We estimate the voltage of the discharges by calculating the mean free path (MFP). The Chemical Rubber Company (CRC) handbook lists the MFP for neutral Helium at Standard Temperature and Pressure (STP) to be cm. Since MFP , where is the density, we multiplied by a factor of 7600 for a 100mTorr pressure giving a 2mm MFP. Now for a lightning discharge to form, the electrons in the leader of the lightning stroke must be accelerated to at least 24.5 eV before they can ionize the Helium, and this must occur in less than 2mm. This gives a value of V/m for an electric field. If the discharge begins at the equator and proceeds in both directions down to the magnet, the distance could be as much as 4 cm, resulting in a voltage greater than 500V. At a density of 200mTorr, the same calculation gives 1 kV, and at 400 mTorr, we would produce 2 kV.
However, we also note that the discharges are brightest at the magnet and fade as they approach the equator. Since our mechanism relies on an electric field scaling as , the fact that the discharges are brightest close to the magnet is consistent with our model. In addition, the electrons diverge as the flux tube expands away from the magnet, resulting in a rapid fading of the glow along the flux tube. However a variable does suggest that our quick calculation above be modified. If the strength of drops below ionization threshold, we would expect the glow to cease within a scattering length or two. So if we start with the dim part of the flux tube and integrate to the brightest spot on the flux tube, one would expect the electric field to increase with some power of , where our quick calculation above used . Thus it would appear that our calculation above is a lower limit for potentials generated by the system.
When we look in detail at the pressure changes, we note that the lower pressure regimes have a ``comet-like'' appearance, with a bright ``coma'' near or on the surface of the magnet, and a dim tail. The high pressure discharges (400 mTorr), lack the presence of the coma, and the intensity of the discharge does not vary along the flux tube. The coma might be the collision of the particles with the magnet and subsequent sputtered plasma glowing more brightly than the neutral gas. Alternatively, it may be that the long tail is a secondary effect of cold accelerated ions colliding with the gas, as proposed for the POLAR/CEPPAD data. Spectroscopic or fast timing images should be able to sort out the evolution of a single discharge, but this is beyond the scope of our preliminary results.
The lack of a coma in the high pressure case may also be explained from the simple theory above. We note that a distinct ``mirror-point'' collection of hot ions occur only for mono-pitchangle particles. At high pressures, we might expect the hot ions to scatter isotropically and therefore not produce as rapidly varying radial electric field.
If the creation of the space charge can be viewed as charging a capacitor in an RC circuit, then it agrees with the observation that the discharges become less frequent and brighter as the pressure is increased. Since a voltage greater than 500V was required experimentally to produce a discharge in 200mTorr He, we estimate that at least half of the 1400 eV of the injected ions is extracted in these discharges, in agreement with the data from space and our simple theory.
Spinning the magnet (panel two of Figure 6) increases the corotation speed of the ions due to the additional -drift. In our experiment, we spun the magnet in the same sense as the -drift for ions, thereby spreading the ions more evenly around the magnet. From our model, this increased the area and hence the capacitance of the space charge, and thus permitted more charge to collect and more intense flashes for the same voltage. Note how evenly the discharges occur in the second panel as compared to the first panel of the Figure 6. In space and astrophysical systems, the more even distribution generated by spinning also has the same effect, allowing the spatial scale of the space charge capacitor to be the size of the system itself. As we calculated for AGN blazars, this permits the longest length scale and highest acceleration potential to produce the 1 GeV astrophysical jets.