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*Astron. Astrophys. 333, 1100-1106 (1998)*
## 4. Spherical harmonic expansions of *w* and
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
,
and , which are the Cartesian STF multipole
moments. The next step is to find their relation with spherical
harmonic coefficients. In Eqs. (21) and (22),
is an explicit factor in the expressions of
*w* and . This means that *w* and
can certainly be expanded in terms of spherical
harmonics. Comparing the expressions of *w* and
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
and
The introduction of a factor in
Eq. (24) is to make coefficients and
dimensionless. In Eqs. (23) and (24),
and are the longitude
and latitude of a field point relating to
respectively, *M* is the BD mass of the Earth,
is the equatorial radius of the Earth. After
taking the 1PN barycenter of the Earth as the origin of spatial
coordinates, i.e. , , all
the terms of in Eqs. (21) and (23),
in Eqs. (22) and (24) will vanish.
Comparing Eq. (21) with Eq. (23), we can determine
and in terms of
, the result is identical with that of Hartmann
et al. (1994). e.g., , 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
under the standard PN gauge can be completely
determined by the BD mass *M*, the equatorial radius
and the projection of mass quadrupole moment
(which equals ) of the
Earth. The 1PN 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
should be explicitly indicated. The first
equation in Eq. (25) provides the simplest definition of the 1PN
. 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
and 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
( and ,
) are needed. Comparing Eq. (22) with
Eq. (24), they are determined in terms of ,
and . Therefore,
and 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 and of the
lowest order ()
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 and
. 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.
© European Southern Observatory (ESO) 1998
Online publication: April 28, 1998
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