Sunspot rotation and the field strengths of subsurface flux tubes

Abstract

Observations show that bipolar magnetic regions (BMRs) have differential rotation profiles that are faster than the local Doppler velocity profiles by about 5%, and thep-spots in the growing sunspot groups rotate faster than thef-spots. Also, the smaller spots rotate faster than the larger ones. We present detailed observations of the functional dependence of the residual rotation of sunspots on the spot size of thep- andf-spots of growing sunspot groups. Through numerical calculations of the dynamics of thin flux tubes we show that flux loops emerging from the bottom of the convection zone acquire a rotation velocity faster than the local plasma velocities, in complete contradiction to what angular momentum conservation would demand. The sunspot flux tubes need not be anchored to regions rotating faster than the surface plasma velocities to exhibit the observed faster rotation; we show that this occurs through a subtle interplay between the forces of magnetic buoyancy and drag, coupled with the important role of the Coriolis force acting on rising flux tubes. The dynamics of rising flux tubes also explains the faster rotation of smaller sunspots; we show that there is no need to evoke a radial differential rotation and anchoring of smaller spots to faster rotating regions. The simulated differential rotation profiles of thep- andf-legs of flux loops emerging in the convection zone, with a latitudinal differential rotation and velocity contours constant along cones, mimic the observed profiles for growing sunspot groups only when the flux loops emerge radially and obey Joy's law. (The ‘legs’ are defined to be the vertical part of the loops.) Also the rotation-size relation of growing sunspots is obeyed only by radially emerging loops which obey Joy's law. This constrains the fields at the bottom of the convection zone that are possible for producing the BMRs we see, to lie between 60 and 160 kG, which is in agreement with previous claims.