  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 which created large numbers of sinks over a large range
in masses to show that the power law exponent of the mass function, $dN/d\log M
\propto M^{\Gamma}$, asymptotically and accurately approaches $\Gamma = -1.$
Simple analytic models show that such a power law can develop if the mass
accretion rate $\dot{M} \propto M^2$, 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.
