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

## 4. Terms of nutation coming from the J4 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 