The Role of Column Density in the Formation of Stars and Black Holes

The stellar mass in disk galaxies scales approximately with the fourth power of the rotation velocity, and the masses of the central black holes in galactic nuclei scale approximately with the fourth power of the bulge velocity dispersion. It is shown here that these relations can be accounted for if, in a forming galaxy with an isothermal mass distribution, gas with a column density above about 8 Msun/pc^2 goes into stars while gas with a column density above about 2 g/cm^2 (10^4 Msun/pc^2) goes into a central black hole. The lower critical value is close to the column density of about 10 Msun/pc^2 at which atomic gas becomes molecular, and the upper value agrees approximately with the column density of about 1 g/cm^2 at which the gas becomes optically thick to its cooling radiation. These results are plausible because molecule formation is evidently necessary for star formation, and because the onset of a high optical depth in a galactic nucleus may suppress continuing star formation and favour the growth of a central black hole.