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

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

Jul 1997

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1997apj...484..238h&link_type=abstract

Astrophysical Journal v.484, p.238

Astronomy and Astrophysics

Astronomy

35

Hydrodynamics, Instabilities, Ism: Clouds, Stars: Formation

Scientific paper

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 (t0 - t)- sigma , where t denotes the time. The symbol, t0, denotes an epoch at which the central density increases infinitely [ rho ~ (t0 - 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 sigma = 56.2 and 6.48 x 103 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 sigma <= 1. We have found an unstable mode in which the angular velocity increases as Omega ~ (t0 - t)-4/3. Since this mode grows slowly, the Larson-Penston solution will be realized approximately when the initial rotation is very small.

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