where ξ ξ is the dimensionless radius and ψ ψ is the related to the density of the gas sphere as ρ ρ = ρ ρ c e − − ψ ψ , where ρ ρ c is the density of the gas at the centre. The equation has no known explicit solution. If a polytropic fluid is used instead of an isothermal fluid, one obtains the Lane–Emden equation. The isothermal assumption is usually modeled to describe the core of a star. The equation is solved with the initial conditions,