Dynamics of Self-Gravitating Gaseous Spheres--III: Analytical Results in the Free-fall of Isothermal Cases

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Abstract
For a cold gas, the analytical solutions for collapse in various symmetries give density and velocity profiles at the instant when a singularity develops in the initially densest part. These profiles follow generally from assuming that the density variation is smooth initially. For spherical and planar symmetries we have extended these solutions to a short time before this singularity occurs. The spherical results are given in Sections 2–4 and imply a density law proportional to r–12/7 . The planar results follow in Appendix I and the cylindrical ones in Appendix II. The solutions before the singularity arises are similarity solutions with the density and velocity profiles retaining their shapes while altering only their scales with time. Applying these results we find a spherical collapse to form a galaxy is modified by the rise in central optical depth at a density of ~ 10 –20 g cm –3 . Flattening instabilities are resisted but not overcome by the more rapid growth of the density gradient which occurs in a planar collapse. In Section 5, a similarity solution for the spherical collapse of an isothermal gas is presented. This has a critical point similar to those found in solar wind flows. The density profile when the singularity arises is proportional to r–2 . The collapse of a proto-star is halted by the rise in optical depth when the density reaches 10 –18 g cm –3 . The stability of this solution and the problem of whether all flows converge to it still remain to be settled.

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