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Using an analytic, axisymmetric approach that includes general relativity,coupled to a condition for pair production deduced from simulations, we derivegeneral results about the spatial distribution of pair-producing field linesover the pulsar polar cap. In particular, we show that pair production onmagnetic field lines operates over only a fraction of the polar cap for analigned rotator for general magnetic field configurations, assuming themagnetic field varies spatially on a scale that is larger than the size of thepolar cap. We compare our result to force-free simulations of a pulsar with adipole surface field and find excellent agreement. Our work has implicationsfor first-principles simulations of pulsar magnetospheres, and for explainingobservations of pulsed radio and high-energy emission.

Spatial Distribution of Pair Production over the Pulsar Polar Cap