On the mass function of stars growing in a flocculent medium

Stars form in regions of very inhomogeneous densities and may have chaotic orbital motions. This leads to a time variation of the accretion rate, which will spread the masses over some mass range. We investigate the mass distribution functions that arise from fluctuating accretion rates in non-linear accretion, $\dot{m} \propto m^{\alpha}$. The distribution functions evolve in time and develop a power law tail attached to a lognormal body, like in numerical simulations of star formation. Small fluctuations may be modelled by a Gaussian and develop a power-law tail $\propto m^{-\alpha}$ at the high-mass side for $\alpha > 1$ and at the low-mass side for $\alpha < 1$. Large fluctuations require that their distribution is strictly positive, for example, lognormal. For positive fluctuations the mass distribution function develops the power-law tail always at the high-mass hand side, independent of $\alpha$ larger or smaller than unity. Furthermore, we discuss Bondi-Hoyle accretion in a supersonically turbulent medium, the range of parameters for which non-linear stochastic growth could shape the stellar initial mass function, as well as the effects of a distribution of initial masses and growth times.