Magnetosphere structure of a Kerr black hole: marginally force-free equatorial boundary condition

TL;DR

This paper investigates the influence of marginally force-free equatorial boundary conditions on the structure of Kerr black hole magnetospheres, using numerical solutions that align with recent high-accuracy simulations.

Contribution

It introduces a numerical algorithm for solving the Grad-Shafranov equation with marginally force-free boundary conditions and demonstrates its effectiveness in modeling Kerr black hole magnetospheres.

Findings

The boundary condition significantly affects the shape of the light surface.

Numerical solutions agree well with high-accuracy force-free electrodynamics simulations.

The proposed method is applicable to various magnetic field configurations.

Abstract

The role of equatorial boundary condition in the structure of a force-free black hole magnetosphere was rarely discussed, since previous studies have been focused on the field lines entering the horizon. However, recent high-accuracy force-free electrodynamics (FFE) simulations \cite{East2018} show that there are both field lines entering the horizon and field lines ending up on the equatorial current sheet within the ergosphere for asymptotic uniform field configuration. For the latter field lines, the equatorial boundary condition is well approximated being marginally force-free, i.e., $B^2-E^2\approx 0$, where $B$ and $E$ are the magnetic and electric field strength, respectively. In this paper, we revisit the uniform field solution to the Kerr BH magnetosphere structure and investigate the role of the marginally force-free equatorial boundary condition. We find this boundary condition plays an important role in shaping the BH magnetosphere in various aspects, including the shape of the light surface, the near-horizon field line configuration and the source of the Poynting flux. We also propose an algorithm for numerically solving the Grad-Shafranov equation and self-consistently imposing the marginally force-free equatorial condition. As a result, we find a good agreement between our numerical solutions and the high-accuracy FFE simulations. We also discuss the applicability of the marginally force-free boundary condition and the numerical algorithm proposed in this paper for general magnetic field configurations.