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B Effects of Extendedness of Stars

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 l of a star are roughly (mass ) × (radius of the star)l and (mass ) × (radius of the star)l × (velocity of steller internal motion ). Since we assume slow stellar rotation where (velocity of steller internal motion ) = 𝒪 (ε) and the strong field point particle limit where 2 (radius of the star) = 𝒪 (ε ), the mass multipole moment and the current multipole moment are of order 𝒪 (ε2l+2) and 𝒪 (ε2l+3), respectively. For example, the spin-orbit coupling force which takes a form of (mass) × (orbital velocity) × (spin) appears at 𝒪 (ε4), 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

mai = FNiewton + ε2F i1 PN + ε4F i2 PN + ε4F iSO + ε4FiQO + ε5F iRR 6 i 6 i 6 i 6 i 6 i 6 i 7 + ε F3 PN + ε F1 PNSO + ε F 1 PN,QO + ε FOO + ε FSS + ε FTO + 𝒪 (ε ). (211 )
Here FiNewton, F i1 PN, F i2 PN, F iRR, and F3iPN, 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. Fi SO and i F 1 PNSO are the spin-orbit coupling force and its 1 PN correction, while i FQO and i F 1 PN QO are the quadrupole-orbit coupling force and its 1 PN correction. i FOO, i F SS, and i FTO 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 [148Jump To The Next Citation Point].

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 [37Jump To The Next Citation Point] 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 [13Jump To The Next Citation Point]. To derive the spin-orbit coupling force in the same form as in previous works [46Jump To The Next Citation Point108Jump To The Next Citation Point152Jump To The Next Citation Point], it is sufficient to assume the coordinate transformation of Λ ττ N in the near zone coordinates to the Λ μ′ν′ A in the (generalized) Fermi normal coordinates in the following implicit form ([133757585960] (see also Appendix B.4):

ττ τ 2 τ′τ′ 2 τ τ τ′i′ 2 ΛN = (ΓA τ′) ΛA + 2ε ΓA τ′Γ Ai′Λ A + 𝒪 (ε ), (212 )
with Γ ττ = 1 + 𝒪 (ε2) A and Γ τi = vi + 𝒪 (ε2) A A (think of a Lorentz transformation). The ε2 in front of the second term arises from the body zone coordinates rescaling (xi = ε2αi A A). An explicit expression of μ Γ Aν 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 8h ττ:
[ ] ττ 4 ∑ P τA 2riA i 2 ik k h = 4ε --- + ε -3-(d A + εM A vA ) + ..., (213 ) A=1,2 rA rA
where “...” denotes irrelevant terms and
∫ ′ ′ diA = ε2 d3αA (Γ τAτ′)2αiAΛτA τ, (214 ) B∫A ij 4 3 τ [i j′]τ′ ij mA = 2ε d αA Γ Aτ′αAΛ A 𝒪 (ε) = M A + 𝒪 (ε), (215 ) BA
where M iAj is given by Equation (90View Equation).

Second, we discuss the χ part of our multipole moments. We have used Λ μν = Θ μν + χμναβ,αβ N N N in the definition of our multipole moments since we could not evaluate χ parts of multipole moments separately except for some low order moments. However, we have to take into account carefully the fact that χ 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 Θττ N. In fact the χ part of the energy, Pτ Aχ, appears at 2 PN order. As another example, the χ part of our dipole moment can be evaluated directly as

∫ 3 i Di = ε2 d3 α αiχ τταβ = ε4175-m-1m2-r12 + 𝒪 (ε5). (216 ) Aχ BA A A N ,αβ 18r312
Thus if we define the center of mass of the star not by Di = 0 A but by Di = 0 AΘ, the form of the equations of motion for the two point masses would change19.

Finally, we list the relevant field and Qi 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.

ττ 6 ∑ rkA- k 8 ∑ rkArlA- kl h = 4ε r3D A + 6ε r5 IA A=1,2 A A=1,2 A ∑ rk ∑ rkrl ∑ rk = 4ε6 -A3dkA + 6ε8 -A5A-IkAl+ 4ε8 -A3M AkiviA + 𝒪 (ε9), (217 ) A=1,2 rA A=1,2 rA A=1,2 rA ∑ rk 6h τi = 2 -AM kAi , (218 ) A=1,2r3A ∑ k hij = 4 rAM k(ivj). (219 ) 6 r3A A A A=1,2 i 42m2M--1ik k Q1 = ε 3r3 r12. (220 ) 12

 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

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