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

Abstract Data from the Michelson Doppler Imager (MDI) and Helioseismic and Magnetic Imager (HMI) are analyzed from 1996 to 2023 to investigate tilt angles ( γ ) of bipolar magnetic regions and Joy’s law for Solar Cycles 23 and 24 and a portion of Cycle 25. The HMI radial magnetic field ( B r ) and MDI magnetogram ( B los ) data are used to calculate ( γ ) using the flux-weighted centroids of the positive and negative polarities. Each active region (AR) is only sampled once. The analysis includes only Beta ( β )-class ARs since computing γ of complex ARs is less meaningful. During the emergence of the ARs, we find that the average tilt angle ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mrow> <mml:mi>γ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:math> ) increases from 3.°30 ± 0.75 when 20% of the flux has emerged to 6.°79° ± 0.66 when the ARs are at their maximum flux. Cycle 24 has a larger average tilt, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>γ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mrow> <mml:mn>24</mml:mn> </mml:mrow> </mml:msub> </mml:math> =6.67 ± 0.66, than Cycle 23, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>γ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mrow> <mml:mn>23</mml:mn> </mml:mrow> </mml:msub> </mml:math> = 5.11 ± 0.61. No significant difference is found in the slope of Joy’s law or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mrow> <mml:mi>γ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:math> when sampling the ARs at the time of maximum flux or central meridian crossing. There are persistent differences in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mrow> <mml:mi>γ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:math> in the hemispheres, with the Sun's southern hemisphere having higher <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mrow> <mml:mi>γ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:math> in Cycles 23 and 24, but the uncertainties are such that these differences are not statistically significant.