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 gravitatio nally contracting isothermal sphere by means of a normal mode analysis . We found that a normal mode grows in proportion to (t(0) - t)(-sigma ), where t denotes the time. The symbol, t(0), denotes an epoch at whi ch the central density increases infinitely [rho proportional to (t(0) - t)(-2)]. The Hunter-b and -d solutions are very unstable against sp herical perturbations; the growth rates of the unstable perturbations are as large as sigma = 56.2 and 6.48 x 10(3) for the Hunter-b and -d solutions, respectively. The Hunter solutions are unlikely to be reali zed in astrophysical situations and even in numerical simulations. The Larson-Penston solution is less unstable. Even if an unstable spheric al perturbation exists, the growth rate should be lower than sigma les s than or equal to 1. We have found an unstable mode in which the angu lar velocity increases as Omega infinity(t(0) - t)(-4/3). Since this m ode grows slowly, the Larson-Penston solution will be realized approxi mately when the initial rotation is very small.