L-shell
The L-shell, L-value, or McIlwain L-parameter (after Carl McIlwain) is a parameter describing a particular set of planetary magnetic field lines. Colloquially, L-value often describes the set of magnetic field lines which cross the Earth's magnetic equator at a number of Earth-radii equal to the L-value. For example, "" describes the set of the Earth's magnetic field lines which cross the Earth's magnetic equator two earth radii from the center of the Earth. L-shell parameters can also describe the magnetic fields of other planets. In such cases, the parameter is renormalized for that planet's radius and magnetic field model.
Although L-value is formally defined in terms of the Earth's true instantaneous magnetic field (or a high-order model like IGRF), it is often used to give a general picture of magnetic phenomena near the Earth, in which case it can be approximated using the dipole model of the Earth's magnetic field.
The motions of low-energy charged particles in the Earth's magnetic field (or in any nearly-dipolar magnetic field) can be usefully described in terms of McIlwain's (B,L) coordinates, the first of which, B is just the magnitude (or length) of the magnetic field vector. 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.
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