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

Newtonian orbital phase evolution 347and where Ä denotes the total energy flux (or gravitational ‘luminosity’), deducedto the Newtonian order from the quadrupole formula of Einstein:Ä = G d 3 Q ij d 3 Q ij5c 5 dt 3 dt 3 . (17.41)The quadrupole moment is merely the Newtonian (trace-free) quadrupole of thesource, which reads in the case of the point-particle binary asIn the mass-centred frame (17.36) we getQ ij = m 1 (y1 i y j 1 − 1 3 δij y 2 1 ) + 1 ↔ 2. (17.42)Q ij = µ(y i y j − 1 3 δij r 2 ). (17.43)The third time derivative of Q ij needed in the quadrupole formula (17.41) iseasily obtained. When an acceleration is generated we replace it by the Newtonianequation of motion (17.35). In the case of a circular orbit we getd 3 Q ijdt 3=−4 Gmµr 3 (y i v j + y j v i ) (17.44)(this is automatically trace-free because y · v = 0). Replacing (17.44) into (17.41)leads to the ‘Newtonian’fluxÄ = 32 G 3 m 2 µ 25 c 5 r 4 v 2 . (17.45)A better way to express the flux is in terms of some dimensionless quantities,namely the mass ratio ν given in (17.33), and a very convenient post-Newtonianparameter defined from the orbital frequency ω byx =( Gmωc 3 ) 2/3. (17.46)Notice that x is of formal order O(1/c 2 ) in the post-Newtonian expansion.Thanks to the Kepler law Gm = r 3 ω 2 we transform (17.45) and arrive atÄ = 32 c 55 G ν2 x 5 . (17.47)In this form the only factor having a dimension isc 5G ≈ 3.63 × 1052 W, (17.48)which is the Planck unit of a power, which turns out to be independent of thePlanck constant. (Notice that instead of c 5 /G the inverse ratio G/c 5 appears as