On the maximum pair multiplicity of pulsar cascades

We study electron-positron pair production in polar caps of energetic pulsars to determine the maximum multiplicity of pair plasma a pulsar can produce under the most favorable conditions. This paper complements and updates our study of pair cascades presented in Timokhin & Harding (2015) with more accurate treatment of the effects of ultra strong B>3x10^{12}G magnetic fields and emission processes of primary and secondary particles. We include pairs produced by curvature and synchrotron radiation photons as well as resonant Compton scattered photons. We develop a semi-analytical model of electron-positrons cascades which can efficiently simulate pair cascades with an arbitrary number of microphysical processes and use it to explore cascade properties for a wide range of pulsar parameters. We argue that the maximum cascade multiplicity can not exceed ~ a few x 10^5 and the multiplicity has a rather weak dependence on pulsar period on pulsar period. The highest multiplicity is achieved in pulsars with magnetic field 4x10^{12}<B<10^{13}G and hot surfaces, with T>10^6K. We also derive analytical expressions for several physical quantities relevant for electromagnetic cascade in pulsars which may be useful in future works on pulsar cascades, including the upper limit on cascade multiplicity and various approximations for the parameter \chi, the exponential factor in the expression for photon attenuation in strong magnetic field.