# Density distribution function of a self-gravitating isothermal compressible turbulent fluid in the context of Molecular Clouds ensembles II: the contribution of the turbulent term and the potential of the outer shells

## TL;DR

This paper explores the density profiles and energy balance of self-gravitating turbulent molecular clouds, emphasizing the roles of outer shell gravity and turbulence scaling, revealing two main density solutions with distinct energy characteristics.

## Contribution

It introduces new gravitational terms from outer shells and analyzes turbulence scaling effects, providing deeper physical insight into molecular cloud density profiles.

## Key findings

- Two density profile solutions with slopes -2 and -3/2 were identified.
- Outer shell gravity influences the energy balance of the density solutions.
- Turbulence scaling determines its importance in the cloud's energy dynamics.

## Abstract

In this paper we continue to investigate the energy conservation equation obtained in our previous work. We set ourselves three new goals. The first one is to rewrite the main equations in terms of density profile in order to give more physical insight. The second one is to investigate the significance of two new terms in the energy conservation equation. They originate from the gravity of the outer shells of cloud and the masses outer to the cloud, respectively. The third goal is to investigate the main equation in the case when the kinetic turbulent term scales according to Larson's law and it is independent, formally, of the accretion, in contrast to the previous work. The combination of supersonic turbulence and spherical symmetry raises a caveat which is commented in our conclusions. We obtained two solutions for the density profile. They scale with slopes -2 and -3/2, respectively. The energy balance for the second solution is the same as in the previous paper: this is a free-fall. For the first solution there are two cases. The first one: if the turbulent term does not scale, then it could be important for the energy balance of the cloud. The second one: if the turbulent term does scale, then it is not important for the energy balance of the cloud. The two new gravitational terms don't affect the existence of the two solutions, but the gravitation of the outer masses calibrate the energy balance for the first solution.