The Structure of Isothermal, Self‐gravitating, Stationary Gas Spheres for Softened Gravity

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Abstract
A theory for the structure of isothermal, self-gravitating gas spheres in pressure equilibrium is developed for softened gravity, assuming an ideal gas equation of state. The one-parameter spline softening proposed by Hernquist & Katz is used. We show that the addition of this extra scale parameter implies that the set of equilibrium solutions constitute a one-parameter family, rather than the one and only one isothermal sphere solution for Newtonian gravity, and we develop a number of approximate, analytical or semianalytical solutions. For softened gravity, the structure of isothermal spheres is, in general, very different from the Newtonian isothermal sphere. For example, for any finite choice of softening length and temperature T, it is possible to deposit an arbitrarily large mass of gas in pressure equilibrium and with a nonsingular density distribution inside of r0 for any r0 > 0. Furthermore, it is sometimes claimed that the size of the small-scale, self-gravitating gas structures formed in dissipative Tree-SPH simulations is simply of the order the gravitational softening length. We demonstrate, that this claim, in general, is not correct. The main purpose of the paper is to compare the theoretical predictions of our models with the properties of the small, massive, quasi-isothermal gas clumps (r ~ 1 kpc, M ~ 1010 M☉, and T 104 K) which form in numerical Tree-SPH simulations of the formation of Milky Way-sized galaxies when effects of stellar feedback processes are not included. We find reasonable agreement, despite the neglect of effects of rotational support in the models presented in this paper. We comment on whether the hydrodynamical resolution is sufficient in our numerical simulations of galaxy formation involving highly supersonic, radiative shocks, and we give a necessary condition, in the form of a simple test, that the hydrodynamical resolution in any such simulations is sufficient. Finally, we conclude that one should be cautious, when comparing results of numerical simulations, involving gratitational softening and hydrodynamical smoothing, with reality.
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