1. An Introduction to Helioseismology (Global) 2008 Solar Physics Summer School June 16-20, Sacramento Peak Observatory, Sunspot, NM

2. Special Acknowledgments Rachel Howe Rudi Komm Frank Hill ( National Solar Observatory )

3. Global Helioseismology What is helioseismology? –A bit of early history Basics –p-modes and g-modes –Spherical harmonics Observations –Instrumentation –Networks and spacecraft –Time series –Spectra Methods –Peak finding –Inversions Results –Internal Properties of the Sun –Solar Cycle variations

4. The early “days” of helioseismology Discovered in 1960 that the solar surface is rising and falling with a 5-minute period Many theories of wave physics postulated: –Gravity waves or acoustic waves or MHD waves? –Where was the region of propagation? A puzzle – every attempt to measure the characteristic wavelength on the surface gave a different answer

5. The k-  (diagnostic) diagram Acoustic waves trapped within the internal temperature gradient predicted a specific dispersion relation between frequency and wavelength A wide range of wavelengths are possible, so every early measurement was correct – result depended on aperture size Observationally confirmed in 1975 Max amplitude 20 cm/s

6. Helioseismology: A window to the Sun’s Interior

7. What is helioseismology? Helioseismology utilizes waves that propagate throughout the Sun to measure its invisible internal (and external) structure and dynamics.

8. Three types of modes G(ravity) Modes – restoring force is buoyancy – internal gravity waves P(ressure) Modes – restoring force is pressure F(undamental) Modes – restoring force is buoyancy modified by density interface – surface gravity waves

9. Wave trapping G modes exist where ω < N 2 (Brunt-Väisälä frequency) P modes exist where ω S (Lamb frequency) F modes are analogous to surface water waves

10. Sound Source - Granulation

11. The P modes A p mode is a standing acoustic wave. Each mode can be described by a spherical harmonic. Quantum numbers n (radial order), l (degree), and m (azimuthal order) identify the mode.

12. Spherical Harmonics The harmonic degree, l, indicates the number of node lines on the surface, which is the total number of planes slicing through the Sun. The azimuthal number m, describes the number of planes slicing through the Sun longitudinally. - l ≤ m ≤ l Picture credits: Noyes, Robert, "The Sun", in _The New Solar System_, J. Kelly Beatty and A. Chaikin ed., Sky Publishing Corporation, 1990, pg. 23. l=6 m=0 l=6 m=3 l=6 m=6

13. Temporal Frequency units ν = 1/(Period in seconds), units are Hertz (Hz) ω = 2 π /(Period in seconds), units are radians/sec P = 5 min = 300 sec, ν = 3.33 mHz or 3333.33 μ Hz; ω = 2.1 ´ 10-2 rad/s Spatial Frequency units k h = √(l(l+1)/Rsun (Mm -1 )

14. Turning points

15. Ray Paths Turning points

17. Duvall law Modes turn at depth where sound speed = horizontal phase speed = ν /ℓ So, all modes with same ν /ℓ must take same time to make one trip between reflections

18. Rotational Splitting In absence of rotation, have standing wave pattern and degenerate case – the frequency 0 ( =  /2  ) is independent of m In presence of rotation, prograde and retrograde waves have different Observed frequency n,l,m = 0 + δ where δ is the splitting frequency IF the Sun rotated uniformly --> δ depend linearly on m.

19. An Observational Problem The sun sets at a single terrestrial site, producing periodic time series gaps The solar acoustic spectrum is convolved with the temporal window spectrum, contaminating solar spectrum with many spurious peaks

20. Solutions Antarctica – max 6 month duration Network – BiSON, IRIS, GONG – needs data merging, but maintainable Space – SoHO: MDI, GOLF,VIRGO

21. Networks 6-site network of single-pixel instruments, data since 1976, completed 1992. Modes up to l=4 Run by University of Birmingham, UK

22. Networks Six stations around the world for continual coverage. 256x256 pixels 1995- 2001 1024 pixels since 2001 Run from NSO Tucson.

23. Space Instruments 1996 - Present Coming Soon…. MDI GOLF VIRGOHMI

24. Observing Time Series X X X = = = Σ

25. Observing & processing challenges Image geometry is paramount Image scale affects ℓ-scale Angular orientation mixes m-states Fitting of spectral features not trivial Can only view portion of solar surface, so have spatial leakage

26. Solar Acoustic Spectra - Diagramm- Diagram -m- Diagram

27. Fitting the Spectrum Standard model is a Lorentzian profile

28. Fitting the Spectrum Considerations: –Observations in Velocity and Intensity –Asymmetric Profile –Leakage matrix

29. Inversions Modes are reflected due to density variations. The lower the l, the fewer surface reflections, and the deeper the mode penetrates. Combining information from different modes lets us build up a picture of properties at different depths. Modes of different m cover different latitude ranges, giving latitudinal resolution.

30. Kernel Averaging Kernel Coefficients to be found The (rotation) inversion problem

31. Eigenfunctions & Kernels Inversion kernels constructed from eigenfunctions weighted by density

32. Internal Rotation Tachocline Near-surface shear layer

33. Temporal Evolution

34. Temporal Evolution of Zonal Flows

35. Sound Speed and Density Inversions

36. Good job constraining solar structure & dynamo models BUT ( Neutrino experiment solved ) Solar abundances Standard model pretty good, but still discrepancy below CZ Near surface poorly understood Very few p-modes propagate deep enough into the Sun --> G-modes will be very welcome

37. G modes? Simulation Observation Analysis uses: very long time series (10 years) even period spacing of g modes assumed internal rotation estimated observational SNR Intriguing, but needs verification Garcia et al, Science, June 15, 2007