Paul S Cally (Monash) Charles Lindsey (CoRA/NWRA) Matthias Rempel (HAO)

1. Paul S Cally (Monash)Charles Lindsey (CoRA/NWRA)Matthias Rempel (HAO) Seismic Probing of Sunspot Umbrae Lindsey, Cally & Rempel 2010, “Seismic Discrimination of Thermal and Magnetic Anomalies in Sunspot Umbrae”, ApJ (accepted)

2. What I’m going to tell you • Seismology of sunspots depends on both thermal and magnetic effects • Ingression-control-correlation (holography) maps of sunspots • How to disentangle using travel time maps? • Construct umbral models for fully compressible radiative magnetoconvective simulations • Apply linear acoustic simulations to umbral centre models to probe “thermal anomaly” GONG 2010 / SOHO 24 Aix en Provence

3. Sunspot seismology: deep or shallow? • Increasing evidence that the seismology of sunspots is essentially a shallow (<4 Mm) phenomenon [Lindsey & Braun (2005a,b), Schunker et al (2005); Korzennik (2006); Braun & Birch (2006); Gizon, Hanasoge & Birch (2006), Hanasoge et al (2008)] • Effect ~vanishes at low frequency The phase, φ−, of the ingression control correlation, CLC−, integrated over 1-mHz pass bands centered at—proceeding from left-to-right—2, 3, 4, and 5 mHz is mapped in the neighbourhood of AR9902 on 2004 April 12. The gray scale (right) calibrates the phase of the control correlation in radians. The penumbral and center-umbral phase signatures increase sharply with frequency, ν, from insignificance at 2 mHz to clearly significant at 4 and 5 mHz. [Pupil: 16.5 – 48 Mm] GONG 2010 / SOHO 24 Aix en Provence

4. Sunspot seismology: magnetic or thermal? • Moradi & Cally (2008): travel-time perturbations for high-l rays skipping across penumbrae are predominantly magnetic • Cally (MNRAS, 2009): “A useful rule-of-thumb appears to be that travel-time perturbations in umbrae are predominantly thermal, whereas in penumbrae they are mostly magnetic” • Umbral magnetic field near-vertical • Therefore near-vertical waves (low-l) from below interact only weakly with the field • Use low-l waves to probe umbral “thermal” structure GONG 2010 / SOHO 24 Aix en Provence

5. Near-vertical wave hardly sees near-vertical B GONG 2010 / SOHO 24 Aix en Provence

6. Atmosphere’s taken from time-averaged Rempel et al simulations • Low resolution (96 km × 32 km) 1022 Maxwell sunspot [LRRMC model] • 26 hr simulation in 49 Mm × 8 Mm box (high res can’t run that long) • No penumbra (insufficient resolution) • Average atmosphere over 24-26 hrs: LRRMC-25 • ~700 km Wilson depression • Initiated with unrealistically strong flux tube that results in (probably) too large enthalpy deficit: hence less “extreme” GHSE-07 model with Wilson depression ~400 km GONG 2010 / SOHO 24 Aix en Provence

7. Bottom panel: magnetic field strength in a vertical slice through sunspot centre. Maximum field at base around 10 kG. Evolution between 3 hrs and 15 hrs. [Courtesy Matthias Rempel] GONG 2010 / SOHO 24 Aix en Provence

8. Model umbrae with Wilson depression “Low Resolution Radiative-Convective-Magnetic” model based on Rempel et al (2009) LRRMC-25 model Dashed model is GHSE-07 (“Gravitational-HydroStatic-Equilibrium”) GONG 2010 / SOHO 24 Aix en Provence

9. Temperature deficit relative to QS The thermal anomaly of the LRRMC-25 umbra is represented by solid curves in terms of its temperature deficit, ∆T, with respect to that of the quiet Sun at the same column mass density, m. The vertical blue and green lines mark the bases of the photospheres of the umbra and quiet Sun, respectively. In the thermal deficit, we recognize a “shallow component” (solid red curve) and a “deep component” (solid brown curve). The column mass densities, m to be identified with a selection of depths, z, in the quiet Sun and sunspot umbra are indicated by skewed fiducials connecting number lines labeled “Quiet” (green) and “Umbra” (blue) beneath the temperature-deficit profile. Blue, red, and brown dashed curves in the top plot represent temperature-deficit profiles of composite, shallow, and deep components, respectively, of an empirical model, GHSE-07, fashioned to fit the helioseismic observations. GHSE-07 is shallower GONG 2010 / SOHO 24 Aix en Provence

10. Why does cool umbra lead to reduction in travel time? • Use column mass density m rather than z • Cooling of medium causes collapse of layer ⇒ Wilson depression • Umbra opaque at greater m due to lower opacity of cooler gas, so “surface” even deeper in m in umbrae than in QS; reduces travel time further • Travel time (simplified view) • Reduced T leads to reduced travel time • Greater m(obs) amplifies effect • Also reduced 1/ωac, i.e., increased acoustic cutoff freq. GONG 2010 / SOHO 24 Aix en Provence

11. Travel time deficits in model atmospheres Travel time deficits as a function of frequency. The solid blue curve shows the travel-time deficit, ∆τ, determined by application of acoustic simulations described by Cally (2009) to the LRRMC-25 model represented by the solid blue curve in Figure 7. The dashed blue curve shows ∆τ determined likewise for the GHSE-07 model, represented by the dashed blue curve in Figure 7. The solid and dashed brown curves show the real part of ∆τr, the travel-time deficits estimated by the ray approximation for the LRRMC-25 and GHSE-07 models, respectively. The magenta data points show mean travel-time deficits derived from the centre-umbral phase anomalies of ARs 9905 and 9902. Error bars express the root mean square variation in the individual travel-time-deficit measurements that contribute to the means plotted. GONG 2010 / SOHO 24 Aix en Provence

12. Summary • Sunspots affect wave travel times thermally and magnetically • Probe umbral thermal structure with near-vertical (low l) waves • Shallow anomaly can explain travel time perturbations • Travel times are shorter than in quiet sun, despite lower temperature, due to Wilson depression • Test on models based on the Rempel et al numerical simulations • Can be used to discriminate between umbral models • Inversion …? GONG 2010 / SOHO 24 Aix en Provence

13. The Endthanks GONG 2010 / SOHO 24 Aix en Provence

14. Abstract Helioseismic observations of sunspots show travel times considerably reduced with respect to equivalent quiet-Sun signatures. We note that waves propagating vertically in a vertical magnetic field are relatively insensitive to the magnetic field, while remaining highly responsive to the attendant thermal anomaly. Travel-time measurements for waves with large skip distances into the centres of axially symmetric sunspots are therefore a crucial resource for discrimination of the thermal anomaly beneath sunspot umbrae from the magnetic anomaly. One-dimensional models of sunspot umbrae based on radiative magneto-acoustic simulations by Rempel et al. can be fashioned to fit observed helioseismic travel-time spectra in the centres of sunspot umbrae. These models are based on cooling of the upper 2–4 Mm of the umbral subphotosphere with no significant anomaly beneath 4.5 Mm. The travel-time reductions characteristic of these models are primarily a consequence of a Wilson depression resulting from a strong downward buoyancy of the cooled umbral medium. GONG 2010 / SOHO 24 Aix en Provence

15. Acoustic diffraction (according to Charlie) I'm going to assume that we're talking about ingression control correlations, which are statistics of the correlation between the acoustic ingression and the local acoustic amplitude. Here, then, is my basic understanding of the issue: • First: The resolution of the acoustic ingression is limited by acoustic diffraction. The acoustic diffraction limit is determined by the pupil of the ingression computation. Selection of the pupil is generally subject to a trade-off: The larger the inner radius of the pupil, the more vertical the propagation of waves represented into the umbra, and the better the avoidance of contamination of the pupil by the periphery of the active region; however, the poorer the acoustic resolution. For the pupils we specify, the angle of acoustic illumination of the focus is within 20° from vertical. The diffraction limit is, then, wavelength/sin(20°) ~3 wavelengths. In the case of 5 mHz waves, the wavelength is 200 sec (i.e., 1/[5 mHz]) times 7 km/sec (the photospheric sound speed), i.e., ~4.5 Mm. For comparison, the pixel resolution of the full-disk MDI images is 2.8 Mm (i.e., 2 times the pixel spacing of ~1.4 Mm). So the effect of acoustic diffraction can be likened to smearing of the image to something like 4.5/2.8 = 1.6 times the pixel resolution. Of course, for lower frequencies, the acoustic diffraction limit is proportionately coarser. • Second, and this can be important: The effective resolution of the control correlations is the electromagnetic (i.e., optical) resolution of the local acoustic amplitude by the helioseismometer, even when the acoustic resolution of the ingression is much poorer than the former. A relatively poor acoustic resolution results not in a smeared correlation map but rather a noisier one. If the local acoustic amplitude in a given pixel is well resolved from that of other pixels, then its correlation with a diffraction-smeared incoming acoustic field (the ingression) should faithfully characterize the response of that pixel to the component of the total incoming signal represented by the ingression. Diffraction-smearing of the ingression is basically a manifestation of the fact that the ingression computed over a limited pupil represents only part of the acoustic radiation that is impinging into the pixel in question. We regard whatever else that is impinging into that pixel as statistically independent of the acoustic ingression, i.e., noise that introduces statistical uncertainties that show up in the error bars. This has obvious liabilities, but does not contribute to the control correlation in any systematic way. Thus, while diffraction smearing of the ingression can introduce noise, it cannot introduce contamination in terms of how other pixels respond to acoustic radiation to which they are exposed. GONG 2010 / SOHO 24 Aix en Provence

16. Acoustic diffraction (according to Charlie 2) Laurent's point: Can you get something quantitative on Laurent's estimate of the resolution to compare with my back-of-the-envelope estimate? The pupil that applies to the images in our paper has an inner radius of 15 Mm and an outer radius of about 45 Mm. Keep in mind that the diffraction limit is basically directly proportional to the wavelength. The estimate I sent you was for 5 mHz acoustic radiation. For 2.5 mHz, it would be double, i.e., 9 Mm instead of 4.5 Mm, meaning four times the area. Bear in mind my insistence that the result is correlations that are noisier at lower frequencies rather than smeared. If the helioseismic observations of sunspot umbrae are not significantly contaminated by signal from the sunspot penumbrae, then I do not see a mechanism whereby the correlations could be contaminated with penumbral responses, either to the same acoustic radiation or any other, however poor the resolution of the acoustic ingression. I could offer you any of several analytical resources to support the back-of-the-envelope resolution estimate I sent you. However, I might rather suggest that you stash the attached Figure in your presentation and invite Laurent to raise this point in the session---if this question remains outstanding in Aix. The attachement is Figure 5 from the paper Alina and I presented at the SOHO XIX meeting in Melbourne. It compares an egression power map of one of the more compact acoustic emission kernels we've gotten with a cospatial/cospectral/concurrent white-light power map, in this case from the flare of 2003 October 29. We've tried, so far with limited success in the US, to capitalize on the use of flare acoustic emission as a major control resource for issues such as spatial resolution. This may be a golden opportunity both to emphasize the control utility of flares and confront some nagging doubts as to whether seismic holography has anything approaching the local discrimination we have so ardently claimed. In this instance, the frequency was 6 mHz (as opposed to 5 mHz in the estimate I sent you yesterday), whereby the back-of-the-envelope estimate of the resolution becomes 3.8 Mm (as opposed to 4.5 Mm). Using my screen ruler, I estimate the E-W width of the white-light kernel (left frame of the figure) to be 4 Mm. The width of the 6 mHz egression-power signature is very little greater; I roughly estimate 5 Mm. My general understanding is that diffraction smearing is basically a convolution, meaning that the rms widths should add in quadrature. That would suggest a smearing width of root(52-42) = 3 Mm. The difference between this and 3.8 is securely encompassed by the 1-pixel discrimination provided by the screen ruler. In any case, the full width of the egression power signature is only 5 Mm, so, I do not see how the resolution could be very much broader than the back-of-the-envelope estimate in practice. GONG 2010 / SOHO 24 Aix en Provence

17. Maps of the phase, φ−, of the local ingression control correlation, CLC−, for two nearly circular sunspots over several consecutive days. Top row shows φ− for seismic ingressions in the 3.5–4.5 mHz spectrum computed over an annular pupil whose radial range is 16.5–48.0 Mm centered on the focus. Middle row shows concurrent, cospatial MDI continuum-intensity maps. Bottom row shows concurrent, cospatial line-of-sight MDI mag- netograms. The phase maps generally show a ring-like phase enhancement in φ− marking the penumbra of the sunspot. Inside of this “penumbral phase anomaly” is a significant umbral signature that tends to be accentuated towards the center, suggesting somewhat of a bull’s eye. GONG 2010 / SOHO 24 Aix en Provence

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19. Sunspot seismology: magnetic or thermal? • Cally (2009): travel time perturbations at various frequencies and field orientations θ, ϕ • Ad hoc thermal model with weak Wilson depression • Full curve: numerically derived Δt(B,ΔT) • Short-dashed: Δt(B,0) • Dotted: Δt(0,ΔT) • Long-dashed: Δt(B,0)+Δt(0,ΔT) GONG 2010 / SOHO 24 Aix en Provence

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