Supermassive Black Holes....

Consider a SBH Binary System free from the influence of the stars....This system can be treated as \(2\) Massive Black holes each of Mass \(M\) moving under the influence of the other in a circle of radius \(a\)....Consequently Energy is lost due to emission of gravitational waves.The rate of Energy lost is given by \[dE/dt=-(1024/5)(G/c^5)(\omega ^3 I)^2\].where \(I\) represents the Quadruple Moment \(I=2Ma^2\) of the system.Calculate the time taken by the SBH binary to shrink to a radius \(r(Schwarszchild radius<)<a\).(initial radius). If \(t\) can be expressed as \(t=\alpha a^\gamma c^\beta /\psi G^\phi M^\zeta\).Where \((\alpha ,\psi )=1\).Compute the sum of \(\alpha +\beta +\gamma +\psi +\phi +\zeta \)...We this say the Binary SBH merged.

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