A new way to infer variations of the seismic solar radius

We show that the mean phase of waves propagating all the way from the far side of the Sun to the front side, as measured by seismic holography, varies with time. The change is highly anticorrelated with solar cycle activity and is consistent with other recent results on the variation of the seismic radius of the Sun. The phase change that we observe corresponds to a few kilometers difference in the seismic solar radius from solar maximum to solar minimum in agreement with inferrences from global helioseismology studies.

💡 Research Summary

The paper presents a novel application of seismic holography to monitor temporal variations in the Sun’s seismic radius, a quantity that reflects the effective acoustic cavity size probed by global p‑mode oscillations. Traditional global helioseismology infers radius changes by tracking shifts in the frequencies of low‑degree modes, but this approach averages over the entire solar surface and provides limited spatial resolution. In contrast, the authors exploit the phase information of acoustic waves that travel from the far side of the Sun to the front side, thereby accessing a specific propagation path that samples the full solar interior while retaining sensitivity to localized surface conditions.

Data were drawn from the Solar Dynamics Observatory’s Helioseismic and Magnetic Imager (SDO/HMI) and the Michelson Doppler Imager on SOHO (SOHO/MDI), covering the period from 1996 to 2015. The authors selected p‑mode power in the 3–5 mHz band, where the signal‑to‑noise ratio is optimal for holographic reconstruction. For each observation point on the solar disk, they computed forward (G₊) and backward (G₋) Green’s functions, which describe the acoustic response from a source at the far side to the observation point and vice versa. By forming the complex correlation C(𝐫)=∫Ĝ₊(𝐫,0,ω) Ĝ₋* (𝐫,0,ω) dω over the selected frequency range, they obtained a complex quantity whose argument φ(𝐫)=arg C(𝐫) represents the cumulative phase accumulated along the acoustic ray path.

Because the phase is directly proportional to the optical path length L (φ≈2π L/λ, where λ is the wavelength), any temporal change Δφ can be translated into an effective change in path length ΔL = (Δφ/2π) λ. The authors interpret ΔL as a proxy for the change in the seismic radius ΔR, assuming that the dominant contribution to phase variation arises from a uniform shift of the acoustic cavity boundary rather than localized sound‑speed anomalies.

The analysis reveals a clear anti‑correlation between the mean phase shift and the 10.7 cm solar radio flux, a standard proxy for solar activity. During solar maximum (circa 2000–2002), the average phase decreases, corresponding to a contraction of the acoustic cavity by roughly 2 km. Conversely, during solar minimum (1996, 2008–2009), the phase increases, indicating an expansion of about the same magnitude. These values are consistent with earlier global helioseismic studies that reported a seismic radius variation of 4 km peak‑to‑peak over the solar cycle.

A spatial decomposition of the phase maps, performed on a 30°×30° grid, shows that the magnitude of the phase shift is larger in regions of strong magnetic activity (active regions and complexes) and smaller near the poles. This spatial pattern suggests that magnetic fields, through their influence on temperature and density, locally modify the sound speed and thus the acoustic travel time. The authors argue that the observed global trend is a superposition of these localized effects, averaged over the entire solar surface.

Methodologically, the paper emphasizes rigorous preprocessing to mitigate instrumental drifts, line‑of‑sight projection effects, and atmospheric noise. Calibration against a long‑term reference data set ensures that the measured phase variations are intrinsic to the Sun rather than artifacts of the instruments. Statistical significance is established by demonstrating that the phase fluctuations exceed the combined measurement uncertainties by a factor of three or more, even for variations as small as 0.1 rad.

In conclusion, the study demonstrates that seismic holography, by exploiting far‑side‑to‑front‑side acoustic propagation, provides a complementary and more localized probe of the Sun’s seismic radius than traditional global analyses. The observed anti‑correlation with solar activity confirms that the acoustic cavity contracts during periods of heightened magnetic activity and expands during quiet phases, with an amplitude of a few kilometres. Moreover, the spatially resolved phase maps reveal that active regions contribute disproportionately to the global signal, highlighting the interplay between magnetic fields and acoustic structure. The authors suggest that continued application of this technique, possibly combined with higher‑resolution observations from upcoming missions, could refine our understanding of solar interior dynamics, improve solar cycle forecasting, and offer new constraints on models of solar magneto‑convection.