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 $gamma = dlog P/dlog (rho)$ which measures the turbulent compressibility. When the cooling timescale is smaller than the dynamical timescale, gamma can be evaluated from the derivatives of cooling and heating functions, using the condition of thermal equilibrium. We present calculations of gamma 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 $gamma >= 1$, at large densities, but moderately "soft", $gamma
8 pages, 5 figures, Accepted for Publication in MNRAS