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Fermi Acceleration

Intuition

Bounce a ping-pong ball between two paddles moving slowly together: every bounce off an approaching paddle adds speed. Replace paddles with magnetic mirrors and the ball with a trapped charged particle, and you have Fermi's 1949 mechanism for accelerating cosmic rays to fantastic energies — magnetic clouds in the galaxy acting as drifting mirrors.

The setup

The course's model: a mirror field whose length slowly shrinks,

\[\psi(r, z, t) = B_\text{min}\pi r^2\left[1 + 2\lambda\frac{\zeta^2}{\zeta^4 + 1}\right], \qquad \zeta = \frac{z}{L(t)}, \quad L(t) = L_0 - \alpha t\]

A time-varying \(\mathbf{B}\) induces an electric field (Faraday), computed from the vector potential:

\[\mathbf{A} = \frac{\psi}{2\pi r}\hat{\boldsymbol\phi} \quad\Longrightarrow\quad \mathbf{E} = -\frac{\partial\mathbf{A}}{\partial t} = -\frac{1}{2\pi r}\frac{\partial \psi}{\partial t}\hat{\boldsymbol\phi}\]

This azimuthal E field is what actually does the work — magnetic fields never do.

The physics

  • Each reflection off an approaching mirror boosts \(v_\parallel\) (head-on collision with a moving wall: \(v \to v + 2u\) in the wall frame).
  • The parallel invariant \(J = \oint v_\parallel\, dz \approx v_\parallel L\) is adiabatically conserved: as \(L\) shrinks, \(v_\parallel\) must grow — \(v_\parallel \propto 1/L\).
  • Meanwhile \(\mu = mv_\perp^2/2B\) holds \(v_\perp\) tied to \(B\). The pitch angle therefore decreases as \(v_\parallel\) grows...
  • ...until the particle enters the loss cone (\(\theta < \theta_\text{trap} = \sin^{-1}(\lambda^{-1})\) in the model's notation) and escapes — carrying its stolen energy. Acceleration with a built-in exit door: exactly what a cosmic-ray source needs.

Numerical experiment

Integrate the full orbit with the leapfrog solver, fields from the analytic derivatives of \(\psi\) (chain rule through \(L(t)\)):

\[\frac{\partial\psi}{\partial t} = \frac{\partial\psi}{\partial\zeta}\cdot\left(-\frac{z}{L^2}\right)\cdot(-\alpha)\]

Watch \(v_\parallel\) ratchet up bounce after bounce, then the escape. Track \(\mu\) to confirm the perpendicular invariant holds while \(J\) does the accelerating.

Where it matters

Cosmic-ray acceleration (Fermi's original 2nd-order mechanism, and the 1st-order version at supernova shocks) · magnetic-pumping heating schemes · particle energization in shrinking magnetospheric flux tubes.

Common mistakes

  • Crediting the magnetic field with the energy. The energy comes from whatever moves the mirrors, delivered via the induced E field.
  • Expecting unlimited acceleration. The loss cone always wins eventually; the mechanism accelerates and releases.
  • Simulating with a symplectic-but-static-field mindset: time-dependent fields mean energy is not conserved — that's the whole point; don't "fix" it.

Knowledge graph position

Prerequisites: Magnetic mirror. Leads to: astrophysical particle acceleration, wave heating.

Quiz

Q1 (computational). Mirrors shrink from \(L_0\) to \(L_0/2\). By what factor does a trapped particle's parallel energy grow (adiabatic \(J\))?

Answer

\(v_\parallel \propto 1/L\) doubles ⇒ parallel energy ×4.

Q2 (conceptual). Why does Fermi acceleration preferentially increase energy, when a particle can also bounce off a receding mirror and lose energy?

Answer

Head-on (approaching) encounters are more frequent than overtaking ones — approach speed is higher, so collisions are biased toward gains (rate ∝ relative velocity). Statistically the gains win — 2nd-order Fermi. With converging mirrors (or a shock), every bounce gains — 1st-order Fermi.

Q3 (MCQ). The work on the accelerated particle is done by:

  • (a) the magnetic field directly (b) the induced electric field
  • (c) collisions (d) gravity
Answer

(b). \(\partial\mathbf{B}/\partial t \neq 0\) induces \(\mathbf{E}\) (Faraday); only \(\mathbf{E}\) does work on charges.