### 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 (153). The center-of-mass energy is given by
Eq. (152) and the total flux by Eq. (168). 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
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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
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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 Update
where is another constant of integration.
With the formula (172) the orbital phase is complete up to the 3.5PN order, except for a single linear
combination of the unknown regularization constants and . 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 [91, 90, 108, 136]). 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. (8)).