In this section, we derive the spin-orbit coupling force, the quadrupole-orbit coupling force, the spin-spin coupling force, and the spin geodesic precession equation to the lowest order.

Our order-counting of the multipole couplings may need an explanation. The magnitude of a mass multipole moment and a current multipole moment of order of a star are roughly and . Since we assume slow stellar rotation where and the strong field point particle limit where , the mass multipole moment and the current multipole moment are of order and , respectively. For example, the spin-orbit coupling force which takes a form of appears at , that is, 2 PN order, not at the usual 1.5 PN order where rapid stellar rotation is assumed.

In summary the structure of the equations of orbital motion is written schematically as

Here , , , , and , respectively, are the Newtonian force, the 1 PN force, the 2 PN force, the 2.5 PN radiation reaction force, and the 3 PN force. and are the spin-orbit coupling force and its 1 PN correction, while and are the quadrupole-orbit coupling force and its 1 PN correction. , , and are the octupole-orbit coupling force, spin-spin coupling force, and tidal-orbit coupling force, respectivelyThough formally the effects of the 1 PN spin-orbit coupling, the 1 PN quadrupole-orbit, the octupole-orbit coupling, and the tidal-orbit coupling appear up to 3 PN order in our ordering, we focus our attention onto the lowest order spin-orbit coupling, the spin-spin coupling, and the quadrupole-orbit coupling forces. The 1 PN spin-orbit coupling force was derived by Tagoshi, Ohashi, and Owen [148].

Before investigating the multipole-orbit coupling forces, it is worth noticing that our definition of multipole moments is operational and the relation between these multipole moments and the intrinsic multipole moments of a star has not been given. Let us discuss briefly about this problem.

First, for an appropriate frame for the definition of multipole moments it is natural to define the multipole moments in a frame attached to the star, and nonrotating with respect to an asymptotic inertial frame (see [37] for the case of an earth-satellite system in the solar system). If we do not define the multipole moments in an appropriate frame, for example, an apparent quadrupole moment would be produced by Lorentz contraction caused by the orbital motion of the star and an apparent spin would be produced by the Thomas precession. One realization of an appropriate frame are the (generalized) Fermi normal coordinates [13]. To derive the spin-orbit coupling force in the same form as in previous works [46, 108, 152], it is sufficient to assume the coordinate transformation of in the near zone coordinates to the in the (generalized) Fermi normal coordinates in the following implicit form ([13, 37, 57, 58, 59, 60] (see also Appendix B.4):

with and (think of a Lorentz transformation). The in front of the second term arises from the body zone coordinates rescaling (). An explicit expression of is not required for our purpose. To 2 PN order, which is the sufficient order to derive the lowest order spin-orbit, spin-spin, and quadrupole-orbit coupling forces, the transformation changes only : where “” denotes irrelevant terms and where is given by Equation (90).Second, we discuss the part of our multipole moments. We have used in the definition of our multipole moments since we could not evaluate parts of multipole moments separately except for some low order moments. However, we have to take into account carefully the fact that parts of our higher multipole moments can affect the equations of motion for two point masses. An obvious example can be found from the definition of the energy and the mass. It is natural to define the mass as a volume integral of . In fact the part of the energy, , appears at 2 PN order. As another example, the part of our dipole moment can be evaluated directly as

Thus if we define the center of mass of the star not by but by , the form of the equations of motion for the two point masses would changeFinally, we list the relevant field and up to the required order to derive the lowest order spin-orbit, the spin-spin, and the quadrupole-orbit coupling forces and the spin geodesic precession equation.

B.1 Spin-orbit coupling force

B.2 Spin-spin coupling force

B.3 Quadrupole-orbit coupling force

B.4 Spin geodesic precession

B.5 Remarks

B.2 Spin-spin coupling force

B.3 Quadrupole-orbit coupling force

B.4 Spin geodesic precession

B.5 Remarks

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