A Scaleable Model of Magnetospheric Radiation Belt Dynamics Robert Sheldon, Wheaton College TOP SIDE t=0 Double Dipole, Asymmetric Bias, 100mTorr If instead of giving both magnets the same –400V bias, we lower one magnet to – 100V or ground, the following configuration is observed: Plasma generated by electrons on one magnet, feed into other trapping field due to diffusion though the separatrix or "x-line“, since the plasma is highly diffusive at 100mTorr. Like northward Bz conditions at Earth, this feeding happens at the cusps, as predicted by Siscoe et al. The cusps themselves hold the plasma long enough to glow, showing that the cusp can trap particles itself, not just migrating equatorial particles. That is, not only Shabansky orbits, but Sheldon orbits are supported by the cusps. Are these cusp trapped particles seen in a real Magnetosphere? We believe they are, and in fact, account for an important heated/accelerated plasma population. POLAR/CAMMICE data, Sheldon et al. GRL [1998a] TOP View 0V/-400V SIDE View, 0V/-400V I Tracing Electrons in the Cusp (Sheldon et al [1998b]) Sheldon et al., (GRL 98) traced 1MeV electrons through cusp to demonstrate that they possessed 3 adiabatic invariants, and were permanently trapped, as expected in a quadrupole trap. (A Nobel prize was awarded for trapping single ions in a quadrupole trap.) But can this trap also accelerate them? - Two Nd-B magnets are used with dipoles parallel. This produces a “mirror image” or bi-dipole magnetic configuration as described by Maxwell and Chapman. Both magnets were biassed at 400V, and produced annular plasma traps with a separatrix between them. The cause for this annular plasma glow we believe to be the DC anode glow as modified by a strong magnetic field. The theory, described in Sheldon et al [2001] suggests that space charge may be present in the anode glow resulting from the pancake ions neutralized by streaming electrons. Whatever the cause, the magnetized plasma of both dipoles interact, showing a bend or flip of the equatorial plane. We interpret this flip to be Shabansky orbits, with equatorial trapped ions moving into one or the other high latitude minima (cusps). In our system, this is a bistable state, and can change from N to S poles as the anode glow fluctuates. The view from the top is instructive. The cusp traps are not as continuous with the equatorial dipole trap as expected from the side views, nor are they symmetric. In hindsight, there is every expectation that as the plasma beta increases in these traps they would naturally become asymmetric. Abstract The magnetospheric cusps, present in every magnetosphere in the solar system, are an intriguing quadrupole trap with properties that promote rapid acceleration of trapped solar wind particles by stochastic acceleration. Unlike the dipole trap, the quadrupole trap has the periods of its three invariants very closely spaced, allowing rapid diffusion through phase space via an Arnol'd web. In addition, the weak fields of the cusp allow impulsive entry that greatly enhance the quadrupole trap, and may initiate the positive feedback filling of the trap. These ubiquitous properties lend themselves to scaling laws applicable to all solar system magnetospheres. Normalizing to the Earth's magnetosphere, we predict the properties of the radiation belts of Mercury and the outer planets, and compare these predictions to the observations. As expected, Venus and Mars are predicted to lack radiation belts entirely. History and Problem The origin of energetic particles in magnetospheres is not necessarily a solved problem. Generally the tail is taken to be the acceleration region and subsequent adiabatic compression produces the radiation belts, but actual mechanisms that energize solar wind from 1keV/nuc to 10’s or 100’s of keV (before adiabatic compression) are all speculative and/or inefficient. For example, Speiser orbits occupy a region of phase space that is a set of measure zero, so that the efficiency of such a mechanism for accelerating particles in the Earth’s tail is very minimal. Likewise, shock acceleration only works on a very finite plane in phase space, which also approaches a set of measure zero. In order to achieve rapid, efficient acceleration, we need a mechanism that operates on a large volume in phase space. Recent discoveries at Earth provide a scalable model for hot plasma at all magnetospheres, that we use to predict plasma energies. POLAR discovered a high-altitude MeV electron population trapped in the cusp (Sheldon et al. GRL [1998]). POLAR also saw energetic ions in diamagnetic cavities in the cusp as well (Chen et al. JGR 1998) We scale up these cavities to other magnetospheres to estimate the energy spectrum observed in the outer planets that might accrue from such a mechanism. Maxwell (~1870) / Chapman (1932) The origin of energetic particles in magnetospheres is not necessarily a solved problem. Generally the tail is taken to be the acceleration region and subsequent adiabatic compression produces the radiation belts, but actual mechanisms that energize solar wind from 1keV/nuc to 10’s or 100’s of keV (before adiabatic compression) are all speculative and/or inefficient. For example, Speiser orbits occupy a region of phase space that is a set of measure zero since they have very precise position/velocity requirements that map to a surface in phase space. This reduces the efficiency of such a mechanism for accelerating particles. What we need is a mechanism that accelerates a volume in phase space. Recent discoveries at Earth provide a scalable model for hot plasma at all magnetospheres, that we use to predict plasma energies at all the planets. The model is based on a POLAR discovery of a high- altitude MeV electron population trapped in the cusp (Sheldon et al. GRL [1998a]). POLAR also saw energetic ions in these diamagnetic cavities in the cusp (Chen et al. JGR [1998]) We scale these cavities to other magnetospheres to estimate the energy spectrum observed in the planets & heliosphere. Two Parallel Dipole Magnets, -400V symmetric bias, 100mTorr SIDE t=1s SIDE t=1.5s SIDE t=2.5s SIDE t=3.0s B B D If plasma is trapped, as we demonstrated by tracing the particles in the cusp, then it reacts back on the magnetic field, much as the ring current produces a reduction in field strength seen on the surface of the earth. This diamagnetic effect grows with the energy density of the plasma, and at sufficiently high trapped density, can expel the external trapping magnetic field. This can produce a “bubble” of close to zero field strength. Since the cusp is a region of very small field strength, these bubbles are likely to form there. Chen et al (JGR 98) showed that indeed, diamagnetic cavities were observed in the cusp. These cavities were filled with mirror mode waves and high turbulence, such that dB ~ B. This makes diamagnetic cavities ideal places for stochastic acceleration, and he found energized ions in these cavities. Note that often it not the energetic plasma, but the cold, dense plasma that maintains the cavity. Chen (JGR 98) showed that these cusp diamagnetic cavities occur near the outer cusp. The left panel shows a X-Y plot of these observations, the right panel shows the X-Z distribution. The CDC cluster in the cusp, which is not surprising, because the cusp is the region of weakest field. The cusp is also a diverging field. It is known that diamagnetic cavities have a buoyant force, so what would keep these cavities in the m’sphere? Wouldn’t they squirt out like a watermelon seed? The University of Nijmegen shows how all substances are diamagnetic, and can be levitated harmlessly by the diverging (cusp-like) field in the 32mm bore of a 16 T Bitter magnet. The buoyant force balances the gravitational force in this case. Algorithm for mapping stability minimum of T96 magnetic field + diamagnetic bubble Place small dipole in the cusp, anti-aligned = a cusp diamagnetic cavity (CDC). Calculate B = B DIPOLE + B t96 for a 1 Re bubble around the little dipole. Since E=mB 2 and F X = dE/dx, we repeat this calculation for a little dx, dy, dz motion and take differences to get F. (We also get dF/dx too.) Finally we adust the strength of B DIPOLE until we can get a zero force. We plot these quantities to find a force free solution Black lines are contours of B 2. Green line shows F=dE/dx = 0. Red line show where d 2 E/dx 2 = dF/dx = 0, showing that green lines are in regions of stability (dF/dx < 0). We interpret this to show several things about CDC: a) The CDC do not squirt out of the magnetosphere, but find a stable location inside. b) This stable position moves inward as the size of the CDC grows larger c) This is understood to be caused by a “wall” of magnetic field generated by the Chapman-Ferraro currents at the magnetopause, as shown schematically in the figure here. B rad ~ B surface = B 0 B cusp ~ B 0 /R stag 3 E cusp ~ v 2 perp ~ (B cusp  ) 2 ~ [(B 0 /R stag 3 )R stag ]  E/B is constant E rad = 5 MeV for Earth E rad-planet ~(R stag-Earth /R stag-planet )(B 0-planet /B 0-Earth ) 2 E rad-Earth Sheldon et al. (GRL98a), observed 1 MeV electrons at L~12, adiabatically but not diffusively contiguous with the radiation belts, as can be seen by connecting the black dots, which correspond to an adiabatic phase space density. At L=12 and L=3 is large density, but a minima occurs in between. This suggests that the two populations are not currently connected diffusively. This MeV population has trapped, 90 deg pitchangle dependence, as shown in the inset. Phase space density is plotted in color according to color bar, pink shows pitchangles. If not currently diffusively connected now, it is still possible that at times of high activity, the diffusion rate will greatly increase and connect these regions. Heuristically, we have two ponds separated by a low wall, so that wave action can spill over. So who supplies who? Careful examination shows that each region has the same color, indicating comparable phase space densities. However many searches in the radiation belts have not uncovered a source for these MeV electrons, suggesting that the outer region may be the source for the inner region. Supporting this conclusion is the high variability and potentially fast acceleration for these cusp trapped electrons. Scaling Laws for CDC  Radiation Belts If these stable CDC are responsible for accelerating the energetic particles in a planetary magnetosphere, we want to formulate a scaling law for all magnetospheres. 1) We take the magnetic field of the radiation belts of the planetary magnetosphere to be approximately surface field. (Sun’s radiation belt = anomalous cosmic rays) 2) We take the cusp field to be diminished by distance to the stagnation point at the nose. 3) We set the maximum energy in the cusp to be determined by the gyroradius of an energetic particle in a CDC. 4) Calibrating with Earth (5MeV) we estimate the adiabatic energization from cusp to rad- belt, giving last equation. M’sphere Mercury Earth Mars Jupiter Saturn Uranus Neptune Sun E RAD (keV) < , ?? R STAG B 0 (nT) ,000 < 6 430,000 21,000 23,000 14, Using the scaling law derived from Earth, we get the above estimates for the energy of the radiation belts at all the planets. For all planets that have been explored, this prediction is relatively accurate, predicting 100Mev for Jupiter. We calculate the radiation belt at Saturn (1200 as well as the plasma temperature for Saturn’s rings at Rs=7, finding a warm plasma capable of 1kV electrostatic field. The value for the Sun is calculated by using the Heliopause- Bow shock estimate for the cusp size. Conclusions Cusp Diamagnetic Cavities are stable in the quadrupole cusp of all magnetospheres due to Chapman Ferraro currents. These CDC are filled with waves and make excellent accelerators. We observe these CDC at Earth simultaneous with energized particles. Scalings based on a CDC at Earth produce a reasonable estimate of Jupiter’s radiation belt energy, predict that Mars will not have a radiation belt, and lead to predictions for the other planets. Scaling this CDC warm plasma to Saturn’s E-ring, we predict a 1-2 kV electrostatic potential well for trapping charged dust.This would account for the thinness of Saturn’s E-ring, a topic not addressed by current models (as discussed in the oral talk) It also predicts the correct energy range for anomalous cosmic rays in the heliosphere, and may account for galactic cosmic rays. Plasma is diamagnetic