RTLI Variance-Geometry Diagnostics in Planetary Plasma Tori: Eigenvalue Structure and √eThresholds at Jupiter and Saturn

This technical note applies eigenvalue-based variance-geometry diagnostics to two planetary plasma torus systems: Jupiter’s Io plasma torus using 33 years of observational brightness data, and Saturn’s Enceladus plasma torus using a diffusive equilibrium magnetospheric model. The analysis employs the eigenvalue ratio \lambda_e = \sigma_1^2 / \sigma_2^2 and the Intrinsic Variance Ratio (IVR) to identify structural transition layers in spatially resolved datasets. For Jupiter, latitude-resolved IVR analysis reveals a transition layer at planetocentric latitudes −2° to −3°, where IVR ≈ 1.627, within 1.3% of \sqrt{e} \approx 1.649. This layer separates temporally stable northern emission from dynamically variable southern emission and is accompanied by a statistically significant north–south brightness asymmetry (mean ratio 2.63, p < 10⁻¹⁵⁹). Null tests confirm that this structure is not a statistical artifact. For Saturn, analysis of a symmetric diffusive equilibrium model identifies angular transition layers at ±19°–21°, consistent with the RTLI coherence angle \theta_c \approx 20.2^\circ, and a radial torus width equal to 0.60 times the Enceladus orbital radius, within 1% of 1/\sqrt{e} \approx 0.6065. Due to the model’s symmetry and lack of temporal variability, IVR and asymmetry diagnostics cannot be computed, and results are presented as structural predictions rather than observational confirmations. Across both systems, √e and 1/√e emerge as empirical organizing thresholds in plasma torus geometry, despite differing plasma compositions and source mechanisms. The framework makes no claims regarding new physical mechanisms, but provides a scale-independent diagnostic for locating structural transitions. The analysis generates falsifiable predictions for future observational testing using archival Cassini data, future outer-planet missions, and laboratory plasma confinement systems.