The index that determines this equation is given by the equation of state "pressure equals a constant times density to the quantity (1 + 1 over n)" and is called the polytropic index.
The standard derivation of this equation is achieved by multiplying the equation for hydrostatic equilibrium by "r squared" and differentiating with respect to r before combining it with the mass continuity equation. The parameter w in this equation is computed through the relation "density equals critical density times w to the n" and in the equation its derivative is multiplied by the inverse square of a dimensionless radius parameter. While there is no general solution to this equation, it can be solved analytically for the indices 0, 1, and 5, where the index 0 corresponds to a homogeneous gas sphere (*) with density equal to the critical density and the index 1 yields a solution in which the radius is independent of the mass. The index that determines this equation is given by the equation of state "pressure equals a constant times density to the quantity (1 + 1 over n)" and is called the polytropic index. For 10 points, name this form of Poisson's equation, a second order differential equation that is used to construct a polytropic stellar model in which pressure is a known function of density.
The standard derivation of this equation is achieved by multiplying the equation for hydrostatic equilibrium by "r squared" and differentiating with respect to r before combining it with the mass continuity equation.
Lane-Emden equation