Van Allen Radiation Belts

There are, in fact, two radiation belts surrounding the Earth (Baumjohan and Treumann 1996). The inner belt, which extends from about 1-3 Earth radii in the equatorial plane, is mostly populated by protons with energies exceeding MeV. The origin of these protons is thought to be the decay of neutrons that are emitted from the Earth's atmosphere as it is bombarded by cosmic rays. The inner belt is fairly quiescent. Particles eventually escape due to collisions with neutral atoms in the upper atmosphere above the Earth's poles. However, such collisions are sufficiently uncommon that the lifetime of particles in the inner belt range from a few hours to 10 years. Obviously, with such long trapping times, only a small input rate of energetic particles is required to produce a region of intense radiation.

The outer belt, which extends from about 3-9 Earth radii in the equatorial plane, consists mostly of electrons with energies below MeV. These electrons originate via injection from the outer magnetosphere. Unlike the inner belt, the outer belt is very dynamic, changing on timescales of a few hours in response to perturbations emanating from the outer magnetosphere.

In regions not too far distant (that is, less than 10 Earth radii) from the Earth, the geomagnetic field can be approximated as a dipole field,

where we have adopted conventional spherical coordinates aligned with the Earth's dipole moment, whose magnitude is (Baumjohan and Treumann 1996). It is convenient to work in terms of the latitude, , rather than the polar angle, . An individual magnetic field-line satisfies the equation

where is the radial distance to the field-line in the equatorial plane ( ). It is conventional to label field-lines using the

where is the equatorial magnetic field-strength on the Earth's surface (Baumjohan and Treumann 1996).

Consider, for the sake of simplicity, charged particles located
on the equatorial plane (
)
whose velocities are predominately directed perpendicular to the magnetic
field. The proton and electron gyrofrequencies are written^{2.1}

(2.94) |

and

(2.95) |

respectively. The proton and electron gyroradii, expressed as fractions of the Earth's radius, take the form

(2.96) |

and

(2.97) |

respectively. Thus, MeV energy charged particles in the inner magnetosphere (that is, ) gyrate at frequencies that are much greater than the typical rate of change of the magnetic field (which varies on timescales that are, at most, a few minutes). Likewise, the gyroradii of such particles are much smaller than the typical variation lengthscale of the magnetospheric magnetic field. Under these circumstances, we expect the magnetic moment to be a conserved quantity. In other words, we expect the magnetic moment to be a good adiabatic invariant. It immediately follows that any MeV energy protons and electrons in the inner magnetosphere that have a sufficiently large magnetic moment are trapped on the dipolar field-lines of the Earth's magnetic field, bouncing back and forth between mirror points located just above the Earth's poles.

It is helpful to define the *pitch-angle*,

(2.98) |

of a charged particle in the magnetosphere. If the magnetic moment is a conserved quantity then a particle of fixed energy drifting along a field-line satisfies

where is the

The mirror points correspond to (i.e., ). It follows from Equations (2.93) and (2.99) that

where is the magnetic field-strength at the mirror points, and the latitude of the mirror points. It can be seen that the latitude of a particle's mirror point depends only on its equatorial pitch-angle, and is independent of the -value of the field-line on which it is trapped.

Charged particles with large equatorial pitch-angles have small parallel
velocities, and mirror points located at relatively low latitudes. Conversely,
charged particles with small equatorial pitch-angles have large parallel velocities,
and mirror points located at high latitudes. Of course, if the pitch-angle
becomes too small then the mirror points enter the Earth's atmosphere, and
the particles are lost via collisions with neutral particles.
Neglecting the thickness of the atmosphere with respect to
the radius of the Earth, we can say that all particles whose mirror points
lie inside the Earth are lost via collisions. It follows from
Equation (2.100) that the *equatorial loss cone* is of approximate width

(2.101) |

where is the latitude of the point at which the magnetic field-line under investigation intersects the Earth. All particles with and lie in the loss cone. According to Equation (2.92),

(2.102) |

It follows that

(2.103) |

Thus, the width of the loss cone is independent of the charge, the mass, or the energy of the particles drifting along a given field-line, and is a function only of the field-line radius on the equatorial plane. The loss cone is surprisingly small. For instance, at the radius of a geostationary satellite orbit ( ), the loss cone is less than wide. The smallness of the loss cone is a consequence of the very strong variation of the magnetic field-strength along field-lines in a dipole field. [See Equations (2.90) and (2.93).]

The *bounce period*,
, is the time it takes a charged particle to move
from the equatorial plane to one mirror point, through the equatorial plane to the other mirror point, and then back
to the equatorial plane. It follows that

(2.104) |

where is an element of arc-length along the field-line under investigation, and . The previous integral cannot be performed analytically. However, it can be solved numerically, and is conveniently approximated as (Baumjohan and Treumann 1996)

(2.105) |

Thus, for protons

(2.106) |

while for electrons

(2.107) |

It follows that MeV electrons typically have bounce periods that are less than a second, whereas the bounce periods for MeV protons usually lie in the range 1 to 10 seconds. The bounce period only depends weakly on equatorial pitch-angle, because particles with small pitch angles have relatively large parallel velocities but a comparatively long way to travel to their mirror points, and vice versa. Naturally, the bounce period is longer for longer field-lines (that is, for larger ).