Whipple's Second Equilibrium

A second solution is possible if one recognizes that the RHS of equation (9) can be very large as $q\Phi \rightarrow (K - \mu B)$. In this case we need to expand the exponentials around $y \equiv (K_b-\mu_b B)/kT - x$ giving,

$$e^{-y}e^{K_{\vert\vert}/kT}=(1-r)e^{y}e^{-K_{\vert\vert}/kT} + \frac{rB}{B_0} \sqrt{\frac{K_{\vert\vert}}{y kT}}$$ (14)

$$\gamma \equiv e^{2K_{\vert\vert}/kT}$$ (15)

$$\frac{r^2 B^2 K}{B^2_0 kT} \cong y [\gamma + 2r-2 +(r-1)^2/\gamma]$$ (16)

$$\frac{r^2 B^2 K}{B^2_0 kT\gamma}\cong y=\frac{K_{\vert\vert}}{kT} - x$$ (17)

where we have kept only first order in $y$. Since $\gamma \gg 1$, the first term in the denominator of the RHS of equation (16) is the only contributer. This leads to,

$$x=\frac{K_{\vert\vert}}{kT}\left(1-\frac{r^2B^2}{B^2_0}\exp(-2K_{\vert\vert}/kT)\right)$$ (18)

$$q\Phi \cong K_{\vert\vert} = (K_b - \mu_b B)$$ (19)

which is a good approximation since the last term is very small. Note that the potential is a strong function of the parallel energy of the ions, $K_{\vert\vert}=K_b - \mu_b B$, and vanishes not at the equator, but at the mirror point of the ions. One can heuristically understand this potential as a maximum at the equator to retard the speed of the ions, and thereby spread the distribution of ions more evenly along the field line rather than concentrate the distribution at the mirror points, (e.g., a square-wave spatial ion distribution that minimizes the self-energy). Since the potential can be offset by a scalar without loss of generality, we can select the offset to be the equatorial parallel energy, $K_b - \mu_b B_0$. If we then plug in this potential for the LHS of equation (9) we retrieve a reasonable density of electrons at the equator, and an exponentially decreasing density as we move away from the equator. Once we have travelled a short distance from the equator, we find unshielded ions around the mirror points resulting in the establishment of a space charge potential.

The transition from solution 1 to solution 2 is rather abrupt, since the potential jumps from a few volts to a few kV discontinuously. That is, these are the only two static equilibrium potential solutions, so the intermediate potentials are dynamic, nonequilibrium voltages. One can understand these two states by considering the fate of the cold plasma in these two equilibria. After the initial appearance of hot ions on a particular flux tube, the first eV solution is initially found by the cold plasma, which attempts to shield the hot ions around their density spikes at the mirror point. When the flux tube runs out of cold electrons, occuring first near the equator, the potential rapidly jumps to the second solution as a wave of high space charge potential radiates outward from the equator where the hot ions are ``stripped'' of their shielding electrons.

We call this violation of quasi-neutrality the ``quasi-neutrality catastrophe'' (QNC). Note that in this second equilibrium the parallel electric field is opposed to the mirror force, and therefore attempts to exclude the ions from the equator. Integrating the electric field from the equator to the mirror point shows that the total potential drop, $\Phi \propto K_{0\vert\vert}$, where $K_{0\vert\vert}$ is the parallel component of the kinetic energy at the equator. In other words, QNC is a transducer converting parallel hot ion energy into parallel potential, which can be much larger than the cold electron thermal energy.