Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 314 / 4

The given velocity is a homologous spherical flow,with the spatially uniform velocity divergenceEvery shocked fluid element therefore expands rather than undergoing continued compression, and the velocity fills the remnant smoothly instead of concentrating its mass in a thin cooling shell. In the absence of radiative losses, the entropy advection equation then makes each element follow an adiabatic process after its one entropy-producing passage through the shock. This broad expanding structure is the expected adiabatic phase of a supernova remnant.

The Strong-shock Rankine-Hugoniot conditions give the compression ratiofor $\gamma =5/3$. ThusIn the shock frame, mass conservation saysUsing ${\rho }_2=4{\rho }_0$ gives

Resolve the upstream uniform field ${\mathbf{B}}_0=B_0{\mathbf{e}}_z$ into components normal and tangential to the spherical shock:The ideal magnetohydrodynamic shock conditions preserve the normal magnetic component and multiply the tangential component of a weak passive field by the gas compression ratio. Part (b) therefore gives

For the homologous spherical flow $\mathbf{u}=q\mathbf{r}$, the ideal magnetohydrodynamic induction equation and $\nabla \cdot \mathbf{B}=0$ give the material derivativeLet $x=r/R(t)$ and use the self-similar ansatzPart (b) gives $q=3\dot{R}/(4R)$, and thereforeThe radial induction equation becomes $x{f}^{\prime }=6f$. The boundary value $f(1)=1$ gives $f=x^6$. The solenoidal vector field condition requireswhich also matches the tangential shock value in part (c). Hence the interior field is

Outside the shock, the uniform-field lines obey $r\sin \theta =\text{constant}$. Inside, the magnetic-field-line equation givessoA sketch therefore shows straight exterior lines refracting at the spherical shock into north-south symmetric curves that bow toward the equatorial interior before leaving through the opposite hemisphere. The field strength falls as $r^6$ toward the centre.