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We obtain two equations (following from two different approaches) for the density profile in a self-gravitating polytropic cylindrically symmetric and rotating turbulent gas disc. The adopted physical picture is appropriate to describe the conditions near to the cloud core where the equation of state of the gas changes from isothermal (in the outer cloud layers) to one of 'hard polytrope', and the symmetry changes from spherical to cylindrical. 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 describes the balance of the kinetic, thermal, and gravitational energy of a fluid element. We also introduce a method for approximating density profile solutions (in a power-law form), leading to the emergence of three different regimes. We apply, as well, dynamical analysis of the motion of a fluid element. Only one of the regimes is in accordance with the two approaches (energy and force balance). It corresponds to a density profile of a slope −2, polytropic exponent 3/2, and sub-Keplerian rotation of the disc, when the gravity is balanced by the thermal pressure. It also matches with some observations and numerical works and, in particular, leads to a second power-law tail (of a slope ∼−1) of the density distribution function in dense, self-gravitating cloud regions. [ABSTRACT FROM AUTHOR]

In this work, we apply virial analysis to the model of self-gravitating turbulent cloud ensembles introduced by Donkov & Stefanov in two previous papers, clarifying some aspects of turbulence and extending the model to account not only for supersonic flows but for trans- and subsonic ones as well. Making use of the Eulerian virial theorem at an arbitrary scale, far from the cloud core, we derive an equation for the density profile and solve it in approximate way. The result confirms the solution ϱ(ℓ) = ℓ−2 found in the previous papers. This solution corresponds to three possible configurations for the energy balance. For trans- or subsonic flows, we obtain a balance between the gravitational and thermal energy (Case 1) or between the gravitational, turbulent, and thermal energies (Case 2) while for supersonic flows, the possible balance is between the gravitational and turbulent energy (Case 3). In Cases 1 and 2, the energy of the fluid element can be negative or zero; thus the solution is dynamically stable and shall be long lived. In Case 3, the energy of the fluid element is positive or zero, i.e. the solution is unstable or at best marginally bound. At scales near the core, one cannot neglect the second derivative of the moment of inertia of the gas, which prevents derivation of an analytic equation for the density profile. However, we obtain that gas near the core is not virialized and its state is marginally bound since the energy of the fluid element vanishes. [ABSTRACT FROM AUTHOR]

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. 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 Γ = 1 to one of 'hard polytrope' with exponent Γ > 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 Γ = 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. [ABSTRACT FROM AUTHOR]