We suggest that the intrinsic , stellar initial mass function ( IMF ) follows a power-law slope \gamma = 2 , inherited from hierarchical fragmentation of molecular clouds into clumps and clumps into stars .
The well-known , logarithmic Salpeter slope \Gamma = 1.35 in clusters is then the aggregate slope for all the star-forming clumps contributing to an individual cluster , and it is steeper than the intrinsic slope within individual clumps because the smallest star-forming clumps contributing to any given cluster are unable to form the highest-mass stars .
Our Monte Carlo simulations demonstrate that the Salpeter power-law index is the limiting value obtained for the cluster IMF when the lower-mass limits for allowed stellar masses and star-forming clumps are effectively equal , m _ { lo } = M _ { lo } .
This condition indeed is imposed for the high-mass IMF tail by the turn-over at the characteristic value m _ { c } \sim 1 \mbox { $ { M } _ { \odot } $ } .
IMF slopes of \Gamma \sim 2 are obtained if the stellar and clump upper-mass limits are also equal m _ { up } = M _ { up } \sim 100 \mbox { $ { M } _ { \odot } $ } , and so our model explains the observed range of IMF slopes between \Gamma \sim 1 to 2 .
Flatter slopes of \Gamma = 1 are expected when M _ { lo } > m _ { up } , which is a plausible condition in starbursts , where such slopes are suggested to occur .
While this model is a simplistic parameterization of the star-formation process , it seems likely to capture the essential elements that generate the Salpeter tail of the IMF for massive stars .
These principles also likely explain the IGIMF effect seen in low-density star-forming environments .