Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 314 / 4 / d

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- Solution d

Solution

The post-shock density is

χ ρ_{0}

, while part c gives

∣ B_{ps} ∣^{2} = B_{0}^{2} (cos^{2} θ + χ^{2} sin^{2} θ) .

Using the Alfvén speed

v_{A} = B / 4 π ρ

in Gaussian units,

\frac{v _{A, ps}}{v _{A, 0}} = (\frac{cos ^{2} θ + χ ^{2} sin ^{2} θ}{χ})^{1/2} .

The ratio is largest in the equatorial plane, where the field is tangential to the shock:

(\frac{v _{A, ps}}{v _{A, 0}})_{m a x} = χ = \frac{γ + 1}{γ - 1} .

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