Abstract
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.
Highlights
Star forming regions typically show a very inhomogeneous structure with large variations in the gas density due to turbulence and filaments
The above findings can be summarized in the rule of thumb that both the average accretion rate a and the level of fluctuations b/a have to be of order unity for unit initial mass and unit growth time in order to sufficiently populate the power-law tail
We investigate the consequences of fluctuations in the accretion rates of non-linearly accreting stars by the means of a non-linear multiplicative stochastic process and find the following: (i) Non-linear accretion, m ∝ mα, with fluctuations in the accretion rates lead to power-law tails in the distribution function of the final masses
Summary
Star forming regions typically show a very inhomogeneous structure with large variations in the gas density due to turbulence and filaments. Fragmentation alone seems not to embrace the whole star formation process, as the initial mass function has a power-law tail at the massive end This deviation from lognormality has been explained, amongst other ideas, by competitive accretion, which is non-linear accretion of the fragments without fluctuations in the accretion rate (Larson 1978; Zinnecker 1982; Bonnell et al 1997, 2001a,b; Bate et al 2003). In a series of papers, Myers (2000, 2008, 2009, 2011, 2012) published increasingly elaborate models of the star formation process Their main components are an exponential distribution of growth times, accretion with a massindependent and a (non-linearly) mass-dependent contribution, and a constant initial mass (a distribution of initial masses is considered in Myers 2009).
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