THE PROPERTIES AND STABILITY OF SELF-GRAVITATING, POLYTROPIC SPHERES WITH γ = 1 TO 1.4 SPECIFIC HEAT RATIOS

We study self-gravitating, hydrostatic spheres with a polytropic equation of state P ∝ ρ^γ (where γ is the specific heat ratio of the gas), considering structures with γ ≈ 1 as a model for molecular cloud cores with small departures from isother- mality. We derive the properties (i.e., mass, radius and center to edge density ratio) as a function of γ for the maximal stable sphere through an application of “Bonnor’s stability criterion”. We find that in the γ = 1 → 4/3 range the mass of the maximal sphere (for a given central temperature) is almost constant, and that its radius and center to edge density ratio are growing functions of γ. We therefore have maximal stable, self-gravitating spheres with similar masses, but with increasing center to edge density contrasts for increasing departures from isothermality.