Diagnostic Disagreement as an Information-Projection Divergence: An Information-Theoretic Reading of the Quiet-Sun Temperature Ratio

Abstract

The quiet-Sun coronal electron-temperature ratio $R \equiv T_\mathrm{EUV}/T_B \approx 2.4$, stable across an eight-year solar cycle, is read here as a measurement of relative entropy between two diagnostic projections of the coronal electron distribution onto the one-parameter Maxwellian family. The EUV ionization temperature is a moment-matching projection against a Bethe-type ionization kernel; the radio brightness temperature is the Rayleigh-Jeans source function of thermal bremsstrahlung. For a kappa distribution in the mean-energy convention, Fleishman & Kuznetsov (2014) give the radio-side projection in closed form as $T_B = T_\mathrm{core}$; the EUV side returns $T_\mathrm{eff}$ up to a shape-dependent correction within the Dud\'{\i}k et al. (2014) intensity-ratio envelope. At $\kappa = 2.5$ the Kullback-Leibler divergences between the true distribution and its two Maxwellian projections evaluate to $0.32$ and $1.20$ nats, and their difference satisfies $\Delta D_\mathrm{KL} = (3/2)[R_0 - \ln R_0 - 1] = (3/2) d_\mathrm{IS}(T_\mathrm{eff}, T_\mathrm{core})$, where $R_0 \equiv \kappa/(\kappa - 3/2)$ is the ideal closed-form ratio and $d_\mathrm{IS}$ is the Itakura-Saito distance. The identity is offered as an analytical reference for observational systems in which two diagnostics project different moments of a common non-equilibrium distribution; the eight-year stability of $R$ expresses a stability of that projection structure.