Lane-Emden Stellar Structure

Polytropic stars · pressure/density profiles · stellar radii and masses
θ (dimensionless density)
Pressure P ∝ θⁿ⁺¹
Enclosed mass m(ξ)
Temperature T ∝ θ
Polytropic index n n = 1.50
Central density ρ_c (kg/m³) 10⁵·⁰
Stellar type
The Lane-Emden equation governs a self-gravitating polytropic gas sphere: (1/ξ²) d/dξ[ξ² dθ/dξ] + θⁿ = 0, where ξ is dimensionless radius and θ is dimensionless temperature/density. The surface is where θ(ξ₁) = 0. n=1.5 describes convective zones and red giants. n=3 (Eddington standard model) approximates the Sun. n=5 (Schuster sphere) has infinite radius but finite mass. n→∞ is isothermal. The Chandrasekhar limit for white dwarfs uses n=3 (ultra-relativistic) and n=1.5 (non-relativistic) polytropes.