Scale-free gravitational collapse as the origin of rho sim r⁻² density profile -- a possible role of turbulence in regulating gravitational collapse

Astrophysical systems, such as clumps that form star clusters share a density profile that is close to $\rho \sim r^{-2}$. We prove analytically this density profile is the result of the scale-free nature of the gravitational collapse. Therefore, it should emerge in many different situations as long as gravity is dominating the evolution for a period that is comparable or longer than the free-fall time, and this does not necessarily imply an isothermal model, as many have previously believed. To describe the collapse process, we construct a model called the turbulence-regulated gravitational collapse model, where turbulence is sustained by accretion and dissipates in roughly a crossing time. We demonstrate that a $\rho \sim r^{-2}$ profile emerges due to the scale-free nature the system. In this particular case, the rate of gravitational collapse is regulated by the rate at which turbulence dissipates the kinetic energy such that the infall speed can be $20 - 50 \%$ of the free-fall speed(which also depends on the interpretation of the crossing time based on simulations of driven turbulence). These predictions are consistent with existing observations, which suggests that these clumps are in the stage of turbulence-regulated gravitational collapse. Our analysis provides a unified description of gravitational collapse in different environments.