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Asteroseismic forward modelling of 36 $\beta$ Cep pulsators and inferences on their internal differential rotation

This study employs a novel forward modelling approach incorporating second-order rotation effects to analyze 36 $\beta$ Cep pulsators, revealing that radial differential rotation is common and often non-monotonic, thereby providing critical constraints on the efficient angular momentum transport mechanisms within massive main-sequence stars.

Original authors: Mathijs Vanrespaille, Dario J. Fritzewski, Vincent Vanlaer, Conny Aerts

Published 2026-07-14

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CC BY 4.0

Original authors: Mathijs Vanrespaille, Dario J. Fritzewski, Vincent Vanlaer, Conny Aerts

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 inside of a massive, spinning star as a giant, cosmic spinning top. For a long time, astronomers thought these tops spun like solid wooden toys—every layer, from the very center to the fluffy outer skin, turning at the exact same speed. But a new study suggests the reality is far more chaotic and interesting: these stars are actually like layers of a spinning onion, where the core might be doing the splits while the outer layers are just cruising along.

The paper, led by a team of astronomers, took a massive leap forward by looking at 36 of these spinning giants, known as β Cephei (β Cep) stars. Before this, only a handful (fewer than ten) had been studied in this much detail. Think of it like finally getting a full class photo instead of just a blurry snapshot of three students.

The Big Discovery: The Stars are Wobbly
The team used a clever trick called "asteroseismology," which is like listening to the star's heartbeat to figure out what's happening inside. By analyzing the star's vibrations, they found that 17 of these stars are definitely spinning at different speeds at different depths. This is called "radial differential rotation."

Here is the kicker: in 14 of those 17 stars, the difference in speed is huge—more than 10%. In some cases, the core is spinning nearly 2.5 times faster than the surface. It's as if the center of a spinning pizza dough is whirling around like a race car while the crust is barely moving.

The Twist: It's Not Just "Fast Inside, Slow Outside"
Usually, scientists expected the core to be fast and the outside to be slow, like a figure skater pulling their arms in. But this study found something even stranger. In 10 of the stars, the speed drops from the core to the surface (fast to slow). But in 4 of them, the speed actually increases as you go outward!

This suggests the rotation profile isn't a simple straight line; it's "non-monotonic." Imagine a rollercoaster that goes up, then down, then up again. The authors suggest this weird shape might happen because the star swallowed a companion star in the past (accretion) or because invisible waves inside the star are pushing the spin around.

What They Ruled Out
The study explicitly argues against the idea that these stars spin like solid, rigid balls. While some stars in the universe might do that, the data from these 36 β Cep stars shows that rigid rotation is the exception, not the rule. They also found that the old way of calculating spin (using a simple, first-order math trick) wasn't good enough for these fast-spinners. They had to use a much more complex, "second-order" approach to get the numbers right, especially for stars spinning faster than 10% of their maximum possible speed.

How Sure Are They?
The team is very confident about the existence of this differential rotation because they found it in 14 out of 17 stars where they could measure it. However, they are careful to say that the exact shape of the spin profile (whether it goes up and down) is "suggested" by the data. They found evidence for it, but they admit that their models assume the stars spin like solid balls, which makes it hard to map the exact curve perfectly. They also note that in 7 of the original 38 stars they tried to model, they couldn't get a good fit, which might mean those stars have secret ingredients (like magnetic fields or binary interactions) that their models didn't account for.

The Bigger Picture
Why does this matter? It turns out that moving energy and spin around inside a star is much more efficient than anyone thought. The authors found that as these stars age (measured by how much hydrogen is left in their core, denoted as Xc), the core overshooting (a mixing process) gets weaker, and the overall spin slows down.

In short, the universe is full of spinning stars that aren't just solid balls of gas. They are complex, wobbly, and sometimes even spin backward in their layers. This study gives us a much clearer map of how these massive stars behave, proving that the inside of a star is a lot more dynamic than we ever imagined.

Technical Summary: Asteroseismic Forward Modelling of 36 β Cep Pulsators and Inferences on Their Internal Differential Rotation

Problem and Motivation
Asteroseismic observations have revealed that angular momentum transport in main-sequence stars is significantly more efficient than standard stellar evolution models predict. While radial differential rotation has been detected in a small number of massive $\beta$ Cephei ( $\beta$ Cep) pulsators, fewer than ten such stars have been subjected to detailed asteroseismic modelling, and never in a consistent population-level study. Current models struggle to explain the transport mechanisms responsible for efficient angular momentum redistribution, leading to systematic uncertainties in stellar evolution. Furthermore, previous forward modelling efforts for $\beta$ Cep stars often relied on first-order perturbative treatments of rotation, which are suboptimal for the low-order pressure and gravity modes characteristic of these stars. The lack of a large, homogeneously modelled sample limits the ability to constrain transport mechanisms in the high-mass regime ( $8-30 M_\odot$ ).

Methodology
The authors present a population-level study of 36 $\beta$ Cep stars, expanding the sample of asteroseismically modelled targets. The methodology involves a novel forward modelling approach that consistently accounts for second-order rotation effects.

Target Selection: The sample was drawn primarily from Fritzewski et al. (2025), who identified mode degrees ( $l$ ) for over 200 $\beta$ Cep stars using TESS and Gaia photometry. The authors applied stricter selection criteria, requiring at least three observed frequencies with identified degrees and azimuthal orders ( $m$ ). Candidates were validated by searching for rotationally split multiplets that satisfied specific conditions regarding splitting magnitude, symmetry, and consistency with multi-colour degree identifications. Six additional literature stars with established mode identifications were included for validation.
Stellar Model Grid: A new grid of non-rotating stellar models was computed using MESA (Modules for Experiments in Stellar Astrophysics). The grid covered masses from 7 to $30 M_\odot$ , varying central hydrogen fraction ( $X_c$ ), core overshoot ( $f_{ov}$ ), and envelope mixing ( $D_{mix,0}$ ). The initial metallicity was fixed at $Z=0.014$ , with systematic uncertainties assessed by varying $Z$ .
Oscillation Computations: Pulsation frequencies were calculated using StORM (Stellar Oscillations with Rotation), a code optimised for $\beta$ Cep modes. Crucially, StORM includes second-order rotation effects (Coriolis force and rotational deformation) via a Chandrasekhar-Milne expansion. This allows for the prediction of asymmetric rotational splitting, a feature absent in first-order treatments.
Forward Modelling Algorithm: The authors developed a five-dimensional optimisation problem (mass, age, overshoot, envelope mixing, rotation rate).
- A "fixed mode" (typically a radial mode or the highest frequency zonal mode) was selected to determine the stellar age for each evolutionary track.
- The rotation rate ( $f_{rot}$ ) was optimised simultaneously by minimising a $\chi^2$ function based on observed rotational splittings ( $\Delta f$ ).
- Statistical parameter estimates were derived from a weighted average of models within the $2\sigma$ error ellipse in the Hertzsprung-Russell diagram (HRD), with uncertainties propagated from grid steps and metallicity variations.
Differential Rotation Constraints: For 17 stars with multiple identified rotationally split multiplets, the authors re-modelled each star individually for each multiplet. By comparing the rotation rates derived from modes sensitive to different stellar radii (via sensitivity kernels), they constrained the internal differential rotation profile without assuming rigid rotation.

Key Results

Successful Modelling: The authors successfully modelled 36 $\beta$ Cep stars, constraining initial mass, internal rotation frequency, convective core mass, and age. The results for six validation stars agreed with literature values within $1\sigma$ , validating the methodology.
Second-Order Effects: The inclusion of second-order rotation effects was found to be essential for stars rotating faster than $\sim 10\%$ of the critical rotation rate ( $f_{crit}$ ). First-order treatments failed to reproduce multiplet asymmetries and led to discrepancies in derived rotation rates, particularly for dipole ( $l=1$ ) modes.
Convective Core Evolution: The study calibrated the evolution of the relative convective core mass ( $M_{cc}/M$ ). Results indicate that $M_{cc}/M$ decreases as stars evolve (decreasing $X_c$ ) and correlates with mass. Additionally, the core overshoot parameter ( $f_{ov}$ ) was found to weaken with age, suggesting a time-dependent overshooting scheme is necessary for accurate modelling.
Rotation Profiles and Differential Rotation:
- The internal rotation rate generally decreases as $\beta$ Cep stars evolve along the main sequence, consistent with intermediate-mass stars.
- Radial differential rotation was constrained in 17 stars. In 14 of these, the rotation rate varied by more than 10%.
- Ten stars exhibited a rotation rate decreasing from the core to the surface. Conversely, four stars showed an increase in rotation rate from the core to the surface (outward gradient).
- The data suggests that the typical $\beta$ Cep rotation profile may be non-monotonic, featuring a rapidly rotating core, a slower envelope, and potentially an outward increase in rotation rate in the outer envelope. This non-monotonicity is supported by the detection of both positive and negative rotation gradients in stars with multiple multiplets probing different layers.
Specific Angular Momentum: The specific angular momentum ( $J/M$ ) of the sample obeys the upper limits established for intermediate-mass stars, despite the presence of differential rotation.

Significance and Claims
The paper claims to provide the first population-level asteroseismic forward modelling of $\beta$ Cep pulsators. By consistently incorporating second-order rotation effects, the study offers more reliable constraints on stellar parameters than previous first-order approaches. The detection of strong radial differential rotation in the majority of the subsample (14/17 stars) and the evidence for non-monotonic rotation profiles provide critical observational constraints for theories of angular momentum transport in massive stars. The authors conclude that current transport mechanisms must be capable of explaining these complex, non-monotonic profiles, which may involve accretion history or internal gravity waves. The study highlights the need for future models to include time-dependent overshooting and realistic transport mechanisms to reproduce the observed rotational splitting. The authors modestly note that while their results are robust, the assumption of rigid rotation in the oscillation computations limits the precision of the differential rotation profiles, and future work incorporating magnetic fields and 2D/3D effects is required to fully disentangle the physics.

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