Starting from Poisson's equation for the gravitational potential,

PHYSICS · Subjective · Fermi University Physics · Astrophysics & Cosmology · Stellar structure, energy generation and evolution · 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

Starting from Poisson's equation for the gravitational potential, ∇2Φ=4πGρ\nabla^2\Phi = 4\pi G\rho, and hydrostatic equilibrium written in terms of the potential as dPdr=−ρdΦdr\dfrac{dP}{dr} = -\rho\dfrac{d\Phi}{dr} (equivalent to the usual dP/dr=−Gm(r)ρ/r2dP/dr = -Gm(r)\rho/r^2, since dΦ/dr=Gm(r)/r2d\Phi/dr = Gm(r)/r^2 for a spherical mass distribution), together with the polytropic relation P=Kρ1+1/nP = K\rho^{1+1/n} for fixed index nn, derive the Lane-Emden equation 1ξ2ddξ(ξ2dθdξ)=−θn\frac{1}{\xi^2}\frac{d}{d\xi}\left(\xi^2\frac{d\theta}{d\xi}\right) = -\theta^n without ever writing down the mass-continuity equation dm/dr=4πr2ρdm/dr = 4\pi r^2\rho. Instead, integrate the hydrostatic-equilibrium relation once to express ρ\rho in terms of Φ\Phi, substitute the result into Poisson's equation, and only then introduce the dimensionless density θ\theta (defined so that ρ=ρcθn\rho = \rho_c\theta^n, with ρc\rho_c the central density) and the dimensionless radius ξ=r/α\xi = r/\alpha. Give α\alpha in terms of KK, nn, GG and ρc\rho_c, and state the two conditions on θ(ξ)\theta(\xi) that follow from regularity at the centre.

Browse the problem library