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

5. Expansion of under the rigidity approximation

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

Because the vector potential only generate the 1PN effects, all the physical quantities in the expression of 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 , neglecting the non-rigidity correction to the real Earth. Under this assumption, the vector potential can be expressed in terms of and . It is preferable that, we think, in place of would make the expression of independent of time-derivatives of Cartesian multipole moments. Moreover, the multipole moments in are also reduced to two sets ( and replace , and ).

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

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

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

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

where . Applying the rigidity approximation, i.e.

and using the following identity

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

© European Southern Observatory (ESO) 1998

Online publication: April 28, 1998

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