Third Adiabatic Invariant

It is clear, by now, that there is an adiabatic invariant associated with every periodic motion of a charged particle in an electromagnetic field. We have just demonstrated that, as a consequence of $J$-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 non-axisymmetric. Thus, there must be a third adiabatic invariant associated with the precession of particles around the Earth. Just as we can define a guiding center associated with a particle's gyromotion around field-lines, we can also define a bounce center associated with a particle's bouncing motion between mirror points. The bounce center lies on the equatorial plane, and orbits the Earth once every drift period, $\tau_d$. We can write the third adiabatic invariant as

$$K \simeq \oint p_\phi\,ds,$$ (2.119)

where the path of integration is the trajectory of the bounce center around the Earth. Incidentally, the drift trajectory effectively collapses onto the trajectory of the bounce center in the limit that $\rho/L\rightarrow 0$, because all of the particle's gyromotion and bounce motion averages to zero. Now, $p_\phi = m\,v_\phi + e\,A_\phi$ is dominated by its second term, as the drift velocity $v_\phi$ is very small. Thus,

$$K \simeq e\oint A_\phi\,ds = e\,{\mit\Psi},$$ (2.120)

where ${\mit\Psi}$ is the total magnetic flux enclosed by the drift trajectory (that is, the flux enclosed by the orbit of the bounce center around the Earth). The previous “proof” of the invariance of ${\mit\Psi}$ is, again, not particularly rigorous. In fact, the invariance of ${\mit\Psi}$ was first demonstrated rigorously by Northrop (Northrop 1963). Of course, ${\mit\Psi}$ 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 drift period, $\tau_d$. Because 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 (in other words, when there are no geomagnetic storms in progress).

The invariance of ${\mit\Psi}$ 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 (that is, on timescales significantly longer than the drift period), thereby, compressing the Earth's magnetic field. The invariance of ${\mit\Psi}$ would cause the charged particles that constitute the Van Allen belts to move radially inwards, toward 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.