Philip R. Goode Big Bear Solar Observatory New Jersey Institute of Technology

1. What Dynamic Changes in the Sun Drive the Evolution of Oscillation Frequencies through the Activity Cycle? Philip R. Goode Big Bear Solar Observatory New Jersey Institute of Technology

2. The Sun’s Irradiance Variations are Small, but Hard to Explain Big Bear Solar Observatory

3. The Solar Cycle Changes in the Sun • Naively, luminosity varies like irradiance and the Sun is a blackbody. Then, allowed size of the changes in the Sun’s output -In truth, we have cast the Sun’s changing magnetic field, thermal structure and turbulent velocity as temperature changes. Further, is there a role for a changing solar radius? • Principal Collaborator: -Wojtek Dziembowski, University of Warsaw Big Bear Solar Observatory

4. Variations in the Solar Radius • Brown & Christensen-Dalsgaard (1998): 1981-1987 annual average Rs each year the same within measurement errors (±37 km) --comparable to those of Wittman (1997) for same time interval --much smaller than annual changes reported by Ulrich & Bertello (1995) -- sun grows with growing activity -- and Laclare et al. (1996) – sun shrinks with growing activity -- for the same interval (200 Km from maximum to minimum) Big Bear Solar Observatory

5. Variations in the Solar Radius from Space • Emilio et al. (2000) MDI limb observations: annual radius increase with increasing activity of 5.9±0.7 Km/year Big Bear Solar Observatory

6. Normal Modes of the Sun( ) Big Bear Solar Observatory

7. Helioseismic Radius Using MDI f-modes • l2l(l+1)GMs/Rl3 : Asymptotically surface waves, but f-modes see different effective gravities – depending on l • l /l=-3/2 Rs/Rs:  means model value minus true value • From this, Schou et al. (1997) determined Rs= 695.68±0.03 Mm Big Bear Solar Observatory

8. The Surface Radius of the Sun • Auwers (1891): Rs=Ds/2=695.99±0.04 Mm --standard value for 100 years • Schou et al. (1997): Rs= 695.68±0.03 Mm --MDI f-modes • Brown & Christensen-Dalsgaard (1998): Rs=Ds/2=695.508±0.026 Mm --HAO Solar Diameter Monitor Big Bear Solar Observatory

9. Must look more carefully at this form of the equation because there are problems with using it to determine the considerably smaller changes the (f-mode) radius from year to year: Even though RlRs , cannot expect RlRs . The problem is we must contemplate changes beyond a simple radius change because near surface effects also contribute to frequency changes — turbulence (Brown 1984) and/or magnetic fields (Evans & Roberts 1990). Solar Cycle Variations in the f-mode Radius Big Bear Solar Observatory

10. To account for near-surface contribution to the frequency change we add a second term — Il 0, as l  , lweak function of l Rate of Shrinking from f-modes Big Bear Solar Observatory

11. With : dRf/dt = -1.51±0.31 km/y Without : dRf/dt = -1.82±0.64 km/y Results imply at a depth of 6-10 Mm, the sun shrank by some 5 km during the rising phase of this activity cycle df/dt= 0.180±0.051 Hz/y, noisy with some cross-talk As small as it is, a shrinking sun is not easy to explain Evolution of f-mode Radius Big Bear Solar Observatory

12. Near-Surface Terms from p-modes • Note that rise of cycle starts early 1997, so fits for dR/dt start then • Note g0 systematically rises, but behavior is much more muted than anisotropic terms — this is a clue to the shrinking of outermost layers of the sun Big Bear Solar Observatory

13. f-mode Radius Changes -- Entropy and Random R.M.S. Field, Goldreich et al. (1991) Big Bear Solar Observatory

14. Big Bear Solar Observatory

15. How Big Can the Shrinking Be? Big Bear Solar Observatory

16. More Acceptable: Variation in the R.M.S. Magnetic Field • For a purely radial random field (=-1), an increasing field implies contraction. • For an isotropic random field ( =1/3), an increasing field implies expansion! A non-trivial constraint! Big Bear Solar Observatory

17. What about the Radius? • Need to have accurate information about the outer ~4 Mm to combine with the shrinking beneath • This outermost region is where one expects the largest activity induced changes — because of rapid decline of gas pressure — thermal structure most susceptible to field induced changes in convective energy transport efficiency Big Bear Solar Observatory

18. p-mode ’s for the Last 4 Mm • p-mode spectrum is >10x richer than that for f-modes — reminder can’t use p-modes for radius (R-1.5 valid only if changes homologous throughout whole sun) • For f-modes: df/dt= 0.180±0.051 Hz/y • For p-modes: dp,0/dt= 0.149±0.008 Hz/y — A much more precise value to describe the last 4Mm Big Bear Solar Observatory

19. Consider Radial R.M.S. Random Field Growth (=-1), then T/T as Source • Solid lines and dotted lines fit , other forms possible • <Bph> < 100 G • <B> < 300 G at 4 Mm • T/T too large at surface • cannot exclude T/T at 10-3 level in subphotospheric layers. Big Bear Solar Observatory

20. Isotropic Random R.M.S. Field in Outermost Layers? • =1/3 is precluded in f-mode layer because that layer shrinks how can it be present above when field above wants to be radial? Radial random field is the most economical to account for frequency changes. • For=-1, the splitting kernels (k>0) are much larger than those for the isotropic part. This is consistent with the anisotropic ’s (like 3) being much larger than for isotropic ’s (0). Big Bear Solar Observatory

21. Is the Sun Hotter or Cooler at Activity Maximum? • Required changes in turbulent flows are probably too large to account for frequency changes • Limiting problem to magnetic field and temperature alone, for aspherical part can use condition of mechanical equilibrium to pose problem for field or temperature change • Spherical part goes through thermal equilibrium condition – much harder to treat Big Bear Solar Observatory

22. Small-scale Aspherical Random R.M.S. Field Each component, k, gives rise to P2kdistortion of sun’s shape and for k>0 temperature pert. Big Bear Solar Observatory

23. For k>0, Eliminate DT/T, and Big Bear Solar Observatory

24. Is the Sun Hotter at Activity Minimum? Big Bear Solar Observatory

25. What Next ? • Treat condition of thermal equilibrium to constrain surface averaged temperature change because of the sharper minimum in c2 for k=0 • More thorough analysis of MDI and GONG data – more years, etc. • Use BBSO Ca II K and Disk Photometer data to constrain the field to link irradiance and luminosity • Three color photometry from Disk Photometer • Use formalism to probe for buried magnetic field Big Bear Solar Observatory