Past exam of the kinetic theory course of the University of Oxford / 2024 / 3 / e

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- Solution e

Solution

Represent the perturber by the periodic Dirac delta function at

x_{p} + v_{p} t

. With spatial Fourier series coefficient normalized by

L^{- 3}

, its nonzero density coefficients are

ρ_{p, k} (t) = \frac{m _{p}}{L ^{3}} e^{- i k \cdot x_{p}} e^{- i k \cdot v_{p} t}, t \geq 0.

The mean density is subtracted as in d. The Poisson equation for Newtonian gravity gives

δ Φ_{k}^{ext} (t) = C_{k} e^{- ia t}, C_{k} = - \frac{4 π G m _{p}}{L ^{3} k ^{2}} e^{- i k \cdot x_{p}}, a = k \cdot v_{p} .

The perturbation is zero before switch-on. For

Im ω > 0

, its Laplace transform is

\int_{0}^{\infty} e^{i (ω - a) t} d t = - \frac{1}{i ( ω - a )} = \frac{1}{i ( a - ω )} .

Hence

δ Φ_{k}^{ext} (ω) = - \frac{4 π G m _{p}}{L ^{3} k ^{2}} e^{- i k \cdot x_{p}} \frac{1}{i ( k \cdot v _{p} - ω )} .

The switch-on creates a forcing pole at the real frequency

a

, approached causally from the original upper inversion contour.

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