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

Using apsidal motion observations of the massive binary HD 152248 to constrain internal stellar structure, this study demonstrates that despite testing various physical mechanisms such as magnetic angular momentum transport and extreme mass-loss rates, current stellar models systematically fail to reproduce observed internal density stratification and the structure constant k2, necessitating large convective boundary mixing to partially match stellar parameters.

Original authors: Sophie Rosu, Luca Sciarini, Sylvia Ekström, Patrick Eggenberger, Joris Josiek, Cyril Georgy

Published 2026-05-21

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CC BY 4.0

Original authors: Sophie Rosu, Luca Sciarini, Sylvia Ekström, Patrick Eggenberger, Joris Josiek, Cyril Georgy

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 massive, double-star system called HD 152248 as a cosmic dance partner pair. These two stars are so massive they are like heavyweight boxers, and they are locked in a tight, elliptical orbit around each other. Because they are so close, they tug on each other, causing their orbit to slowly wobble and rotate over time. This wobble is called apsidal motion.

The authors of this paper treat this wobble like a medical X-ray. By measuring exactly how fast the orbit rotates, they can deduce what the stars look like on the inside. Specifically, they are looking at a number called $k_2$ , which tells us how the mass is distributed inside the star—how dense the core is compared to the fluffy outer layers.

Here is the story of what they found, explained simply:

The Mystery: The "Ghost" Inside the Stars

The scientists built computer models to simulate how these stars should evolve. They used two different rulebooks for how the stars spin and mix their insides:

The Hydro Rulebook: Based purely on fluid dynamics (like water swirling in a bathtub).
The Magnetic Rulebook: Includes magnetic fields acting like invisible gears that lock the star's rotation together.

The Problem: When they compared their computer models to the real observations, the models failed. The models predicted stars that were too "squishy" inside. They predicted a core that wasn't dense enough and an outer layer that wasn't spread out enough. In other words, the models' "X-rays" ( $k_2$ ) didn't match the real "X-rays" from the telescope. The real stars have a much sharper contrast between their heavy core and their light outer skin than the models predicted.

The Investigation: Trying Different Ingredients

The team tried to fix their models by changing various "ingredients" to see if they could make the computer stars look like the real ones. They tested:

Metal content: Changing how much "heavy stuff" (like iron) is in the star.
Helium content: Changing the initial fuel mix.
Mass loss: Changing how fast the stars blow off gas into space.
Mixing length: Tweaking how heat moves through the star's outer layers.

The Result: None of these changes worked. Whether they made the stars heavier, lighter, or changed their chemical makeup, the models still couldn't reproduce the correct internal structure ( $k_2$ ).

The Breakthrough: The "Overshooting" Solution

The key to solving the mystery was a parameter called overshooting ( $\alpha_{ov}$ ).

Think of a star's core as a pot of boiling soup. The "convective core" is the part where the soup is churning violently. Standard physics says the churning stops exactly at the edge of the pot. However, in reality, the churning soup often splashes a bit over the rim before settling down. This "splash" is overshooting.

The paper found that to match the real stars, the "splash" had to be massive.

Standard models usually assume a small splash (about 0.2 times the size of the pot).
To match HD 152248, the models needed a splash 1.2 to 1.3 times the size of the pot.

Why does this help?
When the core mixes more vigorously and extends further out (the big splash), it keeps the star burning its fuel longer and prevents the star from expanding too quickly. This gives the star more time to build up a very dense, heavy core while the outer layers remain light and extended. This creates the sharp density contrast that the real stars show.

The "Twin" Factor

The paper also highlighted a crucial detail: these stars are in a binary system (a pair).

Single Star: A lonely star spins down over time as it loses gas, mixing its insides efficiently.
Binary Star: Because these two stars are dancing so close, their gravity locks them together (tidal locking). They spin at the same speed as they orbit. This "braking" effect stops the internal mixing that usually happens in single stars.

The authors showed that if you ignore the fact that they are a pair and model them as lonely stars, you get the wrong answer. You must account for the fact that they are dancing together to get the physics right.

The "Misalignment" Check

Could the problem be that the stars are tilted? Maybe the stars are spinning on an axis that is tilted relative to their orbit, which would mess up the math?
The authors calculated that even if the stars were tilted at a wild angle (up to 45–50 degrees), it wouldn't fix the problem. The internal structure would still be wrong. The only way to fix the math is to increase that "overshooting" (the big splash of mixing).

The Conclusion

The paper concludes that massive stars are much more "messy" on the inside than we thought. They need a huge amount of mixing at the boundary between their core and their outer layers to explain what we see.

The authors call this method "The Beauty of $k_2$ ." Just as a doctor uses an X-ray to see a broken bone, astronomers can use the wobble of a binary star's orbit to see the hidden, dense heart of a massive star. In this case, the "bone" is the star's internal density, and the "X-ray" revealed that our current recipes for how stars cook need a lot more mixing than we previously believed.

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

Problem Statement
Despite their critical role in shaping the Universe, the internal structure and evolution of massive binaries remain poorly understood. A primary source of uncertainty in predicting the fate of massive binaries (e.g., merger outcomes, supernova types) lies in the treatment of convective boundary mixing and internal angular momentum transport. Previous studies have indicated a need for larger convective boundary mixing to reproduce observed properties, such as the main-sequence width in the Hertzsprung-Russell diagram and the properties of massive eclipsing binaries. However, stars more massive than 20 $M_\odot$ lack a quantitative prescription for convective boundary mixing and an unequivocal constraint on internal mixing mechanisms. While the measurement of apsidal motion in binary systems offers a powerful, independent diagnostic of the internal density stratification via the internal structure constant $k_2$ , previous attempts to model these systems using codes like Clés lacked the inclusion of stellar rotation and tidal effects, limiting their physical interpretability.

Methodology
The authors utilize the massive twin binary HD 152248 as a benchmark to constrain internal mixing. They construct stellar evolution models using the GENeva Evolutionary Code (GENEC), incorporating both stellar rotation and tidal interactions. The study employs two distinct angular momentum transport schemes:

Purely Hydrodynamic Models: Accounting for shear instabilities and meridional circulation.
Magneto-Diffusive Models: Incorporating the Tayler-Spruit dynamo in addition to hydrodynamic processes.

The authors confront single-star and binary-star models to assess the impact of tidal locking. They systematically investigate the influence of various input parameters on stellar evolution, including the step-overshoot parameter ( $\alpha_{ov}$ ), metallicity ( $Z$ ), initial helium abundance ( $Y$ ), initial mass ( $M_{ini}$ ), mass-loss rates (accounting for wind clumping), and the mixing length parameter ( $\alpha_{MLT}$ ). To ensure the accuracy of the stellar radius determination, the authors supplement the GENEC structures with detailed atmosphere models generated by the Potsdam Wolf-Rayet (PoWR) code. Furthermore, they explore the potential impact of a misalignment between the stellar rotation axes and the orbital plane normal on the interpretation of the apsidal motion rate.

Key Results

The $k_2$ -Discrepancy: Models computed with standard assumptions systematically predict $k_2$ values that are significantly larger than the observed value ( $k_{2,\star} = 0.0010 \pm 0.0001$ ). This discrepancy indicates that standard models predict stars with insufficient density contrast between their convective cores and radiative envelopes.
Role of Tides: Binary-star models initialized with pseudo-synchronisation suppress rotational mixing compared to single-star counterparts because the surface angular velocity is fixed by the orbital motion. This suppression leads to larger predicted $k_2$ values in binary models, reinforcing the necessity of using binary-specific models for short-period systems.
Overshooting Requirement: To simultaneously reproduce the observed stellar parameters (mass, radius, effective temperature, luminosity, surface velocity, nitrogen abundance) and the observed $k_2$ $k_{2}$ , the models require significantly enhanced convective boundary mixing.
- Hydrodynamic models require a step-overshoot parameter of $\alpha_{ov} \geq 1.2$ .
- Magneto-diffusive models require $\alpha_{ov} \geq 1.3$ .
- At these high $\alpha_{ov}$ values, the extended main-sequence lifetime and reduced rate of radius expansion allow the star to develop a steeper density contrast between the core and envelope.
Insensitivity to Other Parameters: Variations in metallicity, initial helium abundance, initial mass, and the mixing length parameter have negligible or limited impacts on resolving the $k_2$ discrepancy. Similarly, even if the mass-loss rate is underestimated by a factor of ~1.78 due to wind clumping, the effect on the evolution of stellar parameters and $k_2$ is negligible.
Misalignment Constraints: The authors investigated whether a misalignment of stellar rotation axes could explain the $k_2$ discrepancy. While a large misalignment angle could theoretically increase the required $k_2$ , the efficiency of tidal forces in synchronizing short-period systems like HD 152248 constrains the misalignment angle to a maximum of $45^\circ - 52^\circ$ . Even within this allowed range, misalignment does not resolve the discrepancy; a large overshooting parameter ( $\alpha_{ov} \geq 1.0$ ) remains necessary.

Significance and Conclusions
This paper demonstrates that apsidal motion is a robust and powerful tool for probing the interiors of massive stars. The study concludes that reproducing the observed $k_2$ and apsidal motion rate simultaneously with other stellar parameters necessitates enhanced core boundary mixing, specifically large step-overshoot parameters ( $\alpha_{ov} \approx 1.2 - 1.3$ ). This finding suggests that the core-envelope density profile in massive stars is more pronounced than predicted by standard models. The authors posit that this enhanced mixing might be driven by mechanisms such as gravity waves excited by tidal friction, though the exact physical nature (extension of the convective core vs. strong mixing in the radiative zone) remains to be determined. The work establishes HD 152248 as a critical benchmark for calibrating stellar evolution codes, highlighting the necessity of including tidal effects and large overshooting to accurately model massive binary evolution.