  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.
