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An electron-positron cascade in the magnetospheres of Kerr Black Holes (BH)is a fundamental ingredient to fueling the relativistic $\gamma$-ray jets seenat the polar regions of galactic supermassive BHs (SMBH). This leptonic cascadeoccurs in the "spark gap" region of a BH magnetosphere where the unscreenelectric field parallel to the magnetic field is present, hence it is affectedby the magnetic field structure. A previous study explored the case of a thinaccretion disk, representative of Active Galactic Nuclei (AGN). Here we explorethe case of a quasi-spherical gas distribution, as is expected to be presentaround the SMBH Sgr A* in the center of our Milky Way galaxy, for example. Theproperties and efficiency of the leptonic cascade are studied. The findings ofour study and the implications for SMBH systems in various spectral andaccretion states are discussed. The relationships and scalings derived fromvarying the mass of the BH and background photon spectra are further used toanalyze the leptonic cascade process to power jets seen in astronomicalobservations. In particular, one finds the efficiency of the cascade in aquasi-spherical gas distribution peaks at the jet axis. Observationally, thisshould lead to a more prominent jet core, in contrast to the thin diskaccretion case, where it peaks around the jet-disk interface. One also findsthe spectrum of the background photons to play a key role. The cascadeefficiency is maximum for a spectral index of two, while harder and softerspectra lead to a less efficient cascade.

Pair Plasma Cascade in Rotating Black Hole Magnetospheres with Split Monopole Flux Model