While its electrons remain non-relativistic

PHYSICS · Multi-part · Fermi University Physics · Astrophysics & Cosmology · Stellar structure, energy generation and evolution · Stellar death and compact objects

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

Problem

While its electrons remain non-relativistic, a helium white dwarf can be modelled as an

n=3/2

polytrope with

P=K\rho^{5/3}

K = 1.00\times10^{7}\,\mu_e^{-5/3}

in SI units (

P

in Pa,

\rho

in kg m

^{-3}

), and Lane-Emden constants

\xi_1 = 3.6538

and

A\equiv-\xi_1^2\theta_1'(\xi_1) = 2.7141

. ((a)) Starting from

R=\alpha\xi_1

and

M = 4\pi\alpha^3\rho_c A

, together with

\alpha^2 = \dfrac{(n+1)K}{4\pi G}\rho_c^{1/n-1}

, eliminate

\alpha

and

\rho_c

to derive a mass-radius relation of the form

R = C\,K\,M^{-1/3}

for

n=3/2

, and give the numerical value of

C

in SI units. ((b)) A helium white dwarf orbiting a millisecond pulsar has an inferred mass of

M = 0.40\,M_\odot

. Taking

\mu_e = 2

for fully ionised helium, use the result of part (a) to find the white dwarf's radius, in kilometres.

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