Junwei Zhao¹, S. P. Rajaguru², & Ruizhu Chen¹

1 W. W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305-4085, USA
2 Indian Institute of Astrophysics, Bangalore-34, India

Despite great advances in both global and local helioseismology in the last few decades, helioseismology still faces challenges of some systematic effects. For instance, the helioseismic center-to-limb effect^[1] is an order of magnitude larger than the meridional-circulation-induced signals for certain travel distances^[2,3], posing a great challenge to our inferences of the Sun’s deep meridional circulation. What causes the center-to-limb effect is not exactly known, but it was suggested that it was related to the different heights of the observed oscillations used in the analysis^[4]. Meanwhile, we also must recognize that most of the helioseismic waves we observe are evanescent waves. While it is often expected that phases of evanescent waves no longer change with height in the atmosphere unless perturbed by flows or other physical factors, whether this is exactly the case observationally remains to be examined. Through analyzing a set of well-observed long-duration data with high spatial and spectral resolutions, Zhao et al.^[5] studied how the phases of both evanescent and propagating waves change with atmospheric height. This nugget briefly summarizes this work (Ref. [5]).

Figure 1| (a) A continuum image showing the field of view of the observation with a sunspot at the center of the field. (b) A sample image of Doppler velocities V40, with blue representing blueshift and red representing redshift. The area between the two dashed circles in both panels is used as quiet-Sun region in this study. (c) Comparison of Doppler velocities derived from different intensity levels using the bisector method, taken at a random location for a random 100-min period.

The data used in this study were obtained on 2007 June 8 using the Interferometric BI-dimensional Spectrometer (IBIS) installed at the Dunn Solar Telescope at Sacramento Peak. These observations, as shown in Figure 1, have a spectral resolution of 25 mÅ, spatial resolution of this set of observations is 0.33 arcsec, and temporal cadence of 47.6 sec, lasting for 464 min. For the spectral-line profile acquired at each spatial location, we use 10 bisector levels with equal spacing in line intensity to derive 10 Doppler velocities, corresponding to 0%, 10%, … ,90% intensity levels and relative to the line-core wavelength 6173.34 Å in the rest frame of reference (see Figure 1c). The Doppler velocities derived from different intensity levels form at different optical depths, which correspond to, approximately, different atmospheric heights, with 90%-intensity level slightly above the line’s continuum formation height of 100 km, and 0%-intensity level corresponding to line-core formation height at ∼270 km.

Figure 2| (a) Phase shifts δϕ measured from V₂₀, V₄₀, V₆₀, and V₈₀ relative to V₈₀, displayed as functions of frequency ν. For clarity, standard errors are displayed only for selected points in one of these curves, and the errors for other points in this curve and in other curves are similar. (b) Same as Panel (a) but relative time shifts δτ are displayed.

To measure how phases change in both evanescent and propagating waves above the photosphere, we measure the phase shifts δϕ at each spatial location between the Doppler oscillations obtained at higher atmospheric levels and those at V80 following the formula:
,
where represents the one-dimensional Fourier transform of V_n(x, y, t), † represents the conjugate of the complex numbers, and n represents one of the three numbers: 20, 40, and 60. Figure 2 shows the relative phase shifts δϕ and travel-time shifts δτ measured for all atmospheric levels as functions of ν, averaged from the entire quiet-Sun region where the net vertical flow is presumed to be close to 0. The δτ is computed from δϕ following δτ(ν) = δϕ(ν)/2πν. It can be seen that the δϕ, albeit very small between 1.5 and 4.0 mHz, show positive values below ∼3.0 mHz and negative values above ∼3.0 mHz, indicating that even for evanescent waves in a region with no or little net vertical flows, the measured phases continue to change with height.

Figure 3| (a) Relative phases shifts δϕ, displayed as functions of intensity level for different frequency bands. (b) Same as panel (a) but relative time shifts δτ are displayed.

Figure 3 shows both δϕ and δτ as functions of intensity level (or approximately, atmospheric height), obtained for a few 1.0-mHz-wide frequency bands with their middle frequencies marked in the figure. In most atmospheric layers, the δϕ around 2.0 mHz remain largely flat and positive, and the corresponding δτ are around 1 sec. The δϕ around 3.0 mHz also remain mostly flat, close to 0 across all layers. For all the other frequency bands above 3.0 mHz, the δϕ are negative and have a trend of growth with height, and the δτ values are of an order of 1–2 sec.

Our measurements shown in Figures 2 and 3 pose a great challenge to helioseismic analysis and the interpretation of helioseismic measurements. Any measurements that involve oscillatory signals observed at different atmospheric heights may carry some artificial phase (or travel-time) shifts that are unrelated to flows or interior structures, which were previously thought accounted for those shifts.

For more results and discussions of this work, please refer to Ref. [5].

References

[1] Zhao, J., Nagashima, K., Bogart, R.S., Kosovichev, A.G., & Duvall, T.L. Jr. 2012, ApJ Lett, 749, L5
[2] Rajaguru, S.P., & Antia, H.M. 2015, ApJ, 813, 114
[3] Chen, R., & Zhao, J. 2017, ApJ, 849, 144
[4] Baldner, C.S., & Schou, J. 2012, ApJ Lett, 760, L1
[5] Zhao, J., Rajaguru, S.P., & Chen, R. 2022, ApJ, 933, 109