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Astron. Astrophys. 333, 1100-1106 (1998)

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1. Introduction

The Earth's gravitational potential can be expanded in two forms: (1) spherical harmonics expansion (McCarthy, 1992), which astronomers and geophysicists are familiar with; (2) expansion in terms of rectangular coordinate multipole moments that are Cartesian symmetric and trace-free (STF) tensors, which is in favour with physicists. Two kinds of expressions of the gravitational potential are essentially equivalent, as both are the equivalent irreducible representation of the rotation group [FORMULA]. Various kinds of practical Earth's gravitational field models (e.g. Anderle, 1979, Lerch, 1985, Reigber, 1985) are built in spherical harmonics expansion. But many theoretical physicists trend to use Cartesian STF tensor expansion, which would make theories more elegant, simple and compact in form. A recent investigation (Hartmann et al., 1994) demonstrated that in numerical computation the STF tensors are more efficient than the spherical harmonic coefficients. Thorne (1980) once systematically summarized various kinds of spherical harmonics and STF tensors in gravitational physics. From then on, the STF tensors as a powerful mathematical tool have been extensively used in references on gravitation and general relativistic celestial mechanics, e.g. Kopejkin (1988), Blanchet and Damour (1989), Damour and Iyer (1991a, b), Damour et al. (1991, 1992, 1993, 1994) and Xu et al. (1997). Hartmann et al. (1994) presented an extensive discussion on the application of the STF tensors in classical celestial mechanics.

In the recent two decades, much progress has been made in relativistic celestial mechanics. Especially, a new 1PN celestial mechanics theory (hereafter cited as DSX theory) presented by Damour, Soffel and Xu (1991, 1992, 1993, 1994), is very attractive. The theory has extensive and bright prospects for its application in astronomy and other fields concerned. But at present, efforts have to be made to put DSX theory into its practical use. One difficulty lies in the basic physical quantities in DSX theory, i.e. relativistic mass and spin multipole moments (BD moments, Blanchet, Damour, 1989). Damour et al. (1991) considered BD moments as observable physical quantities. But BD moments are defined in the body-centered coordinate system whose axes could be slowly rotating with an 1PN angular velocity (Damour et al., 1991) and therefore they are generally fast-changing with time due to the irregular figure and the rotation of the body. It would be necessary to introduce the multipole moments that are defined in a co-rotating coordinate system with the body and therefore are slowly-changing with time. Furthermore, the evolution equations of BD moments expressed in terms of themself have not been obtained to our knowledge so far.

Hartmann et al. (1994) has derived a general relation between the spherical harmonic coefficients and the Cartesian STF multipole moments in Newtonian celestial mechanics. It would be also necessary to extend their results to the 1PN celestial mechanics. In the latter case the 1PN gravitational field is described by a scalar potential and a 3-dimensional vector potential, which would change with a gauge transformation in DSX theory. A proper choice of the gauge condition might simplify the relation we are looking for.

This paper is arranged as follows: Sect. 2 briefly states the BD moments expansion of the 1PN gravitational potential, the existing problems in it and the gauge choice. Sect. 3 introduces a co-rotating coordinate system, then projects the above expansion into it, and obtains a time-slowly-changing multipole moments expansion of the 1PN potential. In Sect. 4, we write out the spherical harmonics expansion of the 1PN potential, and derive the relation between the lowest order spherical harmonic coefficients and the relevant Cartesian multipole moments. In Sect. 5, we discuss the Cartesian multipole moments expansion of the 1PN vector potential under the rigidity approximation. In Sect. 6, some main conclusions are summarized.

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© European Southern Observatory (ESO) 1998

Online publication: April 28, 1998