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

TL;DR

This paper derives an equation for the density profile of a self-gravitating, turbulent, polytropic fluid in molecular clouds, connecting theoretical models with observed density distributions and energy balances.

Contribution

It introduces a new equation for density profiles in turbulent, self-gravitating polytropic fluids, applicable to molecular cloud cores, and proposes approximate solutions matching observations.

Findings

Density profile with slope -3 matches observations.

A specific polytropic exponent (Γ=4/3) reproduces observed density distributions.

The model describes energy invariance during steady-state accretion.

Abstract

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 results in four solutions corresponding to different density profiles, polytropic exponents and energy balance equations for a fluid element. One of them, a density profile with slope $-3$ and polytropic exponent $\Gamma=4/3$, matches with observations and numerical works and, in particular, leads to a second power-law tail of the density distribution function in dense, self-gravitating cloud regions.