Magnetic Mirror Ratio and Loss Cone Calculator

A magnetic mirror is a plasma confinement scheme that uses a magnetic field which is strong at both ends of a trap and weaker in the middle. As a charged particle moves from the weak-field midplane toward a strong-field end, its perpendicular velocity increases and parallel velocity decreases (conserving the magnetic moment), and if the field is strong enough, the particle reflects back before reaching the end, effectively 'mirroring' it.

The mirror ratio R = B_max/B_min is the ratio of the peak magnetic field at the mirror throats (the narrow, high-field ends) to the minimum field at the trap's midplane. It is the single number that determines how effective the mirror is at confining particles.

The loss cone is the range of particle pitch angles (the angle between velocity and the magnetic field line) that are too small to be reflected by the mirror. Particles born or scattered into this cone of directions stream straight out along the field lines and are lost from confinement, regardless of their speed.

sinθ_loss = 1/√R, where R is the mirror ratio. Particles with pitch angle θ > θ_loss are trapped and mirror back; particles with θ < θ_loss fall inside the loss cone and escape.

This follows directly from conservation of the magnetic moment μ = mv⊥²/(2B) and total kinetic energy along a particle's trajectory in the mirror field. Combining these two conserved quantities eliminates the particle's speed entirely, leaving a condition that depends only on the field ratio R between the particle's location and the mirror throat.

For a velocity distribution that is isotropic (equal in all directions) at the midplane, the fraction of particles whose pitch angle falls inside the loss cone is 1 − √(1 − 1/R). This fraction is continuously repopulated by collisions scattering trapped particles into the loss cone, so real mirror machines suffer ongoing particle loss even at steady state.

A larger R shrinks 1/√R, narrowing the loss cone half-angle and reducing the fraction of particles that can escape directly. However, R cannot be increased indefinitely in practice, since it is limited by the maximum achievable magnet field strength and engineering constraints at the mirror throats.

Even with a large mirror ratio, collisions continuously scatter trapped particles' pitch angles, refilling the loss cone and causing a steady leak of particles and energy out the ends. This end-loss problem is the central limitation of simple mirror machines, motivating more advanced schemes like tandem mirrors and, ultimately, toroidal (closed-field-line) devices such as tokamaks.

Yes. Earth's dipole magnetic field acts as a natural magnetic mirror for charged particles trapped in the Van Allen radiation belts: particles bounce between mirror points near the poles, where the field is strongest, and only particles whose pitch angle falls inside the loss cone precipitate into the upper atmosphere, contributing to the aurora.

Laboratory mirror machines have historically used mirror ratios roughly in the range of 2 to 10, balancing the confinement benefit of a larger R against the engineering difficulty of producing very high peak fields at the throat coils.