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4.6 Newtonian equations of motion

We first derive the Newtonian equations of motion and the 1 PN correction to the acceleration. The derivation of the 1 PN equations of motion includes some essences of our formalism and shows how it works properly. Thus we shall give a detailed explanation about the derivation, though the calculation is elementary and the resulting equations of motion are well-known.

Now let us derive the Newtonian mass-energy relation first. From Equations (102View Equation, 103View Equation) and the time component of Equation (110View Equation), we have

dPτA- 2 dτ = 𝒪 (ε ). (113 )
Then we define the mass of star A as the integrating constant of this equation,
τ mA ≡ liε→m0 P A. (114 )
here mA is the ADM mass that star A had if it were isolated. (We took ε zero limit in Equation (114View Equation) to ensure that the mass defined above does not include the effect of the companion star and the orbital motion of the star itself. By this limit we ensure that this mass is the integrating constant of Equation (113View Equation).) By definition mA is constant. Then we obtain the lowest order hττ:
∑ mA 4h ττ = 4hτBτ= 4 ----. (115 ) A=1,2 rA

Second, from Equation (91View Equation) with Equations (102View Equation) and (103View Equation) we obtain i QA = 0 at the lowest order. Thus we have the Newtonian momentum-velocity relation PiA = mAviA + 𝒪 (ε2) from Equation (86View Equation) (we set Di = 0 A).

Substituting Equation (104View Equation) with the lowest order ττ h into the general form of equations of motion (111View Equation) and using the Newtonian momentum-velocity relation, we have for star 1:

dvi1 ∮ m1 ----= − dSk (4)[(− g)tikLL] dτ ∂B1( ) ∮ 1-- i k 1-ik ∑ ∑ mAmBylAymB-- = − 4π δlδm − 2δ δlm dSk y3y3 , A=1,2 B=1,2 ∂B1 A B 1 ( 1 ) ∮ ( m2nl nm 2m m n(l(rm )+ εR nm)) = − --- δilδkm − -δikδlm d Ωn1 nk1 --1-1-1-+ ---1--2-A--12------1-1--- 4π 2 (εR1 )2 |⃗r12 + εR1 ⃗n1|3 (εR1 )2m2 (rl + εR1nl )(rm + εR1nm )) + --------2--12-------1---132-------1-- , (116 ) |⃗r12 + εR1 ⃗n1|
where yiA is the integral variable and ⃗yA = ⃗y − ⃗zA. We defined ⃗n1 as the spatial unit vector9 emanating from ⃗z1, ⃗r12 ≡ ⃗z1 − ⃗z2, and ni ≡ ri ∕r12 12 12. We used ⃗r1 = εR1 ⃗n1 and ⃗r2 = ⃗r12 + εR1⃗n1. For details see Figure 3View Image).
View Image

Figure 3: The vectors used in the surface integral over the boundary of the body zone 1.

In the above equation, by virtue of the angular integral the first term (which is singular when the ε zero limit is taken) vanishes. The third term vanishes by letting ε go to zero. Only the second term survives and gives the Newtonian equations of motion as expected. This completes the Newtonian order calculations.

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