go to next pagego upgo to previous page

5.3 Slightly eccentric orbit around a Schwarzschild black hole

Next, we consider a particle in slightly eccentric orbit on the equatorial plane around a Schwarzschild black hole (see [34], Section 7). We define r0 as the minimum of the radial potential R(r)/r4. We also define an eccentricity parameter e from the maximum radius of the orbit rmax, which is given by r = r (1 + e) max 0. These conditions are explicitly given by
| @(R/r4)-| @r |r=r0= 0, and R(r0(1 + e)) = 0. (178)
We assume e « 1. In this case, _O_f is given to 2 O(e ) by
[ 2] 3(1 - 3v2)(1- 8v2) _O_f = _O_c 1- f (v)e , f (v) = --------2--------2-, (179) 2(1 - 2v )(1- 6v )
where _O_c = (M/r30)1/2 is the orbital angular frequency in the circular orbit case. We now present the energy and angular momentum luminosity, accurate to O(e2) and to O(x8) beyond Newtonian order. They are given by
< > ( ) { dE- dE- dt = dt N 1 + (e-independent terms) [ +e2 157-- 6781-x2 + 2335-px3 - 14929x4 - 773px5 24 168 48 189 3 (156066596771 106144 992 2 80464 + --------------- -------g + ----p - ------ln2 69854400 315 ) 9 315 - 234009- ln 3- 106144-ln x x6 - 32443727- px7 ( 560 315 48384 3045355111074427-- 507208- 31271- 2 151336- + - 671272842240 + 245 g- 63 p - 441 ln 2 12887991 507208 ) ]} + --------- ln 3 + -------ln x x8 , (180) 3920 245
and
< > ( ) { dJz dJ ---- = --- 1 + (e-independent terms) dt dt N [ 2 23- 3259- 2 209- 3 1041349- 4 785- 5 +e 8 - 168 x + 8 px - 18144 x - 6 px ( + 91721955203--- 41623g + 389p2 - 24503-ln 2 69854400 210 ) 6 210 78003- 41623- 6 91565- 7 - 280 ln 3- 210 lnx x - 168 px ( 105114325363 696923 4387 2 7051 + - -------------- + -------g - -----p - -----ln 2 72648576 630 ) 18]} 10 + 3986901- ln 3 + 696923-ln x x8 , (181) 1960 630
where (dJ/dt)N is the Newtonian angular momentum flux expressed in terms of x,
(dJz ) 32 ( m )2 ---- = --- --- M x7, (182) dt N 5 M
and the e-independent terms in both <dE/dt > and <dJ/dt> are the same and are given by the terms in the case of circular orbit, Equation (174View Equation).

go to next pagego upgo to previous page