  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", $gamma <= 0.8$, at densities below around $10^4 cm^(-3)$. At low
temperatures the density probability distribution should fall ra pidly for
densities larger than this value, which corresponds physically to the critical
density at which collisional and radiative deexcitation rate s 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.
