### 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 (215). The center-of-mass energy is given by Equation (194) and the total flux by Equation (231). For convenience we adopt the dimensionless time variable
where 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 , which is immediately integrated with the result
The orbital phase is defined as the angle , oriented in the sense of the motion, between the separation of the two bodies and the direction of the ascending node 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 , which translates, with our notation, into , from which we determine
where 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 . For this we get
where is another constant of integration. With the formula (235) 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 (8)).

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 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 [79184152]. We define by

The frequency of the signal at the entrance of the bandwidth is the seismic cut-off frequency of ground-based detectors; the terminal frequency is assumed for simplicity’s sake to be given by the Schwarzschild innermost stable circular orbit. Here 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).

 Newtonian order 1PN 1.5PN (dominant tail) 2PN 2.5PN 3PN 3.5PN

 Table 2: Contributions of post-Newtonian orders to the accumulated number of gravitational-wave cycles (defined by Equation (236)) in the bandwidth of VIRGO and LIGO detectors. Neutron stars have mass , and black holes . The entry frequency is , and the terminal frequency is .