Photon-splitting cascades in gamma-ray pulsars and the spectrum of PSR 1509-58

Magnetic photon splitting gamma --> gamma gamma, a quantum electrodynamics process that becomes important only in magnetic fields approaching the quantum critical value, B-cr = 4.41 x 10(13) G, is investigated as a mechanism for attenuation of gamma-rays emitted near the surface of strongly magnetized pulsars. Since splitting has no threshold, it can attenuate photons and degrade their energies below the threshold for one-photon pair production, and in high enough fields it may dominate photon attenuation above Fair threshold. We model photon-splitting attenuation and subsequent splitting cascades in gamma-ray pulsars, including the dipole field and curved spacetime geometry of the neutron star magnetosphere. We focus specifically on PSR 1509-58, which has the highest surface magnetic field of all the gamma-ray pulsars (B-0 = 3 x 10(13) G). We find that splitting will not be important for most gamma-ray pulsars, i.e., those with B-0 less than or similar to 0.2B(cr), either in competition with pair production attenuation in pair cascades, or in photon escape cutoffs in the spectrum. Photon splitting will be important for gamma-ray pulsars having B-0 greater than or similar to 0.3B(cr), where the splitting attenuation lengths and escape energies become comparable to or less than those for pair production. We compute Monte Carlo spectral models for PSR 1509-58, assuming that either a full photon-splitting cascade or a combination of splitting and pair production (depending on which splitting modes operate) attenuate a power-law input spectrum. We find that photon splitting, or combined splitting and pair production, can explain the unusually low cutoff energy (between 2 and 30 MeV) of PSR 1509 - 58, and that the model cascade spectra, which display strong polarization, are consistent with the observed spectral points and upper limits for polar cap emission at a range of magnetic colatitudes up to similar to 25 degrees.