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ID: past-exam-of-the-astrophysical-fluid-dynamics-course-of-the-university-of-cambridge/2017/3/c/solution

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Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2017 / 3 / c / Solution by

Codex  0 2026-10-18

Let

a_{T}^{2} = k_{B} T / (μ m_{u})

be the constant isothermal sound speed squared. The stationary galactic gas obeys hydrostatic equilibrium in the specified gravitational potential:

a_{T}^{2} \frac{d ρ}{d r} = - ρ \frac{d ψ}{d r}, \frac{d l n ρ}{d r} = - \frac{1}{a _{T}^{2}} \frac{d ψ}{d r} .

Integrating from the pressure-matching radius

r_{0}

gives

ρ (r) = ρ (r_{0}) exp [- (ψ (r) - ψ (r_{0})) / a_{T}^{2}]

. Since

ρ (r_{0}) = p_{0} / a_{T}^{2}

and

ψ = - A r^{- 1/2}

,

ρ (r) = \frac{μ m _{u} p _{0}}{k _{B} T} exp [\frac{A μ m _{u}}{k _{B} T} (r^{- 1/2} - r_{0}^{- 1/2})], 0 < r \leq r_{0} .

The exponent is dimensionless: the quantity in parentheses times

A

is a difference of gravitational potential, of units

m^{2} s^{- 2}

. Density increases inward, as expected. The idealized potential is singular at the centre, so this formal profile should not be extrapolated into a region where the prescribed potential or the neglect of gas self-gravity fails.

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