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On the Origin of the Salpeter Slope for the Initial Mass Function

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 ~ 1 M_sun. IMF slopes of GAMMA ~ 2 are obtained if the stellar and clump upper-mass limits are also equal m_up = M_up ~ 100 M_sun, and so our model explains the observed range of IMF slopes between GAMMA ~ 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.