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

Chapter 7Source calculationsNow that we have the formulae for the radiation from a system, we can use themfor some simple examples.7.1 Radiation from a binary systemThe most numerous sources of gravitational waves are binary stars systems. Injust half an orbital period, the non-spherical part of the mass distribution returnsto its original configuration, so the angular frequency of the emitted gravitationalwaves is twice the orbital angular frequency.We shall calculate here the mass-quadrupole moment for two stars of massesm 1 and m 2 , orbiting in the x–y plane in a circular orbit with angular velocity ,governed by Newtonian dynamics. We take their total separation to be R, whichmeans that the orbital radius of mass m 1 is m 2 R/(m 1 + m 2 ) while that of massm 2 is m 1 R/(m 1 + m 2 ). We place the origin of coordinates at the centre of massof the system. Then, for example, the xx-component of M ij is( )m2 R cos(t) 2 ( )m1 R cos(t) 2M xx = m 1 + m 2m 1 + m 2m 1 + m 2= µR 2 cos 2 (t), (7.1)where µ := m 1 m 2 /(m 1 + m 2 ) is the reduced mass. By using a trigonometricidentity and throwing away the part that does not depend on time (since we willuse only time-derivatives of this expression) we haveM xx = 1 2 µR2 cos(2t). (7.2)By the same methods, the other nonzero components areM yy =− 1 2 µR2 cos(2t),M xy = 1 2 µR2 sin(2t).This shows that the radiation will come out at twice the orbital frequency.71