Evolution of the mass density profile of dense molecular clouds

We set ourselves the goal of obtaining the equations which govern the evolution of the mass density profile of a dense irrotational molecular cloud. We base our study on the notion of "ensemble of molecular clouds", introduced in our previous work. We model the studied clouds making use of the "ensemble abstract representative member"-a spherically symmetric and isotropic cloud. This cloud is isothermal and radially accreting matter from its surroundings. Starting from the equations of hydrodynamics applied to a self-gravitating isothermal spherical gas cloud, we obtain a system of two first-order nonlinear partial differential equations, which govern the evolution of two unknown fields: the exponent of density profile and the accretion velocity. Under the assumption of steady state flow, we get approximate solutions, using the method of leading order terms. Far from the cloud's center we obtain density profile ϱ=ℓ^{-2}, and the accretion velocity is constant, while near to the center we have ϱ=ℓ^{-3/2} and v_{a}∝ℓ^{-1/2}. Through our dynamical equations, the obtained solutions coincide completely with the solutions found by using the equation of energy conservation of a fluid element (in our previous work). Also, combining the equations of energy balance for a fluid element, we arrive at the conclusion that the cloud's layers, far from the center, are in a stable dynamical state if the accretion velocity flow is sub- or trans-sonic, otherwise they are marginally stable (moderately supersonic flow) or unstable (supersonic flow). Both solutions are consistent only if the accretion flow is subsonic, and hence the outer layers are stable. Finally, under the assumption that both the accretion velocity and density scale with ℓ and their power-law exponents are position-independent, we show that the density scaling exponent far away from the center is p=2 and this value is an attractor. Hence, this value should be observable in dense molecular clouds.