Turbulent compressibility of protogalactic gas

The star formation rate in galaxies should be related to the fraction of gas that can attain densities large enough for gravitational collapse. In galaxies with a turbulent interstellar medium, this fraction is controlled by the effective barotropic index γ =d log  P /d log  ρ which measures the turbulent compressibility. When the cooling time‐scale is smaller than the dynamical time‐scale, γ can be evaluated from the derivatives of cooling and heating functions, using the condition of thermal equilibrium. We present calculations of γ for protogalaxies in which the metal abundance is so small that H 2 and HD cooling dominates. For a heating rate independent of temperature and proportional to the first power of density, the turbulent gas is relatively ‘hard’, with γ ≳1 , at large densities, but moderately ‘soft’, γ ≲0.8 , at densities below around 10 4  cm −3 . At low temperatures the density probability distribution should fall rapidly for densities larger than this value, which corresponds physically to the critical density at which collisional and radiative de‐excitation rates of HD are equal. The densities attained in turbulent protogalaxies thus depend on the relatively large deuterium abundance in our Universe. We expect the same physical effect to occur in higher metallicity gas with different coolants. The case in which adiabatic (compressional) heating resulting from cloud collapse dominates is also discussed, and suggests a criterion for the maximum mass of Population III stars.