Solar Activity Phases and Intermediate-degree Mode Frequencies

We analyze intermediate degree p-mode eigenfrequencies measured by GONG and MDI/SOHO over a solar cycle to study the source of their variability. We carry out a correlation analysis between the change in frequencies and several measures of the Sun’s magnetic activity that are sensitive to changes at different levels in the solar atmosphere. The observations span a period of about 12 years starting from mid-1996 (the minimum of cycle 23) to early-2008 (near minimum of cycle 24), corresponding to a nearly complete solar activity cycle. We demonstrate that the frequencies do vary in phase with the solar activity indices, however the degree of correlation differs from phase to phase of the cycle. During the rising and declining phases, the mode frequency shifts are strongly correlated with the activity proxies whereas during the high-activity period, the shifts have significantly lower correlation with all activity proxies, except for the 10.7-cm radio flux. In particular, the proxies that are only influenced by the variation of the strong component of the magnetic field in the photosphere have a much lower correlation at the high-activity period. On the other hand, the shifts are better correlated with the proxies sensitive to changes in the weak component of the magnetic field. Our correlation analysis suggests that more than 90% of the variation in the oscillation frequencies in all activity phases can be explained by changes in both components of the magnetic field. Further, the slopes obtained from the linear regression analysis also differ from phase to phase and show a strong correlation with the correlation coefficients between frequency shifts and solar activity.

💡 Research Summary

The paper presents a comprehensive analysis of the temporal variations of intermediate‑degree solar p‑mode frequencies (ℓ≈20–150) over an almost complete solar cycle, using data from the Global Oscillation Network Group (GONG) and the Michelson Doppler Imager (MDI) aboard SOHO. The authors compiled 108‑day averaged frequency sets spanning from mid‑1996 (the minimum of cycle 23) to early 2008 (the minimum of cycle 24), thereby covering the rising phase, the high‑activity plateau, and the declining phase of the cycle.

Four solar‑activity proxies were selected to represent different magnetic‑field components and atmospheric layers: (i) the 10.7 cm radio flux (F10.7), which is sensitive to both strong and weak magnetic fields in the upper chromosphere and low corona; (ii) the international sunspot number (SN), a classic indicator of strong photospheric magnetic flux; (iii) a full‑disk photospheric magnetic‑field strength index (MF), which emphasizes the strong‑field component; and (iv) the Ca II K line intensity (CaK), which is more responsive to the weak, dispersed magnetic field in the chromosphere.

For each mode the frequency shift Δν was computed relative to the 1996 reference epoch, and a weighted average over all modes was formed, the weights reflecting mode amplitude and measurement uncertainties. Correlation analyses were performed separately for the three activity phases. Pearson correlation coefficients (r) and linear regression slopes (a) were derived for each proxy–Δν pair.

Key findings are:

1. Overall correlation – Across the full 12‑year interval, Δν is positively correlated with all four proxies (r≈0.73–0.82). This confirms the well‑known in‑phase behavior of p‑mode frequencies with solar activity.

2. Phase‑dependent behavior – During the rising (1996‑1999) and declining (2004‑2008) phases, the correlations are exceptionally strong (r≈0.88–0.95) for all proxies. The regression slopes are also large (e.g., a≈0.42 nHz per 10⁻³ W m⁻² for F10.7), indicating that frequency shifts respond linearly and robustly to changes in the strong‑field photospheric component.

3. High‑activity plateau – In the maximum‑activity period (2000‑2003) the correlation with SN and MF drops dramatically (r≈0.45–0.55), while the correlation with F10.7 remains relatively high (r≈0.78) and CaK stays at r≈0.70–0.75. Correspondingly, the regression slopes for SN and MF are reduced by roughly 30–40 % compared with the other phases. This suggests that, when the Sun is saturated with strong magnetic regions, the additional frequency shift contributed by further increases in strong‑field flux is limited, whereas the weak‑field and coronal components continue to modulate the oscillation frequencies.

4. Contribution of strong vs. weak fields – By performing a multiple‑linear regression, the authors estimate that more than 90 % of the observed frequency variation can be explained by a linear combination of the four proxies. In the rising/declining phases, the strong‑field proxies (SN, MF) account for roughly 60 % of the variance, whereas in the high‑activity phase the weak‑field proxy (CaK) and the radio flux together explain about 65 % of the variance.

5. Slope–correlation relationship – The regression slope a for each proxy tracks the corresponding correlation coefficient r very closely; higher r values are always accompanied by steeper slopes. This reinforces the interpretation that the linear sensitivity of p‑mode frequencies to a given proxy is a direct manifestation of the underlying physical coupling.

The authors conclude that the frequency shifts of intermediate‑degree p‑modes are not driven solely by the strong photospheric magnetic field but result from a combined effect of both strong and weak magnetic components distributed throughout the solar atmosphere. The reduced correlation of strong‑field proxies during the activity maximum points to a possible saturation or non‑linear restructuring of the magnetic field, allowing the weaker, higher‑altitude fields to dominate the residual frequency response. The study highlights the importance of employing multiple activity indices that sample different atmospheric layers when interpreting helioseismic frequency changes. Future work is suggested to integrate high‑resolution magnetic‑field modeling and three‑dimensional radiative‑magnetohydrodynamic simulations to disentangle the precise physical mechanisms by which each magnetic component influences p‑mode propagation and frequency.