USNO Circular 179 - U.S. Naval Observatory

MODELING THE EARTH’S ROTATION 65
the CIP; the x-axis is toward σ GCRS , the CIO; and the y-axis is toward n GCRS × σ GCRS . Call the
latter vector y GCRS . Then:
C T
=
σ GCRS y GCRS n GCRS
⎛
σ1
⎜
= ⎝ σ2
y1
y2
n1
n2
⎞ ⎛
σ1
⎟ ⎜
⎠ = ⎝ σ2
y1
y2
X
Y
σ3 y3 Z
σ3 y3 n3
⎞
⎟
⎠ (6.17)
where X, Y , and Z are the CIP coordinates, expressed as dimensionless quantities. As in section
5.4.3, the matrix can also be constructed using only X and Y , together with the CIO locator, s:
C T =
⎛
⎜
⎝
1 − bX 2 −bXY X
−bXY 1 − bY 2 Y
−X −Y 1 − b(X 2 + Y 2 )
⎞
⎟
⎠ R3(s) (6.18)
where b = 1/(1 + Z) and Z = √ 1 − X 2 − Y 2 . The latter form is taken from the IERS Conventions
(2003), Chapter 5, where C T is called Q(t). The two constructions of C T are numerically the same.
6.5.4 Hour Angle
The local hour angle of a celestial object is given by
Equinox-based formula: h = GAST − α Υ + λ
CIO-based formula: h = θ − ασ + λ (6.19)
where GAST is Greenwich apparent sidereal time, θ is the Earth Rotation Angle, and λ is the
longitude of the observer. The quantities involved can be expressed in either angle or time units
as long as they are consistent. The right ascension in the two cases is expressed with respect to
different origins: α Υ is the apparent right ascension of the object, measured with respect to the true
equinox, and ασ is the apparent right ascension of the object, measured with respect to the CIO.
That is, the coordinates of the object are expressed in system E Υ in the equinox-based formula
and in system Eσ in the CIO-based formula. Since both systems share the same equator — the
instantaneous equator of date, orthogonal to the CIP — the apparent declination of the object is
the same in the two cases.
The two formulas in 6.19 are equivalent, which can be seen by substituting, in the equinox-based
formula, GAST = θ − Eo and α Υ = ασ − Eo, where Eo is the equation of the origins.
The longitude of the observer, λ, is expressed in the Eϖ system, that is, it is corrected for polar
motion. Using the first-order form of the matrix W, given in eq. 6.15 (and assuming s ′ =0), it is
straightforward to derive eq. 2.16 for λ. Using notation consistent with that used in this chapter,
this equation is
λ ≡ λEϖ = λITRS + xp sin λITRS + yp cos λITRS tan φITRS /3600 (6.20)
where λ ITRS and φ ITRS are the ITRS (geodetic) longitude and latitude of the observer, with λ ITRS
in degrees; and xp and yp are the coordinates of the pole (CIP), in arcseconds. This formula is
approximate and should not be used for places at polar latitudes.
The corresponding equation for the latitude, φ, corrected for polar motion is
φ ≡ φEϖ = φITRS + xp cos λITRS − yp sin λITRS /3600 (6.21)