5.4 Slightly eccentric orbit around a Schwarzschild black hole

Next, we consider a particle in slightly eccentric orbit on the equatorial plane around a Schwarzschild black hole (see [71], Section 7). We define r0 as the minimum of the radial potential R(r)∕r4. We also define an eccentricity parameter e from the maximum radius of the orbit rmax, which is given by rmax = r0(1 + e). These conditions are explicitly given by
| ∂(R-∕r4)| ∂r |r=r0= 0, and R (r0(1 + e)) = 0. (188 )
We assume e ≪ 1. In this case, Ωφ is given to 2 𝒪 (e ) by
[ 2] 3(1 − 3v2)(1 − 8v2) Ω φ = Ωc 1 − f (v)e , f (v) = --------2--------2-, (189 ) 2(1 − 2v )(1 − 6v )
where Ωc = (M ∕r30)1∕2 is the orbital angular frequency in the circular orbit case. We now present the energy and angular momentum luminosity, accurate to 𝒪 (e2) and to 𝒪 (x8) beyond Newtonian order. They are given by
⟨ ⟩ ( ) { dE- dE- dt = dt N 1 + (e-independent terms) [ +e2 157-− 6781-x2 + 2335-πx3 − 14929x4 − 773πx5 24 168 48 189 3 (156066596771 106144 992 2 80464 + --------------− -------γ + ----π − ------ln2 69854400 315 ) 9 315 − 234009- ln 3 − 106144-ln x x6 − 32443727- πx7 ( 560 315 48384 3045355111074427-- 507208- 31271- 2 151336- + − 671272842240 + 245 γ − 63 π − 441 ln 2 12887991 507208 ) ]} + --------- ln 3 + -------ln x x8 , (190 ) 3920 245
and
⟨ ⟩ ( ) { dJz dJ ---- = --- 1 + (e-independent terms) dt dt N [ 2 23- 3259- 2 209- 3 1041349- 4 785- 5 +e 8 − 168 x + 8 πx − 18144 x − 6 πx ( + 91721955203--− 41623γ + 389π2 − 24503-ln 2 69854400 210 ) 6 210 78003- 41623- 6 91565- 7 − 280 ln 3 − 210 lnx x − 168 πx ( 105114325363 696923 4387 2 7051 + − -------------- + -------γ − -----π − -----ln 2 72648576 630 ) 18] } 10 + 3986901- ln 3 + 696923-ln x x8 , (191 ) 1960 630
where (dJ∕dt)N is the Newtonian angular momentum flux expressed in terms of x,
( dJz) 32 ( μ )2 ---- = --- --- M x7, (192 ) dt N 5 M
and the e-independent terms in both ⟨dE ∕dt⟩ and ⟨dJ ∕dt⟩ are the same and are given by the terms in the case of circular orbit, Equation (178View Equation).
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