USNO Circular 179 - U.S. Naval Observatory

28 CELESTIAL REFERENCE SYSTEM
the precession algorithm assumes a dynamical coordinate system. That is, the above transformation
is a necessary step in obtaining coordinates with respect to the mean equator and equinox of date,
if one starts with ICRS reference data. See Chapter 5 for more information.
The matrix B is, to first order,
⎛
⎞
B =
⎜
⎝
1 dα0 −ξ0
−dα0 1 −η0
ξ0 η0 1
⎟
⎠ (3.3)
where dα0 = −14.6 mas, ξ0 = −16.6170 mas, and η0 = −6.8192 mas, all converted to radians
(divide by 206 264 806.247). The values of the three small angular offsets are taken from the
IERS Conventions (2003). They can be considered adopted values; previous investigations of the
dynamical-ICRS relationship obtained results that differ at the mas level or more, depending on
the technique and assumptions. See the discussion in Hilton & Hohenkerk (2004). The angles ξ0
and η0 are the ICRS pole offsets, and dα0 is the offset in the ICRS right ascension origin with
respect to the dynamical equinox of J2000.0, as measured in an inertial (non-rotating) system.
The above matrix can also be used to transform vectors from the ICRS to the FK5 system at
J2000.0. Simply substitute dα0 = −22.9 mas, ξ0 = 9.1 mas, and η0 = −19.9 mas. However, there
is also a time-dependent rotation of the FK5 with respect to the ICRS (i.e., a slow spin), reflecting
the non-inertiality of the FK5 proper motions; see Mignard & Frœschlé (2000).
Although the above matrix is adequate for most applications, a more precise result can be
obtained by using the second-order version:
⎛
⎞
B =
⎜
⎝
1 − 1
2 (dα2 0 + ξ2 0 ) dα0 −ξ0
−dα0 − η0ξ0
1 − 1
2 (dα2 0 + η2 0
) −η0
ξ0 − η0dα0 η0 + ξ0dα0 1 − 1
2 (η2 0 + ξ2 0 )
⎟
⎠ (3.4)
The above matrix, from Slabinski (2005), is an excellent approximation to the set of rotations
R1(−η0)R2(ξ0)R3(dα0), where R1, R2, and R3 are standard rotations about the x, y, and z axes,
respectively (see “Abbreviations and Symbols Frequently Used” for precise definitions).