Equilibrium of Gravitating System of Spherical Gas-Dust Cloud

In this paper, we take several different analytical and numerical approaches to studying the equilibrium of a gravitating system of a gas-dust cloud. We consider one-dimensional self-gravitating spherically symmetric fluid flow of a gas-dust cloud. We discuss the equilibrium of the system of gas-dust cloud by using simplified analytic stellar, polytropic models. A condition for the equilibrium of the cloud in the form of a differential equation is used. Using mass density in the cloud as a given function, we obtained the corresponding pressure analytically within the cloud and determined the central pressure. In dealing with a polytropic model, we found the analytical and numerical solution of the Lane-Emden equation for the various values of the polytropic index after that, central density and central pressure were obtained. Finally, the density and pressure of the cloud for the various values of the polytropic index are calculated. The result found by the simplified analytical method and the polytropic method is compared.