What Can We Learn About Electron Distributions From Measurements of Weak Bipolar Fields?$^*$

A given bipolar field that is stationary in a co-moving frame can correspond either to an ion soliton or an electron phase-space hole. In the limit of weak potential, $\phi$, with $e\phi_{\mathrm{max}}/T_e\ll1$, either of these structures can have the asymptotic shape $\phi= \phi_{\mathrm{max}}\mathrm{sech}^4(x/\alpha)$. For ion solitons, the half width ($\propto\alpha$) depends on $\phi_{\mathrm{max}}$, whereas for electron holes the half-width is independent of $\phi_{\mathrm{max}}$. We show analytically for holes in this limit that $\phi_{\mathrm{max}}$ depends on the (finite) energy derivative of the trapped distribution at the separatrix, while $\alpha$ depends only on a ``screening'' integral over the untrapped distribution. Idealized trapped and passing electron distributions are shown to be \textit{inferrable} from the speed, amplitude, and shape of weak bipolar waveform measurements. For measurements$^1$ of hundreds of weak bipolar field events in Earth's cusp, the theory is shown to be consistent with the most frequently observed \textit{half-width} between bipolar field peaks, and with various other features of the measured$^1$ distribution of hole velocities vs hole half-widths. \\ $^*$ Work supported by NSF, NASA, and DOE, and submitted in part to \textit{Phys.~Rev.~Lett.} as ``Theory of Weak Bipolar Fields and Electron Holes with Space Applications,'' (2007). \\ $^1$ Franz, J.R., \textit{et al}., \textit{J.~Geophys.~Res.} \textbf{110}, A09212, 2005.