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Astron. Astrophys. 318, 639-652 (1997)

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5. Direct planetary effects on the nutation

Vondrak (1983) calculated the direct influence of the planets on the nutation, and showed that it could reach the 0.1 milliarcsecond level for individual coefficients. Kinoshita & Souchay (1990), by choosing an Hamiltonian formalism, as well as for the lunisolar case, and by using the analytical developments for the planetary parameters given in VSOP82 (Bretagnon 1982), found results very close to those of Vondrak after converting Vondrak's arguments to their own ones. Their count is 25, 2, 6 and 2 terms respectively for Venus, Mars, Jupiter and Saturn influence, down to 5 µas.

Recently, Williams (1995), calculated all the coefficients related to the direct influence of the planets, up to 0.5 µas, both for [FORMULA] and [FORMULA]. At this level the influence of Mercury and Uranus has to be taken into account, and the count is 1, 103, 26, 22, 5 and 1 terms respectively for the Mercury, Venus, Mars, Jupiter, Saturn and Uranus contributions.

In the following we are calculating the direct planetary effects on the nutation starting from the same canonical equations as in Kinoshita & Souchay (1990), but by using recent Fourier developements of the rectangular coordinates of the planets, as they can be found in the ephemeris VSOP87 (Bretagnon & Francou, 1988). The geocentric coordinates X, Y, and Z with respect to the mean ecliptic and equinox of J2000.0 are then converted in the corresponding coordinates with respect to the mean ecliptic and equinox of the date, which are the conventional references to measure the nutation. Then the spherical coordinates [FORMULA], [FORMULA] and r of the perturbing planet can be expressed in a very straightforward and classical manner, with the trivial transformations:

[EQUATION]

The perturbing function can be calculated easily from these last expressions. The big advantage of this procedure when compared with the procedure in Kinoshita & Souchay (1990) is that they started from the initial solution VSOP82, which means that they firstly had to calculate X, Y, Z starting from the Fourier series of a, [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA] (Bretagnon 1982).

The algorithm to calculate the coefficients of the nutation due to the direct action of the planets starting from Hamiltonian formalism is explained in detail in Kinoshita & Souchay (1990). We follow exactly the same procedure but the quality of our calculations with respect to these last work is improved by two facts: at first we keep all the terms in the intermediate Fourier series of our calculations with a relative [FORMULA] instead of [FORMULA]. At second, our level of truncation in the determination of the coefficients is 0.5 µas for [FORMULA] and [FORMULA] instead of 5 µas for [FORMULA] and [FORMULA] as it was the case for Kinoshita & Souchay (1990).

Tables 3.1-3.2, 4.1-4.2, and 5.1-5.2 show our results concerning respectively the direct action of Venus, Mars and Jupiter, both on [FORMULA] and [FORMULA], whereas tables 6.1-6.2 show the contribution due to the other planets (Mercury, Saturn and Uranus). Our coefficients are compared with Williams (1995) results. We can thus remark the quasi perfect agreement between them and the present calculations. The difference between the coefficients listed in the tables above never exceeds 0.1 µas, except for a few very long periodic terms, as we can observe for the two first coefficients of Table 3.1. We do not present here the comparison with the values found by Hartmann & Soffel (1994), but notice that they were also very closed to Williams'ones. Moreover we can remark that in the case of Jupiter (Tables 5.1 and 5.2) some coefficients were not present in Williams calculations, although their value is bigger than 0.5 µas.


[TABLE]

Table 3.1. Coefficients of rigid Earth nutation due to the direct action of Venus, longitude part. Comparison of new results (this paper) is made with results of Kinoshita and Souchay (1990) and Williams (1995)



[TABLE]

Table 3.1. (continued)



[TABLE]

Table 3.2. Coefficients of rigid Earth nutation due to the direct action of Venus, obliquity part. Comparison of new results (this paper) with results of Kinoshita & Souchay (1990) and Williams (1995)



[TABLE]

Table 3.2. (continued)



[TABLE]

Table 4.1. Coefficients of rigid Earth nutation due to the direct action of Mars, longitude part. Comparison of new results (this paper) is made with results of Kinoshita & Souchay (1990) and Williams (1995)



[TABLE]

Table 4.2. Coefficients of rigid Earth nutation due to the direct action of Mars, obliquity part. Comparison of new results (this paper) is made with results of Williams (1990)



[TABLE]

Table 5.1. Coefficients of rigid Earth nutation due to the direct action of Jupiter, longitude part. Comparison of new results (this paper) is made with results of Kinoshita & Souchay (1990) and Williams (1995)



[TABLE]

Table 5.2. Coefficients of rigid Earth nutation due to the direct action of Jupiter, obliquity part. Comparison of new results (this paper) with results of Kinoshita & Souchay (1990) and Williams (1995)



[TABLE]

Table 6.1. Coefficients of rigid Earth nutation due to the planets Mercury, Saturn and Uranus, longitude part. Comparison of new results (this paper) is made with results of Kinoshita & Souchay (1990) and Williams (1995)



[TABLE]

Table 6.2. Coefficients of rigid Earth nutation due to the planets Mercury, Saturn and Uranus, obliquity part. Comparison of new results (this paper) is made with results of Kinoshita & Souchay (1990) and Williams (1995)


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Online publication: July 8, 1998
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