Global helioseismology has measured the Sun's internal rotation precisely down to about 0.2 R⊙0.2\,R_{\odot}, but the deepest radiative interior and core are dynamically unprobed: low-degree pressure modes (p modes) lose sensitivity there, and gravity modes (g modes), which are trapped in the core, have never been unambiguously detected at the surface. Eff-Darwich, Korzennik, Jiménez-Reyes and García (2008) showed by sensitivity analysis of combined GONG, GOLF and MDI rotational splittings that the rotation profile below 0.2 R⊙0.2\,R_{\odot} cannot be inferred from current data, that the p-mode-only route would require multi-decade time series with controlled systematic biases, and that progress hinges on detecting g modes and measuring their rotational splittings. The problem is to determine the rotation profile of the solar core below 0.2 R⊙0.2\,R_{\odot} — in practice, to detect low-degree solar g modes unambiguously, measure their rotational splittings with realistic uncertainties and controlled systematics, and invert them with averaging kernels localized in the core. A 2017 claim of exactly this measurement (core rotation about 3.8 times faster than the radiative envelope) was reanalyzed by three independent groups and remains unconfirmed; the community consensus as of 2026 is that g modes still elude detection, so the problem is open.

Background

The Sun oscillates in millions of global modes. The observed five-minute oscillations are pressure modes (p modes), acoustic waves for which pressure is the restoring force; their frequencies are split by rotation, and measuring these rotational splittings for many modes allows an inversion — a regularized fit using, e.g., the Optimal Localized Averages (Backus–Gilbert) technique — to reconstruct the internal rotation rate Ω(r,θ)\Omega(r,\theta) as a function of radius and latitude. Key instruments are the ground-based network GONG (Global Oscillation Network Group) and the SOHO spacecraft instruments MDI (Michelson Doppler Imager) and GOLF (Global Oscillations at Low Frequencies), which provide multi-year Sun-as-a-star Doppler time series. Inversions of p-mode splittings established that the radiative interior rotates almost rigidly at about 430 nHz down to r≈0.2 R⊙r\approx 0.2\,R_{\odot}, but p-mode averaging kernels cannot be localized below that radius. Gravity modes (g modes) are nonradial oscillations driven by buoyancy as the restoring force; they are trapped in the radiative interior with amplitudes largest in the core and are evanescent through the convective envelope, so at the surface their amplitudes are expected to be extremely small and no individual g mode has ever been unambiguously detected. Eff-Darwich, Korzennik, Jiménez-Reyes and García (2008, ApJ 679, 1636; the source of this problem) inverted a combined MDI–GOLF–GONG set of rotational splittings covering 1.11.1–3.93.9 mHz and performed a sensitivity analysis with artificial data sets. They concluded that (i) the Sun rotates as a rigid body through most of the radiative interior but that nothing can be derived about the dynamics below r≈0.2 R⊙r\approx 0.2\,R_{\odot} because the low-degree modes that reach the core have too-large observational uncertainties; (ii) reducing low-degree p-mode splitting uncertainties to the level of the high-degree modes would require multi-decade uninterrupted time series and assumes residual systematic biases (from fitting techniques, spatial filters, leakage matrices, multi-station merging, and solar activity) can be reduced, which they judged unlikely in the near future; and (iii) recovering the core in their experiments required either unrealistically small errors or the addition of nominal g-mode splittings — g modes being modes that have 'yet to be unambiguously observed'. Just before that work, García et al. (2007, Science 316, 1591) had inferred from candidate dipole g-mode splittings in GOLF data that the core rotates 3–5 times faster than the envelope, a claim that remained unconfirmed; Appourchaux et al. (2010, A&A Rev. 18, 197) set stringent upper limits on g-mode amplitudes and documented the non-detection. Determining the core rotation matters beyond the Sun: Eggenberger, Buldgen and Salmon (2019, A&A 626, L1) showed that the core rotation rate is the key missing constraint for discriminating between competing magnetic angular-momentum transport mechanisms in stellar-interior models.

Problem Statement

Determine the rotation profile Ω(r)\Omega(r) of the solar core below r≈0.2 R⊙r\approx 0.2\,R_{\odot} from helioseismic observations. As established by Eff-Darwich et al. (2008), current low-degree p-mode rotational splittings do not constrain this region, and reaching the required precision with p modes alone would demand multi-decade uninterrupted time series together with control of systematic biases in low-degree splitting measurements. The source work identifies the observational and data-analysis objective through which this is in practice achieved: (1) unambiguous detection of low-degree solar g modes (whether as individual spectral peaks or through a collective signature that exploits the asymptotic property that low-frequency g-mode periods are nearly equally spaced, and that is robust to analysis choices; or of another rotation-sensitive mode family that reaches the core, such as the mixed f/g modes proposed by Leclerc et al. 2026); (2) measurement of their rotational splittings with realistic observational uncertainties and a characterized systematic-error budget (including fitting-technique differences, spatial filtering, leakage, multi-station merging, and solar-cycle effects); and (3) use of those splittings in an inversion whose averaging kernels (Backus–Gilbert resolution vectors) are genuinely localized below 0.2 R⊙0.2\,R_{\odot}, so that the inferred core rotation — including whether the core rotates faster than the radiative envelope, and by how much — is determined by the data rather than by regularization extrapolation. Robustness of the detection and of the inferred rotation to reasonable changes in the analysis (e.g., data calibration, time-series start time, length, and instrument) is essential, as independent reanalyses of the Fossat et al. (2017) claim showed these to be the failure modes.

References

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