Example application of the IAU 2000 resolutions
concerning Earth orientation and rotation
Patrick Wallace
HMNAO/CCLRC/RAL, p.t.wallace@rl.ac.uk
(Original version 20 July 2004, revised 29 July
2004;
this reformatted and corrected version 20
January 2006)
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Here is
an example application of the IAU 2000 resolutions concerning Earth orientation
and rotation. The objective is to
predict the topocentric apparent direction of a star.
The
nomenclature scheme I am using is as follows.
Coordinates are in upper case;
transformations and corrections are in mixed case.
CATALOG RA,DEC
|
proper motion, catalog epoch to J2000
|
ICRS RA,DEC, EPOCH J2000 <- example starts
here
|
proper motion, J2000 to date
|
BCRS RA,DEC
|
parallax
|
ASTROMETRIC RA,DEC
|
light deflection
annual aberration
|
GCRS RA,DEC
|
frame bias
precession
nutation
|
CIRS RA,DEC
|
Earth rotation
|
TIRS
|
polar motion
|
ITRS
|
longitude
|
diurnal aberration and parallax
|
TOPOCENTRIC HA,DEC
|
latitude
|
|
refraction
|
OBSERVED AZ,ALT
The
circumstances are as follows:
Site 9.712156 deg E, 52.385639 deg N,
Fictitious "Tycho 2" star, epoch
2000
RA,Dec = 353.22987757, +52.27730247 deg
Proper motions +22.9, -2.1 mas/year
Parallax 23 mas
Radial velocity +25 km/s
UTC 2003/08/26 00:37:38.973810
IERS X,Y,DUT1 corrections: +0.038,-0.118 mas, -0.349535 s
IERS polar motion x_p = +0.259371, y_p = +0.415573 arcsec
The
results in summary form are as follows (in degrees):
RA,Dec, ICRS epoch 2000 353.22987757000, +52.27730247000
RA,Dec, BCRS 353.22991549972,
+52.27730034185
RA,Dec, astrometric 353.22991889091, +52.27730584235
RA,Dec, GCRS 353.23789320667,
+52.27695262534
RA,Dec, CIRS 353.23300208264,
+52.29554173960
HA,Dec, topocentric -0.29507962185, +52.29549062657
Az,Alt, topocentric 116.44983979538, +89.79843387822
Az,Alt, observed 116.44983979538, +89.79848801461
BCRS
stands for baryentric celestial reference system. GCRS stands for geocentric celestial
reference system. CIRS stands for
celestial intermediate reference system; this RA,Dec is the IAU 2000
counterpart to geocentric apparent place, and is referred to the celestial intermediate
origin (CIO) instead of the equinox. The
coordinates HA,Dec and Az,Alt are not affected by the introduction of the IAU
2000 methods.
The key
variables and some of the workings are presented below.
Before
we apply the IAU 2000 celestial-to-terrestrial transformation, we must first
estimate the geocentric direction of the star by applying the usual corrections
for proper motion, light deflection and annual aberration.
For the
example date:
TT
52877.0268884005800000 (MJD)
we will
use the following (approximate) Earth ephemeris data:
heliocentric posn +0.895306712607 -0.430362177777 -0.186583142292 AU
barycentric posn +0.898130398596 -0.433663195906 -0.188058184682 AU
barycentric vel +0.007714484109 +0.013933051305 +0.006040258850 AU/day
The ICRS
RA,Dec at the catalog epoch (2000.0) is:
RA,Dec, ICRS epoch 2000 353.22987757000, +52.27730247000
Applying
space motion and parallax, we obtain the astrometric place - the ICRS
coordinates of an infinitely distant star coincident with the
one
under study:
astrometric place 353.22991889091, +52.27730584235
The
light deflection from the Sun (we will neglect the other solar-system bodies)
takes us to:
with light deflection 353.22991848163, +52.27730517509
Annual
aberration produces the proper direction, which is the GCRS RA,Dec:
with aberration 353.23789320667, +52.27695262534
This
completes the preliminaries; we are now
ready to apply the IAU 2000 celestial-to-terrestrial transformations that are
the purpose of this demonstration.
To
predict the orientation of the Earth we can either implement the IAU resolutions
literally, working via classical methods, or we can use the direct series
supplied by IERS. The former is somewhat
more computationally efficient (and more instructive), the latter easier to get
right.
To
demonstrate the classical methods, we start with the frame bias:
dpsi
-41.7750 mas
depsilon
-6.8192 mas
dRA
-14.6000 mas
giving
an ICRS to mean J2000 rotation matrix of:
+0.99999999999999420000 -0.00000007078279744199 +0.00000008056217146976
+0.00000007078279477857 +0.99999999999999690000 +0.00000003306041454222
-0.00000008056217380987 -0.00000003306040883981 +0.99999999999999620000
The IAU
2000 corrections to the IAU 1976 precession, for the current date, are:
dpsi
-10.932010 mas
depsilon
-0.920821 mas
giving
an obliquity of date of:
epsilon_A
+84379.739145661 arcsec
and
ultimately a J2000-to-date classical precession matrix:
+0.99999960442692650000 -0.00081577397935781730 -0.00035448385722876160
+0.00081577398094000060 +0.99999966725634080000 -0.00000014012603875794
+0.00035448385358768210 -0.00000014905272408423 +0.99999993717058570000
Using
the IAU 2000A model, we obtain the classical nutation components
dpsi and
depsilon:
luni-solar
-12.687774156,
+5.669802082 arcsec
planetary +0.000048676, +0.000119415 arcsec
which
combine to give:
nutation, IAU 2000A -12.687725480, +5.669921497 arcsec
and the
following classical mean-to-true nutation matrix:
+0.99999999810814740000 +0.00005643620233914664 +0.00002446753280028101
-0.00005643552974070033 +0.99999999802968630000 -0.00002748924554327810
-0.00002446908414069593 +0.00002748786465307690 +0.99999999932284070000
Combining
the rotation matrices, we obtain for the classical NxPxB matrix:
+0.99999965722043850000 -0.00075940856976379120 -0.00032993579616347590
+0.00075939951242126470 +0.99999971127592400000 -0.00002757624279218965
+0.00032995664253816620 +0.00002732568025683818 +0.99999994519095910000
This
matrix could be used to transform the geocentric place into the true place of
date, with the CIP as the pole and the true equinox of date as the RA
origin. The transformation to
terrestrial coordinates would then require the mean sidereal time and the
equation of the equinoxes, complicated functions involving both UT and TT. As we are following IAU 2000 methods, we are
instead going to work via the celestial intermediate system, so that the
transformation to terrestrial coordinates requires only the Earth rotation
angle, a simple linear function of UT.
We
extract the CIP X,Y coordinates from the matrix (elements (3,1) and
(3,2)):
CIP X,Y +0.000329956642538, +0.000027325680257
(As
mentioned earlier we can, if we wish, generate X,Y directly from series. For the date in question we obtain:
from series +0.000329956644592, +0.000027325684592
Note the
good agreement.)
If the
utmost accuracy is needed, the next step is to add the small IERS corrections
(which come from VLBI observations):
IAU 2000A +0.000329956642538, +0.000027325680257
corrected +0.000329956826767 +0.000027325108177
Having
obtained X,Y, we compute the small quantity s using the series for s+XY/2:
s
-2.900355 mas
From X,
Y and s we can obtain the rotation matrix that transforms directions from the
celestial reference system into the celestial intermediate system of date:
+0.99999994556424450000 +0.00000000955326444293 -0.00032995682715159210
-0.00000001856936992539 +0.99999999962666920000 -0.00002732510353706674
+0.00032995682676736510 +0.00002732510817669448 +0.99999994519091400000
Earlier,
we computed the RA,Dec in the celestial reference system at which the star
appears:
GCRS
353.23789320667, +52.27695262534
The
rotation matrix transforms this into the celestial intermediate system:
CIRS
353.23300208264, +52.29554173960
We are
now ready to move from coordinates on the celestial sphere into coordinates on
the Earth. This involves Earth rotation
together with three small effects:
diurnal aberration << 1 arcsec
s' << 0.1 mas
polar motion << 1 arcsec
Because
these effects are small, the precise way they are applied is of little
consequence; individual applications
thus have some leeway. For example, in some applications it may be more
convenient to deal with the diurnal aberration at the same time as annual
aberration and to eliminate the geocentric stage completely. In a similar way, some applications will need
the star direction in terms of the geographical coordinates of the sub-star
point whereas others (and this example) will express the same information in
the form of a local hour angle and declination.
In all
cases, the Earth rotation angle is the main element in the transformation. The UT1 is:
UT1
52877.02614148466 (MJD)
and the
corresponding ERA is:
ERA
343.2256920994647 degrees
The IERS
Conventions set out the CIRS-to-ITRS transformation as a large rotation about
the z-axis corresponding to the ERA, followed by small z, y and x rotations
that take into account s' and polar motion.
The tiny
quantity s' for the given date is:
s'
-0.001714681 mas
The
rotation matrix for the s' and polar motion portion is:
+0.99999999999920940000 -0.00000000000831300878 +0.00000125746609283028
+0.00000000001084649458 +0.99999999999797040000 -0.00000201475475899438
-0.00000125746609281098 +0.00000201475475900642 +0.99999999999717980000
Allowing
for diurnal aberration, Earth rotation, site longitude and polar motion
produces the following local HA,Dec:
HA,Dec, topocentric -0.29507962185, +52.29549062657
or,
rotated into local horizon coordinates:
Az,Alt, topocentric 116.44983979538, +89.79843387822
For this
site and time, the star has just passed almost overhead. Note that this is the "topocentric"
rather than "observed" position;
if pointing a real telescope or antenna, the next correction would be
for atmospheric refraction.
Note
that some numbers in this report are quoted to a number of decimal places
beyond that corresponding to the floating-point precision of the computer used
to generate them. Repeating the
calculations on a different platform may produce slightly different results.
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