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

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4. Terms of nutation coming from the J4 geopotential

As it can be expected the influence of the coefficient [FORMULA] 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 [FORMULA] on the precession, and found the value: [FORMULA] which is in agreement with Kinoshita's value: [FORMULA]. This value is taken into account for when studying the connection between the scaling factors [FORMULA] and [FORMULA], 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 [FORMULA] concerns the leading component at the first-order, that is to say the [FORMULA] component. Contrary to the general case, the amplitude in obliquity: [FORMULA] is relatively much larger than the amplitude in longitude: [FORMULA].

In the following we propose to calculate the values of the nutation due to the [FORMULA] 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: [FORMULA].

The solar part of the potential related to [FORMULA] being completely negligible, only the lunar part is considered here, which can be written classically as follows:

[EQUATION]

Rigorously, [FORMULA] can be expressed in function of the modified Jacobi polynomials (Kinoshita et al. 1974):

[EQUATION]

The modified Jacobi polynomials [FORMULA] are given by:

[EQUATION]

Thus, we have:

[EQUATION]

For [FORMULA], we can neglect the polynomials [FORMULA] because they have [FORMULA] as a factor in their developement, and [FORMULA] is very small, its value being of the order of a few [FORMULA] (J represents the angle between the axis of figure and the axis of angular momentum). Thus the Legendre polynomial [FORMULA] can be restricted to the following expression:

[EQUATION]

where the Legendre polynomials [FORMULA] have the following expressions:

[EQUATION]

There we have neglected the terms for which: [FORMULA] 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:

[EQUATION]

Then the determining function [FORMULA] related to [FORMULA] can be written:

[EQUATION]

And the coefficients of the nutation are given by:

[EQUATION]

[EQUATION]

[EQUATION]

By using the analytical developements in Fourier series of the coordinates [FORMULA], [FORMULA] and [FORMULA], we find the following values for the nutation related to the [FORMULA] geopotential, given in microarcseconds:

[EQUATION]

They are quite in agreement with the values above calculated by Hartmann et al. (1995). Moreover, our determination of the precession rate related to [FORMULA] is: [FORMULA], 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.

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© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998
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