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Formation of classical Be-stars of the early spectral subclass in the case of nonconservative mass transfer in close binary systems

This paper demonstrates that nonconservative mass transfer in close binary systems can spin up a 16 solar-mass accretor to the rapid rotation characteristic of early Be-stars, provided the accreted mass exceeds 30% of the star's total mass, a result that remains robust regardless of the accretor's initial rotation, boundary layer angular momentum, disk sub-Keplerian velocities, or internal turbulence efficiency.

Original authors: Evgeny Staritsin

Published 2026-04-23

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CC BY 4.0

Original authors: Evgeny Staritsin

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

The Cosmic Dance: How Stars Get a Spin

Imagine two stars dancing in a tight embrace, a binary system. One star (the Donor) is older and tired; it's expanding like a balloon being blown up. The other star (the Accretor) is younger and hungry.

As the Donor gets too big, it spills its outer layers of gas over to the Accretor. This isn't just a gentle pour; it's a massive transfer of material. The paper asks a simple but profound question: How much gas does the hungry star need to eat to start spinning so fast that it becomes a "Be-star"?

Be-stars are a special type of star that spin incredibly fast—so fast they often fling their own atmosphere into a disk around them, creating a glowing ring. Scientists have been trying to figure out exactly how they get that speed.

The Main Idea: The "Spin-Up" Recipe

The author, Evgeny Staritsin, ran a complex computer simulation to see what happens when the hungry star eats gas from its neighbor. He tested different scenarios:

What if the star eats a little bit (5% more mass)?
What if it eats a lot (100% more mass)?
What if the gas comes in a specific way?

He found a "magic number" for the recipe.

The Magic Number: 30%

The study concludes that for a star to become a fast-spinning Be-star, it needs to gain at least 30% of its own weight in new material from its partner.

If it eats less than 30%: It gets a little faster, but not fast enough to be a true Be-star. It's like a figure skater who pulls their arms in a little bit; they speed up, but not enough to do a triple axel.
If it eats more than 30%: It spins up to the "critical speed." This is the speed where the star is spinning so fast that if it went any faster, it would fly apart. This is the Be-star zone.

How the Physics Works (The "Spinning Top" Analogy)

You might think, "If I dump a bucket of water onto a spinning top, it should just slow down because of the weight." But stars are different.

The Delivery Truck: The gas falling onto the star isn't just dropping straight down; it's swirling around like water going down a drain. It has a lot of angular momentum (spin energy).
The Boundary Layer: When this swirling gas hits the star, there's a friction zone (the boundary layer). Sometimes this zone acts like a slippery slide, letting the spin energy pass through easily. Sometimes it's sticky. The paper found that it doesn't really matter if the slide is slippery or sticky, as long as the star eats enough gas (the 30% rule).
The Internal Mixer (Meridional Circulation): This is the coolest part. Once the gas lands on the star, the star has to mix it in. Imagine the star is a giant pot of soup. When you pour hot broth in, you need to stir it so the temperature is even.
- The star has a natural "stirring spoon" called meridional circulation. It's a giant current that moves gas from the equator to the poles and back down.
- This circulation acts like a conveyor belt. It takes the fast-spinning gas from the surface and pushes it deep inside the star, while bringing slower gas from the core up to the surface.
- The Result: The star doesn't just spin up on the outside; the whole star starts spinning faster together.

The "Leaky Bucket" Effect

Here is a surprising twist: The star actually loses a lot of the spin energy it gains.

When the star gets too fast (reaching that critical speed), it acts like a centrifuge. It starts flinging excess spin energy back out into the disk of gas surrounding it.
The paper found that if the star gains more than 10% of its mass, it will lose 50% to 80% of the spin energy it tried to keep.
Why does it still spin fast? Because it keeps eating! Even though it's losing energy, the continuous stream of new, fast-spinning gas keeps pushing it to the limit. It's like a child on a swing: you have to keep pushing (accreting mass) to keep them high, even if air resistance (losing angular momentum) is trying to slow them down.

Why This Matters

Before this study, scientists were guessing how much mass a star needed to gain to become a Be-star. Some thought it was a tiny bit (5%), others thought it was a lot.

This paper says: "It's the 30% rule."

If a star gains 30% or more: It will almost certainly become a fast-spinning Be-star, no matter how the gas was transferred or how turbulent the inside of the star is.
If a star gains less: It's a coin toss. It might become a Be-star, or it might just be a normal, moderately spinning star. It depends on many other tiny details (like how fast it was spinning to begin with).

The Bottom Line

Think of the binary system as a cosmic gym. The hungry star is the weightlifter.

If the weightlifter only adds a few small weights (low mass gain), they get a little stronger but not a bodybuilder.
If the weightlifter adds a massive amount of weight (over 30% gain), they transform into a super-athlete (a Be-star).
The way the weights are lifted (the boundary layer) or how much they sweat (turbulence) doesn't change the outcome as much as the total amount of weight they lift.

This helps astronomers understand why we see so many of these fast-spinning stars in the universe and how they are born from the messy, violent interactions of binary star systems.

1. Problem Statement

Classical Be stars are defined by their emission in Balmer lines and rapid rotation, typically exceeding 40–60% of their Keplerian break-up velocity. While the origin of this rapid rotation is debated, a leading hypothesis involves mass transfer in close binary systems. In such systems, a donor star (evolving off the main sequence) fills its Roche lobe, transferring mass to a companion (the accretor).

The central problem addressed is non-conservative mass transfer: a significant portion of the mass lost by the donor leaves the binary system, carrying away angular momentum. The paper investigates how the fraction of mass retained by the accretor ( $\zeta$ ) influences the final rotational velocity of the accretor. Specifically, it seeks to determine the minimum mass gain required for the accretor to achieve the rapid rotation characteristic of early Be stars (spectral types B0–B2, typically $\sim 16 M_\odot$ ), considering complex internal angular momentum transport and boundary conditions at the accretion disk.

2. Methodology

The study employs a detailed 1D evolutionary code (modified from Paczynski 1970) to model the structure and evolution of the accreting component. Key methodological components include:

Physical Framework:
- Angular Momentum Transport: The model accounts for transport within the radiative envelope via meridional circulation and shear turbulence, following the theory of Zahn (1992). The transport is governed by the advection-diffusion equation for angular momentum.
- Boundary Conditions: The interaction between the accretor and the accretion disk is modeled via a boundary layer. Two extreme cases are considered:
  1. Efficient Boundary Layer: Angular momentum is effectively transferred from the disk to the star.
  2. Inefficient Boundary Layer: Angular momentum transfer is blocked; the star only accretes mass with surface-specific angular momentum.
- Critical Rotation: The model assumes that once the surface reaches critical (Keplerian) rotation, the star remains in a state of critical rotation. Excess angular momentum is transported outward to the disk via viscosity, allowing mass accretion to continue without spinning the star up beyond the break-up limit.
Simulation Setup:
- Target Mass: All simulations result in a final accretor mass of $16 M_\odot$ (typical for early Be stars).
- Variable Parameter: The fraction of accreted mass ( $\zeta = \Delta M_a / M_{final}$ ) is varied from 5% to 50%.
- Initial Conditions: The donor completes its main sequence evolution before mass transfer begins. The accretor is initially in synchronous rotation with the orbit.
- Turbulence Efficiency: The study tests the upper limit of turbulence efficiency (setting horizontal turbulent viscosity $\nu_h = K$ , the thermal diffusivity) to ensure robustness of the results.

3. Key Results

A. Angular Momentum Evolution

Critical Rotation Threshold: The accretor reaches critical rotation after a relatively small mass increase (2–4% for efficient boundary layers; 4–7% for inefficient ones).
Mass Loss Mechanism: Once critical rotation is reached, the star enters a regime where it accretes mass but loses angular momentum to the disk.
- If the mass gain ( $\zeta$ ) exceeds 10%, the accretor loses 50–80% of the angular momentum brought in by the accreted mass.
- The majority of the retained angular momentum remains in the outer accreted layers rather than the deep interior.

B. Internal Transport Dynamics

Meridional Circulation: This is the dominant mechanism for transporting angular momentum from the accreted surface layers into the interior.
Circulation Cells:
- An "inner cell" forms immediately upon accretion, transferring angular momentum inward.
- An "outer cell" forms near the surface when the star is critically rotating, transferring excess angular momentum back to the disk.
Turbulence Role: While turbulence aids in deepening the circulation cells, the study finds that the final rotational state is weakly dependent on the efficiency of turbulence. Even with suppressed turbulence, the results hold.

C. Final Rotational State vs. Mass Gain ( $\zeta$ )

The study establishes a clear correlation between the fraction of accreted mass and the final rotation velocity ( $V_{rot}/V_{Kep}$ ):

$\zeta < 10\%$ : The final rotation is highly sensitive to initial conditions (initial rotation speed, boundary layer efficiency, disk rotation speed). It is uncertain if these stars reach Be-star rotation speeds.
$10\% \le \zeta < 30\%$ : The accretor likely reaches rotation speeds typical of Be stars, but this depends on the precise definition of the lower limit for Be-star classification.
$\zeta > 30\%$ : The accretor unconditionally acquires rotation speeds typical of early Be stars ( $>40-60\%$ $> 40 - 60%$ of Keplerian).
- This conclusion is robust: it holds regardless of the initial rotation of the accretor, the efficiency of the boundary layer, the efficiency of internal turbulence, or sub-Keplerian disk rotation.

4. Key Contributions

Quantification of Mass Gain Threshold: The paper provides a specific quantitative threshold ( $\zeta > 30\%$ ) required for a binary accretor to become an early Be star, resolving ambiguities in previous population synthesis studies that assumed a wide range of accretion efficiencies.
Decoupling of Factors: It demonstrates that for significant mass gains, the final rotation is independent of complex, poorly constrained variables such as boundary layer physics and disk sub-Keplerian rotation.
Mechanism Clarification: It details the specific hydrodynamic mechanism (meridional circulation driving angular momentum loss to the disk) that allows a star to accrete massive amounts of material while maintaining critical rotation without breaking apart.
Resolution of X-ray Binary Discrepancy: The results support the hypothesis that Be stars in X-ray binaries must have accreted $>30\%$ of the donor's lost mass to explain their observed mass lower limits (6–8 $M_\odot$ ), contradicting models assuming only 5–10% accretion efficiency.

5. Significance

This work significantly advances the understanding of binary stellar evolution and the formation of Be stars. By establishing that non-conservative mass transfer with high accretion efficiency ( $\zeta > 30\%$ ) is a robust pathway to forming early Be stars, the paper:

Provides a physical explanation for the observed rotation rates of Be stars without requiring fine-tuned initial conditions.
Suggests that population synthesis models must account for high accretion efficiencies to correctly reproduce the mass distribution of Be stars, particularly in X-ray binaries.
Validates the use of simplified boundary conditions in future large-scale simulations, as the final rotational state is insensitive to the detailed physics of the boundary layer once the accreted mass fraction is substantial.

In summary, the paper concludes that the formation of classical early Be stars via binary mass transfer is a natural outcome of non-conservative evolution, provided the accretor retains more than 30% of the mass lost by its donor.