SHOCK-GENERATED VORTICITY IN THE INTERSTELLAR MEDIUM AND THE ORIGIN OF THE STELLAR INITIAL MASS FUNCTION

Observations of the interstellar medium (ISM) and molecular clouds suggest these astrophysical flows are strongly turbulent. The main observational evidence for turbulence is the power-law energy spectrum for velocity fluctuations, E(k) proportional to k(alpha), with alpha is an element of [-1.5,-2.6]. The Kolmogorov scaling exponent, alpha = -5/3, is typical. At the same time, the observed probability distribution function (PDF) of gas densities in both the ISM as well as in molecular clouds is a log-normal distribution, which is similar to the initial mass function (IMF) that describes the distribution of stellar masses. In this paper we examine the density and velocity structure of interstellar gas traversed by curved shock waves in the kinematic limit. We demonstrate mathematically that just a few passages of curved shock waves generically produces a log-normal density PDF. This explains the ubiquity of the log-normal PDF in many different numerical simulations. We also show that subsequent interaction with a spherical blast wave generates a power-law density distribution at high densities, qualitatively similar to the Salpeter power law for the IMF. Finally, we show that a focused shock produces a downstream flow with energy spectrum exponent alpha = -2. Subsequent shock passages reduce this slope, achieving alpha approximate to -5/3 after a few passages. We argue that subsequent dissipation of energy piled up at the small scales will act to maintain the spectrum very near to the Kolomogorov value despite the action of further shocks that would tend to reduce it. These results suggest that fully developed turbulence may not be required to explain the observed energy spectrum and density PDF. On the basis of these mathematical results, we argue that the self-similar spherical blast wave arising from expanding H II regions or stellar winds from massive stars may ultimately be responsible for creating the high-mass, power-law, Salpeter-like tail on an otherwise a log-normal density PDF for gas in star-forming regions. The IMF arises from the gravitational collapse of sufficiently overdense regions within this PDF. Thus, the composite nature of the IMF-a log-normal plus power-law distribution-is shown to be a natural consequence of shock interaction and feedback from the most massive stars that form in most regions of star formation in the galaxy.