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.
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