Particle Acceleration in Pair-starved Pulsars

We investigate the physical situation above the pulsar polar cap (PC) where the accelerating primaries (electrons) are not capable of producing sufficient numbers of electron-positron pairs at low altitudes (within ~1-2 stellar radii above the PC surface) to screen the accelerating electric field, E∥, and continue accelerating up to at least the very high altitudes nearly approaching the light cylinder. We derive an analytic solution for the E∥ valid at high altitudes. The solution is based on the physical condition of asymptotic vanishing of the rotationally induced transverse electric field within the magnetic flux tube. This condition constrains the asymptotic value of the effective space charge that determines the distribution of electrostatic E∥ within the magnetic tube. Our estimates of low- to high-altitude values of E∥ imply the occurrence of a regime of primary acceleration [with the characteristic Lorentz factor up to ~(1-2) × 107] all the way from the PC to the light cylinder limited by curvature radiation reaction. In this model the primary outflow becomes asymptotically force-free and may turn into a relativistic wind beyond the light cylinder. Such a solution will apply to both older pulsars producing only inverse Compton scattering pairs and younger very high B pulsars (magnetars). We suggest that pulsars, which are lying below the pair death line, may be radio-quiet γ-ray sources.