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

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5. Expansion of [FORMULA] under the rigidity approximation

Eq. (14) has shown that, under the standard PN gauge the 1PN vector potential [FORMULA] can be expanded in terms of [FORMULA] and [FORMULA]. We find that the expansion of [FORMULA] is not complicated in form. But [FORMULA], the time-derivative of [FORMULA], is a set of physical quantities that are algebraically independent of [FORMULA]. So it is safer to say that three sets of Cartesian multipole moments [FORMULA], [FORMULA] and [FORMULA] generate together the 1PN potential [FORMULA]. Up to now, there is no explicit expressions of [FORMULA] [FORMULA], i.e. the evolution equations of [FORMULA] [FORMULA]. This makes it difficult to compute [FORMULA]. But in fact, we can use some approximate methods or models for computing [FORMULA].

Because the vector potential [FORMULA] only generate the 1PN effects, all the physical quantities in the expression of [FORMULA] only need to reach the accuracy of Newtonian order. Thus we can adopt the rigidity approximation, i.e. assume that the Earth is a rigid body rotating at an angular velocity [FORMULA], neglecting the non-rigidity correction to the real Earth. Under this assumption, the vector potential can be expressed in terms of [FORMULA] and [FORMULA]. It is preferable that, we think, [FORMULA] in place of [FORMULA] would make the expression of [FORMULA] independent of time-derivatives of Cartesian multipole moments. Moreover, the multipole moments in [FORMULA] are also reduced to two sets ([FORMULA] and [FORMULA] replace [FORMULA], [FORMULA] and [FORMULA]).

According to the rigidity approximation, we have [FORMULA] = [FORMULA]. Inserting it into Eq. (14), and after a tedious calculation (see Appendix), the approximate expression of the vector potential [FORMULA] can be obtained as

[EQUATION]

Inserting Eq. (27) into Eq. (17), we have

[EQUATION]

[FORMULA] can be expanded in terms of [FORMULA] and [FORMULA] [FORMULA] [FORMULA], which are the projections of [FORMULA] and [FORMULA] in the co-rotating system. The expression of [FORMULA] is similar to Eq. (27), it reads

[EQUATION]

The terms containing [FORMULA] in Eq. (28) and [FORMULA] in Eq. (27) are zero as [FORMULA]. Comparing Eq. (28) with Eq. (24), we can obtain the relations between [FORMULA], [FORMULA] and [FORMULA], [FORMULA]. For example, as [FORMULA] we have

[EQUATION]

where [FORMULA]. Applying the rigidity approximation, i.e.

[EQUATION]

and using the following identity

[EQUATION]

it is easy to show that Eq. (29) is consistent with Eq. (26).

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© European Southern Observatory (ESO) 1998

Online publication: April 28, 1998

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