Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 14B / b

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- Solution b

Solution

With the polytropic equation of state and substitutions

P = K ρ_{c}^{1 + 1/ n} θ^{n + 1}, ρ = ρ_{c} θ^{n}, r = a ξ,

we have

\frac{1}{ρ} \frac{d P}{d r} = (n + 1) K ρ_{c}^{1/ n} \frac{d θ}{d r} .

The pressure-support equation therefore becomes

(n + 1) K ρ_{c}^{1/ n} \frac{1}{r ^{2}} \frac{d}{d r} (r^{2} \frac{d θ}{d r}) = - 4 π G ρ_{c} θ^{n} .

Choose

a^{2} = \frac{( n + 1 ) K}{4 π G} ρ_{c}^{(1 - n) / n} .

Changing from

r

to

ξ

then yields the Lane-Emden equation

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

These are the Lane-Emden variables for a stellar polytrope.

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