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

164 Detection of scalar gravitational wavesP j=1n =4 m 2 µ 2 G 3 (23(ω BD + 2) a 4 c 3 − ) 21m 2 m 1× n 2 [J ′2n (ne) + 1 e 2 (1 − e2 )J 2 n (ne) ]4 m 2 µ 2 G 3 (2=3(ω BD + 2) a 4 c 3 − ) 21d(n; e) (11.59)m 2 m 1Pn j=2 8 m 3 µ 2 G 4=15(ω BD + 2) a 5 c 5 g(n; e). (11.60)The total power radiated in scalar GWs by a binary system is the sum ofthree termsP scal = P j=0 + P j=1 + P j=2 (11.61)whereP j=0 16=9(ω BD + 2)P j=1 =2ω BD + 2G 4 m 2 1 m2 2 m( )e 2c 5 a 5 (1 − e 2 ) 7/2 1 + e24(2− ) (21 G 3 m 2 1 m2 2 1m 2 m 1 c 3 a 4 (1 − e 2 ) 5/2 1 + e22)(11.62)(11.63)P j=2 8 G 4 m 2 1=m2 2 m(115(ω BD + 2) c 5 a 5 (1 − e 2 ) 7/2 1 + 7324 e2 + 37 )96 e4 . (11.64)Note that P j=0 , P j=1 , P j=2 all go to zero in the limit ω BD →∞.11.3.4 Scalar GWsWe now give the explicit form of the scalar GWs radiated by a binary system. Tothis end, note that the major semi-axis, a, is related to the total energy, E, ofthesystem through the following equationa =− Gm 1m 22E . (11.65)Let us consider the case of a circular orbit, remembering that in the last phaseof evolution of a binary system this condition is usually satisfied. Furthermorewe will also assume m 1 = m 2 . With these positions only the quadrupole term,(11.60), of the gravitational radiation is different from zero. The total powerradiated in GWs, averaged over time, is then given by (11.62)–(11.64)P =8 G 4 m 2 1 m2 2 m15(ω BD + 2) c 5 d 5 [6(2ω BD + 3) + 1] (11.66)where d is the relative distance between the two stars. The time variation of d inone orbital period isḋ =− Gm 1m 22E 2 P (11.67)