Opacity-Limited Hierarchical Fragmentation and the Masses of Protostars

Hierarchical fragmentation of a collapsing cloud terminates when opacity prevents individual sub-clouds from radiating their gravitational binding energy on a free-fall time scale. The approximate mass of the final fragments is derived in a form which clarifies why it is insensitive to physical conditions in the cloud. This mass is |~ 10⁻³\,T1/4f-1/2| times the Chandrasekhar mass, where T (K) is the gas temperature when the fragments first significantly reabsorb their own radiation, and f is the radiative efficiency of a fragment at that stage relative to a black body.