Quasi-normal acoustic oscillations in the transonic Bondi flow

In recent work, we analyzed the dynamics of spherical and nonspherical acoustic perturbations of the Michel flow, describing the steady radial accretion of a relativistic perfect fluid into a nonrotating black hole. We showed that such perturbations undergo quasi-normal oscillations and computed the corresponding complex frequencies as a function of the black hole mass M and the radius r_c of the sonic horizon. It was found that when r_c is much larger than the Schwarzschild radius r_H = 2GM/c^2 of the black hole, these frequencies scale like the surface gravity of the analogue black hole associated with the acoustic metric. In this work, we analyze the Newtonian limit of the Michel solution and its acoustic perturbations. In this limit, the flow outside the sonic horizon reduces to the transonic Bondi flow, and the acoustic metric reduces to the one introduced by Unruh in the context of experimental black hole evaporation. We show that for the transonic Bondi flow, Unruh's acoustic metric describes an analogue black hole and compute the associated quasi-normal frequencies. We prove that they do indeed scale like the surface gravity of the acoustic black hole, thus providing an explanation for our previous results in the relativistic setting.