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

monthly notices of the royal astronomical society

Scale-free gravitational collapse as the origin of $\rho \sim r^{-2}$  density profile -- a possible role of turbulence in regulating gravitational  collapse