Comprehensive Analyses of Cumulative Frequency Distributions for Solar X-Ray Flares

Abstract We investigate the temporal evolution of cumulative flare frequency (CFF) distributions using solar X-ray data (0.5–4.0 Å) from Geostationary Operational Environmental Satellites between 1974 and 2020. Beyond the conventional power-law fitting, we introduce a novel modelling approach based on the two-phase exponential association (TPEA) function. This method enables a more robust characterization of distribution shapes and allows for a precise derivation of the power-law index (∝) from the linear component of the model. Our analysis reveals that CFF distributions vary in both structure and scale over time. While the frequency axis reflects clear 11 yr solar activity cycle modulations, we identify a secondary, long-term variation in flare energy that is independent of the activity cycle. The ∝ values derived from the TPEA model exhibit sinusoidal oscillations superimposed on an increasing long-term trend, with occasional values exceeding the critical threshold of ∝ = 2. These findings suggest that both the frequency and energy characteristics of solar flares evolve systematically, and that flare energetics may follow a separate, longer-term modulation beyond the sunspot cycle. This study provides a refined method for tracking flare activity and uncovers new insights into the long-term variability of solar magnetic phenomena.