Criteria for gravitational instability and quasi-isolated gravitational collapse in turbulent medium

We study the evolution of structures in turbulent, self-gravitating media, and present an analytical criterion $M_{\rm crit} \approx \epsilon_{\rm cascade}^{2/3} \eta^{-2/3} G^{-1} l^{5/3}$(where$M_{\rm crit}$ is the critical mass, $l$ is the scale, $\epsilon_{\rm cascade}\approx \eta \sigma_{\rm v}^3 /l $ is the turbulence energy dissipation rate of the ambient medium, $G$ is the gravitational constant, ${\sigma }_{\mathrm{v}}$ is the velocity dispersion, $l$ is the scale and $\eta \approx 0.2$ is an efficiency parameter) for an object to undergo quasi-isolated gravitational collapse. The criterion also defines the critical scale ($l_{\rm crit} \approx \epsilon_{\rm cascade}^{1/2} \eta^{-1/2} G^{-3/4} \rho^{-3/4}$) for turbulent gravitational instability to develop. The analytical formalism explains the size dependence of the masses of the progenitors of star clusters ($M_{\rm cluster} \sim R_{\rm cluster}^{1.67}$) in our Galaxy.