A generalized Townsend's theory for Paschen curves in planar, cylindrical, and spherical geometries

TL;DR

This paper extends Townsend's Paschen curve theory from parallel plates to spherical and cylindrical geometries, providing insights into plasma breakdown in planetary atmospheres and multiphase flows with complex electrode shapes.

Contribution

It generalizes Townsend's classical theory to non-parallel geometries, deriving explicit solutions for critical voltages and breakdown conditions in spherical and cylindrical configurations.

Findings

Glow coronae form more easily in Mars's low-pressure atmosphere.

Critical voltage minima occur at 0.5 cm·Torr across geometries.

Breakdown criteria are established for Titan and Venus atmospheres.

Abstract

In 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 the gaseous medium. At low currents and low temperatures, this process can create luminescent emissions: the so-called glow and corona discharges. The parallel plate geometry used in Townsend's (1900) 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 grains or ice particles in multiphase flows. 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 represents rounded or blunted tips. Similarly, in a cylindrical case, a small radius would correspond to a thin wire. We solve continuity equations in the gap and estimate a critical radius and minimum breakdown voltage that allows ionization of neutral gas and formation of a glow discharge. We show that glow coronae form more easily in Mars's low-pressure, $CO_2$-rich atmosphere than in Earth's high-pressure atmosphere. Additionally, we present breakdown criteria for Titan and Venus. We further demonstrate that critical voltage minima occur at 0.5 cm$\cdot$Torr for all three investigated geometries, suggesting easier initiation around millimeter-size particles in dust and water clouds and could be readily extended to examine other multiphase flows with inertial particles.