Origin of the initial mass function Lecture 7 supplementary slides

The morphology and kinematics of molecular clouds (MCs) are best explained as the consequence of supersonic turbulence. Supersonic turbulence fragments MCs into dense sheets, filaments, and cores and large low-density ‘‘ voids,’’ via the action of highly radiative shocks. We refer to this process as turbulent fragmentation. In this work we derive the mass distribution of gravitationally unstable cores generated by the process of turbulent fragmentation. The mass distribution above 1M! depends primarily on the power spectrum of the turbulent flow and on the jump conditions for isothermal shocks in a magnetized gas. For a power spectrum index ! 1⁄4 1:74, consistent with Larson’s velocity dispersion–size relation as well as with new numerical and analytic results on supersonic turbulence, we obtain a power-lawmass distribution of dense cores with a slope equal to 3=ð4$ !Þ 1⁄4 1:33, consistent with the slope of the stellar initial mass function (IMF). Below 1 M!, the mass distribution flattens and turns around at a fraction of 1 M!, as observed for the stellar IMF in a number of stellar clusters, because only the densest cores are gravitationally unstable. The mass distribution at low masses is determined by the probability distribution of the gas density, which is known to be approximately lognormal for an isothermal turbulent gas. The intermittent nature of the turbulent density distribution is thus responsible for the existence of a significant number of small collapsing cores, even of substellar mass. Since turbulent fragmentation is unavoidable in supersonically turbulent molecular clouds, and given the success of the present model in predicting the observed shape of the stellar IMF, we conclude that turbulent fragmentation is essential to the origin of the stellar IMF. Subject headings: ISM: kinematics and dynamics — stars: formation — stars: luminosity function, mass function — turbulence