 Solution

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

The smaller galaxy is treated as a test particle released from rest at

r_{0}

\frac{1}{2} \dot{d}^{2} + ψ (d) = ψ (r_{0}), \dot{d}^{2} = 2 A (d^{- 1/2} - r_{0}^{- 1/2}) .

The galactic gas is stationary in its hydrostatic model, so this is the relative speed squared. Write

Δ (d) = d^{- 1/2} - r_{0}^{- 1/2} \geq 0

on the inward path. The astrophysical ram pressure is the incident normal momentum flux

ρ (d) \dot{d}^{2}

p_{ram} (d) = \frac{2 A μ m _{u} p _{0}}{k _{B} T} Δ (d) exp [\frac{A μ m _{u}}{k _{B} T} Δ (d)], 0 < d \leq r_{0} .

It vanishes at release and increases during infall. This ram pressure convention is

ρ v^{2}

, not the

ρ v^{2} /2

sometimes called dynamic pressure. Gas drag and the object's response are neglected consistently with the prescribed ballistic motion.

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