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

Evoln of orbital elements under GRR
The three expressions obtained here are the 3PN generalizations of
the expressions given in Peters which are at the lowest quadrupolar
order. They could be used to provide 3PN extensions of n(e) and a(e)
relations in the future.
The above results have to be supplemented by the computation of
hereditary terms at 2.5PN and 3PN for completion. These hereditary
terms include the tails at 2.5PN and tail of tails and tail-square terms
at 3PN.
Formally one can analytically solve the coupled evolution system by
successive approximations, reducing it to simple quadratures. Eg, at
the leading order O(c −5 ) one can first eliminate t by dividing d¯n/dt
by dēt/dt, thereby obtaining an equation of the form
d ln ¯n = f0(ēt)dēt. Integration of this equation yields
¯n(ēt) = ni
e 18/19
i
(304 + 121 e 2 i ) 1305/2299
(1 − e 2 i )3/2
e 18/19
t
(1 − e 2 t ) 3/2
(304 + 121 e2 ,
t ) 1305/2299
ei is the value of et when n = ni. First obtained by Peters 64.
BRI-IHP06-I – p.93/??