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

11 March 2002Big Bear Solar Observatory The Sun’s Irradiance Variations are Small, but Hard to Explain

11 March 2002Big Bear Solar Observatory Sunspot Number  Sunspot number 1620 – 2000

11 March 2002Big Bear Solar Observatory 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? -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 -Wojtek Dziembowski, University of Warsaw

11 March 2002Big Bear Solar Observatory Variations in the Solar Radius  Brown & Christensen-Dalsgaard (1998): annual average R s 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)

11 March 2002Big Bear Solar Observatory 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

11 March 2002Big Bear Solar Observatory Normal Modes of the Sun ( )

11 March 2002Big Bear Solar Observatory Helioseismic Radius Using MDI f-modes   l 2  l(l+1)GM s /R l 3 : Asymptotically surface waves, but f-modes see different effective gravities – depending on l   l / l =-3/2  R s /R s :  means model value minus true value  From this, Schou et al. (1997) determined R s = ±0.03 Mm

11 March 2002Big Bear Solar Observatory The Surface Radius of the Sun  Auwers (1891): R s =D s /2=695.99±0.04 Mm --standard value for 100 years --standard value for 100 years  Schou et al. (1997): R s = ±0.03 Mm --MDI f-modes --MDI f-modes  Brown & Christensen-Dalsgaard (1998): R s =D s /2= ±0.026 Mm --HAO Solar Diameter Monitor --HAO Solar Diameter Monitor

11 March 2002Big Bear Solar Observatory Solar Cycle Variations in the f-mode Radius  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 R l  R s, cannot expect  R l  R s. 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).

11 March 2002Big Bear Solar Observatory Still More Problems in Applying Formula to Determine  R More problems with using the equation to determine evolution of radius:  Each mode has its own radius, and so the induced modifications are quite non-uniform  f-modes are not discontinuity modes that see the same gravity, each mode sees a gravity depending on l.

11 March 2002Big Bear Solar Observatory Variational Principle for l,— from Definition of Radius  Relative departures small (2  at l=100 to 8  at l=300), so R l  R s, but not  R l  R s  For high degree modes, f-mode radii close to R s, R l /R s range from at l=100 to at l=300 — can expect R l  R s, but cannot expect  R l  R s — can expect R l  R s, but cannot expect  R l  R s — f-mode changes,  R l, refers to a region beneath the surface — f-mode changes,  R l, refers to a region beneath the surface

11 March 2002Big Bear Solar Observatory Rate of Shrinking from f-modes  To account for near-surface contribution to the frequency change we add a second term — I l  0, as l  ,  l weak function of l — I l  0, as l  ,  l weak function of l

11 March 2002Big Bear Solar Observatory  vs., as Activity Rises   data – model  Solid line is fit to data from two term equation  Dashed line ignores second term  R s increased by 20 km (within errors)

11 March 2002Big Bear Solar Observatory Evolution of f-mode Radius  With  : dR f /dt = -1.51±0.31 km/y  Without  : dR f /dt = -1.82±0.64 km/y  Results imply at a depth of 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

11 March 2002Big Bear Solar Observatory Near-Surface Terms from p- modes  Note that rise of cycle starts early 1997, so fits for dR/dt start then  Note   systematically rises, but behavior is much more muted than anisotropic terms — this is a clue to the shrinking of outermost layers of the sun

11 March 2002Big Bear Solar Observatory Relative Contributions of Radius Change and Near-Surface Term to Frequency Change l l [mHz] l [mHz] Il Il Il Il  R [  Hz]   [  Hz]

11 March 2002Big Bear Solar Observatory f-mode Radius Changes -- Entropy and Random R.M.S. Field, Goldreich et al. (1991)

11 March 2002Big Bear Solar Observatory

11 March 2002Big Bear Solar Observatory How Big Can the Shrinking Be?

11 March 2002Big Bear Solar Observatory 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!

11 March 2002Big Bear Solar Observatory 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 — because of rapid decline of gas pressure — thermal structure most susceptible to field induced changes in convective energy transport efficiency — thermal structure most susceptible to field induced changes in convective energy transport efficiency

11 March 2002Big Bear Solar Observatory 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 — A much more precise value to describe the last 4Mm

11 March 2002Big Bear Solar Observatory Frequency Dependence of  p — Constraint on Source Location  Goldreich et al. (1991) explained frequency evolution with field and temperature changes. Field dominates and temperature increase used to account for high frequency reversal of increasing trend in  ( ) BBSO data.

11 March 2002Big Bear Solar Observatory Consider Radial R.M.S. Random Field Growth (  =-1), then  T/T as Source  Solid lines and dotted lines fit , other forms possible  < 100 G  < 300 G at 4 Mm   T/T too large at surface  cannot exclude  T/T at level in subphotospheric layers level in subphotospheric layers.

11 March 2002Big Bear Solar Observatory Role of Temperature Decrease  Decrease in  T/T at level would be significant contributor to increasing. Brüggen and Spruit (2000) argue that an increasing magnetic field would induce a lower subsurface temperature, thereby reducing the required increase in magnetic field in subsurface layers to match frequency increase.

11 March 2002Big Bear Solar Observatory Role of Decrease in Turbulent Velocity  Roughly, relative change in turbulent velocity,  v t /v t =q, has same effect as relative temperature change  T/T =0.5qM 2, where M is the turbulent Mach number.  Effect may be significant because we can expect a decrease in v t with increasing activity, because the magnetic field should inhibit convection.

11 March 2002Big Bear Solar Observatory Expanding or Shrinking in the Outermost Layers  Isotropic random r.m.s. field implies increase (as before), but such a field seems unlikely. The changes in radius refer to a mass point corresponding to the photosphere at solar minimum.  [G] [G] (dR/dt) ph km/y km/y 1/ km/y 2.3 km/y

11 March 2002Big Bear Solar Observatory 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 ).

11 March 2002Big Bear Solar Observatory Net Radius Change/ Year  Don’t expect non-magnetic contributors would significantly alter results in table  (dR/dt) ph (dR/dt) f (dR/dt) T -1.3 km/y -1.5 km/y ~-3 km/y -1.1 km/y -1.5 km/y ~-3 km/y 1/3 2.3 km/y 2.3 km/y -1.5 km/y ~+1 km/y

11 March 2002Big Bear Solar Observatory Conclusions  f-mode data imply contraction, ~1.5 km/y, in the sun’s outermost layers (4-8 Mm beneath surface) during rising activity. Growing, but modest radial field implies shrinking, but growing random field implies expansion.  To account for p-mode changes with magnetic field requires an inward increase of the growing field. Near-surface layers may either slightly expand or contract depending on form of the perturbation.  Temperature decrease and/or decreasing turbulent velocities may play a role.  Radius changes of about –3 km/y are inferred during the rising phase of activity. Perhaps some part of difference with Emilio et al. (5.9  0.7 km/y) due to definition of photosphere — both imply a negligible contribution of the radius change to solar irradiance variations (  L/L= 4  T/T+ 2  R/R, over half a cycle  L/L  10 -3,but  R/R  ). — both imply a negligible contribution of the radius change to solar irradiance variations (  L/L= 4  T/T+ 2  R/R, over half a cycle  L/L  10 -3,but  R/R  ).

11 March 2002Big Bear Solar Observatory 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

11 March 2002Big Bear Solar Observatory Small-scale Aspherical Random R.M.S. Field Each component, k, gives rise to P 2k distortion of sun’s shape and for k>0 temperature pert.

11 March 2002Big Bear Solar Observatory For k>0, Eliminate  T/T, and

11 March 2002Big Bear Solar Observatory Is the Sun Hotter at Activity Minimum?

11 March 2002Big Bear Solar Observatory What Next ?  Treat condition of thermal equilibrium to constrain surface averaged temperature change because of the sharper minimum in  2 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

11 March 2002Big Bear Solar Observatory Is the Sun Hotter at Activity Maximum?  Radial random field increase in outer layers leads to frequency increase, while horizontal field increase likely to have opposite effect  Most economical requirement is a purely radial field