The magnetron instability in a pulsar’s cylindrical electrosphere

Context. The physics of the pulsar magnetosphere near the neutron star surface remains poorly constrained by observations. Although about 2000 pulsars have been discovered to date, little is known about their emission mechanism, from radio to high-energy X-ray and gamma-rays. Large vacuum gaps probably exist in the magnetosphere, and a non-neutral plasma partially fills the neutron star surroundings to form an electrosphere. Aims. In several previous works, we showed that the differentially rotating equatorial disk in the pulsar’s electrosphere is diocotron unstable and that it tends to stabilise when relativistic effects are included. However, when approaching the light cylinder, particle inertia becomes significant and the electric drift approximation is violated. In this paper, we study the most general instability, i.e. by including particle inertia effects, as well as relativistic motions. Electromagnetic perturbations are described in a fully self-consistent manner by solving the cold-fluid and Maxwell equations. This general non-neutral plasma instability is called the magnetron instability by plasma physicists. Methods. We linearise the coupled relativistic cold-fluid and Maxwell equations. The non-linear eigenvalue problem for the perturbed azimuthal electric field component is solved numerically with standard techniques for boundary-value problems like the shooting method. The spectrum of the magnetron instability in a non-neutral plasma column confined between two cylindrically conducting walls is computed for several cylindrical configurations. For a pulsar electrosphere, no outer wall exists. In this case, we allow for electromagnetic wave emission propagating to infinity. Results. First we checked our algorithm in the low-density limit. We recover the results of the relativistic diocotron instability. When the self-field induced by the plasma becomes significant, it can first increase the growth rate of the magnetron instability. However, equilibrium solutions are only possible when the self-electric field, measured by the parameter se and tending to disrupt the plasma configuration, is bounded to an upper limit, se,max. For se close to but smaller than this value se,max, the instability becomes weaker or can be suppressed as was the case in the diocotron regime. Conclusions. When approaching the light-cylinder, particle inertia becomes significant in the equatorial disk of the electrosphere. Indeed, the rest-mass energy density of the plasma becomes comparable to the magnetic energy density. The magnetron instability sets in and takes over the destabilisation of the stationary flow initiated by the diocotron instability close to the neutron star surface. As a consequence, the flow in the pulsar inner magnetosphere is highly unstable, leading to particle diffusion across the magnetic field line. Therefore, an electric current can circulate in the closed magnetosphere and feed the wind with charged particles.