Gravitational Waves from Inspiralling Compact Binaries in ... - LUTH

Secular variations
Regarding secular variation of cl and cλ, namely, ¯cl and ¯cλ, we find that there
are no secular evolutions for both cl and cλ to the 1PN order of radiation
reaction. ¯ Gl = 0 = ¯ Gλ, where Gl = Fl/n and Gλ = Fλ/n, respectively,
In this case RHS are functions of the form sin u × f(cos u) and
(v − u) × f(cos u), respectively, and hence they are odd under u → −u.
Therefore, their average over dl (1 − et cos u)du exactly vanishes, leading to
¯Gl = 0 = ¯ Gλ to the 3.5PN order.
Related to time-odd character of the perturbing force A ′ , ∂c1/∂v i , and
∂c2/∂v j , respectively, ending up with the conclusion that dcl/dt and dcλ/dt
are time odd.
d¯cl
dt
= 0 ;
¯cl(t) = ¯cl(t0) ,
d¯cλ
dt
= 0 ;
¯cλ(t) = ¯cλ(t0) .
BRI-IHP06-I – p.136/??