Pair production multiplicities in rotation-powered pulsars

We discuss the creation of electron-positron cascades in the context of pulsar polar cap acceleration models and derive several useful analytic and semianalytic results for the spatial extent and energy response of the cascade. Instead of Monte Carlo simulations, we use an integrodifferential equation that describes the development of the cascade energy spectrum in one space dimension quite well, when it is compared to existing Monte Carlo models. We reduce this full equation to a single integral equation, from which we can derive useful results, such as the energy loss between successive generations of photons and the spectral index of the response. We find that a simple analytic formula represents the pair cascade multiplicity quite well, provided that the magnetic field is below 10(12) G and that an only slightly more complex formula matches the numerically calculated cascade at all other field strengths. Using these results, we find that cascades triggered by gamma -rays emitted through inverse Compton scattering of thermal photons from the neutron star's surface, both resonant and nonresonant, are important for the dynamics of the polar cap region in many pulsars. In these objects, the expected multiplicity of pairs generated by a single input particle is lower than previously found in cascades initiated by curvature emission, frequently being on the order of 10 rather than similar to 1000 as usually quoted. Such pulsars also are expected to be less luminous in polar cap gamma -rays than when curvature emission triggers the cascade, a topic that will be the subject of a subsequent paper.