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

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4. Spherical harmonic expansions of w and [FORMULA]

Adopting the standard PN gauge expressed by Eq. (12), Eqs. (17), (21) and (22) determine the expression of the 1PN potential in terms of time-slowly-changing multipole moments [FORMULA], [FORMULA] [FORMULA] and [FORMULA], which are the Cartesian STF multipole moments. The next step is to find their relation with spherical harmonic coefficients. In Eqs. (21) and (22), [FORMULA] is an explicit factor in the expressions of w and [FORMULA]. This means that w and [FORMULA] can certainly be expanded in terms of spherical harmonics. Comparing the expressions of w and [FORMULA] in terms of the spherical harmonics with those in terms of the Cartesian multipole moments, we can immediately get the corresponding relations between the Cartesian multipole moments and the spherical harmonic coefficients.

The spherical harmonics expansions corresponding to Eqs. (21) and (22) can be written as




The introduction of a factor [FORMULA] in Eq. (24) is to make coefficients [FORMULA] and [FORMULA] dimensionless. In Eqs. (23) and (24), [FORMULA] and [FORMULA] are the longitude and latitude of a field point relating to [FORMULA] respectively, M is the BD mass of the Earth, [FORMULA] is the equatorial radius of the Earth. After taking the 1PN barycenter of the Earth as the origin of spatial coordinates, i.e. [FORMULA], [FORMULA], all the terms of [FORMULA] in Eqs. (21) and (23), [FORMULA] in Eqs. (22) and (24) will vanish.

Comparing Eq. (21) with Eq. (23), we can determine [FORMULA] and [FORMULA] in terms of [FORMULA], the result is identical with that of Hartmann et al. (1994). e.g., [FORMULA], we have


Now we can find that, under the standard PN gauge Eq. (12), the expansion of the 1PN scalar potential in terms of the Cartesian STF multipole moments or the spherical harmonics is formally similar to that of the Newtonian potential in classical mechanics. As an astronomical constant, the Earth dynamical form-factor [FORMULA] under the standard PN gauge can be completely determined by the BD mass M, the equatorial radius [FORMULA] and the projection of mass quadrupole moment [FORMULA] (which equals [FORMULA]) of the Earth. The 1PN [FORMULA] is generally non-unique but gauge-dependent. This conclusion is the same as that of Brumberg et al. (1996). So the adopted gauge connected with the 1PN [FORMULA] should be explicitly indicated. The first equation in Eq. (25) provides the simplest definition of the 1PN [FORMULA]. That results from the choice of the standard PN gauge. Actually, it is necessary to choose the standard PN gauge in the spherical harmonics expansion of the 1PN scalar potential, otherwise the spherical harmonic coefficients [FORMULA] and [FORMULA] will depend on r because the 1PN scalar potential does not satisfy the Poisson equation. In other words, the 1PN scalar potential is exactly the solution of the Poisson equation under the standard PN gauge.

Eqs. (22) and (24) show that the expansion of the vector potential is much more complicated than that of the scalar potential. For the vector potential, three sets of expansion coefficients ([FORMULA] and [FORMULA], [FORMULA]) are needed. Comparing Eq. (22) with Eq. (24), they are determined in terms of [FORMULA], [FORMULA] and [FORMULA]. Therefore, [FORMULA] and [FORMULA] depend on the radius r. The reason is that the 1PN vector potential does not follow the Poisson equation under the standard PN gauge.

As a useful example, here we give the expressions of the coefficients [FORMULA] and [FORMULA] of the lowest order ([FORMULA])


In many practical problems, it is usually accurate enough to consider only the largest one of the 1PN terms in the gravitational field of the Earth, i.e. the 1PN term generated by the mass of the Earth. In this case the 1PN vector potential is completely negligible. When a higher accuracy is needed, it would be enough to consider the 1PN terms generated by the mass, mass quadrupole moment and spin of the Earth. In this case, the right-hand side of Eq. (24) only contains the terms of [FORMULA] and [FORMULA]. Generally speaking, if we set up the model of the 1PN gravitational field of the Earth, the vector potential should certainly be taken into account. Only a few terms in Eq. (24) need to be retained, the terms with large values of l are actually negligible. In other words, for a practical model of the gravitational field, the expressions of the scalar and vector potentials must be truncated at some value of l, which is determined according to the practical accuracy of observation.

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© European Southern Observatory (ESO) 1998

Online publication: April 28, 1998