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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,
respectively18.
Though 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):
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
but by
, the form of the equations of motion for the two point masses would
change19.
Finally, 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.
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