Magnetic Mirror and Loss Cone

In an adiabatic guiding-center model, a charged particle moving in a slowly varying magnetic field conserves both its total kinetic energy and its magnetic moment (μ ~ v_perp² / B). As the particle approaches a region of stronger magnetic field (a "magnetic mirror"), its perpendicular kinetic energy increases while its parallel kinetic energy decreases to conserve total energy. If the parallel kinetic energy reaches zero, the particle is reflected. This defines the concept of the loss cone: particles with a pitch angle smaller than a critical value will not be reflected and will escape the trap.

worked-example

Consider a particle with a normalized total energy of 1. It enters a magnetic field with a Mirror ratio (R) of 4, a Pitch angle (α) of 30 degrees, and a Probe fraction of 1 (evaluating at the maximum mirror field). We can determine if it turns before reaching the probe and what its energy fractions are at the probe.

Calculate the critical pitch angle (α_c): α_c = arcsin(1 / sqrt(R)) = arcsin(1 / sqrt(4)) = arcsin(0.5) = 30 degrees.
Determine the maximum accessible field ratio before turning: b_turn = 1 / sin²(30°) = 1 / 0.25 = 4.
The probe evaluates the field ratio at the given fraction: b = 1 + fraction * (R - 1) = 1 + 1 * (4 - 1) = 4.
Since b = b_turn, the particle is exactly at the turning point.
Calculate the perpendicular energy fraction at the probe: E_perp = b * sin²(α) = 4 * 0.25 = 1.
Calculate the parallel energy fraction at the probe: E_parallel = 1 - E_perp = 1 - 1 = 0.

experiment

To observe the adiabatic reflection of particles in a magnetic mirror, use the Interactive visualization below to adjust the initial conditions and track the particle's energy components. Follow the link to open the module: https://riano.app/module/magnetic-mirror-loss-cone.

Set the Mirror ratio to 4 and observe the critical pitch angle calculated by the model.
Change the Pitch degrees to 45 (greater than the critical angle). Note that the particle is reflected before reaching the maximum probe point, showing its energy components as "Unavailable" at the probe fraction of 1.
Adjust the Probe fraction to 0.5 to investigate the particle's state halfway to the mirror point. Notice how the perpendicular energy fraction increases as the probe moves deeper into the stronger field, at the expense of the parallel energy fraction.

limitations

This model makes several simplifying assumptions about the particle's motion and the magnetic field.

Assumes a slowly varying magnetic field relative to the particle's Larmor radius, ensuring the magnetic moment is an adiabatic invariant.
No electric work or collisions are considered in this model.
The model uses a guiding-center approximation and does not resolve the full Lorentz orbit of the particle.

references

For more detailed mathematical derivations and physical context, consult the following resources:

Magnetic Mirrors - UT Austin Plasma Physics

Magnetic Mirror and Loss Cone