2. Terms of nutation coming from the J3 geopotential
As explained by Kinoshita (1977) the coefficient of the geopotential is small, so that when calculating its contribution to the nutation due to the Moon, the motion of the Moon can be considered as Keplerian. Nevertheless, we recompute here these coefficients by including all the perturbations in the coordinates , , and of the Moon given by their analytical expressions in the form of Fourier series (Chapront-Touzé and Chapront 1988). Notice that Kinoshita (1977) did not calculate the terms coming from the J3 geopotential, because they could be considered as negligible at that time. Notice also that the geopotential does not bring any significant contribution to the precession (see Hartmann et al. 1996).
In fact, Kinoshita & Souchay (1990) showed that only one term in longitude, that is to say: , of period 8.85 y, is above the level of truncation of the series of Kinoshita (1977), that is to say 0.1 milliarcsecond (mas), whereas they found several other coefficients up to the level of truncation of their own series, that is to say 5 microarcseconds. Recently Hartmann et al. (1995) computed the nutations and for the figure axis, starting from the harmonic tidal development of the torque exerted by the Moon, as given by Hartmann & Wenzel (1995). Their results is in very good agreement with those of Kinoshita & Souchay (1990), with a 3.3 and 2.6 microarecseconds rms respectively in longitude and in obliquity. By contrast, their results do not fit with those given by Zhu & Groten (1989), with a respective 41.7 and 86.8 microarcseconds rms, although the method used is quite equivalent. This is due in some errors in Zhu & Groten calculations, as explained by Hartmann et al. (1995), who could also avoid the difficulty to compute long-periodic components of the nutation starting from tidal quasi-diurnal harmonics, as explained by Souchay (1993). The expression of the lunar potential related to the harmonic can be expressed as follows (Kinoshita & Souchay 1990):
is the Gauss constant, and are respectively the masses of the Moon and of the Sun, and is the mean equatorial radius of the Earth.
The Legendre polynomial can be expressed itself as a function of the associated Legendre polynomials , where is the declination of the Moon, its latitude with respect to the equator, and the distance between the center of mass of the Earth and the center of mass of the Moon. Thus we have:
The polynomials have the classical following expressions:
The variable I is related to the canonical variables H and G in the Hamitonian theory (Kinoshita 1977), by the equation: , where G is the angular momentum of the rotation of the Earth, and H its projection along the axis of the ecliptic of the epoch. I corresponds to , where is the mean value of the obliquity of the axis of angular momentum with respect to the axis of the ecliptic of the date.
In a similar way, the canonical variable h represents the opposite of the general precession in longitude of the plane perpendicular to the angular-momentum vector (Kinoshita 1977): , where p is the linear component of the general precession in longitude (Lieske et al. 1977).
Following the equations above, we calculate the determining function by the way of integration:
Then the coefficients of the nutation coming from are calculated by the direct partial derivatives (Kinoshita & Souchay 1990):
For our calculations, the constant term , which serves as a scaling factor, is given by the following formula (Kinoshita & Souchay 1990):
is the semi-major axis of the Earth. The direct correspondence between the value of the general precession in longitude and the scaling factors and which enable calculating the coefficients of the nutation due respectively to the Moon and the Sun has been studied in detail by Kinoshita & Souchay (1990) by using the conventional value of (Lieske et al. 1977). Williams (1994) and Souchay & Kinoshita (1996) repeated this study by choosing an updated value of the linear component , as confirmed by various authors (see Williams 1994), and corresponding to a correction of /cy. Notice that the consequence is a respective change of about 1 mas for the amplitude of leading term of nutation of argument (Souchay & Kinoshita 1996) due to .
being the scaling factor related to the first-order nutation coefficients (Kinoshita 1977; Kinoshita & Souchay 1990) In Souchay & Kinoshita (1996) we have calculated a new value of by taking into account a recent evaluation of the general precession in longitude (Williams 1994) different from the conventional one (Lieske et al. 1977). This new value is used in our calculations of starting from (6), that is to say: .
In Table 1 we are listing our results and and comparing them with those of Kinoshita & Souchay (1990) and Hartmann et al. (1996), by retaining all the coefficients bigger than 0.5 microarcseconds () both for and . We can observe the remarkable agreement between the authors, and especially between our results and those of Hartmann et al. (1995): the amplitude difference is bigger than 0.1 for only 3 terms among the 20 terms to be compared, and anyway does not exceed 0.6 . The comparison is all the more convincing since the computation method used by Hartmann et al. (1995) is quite different from ours, that is to say based on a recent expansion on the tesseral part of the tidal potential (Hartmann & Wenzel 1995).
Table 1. List of the coefficients of rigid Earth nutation coming from the geopotential. Comparison of new results (this paper) is made with results of Kinoshita & Souchay (1990) and Hartmann et al. (1995), both in longitude and in obliquity
© European Southern Observatory (ESO) 1997
Online publication: July 8, 1998