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=2n = 2, 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=0.10h = 0.10: θ(4.30)=0.0067\theta(4.30) = 0.0067, θ(4.40)=−0.0060\theta(4.40) = -0.0060. Step size h=0.05h = 0.05 (halved): θ(4.30)=0.0068\theta(4.30) = 0.0068, θ(4.40)=−0.0059\theta(4.40) = -0.0059. ((a)) For each step size, use linear interpolation between the two tabulated points to estimate ξ1\xi_1, the first zero of θ\theta. Do the two step sizes agree to three significant figures? ((b)) Using the more accurate (h=0.05h = 0.05) data, estimate the slope θ1′=dθ/dξ\theta_1' = \mathrm{d}\theta/\mathrm{d}\xi at ξ1\xi_1 by a finite difference between the two tabulated points, 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').

Browse the problem library