On the Power-Law Tail in the Mass Function of Protostellar Condensations and Stars

We explore the idea that the power-law tail in the mass function of protostellar condensations and stars arises from the accretion of ambient cloud material on to a condensation, coupled with a nonuniform (exponential) distribution of accretion lifetimes. This model allows for the generation of power-law distributions in all star-forming regions, even if condensations start with a lognormal mass distribution, as may be expected from the central limit theorem, and supported by some recent numerical simulations of turbulent molecular clouds. For a condensation mass $m$ with growth rate $dm/dt \propto m$, an analytic three-parameter probability density function is derived; it resembles a lognormal at low mass and has a pure power-law high-mass tail. An approximate power-law tail is also expected for other growth laws, and we calculate the distribution for the plausible case $dm/dt \propto m^{2/3}$. Furthermore, any single time snapshot of the masses of condensations that are still accreting (and are of varying ages) also yields a distribution with a power-law tail similar to that of the IMF.