Electron-positron pair production by gamma-rays in an anisotropic flux of soft photons, and application to pulsar polar caps

Electron-positron pair production by collision of photons is investigated in view of application to pulsar physics. We compute the absorption rate of individual gamma-ray photons by an arbitrary anisotropic distribution of softer photons, and the energy and angular spectrum of the outgoing leptons. We work analytically within the approximation that 1 >> mc(2)/E > epsilon/E, with E and epsilon the gamma-ray and soft-photon maximum energy and mc(2) the electron mass energy. We give results at leading order in these small parameters. For practical purposes, we provide expressions in the form of Laurent series which give correct reaction rates in the isotropic case within an average error of similar to 7 per cent. We apply this formalism to gamma-rays flying downward or upward from a hot neutron star thermally radiating at a uniform temperature of 10(6) K. Other temperatures can be easily deduced using the relevant scaling laws. We find differences in absorption between these two extreme directions of almost two orders of magnitude, much larger than our error estimate. The magnetosphere appears completely opaque to downward gamma-rays while there are up to similar to 10 per cent chances of absorbing an upward gamma-ray. We provide energy and angular spectra for both upward and downward gamma-rays. Energy spectra show a typical double peak, with larger separation at larger gamma-ray energies. Angular spectra are very narrow, with an opening angle ranging from 10(-3) to 10(-7) radians with increasing gamma-ray energies.