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10.3 Orbital phase evolution

We shall now deduce the laws of variation with time of the orbital frequency and phase of an inspiralling compact binary from the energy balance equation (153View Equation). The center-of-mass energy E is given by Eq. (152View Equation) and the total flux L by Eq. (168View Equation). For convenience we adopt the dimensionless time variable24
nc3 Q =_ -----(tc- t), (169) 5Gm
where tc denotes the instant of coalescence, at which the frequency tends to infinity (evidently, the post-Newtonian method breaks down well before this point). We transform the balance equation into an ordinary differential equation for the parameter x, which is immediately integrated with the result UpdateJump To The Next Update Information
( ) ( ) 1 -1/4{ 743 11 - 1/4 1 -3/8 19583 24401 31 2 -1/2 x = 4Q 1 + 4032-+ 48n Q - 5-pQ + 254016- + 193536-n + 288-n Q ( ) 11891- -109- -5/8 + - 53760 + 1920 n pQ [ ( ) + - 10052469856691--+ 1p2 + 107-C - -107- ln -Q-- 6008596070400 6 420 3360 256 (15335597827 451 77 11 ) 15211 25565 ] + -------------- ----p2 - --c + ---h n - -------n2 + -------n3 Q -3/4 ( 3901685760 3072 72 24) 442368( ) 331776 113868647-- -31821- 294941-- 2 -7/8 1- } + - 433520640 - 143360 n + 3870720 n pQ + O c8 . (170)
The orbital phase is defined as the angle f, oriented in the sense of the motion, between the separation of the two bodies and the direction of the ascending node N within the plane of the sky, namely the point on the orbit at which the bodies cross the plane of the sky moving toward the detector. We have df/dt = w, which translates, with our notation, into 3/2 df/dQ = -5/n .x, from which we determine UpdateJump To The Next Update Information
{ ( ) ( ) f = - 1Q5/8 1 + 3715-+ 55n Q -1/4- 3pQ - 3/8 + 9275495-- + 284875-n + 1855-n2 Q -1/2 n 8064 96 4 14450688 258048 2048 ( 38645 65 ) ( Q ) + - ------- + -----n pQ - 5/8 ln --- [ 172032 2048 Q0 ( ) 831032450749357 53 2 107 107 Q + ----------------- - ---p - ---C + ----ln ---- 57(682522275840 40 56 448 2)56 123292747421-- 2255- 2 385- 55- + - 4161798144 + 2048 p + 48 c - 16h n ] + 154565--n2- 1179625-n3 Q- 3/4 1835008 1769472 (188516689 488825 141769 ) ( 1 )} + ----------+ -------n - -------n2 pQ - 7/8 + O -8 , (171) 173408256 516096 516096 c
where Q 0 is a constant of integration that can be fixed by the initial conditions when the wave frequency enters the detector’s bandwidth. Finally we want also to dispose of the important expression of the phase in terms of the frequency x. For this we get Update
{ ( ) ( ) x--5/2 3715- 55- 3/2 15293365- 27145- 3085- 2 2 f = - 32n 1 + 1008 + 12n x- 10px + 1016064 + 1008 n + 144 n x ( ) ( ) + 38645-- 65n px5/2ln -x- 1344 16 x0 [12348611926451 160 1712 856 + ----------------- ---p2 - -----C - ----ln(16 x) 18(776862720 3 21 21 ) ] 15335597827-- 2255- 2 3080- 440- 76055- 2 127825- 3 3 + - 12192768 + 48 p + 9 c - 3 h n + 6912 n - 5184 n x ( ) ( )} + 77096675- + 378515-n - 74045-n2 px7/2 + O 1- , (172) 2032128 12096 6048 c8
where x0 is another constant of integration. With the formula (172View Equation) the orbital phase is complete up to the 3.5PN order, except for a single linear combination of the unknown regularization constants c and h. More work should be done to determine these constants. The effects due to the spins of the particles, i.e. spin-orbit coupling from the 1.5PN order for compact bodies and spin-spin coupling from the 2PN order, can be added if necessary (they are known up to the 2.5PN order [9190108136]). On the other hand, the contribution of the quadrupole moments of the compact objects, which are induced by tidal effects, is expected to come only at the 5PN order (see Eq. (8View Equation)).
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