Circular orbit around a Schwarzschild black hole

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 [100

, 105 ]. In this case, $Ωφ$ is given by $Ωφ = (M ∕r30)1∕2$ $≡ Ωc$ , where $r0$ is the orbital radius, in standard Schwarzschild coordinates. The luminosity to $𝒪 (x11)$ is given by

⟨ dE ⟩ ( dE ) --- = --- dt [dt N 1247 2 3 44711 4 8191 5 × 1 − -336-x + 4πx − 9072-x − 672-πx ( ) 6643739519-- 1712- 16-2 3424- 1712- 6 16285- 7 + 69854400 − 105 γ + 3 π − 105 ln 2 − 105 ln x x − 504 πx ( 323105549467 232597 1369 + − --------------+ -------γ − ----π2 3178375200 4410 126 ) + 39931-ln 2 − 47385-ln 3 + 232597-lnx x8 294 1568 4410 ( 265978667519 6848 13696 6848 ) + --------------π − -----πγ − ------π ln2 − -----π ln x x9 ( 745113600 105 105 105 2500861660823683-- 916628467-- 424223- 2 + − 2831932303200 + 7858620 γ − 6804 π 83217611 47385 916628467 ) 10 − ---------ln2 + ------ln 3 + -----------ln x x ( 1122660 196 7858620 + 8399309750401--π + 177293π γ 101708006400 1176 ] 8521283- 142155- 177293- ) 11 + 17640 πln 2 − 784 π ln 3 + 1176 π ln x x , (178 )

where $(dE ∕dt)N$ is the Newtonian quadrupole luminosity given by

This is the 5.5PN formula beyond the lowest, Newtonian quadrupole formula. We can find that our result agrees with the standard post-Newtonian results up to $𝒪 (x5)$ [13 ] in the limit $μ ∕M$ $≪ 1$ .