Trapped Radiation and the South Atlantic Anomaly
It is difficult for any electrically charged particles originating from outside of the Earth's magnetosphere to enter inside it, as they tend to be deflected away via the Lorentz force, i.e. the force F on a moving charge with a velocity v in a magnetic field of strength B is given by F = qvB with a direction orthogonal to both v and B.
However, the tendency to be deflected is opposed to some extent by the particles' momentum. Thus, the ability of a particle to penetrate into the geomagnetic field actually depends upon a quantity called the particle's magnetic rigidity, P:
The rigidity parameter is extremely useful in describing the motion of particles in the geomagnetic field. This is because particles injected into the field with the same rigidity will follow identical trajectories, whereas particles with the same momentum or energy, but different charges, will not.
For each point in the magnetosphere there will be a minimum rigidity (called the cut-off or threshold rigidity) required to reach that point. Particles with less rigidity than the cut-off will be deflected before they reach the point, whereas those with more than the cut-off will penetrate to it.
The cut-off rigidity required for a particle to reach a given position above the Earth, ie. a point with a magnetic latitude , and geocentric radius R, can be easily estimated using an equation called Störmer's theory:
Several very important conclusions about the distribution of particles and their energies in the magnetosphere can be drawn from this equation. For a particle to penetrate the Earth's field successfully, the cut-off rigidity must be low. The equation above shows that for vertical arrival of the particle (a = 90°), the cut-off rigidity goes as , where is the magnetic latitude. Thus, it is easier for particles to penetrate at high magnetic latitudes (where is minimised) than near to the magnetic equator. The equation also shows the asymmetry in cut-off rigidity with respect to arrival direction. For example, for a positive ion, it is easiest to penetrate from the West (a = 0°).
Cut-off rigidity is also inversely proportional to the square of geocentric radius. Therefore, at a given latitude, penetration to lower altitudes requires a greater rigidity. In other words, at a given latitude, the particles with the highest values of rigidity will be at the lowest altitude, and the particles of lowest rigidity will be at the highest altitude.
The most shielded location lies on the magnetic equator at the Earth's surface. For a positive ion arriving here (ignoring the atmosphere) from the least favoured direction, East (a = 180°), the cut-off rigidity has a maximum value: