Adiabatic evolution of constants of motion for orbits with generic inclination angle and with small eccentricity around a Kerr black hole

5.9 Adiabatic evolution of constants of motion for orbits with generic inclination angle and with small eccentricity around a Kerr black hole

Update The calculation in Section 5.8 was extended to orbits with generic inclination angle by Ganz et al. [48

]. We specify the geodesics by the semi-latus rectum $p$ and the eccentricity $e$ and a dimensionless inclination parameter $′ y$ . The outer and inner turning point of the radial motion is here define as

The inclination parameter is defined by

which is the same as Equation (210

). We define $∘ ----- v = M ∕p$ . By solving these equations with respect to $ˆ ℰ$ and $ˆ lz$ , we obtain

{ } ˆℰ = 1 − 1-v2 + 3v4 − qY v5 + e2 1v2 − 3-v4 + 2qY v5 , 2 8 2 4 ˆlz 3Y ( 27Y ) -- = vY + ---v3 − 3qY 2v4 + q2Y 3 + ----- v5 p { 2 ( )8 } +e2 Y-v3 − qY 2v4 + q2Y 3 + 9Y- v5 , (225 ) 2 4

where

The average rate of change of $ℰ$ , $l z$ and $C$ become up to $O (e2,v5)$ ,

⟨ ⟩ [( ) ( ) dℰ 32 ( μ )2 10 2 3∕2 73 2 1247 9181 2 2 dt- = − -5- M-- v (1 − e ) 1 + 24e − -336- + -672-e v ( ) ( ) − 73Y--+ 823Y--e2 qv3 + 4 + 1375-e2 πv3 ( 12 24 ) 48 44711 172157 2 4 − -9072-+ -2592--e v ( 2 { 2} ) − 329-− 527Y--+ 4379-− 6533Y--- e2 q2v4 ( 96 96 ) 192 ( 192 ) ] 8191- 44531- 2 5 3749Y-- 1759Y-- 2 5 − 672 + 336 e πv + 336 + 56 e qv , ⟨ ⟩ ( )2 [( ) ( ) dlz = − 32- -μ- M v7(1 − e2)3∕2 Y + 7Y-e2 − 1247Y--+ 425Y-e2 v2 dt 5( M { } 8) ( 336 33)6 61- 61Y-2 63- 91Y-2 2 3 97Y--2 3 + 24 − 8 + 8 − 4 e qv + 4Y + 8 e πv ( ) − 44711Y--+ 302893Y-e2 v4 ( 9072 604{8 } ) 57Y-- 45Y-3 201Y-- 37Y-3 2 2 4 − 16 − 8 + 16 − 2 e q v ( ) − 8191Y--+ 48361Y-e2 πv5 ( 672 1344 { } ) ] 2633- 4301Y-2- 66139- 18419Y-2- 2 5 − 224 − 224 + 1344 − 448 e qv , ⟨ ⟩ ( )3 ( )[( ) d𝒞- = − 64- μ-- M 3v6(1 − e2)3∕2 1 − Y 2 1 + 7e2 dt 5( M ) ( ) 8 743- 23-2 2 85Y-- 211Y-- 2 3 − 336 − 42e v − 8 + 8 e qv ( 97 ) ( 129193 84035 ) + 4 + --e2 πv3 − -------+ ------e2 v4 ( 8 { 18144 1}728) 329- 53Y-2 929- 163Y-2- 2 2 4 − 96 − 8 + 96 − 8 e q v ( 2553Y 553Y ) ( 4159 21229 ) ] + -------− -----e2 qv5 − -----+ ------e2 πv5 . (227 ) 224 192 672 1344

Here, a term $(1 − e2)3∕2$ is factored out. We can express $v$ in terms of $x ≡ (M Ω )1∕3 φ$ by using Equation (3.15) in [48

] as

[ ( ) ( 2) v = x 1 + qx3 − 2-+ Y + q2x4 1-+ Y-− 3Y-- ( 3 ( )4 2( 4 ( )) 2 1- 2 3 4- 4 2 5- 5Y- 19Y-2 +e 2 − x + qx − 3 + 3Y + x − 3 + q 8 + 4 − 8 )] +qx5 (3 + 3Y ) . (228 )

The average rate of change of $ℰ$ , $lz$ and $C$ are rewritten as

⟨ ⟩ ( ) dℰ 32 ( μ )2 ( 2)3∕2 10[ 193e2 1247 30865e2 2 --- = − --- --- 1 − e x 1 + ------+ − -----− -------- x dt 5( M ( 24 336 672)) +x3 4π − 20q-+ 47qY--+ e2 2623-π − 1145q-+ 379qY-- ( 3 12 48 18 12 4 44711 89q2 2 193q2Y 2 +x − ------− -----+ 5q Y − --------- ( 9072 96 2 2 96 2 2 )) +e2 − 522439- − 4165q--+ 1205q-Y--− 503q--Y-- ( 6048 192 24 64 5 8191 π 1247q 11215qY +x − ------ + ------− --------- 67(2 42 336 ) ) ] +e2 − 370877π-+ 17723q-− 39199qY-- . (229 ) 1344 42 96

⟨ dl ⟩ 32 ( μ )2 ( )3∕2 [ 35e2Y ( 1247Y 16777e2Y ) --z = − --- --- M x7 1 − e2 Y + ------ + x2 − -------− ---------- dt 5( M 2 8 ( 336 672 2) ) 3 61q- 14qY-- 5qY-- 2 247q- 257πY-- 329qY-- 51qY--- +x 24 + 4πY − 3 − 8 + e 12 + 8 − 12 − 4 ( 44711Y 29q2Y 7q2Y 2 3q2Y 3 +x4 − --------− ------+ ------+ ------ 9(072 16 2 8 ) ) 2 83963Y-- 2 357q2Y-2- 399q2Y-3- +e − 1296 − 21q Y + 16 + 32 ( 2633q 8191πY 1247qY 3181qY 2 +x5 − ------− --------+ --------− --------- 22(4 672 56 224 )) ] 2 65029q- 200413-πY- 10651qY-- 15065qY--2 +e − 448 − 1344 + 56 − 448 . (230 )

⟨ ⟩ ( ) dC- 64-(-μ-)3 3 6 ( 2)3∕2( 2) [ 31e2- 743- 1201e2- 2 dt = − 5 M M x 1 − e 1 − Y 1 + 8 + − 336 − 84 x ( 37qY ( 241π 43q 575qY )) +x3 4π − 4q − ------+ e2 -----− ----− ------- ( 8 8 2 16 4 129193- 185q2- 2 17q2Y-2 +x − 18144 − 96 + 3q Y + 8 ( 438271 141q2Y 385q2Y 2) ) +e2 − ------- − 18q2 + --------+ --------- ( 5184 8 16 5 4159π- 743q- 4229qY-- +x − 672 + 63 − 672 ( 19227 π 1799q 3151qY ) ) ] +e2 − ------- + ------+ -------- . (231 ) 224 18 96

If we assume that the inclination angle is small and $′ ′2 Y = 1 − y ∕2 + O (y )$ , we find that Equations (229 ) – (231 ) reduce respectively to (220 ) – (222 ) in Section 5.8. As discussed in [48 ], in the case of largely inclined orbits, the fundamental frequency of gravitational waves is expressed not only with $Ω φ$ but also the frequency of $𝜃$ -ocillation, $Ω𝜃$ .