A Unified Model for Shock Interaction and $\gamma$-Ray Emission in Classical Novae

We present a parameterized ("toy") model for shock interaction and $\gamma$-ray emission in classical novae, in which a white dwarf envelope of mass $M_{\rm env}$ is removed over a timescale $\tau$ (proportional to the nova speed class, $t_{2}$) in an outflow that accelerates on the same timescale to a terminal speed $v_{\rm f}$. Particle acceleration occurs at the reverse shock generated when the outflow collides with a thin, dense shell of slower material released earlier. Accelerated protons are then advected into the shell, where for typical ${ M_{\rm env}, \tau, \text{and } v_{\rm f}}$ they radiate in the calorimetric limit, consistent with correlated optical and $\gamma$-ray emission seen in well-sampled novae. The maximum proton energy, set by a Hillas-like argument, scales with the thickness of the hot post-shock region. Recent work shows turbulent mixing of hot post-shock gas with cooler dense gas may limit this thickness to $\lesssim 10^{-4}$ of the shock radius, explaining low X-ray luminosities. Using this empirically motivated thickness, and assuming efficient magnetic amplification, we predict maximum proton energies $E_{\rm max} \sim 10$ GeV, consistent with $\gamma$-ray spectra of Fermi-detected novae near optical peak ($\sim \tau$). However, as the shock and post-shock layer expand, $E_{\rm max}$ can grow to $\gtrsim 10$ TeV on timescales of a few $\tau$, enabling potential detection by atmospheric Cherenkov telescopes. We encourage TeV follow-up of Fermi-detected novae weeks to months after the optical/GeV peak and quantify the most promising events.