The Dependence of Joy's Law and Mean Tilt as a Function of Flux Emergence Phase

Data from the Michelson Doppler Imager (MDI) and Helioseismic and Magnetic Imager (HMI) are analyzed from 1996 to 2023 to investigate tilt angles (\(\gamma\)) of bipolar magnetic regions and Joy's Law for Cycles 23, 24, and a portion of 25. The HMI radial magnetic field (\(B_{r}\)) and MDI magnetogram (\(B_{los}\)) data are used to calculate (\(\gamma\)) using the flux-weighted centroids of the positive and negative polarities. Each active region (AR) is only sampled once. The analysis includes only Beta (\(\beta\))-class active regions since computing \(\gamma\) of complex active regions is less meaningful. During the emergence of the ARs, we find that the average tilt angle (\(\bar{\gamma}\)) increases from 3.30\(^{\circ}\pm\)0.75 when 20\% of the flux has emerged to 6.79\(^{\circ}\pm\)0.66 when the ARs are at their maximum flux. Cycle 24 had a larger average tilt \(\bar{\gamma}_{24}\)=6.67\(\pm\)0.66 than Cycle 23, \(\bar{\gamma}_{23}\)=5.11\(\pm\)0.61. No significant difference is found in the slope of Joy's law or \(\bar{\gamma}\) when sampling the ARs at the time of maximum flux or central meridian crossing. There are persistent differences in \(\bar{\gamma}\) in the hemispheres, with the southern hemisphere having higher \({\bar{\gamma}}\) in Cycles 23 and 24, but the uncertainties are such that these differences are not statistically significant.