*Astron. Astrophys. 318, 639-652 (1997)*
## 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 *µ*as.
Recently, Williams (1995), calculated all the coefficients related
to the direct influence of the planets, up to 0.5 *µ*as,
both for and . 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 ,
and *r* of the perturbing planet can be
expressed in a very straightforward and classical manner, with the
trivial transformations:
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*, , ,
, and
(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 instead of
. At second, our level of truncation in the
determination of the coefficients is 0.5 *µ*as for
and instead of
5 *µ*as for and
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
and , 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 *µ*as, 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 *µ*as.
**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 3.1.** (continued)
**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 3.2.** (continued)
**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 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 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 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 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 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)
© European Southern Observatory (ESO) 1997
Online publication: July 8, 1998
helpdesk.link@springer.de |