Electrostatic Instability in Electron-Positron Pairs Injected in an External Electric Field

Motivated by the particle acceleration problem in pulsars, we numerically investigate electrostatic instability of electron-positron pairs injected in an external electric field. The electric field is expected to be so strong that we cannot neglect effects of spatial variation in the 0-th order distribution functions on the scale of the plasma oscillation. We assume that pairs are injected mono-energetically with 4-velocity $u_0>0$ in a constant external electric field by which electrons (positrons) are accelerated (decelerated). By solving linear perturbations of the field and distribution functions of pairs, we find a new type of electrostatic instability. The properties of the instability are characterized by $u_0$ and the ratio $R$ of the braking time-scale (determined by the external electric field) to the time-scale of the plasma oscillation. The growth rate is as large as a few times the plasma frequency. We discuss the possibility that the excited waves prevent positrons from returning to the stellar surface.