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

Gravitational wave radiation in the JBD theory 163Substituting (11.44) in (11.43), considering the expressions (11.47) and(11.49) and averaging over time, one finally obtainsP scal = (2ω BD + 3)c 48πϕ 0∑|X jm | 2 . (11.51)To compute the power radiated in scalar GWs, one has to determine thecoefficients X jm ,defined in (11.50). The detailed calculations can be foundin appendix A of the third reference in [6], while here we only give the finalresults. Introducing the reduced mass of the binary system µ = m 1 m 2 /m and thegravitational self-energy for the body a (with a = 1, 2) a =− 1 ∫ρ(⃗x)ρ(⃗x ′ )2 V a|⃗x −⃗x ′ d 3 x d 3 x ′ (11.52)|one can write the Fourier components with frequency nω 0 in the Newtonianapproximation(X 00 ) n =− 16√ 2π iω 0 ϕ 0 mµ3 ω BD + 2 a nJ n(ne) (11.53)√2π 2iω 2 (0 ϕ 0 2(X 1±1 ) n =−− )1µa3 ω BD + 2 m 2 m 1[× ±J n ′ (ne) − 1 ]e (1 − e2 ) 1/2 J n (ne)(11.54)(X 20 ) n = 2 √ π iω 3 0 ϕ 03 5 ω BD + 2 µa2 nJ n (ne) (11.55)√ π iω 3 0 ϕ 0(X 2±2 ) n =∓230 ω BD + 2 µa2× 1 n {(e2 − 2)J n (ne)/(ne 2 ) + 2(1 − e 2 )J ′ n (ne)/ejm∓ 2(1 − e 2 ) 1/2 [(1 − e 2 )J n (ne)/e 2 − J n ′ (ne)/(ne)]}. (11.56)Substituting these expressions in (11.51), leads to the power radiated in scalarGWs in the nth harmonic(P scal ) n = P j=0n + P j=1n + P j=2n (11.57)where the monopole, dipole and quadrupole terms are, respectively,Pn j=0 64 m 3 µ 2 G 4=9(ω BD + 2) a 5 c 5 n 2 Jn 2 (ne)64 m 3 µ 2 G 4=9(ω BD + 2) a 5 c 5 m(n; e) (11.58)