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3.4 Newtonian equations of motion for extended bodies

Before ending this section, we present some equations for Newtonian extended bodies (stars). These equations will give a useful guideline when we develop our formalism.

The basic equations are the equation of continuity, the Euler equation, and the Poisson equation, respectively:

i ∂-ρ+ ∂ρv--= 0, (48 ) ∂ τ ∂xi Dvi ∂Φ ρ D-τ-= ρ ∂x-, (49 ) i Δ Φ = 4 πρ. (50 )
We define the mass, the dipole moment, the quadrupole moment, and the momentum of the star A as
∫ 3 mA ≡ d x ρ, (51 ) ∫BA Di ≡ d3x ρ(xi − zi ), (52 ) A BA A ∫ IiAj≡ d3x ρ(xi − ziA)(xj − zjA), (53 ) ∫BA i 3 i P A ≡ d x ρv . (54 ) BA
Here ziA is a representative point of the star A. The time derivative of the mass vanishes. Setting the time derivative of the dipole moment to zero gives the velocity momentum relation and a definition of the center of mass,
i dD-- = P i − m vi = 0, (55 ) dτ A A A
where vi = dzi ∕dτ A A. Using the velocity momentum relation, we calculate the time derivative of the momentum,
d2zi1- i m1 dτ2 = F 1, (56 )
where i F1 is defined by Equation (43View Equation). The Newtonian potential can be expressed by the mass and multipole moment as
∑ [ 2 ( )] Φ = ---mA--- + 1-Iij---∂---- ---1---- . (57 ) |⃗x − ⃗zA| 2 A∂xi ∂xj |⃗x − ⃗zA| A=1,2
Substituting Φ into Equation (56View Equation), we obtain the equations of motion,
d2zi ∂ m m 1 ∂3 ( 1 ) m1 ---1-= ------1--2- + --(Iij1 + Ii2j)-----j---- -------- . (58 ) d τ2 ∂zi1|⃗z1 − ⃗z2| 2 ∂zi1∂z2∂zk2 |⃗z1 − ⃗z2|
Here we ignored the mass multipole moments of the stars that are of higher order than the quadrupole moments.

Actually, it is straightforward to formally include all the Newtonian mass multipole moments in the surface integral approach,

d2zi ∑ ∑ (− 1)p+1(2p + 2q + 1)!!I ⟨Mp ⟩I ⟨Nq⟩ m1 --21= -----------------------1-p+q2+2--N ⟨iMpNq ⟩, (59 ) dτ p=0 q=0 p!q! r12
where ⃗r12 = ⃗z1 − ⃗z2, i i N = r12∕r12, Mp = m1m2 ...mp is a corrective index, ⟨...⟩ denotes the symmetric-tracefree operation on the indices between the brackets, and IMAp are the Newtonian mass multipole moments of order 2p.


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