A quantitative, comprehensive analytical model for ``fast'' magnetic reconnection in Hall MHD*

Magnetic reconnection in nature usually happens on fast (e.g. dissipation independent) time scales. While such scales have been observed computationally [1], a fundamental analytical model capable of explaining them has been lacking. Here, we propose such a quantitative model for 2D Hall MHD reconnection without a guide field. The model recovers the Sweet-Parker and the electron MHD [2] results in the appropriate limits of the ion inertial length, $d_i$, and is valid everywhere in between [3]. The model predicts the dissipation region aspect ratio and the reconnection rate $E_z$ in terms of dissipation and inertial parameters, and has been found to be in excellent agreement with non-linear simulations. It confirms a number of long-standing empirical results and resolves several controversies. In particular, we find that both open X-point and elongated dissipation regions allow ``fast'' reconnection and that $E_z$ depends on $d_i$. Moreover, when applied to electron-positron plasmas, the model demonstrates that fast dispersive waves are not instrumental for ``fast'' reconnection [4]. [1] J. Birn {\it et al.}, {\it J. Geophys. Res.} {\bf 106}, 3715 (2001). [2] L. Chac\'{o}n, A. N. Simakov, and A. Zocco, {\it Phys. Rev. Lett.} {\bf 99}, 235001 (2007). [3] A. N. Simakov and L. Chac\'on, submitted to {\it Phys. Rev. Lett.} [4] L. Chac\'on, A. N. Simakov, V. Lukin, and A. Zocco, {\it Phys. Rev. Lett.} {\bf 101}, 025003 (2008).