The post-shock density is $\chi\rho_0$, while part c gives
$$
|\mathbf B_{\rm ps}|^2
=B_0^2(\cos^2\theta+\chi^2\sin^2\theta).
$$
Using the <Alfvén speed> $v_A=B/\sqrt{4\pi\rho}$ in Gaussian units,
$$
\boxed{
\frac{v_{A,\rm ps}}{v_{A,0}}
=\left(\frac{\cos^2\theta+\chi^2\sin^2\theta}{\chi}\right)^{1/2}}.
$$
The ratio is largest in the equatorial plane, where the field is tangential to the shock:
$$
\boxed{
\left(\frac{v_{A,\rm ps}}{v_{A,0}}\right)_{\max}
=\sqrt\chi
=\sqrt{\frac{\gamma+1}{\gamma-1}}}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-314/4/d/solution