Stellar mass spectrum within massive collapsing clumps I. Influence of
  the initial conditions
  We conduct numerical experiments in which we systematically vary the initial
density over four orders of magnitude and the turbulent velocity over a factor
ten. In a companion paper, we investigate the dependence of this distribution
on the gas thermodynamics. We performed a series of hydrodynamical numerical
simulations using adaptive mesh refinement, with special attention to numerical
convergence. We also adapted an existing analytical model to the case of
collapsing clouds by employing a density probability distribution function
(PDF) $\propto \rho^{-1.5}$ instead of a lognormal distribution. Simulations
and analytical model both show two support regimes, dominated by either thermal
energy or turbulence. For the first regime, we infer that $dN/d \log M \propto
M^0$, while for the second, we obtain $dN/d \log M \propto M^{-3/4}$. This is
valid up to about ten times the mass of the first Larson core, as explained in
the companion paper, leading to a peak of the mass spectrum at $\sim 0.2
M_\odot$. From this point, the mass spectrum decreases with decreasing mass.
Although the mass spectra we obtain for the most compact clouds qualitatively
resemble the observed initial mass function, the distribution exponent is
shallower than the expected Salpeter exponent of -1.35. Nonetheless, we observe
a possible transition toward a slightly steeper value that is broadly
compatible with the Salpeter exponent for masses above a few solar masses. This
change in behavior is associated with the change in density PDF, which switches
from a power-law to a lognormal distribution. Our results suggest that while
gravitationally induced fragmentation could play an important role for low
masses, it is likely the turbulently induced fragmentation that leads to the
Salpeter exponent.
