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

Log terms in total energy flux
At the lowest Newtonian order Uij reduces to the second
time derivative of Iij, and where the dots indicate all the
terms which do not depend on r0.
Trivial to deduce that the corresponding dependence of the
tail part of the energy flux on r0 is given by
Ftail = · · · − 428
525 M 2 〈I (4)
ij I(4)
ij 〉 ln r0 + · · · ,
where inside the time average operation 〈〉 one can freely
operate by parts the time derivatives. Hence,
〈I (3)
ij I(5) ij 〉 = −〈I(4) ij I(4) ij 〉
Thus, the effect looks like a “quadrupole formula” but where
the third time derivative of the moment is replaced by the
fourth one.
BRI-IHP06-I – p.59/??