McCollough et al.  suggested a means for temperature anisotropy to develop in response to a bifurcated Bmin plane that assumed qualitatively different particle behaviors for particles of different initial equatorial pitch angle. This is supported by the densities plotted in Figure 5: the density on the left is for lower initial equatorial pitch angles (αeq0 < 30°), and is almost exclusively confined to small radial distances where the field minima do not bifurcate (i.e., there is one minimum along a field line), while the density on the right (αeq0 > 60°) extends along the bifurcated Bmin-plane to higher latitudes.
 With the analytic field, we followed test particles with different initial equatorial pitch angles and computed several quantities of interest: the kinetic energy KE of the particle, the first adiabatic invariant μ, the field-geometric integral I = J/2p (with J the second invariant and p the mechanical momentum), and the magnetic field along the guiding-center field line. On the basis of these studies, we separated the particles into four types, according to the characteristics of their trajectories. Each type and its role in the resulting temperature anisotropy is described below, in the context of sample trajectories of particles launched from the same nightside location (R0 = 5.5, ϕ0 = 150°) with different pitch angles.
3.1. Non-Shabansky Particles
 Non-Shabansky particles execute dayside drifts without passing through the bifurcated region of the magnetic field. A non-Shabansky particle trajectory in a b2 = 8 field is displayed along with I in Figure 7. The first invariant μ = 31 MeV/G is conserved within 10% and the kinetic energy KE = 200 keV is conserved within 0.02%. Thus I, μ, and KE are all conserved for this particle throughout the simulation. The drift-periodic “fuzziness” in I and μ is due to strong magnetic field curvature near midnight [Young et al., 2002], and the small-amplitude higher-frequency fluctuations are due to numerical errors in calculating μ. The only way these particles contribute to anisotropy is through DSS-induced anisotropy, since their motion is adiabatic. This is due to the particle drifting without passing through the bifurcated fields.
Figure 7. I, trajectory projections, and magnetic field profiles for a non-Shabansky particle. The mirror field is indicated by the dashed line in the field profile.
Download figure to PowerPoint
 Figure 7 shows the magnetic field profile just outside (Figure 7, left) and just inside (Figure 7, right) the Shabansky region. The mirror field strength,
is indicated by the dashed line. The last expression is only accurate for nonrelativistic energies, which is an appropriate approximation for the populations used here. Recalling the expression for the mirror force [Northrop, 1963]:
it is obvious that the field strength along a field line B(s) functions as a potential, so in the lower portion of Figure 7, a particle will move along the field line up to Bm, where the particle slows to a stop and turns around, as in a graviational well. It is clear from this that particle motion in the Shabansky region is unchanged in the adiabatic sense from motion outside the region, since there is only one minimum and it is at the equator.
3.2. Shabansky I Particles
 Shabansky I particles pass through both hemispheres along field lines with bifurcated minima in the Shabansky region. A Shabansky I particle trajectory is shown in Figure 8. The first invariant μ = 125 MeV/G is conserved within 10% and the kinetic energy KE = 200 keV is conserved within 0.02%. Thus like the non-Shabansky particle, this particle also conserves I, μ, and KE. I has some short dips in the Shabansky region (between times t = 20 min and t = 30 min) that correspond to numerical limitations of computing I. These artifacts are due to the fact that the mirror field strength for this particle is only slightly greater than the field strength at the equator. The small bounce-frequency error in μ mentioned in section 3.1 is reflected in the Bm value. Near the equator this Bm can fall very close or even below the total field strength, which leads to an incorrect value for I near zero. It is apparent, aside from this numerical artifact, that I is conserved throughout the simulation period.
Figure 8. I, trajectory projections, and magnetic field profiles for a Shabansky I particle. The mirror field is indicated by the dashed line in the field profile.
Download figure to PowerPoint
 Although I, μ, and KE are conserved, these particles contribute to the temperature anisotropy near the equator, which can be understood by considering the following. Figure 8 (bottom) shows the magnetic field profile for a Shabansky I particle in the same manner as Figure 7. Inside the Shabansky region, there is a “hump” near the equator where magnetic field increases along the field line. As a particle bounces along this field line near the equator, it transfers parallel energy into perpendicular energy in order to conserve μ and KE, since the magnetic field increases to a local maximum at the equator. All Shabansky I particles must do this, and the result is a net increase in perpendicular temperature at the cost of parallel temperature. Thus a temperature anisotropy arises near the equator in the Shabansky region. In addition, since these particles are conserving μ and I, they will contribute to DSS-induced anisotropy.
3.3. Shabansky II Particles
 Shabansky II particles execute “classic” Shabansky orbits, spending most of their time in the Shabansky region at high latitudes in one hemisphere. Figure 9 displays a Shabansky II trajectory. The first invariant μ = 155 MeV/G is conserved within 10% and the kinetic energy KE = 200 keV is conserved within 0.02%. A key feature of Shabansky II orbits is the breaking of the second adiabatic invariant, evidenced by the sudden change in value of I as the particle enters (near t = 13 min) and leaves (t = 28 min) the Shabansky region. This shift to a lower value of I, and thus J, indicates a loss of parallel momentum (since J = ∮ p∥ds) during the off-equatorial portion of the particle trajectory; since KE is conserved, this implies a shift in energy from the parallel component to the perpendicular component.
Figure 9. I, trajectory projections, and magnetic field profiles for a Shabansky II particle. The mirror field is indicated by the dashed line in the field profile.
Download figure to PowerPoint
 The loss of p∥ can be thought of in another way: since μ and KE are conserved and these ions are at nonrelativistic energies, Bm is constant (see equation (8)). Figure 9 shows that as the particle moves into the Shabansky region, it is forced into a region with higher Bmin than before. This results in a higher minimum pitch angle and thus more perpendicular energy at the cost of parallel energy.
 Shao et al.  studied the time-evolution of J for warm protons in LFM-generated electromagnetic fields and saw similar features. In addition, we have noticed that these Shabansky II particles tend to have approximately the same I value after leaving the Shabansky region as before entering, which is different from Shabansky III particles. This “conservation” of I in a drift sense was also noted by Shao et al. .
 The breaking of the second invariant can be seen clearly in Figure 9 (bottom). Outside the Shabansky region, the particle mirrors in a “single well”. Inside, it finds itself stuck in one of two wells, with the mirror field closer to the minimum field strength. This lowers J, which leads to a transfer of parallel energy to perpendicular energy and thus increased temperature anisotropy.
3.4. Shabansky III Particles
 Shabansky III particles are initially near-equatorial (αeq0 ≳ 85°) particles that undergo Shabansky orbits but do not enhance anisotropy. A Shabansky III trajectory is shown in Figure 10. The first invariant μ = 157 MeV/G is conserved within 10% and the kinetic energy KE = 200 keV is conserved within 0.02%. The key difference between type III and type II particles is what happens to I as the particle enters the Shabansky region. While a type II particle lowers I as it enters the Shabansky region, these near-equatorial particles have such small initial I values that I is no longer lowered but is broken and can increase in value or stay close to the same value as before. This change in behavior can be thought of as a result of a “zero bound problem”: I cannot be negative. Öztürk and Wolf  discussed this distinction between “equatorial” and “nonequatorial” particles, corresponding to types III and II, respectively.
Figure 10. I, trajectory projections, and magnetic field profiles for a Shabansky III particle. The mirror field is indicated by the dashed line in the field profile.
Download figure to PowerPoint
 The difference between types III and II is evident in Figure 10. Similar to type II particles (Figure 9), these particles find themselves trapped in one well inside the Shabansky region. In this case, Bmin is lowered, so they can have parallel momentum (or KE) enhanced.
 Breaking I leads to breaking the third adiabatic invariant which keeps μ and KE fixed. Evidence of this is seen in the SM X-Y projection of the trajectory in Figure 10. One can see on the second orbit that the guiding center on the nightside is further out in radius. This is an explanation for why Bmin is raised for type II particles and lowered for type III particles: To keep μ fixed, type II particles must move inward in L*, corresponding to stronger fields, and type III particles must move outward in L*, corresponding to weaker fields. By increasing their I value significantly in the Shabansky region, type III particles can decrease the anisotropy there.
 The lack of “conserving” I in the sense of section 3.3 appears to be a diffusive process. By breaking I, the particles must break the third invariant as well; in an asymmetric field this will lead to radial diffusion. Diffusion in phase space is an important component in the dynamics of energetic partices in the magnetosphere and can be responsible for energization and loss (see Schulz and Lanzerotti  for a detailed treatment). The implications and study of this potentially new diffusive process is the subject of future research.
3.5. Features of Anisotropy Profiles
 To summarize the different particle behaviors, we have the following types:
 1. Non-Shabansky particles execute dayside drifts without passing through the bifurcated region of the magnetic field. They conserve μ, I, and KE throughout an orbit.
 2. Shabansky I particles pass through both hemispheres in the presence of a bifurcated field. They also conserve μ, I, and KE throughout an orbit.
 3. Shabansky II particles stay in one hemisphere while drifting through the bifurcation. They conserve μ and KE. I is lower while in the Shabansky region than during the rest of the drift motion. The value of I returns to its approximate initial value upon exiting the Shabansky region.
 4. Shabansky III particles also stay in one hemisphere while drifting through the bifurcation but have an increased I while in the Shabansky region. They do not recover the initial I value after executing a complete orbit.
 A key discriminator between different Shabansky types is the initial value of I: type I have the largest I (corresponding to lower values of αeq0), type II have smaller values, and type III have near-zero values. We can thus make the correspondence between low, high, and near-equatorial αeq0 with Shabansky I, II, and III, respectively.
 With both Shabansky II and III particles present, it is prudent to ask if the net effect is an increase in anisotropy. The answer comes from the amount of type III particles relative to type II particles: with an initially isotropic distribution, the number of particles is constant for different values of αeq0. The number of particles that would undergo type II particle motions is significantly larger than the near-equatorial range required for type III behavior, so the total anisotropy increases in these off-equatorial regions.
 Looking back at the left-hand sides of Figures 2–4, we can begin to understand how the anisotropy arises. We can see the anisotropy due to the lack of isotropizing process in the inner nonbifurcated region, getting weaker at further radial distances. In addition, Figures 3 and 4 show anisotropy in the bifurcated region and at high latitudes for bifurcated field lines due to Shabansky II particles. Figure 4 shows anisotropy near the equator in the bifurcated region from Shabansky I “hump”-induced anisotropy. This is happening for b2 = 3, but is off the plotting area of Figure 3.
 The right-hand side, with DSS-induced anisotropy “switched on” using AP-8 instead of a flat flux profile, enhances the anisotropy in a straightforward manner. For the b2 = 1 case (Figure 2), almost all the anisotropy at higher radial distances is DSS-induced and is centered at the equator as one would expect. In Figures 3 and 4, it is easy to see the DSS-induced anisotropy enhances the total anisotropy at all radial distances near the equator, but the anisotropy seems to decrease slightly at very high latitudes along the bifurcated Bmin-plane. This is likely due to DSS occurring outside the Shabansky region, which is most significant for particles with the highest pitch angles and the largest drift radii. Because of this, the particles that find themselves at the highest latitudes in the XZ-plane are disproportionately Shabansky III particles which do not increase the anisotropy.