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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 [58]. In this case, the orbital angular frequency _O_ f is given by
[ 3 2 6 9 ] _O_f = _O_c 1 - qv + q v + O(v ) , (176)
where _O_ c is the orbital angular frequency of the circular orbit in the Schwarzschild case, v = (M/r )1/2 0, q = a/M, and r0 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 O(x8) (4PN order) by
&lt;dE &gt; (dE ) [ 11 33 59 --- = --- 1 + (q-independent terms) - --qx3 + ---q2x4 - ---qx5 dt dt N ( ) ( 4 16 16 ) 65- 611-2 6 162035- 65- 2 71- 3 7 + - 6 pq + 504q x + 3888 q + 8 pq - 24 q x ( ) ] 359- 22667- 2 17-4 8 + 14 pq + 4536 q + 16q x . (177)


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