A Generalized Townsend's Theory for Paschen Curves in Planar, Cylindrical, and Spherical Geometries in Planetary Atmospheres

Abstract

AbstractIn this work, we focus on plasma discharges produced between two electrodes with a high potential difference, resulting in the ionization of the neutral particles supporting a current in a gaseous medium. At low currents and low temperatures, this process can create luminescent emissions: glow and corona discharges. The parallel plate geometry used in Townsend's theory lets us develop a theoretical formalism, with explicit solutions for the critical voltage effectively reproducing experimental Paschen curves. However, most discharge processes occur in non‐parallel plate geometries, such as discharges between particles in multiphase systems and between cylindrical conductors. Here, we propose a generalization of the classic parallel plate configurations to concentric spherical and coaxial cylindrical geometries in Earth, Mars, Titan, and Venus atmospheres. In a spherical case, a small radius effectively represents a sharp tip rod, while larger, centimeter‐scale radii represent blunted tips. In cylindrical geometries, small radii resemble thin wires. We solve continuity equations in the gap and estimate a critical radius and minimum breakdown voltage that allows the formation of a glow discharge. We show that glow coronæ form more easily in Mars's low‐pressure, CO2‐rich atmosphere than in Earth's high‐pressure, N2‐rich atmosphere. Additionally, we present breakdown criteria for Titan and Venus, two planets where discharge processes have been postulated. We further demonstrate that critical voltage minima occur at 0.5 cm⋅Torr for all three investigated geometries, suggesting easier initiation around millimeter‐size particles in dust and water clouds. This approach could be readily extended to examine other multiphase flows with inertial particles.