Second Adiabatic Invariant
We have seen that there is an adiabatic invariant associated with the
periodic gyration of a charged particle around magnetic fieldlines.
Thus, it is reasonable to suppose that there is a second adiabatic invariant
associated with the periodic bouncing motion of a particle trapped
between two mirror points on
a magnetic fieldline. This is indeed the case.
Recall that an adiabatic invariant is the lowest order approximation
to a PoincarĂ© invariant:

(2.115) 
In this case, let the curve correspond to the trajectory of
a guiding center as a charged
particle trapped in the
Earth's magnetic field
executes a bounce orbit. Of course, this trajectory does not quite close,
because of the slow azimuthal drift of particles around the Earth. However,
it is easily demonstrated that
the azimuthal displacement of the end point of the trajectory, with respect to
the beginning point, is similar in magnitude to the gyroradius. Thus, in the limit
in which the ratio of the gyroradius, , to the variation lengthscale of the
magnetic field, , tends to zero, the trajectory of the guiding center
can be regarded as being approximately closed,
and the actual particle trajectory conforms very closely
to that of the guiding center. Thus, the adiabatic invariant associated with
the bounce motion can be written

(2.116) 
where the path of integration is along a fieldline, from the equatorial plane to
the upper mirror point, back along the fieldline to the lower mirror point, and
then back
to the equatorial plane. Furthermore, is an element of arclength along the
fieldline, and
.
Using
, the previous
expression yields

(2.117) 
Here,
is the total magnetic flux enclosed by the curve—which,
in this case, is obviously zero. Thus, the socalled second adiabatic invariant, or
longitudinal adiabatic invariant, takes the form

(2.118) 
In other words, the second invariant is proportional to the loop integral
of the parallel (to the magnetic field) velocity taken over a bounce orbit.
The preceding “proof” of the invariance of is not particularly rigorous. In fact, the rigorous proof
that is an adiabatic invariant was first given by Northrop and Teller (Northrop and Teller 1960). Of course, is only a constant of the motion
for particles trapped in the inner magnetosphere provided the
magnetospheric
magnetic field varies on timescales that are much longer than the bounce time,
. Because the bounce time for MeV energy protons and electrons is,
at most, a few seconds, this is not a particularly onerous constraint.
Figure 2.4:
Schematic diagram showing the distortion of the Earth's magnetic field by the solar wind.

The invariance of is of great importance for charged particle
dynamics in the Earth's inner magnetosphere. It turns out that the
Earth's magnetic field is distorted from pure axisymmetry by the action of
the solar wind, as illustrated in Figure 2.4. Because of this asymmetry, there
is no particular reason to believe that a particle will return to its
earlier trajectory as it makes a full circuit around the Earth. In other words,
the particle may well end up on a different fieldline when it returns to
the same azimuthal angle. However, at a given azimuthal angle, each
fieldline has a different length between mirror points, and a different
variation of the fieldstrength, , between the mirror points (for a particle
with given energy, , and magnetic moment, ). Thus, each fieldline
represents a different value of for that particle. So, if is
conserved, as well as and , then the particle
must return to the same fieldline after precessing around the Earth.
In other words, the conservation of prevents charged particles from
spiraling radially in or out of the Van Allen belts as they rotate around the Earth.
This helps to explain the persistence of these belts.