Lower bounds of altitudes for pulsar $gamma$-ray radiation

Determining radiation location observationally plays a very important role in testing the pulsar radiation models. One-photon pair production in the strong magnetic field, $gamma-e^{+}e^{1}$, is one of the important physical processes in pulsar radiation mechanisms. Photons near pulsar surface with sufficient energy will be absorbed in the magnetosphere and the absorption optical depth for these GeV $gamma$-ray photons is usually large. In this paper, we include the aberrational, rotational and general relativistic effects and calculate the $gamma$-B optical depth for $gamma$-ray photons. Then we use the derived optical depth to determine the radiation altitude lower bounds for photons with given energies. As a case study, we calculate the lower bounds of radiation altitudes of Crab pulsar for photons with energy from 5 GeV to 1 TeV.