(ed.). Gravitational waves (IOP, 2001)(422s).

346 Post-Newtonian computation of binary inspiral waveforms17.4 Newtonian orbital phase evolutionLet y 1 (t) and y 2 (t) be the two trajectories of the masses m 1 and m 2 , andy = y 1 − y 2 be their relative position, and denote r =|y|. The velocities arev 1 (t) = dy 1 /dt, v 2 (t) = dy 2 /dt and v(t) = dy/dt. The Newtonian equations ofmotion read asdv 1dt=− Gm 2r 3 y;dv 2dt= Gm 1r 3 y. (17.34)The difference between these two equations yields the relative acceleration,dvdt=− Gm y. (17.35)r 3We place ourselves into the Newtonian centre-of-mass frame defined bym 1 y 1 + m 2 y 2 = 0, (17.36)in which frame the individual trajectories y 1 and y 2 are related to the relative oney byy 1 = m 2m y; y 2 =− m 1y.m(17.37)The velocities are given similarly byv 1 = m 2m v; v 2 =− m 1v. (17.38)mIn principle, the binary’s phase evolution φ(t) should be determined froma knowledge of the radiation reaction forces acting locally on the orbit. Atthe Newtonian order, this means considering the ‘Newtonian’ radiation reactionforce, which is known to contribute to the total acceleration only at the 2.5PNlevel, i.e. 1/c 5 smaller than the Newtonian acceleration (where 5 = 2s + 1, withs = 2 the helicity of the graviton). A simpler computation of the phase is todeduce it from the energy balance equation between the loss of centre-of-massenergy and the total flux emitted at infinity in the form of waves. In the case ofcircular orbits one needs only to find the decrease of the orbital separation r andfor that purpose the balance of energy is sufficient. Relying on an energy balanceequation is the method we follow for computing the phase of inspiralling binariesin higher post-Newtonian approximations (see section 17.6). Thus, we writedEdt=−Ä, (17.39)where E is the centre-of-mass energy, given at the Newtonian order byE =− Gm 1m 2, (17.40)2r