1 Introduction

  The theory of the nutation for a rigid Earth model is of special interest. It still serves as a basis for the nutation of more complicated and realistic Earth models including elastic and anelastic effects of the mantle, effects from outer and inner core, effects from the oceans and the atmosphere etc. (see e.g. the ZMOA series by Herring 1991; or Mathews et al. 1991a,b; Dehant & Defraigne 1997; Herring 1996). Another important reason for improving the theoretical nutation series of a rigid Earth model is the increasing accuracy of modern space techniques such as Very Long Baseline Interferometry (VLBI) or the Global Positioning System (GPS). In particular with VLBI the nutation angles are observed with an accuracy of the order of 0.1 mas. Therefore present theoretical models for the nutation of a rigid Earth model (or briefly nutation models/series in the following) should deal with amplitudes much less than the observational accuracy limit in order not to contribute significantly to the total error budget. Note, that an accuracy of 1 $\mathrm \mu$as corresponds to 0.031 mm on the Earth's surface.

For an overall threshold of a nutation series of about 1 $\mathrm \mu$as, many different effects have to be taken into account. Although they depend somewhat on the computation method (e.g. the so-called second order terms, see below) one finds that the following contributions have to be included: the main effect (first and second order) from the Moon and the Sun, the direct and indirect influence of the planets (see e.g. Vondrák 1982, 1983a,b; Hartmann & Soffel 1994 and Souchay & Kinoshita 1997a, for the former) and the influence of some geophysical properties of the Earth like the spherical harmonics J₃, J₄ (see Hartmann et al. 1996) and the triaxiality coming due to the geopotential coefficients C₂₂ and S₂₂. In addition also a relativistic contribution called geodesic nutation (Fukushima 1991) has to be considered.

One of the first precise nutation theories has been set up by Woolard (1953). That series is fairly complete, although the truncation level is 200 $\mathrm \mu$as only. It was followed by the theory of Kinoshita (1977), or briefly K77, with a threshold of 100 $\mathrm \mu$as and 106 terms in total. This nutation model was modified for a deformable Earth by Wahr (1981) and is usually referred to as the standard IAU 1980 nutation series. Later, the rigid Earth nutation was improved by Zhu & Groten (1989), denoted by ZG89, and independently by Kinoshita & Souchay (1990), denoted by KS90, by one order of magnitude to 10 $\mathrm \mu$as for ZG89 and 5 $\mathrm \mu$as for KS90, so that they came up with 262 and 277 nutation terms, respectively. The detailed comparison and discussion between these two models can be found in Souchay (1993). Both series contain for the first time second order terms. Hartmann & Soffel (1994) and Williams (1995) treated a special part of the nutation model, namely the direct effect of the planets with thresholds of 2.5 $\mathrm \mu$as and 0.5 $\mathrm \mu$as, respectively.

New theories of the nutation for a rigid Earth model (containing all of the faint effects mentioned above) appeared in 1997, to be prepared for the General Assembly of IAU held in Kyoto, Japan at August 1997. There is an updated version of KS90's series, see Souchay & Kinoshita (1996, 1997a) where the corrections to the KS90 series are presented in parts due to some errors and change of the threshold to 0.5 $\mathrm \mu$as but mainly due to an updated precession constant which is a scale factor for the nutation. Then Souchay & Kinoshita (1997b) extended their solution to the threshold 0.1 $\mathrm \mu$as. That solution called SK96.2 by the authors contains nearly 1500 terms. The final version will be called REN-2000 and will be published in Souchay et al. (1998). The new theory of Roosbeek & Dehant (1997) denoted as RDAN97 with the same threshold 0.1 $\mathrm \mu$as containing 1553 terms has appeared recently. This theory is based on the torque approach where the torque is derived from ephemerides in the frequency domain. Finally, Bretagnon et al. (1998) made an exhaustive analytical solution of the motion of the figure axis (and also of the rotation and angular momentum axes) of the Earth, based on the analytical theories of the motion of the Moon, the Sun and the planets of the Bureau des Longitudes, called SMART97 by the authors with the threshold of 0.01 $\mathrm \mu$as and containing nearly 5000 terms that completes this list of nutation series.

The goal of this paper is to present a rigid Earth nutation theory called H96NUT computed independently and exclusively by a completely geophysical approach. As emphasized by Melchior & Georis (1968) the luni-solar torque, which is the driving force of nutation and precession, is directly related to the tesseral part of the tidal potential. This method of computation described therein was greatly extended and applied rigorously to derive a highly accurate nutation series. Since Hartmann & Wenzel (1995a,b) recently improved and drastically simplified the tidal potential development (called HW95) to a very high precision, all tools were at hand to start this investigation. Note that also ZG89 used this method partly, namely to compute their 156 additional terms to those 106 ones of the IAU nutation series. Obviously, this is not a homogeneous method of computation (as was already argued by Souchay 1993). As explained in Hartmann et al. (1996) all the terms in ZG89 due to J₃ are erroneous. Furthermore, the tidal potential used by ZG89 does not contain any planetary influence (in contrast to HW95 which contains both, the direct and the indirect planetary effects) and is inferior to HW95 by two orders of magnitude in truncation. All this provided another motivation to extend the geophysical approach.

The organization of this paper is as follows: in Sect. 2 the new nutation theory is presented. The computation of the tidal development is briefly described there, the different coordinate systems are discussed, the relationship between the torque and the tidal potential is given, the nutation theory is developed in time domain using a perturbation theory up to second order and some principle aspects of the new computation method are discussed. Then, in Sect. 3 some of the results of this procedure are given. Section 4 tries to give some error estimates for the individual nutation terms. Then, a comparison with the theories of other authors is given in Sect. 5. Finally, some conclusions are drawn in Sect. 6.