Time-dependent pair cascades in magnetospheres of neutron stars - I. Dynamics of the polar cap cascade with no particle supply from the neutron star surface

I argue that the problem of electromagnetically driven electron-positron cascades in magnetospheres of neutron stars must be addressed starting from first principles. I describe a general numerical algorithm for doing self-consistent kinetic simulations of electron-positron cascades - wherein particle acceleration, pair creation and screening of the electric field are calculated simultaneously - and apply it to model the Ruderman & Sutherland cascade in one dimension. I find that pair creation is quite regular and quasi-periodic. In each cycle a blob of ultra-relativistic electron-positron plasma is generated; it propagates into the magnetosphere leaving a tail of less relativistic plasma behind, and the next discharge occurs when this mildly relativistic plasma leaves the polar cap. A short burst of pair formation is followed by a longer quiet phase when accelerating electric field is screened and no pairs are produced. Some of freshly injected electron-positrons pairs get trapped in plasma oscillations creating a population of low-energy particles. The cascade easily adjusts to the current density required by the pulsar magnetosphere by reversing some of the low-energy particles. Each discharge generates a strong coherent superluminal electrostatic wave, which may be relevant for the problem of pulsar radioemission.