L-shell

[2] This description is most valuable when the gyroradius of the charged particle orbit is small compared to the spatial scale for changes in the field.

Then a charged particle will basically follow a helical path orbiting the local field line.

In a local coordinate system {x,y,z}  where z  is along the field, the transverse motion will be nearly a circle, orbiting the "guiding center", that is the center of the orbit or the local B  line, with the gyroradius and frequency characteristic of cyclotron motion for the field strength, while the simultaneous motion along z  will be at nearly uniform velocity, since the component of the Lorentz force along the field line is zero.

Since the Lorentz force is strictly perpendicular to the velocity, it cannot change the energy of a charged particle moving in it.

Then the longitudinal motion is stopped and reversed, and the particle is reflected back towards regions of weaker field, the guiding center now retracing its previous motion along the field line, with the particle's transverse velocity decreasing and its longitudinal velocity increasing.

The result is that, as the particle orbits its guiding center on the field line, it bounces back and forth between the north mirror point and the south mirror point, remaining approximately on the same field line.

Particles with too-small pitch angles may strike the top of the atmosphere if they are not mirrored before their field line reaches too close to the Earth, in which case they will eventually be scattered by atoms in the air, lose energy, and be lost from the belts.

[4] However, for particles which mirror at safe altitudes, (in yet a further level of approximation) the fact that the field generally increases towards the center of the Earth means that the curvature on the side of the orbit nearest the Earth is somewhat greater than on the opposite side, so that the orbit has a slightly non-circular, with a (prolate) cycloidal shape, and the guiding center slowly moves perpendicular both to the field line and to the radial direction.

The guiding center of the cyclotron orbit, instead of moving exactly along the field line, therefore drifts slowly east or west (depending on the sign of the charge of the particle), and the local field line connecting the two mirror points at any moment, slowly sweeps out a surface connecting them as it moves in longitude.

The L  parameter is traditionally labeled in Earth-radii, of the point where the shell crosses the magnetic Equator, of the equivalent dipole.

For instance, auroral light displays are most common around L=6, can reach L=4 during moderate disturbances, and during the most severe geomagnetic storms, may approach L=2.

Plot showing field lines (which, in three dimensions would describe "shells") for L-values 1.5, 2, 3, 4 and 5 using a dipole model of the Earth's magnetic field
Map of L-shell field line locations on the surface of the Earth. The real terrestrial field is approximately dipolar, but misaligned with the rotation axis, and offset a few hundred km in the direction opposite to the South Atlantic Anomaly .