Newtonian equations of motion

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 (102 , 103 ) and the time component of Equation (110 ), we have

Then we define the mass of star $A$ as the integrating constant of this equation,

here $mA$ is the ADM mass that star $A$ had if it were isolated. (We took $ε$ zero limit in Equation (114

) 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 (113

).) By definition $mA$ is constant. Then we obtain the lowest order $hττ$ :

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

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

dvi1 &#x222E; m1 ----= &#x2212; dSk &#x0028;4&#x0029;&#x005B;&#x0028;&#x2212; g&#x0029;tikLL&#x005D; d&#x03C4; &#x2202;B1&#x0028; &#x0029; &#x222E; 1-- i k 1-ik &#x2211; &#x2211; mAmBylAymB-- = &#x2212; 4&#x03C0; &#x03B4;l&#x03B4;m &#x2212; 2&#x03B4; &#x03B4;lm dSk y3y3 , A=1,2 B=1,2 &#x2202;B1 A B 1 &#x0028; 1 &#x0029; &#x222E; &#x0028; m2nl nm 2m m n&#x0028;l&#x0028;rm &#x0029;+ &#x03B5;R nm&#x0029;&#x0029; = &#x2212; --- &#x03B4;il&#x03B4;km &#x2212; -&#x03B4;ik&#x03B4;lm d &#x03A9;n1 nk1 --1-1-1-+ ---1--2-A--12------1-1--- 4&#x03C0; 2 &#x0028;&#x03B5;R1 &#x0029;2 &#x007C;&#x20D7;r12 + &#x03B5;R1 &#x20D7;n1&#x007C;3 &#x0028;&#x03B5;R1 &#x0029;2m2 &#x0028;rl + &#x03B5;R1nl &#x0029;&#x0028;rm + &#x03B5;R1nm &#x0029;&#x0029; + --------2--12-------1---132-------1-- , &#x0028;116 &#x0029; &#x007C;&#x20D7;r12 + &#x03B5;R1 &#x20D7;n1&#x007C;

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 3

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.