Modification of the loss cone for energetic particles

Abstract

The optimal pitch angle which maximizes the penetration distance, along the magnetic field, of relativistic charged particles injected from the midplane of an axisymmetric field is investigated analytically and numerically. Higher-order terms of the magnetic moment invariant are necessary to correctly determine the mirror point of trapped energetic particles, and therefore the loss cone. The modified loss cone resulting from the inclusion of higher-order terms is no longer entirely defined by the pitch angle but also by the phase angle of the particle at the point of injection. The optimal orientation of the injection has a nonzero component perpendicular to the magnetic field line, and is in the plane tangential to the flux surface. Numerical integration of particle orbits were carried out for a relativistic electron in a dipole field, showing agreement with analytic expressions. The results are relevant to experiments, which are concerned with injection of relativistic beams into the atmosphere from aboard a spacecraft in the magnetosphere.

Relativistic Particle Beams as a Resource to Solve Outstanding Problems in Space Physics

“…The denominator, L B , of the small parameter is the characteristic gradient length scale of the mangetic field L −1 B = |∇ lnB|. For relativistic electrons Porazik et al (2014) showed that higherorder terms of the magnetic moment invariant are necessary to correctly determine the mirror point of trapped energetic particles, and therefore the loss cone. Figure 6 (left) shows the pitch angles (δ) that would lead to precipitation for different azimuthal injection angles (λ) as a function of electron energy at 10 R e .…”

“…Edges of loss cones for (Middle) an electron initialized from 10 Re at the equatorial plane of a dipole field for different energies and (Right) for a 7-MeV electron initialized from different distances at the equatorial plane of a dipole field. The black dashed line corresponds to the unmodified loss cone for injection from the equatorial plane (Adapted from Porazik et al, 2014). FIGURE 7 | Solution to the envelope equations without beam perveance (blue lines).…”

Effect of Field-Line Curvature on the Ionospheric Accessibility of Relativistic Electron Beam Experiments

“…In this case, λ and β are azimuthal and lateral angles for a spherical coordinate system with the z-axis aligned with v 0 . As described by Porazik et al (2014), v 0 is not exactly aligned with the magnetic field and will always have a perpendicular component that is aligned with the drift. The transformation T is therefore simply a rotation of the system about the normal axisN by an angle α 0 such that cos(α 0 ) = v 0 · B/vB.…”

“…It is also suggested by simple linear analysis that relativistic beams traveling through the magnetosphere are stable to two-stream instabilities (Galvez and Borovsky, 1988), and relativistic beams entering the ionosphere are stable to resistive hose, ion hose, and filamentation instabilities (Gilchrist et al, 2001). Nevertheless, relativistic beams do come with their own issues, as discussed by Porazik et al (2014), due to the fact that the first adiabatic invariant will not necessarily be conserved to zeroth order. Using a second order asymptotic expansion derived by Gardner (1966), it was shown that the dependence on field-line curvature in the higher order terms of µ has a substantial effect on the loss cone.…”

“…For practical purposes, the use of relativistic electron beams with kinetic energy on the order of 1 MeV would be ideal for detectability. However, Porazik et al (2014) has shown that, for relativistic particles, higher order terms of the magnetic moment are necessary for consideration of the ionospheric accessibility of the beams. These higher order terms are related to gradients and curvature in the magnetic field and are typically unimportant unless the beam is injected along the magnetic field direction, such that the zero order magnetic moment is small.…”