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On the feasibility of inverting the rotation of the solar core with mixed f/g modes

This paper proposes a novel method to constrain the rotation of the solar core by utilizing mixed f/g modes, which couple the radiative interior with the surface, thereby enabling the inversion of core rotation rates without requiring the direct detection of elusive g modes.

Original authors: Armand Leclerc, Arthur Le Saux, J. M. Joel Ong, Rafael A. Garcia

Published 2026-07-07

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CC BY 4.0

Original authors: Armand Leclerc, Arthur Le Saux, J. M. Joel Ong, Rafael A. Garcia

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the Sun as a giant, glowing musical instrument. For decades, astronomers have been listening to its "notes" (sound waves) to figure out how it spins inside. This field is called helioseismology.

Here is the problem: The Sun has a deep, dense core (the center) and a noisy, churning outer layer. The sound waves we can currently hear easily (called p-modes) are like high-pitched whistles that travel mostly through the outer layers. They tell us how the outer Sun spins, but they bounce off before they can reach the very center.

The waves that could reach the center (called g-modes) are like deep, heavy bass notes. The problem is that by the time these bass notes try to reach the surface to be heard, they fade away completely, making them impossible to detect with our current microphones.

The New Idea: The "Hybrid" Wave
This paper proposes a clever workaround. The authors suggest looking for a special type of "hybrid" wave, which they call a mixed f/g mode.

Think of it like a duet between a high-pitched whistle and a deep bass note.

Normally, the whistle (f-mode) stays near the surface, and the bass (g-mode) stays trapped in the core.
However, the authors found that under certain conditions, these two waves can "shake hands" and merge into a single hybrid wave.
This hybrid wave is unique because it has the reach of the bass (it goes deep into the core) but the strength of the whistle (it is loud enough to be heard at the surface).

What They Found
The researchers used a computer model of the Sun to hunt for these hybrid waves. They discovered:

There are about 6 to 12 of these hybrid waves waiting to be found, each vibrating at a slightly different pitch.
These waves are sensitive enough to tell us exactly how fast the Sun's core is spinning.
By listening to these specific hybrid waves, astronomers could finally "invert" the data—meaning they could work backward from the sound to map the rotation speed deep inside the Sun.

Why Does This Matter?
Right now, scientists have two very different theories about how the Sun's core spins:

Theory A: The core spins at the exact same speed as the rest of the Sun (like a solid spinning top).
Theory B: The core spins much faster than the rest of the Sun (like a figure skater pulling in their arms).

Currently, we can't tell which theory is right because we can't hear the core. The authors show that if we can detect just a few of these hybrid waves, we will have enough information to settle the debate. Even if we only find the one loudest hybrid wave, it should be precise enough to tell us if the core is spinning fast or slow.

The Challenge
The paper admits that finding these waves is hard. The Sun's surface is very noisy (like trying to hear a whisper in a hurricane), and these hybrid waves are often drowned out by that noise. However, the authors suggest that if we look at the Sun using different types of light (not just visible light), we might be able to cut through the noise and finally hear these "hybrid notes."

In Summary
This paper doesn't claim we have already found these waves. Instead, it provides a roadmap. It tells astronomers: "Stop looking for the impossible deep bass notes. Instead, listen for these specific hybrid notes that travel from the core to the surface. If you find them, you will finally know how the Sun's heart spins."

Technical Summary: On the feasibility of inverting the rotation of the solar core with mixed f/g modes

Problem Statement
Helioseismology has successfully mapped the Sun's internal rotation profile down to approximately $0.2 R_\odot$ using acoustic ( $p$ ) modes. However, the innermost 20% of the solar radius, including the core, remains inaccessible to current observations. Pure gravity ( $g$ ) modes, which are sensitive to the core, are evanescent in the convective envelope and have eluded definitive detection. This observational gap prevents astronomers from distinguishing between competing theories of angular momentum transport in stellar interiors. Specifically, models based on the Tayler-Spruit instability predict conflicting rotation rates for the solar core: Fuller et al. (2019) predict solid-body rotation extending to the center, while Eggenberger et al. (2019) predict a core spinning approximately three times faster than the surrounding radiative zone. Resolving this tension requires a direct measurement of the core's rotation rate.

Methodology
The authors propose utilizing "mixed $f/g$ modes" to probe the solar core without requiring the direct detection of pure $g$ modes. These modes arise from the coupling between fundamental ( $f$ ) modes in the outer stellar layers and trapped $g$ modes in the radiative interior.

Mode Identification and Characterization: Using a 1D solar model calibrated with the MESA code and eigenfunctions computed via GYRE, the authors systematically searched for mixed modes across angular degrees $\ell$ from 1 to 20. They employed a diagnostic based on period spacing patterns ( $\Delta P$ ) to identify dips caused by $f$ -mode coupling.
Coupling and Sensitivity Analysis: For identified modes, the authors calculated:
- Mixing fraction ( $\zeta$ ): The contribution of the pure $g$ -mode component to the mixed mode.
- Resonance width ( $\Gamma$ ): To determine the strength of the coupling.
- Sensitivity parameter ( $s$ ): Defined as the derivative of the mode frequency with respect to the core rotation rate ( $\Omega_c$ ), integrated over the core ( $r < 0.2 R_\odot$ ).
- Normalized Mode Inertia ( $E_{norm}$ ): A proxy for detectability, where lower values indicate easier detection.
Rotational Inversion: The authors applied the Optimal Localized Averages (OLA) method to construct averaging kernels. They combined the rotational kernels of standard $p$ -modes with those of the identified mixed $f/g$ modes to create localized sensitivity kernels ( $K_{eff}$ ) targeting specific radii within the core ( $r=0.07 R_\odot$ and $r=0.2 R_\odot$ ).
Scenario Discrimination: The authors simulated inversions using the rotation profiles predicted by Fuller et al. (2019) and Eggenberger et al. (2019) to test if the proposed mixed modes could statistically distinguish between these two scenarios given realistic observational uncertainties.

Key Contributions and Results

Discovery of Mixed Modes: The study identifies that mixed $f/g$ modes exist for angular degrees $\ell$ between 3 and 8. While Le Saux et al. (2025) previously identified a mode at $\ell=4$ , this work expands the search to find up to 12 such modes (two per degree for $\ell=3$ to $8$). Modes with $\ell \ge 9$ are found to be essentially pure $f$ or $g$ modes with negligible coupling.
Selection of Viable Modes: The authors select modes with $\ell \in [3, 8]$ that possess a sensitivity $s > 0.2$ and low mode inertia. Notably, the mode at $\ell=7$ with a frequency of $\nu = 305 \, \mu\text{Hz}$ is highlighted as a promising candidate. It is strongly $f$ -dominated (favoring surface amplitude) yet retains sufficient mixing ( $\zeta \approx 0.11$ ) to provide a sensitivity of $s=0.06$ .
Inversion Capability: By incorporating these mixed modes into the OLA procedure, the authors demonstrate that it is possible to construct averaging kernels that peak deep within the solar core ( $r \approx 0.07 R_\odot$ ). In contrast, kernels constructed using only $p$ -modes peak at $r \approx 0.2 R_\odot$ and fail to resolve the inner core.
Discrimination of Theories: The authors estimate that if the $\ell=7$ mode could be measured with a precision of $25 \, \text{nHz}$ (five times the current BiSON uncertainty), the resulting uncertainty on the core rotation rate would be approximately $420 \, \text{nHz}$ . This precision is sufficient to distinguish between the solid-body rotation prediction of Fuller et al. ( $\sim 440 \, \text{nHz}$ ) and the rapid rotation prediction of Eggenberger et al. ( $> 1000 \, \text{nHz}$ ).

Significance and Claims
The paper claims that mixed $f/g$ modes offer a viable pathway to constrain the rotation of the solar core without the need for the direct detection of elusive pure $g$ modes. The authors assert that these modes are simultaneously sensitive to the core dynamics and capable of reaching the surface with significant amplitude.

The study concludes that the detection of a small number of these mixed modes (specifically the $\ell=7$ mode or a combination of modes from $\ell=3$ to $8$) would allow for the construction of independent rotation measurements at $r=0.07 R_\odot$ and $r=0.2 R_\odot$ . This capability would resolve the current tension between angular momentum transport theories.

However, the authors remain modest regarding observational feasibility. They acknowledge that the primary obstacle is the high level of convective noise at the low frequencies ( $\sim 300 \, \mu\text{Hz}$ ) where these modes reside. They note that reaching the necessary detection threshold with current continuous observations might require 30 to 40 additional years of data. As a potential alternative, they suggest that multi-wavelength observations (e.g., in the near-UV via the SDO/AIA instrument) might offer a route to detect these modes sooner by probing regions with weaker granulation noise.

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