Precession and nutation according the non-classical procedure of Capitaine-Guinot

Memorandum N 3
25.06.997

Precession and nutation according the non-classical procedure of Capitaine-Guinot (the IAU 1980 theory)

We construct both variants of the precession and nutation calculation: classical procedure using the notion of ecliptics and non-classical procedure of Captaine Capitaine 1990) using the notion of "non-rotating origin" Guinot (Guinot 1979). Regrettably, on the request on the debugged variant of program, which we directed in Central Buro of IERS in 1997, we have not got an answer up to now. So it is necessary to write both procedures for debugging by the way of comparison. We begin with the non-classical procedure, since any sensible physicist or mathematician will prefer exactly it by reason of incomparably greater mathematical rigor and transparency. Both ways give the results, complying with accuracy ±0.05 milliseconds of arcs in interval of several hundred years (Capitaine and Chollet 1991, Capitaine and Gontier 1991)v.

Temporal argument. Generally accepted definition:
t=(TT–2000 January 1d 12h TT) in days/36525.
According to our notation this formula has the form:
t=(TT–0.5)/36525.

Coordinates CEP (celestial ephemeris pole) in CRS (celestial reference system).
X=2004.3109"t – 0.42665"t² – 0.198656"t³ + 0.0000140"t⁴+0.00006"t²cosW
+sine₀{S[(A_i+A'_it)sin(ARGUMENT)+A''_itcos(ARGUMENT)]}+0.00204"t²sinW +0.00016t²sin2(F-D+W),
Y =– 0.00013"–22.40992"t²+0.001836"t³+0.0011130"t⁴+ S[(B_i+B'_it)cos(ARGUMENT)+B"_itsin(ARGUMENT)]–0.00231"t²cosW –0.00014"t²cos2(F-D+ W),
where the argument (ARGUMENT) is given by
ARGUMENT=S N_iF_i,
the N_i, (i=1,…,5) being integers multiplying the Fundamental Arguments F_i of nutation theory, namely,
F₁şl= Mean Anomaly of the Moon
= 134.96340251°+ 1717915923.2178"t+ 31.8792"t²+0.051635"t³–0.00024470"t⁴F₂ şl'= Mean Anomaly of the Sun
=357.52910918° + 129596581.0481"t–0.5532"t²+0.000136"t³–0.00001149"t⁴F₃şF=L–W

=93.27209062°+1739527262.8478"t–12.7512"t²–0.001037"t³+0.00000417"t⁴
F₄ şD= Mean Elongation of the Moon from the Sun
=297.85019547°+1602961601.2090"t–6.3706"t²+0.006593"t³–0.00003169"t⁴
F₅şW= Mean Longitude of the Ascending Node of the Moon
=125.04455501°–6962890.2665"t+7.4722"t²+0.007702"t³–0.00005939"t⁴
(Simon et al. 1994). L is the Mean Longitude of the Moon.

To totalize the folowing table from the end!!!

Period LONGITUDE(0.0001") OBLIQUITY(0.0001")

l l' F D W A_i A'_i A"_i B_i B'_i B"_i

0 0 0 0 1 -6798.4 -171996 -84.2 5173.2 92025 8.9 1529.9

0 0 2 -2 2 182.6 -13187 5.3 322.2 5736 -3.1 117.3

0 0 2 0 2 13.7 -2274 1.0 54.8 977 -0.5 20.2

0 0 0 0 2 -3399.2 2053.2 -1.0 -50.5 -893 .7 0.5 -18.3

0 -1 0 0 0 -365.3 -1426 4.3 3.0 54 -0.1 12.7

1 0 0 0 0 27.6 712 0.1 0.0 -7 0.0 -6.3

0 1 2 -2 2 121.7 -517 1.5 12.6 224 -0.6 4.6

0 0 2 0 1 13.6 -386 -0.4 11.3 200 0.0 3.4

1 0 2 0 2 9.1 -301 0.0 7.3 129 -0.1 2.7

0 -1 2 -2 2 365.2 217 -0.5 -5.3 -95 0.3 -1.9

-1 0 0 2 0 31.8 158 0.0 0.0 -1 0.0 -1.4

0 0 2 -2 1 177.8 129 0.1 -4.0 -70 0.0 -1.2

-1 0 2 0 2 27.1 123 0.0 -3.0 -53 0.0 -1.1

1 0 0 0 1 27.7 63 0.1 -1.8 -33 0.0 -0.6

0 0 0 2 0 14.8 63 0.0 0.0 -2 0.0 -0.6

-1 0 2 2 2 9.6 -59 0.0 1.5 26 0.0 0.5

-1 0 0 0 1 -27.4 -58 -0.1 1.8 32 0.0 0.5

1 0 2 0 1 9.1 -51 0.0 1.5 27 0.0 0.5

-2 0 0 2 0 -205.9 -48 0.0 0.0 1 0.0 0.0

-2 0 2 0 1 1305.5 46 0.0 -1.3 -24 0.0 0.0

0 0 2 2 2 7.1 -38 0.0 0.0 16 0.0 0.0

2 0 2 0 2 6.9 -31 0.0 0.0 13 0.0 0.0

2 0 0 0 0 13.8 29 0.0 0.0 -1 0.0 0.0

1 0 2 -2 2 23.9 29 0.0 0.0 -12 0.0 0.0

0 0 2 0 0 13.6 26 0.0 0.0 -1 0.0 0.0

0 0 2 -2 0 173.3 -22 0.0 0.0 0 0.0 0.0

-1 0 2 0 1 27.0 21 0.0 0.0 -10 0.0 0.0

0 2 0 0 0 182.6 17 -0.1 0.0 0 0.0 0.0

0 2 2 -2 2 91.3 -16 0.1 0.0 7 0.0 0.0

1 0 0 2 1 32.0 16 0.0 0.0 -8 0.0 0.0

0 1 0 0 1 386.0 -15 0.0 0.0 9 0.0 0.0

1 0 0 -2 1 -31.7 -13 0.0 0.0 7 0.0 0.0

0 -1 0 0 1 -346.6 -12 0.0 0.0 6 0.0 0.0

2 0 -2 0 0 -1095.2 11 0.0 0.0 0 0.0 0.0

-1 0 2 2 1 9.5 -10 0.0 0.0 5 0.0 0.0

1 0 2 2 2 5.6 -8 0.0 0.0 3 0.0 0.0

0 -1 2 0 2 14.2 -7 0.0 0.0 3 0.0 0.0

0 0 2 2 1 7.1 -7 0.0 0.0 3 0.0 0.0

1 1 0 -2 0 -34.8 -7 0.0 0.0 0 0.0 0.0

0 1 2 0 2 13.2 7 0.0 0.0 -3 0.0 0.0

-2 0 0 2 1 -199.8 -6 0.0 0.0 3 0.0 0.0

0 0 0 2 1 14.8 -6 0.0 0.0 3 0.0 0.0

2 0 2 -2 2 12.8 6 0.0 0.0 -3 0.0 0.0

1 0 0 2 0 9.6 6 0.0 0.0 0 0.0 0.0

1 0 2 -2 1 23.9 6 0.0 0.0 -3 0.0 0.0

0 0 0 -2 1 -14.7 -5 0.0 0.0 3 0.0 0.0

0 -1 2 -2 1 346.6 -5 0.0 0.0 3 0.0 0.0

2 0 2 0 1 6.9 -5 0.0 0.0 3 0.0 0.0

1 -1 0 0 0 29.8 5 0.0 0.0 0 0.0 0.0

1 0 0 -1 0 411.8 -4 0.0 0.0 0 0.0 0.0

0 0 0 1 0 29.5 -4 0.0 0.0 0 0.0 0.0

0 1 0 -2 0 -15.4 -4 0.0 0.0 0 0.0 0.0

1 0 -2 0 0 -26.9 4 0.0 0.0 0 0.0 0.0

2 0 0 -2 1 212.3 4 0.0 0.0 -2 0.0 0.0

0 1 2 -2 1 119.6 4 0.0 0.0 -2 0.0 0.0

1 1 0 0 0 25.6 -3 0.0 0.0 0 0.0 0.0

1 -1 0 -1 0 -3232.9 -3 0.0 0.0 0 0.0 0.0

-1 -1 2 2 2 9.8 -3 0.0 0.0 1 0.0 0.0

0 -1 2 2 2 7.2 -3 0.0 0.0 1 0.0 0.0

1 -1 2 0 2 9.4 -3 0.0 0.0 1 0.0 0.0

3 0 2 0 2 5.5 -3 0.0 0.0 1 0.0 0.0

-2 0 2 0 2 1615.7 -3 0.0 0.0 1 0.0 0.0

1 0 2 0 0 9.1 3 0.0 0.0 0 0.0 0.0

-1 0 2 4 2 5.8 -2 0.0 0.0 1 0.0 0.0

1 0 0 0 2 27.8 -2 0.0 0.0 1 0.0 0.0

-1 0 2 -2 1 -32.6 -2 0.0 0.0 1 0.0 0.0

0 -2 2 -2 1 6786.3 -2 0.0 0.0 1 0.0 0.0

-2 0 0 0 1 -13.7 -2 0.0 0.0 1 0.0 0.0

2 0 0 0 1 13.8 2 0.0 0.0 -1 0.0 0.0

3 0 0 0 0 9.2 2 0.0 0.0 0 0.0 0.0

1 1 2 0 2 8.9 2 0.0 0.0 -1 0.0 0.0

0 0 2 1 2 9.3 2 0.0 0.0 -1 0.0 0.0

1 0 0 2 1 9.6 -1 0.0 0.0 0 0.0 0.0

1 0 2 2 1 5.6 -1 0.0 0.0 1 0.0 0.0

1 1 0 -2 1 -34.7 -1 0.0 0.0 0 0.0 0.0

0 1 0 2 0 14.2 -1 0.0 0.0 0 0.0 0.0

0 1 2 -2 0 117.5 -1 0.0 0.0 0 0.0 0.0

0 1 -2 2 0 -329.8 -1 0.0 0.0 0 0.0 0.0

1 0 -2 2 0 32.8 -1 0.0 0.0 0 0.0 0.0

1 0 -2 -2 0 -9.5 -1 0.0 0.0 0 0.0 0.0

1 0 2 -2 0 32.8 -1 0.0 0.0 0 0.0 0.0

1 0 0 -4 0 -10.1 -1 0.0 0.0 0 0.0 0.0

2 0 0 -4 0 -15.9 -1 0.0 0.0 0 0.0 0.0

0 0 2 4 2 4.8 -1 0.0 0.0 0 0.0 0.0

0 0 2 -1 2 25.4 -1 0.0 0.0 0 0.0 0.0

-2 0 2 4 2 7.3 -1 0.0 0.0 1 0.0 0.0

2 0 2 2 2 4.7 -1 0.0 0.0 0 0.0 0.0

0 -1 2 0 1 14.2 -1 0.0 0.0 0 0.0 0.0

0 0 -2 0 1 -13.6 -1 0.0 0.0 0 0.0 0.0

0 0 4 -2 2 12.7 1 0.0 0.0 0 0.0 0.0

0 1 0 0 2 409.2 1 0.0 0.0 0 0.0 0.0

1 1 2 -2 2 22.5 1 0.0 0.0 -1 0.0 0.0

3 0 2 -2 2 8.7 1 0.0 0.0 0 0.0 0.0

-2 0 2 2 2 14.6 1 0.0 0.0 -1 0.0 0.0

-1 0 0 0 2 -27.3 1 0.0 0.0 -1 0.0 0.0

0 0 -2 2 1 -169.0 1 0.0 0.0 0 0.0 0.0

0 1 2 0 1 13.1 1 0.0 0.0 0 0.0 0.0

-1 0 4 0 2 9.1 1 0.0 0.0 0 0.0 0.0

2 1 0 -2 0 131.7 1 0.0 0.0 0 0.0 0.0

2 0 0 2 0 7.1 1 0.0 0.0 0 0.0 0.0

2 0 2 -2 1 12.8 1 0.0 0.0 -1 0.0 0.0

2 0 -2 0 1 -943.2 1 0.0 0.0 0 0.0 0.0

1 -1 0 -2 0 -29.3 1 0.0 0.0 0 0.0 0.0

-1 0 0 1 1 -388.3 1 0.0 0.0 0 0.0 0.0

-1 -1 0 2 1 35.0 1 0.0 0.0 0 0.0 0.0

0 1 0 1 0 27.3 1 0.0 0.0 0 0.0 0.0

0 0 2 -2 3 177.8 -1 .2 0.0 0.0 0 0.0 0.0

Angular arguments connected with X and Y. As X=sindcosE and Y=sindsinE, then E=arctg(Y/X), ŕ d=arcsin((X²+Y²)^˝). No need to think about signs and quadrants, since the values of both angles are less 90ş. Besides
S=–XY/2+0.00385"t–0.07259"t³–0.00264"sinW–0.00006"sin2W +0.00074"t²sinW+0.00006"t²sin 2(F–D+W)

"Celestial pole offsets". Observable corrections to the nutation theory of 1980 are made by formulae
dX=dysine₀
dY=de
dy and de can be taken from the files eop*c*04.**. It is necessary to add these corrections to the current values of X č Y. e₀=23°26'21.448" or sine₀=0.39777716.

Geodesic nutation. It is an "ether drag" (of inertial reference system) by the rotating Earth. Described by the correction:
DX_g=(–0.0000609"sinl'–0.0000008"sin2l') sine₀. The correction would be added to the uncorrected determination of X.

Passage from CRS to TRS(terrestrial reference system). Adduce formula, the commentary to which will be given below. The transformation is carried by the formula:
[1:2:3]_TRS=R₂(–x_p)R₁(–y_p)R₃(–E–S+q+s')R₂(d)R₃(E)[1:2:3]_CRS
In our work we put s'=0. Its value is accumulated by 0.0001" by century (Capitaine et al. 1986). It is not clear, where to take data on it, so it is more simple to place it equal to zero. x_p and y_p are the coordinates of the Earth's axis of rotation (pole) in the Earth's body, which are available from the beginning of XIX century, and on recent years available in the files eop*c*04.**. It remains to write q — the stellar angle.
q(d_u)=2p(0.779057273264+1.00273781191135448d_u), where d_u is UT1 in units of days, counted from 12h UT1 1 January 2000. Under 2p we understand the angle equal to the complete turn (360°).

Rotation around axes in the arbitrary reference frame. In difference from the memorandum N 2, in which the rotations are given in the basic coordinate system (CRS), under the rotation here we understand the rotation around one of the axes of the current reference frame obtained by the preceding rotation of the preceding reference frame. Then, unlike the memorandum N 2, under a rotation, for instance, R₃(e) we understand (such understanding of rotation we have denoted by the italic letter R)
R₃(e)x=(x₁cose–x₂sine)1+(x₁sine+x₂cose)2+x₃3
However in reality we'll seldom use such formula, since we rotate the elements of trihedron, rather then an arbitrary oriented vector.

Agreement on the trihedron rotation in the current reference frame. Let us denote by R (bold) a transformation of reference frame by rotation around corresponding axis of the current reference frame possibly obtained by a preceding rotation. Exactly in such sense the notion of rotation was used in the item on the transformation between CRS and TRS. Adduce all three formulae for bold R.

R₃(e): [1;2;3]Ţ[cose1+sine2;–sine1+cose2;3]

R₂(e): [1;2;3]Ţ[cose1–sine3;2;sine1+cose3]

R₁(e): [1;2;3]Ţ[1;cose2+sine3;–sine2+cose3]

Capitaine, N. and Chollet, F., 1991, "The use of the nonrotating origin in the computation of apparent places of stars for estimating Earth Rotation Parameters", In: Reference Systems, J. A. Hughes, C. A. Smith, and G. H. Kaplan (eds), pp. 224--227.

Capitaine, N. and Gontier A. -M., 1991, "Procedure for VLBI estimates of Earth Rotation Parameter referred to the nonrotating origin", In: Reference Systems, J. A. Hughes, C. A. Smith, and G. H. Kaplan (eds), pp. 77--84.

This definition is consistent with Resolution C7 passed at the 1994 Hague General Assembly of the IAU which recommends that J2000.0 be defined at the geocenter and at the date 2000.0 January 1.5 TT = Julian Date 2451545.0 TT.

Simon, J. L., Bretagnon, P., Chapront, J., Chapront-Touz\'e, M., Francou, G., Laskar, J., 1994, ``Numerical Expressions for Precession Formulae and Mean Elements for the Moon and Planets", Astron. Astrophys., 282, pp. 663–683.

Capitaine, N., Guinot, B., and Souchay, J., 1986, ``A Non-rotating Origin on the Instantaneous Equator: Definition, Properties and Use", Celest. Mech., 39, pp. 283--307.