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arxiv: 2605.21236 · v2 · pith:6QVSRYGHnew · submitted 2026-05-20 · 🌌 astro-ph.SR

The Beauty of k2: Probing Stellar Interiors Using Apsidal Motion. I. The Benchmark Massive Binary HD 152248

Sophie Rosu , Luca Sciarini , Sylvia Ekstr\"om , Patrick Eggenberger , Joris Josiek , Raphael Hirschi , Cyril Georgy This is my paper

Pith reviewed 2026-05-22 08:49 UTC · model grok-4.3

classification 🌌 astro-ph.SR
keywords apsidal motionmassive binary starsconvective overshootingstellar structure constantsk2density stratificationGENEC stellar models
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The pith

Apsidal motion in the massive binary HD 152248 requires large convective overshooting to match observed stellar structure constants k2.

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

The paper examines the massive twin binary HD 152248 to use its observed apsidal motion as a probe of the stars' internal structure. Apsidal motion rates depend on the structure constants k2, which reflect the density distribution inside the stars. Models using both hydrodynamic and magnetic treatments of angular momentum transport require a large step overshoot of 1.2 to match the observed k2 values and other stellar properties. This finding indicates that current models underestimate the density contrast between the convective core and outer layers in stars above 20 solar masses. The result holds even when varying metallicity, helium abundance, or mass-loss rates, establishing apsidal motion as a robust diagnostic for convective boundary mixing.

Core claim

Models of the stars in HD 152248, computed with the GENEC code under both purely hydrodynamic and magneto-diffusive angular momentum transport, yield k2 values systematically larger than those inferred from the observed apsidal motion. To reproduce the stellar parameters including the apsidal motion, both model families require a step-overshoot parameter of 1.2 pressure scale heights. This points to an insufficient density contrast between the core and external layers in the models. Other parameters such as initial mass, metallicity, helium content, and mass-loss rate have negligible impact on the evolution of k2. The assumption of pseudo-synchronization due to efficient tidal locking is a

What carries the argument

The internal structure constant k2, which quantifies the departure from uniform density and enters the expression for the apsidal motion rate in close binaries.

If this is right

  • Massive star models need significantly larger convective overshooting than currently standard to match internal structure.
  • The k2 discrepancy suggests current models have insufficient core-envelope density contrast.
  • Apsidal motion observations can constrain mixing physics independently of other parameters like mass loss.
  • For this system, pseudo-synchronization due to tides is a valid assumption, limiting misalignment angles to about 50 degrees.

Where Pith is reading between the lines

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

  • This approach could be applied to other massive binaries to build a larger sample of k2 measurements.
  • If the large overshoot is confirmed, it would affect predictions for supernova progenitors and chemical enrichment.
  • The result highlights the value of binary systems for testing single-star evolution assumptions.

Load-bearing premise

The observed apsidal motion rate is determined solely by the internal structure constants k2 of the two stars, with negligible contributions from unmodeled tidal or rotational effects.

What would settle it

A high-precision asteroseismic measurement of the core size or density profile in one of the stars would directly test whether the large overshoot value of 1.2 is required.

Figures

Figures reproduced from arXiv: 2605.21236 by Cyril Georgy, Joris Josiek, Luca Sciarini, Patrick Eggenberger, Raphael Hirschi, Sophie Rosu, Sylvia Ekstr\"om.

Figure 1
Figure 1. Figure 1: Evolution of stellar parameters with stellar radius for hydro (plain lines) and magnetic (dashed lines) single-star models for different values of 𝑣ini (colours). The models have 𝑀ini = 30.2𝑀⊙, 𝛼ov = 0.5, 𝑍 = 0.0183, 𝑌⊙, and Krtička et al. (2024) mass-loss rate. Binary-star models initialised synchronised, all other parameters identical, are shown for comparison. Observational values and their error bars a… view at source ↗
Figure 2
Figure 2. Figure 2: Evolution of the surface angular velocity Ωsurf (left panel) and surface velocity 𝑉surf (right panel) with stellar age for single and binary￾star models. The models have 𝑀ini = 30.2𝑀⊙, 𝛼ov = 0.8, 𝑍 = 0.0183, 𝑌⊙, and Krtička et al. (2024) mass-loss rate. Models that reproduce 𝑅★ are in the gray boxes. Therefore, models with larger 𝛼ov have slightly larger 𝑉surf. 𝑁/𝐻 is not significantly nor directly affecte… view at source ↗
Figure 4
Figure 4. Figure 4: Density stratification (top panels) and density function (Eq. (5), bottom panels) in binary-star models as a function of the radial distance 𝑟 in the star. Models have 𝛼ov = 0.5 (left panels) or 𝛼ov = 1.4 (right panels), 𝑍 = 0.0183, 𝑌⊙, and Krtička et al. (2024) mass-loss rate. Mod￾els have increasing age from dark to light shades (the lightest shade models have 𝑅★). The location of the border of the conve… view at source ↗
Figure 5
Figure 5. Figure 5: Density stratification (top panels) and density function (Eq. (5), bottom panels) in binary-star models as a function of the normalised radial distance 𝑟/𝑅 in the star for the same models as in [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Evolution of stellar parameters with stellar radius for hydro (plain lines) and magnetic (dashed lines) binary-star models of 𝑀ini = 30.2𝑀⊙, 𝛼ov = 0.5, and Krtička et al. (2024) mass-loss rate for different values of 𝑍 and𝑌. A clumped model with 𝑀ini = 31.0𝑀⊙, 𝛼ov = 0.5, 𝑍 = 0.0183, 𝑌⊙, and Krtička et al. (2024) mass-loss rate multiplied by 𝜉 = 1.78 is also shown. Observational values and their error bars … view at source ↗
Figure 7
Figure 7. Figure 7: Density stratification (top panels) and density function (Eq.(5), bottom panels) in binary-star models with 𝑅 = 𝑅★ as a function of the normalised radial distance 𝑟/𝑅 in the star. The reference model (fuchsia) has 𝛼ov = 0.5, 𝑍 = 0.0183, 𝑌⊙, and Krtička et al. (2024) mass-loss rate. Models with 𝑍 = 0.0215, 𝑌⊙ + 0.10, and Krtička et al. (2024) mass-loss rate multiplied by 1.78, all other parameters identical… view at source ↗
Figure 8
Figure 8. Figure 8: Evolution of 𝑘2 with stellar radius for hydro (plain lines) and magnetic (dashed lines) binary-star models of 𝑀ini = 30.2𝑀⊙, 𝛼ov = 0.5, 𝑍 = 0.0183,𝑌⊙, and Krtička et al.(2024) mass-loss rate for different values of 𝛼MLT. Observational values and their error bars are shown (plain black line and grey shaded area). 0.8 1.0 1.2 1.4 1.6 1.8 k2 (10 − 3) Hydro Magnetic Standard Clumped 28.8 29.0 29.2 29.4 29.6 29… view at source ↗
Figure 10
Figure 10. Figure 10: Dependence of − 𝐹𝛼 sin2 𝑖 (from Eq. (11), top panel), 𝑉surf (middle panel), and 𝑘2,mis (from Eq. (10), bottom panel) with 𝛼 for the specific case of 𝜃 = 0 ◦ (blue) and 𝜃 = 180◦ (pink) and with 𝑖 = 67.6 ◦ . The forbid￾den region of 𝑉surf > 𝑣cr (in orange in the middle panel) translates into forbidden regions for 𝛼 (in blue and pink delimited by dotted lines for 𝜃 = 0 ◦ and 180◦ , respectively). Consequentl… view at source ↗
Figure 9
Figure 9. Figure 9: Best-fit models in terms of radius: Stellar parameters as a function of 𝛼ov. Models that fit 𝑘2,★, 𝑀★, 𝑇eff,★, 𝐿bol,★, 𝑉surf,★, and 𝑀¤★ are shown with filled symbols. Error bars include the error bars on 𝑅★ and 𝑍★ (the latter dominates over the former). Observational values and their error bars are shown (plain and dashed black lines). −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 − Fα /sin 2 i Forbidden Vsurf > … view at source ↗
read the original abstract

Over the last decades, several independent studies have shown the need for large convective boundary mixing (CBM) and convective core sizes in massive stars to reproduce a variety of their observed properties. Yet, stars more massive than 20Msun lack a quantitative prescription for CBM as well as an unequivocal constraint on the internal mixing mechanisms acting in them. We use the apsidal motion observed in the twin binary HD152248 - linked to the internal stellar structure constants k2 of the stars - to constrain massive stars' internal density stratification and CBM. We build GENEC stellar models assuming two different angular momentum transports: purely hydrodynamic (hydro) and magneto-diffusive (magnetic). We confront single- and binary-star models to assess the impact of tidal locking on the star's evolution. We investigate the impact of CBM (overshooting), metallicity, initial helium abundance and mass, mass-loss rate, and mixing length parameter on the evolution of stellar parameters. We highlight that k2 from the models are systematically larger than observed ones, the so-called k2-discrepancy. Models predict stars with too low a density contrast between their core and external layers. Both hydro and magnetic models require large step-overshoot of 1.2 to reproduce stellar parameters, including k2. Other parameters have almost no impact. Given the efficiency of tides to synchronise systems, the assumption of pseudo-synchronisation is sound for this system. It sets an upper limit on the misalignment angle of stellar rotation axes of ~50{\deg}. Even with such unexpected large angles, the k2-discrepancy is not solved. Even if the mass-loss rate was underestimated by a factor two, it would have no impact on stellar parameters evolution, including k2. It demonstrates that the apsidal motion is a powerful, robust means to probe stellar interiors.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript models the massive twin binary HD 152248 with the GENEC code under purely hydrodynamic and magneto-diffusive angular-momentum transport. By comparing predicted apsidal-motion constants k2 to the value inferred from the observed apsidal rate, the authors conclude that a step-overshoot parameter of 1.2 is required to reproduce both the stellar parameters and k2, that models systematically under-predict the core-to-envelope density contrast (the k2-discrepancy), and that apsidal motion remains a robust interior probe even after allowing for misalignment angles up to ~50°.

Significance. If the mapping from observed apsidal rate to k2 is robust, the work supplies a rare quantitative anchor for convective-boundary mixing in stars above 20 solar masses, where prescriptions remain largely unconstrained. The side-by-side hydro versus magnetic comparison and the systematic exploration of mass, metallicity, helium, mass-loss, and mixing-length variations are positive features that isolate overshoot as the dominant degree of freedom.

major comments (2)
  1. [Abstract and results section describing the overshoot scan] The central numerical result (step-overshoot = 1.2) is obtained by tuning the overshoot parameter until the model k2 matches the observed value. This procedure converts the exercise into a calibration rather than an independent test of the internal-structure prediction; the k2-discrepancy is therefore an empirical finding only after the fit has been performed.
  2. [Discussion of pseudo-synchronization and misalignment limit] The assumption that the observed apsidal rate is determined solely by the two stars' k2 values, with negligible contributions from GR precession, higher-order tidal terms, or residual misalignment, is stated but not quantified relative to the measurement precision. The manuscript checks misalignment up to ~50° and invokes efficient tidal synchronization, yet does not report the magnitude of the omitted terms or propagate them into the uncertainty on the inferred k2.
minor comments (2)
  1. [Parameter-variation subsection] Explicit tables or figures showing the change in k2 when each secondary parameter (metallicity, initial helium, mass-loss rate, mixing length) is varied would strengthen the claim that these quantities have 'almost no impact'.
  2. [Observational constraints paragraph] The observational error budget on the apsidal rate, the data-reduction steps, and the precise fitting procedure used to extract the observed k2 should be stated with numerical values rather than summarized qualitatively.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. Below we respond point by point to the two major comments, indicating the changes we will make to the manuscript.

read point-by-point responses

  1. Referee: [Abstract and results section describing the overshoot scan] The central numerical result (step-overshoot = 1.2) is obtained by tuning the overshoot parameter until the model k2 matches the observed value. This procedure converts the exercise into a calibration rather than an independent test of the internal-structure prediction; the k2-discrepancy is therefore an empirical finding only after the fit has been performed.

    Authors: We agree that the specific value of 1.2 is identified by matching the model k2 to the observed apsidal-motion constant. Our systematic parameter study nevertheless shows that overshoot is the only input that substantially alters k2, while changes in mass, metallicity, initial helium, mass-loss rate and mixing length produce negligible shifts. The k2-discrepancy therefore remains a genuine finding: models computed with the smaller overshoot values commonly adopted for lower-mass stars systematically under-predict the core-to-envelope density contrast required by the data. We will revise the abstract and the results section that describes the overshoot scan to frame the exercise explicitly as a determination of the convective-boundary-mixing parameter needed to reproduce both the stellar parameters and k2, rather than as an a-priori prediction. revision: yes

  2. Referee: [Discussion of pseudo-synchronization and misalignment limit] The assumption that the observed apsidal rate is determined solely by the two stars' k2 values, with negligible contributions from GR precession, higher-order tidal terms, or residual misalignment, is stated but not quantified relative to the measurement precision. The manuscript checks misalignment up to ~50° and invokes efficient tidal synchronization, yet does not report the magnitude of the omitted terms or propagate them into the uncertainty on the inferred k2.

    Authors: We acknowledge that the manuscript does not yet provide explicit estimates of the general-relativistic precession term, higher-order tidal contributions, or the residual misalignment effect expressed relative to the observational uncertainty on the apsidal rate. The present text shows only that misalignment angles up to 50° leave the k2-discrepancy intact and that tidal synchronisation is expected to be efficient. In the revised version we will calculate the magnitude of the GR term for the observed orbital period and masses, estimate the size of higher-order tidal corrections, and propagate these contributions into the uncertainty budget on the inferred k2. This will allow a direct comparison with the measurement precision and will strengthen the claim that the observed apsidal motion remains a robust interior diagnostic. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives an observed k2 value from the measured apsidal motion rate of HD 152248 and then adjusts the step-overshoot parameter in GENEC models (both hydro and magnetic) until the model k2 matches the observed value. This is presented as a constraint on convective boundary mixing rather than a first-principles prediction. The abstract and text explicitly frame the large overshoot of 1.2 as required 'to reproduce stellar parameters, including k2' and highlight the k2-discrepancy as a comparison between model outputs and external observational data. No equation reduces to its input by construction, no fitted quantity is relabeled as an independent prediction, and no self-citation chain is load-bearing for the central result. The derivation remains self-contained against the external benchmark of the observed apsidal motion once the stated assumptions (pseudo-synchronization, negligible higher-order tides) are granted.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the interpretation of observed apsidal motion as a direct probe of internal density stratification and on the ability of the GENEC code to isolate the effect of convective boundary mixing from all other physics.

free parameters (1)
  • step-overshoot = 1.2
    Adjusted to 1.2 to bring model k2 into agreement with the observed value.
axioms (1)
  • domain assumption Observed apsidal motion constant k2 directly reflects the internal density stratification of the stars
    This link is invoked to translate the measured apsidal motion into a constraint on core size and mixing.

pith-pipeline@v0.9.0 · 5906 in / 1438 out tokens · 43825 ms · 2026-05-22T08:49:46.492615+00:00 · methodology

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    Zahn, J. P. 1992, A&A, 265, 115 Article number, page 12 Rosu et al.: The Beauty of𝑘2: Probing Stellar Interiors Using Apsidal Motion Appendix A: Additional figures 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 k2 (10□3) Solid: Hydro Dashed: Magnetic Reference Period free vini/vcr = 0.1 vini/vcr = 0.2 28.8 29.0 29.2 29.4 29.6 29.8 30.0 30.2 M (M⊙) 32.0 32.5 33.0...

    work page 1992