where ξ ξ is a dimensionless radius and θ θ is related to the density, and thus the pressure, by ρ ρ = ρ ρ c θ θ n for central density ρ ρ c . The index n is the polytropic index that appears in the polytropic equation of state, P = K ρ ρ 1 + 1 n where P and ρ ρ are the pressure and density, respectively, and K is a constant of proportionality. The standard boundary conditions are θ θ ( 0 ) = 1 and θ θ ′ ( 0 ) = 0 . Solutions thus describe the run of pressure and density with radius and are known as polytropes of index n . If an isothermal fluid (polytropic index tends to infinity) is used instead of a polytropic fluid, one obtains the Emden–Chandrasekhar equation.