w / (sqrt(1.0-w) + 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
}
// Scale observed tangential velocity vector into inertial (au/d). */
Sxp(d, ust, &ut)
/* Compute inertial radial velocity vector (au/d). */
Sxp(DC*(d*betsr+del), pu, &ur)
/* Combine the two to obtain the inertial space velocity vector. */
Ppp(ur, ut, &pv[1])
/* Return the status. */
return iwarn
}
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