Evolution and Instability of Bondi Accretion

Stationary spherically symmetric transonic (Bondi) accretion is a paradigm in

astrophysical fluid dynamics. We, however, show that in the stationary regime this process

cannot be physically realisable because its mathematical solution lies along a separatrix and

passes through a saddle point in the phase portrait of the flow. We argue that the transonic

solution is owed actually to the dynamics. In studying the dynamic effects, we subject the

stationary flow to a time-dependent radial perturbation, with the equation of the perturbation

containing nonlinearity up to any arbitrary order. Casting the perturbation as a standing wave

on subsonic solutions, and maintaining nonlinearity in it up to the second order, we get the

time-dependence of the perturbation in the form of a Li\'enard system. A dynamical systems

analysis of the Li\'enard system reveals a saddle point in real time, with the implication that

instabilities will develop in the accreting system when the perturbation is extended into the

nonlinear regime. The instability of initial subsonic states also adversely affects the temporal

evolution of the flow towards a final and stable transonic state. We provide numerical support in

favour of this claim.