  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.
