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Background Information | Background Information | |
Invariant coordinates |
In a dipole magnetic field, the total field strength is given by:
B = MR^{-3} [1+3 sin^{2}(Lambda)]^{1/2} | (1) |
The radial and latitudinal components of the field are given in polar coordinates by:
B_{R} = 2 MR^{-3} sin(Lambda) | (2) |
B_{Lambda} = - MR^{-3} cos(Lambda) | (3) |
By definition, McIlwain's (1961) L-parameter for a dipole field is the radial distance of the intersection of the field line with the magnetic equator; (Note: this is not true for a real field, where L is defined by means of a function of the adiabatic integral invariant I). In this definition of L, McIlwain used the value 0.311653 gauss R_{E} for M.
B is the magnetic field strength, determining the position along the fieldline from the minimum B_{0} = M L^{-3} at the magnetic equator.
The equation for a dipole magnetic field line in spherical coordinates is:
[1/R][dR/d(Lambda)] = B_{R}/B_{Lambda} = -2 sin(Lambda) /cos(Lambda) | (4) |
Integration of this equation gives:
R = L cos^{2}(Lambda) | (5) |
In a non-dipole magnetic field, L is defined as a function of the adiabatic integral invariant I. R, Lambda coordinates can now be defined as:
B = MR^{-3} [1+3 sin^{2}(Lambda)]^{1/2} = MR^{-3} [ 4 - 3 RL^{-1}]^{1/2} | (6) |
R = L cos^{2}(Lambda) | (7) |
B_{0} = ML^{-3} | (8) |
It is clear that it is easy to compute B and L given R and Lambda. The inverse transformation however can not be accomplished in terms of well-known functions.
Substituting Equations 8 and 9 in Equation 7 gives:
(BB_{0}^{-1})^{2} (RL^{-1})^{6} - (4 - 3 RL^{-1}) = 0 | (9) |
This equation can be solved numerically (e.g. with the Newton-Raphson algorithm) for RL^{-1}. Substitution of RL^{-1} in Equation 8 then gives Lambda.
Note that in a non-dipolar field R and Lambda are different from the radial distance and the magnetic latitude, respectively.
Figure 1. Invariant coordinate map of the AE-8 MAX electron flux distribution for E > 2 MeV. |
McIlwain, C. E., Coordinates for Mapping the Distribution of Magnetically Trapped Particles, J. Geophys. Res., 66, pp. 3681-3691, 1961.