A first-principles analytical theory for 2D magnetic reconnection in electron and Hall MHD.

While the relevance of two-fluid effects in fast magnetic reconnection is well-known,\footnote{J. Birn et al., {\em J. Geophys. Res.}, {\bf 106} (A3), pp. 3715--3719 (2001) } a first-principles theory --akin to Sweet and Parker's in resistive MHD-- has been elusive. Here, we present such a first principles steady-state theory for electron MHD,\footnote{L. Chac\'on, A. N. Simakov, A. Zocco, {\em Phys. Rev. Lett.}, submitted} and its extension to Hall.\footnote{A. N. Simakov, L. Chac\'on, in preparation} The theory discretizes the extended MHD equations at the reconnection site, leading to a set of time-dependent ODEs. Their steady-state analysis provides predictions for the scaling of relevant quantities with the dissipation coefficients (e.g, resistivity and hyper-resistivity) and other relevant parameters. In particular, we will show that EMHD admits both elongated and open-X point configurations of the reconnection region, and that the reconnection rate $E_z$ can be shown not to scale explicitly with the dissipation parameters. This analytic result confirms earlier computational work on the possibility of fast (dissipation-independent) magnetic reconnection in EMHD. We have extended the EMHD results to Hall MHD, and have found a general scaling law for the reconnection rate (and associated length scales) that bridges the gap between resistive and EMHD.