We study the theoretical properties of relativistic, transverse, magnetosonic collisionless shock waves in electron-positron-heavy ion plasmas of relevance to astrophysical sources of synchrotron radiation. We use both one-dimensional electromagnetic particle-in-cell simulations and quasi-linear theory to examine the spatial and kinetic structure of these nonlinear flows. All the upstream ions are electromagnetically reflected from the shock front, causing the magnetic field strength in the shock to overshoot its final downstream value in a series of long compressional oscillations with wavelengths comparable to the Larmor radius of ions with the rigidity appropriate to the downstream flow. We describe a new process of shock acceleration of nonthermal positrons, in which the gyrating reflected heavy ions dissipate their energy in the form of collectively emitted, left-handed magnetosonic waves which are resonantly absorbed by the positrons immediately behind the ion reflection region. This absorption gives rise to an ultrarelativistic downstream positron spectrum The positrons with energy between gamma-m(e) + c2 and (gamma + d-gamma)m(e) + c2 have a spectrum N(gamma)d-gamma is-proportional-to gamma(-s) with s approximately 2 when the upstream flow energy of the heavy ions exceeds that of the electrons and positrons. When the Lorentz factor of the upstream flow is gamma-1, the power-law part of the downstream positron spectrum extends from gamma = gamma-1, to gamma = (m(ion)/Zm(e +/-)gamma-1. The highest energy positrons have Larmor radii comparable to the thickness of the shock. The efficiency of conversion of flow energy into these highly accelerated particles is about 10% approximately 20%. We argue that the nonthermal acceleration of positrons found here also applies to the partial acceleration of electrons. We also point out another acceleration mechanism which may be important in oblique shocks. We briefly outline applications of these results to the termination shocks of pulsar winds and to the termination shocks of jets emanating from active galactic nuclei. Our kinetic shock acceleration model can account for the spectral range of the synchrotron radiation observed in the Crab Nebula from the infrared through high-energy gamma rays, as well as for the radio emission seen from the hot spots in extragalactic radio sources. We propose that the "wisps" seen in the optical emission from the Crab Nebula are brightness enhancements formed in the compressonal magnetic overshoots formed as part of the shock, so that the structure of the shock terminating the relativistic wind from the Crab pulsar is spatially resolved in this system. This model implies the wind from the pulsar has its energy flux largely in partially ionized iron ions, although pairs form the dominant species by number, with the bulk flow velocity corresponding to Lorentz factor approximately 10(6), and an energy per iron ion corresponding to approximately 20% of the electric potential energy associated with the open field lines of the pulsar.
