Adiabatic evolution of Carter constant for orbit with small eccentricity and small inclination angle around a Kerr black hole

5.8 Adiabatic evolution of Carter constant for orbit with small eccentricity and small inclination angle around a Kerr black hole

In the Schwarzschild case, the particle’s trajectory is characterized by the energy $E$ and the z-component angular momentum $Lz$ . In the adiabatic approximation, the rates of change of $E$ and $Lz$ are equated with those radiated to infinity as gravitational waves or with those absorbed into the black hole horizon, in accordance with the conservation of $E$ and $L z$ . On the other hand, in the Kerr case, the Carter constant, $Q$ , is also necessary to specify the particle’s trajectory. In this case, the rate of change of $Q$ is not directly related to the asymptotic gravitational waves. Mino [70 ] proposed a new method for evaluating the average rate of change of the Carter constant by using the radiative field in the adiabatic approximation. He showed that the average rate of change of the Carter constant as well as the energy and angular momentum can be obtained by the radiative field of the metric perturbation. The radiative field is defined as half the retarded field minus half the advanced field. Mino’s work gave a proof of an earlier work by Gal’tsov [46 ] in which the radiative field is used to evaluate the average rate of change of the energy and the angular momentum without proof. Inspired by this new development, it was demonstrated in [53 ] and [31 ] that the time-averaged rates of change of the energy and the angular momentum can be computed numerically for generic orbits. A first step toward explicit calculation of the rate of change of the Carter constant was done in the case of a scalar charged particle in [30 ]. After that a simpler formula for the average rate of change of the Carter constant was found in [90 , 89

]. This new formula relates the rate of change of the Carter constant to the flux evaluated at infinity and on the horizon. Based on the new formula, in [89

], the rate of change of the Carter constant for orbits with small eccentricities and inclinations is derived analytically up to $O (v5)$ by using the MK method discussed in Section 4. In Ref. [48

], the method was extended to the case of the orbits with small eccentricity but arbitrary inclination angle.

First we show the results for the small inclination case [89 ]. We define the radius $r0$ and the eccentricity $e$ as

In Ref. [89

], the parameter which expresses a small inclination from the equatorial plane is defined as

Note that this definition of $y′$ is different from $y$ in Equation (197

). By solving Equation (209

) with respect to $E$ and $ˆl z$ , we obtain

2 4 ( 2 4 ) ˆℰ = 1 − v--+ 3v--− qv5 − e2 v--− v--+ 2qv5 + 1qv5y ′ + qv5e2y′, (211 ) 2 8 2 4 2 [ 3v2 27v4 15qv5 ˆlz = r0v 1 + ----− 3qv3 + -----+ q2v4 − ------ ( 2 2 8 4 2 42 5) +e2 − 1 + 3v--− 6qv3 + 81v--+ 7q-v--− 63qv-- ( 2 8 2 2 ) ′ 1 3v2 3 27v4 3q2v4 15qv5 +y − 2 − -4--+ 3qv − -16--− --2---+ --2--- ( 2 4 2 4 5) ] +e2y ′ 1-− 3v--+ 6qv3 − 81v--− 19q-v--+ 63qv-- . (212 ) 2 4 16 4 2

By using Equations (4.9) and (4.12) in [89 ], we find the azimuthal angular frequency $Ω φ$ as

[ ( 2 3 ) Ω φ = Ωc 1 − qv3 + e2 3-+ 9v--− 21qv-- + 18v4 + 3q2v4 − 60qv5 ( )2 2( 2 ) 3qv3- 3q2v4- ′ 39qv3- 51q2v4- 75qv5- 2 ′] + 2 − 2 y + 4 − 4 + 2 e y , (213 )

where $v = (M ∕r0)1∕2$ and $Ωc = (M ∕r30)1∕2$ . From the definition of $x ≡ (M Ω φ)1∕3$ , the parameter $v$ is expressed with $x$ as

[ qx3 ( 1 3x2 7qx3 31qx5 ) v = x 1 + ----+ e2 − -− ----+ ----- − 6x4 − q2x4 + ------ ( 3 ) 2 2 ( 3 2) ] qx3 q2x4 ′ 2 ′ 3qx3 9q2x4 23qx5 + − ----+ ----- y + e y − -----+ ------− ------ . (214 ) 2 2 2 4 4

By using the above formula, we can express $y$ defined in Equation (197

) in terms of $y′$ above as

Cˆ y = --------------- ˆl2z + a2(1 − ℰˆ2) ˆl2zy′ = ˆ2----2-----ˆ2- l(z + a2(14− ℰ )2 6) ′ 2( 2 4 2 6) ′ = 1 − q x + 4q x y + e − q x + 12q x y . (215 )

The average rate of change of $ℰ$ , $lz$ and $Q$ become

⟨ ⟩ dℰ 32 ( μ )2 10 dt- = − 5-- M-- v [ 1247 ( 73 ) × 1 − ----v2 − --q − 4π v3 ( 336 12) ( ) − 44711-− 33q2 v4 + 3749-q − 8191π v5 { 9072 16 ( 336 67)2 + 277-− 4001v2 + 3583π − 457q v3 24( 84 )48 ( 4 ) } + 42q2 − 1091291- v4 + 58487-q − 364337-π v5 e2 ( 9072 67)2 1344 73- 3 527- 2 4 3749- 5 ′ + 24qv − 96 q v − 672 qv y ( 457- 3 5407-2 4 58487- 5) 2 ′] + 8 qv − 48 q v − 1344 qv e y , (216 ) ⟨ dl ⟩ 32 ( μ )2 --z = − --- --- M v7 dt [5 M ( ) × 1 − 1247v2 − 61q − 4π v3 ( 336 12) ( ) 44711- 33-2 4 417- 8191- 5 − 9072 − 16q v + 56 q − 672 π v { 51- 17203- 2 ( 781- 369- ) 3 + 8 − 672 v + − 12 q + 8 π v (929- 2 1680185-) 4 ( 1809- 48373- ) 5} 2 + 32 q − 18144 v + 224 q − 336 π v e { 1 1247 2 (61 ) 3 + − 2-+ -672-v + -8-q − 2 π v (213 2 44711 ) 4 ( 4301 8191 ) 5} ′ − ----q − ------ v − ----q − ----π v y { 3521 1721083144 (1513 224 369 1)344 + − ---+ ------v2 + -----q − ----π v3 (1166801851344 5981 ) 16 ( 16 48373 ) } ] + -------- − ----q2 v4 − 168q − ------π v5 e2y ′, (217 ) ⟨ ⟩ ( )36288 64 672 dQ- 64- -μ- 3 3 6 dt = − 5 M M v [ 743 2 ( 1637 ) 3 × 1 − qv − ---v − ----q − 4π v ( 439 336129193 336 ) (151765 4159 33 ) + ---q2 − ------- − 4πq v4 + -------q − -----π − --q3 v5 { 4483 51 181442425 (14869 18144337 ) 672 16 + ---− ---qv − -----v2 − ------q − ----π v3 8(4536801 3623214 369224) 8 − ------- − -----q2 + ----πq v4 ( 4536 32 8 ) } + 141049-q − 38029π − 929q3 v5 e2 { 9072 672( 32 ) + 1qv + 1637-qv3 − 1355q2 − 2πq v4 2( 672 )96 } − 151765-q − 213q3 v5 y′ { 36288 32 ( ) 51- 14869- 3 369- 33257- 2 4 + 16qv + 448 qv + 16 πq − 192 q v ( 141049- 5981- 3) 5} 2 ′] + − 18144 q + 64 q v e y . (218 )

The rate of change of $C$ becomes

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ dC- dQ- ( d-ℰ dlz- ) dt = dt − 2(aℰ − lz) a dt − dt ( )3 [ ( ) = − 64- μ-- M 3v6y′ 1 − 743-v2 − 85-q − 4π v3 5 M ( 336 ) 8 ( ) − 129193- − 307-q2 v4 + 2553q − 4159π v5 { 18144 96 ( 224 ) 672 + 43− 2425-v2 + 337-π − 1793-q v3 8( 224 8 ) 16 453601- 7849- 2 4 − 4536 − 192 q v ( 3421- 38029- ) 5} 2] + 224 q − 672 π v e . (219 )

We can use Equation (214 ) to express these equations in terms of $x$ . We obtain

⟨ ⟩ ( ) dℰ 32 ( μ )2 10[ 1247x2 11q 3 --- = − --- --- x 1 − ------- + 4 π − ---- x dt (5 M 2) 336 ( 4 ) + − 44711-+ 33q-- x4 + − 8191π- − 59q- x5 9072 16 ( 672 )16 2{ 157 6781x2 2335π 2009q 3 +e ----− ------- + ------− ------ x ( 24 168 2) 48( 72 ) } + − 14929-+ 281q-- x4 + − 773π- + 3223q- x5 ( 189 16 3) 168 47qx3 47q2x4 11215qx5 ′ + − ------− -------+ --------- y ( 24 3 96 2 6472 5) ] +e2y ′ − 617qx--− 1585q-x--+ 60187qx-- , (220 ) 48 96 336

⟨ ⟩ ( ) dlz 32 ( μ )2 7[ 1247x2 11q 3 --- = − --- --- M x 1 − ------- + 4π − ---- x dt 5( M 2) 336( 4 ) + − 44711-+ 33q-- x4 + − 8191-π − 59q- x5 9072 16 ( 672 ) 16 2{ 23 3259x2 209 π 371q 3 +e ---− -------+ ----- − ----- x ( 8 168 2) 8 ( 24 ) } + − 1041349-+ 171q-- x4 + − 785π-+ 949q- x5 18144 16 ( 6 5)6 2 ′{ 23 3259x2 209 π 1057q 3 +e y − ---+ ------- + − ----- + ------ x ( 16 3362) ( 16 48 ) } + 1041349-− 825q-- x4 + 785-π − 925q- x5 36288 32 ( 12) 14 { 1 1247x2 71q 3 + − --+ ------- + − 2 π +---- x ( 2 672 2) ( 24 ) } ] + 44711-− 101q-- x4 + 8191-π + 687q- x5 y′ , (221 ) 18144 32 1344 224

⟨ ⟩ ( )3 [ 2 ( ) dC- = − 64- μ-- M 3x6y ′1 − 743x--+ 4π − 69q- x3 dt 5( M ) 336( 8 ) 129193 307q2 4 4159 π 11089q 5 + − ------- + ------ x + − ------ + ------- x { 18144 296 ( 6)72 2016 +e2 19-− 7379x-- + 193π-− 89q- x3 ( 8 672 )8 (2 ) 1340159 1209q2 4 34295π 502051q 5} ] + − -------- + ------- x + − -------+ -------- x . (222 ) 18144 64 448 4032

We note that if we set $y′ = 0$ , $⟨dℰ∕dt ⟩$ and $⟨dlz∕dt⟩$ agree, respectively, with the rate of emission of the energy and the azimuthal angular momentum radiated to infinity, Equations (195

) and (196