E.5             Geomagnetic coordinate system – B and L

Geomagnetic coordinates are useful or necessary for a number of applications where charged particle morphology or behaviour needs to be described in the magnetosphere. The most important application is in models of the Earth’s radiation­belt environment (see clause 9). These particle models give fluxes of trapped energetic particles as functions of particle energy and of McIlwain’s geomagnetic co­ordinates L and B/B0[RD.102].

The kinetic energy of a charged particle trapped in a geomagnetic field model is conserved (a constant of motion) provided the B-field is stationary (independent of time), and provided that the acceleration by magnetospheric electric fields can be neglected. This is a satisfactory approximation for particles whose kinetic energy is larger than 500 keV.

When the kinetic energy of the particles is smaller than 500 MeV their motion can be described as the superposition of a gyration about the magnetic field lines, a latitudinal oscillation between two conjugate mirror points and an azimuthal drift around the Earth. Three adiabatic invariants (μ, I and Ф) can be associated respectively with these three periodic motions provided certain conditions are satisfied. The approximate conservation of the adiabatic invariants, contributes to the definition of invariant coordinates for mapping directional and omni-directional fluxes of particles trapped in the Earth’s Radiation Belts.

Two invariant coordinates, e.g. mirror point magnetic field (Bm) and I (E-4), are required to define a drift shell, i.e. the surface formed by the segments of geomagnetic field lines between conjugate mirror points of particles.

(E-4)

Bm  is the magnetic field strength at the mirror points (the low altitude edge of a drift shell); since the magnetic moment μ of trapped particles is the first adiabatic invariant of motion of trapped particles, Bm is also an invariant coordinate characterizing a drift shell;

I   is the second invariant coordinate required to identify uniquely the drift shell. The integral in (E-4) is evaluated along the field line between both conjugate mirror points l1 and l2 .

Since I is not a visually suggestive coordinate, McIwain [RD.101], [RD.102], [RD.118] introduced his L parameter which is approximately (but not exactly) equal to the equatorial distance of the magnetic field line passing across the point of an observational measurement.

When Bm is determined by using a geomagnetic field model (e.g. IGRF) the value of L is uniquely determined by the mathematical transformation (E-5):

(E-5)

In this equation I is computed by numerical integration of (E-4) ausing the same magnetic field model (IGRF).

Md                    = 31 165,3 nT RE3 is the fixed value of the magnetic moment of the reference dipole adopted in 1961 by McIwain to map the measured fluxes of trapped of radiation belt particles [RD.102], [RD.118].

L                       is the second invariant coordinate used (instead of I) to label drift shells.

The pair of invariant coordinates (Bm, L) uniquely defines a drift shell. It should be emphasized that different points along the same geomagnetic field lines may be characterized by different values L and by different drift shells.

The function f (E-5) was calculated by McIlwain [RD.102] and a simple approximation for f was found by Hilton [RD.103].

Note that a drift shell can also be characterized by Ф the third invariant of motion (the flux invariant), and an associated L* parameter. In general L and L* are not equal, except for a dipole geomagnetic field. However, the invariance of Ф or L* requires that the geomagnetic field distribution does not change significantly over a time period longer than the azimuthal drift period (> 10 minutes), while the adiabatic invariance of I or L requires that the B-field distribution is independent of time over only several bounce periods (> 1 second).

Other pairs of invariant coordinates derived from Bm and L have been proposed and happen to be more appropriate in certain cases: e.g. the invariant latitude (Λ), the invariant radius (R), or the invariant altitude (h_inv) [RD.120] which is quite convenient to bin/map fluxes measurements at low-altitudes in the Radiations Belts .