Magnetic mirrors and loss cones

Consider a static mirror field with minimum magnitude $B_0$ at the midplane and equal maxima $B_{\max}$ at its two ends. Neglect electric fields, collisions, and the small drift kinetic energy. Total kinetic energy $K$ and adiabatic scalar moment $\mu$ are then conserved. Parallel kinetic energy at any sampled field is $K-\mu B$. A turning point occurs when that expression reaches zero; stronger fields would require an impossible negative parallel energy.

At the midplane let pitch angle $\alpha_0$ measure velocity direction from the local field, so perpendicular energy is $K\sin^2\alpha_0$. Substitution gives the turning field $B_m=B_0/\sin^2\alpha_0$. A particle launched almost parallel to the field has a small perpendicular fraction and needs an enormous turning field. A perpendicular launch is already at zero parallel velocity. These limiting cases are useful checks on the direction of the inequality.

Define mirror ratio $R_m=B_{\max}/B_0>1$. Reflection before the maximum requires $\sin^2\alpha_0>1/R_m$. Directions closer to either parallel axis fall inside a loss cone and escape toward the corresponding end. The equality is the separatrix between passing and trapped orbits; a trajectory can asymptotically approach a smooth maximum there, so it should not be counted as a robustly trapped finite-period orbit.

For isotropically distributed velocity directions at the midplane, the population fraction in a cone is its solid angle divided by $4\pi$. Two opposite cones together occupy fraction $1-\cos\alpha_c$, not simply $2\alpha_c/\pi$. The sine weighting of spherical area matters. This is the fraction of an initially isotropic population in loss directions; the particle flux through an aperture also weights parallel speed and is therefore a different quantity.

The parallel force in the averaged description is $-\mu\,\partial B/\partial s$, directed away from stronger field. It acts on either charge sign in this approximation because scalar moment is positive. Real divergence-free mirror geometry also changes transverse field components; a purely axial field whose magnitude varies along its own straight direction everywhere would violate the magnetic divergence constraint. The one-dimensional formula follows motion along a field line, rather than specifying a complete field configuration.

An electrostatic potential adds $q\phi$ to the conserved energy and can either improve or weaken confinement depending on charge. Collisions change pitch angle and can refill a loss cone after its initial particles escape. Rapid field variation, large gyroradius, field nulls, and collective instabilities further limit the ideal picture. A calculated trapped fraction therefore describes accessible single-particle orbit classes under stated assumptions, not a demonstration of indefinitely stable plasma confinement or a predicted fusion performance.