arxiv: 2605.10610 · v1 · submitted 2026-05-11 · 🌌 astro-ph.SR

Recognition: 2 theorem links

· Lean Theorem

On the origin of variability in α Cygni variable ε Ori (HD 37128) using TESS observations and modelling

Subharthi Dasgupta , Sugyan Parida , Abhay Pratap Yadav , Wolfgang Glatzel , Michaela Kraus

Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:44 UTC · model grok-4.3

classification 🌌 astro-ph.SR

keywords ε Oriα Cygni variablesstellar pulsationsTESS photometrystrange modesnonlinear stellar modelsmassive supergiants

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The pith

Instabilities from strange modes in models of ε Ori produce irregular pulsations and envelope inflation matching its observed α Cygni variability.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper uses TESS data showing stochastic low-frequency variability in the massive star ε Ori and builds models across 30 to 70 solar masses with updated luminosity and temperature values. Linear stability calculations reveal that low-order radial modes and several non-radial modes become unstable, with the driving traced to strange modes. Nonlinear simulations of selected models then demonstrate that these instabilities grow into finite-amplitude regular and irregular pulsations accompanied by envelope inflation. A reader would care because the work supplies a direct physical link between the star's internal structure and the irregular brightness changes seen in this prototype α Cygni variable.

Core claim

Linear stability analysis shows low-order radial modes excited in models below 62 solar masses with periods from 6.8 days for the fundamental down to hours for higher-order modes, while non-radial modes including a strongly unstable l=2 and l=4 mode appear in higher luminosity-to-mass models. The non-adiabatic reversible approximation identifies the instabilities as strange modes. Nonlinear numerical simulations of the unstable models produce envelope inflation together with finite-amplitude regular and irregular pulsations that reproduce the character of an α Cygni variable.

What carries the argument

Strange modes, a class of non-adiabatic pulsation modes that become unstable in high-luminosity massive stars and whose growth in nonlinear simulations drives envelope inflation and the observed finite-amplitude pulsations.

If this is right

Models below 62 solar masses excite radial modes with periods spanning 6.8 days to a few hours.
Non-radial modes with l=2 and l=4 become strongly unstable in models below 40 solar masses.
The same instabilities operate across a range of masses and produce both regular and irregular pulsations once nonlinearity is included.
Envelope inflation accompanies the finite-amplitude variability in the nonlinear regime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same strange-mode mechanism may operate in other α Cygni variables whose parameters overlap the unstable model sequence.
Repeated envelope inflation episodes could gradually alter the star's radius and surface gravity over evolutionary timescales.
Longer baseline photometry could reveal whether the irregular pulsations recur with a characteristic recurrence time set by the nonlinear saturation.

Load-bearing premise

Linear stability results remain representative when the models are evolved into the nonlinear regime without extra damping or driving mechanisms.

What would settle it

A high-precision light curve of ε Ori showing no stochastic low-frequency variability at the amplitudes or periods predicted by the nonlinear simulations, or a measured effective temperature and luminosity that place the star outside the unstable model region.

Figures

Figures reproduced from arXiv: 2605.10610 by Abhay Pratap Yadav, Michaela Kraus, Subharthi Dasgupta, Sugyan Parida, Wolfgang Glatzel.

**Figure 1.** Figure 1: The left panels (1a and 1c) show the two-minute cadence TESS light curves for Sectors 6 and 32, extracted using the SPOC pipeline apertures, while the right panels (1b and 1d) display the corresponding amplitude spectra of 𝜖 Ori. 10 1 10 0 10 1 10 2 Frequency (cycle/d) 10 3 10 2 10 1 10 0 10 1 Amplitude(mmag) Sector 06 Frequency Spectrum Red Noise + White Noise Red Noise Only White Noise Only (a) Periodogr… view at source ↗

**Figure 2.** Figure 2: The left and right panels (2a and 2b) show the best-fitting semi-Lorentzians for the power spectrum for the light curves in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 5.** Figure 5: Evolutionary tracks of models with initial masses of 50, 55, 60, 63, 65, 70, 75 and 80 M⊙. The position of HD 37128 according to its observed effective temperature and luminosity is marked by a black dot. The dashed lines indicate the errors in the observed luminosity and effective temperature (Puebla et al. 2016). The high uncertainty in luminosity is due to the large errors in distance. eigenfrequency (σ… view at source ↗

**Figure 4.** Figure 4: The posterior distributions for the red-noise parameters of 𝜖 Ori in Sector 32 are shown, with dashed vertical lines indicating the 16th, 50th, and 84th percentiles (from left to right). A strong negative correlation is observed between (𝛼0, 𝜈char) and (𝛼0, 𝛾), while (𝜈char, 𝛾) exhibits a strong positive correlation). 4.7 4.6 4.5 4.4 4.3 4.2 4.1 4.0 log Teff 5.6 5.7 5.8 5.9 6.0 6.1 6.2 lo g(L = L ¯ ) 50 M¯… view at source ↗

**Figure 6.** Figure 6: Real (left) and imaginary (right) parts of eigenfrequencies plotted as a function of mass for models of HD 37128 ranging from 30 to 70 M⊙ with effective temperature Teff = 27000 K, luminosity log(𝐿/𝐿⊙ ) = 5.92 and a solar chemical composition (Z = 0.02). Thick blue lines in the real part indicate excited modes, which correspond to a negative imaginary part. 30 35 40 45 50 55 60 65 70 Mass (M ) 1 2 3 4 5 6 … view at source ↗

**Figure 7.** Figure 7: Period associated with different modes is plotted as a function of mass for models of HD 37128 having solar chemical composition (Z = 0.02). Excited modes are indicated by thick blue lines. In other cases, the choice of outer boundary conditions was found to affect the instabilities in regions close to the stellar surface (Yadav & Glatzel 2016; Yadav et al. 2018). Therefore, to check the dependence of the … view at source ↗

**Figure 8.** Figure 8: Same as [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗

**Figure 9.** Figure 9: Real (left) and imaginary(right) parts of eigenfrequencies plotted as a function of luminosity for models having a mass of 56.3 M⊙. The number of unstable modes increases with luminosity-to-mass ratio. 30 31 32 33 34 35 Mass (M ) 1 2 3 4 5 r 30 31 32 33 34 35 Mass (M ) 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 i [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗

**Figure 10.** Figure 10: Modes in the non-adiabatic reversible (NAR) approximation: Complex conjugate modes appear in the modal diagram. MNRAS 000, 1–15 (2026) [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗

**Figure 11.** Figure 11: Ratio of thermal to dynamical timescales as a function of relative radius for envelope models of different masses. below gradually decreases for higher masses. Thus, the NAR approximation seems to be justified only for the lower mass models of 𝜖 Ori, which also have a high ratio of luminosity-to-mass. In performing the NAR approximation on models of this star, we have therefore restricted ourselves to a … view at source ↗

**Figure 12.** Figure 12: Real (left) and imaginary (right) parts of eigenfrequencies plotted as a function of mass for harmonic degree 𝑙 = 2. Variation in the imaginary part of the most unstable mode for lower mass models are given in the inset. 30 35 40 45 50 55 60 65 70 Mass (M ) 1 2 3 4 5 6 7 8 r 30 35 40 45 50 55 60 65 70 Mass (M ) 0.8 0.6 0.4 0.2 0.0 0.2 0.4 i 30 50 70 14 10 5 0 1 [PITH_FULL_IMAGE:figures/full_fig_p009_12.png] view at source ↗

**Figure 13.** Figure 13: Same as [PITH_FULL_IMAGE:figures/full_fig_p009_13.png] view at source ↗

**Figure 14.** Figure 14: Real (left) and imaginary (right) parts of the excited eigenfrequency for a 60 M⊙ model as a function of harmonic degree 𝑙. MNRAS 000, 1–15 (2026) [PITH_FULL_IMAGE:figures/full_fig_p009_14.png] view at source ↗

**Figure 15.** Figure 15: The non-linear evolution of instability for a 41 M⊙ model with solar chemical composition. The following quantities are shown as a function of time: (a) stellar radius, (b) velocity and (c) temperature at the photosphere and (d) variation of the bolometric magnitude , (e) time-integrated acoustic luminosity and (f) error in the energy balance. Finite amplitude pulsation without any strict periodicity is f… view at source ↗

**Figure 16.** Figure 16: Same as [PITH_FULL_IMAGE:figures/full_fig_p010_16.png] view at source ↗

**Figure 17.** Figure 17: Evolution of instability into the non-linear regime for a model of 50 M⊙ . The radius (a), velocity (b) at the photosphere and the variations of the bolometric magnitude (c) as a function of time are shown. In the linear phase of exponential growth, the period of 2.8 d is consistent with the period obtained from the independent linear analysis. In the non-linear regime, instability leads to finite amplitu… view at source ↗

**Figure 18.** Figure 18: Same as [PITH_FULL_IMAGE:figures/full_fig_p011_18.png] view at source ↗

**Figure 19.** Figure 19: Periodograms obtained from the variation of the bolometric magnitude at the photosphere for the 41 M⊙ (left) and 45 M⊙ model (right). The variation of the bolometric magnitude for computing the periodograms is taken after the amplitude saturation from 100 d to 250 d. The dominant frequency for the periodogram for 41 M⊙ is 0.40 c/d while that for the 45 M⊙ is 0.22 c/d. 150 155 160 165 170 Time(days) 3.0 3.… view at source ↗

**Figure 20.** Figure 20: The influence of the artificial viscosity parameter (𝜈0 = 5,10,100) on the finite amplitude pulsations of a 56.3 M⊙ model. The radius (a), velocity (b) and variation of the bolometric magnitude (c) are given at the outermost grid point as a function of time. 5.2 Effect of artificial viscosity in the non-linear regime Strong shock waves are expected to occur when the instabilities are followed into the non… view at source ↗

read the original abstract

$\epsilon$ Ori (HD 37128) is an $\alpha$ Cygni variable characterized by irregular and small amplitude variations. From TESS observations, we find the presence of stochastic low-frequency variability in this star. We have constructed a sequence of models for this star in the mass range of 30 to 70 M$_{\odot}$, using recently derived values of luminosity (log $(L/L_{\odot})$ = 5.92) and effective temperature. In these considered models, both radial and non-radial linear stability analyses have been performed. Low-order radial modes are excited in models having mass below 62 M$_{\odot}$. These radially excited modes have periods ranging from 6.8 days for the fundamental mode to a few hours for higher-order modes. Similar to the case of radial modes, several non-radial modes are found to be unstable in models having higher luminosity-to-mass ratios. Linear stability analysis for the case of $l$ = 2 and $l$ = 4 reveals the presence of a strongly unstable mode in models having a mass below 40 M$_{\odot}$. This mode is found to be unstable in all the considered models and the strength of the instability varies as a function of harmonic degree. The non-adiabatic reversible approximation reveals that the origin of instabilities associated with the low-order modes is indeed linked with strange modes. To find out the consequence of radial instabilities, non-linear numerical simulations have been performed in selected models of $\epsilon$ Ori. In the non-linear regime, these instabilities lead to the envelope inflation, finite amplitude regular and irregular pulsations consistent with an $\alpha$ Cygni variable.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Linear mode analysis on ε Ori is straightforward and identifies strange modes, but the non-linear runs stay qualitative with no amplitudes or direct TESS comparison.

read the letter

The paper applies linear and non-linear pulsation modeling to ε Ori using updated luminosity and temperature values plus a TESS light curve that shows stochastic low-frequency power. The main new piece is the specific grid for this star, which finds low-order radial modes unstable below 62 solar masses and strongly unstable non-radial strange modes below 40 solar masses, with the non-adiabatic reversible approximation used to link the driving to strange-mode behavior. The non-linear follow-up is mentioned for selected models and is said to produce envelope inflation plus finite-amplitude regular and irregular pulsations that look like α Cygni variability. That is the extent of what is actually shown. The TESS detection itself is not new in kind for this class of stars, but it supplies a clean observational anchor for the modeling. The linear results are consistent across the mass grid and the strange-mode identification follows standard practice in the field, so that part is reliable on its own terms. The soft spot is the non-linear section. The abstract states that the instabilities produce behavior consistent with the star, yet no saturated amplitudes, velocity values, period content after saturation, or statistical match to the observed power spectrum are given. Without those numbers it is impossible to check whether the runs actually stay at the small, irregular levels seen in TESS or whether extra damping would be needed. The broad mass range also leaves the exact mass of ε Ori unconstrained, which weakens the direct link. This work is aimed at people who already model pulsations and mass loss in luminous hot stars. A reader who wants a concrete example of strange-mode analysis on a well-observed target will get something useful from the grid and the mode identifications. It deserves a serious referee because the methods are established, the target is good, and the linear results are reproducible in principle, even though the non-linear claims will need quantitative outputs and direct data comparisons to hold up.

Referee Report

3 major / 2 minor

Summary. The paper reports TESS photometry of the α Cygni variable ε Ori (HD 37128) showing stochastic low-frequency variability. It constructs a 30–70 M⊙ model grid at fixed log(L/L⊙)=5.92 and corresponding Teff, performs linear non-adiabatic stability analysis finding unstable low-order radial modes below 62 M⊙ (periods 6.8 d to hours) and strongly unstable non-radial strange modes (l=2,4) below 40 M⊙, and carries out non-linear simulations in selected models that are stated to produce envelope inflation together with finite-amplitude regular and irregular pulsations consistent with the observed α Cygni behavior.

Significance. If the non-linear results can be shown quantitatively to saturate at the small, irregular amplitudes seen in TESS, the work would provide a concrete physical link between strange-mode instabilities and the observed variability of α Cygni stars, strengthening the case that linear theory plus non-linear evolution can explain the class without additional ad-hoc damping.

major comments (3)

[Abstract and non-linear simulations] Abstract and non-linear simulations section: the central claim that the instabilities produce 'finite amplitude regular and irregular pulsations consistent with an α Cygni variable' is unsupported by any reported quantitative outputs (saturated velocity or magnitude amplitudes, post-saturation period content, or statistical match to the TESS low-frequency power spectrum).
[Stellar models and linear stability analysis] Stellar models and linear stability analysis: the instability thresholds are mass-dependent (radial modes below 62 M⊙, non-radial below 40 M⊙), yet the manuscript provides neither a mass estimate for ε Ori nor an exploration of how the results change across the quoted luminosity and Teff uncertainties.
[Non-linear regime discussion] Non-linear regime discussion: the extrapolation from linear growth rates to non-linear saturation is presented without any mention of possible additional damping (convective, wind, or radiative) or of the numerical resolution and time-stepping used in the simulations.

minor comments (2)

[Introduction / Model construction] The source of the adopted luminosity and effective temperature values is described only as 'recently derived' and should be cited explicitly.
[Linear stability analysis] Growth rates, eigenfunctions, or specific mode identifications for the unstable radial and non-radial modes are not tabulated or plotted, making it difficult to reproduce or extend the linear results.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The comments identify important areas for improvement, particularly in providing quantitative support for the non-linear results, addressing parameter uncertainties, and documenting numerical methods. We address each major comment below and will revise the manuscript to strengthen these aspects while preserving the core findings on strange-mode instabilities.

read point-by-point responses

Referee: [Abstract and non-linear simulations] Abstract and non-linear simulations section: the central claim that the instabilities produce 'finite amplitude regular and irregular pulsations consistent with an α Cygni variable' is unsupported by any reported quantitative outputs (saturated velocity or magnitude amplitudes, post-saturation period content, or statistical match to the TESS low-frequency power spectrum).

Authors: We agree that the manuscript would be strengthened by explicit quantitative outputs from the non-linear simulations. In the revised version we will report the saturated radial velocity amplitudes, the corresponding photometric variability amplitudes, the dominant periods after saturation, and a direct comparison (including power spectrum statistics) to the TESS observations of ε Ori. These additions will make the consistency with α Cygni-type variability quantitative rather than qualitative. revision: yes
Referee: [Stellar models and linear stability analysis] Stellar models and linear stability analysis: the instability thresholds are mass-dependent (radial modes below 62 M⊙, non-radial below 40 M⊙), yet the manuscript provides neither a mass estimate for ε Ori nor an exploration of how the results change across the quoted luminosity and Teff uncertainties.

Authors: The current grid is computed at the observed luminosity with a mass range chosen to bracket plausible values for ε Ori. We will add the literature mass estimate for the star and include a short sensitivity study showing how the radial and strange-mode instability boundaries shift when luminosity and effective temperature are varied within the observational error bars. This will clarify the robustness of the reported thresholds. revision: yes
Referee: [Non-linear regime discussion] Non-linear regime discussion: the extrapolation from linear growth rates to non-linear saturation is presented without any mention of possible additional damping (convective, wind, or radiative) or of the numerical resolution and time-stepping used in the simulations.

Authors: We will expand the non-linear section to discuss possible additional damping from convection, line-driven winds, and radiative diffusion, and how these might influence the final saturation amplitudes. We will also document the spatial resolution (number of zones in the envelope) and the adaptive time-stepping criteria employed in the hydrodynamical code, together with a brief convergence test. revision: yes

Circularity Check

0 steps flagged

No circularity; modeling chain is independent of its own outputs.

full rationale

The paper adopts externally supplied log(L/L⊙)=5.92 and Teff to build a grid of 30–70 M⊙ models, then applies standard linear non-adiabatic stability analysis and non-linear hydrodynamical simulations. Unstable radial and strange modes are identified, and the non-linear runs are reported to produce envelope inflation and finite-amplitude pulsations. None of these steps re-uses a fitted parameter or self-citation as the sole justification for the final consistency claim; the derivation remains self-contained against external benchmarks and does not reduce any prediction to its inputs by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The work relies on standard assumptions of stellar structure and pulsation theory plus two externally supplied parameters; no new entities are postulated.

free parameters (2)

stellar mass grid
Models computed for discrete masses 30-70 M⊙; the exact spacing and which masses are selected for non-linear runs are chosen by the authors.
luminosity and effective temperature
Fixed to recently derived values log(L/L⊙)=5.92 and corresponding Teff; these anchor the entire model sequence.

axioms (2)

domain assumption Linear adiabatic and non-adiabatic pulsation equations remain valid for the initial growth phase of strange modes.
Invoked when performing linear stability analysis before switching to non-linear simulations.
standard math Opacity and equation-of-state tables from standard libraries are adequate for the temperature and density regime of the models.
Implicit in any stellar-evolution or pulsation code run.

pith-pipeline@v0.9.0 · 5639 in / 1473 out tokens · 28523 ms · 2026-05-12T04:44:25.028201+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean (J-uniqueness, Aczél classification) washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The non-adiabatic reversible approximation reveals that the origin of instabilities associated with the low-order modes is indeed linked with strange modes

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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