Threshold energy for confinement of charged particles by an axisymmetric magnetic field.

Motion of the radiation belt particles is confined to a certain area of space by the Earth magnetic field. It may be expected that this confinement is only possible for the particle energies below some threshold, which depends on the magnetic field. Here we study this threshold energy in an axisymmetric longitudinal magnetic field with power law dependence on the radius. A particular example of such fields is the equatorial plane of a magnetic dipole. We derive a transcendental equation for the threshold speed corresponding to the transition between bounded and unbounded trajectories of the particles. The equation for threshold speed can be solved exactly for several specific values of the power exponent, including that corresponding to a magnetic dipole, but in general it requires a numerical treatment. We find that the threshold energy depends on the direction of the initial velocity, and this dependence becomes more prominent as the exponent of the power law increases. Remarkably, if the magnetic field decreases slower than the inverse of the radius, charged particles remain confined no matter how large their energies may be. The results allow us to calculate the theoretical upper boundaries for the possible energies of radiation belt particles.