tokens: of Δ s as a function of time and Mach number. Using the driving routines from Federrath et al. ( 2010 ), we ran an additional M = 3.3 hydro simulation with 50% of the energy in compressible modes, and again found a good fit to the PDF and Δ s evolution with s 0 = 0.7 and Equation ( 7 ) unchanged. Thus τ ev is independent of M and the same for mixed and solenoidal driving, and for all cases it is significantly less than τ eddy . In fact, over the s ≈ s * ± 2 range in which we have made our measurements τ ev ( s ) varies from ≈ τ eddy /4 to ≈ τ eddy /12. Meanwhile, the effect of the Mach number is to increase the width of the PDF and shift its peak as required by mass conservation. 4. Gravity and Star Formation Calculating the rate at which stars form requires extending the model to include gravitational collapse. Current analytic models do this by assigning a critical overdensity, s crit , above which gravity overcomes turbulent motions. In this case, the normalized star formation rate per free fall time becomeswhere φ t is a “replenishment factor” that accounts for the timescale over which the log-normal is assumed to be replenished, and the efficiency factor collapse accounts for the fraction of the mass with density larger than s crit that actually collapses. Various models in the literature make different assumptions about s crit and take s ff = 0 (Krumholz McKee 2005 ), s ff = s crit Padoan Nordlund ( 2011 ) or s ff = s (Hennebelle Chabrier 2011 ; Federrath Klessen 2012 ), calibrating collapse / φ t to direct numerical simulations. Recently, Burkhart ( 2018 ) developed a model in which the log-normal PDF was combined with a high-density power-law tail in a continuous and differentiable way, which has the effect of leading to higher star formation rates for lower power-law slopes, as seen in observations and simulations. This behavior also arises naturally through a simple extension of our Markov model. If we approximate each perturbation as a constant density sphere collapsing from rest from a large initial radius, conservation of energy gives ds/dt = (24 πGρ ) 1/2 . Adding this term to Equation ( 6 ), we arrive at a Langevin equation that describes supersonic, isothermal turbulence including self-gravity:where , and are dimensionless such that and are purely a functions of is the virial parameter with R and M the cloud radius and mass, and is the turbulent driving scale in units of the cloud radius. In the left panel of Figure 3 , we compare the results of this modified Langevin equation with full three-dimensional gravoturbulent simulations (Kritsuk et al. 2010 ). These simulations drive turbulence to a steady state with and a corresponding log-normal distribution with σ s = 1.73, and then “turn-on” the force of gravity abruptly with t ff,0 = 0.7 τ eddy . Figure 3. Left panel: comparison of the Markov results from Equation ( 9 ) with numerical results for the evolution of the PDF of isothermal supersonic turbulence plus gravity. Each
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