Investigation on the Relation between Active Regions' Compliance with Empirical Laws and Flare Productivity

Abstract

It remains evasive whether solar active regions (ARs) obeying or violating Hale's polarity law, Joy's tilt law, and the hemispheric helicity rule (HHR) differ in flare productivity. Here we conduct a comprehensive statistical analysis of ARs during the Solar Cycle 24 and the ascending phase of Cycle 25. ARs are automatically detected from full-disk line-of-sight magnetograms acquired by the Michelson Doppler Imager (MDI) and the Helioseismic and Magnetic Imager (HMI). We calculate tilt angles via flux-weighted polarity centroids, estimate magnetic twist by the force-free parameter $α_{\mathrm{best}}$ from HMI vector magnetograms, and measure flare productivity using the flare index (FI) built from GOES C-class-and-above events. Our results substantiate that the majority of ARs follow the aforementioned three empirical laws. The compliance rate tends to be higher for ARs emerging at higher latitudes or having larger centroid distance, while total unsigned magnetic flux exerts limited influence, with a clear positive correlation only for Hale's law. Overall, FI shows no significant discrepancies across different compliance groups, except that Cycle 24 ARs that satisfy Hale's and Joy's laws but violate the HHR exhibit higher FI than other groups. We also identify empirical thresholds for centroid distance and total unsigned flux, above which the median FI of binned ARs becomes nonzero. Combining the flux and distance thresholds effectively separates flare-productive from flare-quiet ARs. We hence conclude that the flare productivity of ARs is not dependent on the compliance with the empirical laws, but more closely associated with sufficiently large and strong magnetic systems.