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We obtain two equations (following from two different approaches) for thedensity profile in a self-gravitating polytropic cylindrically symmetric androtating turbulent gas disk. The adopted physical picture is appropriate todescribe the conditions near to the cloud core where the equation of state ofthe gas changes from isothermal (in the outer cloud layers) to one of "hardpolytrope", and the symmetry changes from spherical to cylindrical. On theassumption of steady state, as the accreting matter passes through all spatialscales, we show that the total energy per unit mass is an invariant withrespect to the fluid flow. The obtained equation describes the balance of thekinetic, thermal and gravitational energy of a fluid element. We also introducea 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 isin accordance with the two approaches (energy and force balance). Itcorresponds to a density profile of a slope -2, polytropic exponent 3/2, andsub-Keplerian rotation of the disk, when the gravity is balanced by the thermalpressure. It also matches with some observations and numerical works and, inparticular, leads to a second power-law tail (of a slope approx. -1) of thedensity distribution function in dense, self-gravitating cloud regions.

Density profile of a self-gravitating polytropic turbulent fluid in a rotating disk near to the cloud core