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It is clear, by now, that there is an adiabatic invariant associated
with every periodic motion of a charged particle in
an electromagnetic field.
Now, we have just demonstrated that, as a consequence of conservation,
the drift orbit of a charged particle precessing around the Earth is approximately
closed, despite the fact that the Earth's magnetic field is nonaxisymmetric.
Thus, there must be a third adiabatic invariant associated with the
precession of particles around the Earth. Just as we can define a
guiding centre associated with a particle's gyromotion around fieldlines,
we can also define a bounce centre associated with a particle's bouncing
motion between mirror points. The bounce centre lies on the equatorial plane, and
orbits the Earth once every drift period, .
We can write the third adiabatic invariant as

(149) 
where the path of integration is the trajectory of the bounce centre around
the Earth. Note that the drift trajectory
effectively collapses onto the trajectory of the bounce centre
in the limit in which
all
of the particle's gyromotion and bounce motion averages to zero.
Now
is dominated by its second term,
since the drift velocity is very small. Thus,

(150) 
where is the total magnetic flux enclosed by the drift
trajectory (i.e., the flux enclosed by the orbit of the bounce centre
around the Earth). The above ``proof'' is, again, not particularly rigorousthe invariance of is demonstrated rigorously by Northrup.^{} Note, of course, that is only a constant
of the motion for particles trapped in the inner magnetosphere provided that the
magnetospheric magnetic field varies on timescales much longer than
the drift period, . Since the drift period for MeV energy
protons and electrons is of order an hour, this is only likely
to be the case when the magnetosphere is relatively quiescent (i.e.,
when there are no geomagnetic storms in progress).
The invariance of has interesting consequences for charged
particle dynamics in the Earth's inner magnetosphere. Suppose, for instance,
that the
strength of the solar wind were to increase slowly (i.e., on timescales
significantly longer than the drift period), thereby, compressing the Earth's
magnetic field. The invariance of would cause the charged particles
which constitute the Van Allen belts
to move radially inwards, towards the Earth, in order
to conserve the magnetic flux enclosed by their drift orbits. Likewise, a
slow decrease in the strength of the solar wind would cause an outward radial motion
of the Van Allen belts.
Next: Motion in Oscillating Fields
Up: Charged Particle Motion
Previous: Second Adiabatic Invariant
Richard Fitzpatrick
20110331