A solar-flare magnetic loop is anchored at both footpoints
PHYSICS · Multi-part · Fermi University Physics · Plasma Physics · The plasma state and single-particle motion · Fluid and magnetohydrodynamic description
Plasma Physics, Ch From particles to fluids: the distribution function, its moments and the two-fluid model
Problem
A solar-flare magnetic loop is anchored at both footpoints. A parcel of protons near its top starts in a field B0=0.01T with an isotropic Maxwellian distribution, T⊥0=T∥0=T0=100eV. The field threading the parcel is then raised to B1=0.05T, slowly compared with a gyration period and with a proton's bounce time between the mirror points, but fast compared with the proton collision time, while the length of flux tube between the mirror points does not change.
((a)) Using conservation of the magnetic moment μ=mv⊥2/2B for each proton, find T⊥ after the rise.
((b)) State which conserved quantity fixes T∥, and give its value after the rise.
((c)) Give the ratio p⊥/p∥ after the rise, and explain in one sentence why the two-fluid model needs a pressure tensor rather than a scalar pressure to describe this parcel.
((d)) Flux conservation raises the parcel's density by the factor B1/B0. Find the single temperature that the adiabatic closure with γs=5/3 would predict for the same density rise, and state which condition in the problem makes that closure the wrong choice here.