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

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3. Projections of BD moments in the co-rotating coordinate system

The BD moments are the basic physical quantities in DSX theory. They generate the gravitational field in place of the stress-energy tensor of the matter. They are defined in the local DSX coordinate system (e.g. dynamical non-rotating coordinate system) of a body. Due to the rotation of celestial bodies, BD moments generally change with time T. Only physical quantities that are time-independent or slowly-changing with time can be considered as astronomical or physical constants. Therefore, a set of new multipole moments that are time-independent or slowly-changing with time should be defined.

According to Newtonian mechanics, the mass multipole moments of a rigid body in its mass-centered and co-rotating reference system are time-independent. This prompts that the projections of the 1PN mass multipole moments of a body in its co-rotating system would become time-slowly-changing. Moreover, it is easy to infer that the projections of the time-derivatives of [FORMULA] ([FORMULA] and [FORMULA]) in the co-rotating system are also time-slowly-changing.

Two facts make things more complicated. They are: (a) There is no " rigid body" in general relativity; (b) The real celestial body such as the Earth is not rigid under tidal influence and geophysical mass redistribution even in Newtonian mechanics. So there is no unique co-rotating system. We must define the co-rotating system of a body to assure that the spin of the body with respect to this system is as small as possible.

Let [FORMULA] be the co-rotating coordinate system of the Earth, which is connected with the local DSX coordinate system of the Earth by an orthogonal matrix [FORMULA]. The relation between the two coordinate systems is [FORMULA], or [FORMULA]. The matrix [FORMULA] satisfies

[EQUATION]

Apparently, [FORMULA]. The rotational angular velocity [FORMULA] reads

[EQUATION]

where [FORMULA] is the projection of [FORMULA] on the spatial axes of the co-rotating system, [FORMULA] = [FORMULA], [FORMULA] = [FORMULA]. Making use of [FORMULA] and Eq. (15), W and [FORMULA] can be projected into the co-rotating system [FORMULA], and obtain

[EQUATION]

where w and [FORMULA] have expansions formally similar to those of W and [FORMULA]. If the harmonic gauge is adopted: [FORMULA], corresponding to Eqs. (10) and (11), w and [FORMULA] can be written as

[EQUATION]

and

[EQUATION]

respectively, where

[EQUATION]

If the standard PN gauge is adopted, corresponding to Eqs. (13) and (14), w and [FORMULA] read

[EQUATION]

and

[EQUATION]

respectively. In Eqs. (18)-(22), [FORMULA], [FORMULA], [FORMULA] and [FORMULA] are the projections of [FORMULA], [FORMULA], [FORMULA] and [FORMULA], respectively. They are all time-slowly-changing physical quantities. It should be noted that [FORMULA] and [FORMULA] are in general not small quantities, though [FORMULA] and [FORMULA] are small indeed.

One can find that the right-hand side of Eq. (21) is simpler than that of Eq. (18), which is just the reason to choose the standard PN gauge (Eq. (12)) other than the harmonic gauge [FORMULA]. The merits of choosing the gauge (Eq. (12)) have been stated in Sect. 2. One more argument is : if we take up the harmonic gauge [FORMULA], w is expressed by Eq. (18), and it can be formally expanded in terms of spherical harmonics, but, the spherical harmonic coefficients [FORMULA] and [FORMULA] must be related to r. This would cause that [FORMULA] and [FORMULA] are not constants even in the rigid case.

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© European Southern Observatory (ESO) 1998

Online publication: April 28, 1998

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