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 = { dlogP / 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 \ga 1 , at large densities , but moderately “ soft ” , \gamma \la 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 deexcitation 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 due to cloud collapse dominates is also discussed , and suggests a criterion for the maximum mass of Population III stars .