Hall equilibrium of thin Keplerian disks embedded in mixed poloidal and toroidal magnetic fields

TL;DR

This paper develops an analytical model for thin Keplerian disks influenced by mixed magnetic fields, revealing two types of Hall equilibrium disks with diverse magnetic and density structures.

Contribution

It introduces a novel analytical approach to Hall-MHD equilibrium in thin disks, incorporating both azimuthal and poloidal magnetic components and classifying equilibrium types based on disk radius.

Findings

Two types of Hall equilibrium disks identified: small and large radius.

Analytical solutions demonstrate diverse magnetic and density configurations.

Model accounts for primordial and dipole magnetic field effects.

Abstract

Axisymmetric steady-state weakly ionized Hall-MHD Keplerian thin disks are investigated by using asymptotic expansions in the small disk aspect ratio \epsilon. The model incorporates the azimuthal and poloidal components of the magnetic fields in the leading order in \epsilon. The disk structure is described by an appropriate Grad-Shafranov equation for the poloidal flux function \psi that involves two arbitrary functions of \psi for the toroidal and poloidal currents. The flux function is symmetric about the midplane and satisfies certain boundary conditions at the near-horizontal disk edges. The boundary conditions model the combined effect of the primordial as well as the dipole-like magnetic fields. An analytical solution for the Hall equilibrium is achieved by further expanding the relevant equations in an additional small parameter \delta that is inversely proportional to the Hall parameter. It is thus found that the Hall equilibrium disks fall into two types: Keplerian disks with (i) small (Rd ~\delta^0) and (ii) large (Rd > \delta^k, k > 0) radius of the disk. The numerical examples that are presented demonstrate the richness and great variety of magnetic and density configurations that may be achieved under the Hall-MHD equilibrium.