Alexander Kosovichev1 & Jean-Pierre Rozelot2
1. Department of Physics, New Jersey Institute of Technology, Newark, NJ 07102, USA
2. Université de la Côte d’Azur, 77 Chemin des Basses Moulières, F-06130 Grasse, France
First accurate measurements of the size of the Sun performed in 18th and 19th centuries indicated that “the systematically larger diameters correspond to the time when the number of spots and protuberances is lower” (Secchi 1872). The long standing question whether the solar radius is constant or not is still debated. Most recent measurements of the solar limb from the PICARD satellite put an upper limit of 14.5 km (twenty parts in a million) on the solar radius changes during the rising phase of the current sunspot cycle.
Helioseismology provides an alternative measure of the solar radius, so-called “seismic radius”. The seismic radius is determined from frequencies of surface gravity waves (f-modes). The surface gravity waves travel beneath the visible surface of the Sun, and their frequencies are sensitive to the sharp density gradient in the near-surface layers. Therefore, the seismic radius is an attribute of the subsurface stratification of the Sun. It is different from the visual radius; the relationship between them can be made only through modeling. The first helioseismology measurements showed that the Sun’s seismic radius is about 300 km smaller than the predictions of the standard evolutionary solar model calibrated to the visual radius. The discrepancy was explained by an apparent shift of the limb inflection point due to light absorption in the solar atmosphere. This led to a revision of the standard solar radius to the value of 695,700 km by the International Astronomical Union in 2015. Also, initial measurements of the seismic radius from SoHO showed that its changes do not exceed 1-2 km per year.
We use helioseismology data obtained in 1996-2017 from: Solar and Heliospheric Observatory (SoHO) (1996-2010) and Solar Dynamics Observatory (SDO) (2010-2017). For the data analysis we employ theoretical model developed by Dziembowski et al. The model considers two primary contributions to the f-mode frequencies, arising from changes in the stratification of subsurface layers and also from surface effects caused by interaction of f-modes with surface magnetic fields and changes in the near-surface structure. Following this model, we represent the relative frequency variations, , in the following form: , where are relative variations of the seismic radius determined by f-modes of angular degree , and is a function of with coefficient , which describes variations of the surface effects with the solar cycle (so-called, the ‘surface term’), and is the mode inertia. The surface term fits the -dependence quite well meaning that the main part of the frequency variations can be assigned to the ‘surface effects’ which can be due to both, variations of the solar surface structure and interaction of f-modes with magnetic fields at the solar surface. The surface term is subtracted from the observed frequency variations, and the remaining difference represents changes in the seismic radius term.
Figure 1| Solar cycle variations of: a) the mean seismic radius, b) the coefficient, , of the surface perturbation of f-mode frequencies, and c) the sunspot number averaged for the same time intervals as the helioseismology data. The arrow in panel (a) indicates the start of the SDO data set.
Figure 1a shows the temporal variations of the mean seismic radius averaged over all values, relative to the seismic radius in 2009, during the solar deep minimum. Figure 1b shows the variations of the surface term coefficient, . It changes in phase with the solar activity and correlates quite well with the sunspot number (Fig. 1c), except that it decays slower than the sunspot number in the declining phase of the solar cycles. The mean seismic radius changes in anti-correlation with the sunspot number.
Figure 2| Variations with depth of the seismic radius relative to the solar minimum in 2009 for: a) the beginning of Solar Cycle 23, b) maximum of Cycle 23, and c) maximum of Cycle 24. The error bars are uncertainties estimated by Monte-Carlo simulations. The horizontal bars show the depth resolution.
However, the decrease of does not represent simple “shrinking” of the Sun. In fact, the different subsurface layers are displaced by different amount, , and these variations are not homologous. To determine these variations we adopt the helioseismology inversion approach. The inversion results for shown in Fig. 2 reveal that most of the radius displacement was beneath the visible surface in the depth range of 3-8 Mm. The radius variations were not monotonic: above and below of this range the radius increased by 1-2 km. This means that the deeper layers (6-10 Mm) were compressed while the subsurface layers (3-5 Mm) expanded. The most significant variations were at the depth of about Mm, and reached about km in Cycle 23 and km in Cycle 24 (Fig. 3a). Figure 3b shows the variations of displacement (smoothed in time using a Gaussian kernel with the standard deviation of one year to remove the annual instrumental variations) as a function of depth during the whole 21-year period of the helioseismology observations from SoHO and SDO, relative to the deep activity minimum between Cycles 23 and 24 in 2010. Apparently, in Solar Cycle 23 the radius changed substantially greater than in Solar Cycle 24.
Figure 3| Variations of the seismic radius: a) at the depth of 5 Mm; b) with time and depth beneath the solar surface. The dark color (negative values) correspond to contraction and light color (positive values) to expansion of subsurface layers.
The inferred displacements are probably associated with magnetic fields accumulating in the subsurface layers during the solar maxima2. According to the theory of Goldreich et al., the radius change can be explained if the magnetic field is predominantly radial in the subsurface layer located at the depth 5-10 Mm beneath the solar surface and it strength is: kG. This is significantly greater than the strength of magnetic field emerging on the solar surface. As shown by numerical simulations, the subsurface magnetic field becomes predominantly vertical due to convective downdrafts that cause concentration of magnetic field into vertical structures of several kG strength.
For more detals, please see Kosovichev, A. G., & Rozelot, J.-P., 2018, ApJ, 861, 90.
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