 Solution

ID: past-exam-of-the-astrophysical-fluid-dynamics-course-of-the-university-of-cambridge/2021/section-b/5/b/solution

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Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2021 / Section B / 5 / b / Solution by

Codex  0 2026-10-18

ρ = ρ_{c} e^{- Ψ (ξ)}

and

r = a ξ

. Then

d ln ρ / d r = - a^{- 1} Ψ^{'}

and the equation from part (a) becomes

- \frac{1}{a ^{2} ξ ^{2}} \frac{d}{d ξ} (ξ^{2} Ψ^{'}) = - \frac{4 π G ρ _{c}}{K} e^{- Ψ} .

With

a^{2} = K / (4 π G ρ_{c})

, the coefficients cancel, giving the isothermal Lane-Emden equation

\frac{1}{ξ ^{2}} \frac{d}{d ξ} (ξ^{2} Ψ^{'}) = e^{- Ψ} .

The regular central boundary conditions are

Ψ (0) = 0, Ψ^{'} (0) = 0.

The first makes the central mass density equal to

ρ_{c}

. The second follows from smooth spherical symmetry and the vanishing central gravitational acceleration; a preferred radial slope at the origin would be singular. Substitution of a regular power series gives

Ψ (ξ) = \frac{ξ ^{2}}{6} - \frac{ξ ^{4}}{120} + O (ξ^{6}),

which supplies nonsingular starting data for numerical integration. The central conditions select the regular solution, not the singular isothermal sphere.

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