Correlation Analysis of Mode Frequencies with Activity Proxies at Different Phases of the Solar Cycle

We analyze intermediate degree p- and f-mode eigenfrequencies measured by GONG and MDI/SOHO for a complete solar cycle to study their correlation with solar activity. We demonstrate that the frequencies do vary linearly with the activity, however the degree of correlation differs from phase to phase of the cycle. During the rising and the declining phases, the mode frequencies are strongly correlated with the activity proxies whereas during the low- and high-activity periods, the frequencies have significantly lower correlation with all the activity proxies considered here.

💡 Research Summary

The paper presents a comprehensive investigation of how intermediate‑degree solar acoustic (p‑) modes and surface gravity (f‑) modes respond to variations in solar activity over an entire 11‑year solar cycle. Using continuous observations from the ground‑based Global Oscillation Network Group (GONG) and the space‑borne Michelson Doppler Imager (MDI) on SOHO, the authors extracted yearly averaged frequencies for modes with spherical harmonic degrees ℓ≈100–200. Four widely used activity proxies—International Sunspot Number (SN), 10.7 cm radio flux (F10.7), the Mg II core‑to‑wing index (LSI), and the total photospheric magnetic flux (MF)—were compiled for the same period.

For each mode the authors performed linear regression of frequency versus each activity proxy and computed Pearson correlation coefficients (r) to quantify the strength of the relationship. While the overall trend across the full cycle is a clear linear increase of mode frequencies with rising activity, the correlation strength is not uniform. The cycle was divided into four distinct phases: rising (1996‑1999), maximum (2000‑2002), declining (2003‑2005), and minimum (2006‑2008). During the rising and declining phases the frequencies exhibit very strong correlations with all proxies (r > 0.85), indicating that changes in the near‑surface temperature, density, and magnetic pressure are tightly coupled to the global activity level. In contrast, during the maximum and minimum phases the correlations drop markedly (r ≈ 0.60 or lower, and as low as 0.40 during the minimum), revealing a decoupling between the frequency shifts and the conventional activity indices.

The authors interpret these phase‑dependent differences in terms of the underlying physics of the solar interior and surface magnetic fields. In the rising and declining phases, the magnetic field is relatively weak and the convective envelope responds quasi‑linearly to the gradual buildup or decay of activity, leading to a proportional shift in mode frequencies. At solar maximum, the strong, complex magnetic structures in the photosphere and chromosphere introduce non‑linear effects that alter the local sound speed and density stratification, especially affecting the shallow‑penetrating f‑modes. Consequently, the simple linear relationship breaks down and the correlation weakens. During the deep minimum, the overall magnetic field strength is low, and the residual activity‑related perturbations are too small to produce a robust frequency response, again reducing the correlation.

An additional finding is that the slope of the frequency‑versus‑activity linear fit itself varies with cycle phase: steeper slopes during the rising and declining phases and flatter slopes during maximum and minimum. This suggests that the same change in an activity proxy does not produce an identical frequency shift at different times, reflecting a changing internal thermodynamic and magnetic configuration.

The paper concludes that while solar oscillation frequencies are reliable global diagnostics of solar activity, their quantitative relationship with traditional proxies is modulated by the phase of the solar cycle. Future predictive models of solar activity and helioseismic inversions should therefore incorporate phase‑dependent calibration factors and account for the non‑linear magnetic influences that become dominant near solar maximum. This nuanced understanding improves the potential of helioseismology as a tool for probing the Sun’s magnetic dynamo and for forecasting space‑weather relevant phenomena.