Density profile of a self-gravitating polytropic turbulent fluid in the context of ensembles of molecular clouds

We obtain an equation for the density profile in a self-gravitating polytropic spherically symmetric turbulent fluid with an equation of state $p_{\rm gas}\propto \rho^\Gamma$. This is done in the framework of ensembles of molecular clouds represented by single abstract objects as introduced by Donkov et al. (2017). The adopted physical picture is appropriate to describe the conditions near to the cloud core where the equation of state changes from isothermal (in the outer cloud layers) with $\Gamma=1$ to one of `hard polytrope' with exponent $\Gamma>1$. On the assumption of steady state, as the accreting matter passes through all spatial scales, we show that the total energy per unit mass is an invariant with respect to the fluid flow. The obtained equation reproduces the Bernoulli equation for the proposed model and describes the balance of the kinetic, thermal and gravitational energy of a fluid element. We propose as well a method to obtain approximate solutions in a power-law form which yields four solutions corresponding to different density profiles, polytropic exponents and energy balance equations for a fluid element. Only one of them, a density profile with slope $-3$ and polytropic exponent $\Gamma=4/3$, matches with observations and numerical works. In particular, it yields a second power-law tail of the density distribution function in dense cloud regions.