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5.1 Circular orbit around a Schwarzschild black hole

First, we present the gravitational wave luminosity for a particle in a circular orbit around a Schwarzschild black hole [5760]. In this case, _O_ f is given by _O_ = (M/r3 )1/2 f 0 =_ _O_ c, where r 0 is the orbital radius, in standard Schwarzschild coordinates. The luminosity to 11 O(x ) is given by
< dE > (dE ) --- = --- dt [dt N 1247 2 3 44711 4 8191 5 × 1 - -336-x + 4px - 9072-x - 672-px ( ) 6643739519-- 1712- 16-2 3424- 1712- 6 16285- 7 + 69854400 - 105 g + 3 p - 105 ln 2- 105 ln x x - 504 px ( 323105549467 232597 1369 + - --------------+ -------g - ----p2 3178375200 4410 126 ) + 39931-ln 2 - 47385-ln 3 + 232597-lnx x8 294 1568 4410 (265978667519 6848 13696 6848 ) + --------------p - -----pg - ------p ln2 - -----p ln x x9 ( 745113600 105 105 105 2500861660823683-- 916628467-- 424223- 2 + - 2831932303200 + 7858620 g - 6804 p 83217611 47385 916628467 ) 10 - ---------ln2 + ------ln 3 + -----------ln x x ( 1122660 196 7858620 + 8399309750401--p + 177293pg 101708006400 1176 ] 8521283- 142155- 177293- ) 11 + 17640 pln 2- 784 p ln 3 + 1176 p ln x x , (174)
where (dE/dt)N is the Newtonian quadrupole luminosity given by
( ) 2 3 ( )2 dE- = 32m-M--- = 32- m-- x10. (175) dt N 5r50 5 M
This is the 5.5PN formula beyond the lowest, Newtonian quadrapole formula. We can find that our result agrees with the standard post-Newtonian results up to 5 O(x ) [8] in the limit m/M « 1.

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