A dual power-law distribution for the stellar initial mass function

Abstract

We introduce a new dual power law (DPL) probability distribution function for the mass distribution of stellar and substellar objects at birth, otherwise known as the initial mass function (IMF). The model contains both deterministic and stochastic elements, and provides a unified framework within which to view the formation of brown dwarfs and stars resulting from an accretion process that starts from extremely low mass seeds. It does not depend upon a top down scenario of collapsing (Jeans) masses or an initial lognormal or otherwise IMF-like distribution of seed masses. Like the modified lognormal power law (MLP) distribution, the DPL distribution has a power law at the high mass end, as a result of exponential growth of mass coupled with equally likely stopping of accretion at any time interval. Unlike the MLP, a power law decay also appears at the low mass end of the IMF. This feature is closely connected to the accretion stopping probability rising from an initially low value up to a high value. This might be associated with physical effects of ejections sometimes (i.e., rarely) stopping accretion at early times followed by outflow driven accretion stopping at later times, with the transition happening at a critical time (therefore mass). Comparing the DPL to empirical data, the critical mass is close to the substellar mass limit, suggesting that the onset of nuclear fusion plays an important role in the subsequent accretion history of a young stellar object.

A multiple power-law distribution for initial mass functions

“…Following the literature (Basu & Jones 2004;Basu et al 2015;Hoffmann et al 2018) we now make a further simplifying assumption, namely that the mass accretion is proportional to the already accreted mass. In the literature several different accretion growth laws have been proposed, for example the Bondi rate has mass accretion proportional to mass squared.…”

“…Recently, Hoffmann et al (2018) showed that it is possible to produce a peaked IMF with dual power law (DPL) behavior when all protostars start with very small initial masses, thereby also removing the need to start with an an initial lognormal distribution of seed masses to produce a peaked IMF. The derivation in the following sections is a direct sequel of that paper.…”

“…The derivation in the following sections is a direct sequel of that paper. We advise the reader to consult Hoffmann et al (2018) in order to develop a deeper appreciation of the formalism. Hoffmann et al (2018) developed the peaked IMF based on a special case of general evolution equations for the distribution function of protostars still accreting and that for stars that are no longer accreting.…”

“…We advise the reader to consult Hoffmann et al (2018) in order to develop a deeper appreciation of the formalism. Hoffmann et al (2018) developed the peaked IMF based on a special case of general evolution equations for the distribution function of protostars still accreting and that for stars that are no longer accreting. These evolution equations account for the rearrangement in time of the probability to find a (proto)star of a certain mass due to the accretion of mass or other specifiable processes that need not be purely deterministic or stochastic in nature.…”

“…Interestingly, a three power law IMF is shown by Kroupa (2001Kroupa ( , 2002. We follow the same methodology as in Hoffmann et al (2018), employing an evolutionary approach that accounts for both accreting objects and those that have dropped out of the accretion process according to a specified dropout rate. This methodology is quite general and can be expanded to increasing levels of complexity in future work.…”