Invariant coordinates

McIlwain's (B, L) coordinates for a dipole field

In a dipole magnetic field, the total field strength is given by:

B = MR-3 [1+3 sin2(Lambda)]1/2

(1)

with

• Lambda: the magnetic latitude;
• R: the distance to the dipole centre;
• M: the magnetic moment.

The radial and latitudinal components of the field are given in polar coordinates by:

BR = 2 MR-3 sin(Lambda)	(2)
BLambda = - 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 B₀ = M L^-3 at the magnetic equator.

The equation for a dipole magnetic field line in spherical coordinates is:

[1/R][dR/d(Lambda)] = BR/BLambda = -2 sin(Lambda) /cos(Lambda)

(4)

Integration of this equation gives:

R = L cos2(Lambda)

(5)

R, Lambda coordinates

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 sin2(Lambda)]1/2 = MR-3 [ 4 - 3 RL-1]1/2	(6)
R = L cos2(Lambda)	(7)
B0 = 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:

(BB0-1)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.

References

McIlwain, C. E., Coordinates for Mapping the Distribution of Magnetically Trapped Particles, J. Geophys. Res., 66, pp. 3681-3691, 1961.