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Stability and Solution of the Time-Dependent Bondi-Parker Flow

Bondi (1952) and Parker (1958} derived a steady-state solution for Bernouilli's equation in spherical symmetry around a point mass for two cases, respectively, an inward accretion flow and an outward wind. Left unanswered were the stability of the steady-state solution, the solution itself of time-dependent flows, whether the time-dependent flows would evolve to the steady-state, and under what conditions a transonic flow would develop. In a Hamiltonian description, we find that the steady state solution is equivalent to the Lagrangian implying that time-dependent flows evolve to the steady state. We find that the second variation is definite in sign for isothermal and adiabatic flows, implying at least linear stability. We solve the partial differential equation for the time-dependent flow as an initial-value problem and find that a transonic flow develops under a wide range of realistic initial conditions. We present some examples of time-dependent solutions.