The Role of Gravity in Producing Power-Law Mass Functions

Numerical simulations of star formation have found that a power-law mass function can develop at high masses. In a previous paper, we employed isothermal simulations that created large numbers of sinks over a large range in masses to show that the power-law exponent of the mass function, dN/d log M proportional to M-Gamma, asymptotically and accurately approaches Gamma = -1. Simple analytic models show that such a power law can develop if the mass accretion rate <(M) over dot> proportional to M-2, as in Bondi-Hoyle accretion; however, the sink mass accretion rates in the simulations show significant departures from this relation. In this paper, we show that the expected accretion rate dependence is more closely realized provided the gravitating mass is taken to be the sum of the sink mass and the mass in the near environment. This reconciles the observed mass functions with the accretion rate dependencies, and demonstrates that power-law upper mass functions are essentially the result of gravitational focusing, a mechanism present in, for example, the competitive accretion model.