Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 321 / 2 / d

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- Solution d

Solution

A satellite on a circular orbit is stationary in the corotating sheet. Write its tidal forcing and the response as

Ψ = ψ (x) e^{i k_{y} y}, v_{y} = v (x) e^{i k_{y} y} .

Since

D = - i S k_{y} x

, the forced wave equation becomes

- v_{s}^{2} v^{''} + [κ_{r}^{2} + v_{s}^{2} k_{y}^{2} - S^{2} k_{y}^{2} x^{2}] v = (2Ω - S) ψ^{'} - S x k_{y}^{2} ψ .

This ordinary differential equation describes a satellite-forced density wave in an astrophysical disk.

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