USNO Circular 179 - U.S. Naval Observatory

60 MODELING THE EARTH’S ROTATION
the true right ascension of the CIO). The equation of the origins is also the difference θ − GAST.
It therefore equals the accumulated precession-nutation of the equinox in right ascension, given
by the sum of the terms in parentheses from eq. 2.12 and the equation of the equinoxes given in
eq. 2.14 (all times –1). The equation of the origins in arcseconds therefore is:
Eo = − 0.014506 − 4612.156534 T − 1.3915817 T 2 + 0.00000044 T 3
+ 0.000029956 T 4 + 0.0000000368 T 5 − ∆ψ cos ɛ
− 0.00264096 sin (Ω)
− 0.00006352 sin (2Ω)
− 0.00001175 sin (2F − 2D + 3Ω)
− 0.00001121 sin (2F − 2D + Ω) (6.2)
+ 0.00000455 sin (2F − 2D + 2Ω)
− 0.00000202 sin (2F + 3Ω)
− 0.00000198 sin (2F + Ω)
+ 0.00000172 sin (3Ω)
+ 0.00000087 T sin (Ω) + · · ·
where T is the number of centuries of TDB (or TT) from J2000.0; ∆ψ is the nutation in longitude,
in arcseconds; ɛ is the mean obliquity of the ecliptic; and F , D, and Ω are fundamental luni-solar
arguments. All of the angles are functions of time; see Chapter 5 for expressions (esp. eqs. 5.12,
5.15, & 5.19). There is a table of daily values of Eo in Section B of The Astronomical Almanac.
To transform an object’s celestial coordinates from the true equator and equinox of t to the
Celestial Intermediate System (i.e., from E Υ to Eσ), simply add Eo to the object’s true right
ascension. To similarly transform the components of a position vector, apply the rotation R3(−Eo).
Since many existing software systems are set up to produce positions with respect to the equator and
equinox of date, this is a relatively easy way to convert those positions to the Celestial Intermediate
Reference System if desired. Note that in such a case there is no computational difference in using
either the equinox-based or CIO-based methods for computing hour angle: Eo is computed in both
methods and is just applied to different quantities. In the equinox-based method, Eo is subtracted
from θ to form sidereal time; in the CIO-based method, Eo is added to the object’s true right
ascension so that θ can be used in place of sidereal time.
The position of the CIO in the GCRS, σ GCRS , can be established by taking the position vector of
the equinox in the GCRS, Υ GCRS , and rotating it counterclockwise by the angle −Eo (i.e., clockwise
by Eo) about the axis n GCRS . Equivalently, establish the orthonormal basis triad of the equatorand-equinox
system within the GCRS: Υ GCRS , (n GCRS × Υ GCRS ), and n GCRS . Then
σ GCRS = Υ GCRS cos Eo − (n GCRS × Υ GCRS ) sin Eo (6.3)
6.5.1.2 CIO Location from Numerical Integration
As described above, a non-rotating origin can be described as a point on the moving equator whose
instantaneous motion is always orthogonal to the equator. A simple geometric construction based
on this definition yields the following differential equation for the motion of a non-rotating origin:
˙σ(t) = − σ(t) · ˙n(t) n(t) (6.4)