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

Method of variation of constants
At 2PN, one can write down explicit expressions for the functions S(l)
and W (l). GQKR yields:
S(l; c1, c2) = ar(1 − er cos u) ,
W (l; c1, c2) = (1 + k)(v − l) + fφ
c
4 sin 2v + gφ
sin 3v ,
c4 where v and u are some 2PN accurate true and eccentric
anomalies, which must be expressed as functions of l, c1, and c2,
say v = V(l; c1, c2) = V (U(l; c1, c2)) and u = U(l; c1, c2).
ar and er are some 2PN accurate semi-major axis and radial
eccentricity, while fφ and gφ are certain functions, given in terms of
c1 and c2.
v = V (u) ≡ 2 arctan
1 + eφ
1 − eφ
1/2
tan u
.
2
Fn u = U(l) defined by inverting the ‘Kepler equation’ l = l(u)
l = u − et sin u + ft
gt
sin V (u) +
c4 c4 (V (u) − u) . BRI-IHP06-I – p.105/??