USNO Circular 179 - U.S. Naval Observatory

PRECESSION & NUTATION 47
of the coefficients Si, Si, ˙ C ′ i , Ci, Ci, ˙ and S ′ i for the term. Generally it is good practice to sum the
terms from smallest to largest to preserve precision in the sums.
The entire IAU 2000A nutation series is listed at the end of this circular. About the first half
of the series consists of lunisolar terms, which depend only on l, l ′ , F , D, and Ω (= φ10 to φ14).
In all of these terms, the first nine multipliers are all zero. The generally smaller planetary terms
comprise the remainder of the series. As an example of how the individual terms are computed
according to eqs. 5.15 and 5.16, term 6 would be evaluated
∆ψ6 = (−0.0516821 + 0.0001226 T ) sin Φ6 − 0.0000524 cos Φ6
∆ɛ6 = ( 0.0224386 − 0.0000667 T ) cos Φ6 − 0.0000174 sin Φ6
where Φ6 = φ11 + 2φ12 − 2φ13 + 2φ14
since M6,1 through M6,10 are zero, and only φ11 through φ14 are therefore relevant for this term. It
is assumed that all the φj have been pre-computed (for all terms) using the appropriate value of T
for the date and time of interest. A printed version of a 1365-term nutation series is obviously not
the most convenient form for computation; it is given here only for the record, since the full series
has not previously appeared in print. As noted earlier, the series is available as a pair of plain-text
computer files at URL 14, and the SOFA and NOVAS software packages (URL 7, URL 8) include
subroutines for evaluating it. There are also shorter series available where the highest precision is
not required. The IERS web site provides, in addition to the full IAU 2000A series, an IAU 2000B
series, which has only 77 terms and duplicates the IAU 2000A results to within a milliarcsecond for
input times between 1995 and 2050. NOVAS also provides a subroutine that evaluates a truncated
series, with 488 terms, that duplicates the full series to 0.1 milliarcsecond accuracy between 1700
and 2300.
Once the nutation series has been evaluated and the values of ∆ψ and ∆ɛ are available, the nutation
matrix can be constructed. The nutation matrix is simply N(t) = R1(−ɛ ′ ) R3(−∆ψ) R1(ɛ),
where, again, R1 and R3 are standard rotations about the x and z axes, respectively (see “Abbreviations
and Symbols Frequently Used” for precise definitions), and ɛ ′ = ɛ+∆ɛ is the true obliquity
(compute ɛ using eq. 5.12). This formulation is comprised of
1. A rotation from the mean equator and equinox of t to the mean ecliptic and equinox of t.
This is simply a rotation around the x-axis (the direction toward the mean equinox of t) by
the angle ɛ, the mean obliquity of t. After the rotation, the fundamental plane is the ecliptic
of t.
2. A rotation around the new z-axis (the direction toward the ecliptic pole of t) by the angle
−∆ψ, the amount of nutation in longitude at t. After the rotation, the new x-axis is in the
direction of the true equinox of t.
3. A rotation around the new x-axis (the direction toward the true equinox of t) by the angle
−ɛ ′ , the true obliquity of t. After the rotation, the fundamental plane is the true equator of
t, orthogonal to the computed position of the CIP at t.
If we let