5.6 Circular orbit with a small inclination from the equatorial plane around a Kerr black hole

Next, we consider a particle in a circular orbit with small inclination from the equatorial plane around a Kerr black hole [94]. In this case, apart from the energy ℰ and z-component of the angular momentum lz, the particle motion has another constant of motion, the Carter constant C. The orbital plane of the particle precesses around the symmetry axis of the black hole, and the degree of precession is determined by the value of the Carter constant. We introduce a dimensionless parameter y defined by
y = ------ˆC-------. (197 ) ˆl2 + a2(1 − ˆℰ2) z
UpdateJump To The Next Update Information Given the Carter constant and the orbital radius r0, the energy and angular momentum is uniquely determined by R (r) = 0, and ∂R (r)∕∂r = 0. By solving the geodesic equation with the assumption y ≪ 1, we find that y1∕2 is equal to the inclination angle from the equatorial plane. The angular frequency Ω φ is determined to 𝒪 (y) and 4 𝒪 (v ) as
[ ] 3 3 ( 3 2 4) 6 Ω φ = Ωc 1 − qv + 2y qv − q v + 𝒪 (v ) . (198 )

We now present the energy and the z-component angular flux to 𝒪 (v5):

⟨ dE ⟩ ( dE ) [ 1247 ( 73 ( y ) ) ( 44711 33 527 ) --- = --- 1 − -----v2 + 4π − --- 1 − -- q v3 + − ------+ ---q2 − ---q2y v4 dt dt N ( 336 12 2) 9072 16 96 8191 3749 ( y ) 5] + − -----π + -----q 1 − -- v . (199 ) 672 336 2
⟨ ⟩ ( ) { [ ] dJz dJz ( y) 1247 ( y ) 2 ( y) 61( y) 3 -dt- = dt-- 1 − 2- − 336-- 1 − 2- v + 4π 1 − 2- − 12- 1 − 2- q v N [ ( ) ( ) ] 44711- y- 33- 229- 2 4 + − 9072 1 − 2 + 16 − 32 y q v [ ( ) ( ) ] } + − 8191- 1 − y- π + 417-− 4301-y q v5 . (200 ) 672 2 56 224
Using Equation (198View Equation), we can express v in terms of 1∕3 x = (M Ω φ) as
( ) q 3 1 ( 3 2 4) v = x 1 + 3x + 2-y − qx + q x . (201 )
We then express Equations (199View Equation) and (200View Equation) in terms of x as
⟨ dE ⟩ (dE ) { 1247 ( 11 47 ) [ 44711 ( 33 47 ) ] --- = --- 1 − -----x2 + 4π − --q − ---qy x3 + − ------ + ---− ---y q2 x4 dt dt N [ 336 ( 4 24 ) ] } 9072 16 96 8191- 59- 11215- 5 + − 672 π + − 16 + 672 y q x , (202 )
⟨ ⟩ ( ) { [ ( ) ] dJz dJz ( y ) 1247 ( y) 2 ( y ) 7 11 ( y) 3 ---- = ---- 1 − -- − ----- 1 − -- x + 4π 1 − -- − -y + --- 1 − -- q x dt dt N [ 2 (336 ) 2( ) 2] 2 4 2 44711- y- 33- 117- 2 4 + − 9072 1 − 2 + 16 − 32 y q x [ ( ) ( ) ] } + − 8191- 1 − y- π + − 59-+ 687-y q x5 . (203 ) 672 2 16 224

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