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Revealing mixed modes in compressible hydrodynamical simulations of red giant stars

This study presents the first 2D hydrodynamical simulations of a red giant star using the \textsc{music} code to constrain mixed mode amplitudes, revealing that while high-frequency modes match empirical predictions, low-frequency modes possess unexpectedly large interior kinetic energies that could significantly influence angular momentum transport despite having small, hard-to-observe surface velocities.

Original authors: Nils B. de Vries, Arthur Le Saux, Isabelle Baraffe, Thomas Guillet, Richard H. D. Townsend, Armand Leclerc, Adrien Morison

Published 2026-06-08

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CC BY 4.0

Original authors: Nils B. de Vries, Arthur Le Saux, Isabelle Baraffe, Thomas Guillet, Richard H. D. Townsend, Armand Leclerc, Adrien Morison

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 a giant, aging star as a massive, glowing drum. When you hit a drum, it doesn't just make one sound; it vibrates in many different ways at once. Some vibrations are loud and happen near the surface (like the skin of the drum), while others are deep, rumbling waves that travel through the very center. In stars, these vibrations are called "modes."

For decades, astronomers have listened to these stellar drums using space telescopes. They discovered something strange: the deep, rumbling vibrations (called "mixed modes") are telling them that the cores of these stars are spinning much slower than our best computer models predicted. It's as if the star's core is spinning like a figure skater who suddenly slowed down, but no one knows what force is acting as the "brake."

One theory suggests that these very vibrations themselves might be the brake, carrying angular momentum (spin) from the fast core to the slow outer layers. But to prove this, scientists need to know exactly how "loud" or energetic these vibrations are inside the star. Until now, this has been a guessing game.

The Experiment: Building a Digital Star
In this paper, the authors built a digital, 2D simulation of a red giant star (a star about 1.3 times the mass of our Sun) using a supercomputer code called MUSIC. Think of this as creating a virtual, cut-open slice of a star to watch how the fluid inside moves and vibrates.

They ran two versions of this simulation:

RGext90: They cut the simulation off at 90% of the star's radius (leaving the very outer skin out).
RGext98: They cut it off at 98% of the radius (getting much closer to the surface).

What They Found
The team compared their digital star's vibrations against two different mathematical "rulebooks" (linear theory solvers) to see if their simulation was accurate.

The High-Frequency Sounds: For the vibrations that act mostly like sound waves near the surface, the simulation matched the rulebooks perfectly. It was like tuning a guitar and hearing the exact right note.
The Low-Frequency Rumbles: For the deep, gravity-driven vibrations, there were some small mismatches in the middle frequency range. It's as if the digital star was slightly out of tune for the deepest, lowest notes.

The Big Surprise: Where the Energy Hides
The most important discovery concerns where the energy of these vibrations is actually located.

The Old Guess: Based on observations, scientists expected the most energetic vibrations to be a "bell curve" centered around a specific frequency (about 313 microHertz), similar to the loudest part of a song.
The Simulation Reality: The digital star told a different story. The vibrations with the most kinetic energy (the most violent movement of gas) were actually at very low frequencies (below 50 microHertz). These are the deep, slow rumbles.
- Analogy: Imagine a crowd at a concert. The old theory said the loudest cheering happens in the middle of the song. The simulation showed that the loudest cheering is actually happening at the very beginning, but it's happening so deep in the crowd that you can't hear it from the back of the hall.

The Surface vs. The Interior
Here is the tricky part:

Inside the Star: These low-frequency, high-energy rumbles are moving a huge amount of gas deep inside. This suggests they could be very effective at acting as a "brake" to slow down the star's core spin.
At the Surface: When the authors tried to "extrapolate" (project) these deep vibrations out to the star's surface, they found the surface movement was tiny. It's like a massive earthquake deep underground that barely shakes the ground above. Because the surface movement is so small, it is very hard for telescopes to detect these specific waves.

However, in the simulation that went closer to the surface (RGext98), they did see a "bell curve" of energy at higher frequencies (around 700 microHertz). These high-frequency waves had strong surface signals, similar to what we see in real observations, but they weren't the ones carrying the most energy deep inside.

Why This Matters
The paper concludes that while we can't easily see these powerful, low-frequency waves from Earth, they might be the missing piece of the puzzle for how stars slow down their spin. The current models only look at the "loud" surface waves, but the simulation suggests the "quiet" deep waves are doing the heavy lifting for angular momentum transport.

The authors note that their simulation is a simplified 2D slice and doesn't include rotation or magnetic fields yet, so it's not the final answer. But it proves that we can now simulate these complex stellar vibrations in detail, opening the door to understanding the hidden mechanics of dying stars.

Technical Summary: Revealing Mixed Modes in Compressible Hydrodynamical Simulations of Red Giant Stars

Problem Statement
Mixed modes in low-mass evolved stars provide critical diagnostics for core rotation rates, which are observed to be significantly lower than predicted by standard stellar evolution models. While mixed modes have been proposed as a mechanism for angular momentum (AM) transport, quantifying their efficiency requires accurate knowledge of their amplitudes. Current empirical predictions for these amplitudes rely on semi-empirical formalisms that are poorly constrained by observations for evolved red giants. Specifically, it remains uncertain whether mixed modes can transport sufficient AM to reconcile observed core rotation rates with theoretical models.

Methodology
This study presents the first 2D compressible hydrodynamical simulations of mixed modes in the interior of an evolved low-mass star using the music code. The authors simulated a meridional wedge slice of a 1.3 $M_\odot$ red giant (based on the M1 model of Belkacem et al. 2015b). Two distinct simulations were performed to investigate the impact of the outer boundary on mode amplitudes:

RGext90: Outer radial truncation at $r_o/r_\star = 0.90$ .
RGext98: Outer radial truncation at $r_o/r_\star = 0.98$ .

Both models included a large convective region to stochastically excite modes and a substantial stably stratified region to house the gravity-mode component of the mixed modes. The simulations were run until a dynamical steady state was reached, after which spectra were captured.

To validate the simulations, the authors compared the retrieved mode frequencies and eigenfunctions against linear theory predictions generated by two independent solvers: the gyre code (version 8.0.1) and a custom eigenvalue solver written in dedalus. The analysis focused on radial velocity spectra, constructing Échelle diagrams and frequency-radius plots for harmonic degrees $\ell < 5$ .

Key Results

Frequency and Eigenfunction Agreement:
- Excellent agreement was found between the simulations and linear theory for all p-dominated modes.
- Minor discrepancies (a few $\mu$ Hz) were observed for g-dominated modes, particularly in the frequency range $[60, 240]$ $\mu$ Hz.
- Eigenfunction agreement was excellent for all modes except those in the aforementioned frequency range. The authors attribute discrepancies in this range to insufficient resolution of the g-mode part in the simulation compared to the high-order linear theory computations.
Kinetic Energy Distribution:
- Contrary to empirical predictions which place the maximum kinetic energy around $\nu_{max} \approx 312.8$ $\mu$ Hz, the simulations revealed that the largest kinetic energies are located at low frequencies ( $\nu < 50$ $\mu$ Hz).
- These low-frequency modes exhibit kinetic energies orders of magnitude larger than empirical predictions, with a rapid decrease in energy as frequency increases.
- The RGext98 simulation (extending closer to the surface) showed higher overall kinetic energies and the emergence of a secondary, broad Gaussian-like distribution of energy centered around $\nu = 700$ $\mu$ Hz, likely due to surface excitation of p-modes.
Extrapolated Surface Velocities:
- Despite their high internal kinetic energies, the low-frequency modes ( $\nu < 50$ $\mu$ Hz) extrapolate to very small surface velocities (mostly $< 1$ cm s $^{-1}$ ), making them difficult to observe.
- The RGext98 simulation produced surface velocities comparable to empirical predictions for high-frequency modes ($500-800$ $\mu$ Hz), peaking in a bell-shaped curve. However, the peak is shifted to higher frequencies compared to empirical scaling.
- The RGext90 simulation yielded negligible surface velocities for high-frequency modes.

Significance and Claims
The paper claims to provide the first direct constraints on mixed mode amplitudes in 2D hydrodynamical simulations of red giants. The primary significance lies in the discovery of highly energetic, low-frequency mixed modes that are not captured by current empirical scaling relations.

The authors posit that while these low-frequency modes are hard to detect via surface velocities, their large kinetic energies deep within the stellar interior could significantly impact angular momentum transport. This suggests that mixed modes may play a more substantial role in AM transport than previously estimated, potentially explaining the observed core rotation rates in young and very evolved red giants. The study demonstrates the feasibility of using compressible hydrodynamical simulations to capture mixed modes, offering a pathway to quantitatively estimate AM transport efficiency and predict observability for future missions like PLATO, provided that 3D effects, rotation, and magnetic fields are eventually incorporated.

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