A fourth-order Runge-Kutta integration of the Lane-Emden equation for polytropic index

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Astrophysics & Cosmology · Stellar structure, energy generation and evolution

Stars from the Inside, Ch Pressure against gravity: the equation of state, polytropes and the Lane-Emden equation

Problem

A fourth-order Runge-Kutta integration of the Lane-Emden equation for polytropic index n=4n = 4, started from the series solution θ=1−ξ2/6+nξ4/120\theta = 1 - \xi^2/6 + n\xi^4/120 a short distance off the centre, gives near the first zero of θ\theta: Step size h=0.15h = 0.15: θ(14.90)=0.00058\theta(14.90) = 0.00058, θ(15.05)=−0.00063\theta(15.05) = -0.00063. Step size h=0.075h = 0.075 (halved): θ(14.90)=0.00057\theta(14.90) = 0.00057, θ(15.05)=−0.00064\theta(15.05) = -0.00064. ((a)) For each step size, interpolate linearly between the two tabulated points to estimate ξ1\xi_1. Compare the two step sizes. ((b)) Using the h=0.075h = 0.075 data, estimate θ1′\theta_1' by finite difference and hence the mass constant −ξ12θ1′-\xi_1^2\theta_1' and the central concentration ρc/ρˉ=−ξ1/(3θ1′)\rho_c/\bar\rho = -\xi_1/(3\theta_1'). Comment on how these compare with the n=2n=2 values found for a less centrally concentrated polytrope (ξ1≈4.35\xi_1 \approx 4.35, −ξ12θ1′≈2.41-\xi_1^2\theta_1' \approx 2.41, ρc/ρˉ≈11.4\rho_c/\bar\rho \approx 11.4).

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