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Abstract Sharp point electrodes generate significant electric field enhancements where electron impact ionization leads to the formation of electron avalanches that are seeded by photoionization. Photoionization of molecular oxygen due to extreme ultraviolet emissions from molecular nitrogen is a fundamental process in the inception of a positive corona in air. In a positive corona system, the avalanche of electrons in the bulk of the discharge volume is initiated by a specific distribution of photoionization far away from the region of maximum electron density near the electrode where these photons are emitted. Here, we present a new approach to finding the inception conditions for a positive corona, which is based on a differential formulation of the photoionization problem. The proposed iterative solution considers the same inception problem that has been solved in the existing literature by using either an integral approach to photoionization or a differential formulation of photoionization and considering the inception problem as a boundary-value eigenvalue problem. The results are validated by comparisons with previous integral formulations and time dynamic plasma fluid solutions in planar and spherical geometries. The results illustrate ideas advanced in Kaptzov (1950Elektricheskiye Yavleniya v Gazakh i Vacuumep 610) providing a physically transparent connection between an effective secondary electron emission coefficient due to volume photoionization in a positive corona system and the secondary electron emission in conventional Townsend discharge theory. The results also demonstrate the significance of boundary conditions for accurate corona solutions that are based on a differential formulation of photoionization. 

Abstract Terrestrial gamma‐ray flashes are linked to growth of long bidirectional lightning leader system consisting of positive and stepping negative leaders. The spatial extent of streamer zones of a typical lightning leader with tip potential exceeding several tens of megavolts is on the order of 10–100 m. The photoelectric absorption of bremsstrahlung radiation generated by avalanching relativistic runaway electrons occurs efficiently on the same spatial scales. The intense multiplication of these electrons is triggered when the size of the negative leader streamer zone crosses a threshold of approximately 100 m (for sea‐level air pressure conditions) allowing self‐replication of these avalanches due to the upstream relativistic electron seeds generated by the photoelectric absorption. The model results also highlight importance of electrode effects in interpretation of X‐ray emissions from centimeter to meter long laboratory discharges, in particular, a similar feedback effect produced by generation of runaway electrons from the cathode material.