Constraints on the Acceleration of Ultra-High-Energy Cosmic Rays in Accretion-Induced Collapse Pulsars

(Abridged) We have recently proposed that the ultra-high energy cosmic rays (UHECRs) observed above the GZK limit could be mostly protons accelerated in reconnection sites just above the magnetosphere of newborn millisecond pulsars originated by accretion induced collapse (AIC-pulsars). Although the expected rate of AIC sources in our own Galaxy is very small, our estimates have shown that the observed total flux of UHECRs could be obtained from the integrated contribution from AIC-pulsars of the whole distribution of galaxies located within a distance which is unaffected by the GZK cutoff ($\sim 50 $ Mpc). We presently examine the potential acceleration mechanisms in the magnetic reconnection site and find that first-order Fermi acceleration cannot provide sufficient efficiency (due to synchrotron losses). This leaves the one-shot acceleration via an induced electric field within the reconnection region as the only viable process for UHECR acceleration. We formulate the constraints on both the magnetic field topology and strength in order to accelerate the particles and allow them to freely escape from the system. Under fast reconnection, we find that AIC-pulsars with surface B-fields $10^{12} $ G $< B_{\star} \lesssim $ $10^{15}$ G and spin periods 1 ms $\lesssim P_{\star} < $ 60 ms, are able to accelerate particles to energies $\geq 10^{20} $ eV, but the magnetic field just above the Alfv\'en surface must be predominantly toroidal for the particles to be allowed to escape from the acceleration zone without being deflected. Synchrotron losses bring potentially important constraints on the B-field geometry of $any$ UHECR accelerators involving compact sources with strong magnetic fields.