Example · Orbit in a Magnetic Mirror
Problem statement
Integrate an electron orbit in the mirror field derived from the flux function \(\psi = B_0\pi r^2\left(1 + z^2/L^2\right)\) and demonstrate: (a) trapping and reflection, (b) conservation of the magnetic moment \(\mu\), (c) that the guiding center rides a constant-\(\psi\) contour.
Given information
Field components (from \(\mathbf{B} = \frac{1}{2\pi}\nabla\psi\times\nabla\theta\)):
\[B_r = -\frac{B_0\, r z}{L^2}, \qquad B_z = B_0\left(1 + \frac{z^2}{L^2}\right)\]
with \(B_0 = L = 1\) (normalized), no electric field.
Solution strategy
- Implement the field in Cartesian components (\(B_x = B_r x/r\), \(B_y = B_r y/r\)).
- Push with the 3-D leapfrog solver — symplecticity is essential over many bounce periods.
- Launch from mid-plane with chosen pitch angle \(\tan\theta_p = v_\perp/v_\parallel\); try several.
- Diagnostics per step: \(|\mathbf{B}|\) at the particle, \(v_\perp\), \(v_\parallel\), \(\mu = \frac{mv_\perp^2}{2B}\), and \(\psi(r, z)\).
Expected results
- Large pitch angle (mostly perpendicular): the particle bounces between mirror points where \(B = B_\text{min} v^2/v_{\perp 0}^2\) — trapped.
- Small pitch angle (inside the loss cone, \(\sin^2\theta_p < B_\text{min}/B_\text{max}\)): sails through and escapes.
- \(\mu(t)\) stays constant to high accuracy while \(v_\parallel^2\) and \(v_\perp^2\) trade places — textbook adiabatic invariance, computed rather than assumed.
- The guiding center traces a constant-\(\psi\) surface: flux conservation made visible.
Key takeaways
- The bounce motion is the second periodicity of magnetized particles (gyration = first, drift precession = third) — the basis of the three adiabatic invariants (\(\mu\), \(J\), \(\Phi\)) in PC368.
- Total speed never changes (magnetic force does no work); only its parallel/perpendicular split oscillates.
- Set the mirrors in motion and each bounce pumps \(v_\parallel\) — Fermi acceleration, the sequel.