A general theory of turbulent fragmentation

Abstract

We develop an analytic framework to understand fragmentation in turbulent, self-gravitating media.In previous work, we showed how some properties of turbulence can be predicted by application of the excursion-set formalism.Here, we generalize this to understand fully time-dependent gravo-turbulent fragmentation and collapse.We show that turbulent systems are always gravitationally unstable in a probabilistic sense.The fragmentation mass spectrum, size-mass-density-linewidth relations of collapsing objects, their correlation functions and clustering, the range of spatial scales over which fragmentation occurs, and the time-dependent rate of collapse/fragmentation (as a function of size/mass) are analytically predictable.We show how these depend on bulk properties of turbulence; fragmentation is promoted at higher Mach numbers and shallower power spectra.We also generalize the model to properly include rotation, complicated gas equations of state, collapsing/expanding backgrounds, magnetic fields, intermittency and non-normal statistics (with inherently correlated fluctuations).This allows us to formally derive how fragmentation is suppressed with 'stiffer' equations of state (e.g. higher polytropic index γ ) or differently driven turbulence (solenoidal versus compressive).The suppression appears at an 'effective sonic scale' where bM(R s , ρ crit [R s ]) ≈ 1, with ρ crit being the (scale-dependent) critical density for fragmentation.Gas becomes stable against collapse below this scale for γ > 4/3; however, fragmentation still occurs on larger scales.We show that the scale-free nature of turbulence and gravity generically drives mass spectra and correlation functions towards universal shapes (observed in a wide variety of astrophysical phenomena), with weak residual dependence on many properties of the media.We find that correlated fluctuations on different scales, non-Gaussian density distributions and intermittency have surprisingly small effects on the fragmentation process.We demonstrate that this is because fragmentation cascades on small scales are generically 'frozen in' when large-scale fluctuations push the 'parent' region above the collapse threshold; though they collapse, their statistics are only weakly modified by the collapse process.Finally, with thermal or turbulent support, structure develops 'top-down' in time via a fragmentation cascade, but we show that significant rotational/angular momentum support reverses the sense of structure formation to 'bottom-up' growth via mergers of bound clumps, and introduces a characteristic 'maximal instability scale' distinct from the Toomre scale.