USNO Circular 179 - U.S. Naval Observatory

62 MODELING THE EARTH’S ROTATION
N westward to the CIO (on the instantaneous equator) and the length of the arc from N westward
to the GCRS origin of right ascension (on the GCRS equator). The quantity s is called the CIO
locator. If σ represents the CIO and Σ0 represents the right ascension origin of the GCRS (the
direction of the GCRS x-axis), then
s = σN − Σ0N (6.6)
See Fig. 6.3, where the points Σ0 and Σ ′ 0 are equidistant from the node N. The quantity s is seen
to be the “extra” length of the arc on the instantaneous equator from N to σ, the position of the
CIO. The value of s is fundamentally obtained from an integral,
t X(t)
s(t) = −
t0
˙ Y (t) − Y (t) ˙ X(t)
dt + s0
1 + Z(t)
where X(t), Y (t), and Z(t) are the three components of the unit vector, n GCRS (t), toward the
celestial pole (CIP). See, e.g., Capitaine et al. (2000) or the IERS Conventions (2003). The constant
of integration, s0, has been set to ensure that the equinox-based and CIO-based computations of
Earth rotation yield the same answers; s0=94 µas (IERS Conventions 2003). Effectively, the
constant adjusts the position of the CIO on the equator and is thus part of the arc σN. For
practical purposes, the value of s at any given time is provided by a series expansion, given in
Table 5.2c of the IERS Conventions (2003). Software to evaluate this series is available at the
IERS Conventions web site and is also part of the SOFA package. There is a table of daily values
of s in Section B of The Astronomical Almanac.
At any time, t, the the unit vector toward the node N is simply N GCRS = (−Y, X, 0)/ √ X 2 + Y 2
(where we are no longer explicitly indicating the time dependence of X and Y ). To locate the CIO,
we rearrange eq. 6.6 to yield the arc length σN:
(6.7)
σN = s + Σ0N = s + arctan(X/(−Y )) (6.8)
The location of the CIO is then obtained by starting at the node N and moving along the instantaneous
equator of t through the arc σN. That is, σ GCRS can be constructed by taking the position
vector of the node N in the GCRS, N GCRS , and rotating it counterclockwise by the angle −σN (i.e.,
clockwise by σN) about the axis n GCRS . Equivalently,
σ GCRS = N GCRS cos(σN) − (n GCRS × N GCRS ) sin(σN) (6.9)
The three methods for determining the position of the CIO in the GCRS are numerically the
same to within several microarcseconds (µas) over six centuries centered on J2000.0. We now
have formulas in hand for obtaining the positions of the three cardinal points on the sky — the
CIP, the CIO, and the equinox — that are involved in the ITRS-to-GCRS (terrestrial-to-celestial)
transformations. In the following, it is assumed that n GCRS , σ GCRS , and Υ GCRS are known vectors
for some time t of interest.