🔭 astrophysics

Towards inertial-mode helioseismology: Direct sensing of solar rotation at 75 deg latitude and 0.8 Rsun

This study demonstrates that the observed frequency of the $m=1$ high-latitude inertial mode provides a direct constraint on solar rotation at 75° latitude and 0.8 $R_\odot$ , revealing a rotation rate significantly faster than previous p-mode helioseismology estimates.

Original authors: Prithwitosh Dey, Laurent Gizon, Yuto Bekki

Published 2026-05-20

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CC BY 4.0

Original authors: Prithwitosh Dey, Laurent Gizon, Yuto Bekki

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, spinning ball of hot gas. For a long time, scientists have tried to figure out how fast different parts of this ball are spinning. They usually do this by listening to "sound waves" (called p-modes) that bounce around inside the Sun, kind of like how a doctor uses ultrasound to see inside a human body.

However, this "sound wave" method has a blind spot. It works great near the surface and the equator, but it gets very fuzzy when trying to measure the spin deep inside the Sun, especially near the poles. It's like trying to hear a whisper from the other side of a noisy room; the signal gets lost.

The New "Ear": Listening to the Sun's Spin
In this paper, the authors introduce a new way to listen to the Sun. Instead of sound waves, they are listening to "inertial modes." Think of these as giant, slow-motion swirls or currents in the Sun's gas, driven by the Sun's own spin (similar to how the Coriolis force makes hurricanes spin on Earth).

The researchers focused on one specific swirl, called the "HL1 mode," which is most active near the Sun's poles. They found that this swirl is extremely sensitive to how fast the gas is spinning in a specific deep region: about 80% of the way from the center to the surface, and at a latitude of 75 degrees (near the pole).

The Mystery: The Model vs. Reality
The scientists took the best existing map of how the Sun spins (based on the old sound-wave method) and used a super-accurate computer program to predict what the frequency of this "swirl" should be.

Here is the problem: The computer predicted a frequency that didn't match what they actually observed. The real Sun's swirl was spinning at a different speed than the model said it should.

The Detective Work
The team asked: "What is wrong with our map?"
They checked if changing other things in their model (like how hot or sticky the gas is) could fix the mismatch. It didn't. The only thing that fixed the math was changing the rotation speed in that specific deep, high-latitude region.

The Discovery
By using the observed speed of the swirl as a clue, they calculated that the gas at this specific location (75° latitude, 0.8 solar radii deep) is spinning faster than the old sound-wave maps suggested.

The Old Map Said: ~357 nHz (a unit of rotation speed).
The New Measurement Says: ~365 nHz.
The Difference: The gas is spinning about 8 nHz faster than we thought.

Why This Matters
This is the first time scientists have been able to "see" inside the Sun's deep polar regions using a single, specific type of wave. It's like finally finding a way to measure the speed of a car's engine while it's driving through a thick fog, using a new type of radar that wasn't available before.

In Summary
The paper claims that by listening to a specific giant swirl on the Sun, they have directly measured that the deep, polar regions of the Sun's convection zone are spinning faster than our previous best maps indicated. This proves that these "inertial modes" are powerful new tools for mapping the Sun's interior, offering a direct look at places that were previously hidden.

Technical Summary: Towards Inertial-mode Helioseismology

Problem Statement
Solar internal differential rotation at high latitudes and deep within the convection zone remains poorly constrained by traditional p-mode helioseismology. While p-modes effectively map rotation in the bulk of the convection zone, their sensitivity kernels peak near the solar surface, providing limited information about regions within $\sim20^\circ$ of the rotation axis and below $\sim0.2R_\odot$ . Consequently, inversions of p-mode frequency splittings yield significant discrepancies in polar regions. The authors posit that global solar inertial modes, which are restored by the Coriolis force and intrinsically linked to rotation, offer a promising alternative diagnostic for these inaccessible regions. Specifically, the study focuses on the high-latitude inertial mode with azimuthal wavenumber $m=1$ (the 'HL1' mode), which exhibits the largest surface amplitude among observed inertial modes.

Methodology
The study employs a forward modeling approach to constrain solar rotation using the measured frequency of the HL1 mode.

Reference Model: The authors adopt a reference differential rotation profile ( $\Omega_{\text{ref}}$ ) derived from p-mode helioseismology (Larson & Schou 2018) for the period 2010–2024.
Eigenvalue Solver: Using a validated eigenvalue solver (Solver 1), they compute the complex eigenfrequency of the HL1 mode within a computational domain spanning $0.65R_\odot < r < 0.985R_\odot$ . The solver was cross-validated against an independent implementation (Solver 2).
Sensitivity Kernel Calculation: To determine how local changes in rotation affect the mode frequency, the authors numerically compute the linear sensitivity kernel $K(r, \theta)$ . This involves perturbing the rotation rate at specific grid points ( $\delta\Omega$ ) while maintaining thermal wind balance via latitudinal entropy gradient adjustments, and calculating the resulting frequency shift ( $\delta\omega$ ).
Parametric Correction: To move beyond linear approximations, the authors introduce a smooth, latitudinally dependent, radially independent parametric correction ( $\Delta\Omega$ ) to the rotation rate at high latitudes. They recompute eigenfrequencies for finite-amplitude perturbations to map the non-linear response.

Key Results

Model-Observation Mismatch: The model eigenfrequency computed using the p-mode reference rotation ( $\omega_{\text{ref}}/2\pi = -95.6$ nHz) differs significantly from the observed mean frequency in the Carrington frame ( $\omega_{\text{obs}}/2\pi = -87.9 \pm 1.9$ nHz). The discrepancy of $\sim7.7$ nHz cannot be resolved by varying background parameters such as turbulent diffusivity or superadiabaticity, pointing to the rotation profile as the primary source of error.
Localization of Sensitivity: The linear sensitivity kernel for the HL1 mode is highly localized. It peaks at a latitude of $75^\circ$ (colatitude $15^\circ$ ) and a radius of $0.8R_\odot$ , with full widths at half maximum of $7^\circ$ in latitude and $0.13R_\odot$ in radius. The mode shows negligible sensitivity to the radiative interior.
Rotation Rate Inference:
- Local Estimate: By integrating the kernel over the meridional plane, the authors infer a local rotation perturbation of $\delta\Omega/2\pi \approx 8.1 \pm 2.0$ nHz at $(0.8R_\odot, 75^\circ)$ . This implies a local rotation rate of $365.3 \pm 2.0$ nHz, which exceeds the p-mode reference estimate ($357.2$ nHz) by $8.1$ nHz.
- Global Parametric Estimate: Using a smooth parametric correction, the best-fit model yields a polar rotation enhancement of $13.6 \pm 3$ nHz relative to the reference model, corresponding to a polar rotation rate of $364.0$ nHz.
Comparison with p-modes: The inferred rotation rate at the target location is significantly higher than estimates derived from p-mode Optimal Localized Averaging (OLA) inversions, highlighting the distinct and potentially more accurate constraints provided by inertial modes in high-latitude regions.

Significance and Claims
The paper claims to present the first example of spatially resolved inertial-mode helioseismology. The primary significance lies in demonstrating that individual inertial modes can provide direct, absolute constraints on solar rotation in the bulk of the convection zone, well below the surface and at high latitudes where p-modes are insensitive. Unlike p-mode helioseismology, which relies on differential measurements between prograde and retrograde modes, this approach utilizes the absolute frequency of a single retrograde inertial mode. The authors conclude that validated eigenvalue solvers are essential tools for interpreting the full spectrum of inertial modes and that future work will aim to solve inverse problems using multiple modes simultaneously to map the solar differential rotation more comprehensively.

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