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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 (215View Equation). The center-of-mass energy E is given by Equation (194View Equation) and the total flux L by Equation (231View Equation). For convenience we adopt the dimensionless time variable41
nc3 Q =_ -----(tc- t), (232) 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
{ ( ) ( ) 1 -1/4 743 11 -1/4 1 -3/8 19583 24401 31 2 -1/2 x = 4Q 1 + 4032- + 48-n Q - 5-pQ + 254016- + 193536-n + 288-n Q ( ) 11891- 109-- -5/8 + - 53760 + 1920 n pQ [ ( ) + - 10052469856691--+ 1-p2 + 107-C - -107- ln Q--- 6008596070400 6 420 3360 256 (3147553127 451 ) 15211 25565 ] + ------------- -----p2 n - -------n2 + -------n3 Q- 3/4 ( 780337152 3072 442368) 331776 ( )} 113868647-- -31821- 294941-- 2 -7/8 1- + - 433520640 - 143360 n + 3870720 n pQ + O c8 . (233)
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
{ ( ) ( ) 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 - 5/8ln --- [ 172032 2048 Q0 ( ) 831032450749357 53 2 107 107 Q + ------------------ --p - ---C + ----ln ---- 57(682522275840 40 56) 448 256 126510089885-- 2255- 2 + - 4161798144 + 2048p n ] + 154565--n2- 1179625-n3 Q- 3/4 1835008 1769472 (188516689 488825 141769 ) ( 1 )} + -----------+ -------n- -------n2 pQ -7/8 + O -8 , (234) 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
-5/2{ ( ) ( ) x----- 3715- 55- 3/2 15293365- 27145- 3085- 2 2 f = - 32n 1 + 1008 + 12 n x - 10px + 1016064 + 1008 n + 144 n x ( ) ( ) + 38645-- 65-n px5/2 ln x-- 1344 16 x0 [12348611926451 160 1712 856 + ---------------- - ----p2- -----C - ----ln(16 x) 18(776862720 3 2)1 21 ] 15737765635-- 2255- 2 76055- 2 127825- 3 3 + - 12192768 + 48 p n + 6912 n - 5184 n x ( ) ( )} + 77096675-+ 378515-n- 74045-n2 px7/2 + O 1- , (235) 2032128 12096 6048 c8
where x0 is another constant of integration. With the formula (235View Equation) the orbital phase is complete up to the 3.5PN order. The effects due to the spins of the particles, i.e. the spin-orbit (SO) coupling arising at the 1.5PN order for maximally rotating compact bodies and the spin-spin (SS) coupling at the 2PN order, can be added if necessary; they are known up to the 2.5PN order included [14614416820411025]. 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 Equation (8View Equation)).

As a rough estimate of the relative importance of the various post-Newtonian terms, let us give in Table 2 their contributions to the accumulated number of gravitational-wave cycles N in the bandwidth of the LIGO and VIRGO detectors (see also Table I in Ref. [35] for the contributions of the SO and SS effects). Note that such an estimate is only indicative, because a full treatment would require the knowledge of the detector’s power spectral density of noise, and a complete simulation of the parameter estimation using matched filtering [79Jump To The Next Citation Point184152]. We define N by

1[ ] N = p- fISCO - fseismic. (236)
The frequency of the signal at the entrance of the bandwidth is the seismic cut-off frequency fseismic of ground-based detectors; the terminal frequency fISCO is assumed for simplicity’s sake to be given by the Schwarzschild innermost stable circular orbit. Here f = w/p = 2/P is the signal frequency at the dominant harmonics (twice the orbital frequency). As we see in Table 2, with the 3PN or 3.5PN approximations we reach an acceptable level of, say, a few cycles, that roughly corresponds to the demand which was made by data-analysists in the case of neutron-star binaries [7778791835958]. Indeed, the above estimation suggests that the neglected 4PN terms will yield some systematic errors that are, at most, of the same order of magnitude, i.e. a few cycles, and perhaps much less (see also the discussion in Section 9.6).










2× 1.4 Mo . 10Mo . + 1.4 Mo . 2× 10Mo .




Newtonian order 16031 3576 602
1PN 441 213 59
1.5PN (dominant tail) - 211 - 181 - 51
2PN 9.9 9.8 4.1
2.5PN - 11.7 - 20.0 - 7.1
3PN 2.6 2.3 2.2
3.5PN - 0.9 - 1.8 - 0.8









Table 2: Contributions of post-Newtonian orders to the accumulated number of gravitational-wave cycles N (defined by Equation (236View Equation)) in the bandwidth of VIRGO and LIGO detectors. Neutron stars have mass 1.4 Mo ., and black holes 10Mo .. The entry frequency is fseismic = 10 Hz, and the terminal frequency is fISCO = c3/(63/2pGm).


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