Observational Evidence Linking Loop Length and Thermal-Nonthermal Peak Timing in Solar Flares

We investigate how the magnetic loop length of solar flares relates to the timing between their thermal and nonthermal emission signatures. Our study analyzes a sample of 96 C-, M-, and X-class flares observed between 2013 and 2015 with soft X-rays, hard X-rays, and extreme UV. For each event, we determine the time delay {\Delta}t between the hard X-ray and soft X-ray peak, and estimate the flare loop length L from UV footpoints assuming a semicircular geometry. In every case, longer flare loops are consistently associated with larger timing delays. Across the full sample, we find a strong correlation, R = 0.88 between L and {\Delta}t. We also quantify how closely each flare follows the Neupert effect using a coefficient RN, defined as the Pearson correlation between the time derivative of the soft X-ray flux and the hard X-ray light curve. Applying correlation thresholds of RN > 0.5 and RN > 0.8 yields subsets of 87 and 46 events, respectively. In both cases, the linear relationship between loop length and peak delay remains clearly expressed. For the RN > 0.5 subset, the correlation is R = 0.87, while the more selective subset with RN > 0.8 displays an even stronger correlation of R = 0.91. These results show that the overall trend persists across increasingly stringent correlation thresholds. The results provide direct observational confirmation that magnetic loop geometry plays a key role in governing the temporal evolution of energy transport in solar flares.