USNO Circular 179 - U.S. Naval Observatory

46 PRECESSION & NUTATION
∆ɛ =
N
i=1
(Ci + ˙ CiT ) cos Φi + S ′
i sin Φi
where, in each term, Φi =
(5.15)
K
Mi,j φj(T ) (5.16)
For the IAU 2000A model, N=1365 and K=14. The 14 φj(T ) are the fundamental arguments,
which are, except for one, orbital angles. The main time dependence of the nutation series enters
through these arguments. The expressions given below are all taken from Simon et al. (1994) and
all coefficients are in arcseconds.
The first eight fundamental arguments are the mean heliocentric ecliptic longitudes of the
planets Mercury through Neptune:
j=1
φ1 = 908103.259872 + 538101628.688982 T
φ2 = 655127.283060 + 210664136.433548 T
φ3 = 361679.244588 + 129597742.283429 T
φ4 = 1279558.798488 + 68905077.493988 T (5.17)
φ5 = 123665.467464 + 10925660.377991 T
φ6 = 180278.799480 + 4399609.855732 T
φ7 = 1130598.018396 + 1542481.193933 T
φ8 = 1095655.195728 + 786550.320744 T
In all of these expressions, T is the number of Julian centuries of TDB since 2000 Jan 1, 12 h TDB (or,
with negligible error, the number of Julian centuries of TT since J2000.0). In some implementations
it may be necessary to reduce the resulting angles, which are expressed in arcseconds, to radians
in the range 0–2π. The ninth argument is an approximation to the general precession in longitude:
φ9 = 5028.8200 T + 1.112022 T 2
(5.18)
The last five arguments are the same fundamental luni-solar arguments used in previous nutation
theories, but with updated expresssions. They are, respectively, l, the mean anomaly of the Moon;
l ′ , the mean anomaly of the Sun; F , the mean argument of latitude of the Moon; D, the mean
elongation of the Moon from the Sun, and Ω, the mean longitude of the Moon’s mean ascending
node:
φ10 = l = 485868.249036 + 1717915923.2178 T + 31.8792 T 2 + 0.051635 T 3 − 0.00024470 T 4
φ11 = l ′ = 1287104.79305 + 129596581.0481 T − 0.5532 T 2 + 0.000136 T 3 − 0.00001149 T 4
φ12 = F = 335779.526232 + 1739527262.8478 T − 12.7512 T 2 − 0.001037 T 3 + 0.00000417 T 4
φ13 = D = 1072260.70369 + 1602961601.2090 T − 6.3706 T 2 + 0.006593 T 3 − 0.00003169 T 4
φ14 = Ω = 450160.398036 − 6962890.5431 T + 7.4722 T 2 + 0.007702 T 3 − 0.00005939 T 4
The first step in evaluating the series for nutation for a given date is to compute the values of
all 14 fundamental arguments for the date of interest. This is done only once. Then the nutation
terms are evaluated one by one. For each term i, first compute Φi according to eq. 5.16, using the
14 integer multipliers, Mi,j, listed for the term; i.e., sum over Mi,j × φj (where j=1–14). Then the
cosine and sine components for the term can be evaluated, as per eq. 5.15, using the listed values
(5.19)