Stability of Similarity Solutions for a Gravitationally Contracting Isothermal Sphere: Convergence to the Larson‐Penston Solution

We investigated the stability of similarity solutions for a gravitationally contracting isothermal sphere by means of a normal mode analysis. We found that a normal mode grows in proportion to ( t 0 - t ) -σ , where t denotes the time. The symbol, t 0 , denotes an epoch at which the central density increases infinitely [ρ ∝ ( t 0 - t ) -2 ]. The Hunter-b and -d solutions are very unstable against spherical perturbations; the growth rates of the unstable perturbations are as large as σ = 56.2 and 6.48 × 10 3 for the Hunter-b and -d solutions, respectively. The Hunter solutions are unlikely to be realized in astrophysical situations and even in numerical simulations. The Larson-Penston solution is less unstable. Even if an unstable spherical perturbation exists, the growth rate should be lower than σ ≤ 1. We have found an unstable mode in which the angular velocity increases as Ω ∝ ( t 0 - t ) -4/3 . Since this mode grows slowly, the Larson-Penston solution will be realized approximately when the initial rotation is very small.