Inefficient star formation in high Mach number environments I. The turbulent support analytical model

The star formation rate (SFR), the number of stars formed per unit of time, is a fundamental quantity in the evolution of the Universe. While turbulence is believed to play a crucial role in setting the SFR, the exact mechanism remains unclear. Turbulence promotes star formation by compressing the gas, but also slows it down by stabilizing the gas against gravity. Most widely-used analytical models rely on questionable assumptions, including: i) integrating over the density PDF, a one-point statistical description that ignores spatial correlation, ii) selecting self-gravitating gas based on a density threshold that often ignores turbulent dispersion, iii) assuming the freefall time as the timescale for estimating SFR without considering the need to rejuvenate the density PDF, iv) assuming the density PDF to be lognormal. Improving upon the only existing model that incorporates the spatial correlation of the density field, we present a new analytical model. We calculate the time needed to rejuvenate density fluctuations of a given density and spatial scale, revealing that it is generally much longer than the freefall time, rendering the latter inappropriate for use. We make specific predictions regarding the role of the Mach number, M, and the driving scale of turbulence divided by the mean Jeans length. At low to moderate Mach numbers, turbulence does not reduce and may even slightly promote star formation by broadening the PDF. However, at higher Mach numbers, most density fluctuations are stabilized by turbulent dispersion, leading to a steep drop in the SFR as the Mach number increases.