*Astron. Astrophys. 318, 639-652 (1997)*
## 4. Terms of nutation coming from the *J*_{4} geopotential
As it can be expected the influence of the coefficient
of the geopotential on the nutation is very
small. Kinoshita (1977) made a brief study which showed that this
influence was negligible in regards of the accuracy of the
precession-nutation observations, and anyway far under his level of
truncation for his series of nutation, that is to say 0.1
milliarcsecond. Kinoshita & Souchay (1990) calculated only the
influence of on the precession, and found the
value: which is in agreement with Kinoshita's
value: . This value is taken into account for
when studying the connection between the scaling factors
and , and the value of
the general precession in longitude (Souchay & Kinoshita
1996).
Hartmann et al. (1995) showed that the biggest term of nutation due
to concerns the leading component at the
first-order, that is to say the component.
Contrary to the general case, the amplitude in obliquity:
is relatively much larger than the amplitude in
longitude: .
In the following we propose to calculate the values of the nutation
due to the geopotential, by the same way as
precedently, that is to say starting from the theory derived from
Hamiltonian formulation (Kinoshita 1977; Kinoshita & Souchay
1990). We choose the same value as Hartmann et al. (1995), that is to
say: .
The solar part of the potential related to
being completely negligible, only the lunar part is considered here,
which can be written classically as follows:
Rigorously, can be expressed in function of
the modified Jacobi polynomials (Kinoshita et al. 1974):
The modified Jacobi polynomials are given
by:
Thus, we have:
For , we can neglect the polynomials
because they have as a
factor in their developement, and is very
small, its value being of the order of a few
(*J* represents the angle between the axis of figure and the axis
of angular momentum). Thus the Legendre polynomial
can be restricted to the following
expression:
where the Legendre polynomials have the
following expressions:
There we have neglected the terms for which:
in Eq. (15), because their amplitude after
integration becomes very small, the canonical variable *l* having
a quasi-diurnal period.
By the means of (18) and (19.1-5) we finally get:
Then the determining function related to
can be written:
And the coefficients of the nutation are given by:
By using the analytical developements in Fourier series of the
coordinates , and
, we find the following values for the nutation
related to the geopotential, given in
microarcseconds:
They are quite in agreement with the values above calculated by
Hartmann et al. (1995). Moreover, our determination of the precession
rate related to is: ,
which is very close to the values of Kinoshita (1977) and Kinoshita
& Souchay (1990), and exactly the same as the value found by
(Hartmann et al. 1995), with a completely different way of
computation.
© European Southern Observatory (ESO) 1997
Online publication: July 8, 1998
helpdesk.link@springer.de |