Relativistic stabilisation of the diocotron instability in a pulsar “cylindrical” electrosphere

Abstract

Context. The physics of the pulsar inner magnetosphere remains poorly constrained by observations. Although about 2000 pulsars have been discovered to date, only a little is known about their emission mechanism. Large vacuum gaps exist in the magnetosphere, and a non-neutral plasma partially fills the neutron star surroundings to form an electrosphere. Aims. In a previous work, we showed that the differentially rotating equatorial disk in the pulsar's electrosphere is diocotron unstable in the non-relativistic regime. In this paper, we extend these results and study the relativistic and electromagnetic stabilisation effects by including the magnetic field perturbation and allowing for relativistic speeds of the guiding centre, in a self-consistent manner. We use the electric drift approximation, valid for low-density plasmas. Methods. We linearise the coupled relativistic cold-fluid and Maxwell equations in the electric drift approximation. The non-linear eigenvalue problem for the perturbed azimuthal electric field is solved numerically with standard techniques for boundary value problems like the shooting method. The spectrum of the relativistic diocotron instability is computed in a non-neutral plasma column confined between two cylindrically conducting walls. Results. For low-speed motions, we recover the eigenfunctions and eigenspectra of the non-relativistic diocotron instability. Our algorithm is also checked in the relativistic planar diode geometry for which an analytical expression of the dispersion relation is known. As expected, when the relativistic and electromagnetic effects become significant, the diocotron instability tends to stabilise. In cylindrical geometry, for some special rotation profiles, all azimuthal modes l are completely suppressed for sufficiently relativistic flows. However, for the profile relevant to the electrosphere, depending on the exact rotation curves, the growth rates can either significantly decrease till they vanish or persist for moderate l. Conclusions. The non-neutral plasma flowing in the pulsar electrosphere approaches the speed of light when reaching the light-cylinder. Therefore, relativistic and electromagnetic effects are important. They are able to completely suppress the diocotron instability. Nevertheless, results are sensitive to the tail of the rotation curves; therefore, particle diffusion across the magnetic field due to the diocotron instability only works efficiently close to the neutron star surface.