USNO Circular 179 - U.S. Naval Observatory

58 MODELING THE EARTH’S ROTATION
right ascension of the object, measured with respect to the true equinox, and λ is the longitude
of the observer (corrected, where necessary, for polar motion). Obviously these quantities must
all be given in the same units. In the CIO-based scheme, the apparent place would be expressed
in the Celestial Intermediate Reference System (Eσ), and h = θ – ασ + λ, where θ is the Earth
Rotation Angle and ασ is the apparent right ascension of the object, measured with respect to the
CIO. (The recommended terminology is intermediate place for the object position in the Eσ and
intermediate right ascension for the quantity ασ, although some people hold that the term right
ascension should refer only to an origin at the equinox.) In the CIO-based formula, precession and
nutation come into play only once, in expressing the object’s right ascension in the Eσ system. See
section 6.5.4 for more details.
6.5 Formulas
The formulas below draw heavily on the developments presented previously. In particular, the
3×3 matrices P, N, and B represent the transformations for precession, nutation, and frame bias,
respectively, and are taken directly from Chapters 5 and 3. The matrices P and N are functions
of time, t. The time measured in Julian centuries of TDB (or TT) from J2000.0 is denoted T and
is given by T = (JD(TDB) – 2451545.0)/36525. The elementary rotation matrices R1, R2, and
R3 are defined in “Abbreviations and Symbols Frequently Used”. Formulas from Chapter 2 for
sidereal time and the Earth Rotation Angle are used. Explanations of, and formulas for the time
scales UT1, TT, and TDB are also found in Chapter 2.
The ultimate objective is to express “local” Earth-fixed vectors, representing geographic positions,
baselines, instrumental axes and boresights, etc., in the GCRS, where they can be related to
the proper coordinates 5 of celestial objects. As mentioned above, the “GCRS” is a reference system
that can be thought of as the “geocentric ICRS”. Celestial coordinates in the GCRS are obtained
from basic ICRS data (which are barycentric) by applying the usual algorithms for proper place;
see section 1.3. In this chapter we will be working entirely in a geocentric system and the GCRS
will be obtained from a series of rotations that start with an ordinary Earth-fixed geodetic system.
6.5.1 Location of Cardinal Points
We will start by establishing the positions of three cardinal points within the GCRS: the Celestial
Intermediate Pole (CIP), the true equinox of date (Υ), and the Celestial Intermediate Origin (CIO).
The unit vectors toward these points will be designated n GCRS , Υ GCRS , and σ GCRS , respectively. As
the Earth precesses and nutates in local inertial space, these points are in continual motion.
The CIP and the equinox can easily be located in the GCRS at any time t simply by recognizing
that they are, respectively, the z- and x-axes of the true equator and equinox of date system (E Υ )
at t. The unit vectors therefore are:
5 See footnote 1 on page 41.
CIP : n GCRS (t) = B T P T (t) N T (t)
Equinox: Υ GCRS (t) = B T P T (t) N T (t)
⎛
⎜
⎝
⎛
⎜
⎝
0
0
1
1
0
0
⎞
⎟
⎠
⎞
⎟
⎠
(6.1)