Ruizhu Chen^1,2 and Junwei Zhao²
1. Department of Physics, Stanford University, Stanford, CA 94305-4060
2. W. W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305-4085

Time–distance helioseismology measures acoustic travel times of p-mode waves travelling inside the Sun to infer the structure and flow field in the solar interior¹; however, both the measured mean travel times and travel-time differences of oppositely traveling waves suffer systematic center-to-limb variations^{2, 3}. In particular, the center-to-limb variation in travel-time shifts (CtoL effect) has a significant impact on the inference of large-scale flows, such as the Sun’s meridional circulation³. The CtoL effect needs to be removed from the helioseismic measurements, but not much about the physical causes and the observational properties of the CtoL effect are known. Therefore, here we measure the CtoL effect in the Fourier domain and study its dependence on various parameters, with an attempt to provide more clues on the nature of this effect.

In this study, we develop a new analysis approach to calculate the frequency-dependent time shifts (or phase shifts) in the Fourier domain from the cross-correlation functions in the time-distance measurements. Helioseismic waves typically span from 2.0 to at least 7.0 mHz, while previous time-domain methods of fitting the cross-correlation functions measure mostly on the dominant frequency, of about 3.3 mHz. Our new approach can now recover more information on all frequencies, and we thus apply it to the CtoL-effect measurements along the equator using 7 years of Doppler-velocity data from the SDO/HMI.

Figure 1| CtoL effect as functions of ∆ and ν, for selected disk locations: (a) ϕ = 10.8°, (b) ϕ = 21.6°, (c) ϕ = 32.4°, and (d) ϕ = 43.2°.

We study the CtoL effect δτ_CtoL as a function of disk-centric location ϕ (in this case, the apparent longitude without considering B-angle), wave travel distance ∆, and frequency ν. Figure 1 shows the CtoL effect for a few selected ϕ’s, exhibiting a significant frequency dependence. The δτ_CtoL in all panels show a similar pattern, i.e., it varies notably with frequency, reversing signs at a frequency around 5 mHz and reaching maximum around 4 mHz before the sign reversal. The sign-reversal and the maximum-value frequencies both remain roughly flat with ∆, although the values increase substantially with both ∆ and ϕ.

Figure 2| (a) CtoL effect in travel-time shifts, displayed as a function of ν for selected ∆’s and for ϕ = 43.2°. (b) Same as (a), but all curves are linearly scaled to ∆ = 21.6°. (c)(d) Same as (a)(b), respectively, but in phase shifts.

We then study the frequency dependence variation with ∆. Figure 2 shows the profiles of δτ_CtoL for a few ∆’s and at ϕ = 43.2°. The curves in Figure 2a, for evenly-spaced travel distances, are also approximately evenly spaced in the magnitude. The shapes of these curves, i.e., their frequency dependence, are very similar. If we linearly scale all the curves to a same ∆, e.g., 21.6°, by the ratios of their respective travel distances to 21.6°, almost all curves fall onto a similar frequency-dependent profile, as shown in Figure 2b. This is a remarkable property of the CtoL effect that was unknown before. Since different travel distances represent waves traveling to different depths inside the Sun, that the frequency dependence does not rely on the travel distance indicates that the frequency dependence of the CtoL effect is more a surface effect.

Figure 3| (a) Profile of ⟨δτ_CtoL(ϕ, ν)⟩ for an effective travel distance ∆ = 21.6°. The dotted line is a smoothed contour of δτ_CtoL= 0, and the dashed line represents the frequency of 5.4 mHz. (b) Frequency-dependent ⟨δτ CtoL⟩ for selected ϕ’s after a finite-light-speed correction.

Next we study the frequency dependence variation with ϕ. Since ∆ does not change the frequency dependence of δτ_CtoL but only changes its magnitude, we can scale the magnitude linearly by ∆ and then average over all ∆’s to obtain a general ⟨δτ_CtoL(ϕ, ν)⟩ profile, as shown in Figure 3a. The sign-reversal frequency drops sharply from about 5.4 mHz near the disk center to approximately 4.3 mHz at ϕ = 60◦. The curves of ⟨δτ_CtoL⟩ (after a finite-light-speed correction, see Ref 5) for selected ϕ’s are shown in Figure 3b, where there is clear variation among different ϕ’s. The 5.4 mHz sign-reversal frequency is close to the cutoff frequency observed for a quiet Sun⁴, and the variation with ϕ indicates that the CtoL effect may be related to the spectrum line formation height.

We have shown that the CtoL effect has a significant frequency dependence, which varies with disk-centric distance but not much with travel distance. For comparison, we also examined the flow-caused travel time shifts, but there is no clear frequency dependence found.

These new findings may shed light on the cause of the CtoL effect, and possibly lead us to new methods of effect removal when inferring the Sun’s interior flows. For more details of this work, please refer to our recent publication Ref 5.

[1] Duvall, T. L., Jr., Jefferies, S. M., Harvey, J. W., & Pomerantz, M. A. 1993, Nature, 362, 430
[2] Duvall, T. L., Jr. 2003, GONG+ 2002. Local and Global Helioseismology: the Present and Future, 517, 259
[3] Zhao, J., Nagashima, K., Bogart, R. S., Kosovichev, A. G., Duvall, T. L., Jr. 2012, ApJL, 749, L5
[4] Jiménez, A., García R. A., & Pallé, P. L. 2011, ApJ, 743, 99
[5] Chen, R., & Zhao, J. 2018, ApJ, 853, 161