The magnetic gyrofrequency [[omega].sub.H] is a function of the height h above the ground and the geomagnetic colatitude [lambda] [7]:
Reference [8] describes the transformations between the geographic and geomagnetic coordinate systems, respectively, with colatitudes [theta] and [lambda] = [theta] - ([pi]/180[degrees])[DELTA][theta].
First of all we partitioned the phantom surfaces at grid points located at the colatitude angles [[theta].sub.1] = 0 < [[theta].sub.2] < *** < [[theta].sub.[upsilon]] < [[theta].sub.[[upsilon]+1]] = [pi] from the north pole to the south pole and azimuthal angles [[empty set].sub.1] = 0 < [[empty set].sub.2] < *** < [[empty set].sub.h] < [[empty set] sub.[h+1]] = 2[pi] around the phantom surface, where [upsilon] stands for vertical and h for horizontal.
In order to make this process more concrete, we have included a sample surface triangulation with [upsilon] = 5 colatitude angles and h = 8 azimuthal angles in Fig.
The colatitude of the bottom of the North polar cap, [[theta].sub.c], is the spherical radius of a spherical cap of area [V.sub.R].
The colatitude of top of the South polar cap is then [pi] - [[theta].sub.c].
Now note that we can apply a single [S.sup.d-1] rotation to 5d while keeping the [S.sup.d] colatitude fixed.
Spherical polar coordinates describe a point a on [S.sup.d] by using one longitude, [[alpha].sub.1] [member of ] R, considered modulo 2[pi], and d - 1 colatitudes, [[alpha].sub.k], for k [member of] {2, ...
We use [[delta].sub.F] to produce an increasing list of "fitting" colatitudes of caps, defined by
The area of each corresponding "fitting" collar is given by successive colatitudes in this list.