The reconstruction problem for steady symmetrical two-dimensional magnetic reconnection is addressed in the frame of a two-fluid approximation with neglected ion current. This approach yields Poisson's equation for the magnetic potential of the in-plane magnetic field, where the right-hand side contains the out-of-plane electron current density with the reversed sign. In the simplest case of uniform electron temperature and number density and neglecting the electron inertia, Poisson's equation turns to the Grad-Shafranov one. With boundary conditions fixed at any unclosed curve (the satellite trajectory), both equations result in an ill-posed problem. Since the magnetic configuration in the reconnection region is highly stretched, one can make use of the boundary layer approximation; hence, the problem becomes well-posed. The described approach is generalized for the case of nonuniform electron temperature and number density. The benchmark reconstruction of the PIC simulations data has shown that the main contribution for inaccuracy arises from replacing Poisson's equation by the equation of Grad-Shafranov. Under this substitution, the reachable cross-size of the reconstructed region is shrinking down to fractions of the proton inertial length. Artificial smoothing, demanded by solving the ill-posed problem, and boundary layer approximation represent two alternative methods of problem regularization. In terms of the reconstruction error, they perform nearly the same; the second method benefits from the comparative simplicity and less restrictions imposed on the boundary shape.

[1] An analytical model of steady-state magnetic reconnection in a collisionless incompressible plasma is developed using the electron Hall MHD approximation. It is shown that the initial complicated system of equations may be split into a system of independent equations, and the solution of the problem is based on the solution of the Grad-Shafranov equation for a magnetic potential. This equation is found to be fundamental for the whole problem analysis. An electric field potential jump across the electron diffusion region and the separatrices is proved to be the necessary condition for steady-state reconnection. Besides of this fact, it is found that the protons in-plane motion obeys to Bernoulli law. The solution obtained demonstrates all essential Hall reconnection features, namely proton acceleration up to Alfvén velocities and the formation of Hall current systems and a magnetic field structure as expected.

[1] The Sweet-Parker analysis of the inner electron diffusion region of collisionless magnetic reconnection is presented. The study includes charged particles motion near the X-line and an appropriate approximation of the off-diagonal term for the electron pressure tensor. The obtained scaling shows that the width of the inner electron diffusion region is equal to the electron inertial length, and that electrons are accelerated up to the electron Alfvén velocity in X-line direction. The estimated effective plasma conductivity is based on the electron gyrofrequency rather than the binary collision frequency, and gives the extreme (minimal) value of the plasma conductivity similar to Bohm diffusion. The scaling properties are verified by means of Particle-in-Cell simulations. An ad hoc parameter needs to be introduced to the scaling relations in order to better match the theory and simulations.