Superelastic Heating in Treanor-Gordiets Plasmas: A Unified Analytic Closure

Abstract: In non-equilibrium plasmas where the vibrational temperature exceeds the gas temperature, conventional harmonic models underestimate superelastic electron heating rates by an order of magnitude or more. This failure stems from the artificial decoupling of energy modes, which ignores the exponential heating contributions from overpopulated high-lying states characteristic of Treanor-Gordiets distributions. We resolve this limitation by deriving a closed-form, thermodynamically consistent anharmonic gain function based on detailed balance and a second-order Dunham expansion. This formulation serves as a unified governing equation that naturally identifies the kinetic crossover between vibrational-vibrational (V-V) up-pumping and vibrational-translational (V-T) relaxation. This approach accurately predicts the Treanor minimum and recovers the accuracy of full state-to-state benchmarks at a fraction of the computational cost. The model provides a robust closure for predicting electron temperature evolution in applications ranging from hypersonic flows to plasma-assisted combustion.