 Solution

ID: past-exam-of-the-kinetic-theory-course-of-the-university-of-oxford/2025/3/d/solution

The restriction

k_{φ} = \pm m

describes an

m

m = 2

Ω_{p}

, its amplitude peaks at

t_{peak}

, and its Gaussian function envelope has duration of order

τ

u_{k} (J)

describe its strength along different stellar orbits. A real perturbing potential requires

u_{- k} = u_{k}^{*}

Write

g (t) = exp [- (t - t_{peak})^{2} / (2 τ^{2})]

. The product entering c is

δ ϕ_{k} (t^{'}) δ ϕ_{k}^{*} (t) = ∣ u_{k} ∣^{2} e^{i k_{φ} Ω_{p} (t - t^{'})} g (t^{'}) g (t) .

The phase therefore measures detuning from the rotating pattern:

ω_{k} (J) = k \cdot Ω - k_{φ} Ω_{p} = k_{φ} (Ω_{φ} - Ω_{p}) + k_{R} Ω_{R} .

Substitution gives

Q_{0} = - 2 π k_{φ} = \pm m \sum k_{R} \sum k_{φ} \int_{0}^{\infty} d J_{R} ∣ u_{k} ∣^{2} \times \int_{0}^{t} d t^{'} k \cdot \partial_{J} f_{0} (J, t^{'}) e^{- i ω_{k} (t - t^{'})} \times exp [- \frac{( t - t _{peak} ) ^{2}}{2 τ ^{2}} - \frac{( t ^{'} - t _{peak} ) ^{2}}{2 τ ^{2}}] .

The strongest long-lived response occurs near an orbital resonance,

ω_{k} = 0

, where successive force contributions do not rapidly cancel in phase.

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