A stability property of a force-free surface bounding a vacuum gap

TL;DR

This paper proves a general stability property of force-free surfaces (FFS) that bound vacuum gaps, showing they attract like-signed charges and cannot form in uniform or monopolar magnetic fields, with implications for pulsar magnetospheres.

Contribution

It establishes a universal relation between charge density and magnetic field derivatives on FFS, proving their stability and non-existence in certain magnetic configurations.

Findings

FFS attract same-signed charges on the vacuum side.

No vacuum gap can form in uniform or monopolar magnetic fields.

The relation simplifies in axisymmetric pulsar magnetospheres.

Abstract

A force-free surface (FFS) ${\cal S}$ is a sharp boundary separating a void from a region occupied by a charge-separated force-free plasma. It is proven here under very general assumptions that there is on ${\cal S}$ a simple relation between the charge density $\mu$ on the plasma side and the derivative of $\delta=\E\cdot\B$ along $\B$ on the vacuum side (with $\E$ denoting the electric field and $\B$ the magnetic field). Combined with the condition $\delta=0$ on ${\cal S}$, this relation implies that a FFS has a general stability property, already conjectured by Michel (1979, ApJ 227, 579): ${\cal S}$ turns out to attract charges placed on the vacuum side if they are of the same sign as $\mu$. In the particular case of a FFS existing in the axisymmetric stationary magnetosphere of a "pulsar", the relation is given a most convenient form by using magnetic coordinates, and is shown to imply an interesting property of a gap. Also, a simple proof is given of the impossibility of a vacuum gap forming in a field $\B$ which is either uniform or radial (monopolar).