USNO Circular 179 - U.S. Naval Observatory

PRECESSION & NUTATION 43
another date, t. To precess in the opposite direction, the angles are the same but the transpose
of the precession matrix, P T (t), is used. To precess coordinates from one arbitrary date, t1, to
another, t2, it is necessary to precess them from t1 to J2000.0 (using P T (t1)), then from J2000.0
to t2 (using P(t2)). Where high accuracy is not required, and t1 and t2 are not more than a few
years apart, a simpler procedure for precession from t1 to t2 is available and is given at the end of
this subsection.
All expressions given in this subsection are from Section 7 (P03 solution) of Capitaine et al.
(2003) and all coefficients are expressed in arcseconds. This is the theory of precession recommended
by the IAU Working Group on Precession and the Ecliptic (Hilton et al. 2006).
For a given TDB date and time t, let T be the number of Julian centuries of TDB since
2000 Jan 1, 12 h TDB. If the dates and times are expressed as Julian dates, then T = (t −
2451545.0)/36525. TT dates and times can be used equally well — the resulting error in precession
is only a few ×10 −9 arcseconds.
Then the mean obliquity of the ecliptic at J2000.0 (or the equivalent TDB date) is ɛ0 =
84381.406 arcseconds and let
ψA = 5038.481507 T − 1.0790069 T 2 − 0.00114045 T 3 + 0.000132851 T 4 − 0.0000000951 T 5
ωA = ɛ0 − 0.025754 T + 0.0512623 T 2 − 0.00772503 T 3 − 0.000000467 T 4 + 0.0000003337 T 5
χA = 10.556403 T − 2.3814292 T 2 − 0.00121197 T 3 + 0.000170663 T 4 − 0.0000000560 T 5 (5.7)
Equivalently, in notation appropriate for computer programs,
ψA = (((( − 0.0000000951 T
+ 0.000132851 ) T
− 0.00114045 ) T
− 1.0790069 ) T
+ 5038.481507 ) T
ωA = (((( + 0.0000003337 T
− 0.000000467 ) T
− 0.00772503 ) T (5.8)
+ 0.0512623 ) T
− 0.025754 ) T + ɛ0
χA = (((( − 0.0000000560 T
+ 0.000170663 ) T
− 0.00121197 ) T
− 2.3814292 ) T
+ 10.556403 ) T
The precession matrix is then simply P(t) = R3(χA) R1(−ωA) R3(−ψA) R1(ɛ0), where R1 and
R3 are standard rotations about the x and z axes, respectively (see “Abbreviations and Symbols
Frequently Used” for precise definitions). This 4-angle precession formulation is comprised of
1. A rotation from the mean equator and equinox of J2000.0 to the mean ecliptic and equinox of
J2000.0. This is simply a rotation around the x-axis (the direction toward the mean equinox of
J2000.0) by the angle ɛ0, the mean obliquity of J2000.0. After the rotation, the fundamental
plane is the ecliptic of J2000.0.