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

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

Codex  0 2026-10-18

Along a family of equilibria at fixed temperature,

K

and the dimensionless function

Ψ

are fixed. The boundary mass density changes through both

ρ_{c}

and the boundary coordinate

ξ

:

δ ρ = e^{- Ψ} δ ρ_{c} - ρ_{c} e^{- Ψ} Ψ^{'} δ ξ = ρ (\frac{δ ρ _{c}}{ρ _{c}} - Ψ^{'} δ ξ) .

From part (c),

N = C ρ_{c}^{- 1/2} f (ξ)

, where

C

is constant at fixed temperature and

f = ξ^{2} Ψ^{'}

. Set its logarithmic variation to zero:

0 = \frac{δ N}{N} = - \frac{1}{2} \frac{δ ρ _{c}}{ρ _{c}} + \frac{f ^{'}}{f} δ ξ .

The isothermal Lane-Emden equation gives

f^{'} = ξ^{2} e^{- Ψ}

, so the fixed-

N

relation is

\frac{δ ρ _{c}}{ρ _{c}} = \frac{2 e ^{- Ψ}}{Ψ ^{'}} δ ξ .

Substitute back into the boundary-density variation:

δ ρ_{N} = ρ (\frac{2 e ^{- Ψ}}{Ψ ^{'}} - Ψ^{'}) δ ξ .

For a nontrivial variation at a finite positive radius, its vanishing gives the fixed-mass turning point of an isothermal sphere:

[Ψ^{'} (ξ)]^{2} = 2 e^{- Ψ (ξ)} .

This derivative follows a fixed-mass sequence of equilibria; it is not merely the spatial derivative of density inside one sphere. The corresponding radial variation is

\frac{δr}{r} = (\frac{1}{ξ} - \frac{e ^{- Ψ}}{Ψ ^{'}}) δ ξ .

At small

ξ

,

Ψ^{'} ≃ ξ /3

: the boundary density increases while the radius decreases along increasing

ξ

, giving the stated negative

(\partial p / \partial V)_{N, T}

of the stable branch. The first density turning point marks its first loss of radial stability; it need not be confused with a turning point of radius itself.

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