A general theory of turbulent fragmentation

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 (Rₛ, 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.