The Dependence of Joy's Law 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 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. 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 errors are such that these differences are not statistically significant.