Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 317 / 2 / ii

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- Solution ii

Solution

For

P = K ρ^{1 + 1/ n}

, introduce the Lane-Emden equation variables

ρ = λ θ^{n}, r = α ξ, λ = ρ_{c},

α^{2} = \frac{( n + 1 ) K}{4 π G} λ^{1/ n - 1} .

Combining mass conservation with hydrostatic equilibrium gives

\frac{1}{ξ ^{2}} \frac{d}{d ξ} (ξ^{2} \frac{d θ}{d ξ}) = - θ^{n} .

Regularity and normalization impose

θ (0) = 1, θ^{'} (0) = 0.

The stellar surface is the first positive zero

ξ_{1}

of

θ

. Integrating the equation once gives

m (ξ) = - 4 π α^{3} λ ξ^{2} θ^{'} (ξ),

and hence

R = α ξ_{1}, M = - 4 π α^{3} λ ξ_{1}^{2} θ^{'} (ξ_{1}) .

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