Evolution of Circumbinary Disks Following Super-massive Black Hole Mergers

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There has been a growing interest in electromagnetic counterparts to gravitational wave signals. Of particular interest here, are counterparts to gravitational wave signals from super-massive black hole mergers. We consider a circumbinary disk, hollowed out by torques from the binary, and provide an analytic solution to its response following merger. There are two changes to the potential which occur during the merger process: an axisymmetric mass-energy loss and asymmetric recoil kick given to the resulting super-massive black hole. With a brief literature search we argue that, for fiducial disk values and for black hole spins aligned and anti-aligned with the orbital angular momentum, throughout the majority of parameter space the mass loss well dominates the effects of the recoil kicks on the circumbinary disk. This, along with assuming vertical hydrodynamic equilibrium, reduces the problem to one dimension. Using a 1D hydrodynamical code we explore the majority of parameter space and describe the different possible flows. In the 1D case, we give analytic approximations for the locations of the first shocks, their strengths, and the final density after the disk has again reached a steady state. This allows one to determine the temperature jump across the shock front and determine the observability, modulo the yet unknown disk mass.