The fundamental equations governing the self-similar dynamics of 'polytropic' gaseous spheres are derived, and the asymptotic solutions are given. The solutions divide into cases with and without 'critical points' in closed analogy with the solar wind solutions of Holzer and Axford (1970). Properties for solutions with critical points are discussed, and their behavior around the critical point is derived explicitly for n = 1. Numerical examples of self-similar solutions for n = 1 and n = 2 - gamma are presented, and the properties of the solutions are discussed.
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Suto et al. (1988) studied this question.