5.2 Circular orbit on the equatorial plane around a Kerr black hole

Next, we consider a particle in a circular orbit on the equatorial plane around a Kerr black hole [101Jump To The Next Citation Point]. In this case, the orbital angular frequency Ω φ is given by
[ 3 2 6 9 ] Ω φ = Ωc 1 − qv + q v + 𝒪 (v ) , (180 )
where Ωc is the orbital angular frequency of the circular orbit in the Schwarzschild case, v = (M ∕r0)1∕2, q = a ∕M, and r 0 is the orbital radius in the Boyer–Lindquist coordinate. The effect of the angular momentum of the black hole is given by the corrections depending on the parameter q. Here, q is arbitrary as long as |q| < 1. The luminosity is given up to 8 𝒪 (x ) (4PN order) by
⟨ ⟩ ( ) [ dE dE 11 3 33 2 4 59 5 -dt = -dt 1 + (q-independent terms ) − 4-qx + 16-q x − 16-qx N ( ) ( ) 65- 611-2 6 162035- 65- 2 71- 3 7 + − 6 πq + 504q x + 3888 q + 8 πq − 24 q x ( ) ] + − 359πq + 22667-q2 + 17q4 x8 . (181 ) 14 4536 16
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