 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-317/3/ii/b/solution

 Top articles  Latest articles New article in topic

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 317 / 3 / ii / b / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-05

Let

ω = - ξ_{1}^{2} θ^{'} (ξ_{1})

be the Lane-Emden surface mass constant. The Lane-Emden mass formula gives

R = a ξ_{1}, M = 4 π a^{3} ρ_{c} ω, a^{2} = \frac{5 K _{NR}}{8 π G} ρ_{c}^{- 1/3} .

Thus

R \propto ρ_{c}^{- 1/6}

while

M \propto ρ_{c}^{1/2}

at fixed composition. Eliminating central mass density gives the polytropic mass-radius relation

M \propto R^{- 3}, s = - 3.

More explicitly,

R = ξ_{1} (4 πω)^{1/3} \frac{5 K _{NR}}{8 π G} M^{- 1/3} .

Increasing the mass of a nonrelativistic white dwarf decreases its radius. This scaling ceases to apply when relativistic electron degeneracy pressure becomes important.

 Read the full article

 New to topics? Read the docs here!