3. Terms of nutation coming from the sectorial parts C2,2 and S2,2 of the geopotential
A, B, and C being the principal moments of inertia of the Earth, the dimensionless coefficients and of the geopotential are characterzing the dynamical triaxiality of the Earth, by the way of the following equation:
Although the Earth is relatively close to an axisymetric body, with: , the influence of the difference on the nutation of the Earth is not negligible. It was calculated firstly by Kinoshita (1977) for the 3 biggest terms and by Kinoshita & Souchay (1990), up to 0.005 milliarcsecond (mas). The expression of the luni-solar potential related to the triaxiality can be expressed as follows (Kinoshita 1977):
where , and are the geocentric distance, latitude and longitude of the perturbing body (Moon or Sun).
is given by the following formula (Kinoshita 1977):
h, l and g are the canonical variables in Kinoshita (1977) theory. Three planes are involved in order to define these variables. h is defined along the ecliptic of the date from the mean equinox of the date to the intersection with the plane perpendicular to the angular-momentum vector. g is the angle from that intersection along the angular momentum plane and its intersection with the equator. And l is from that last intersection to the prime meridian. Thus the sum () is equivalent to the sidereal angle of rotation .
The polynomials have the following expressions:
Notice that for the Sun, we can adopt the approximations: and: . The coefficients of the nutation due to the triaxiality can be calculated starting from the determining function, by integrating the potential as given by (9):
Then, by the following partial derivatives:
The constant terms and , which serve as a scaling factor respectively for the action of the Moon and of the Sun, are given by the following relationships:
where the value of the constant term has been given in Sect. 2, when calculating the terms of the nutation due to , and where in a similar manner an accurate value of has been recently calculated with an updated value of the general precession in longitude by Souchay & Kinoshita (1996), that is: . We choose for the ratio the same value as in (Hartmann et al. 1995) which is significantly different from the value in Kinoshita & Souchay (1990), that is to say: instead of .
The results of the computations are listed in Table 2. Notice that all the terms have a quasi semi-diurnal period, because of the presence of in their argument. We have 34 coefficients up to 0.1 µas instead of the 7 coefficients up to 5 µas found by Kinoshita & Souchay (1990). A sign error has been detected in this last work, for the term of amplitude as and argument . Moreover, a big correction concerns the amplitude of the term of argument which is as instead of as. Except this term all the other ones have been confirmed.
Table 2. Coefficients of rigid Earth nutation coming from the Earth triaxiality. Comparison of new results (this paper) is made with results of Kinoshita & Souchay (1990), both in longitude and in obliquity
© European Southern Observatory (ESO) 1997
Online publication: July 8, 1998