We study the dynamics of weak discontinuities at the sonic point, and a necessary condition for the stability of Newtonian self-similar spherical isothermal flow (collapse and expansion). This stability criterion depends only on the value of the spatial derivative of the velocity at the sonic point. In collapse, all primary-direction solutions, including the homogeneous-collapse solution, are unstable, while all the secondary-direction solutions, including the Penston–Larson solution, satisfy this stability criterion. The situation is reversed in expansion.
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Ori et al. (1988) studied this question.