We present a simple physical mechanism that can account for the observed stellar mass spectrum for masses M _ { * } \mathbin { \lower 3.0 pt \hbox { $ \hbox to 0.0 pt { \raise 5.0 pt \hbox { $ \char 62 $ } } % \mathchar 29208 $ } } 0.5 { M _ { \odot } } .
The model depends solely on the competitive accretion that occurs in stellar clusters where each star ’ s accretion rate depends on the local gas density and the square of the accretion radius .
In a stellar cluster , there are two different regimes depending on whether the gas or the stars dominate the gravitational potential .
When the cluster is dominated by cold gas , the accretion radius is given by a tidal-lobe radius .
This occurs as the cluster collapses towards a \rho \propto R ^ { -2 } distribution .
Accretion in this regime results in a mass spectrum with an asymptotic limit of \gamma = -3 / 2 ( where Salpeter is \gamma = -2.35 ) .
Once the stars dominate the potential and are virialised , which occurs first in the cluster core , the accretion radius is the Bondi-Hoyle radius .
The resultant mass spectrum has an asymptotic limit of \gamma = -2 with slightly steeper slopes ( \gamma \approx - 2.5 ) if the stars are already mass-segregated .
Simulations of accretion onto clusters containing 1000 stars show that as expected , the low-mass stars accumulate the majority of their masses during the gas dominated phase whereas the high-mass stars accumulate the majority of their massed during the stellar dominated phase .
This results in a mass spectrum with a relatively shallow \gamma \approx 3 / 2 power-law for low-mass stars and a steeper , power-law for high-mass stars -2.5 \mathbin { \lower 3.0 pt \hbox { $ \hbox to 0.0 pt { \raise 5.0 pt \hbox { $ \char 60 $ } } % \mathchar 29208 $ } } \gamma \leq - 2 .
This competitive accretion model also results in a mass segregated cluster .