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Evolution of Massive Main-sequence Stars in Rapid Population Synthesis. I. Framework and Implementation

This paper introduces a new semianalytical framework implemented in the COMPAS code that improves the physical realism of massive main-sequence star evolution in rapid binary population synthesis by accurately modeling convective core growth, rejuvenation, and mergers under arbitrary mass-loss or mass-gain histories, ultimately predicting more massive helium cores, compact radii for stripped stars, and higher black hole masses.

Original authors: Adam Brček, Ryosuke Hirai, Ilya Mandel, Harmony Lower

Published 2026-04-30

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CC BY 4.0

Original authors: Adam Brček, Ryosuke Hirai, Ilya Mandel, Harmony Lower

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 Big Picture: Updating the "Star Map"

Imagine you are trying to predict the future of a city's population. To do this, you need a map that tells you how people age, how they move, and how their families change over time.

For decades, astronomers have used a specific "map" (a computer code called COMPAS) to predict how massive stars and their binary partners (two stars orbiting each other) evolve. This map was good, but it had a major blind spot: it treated stars like they had no memory.

If a star lost a huge chunk of its weight (mass) due to strong winds or a fight with a neighbor, the old map just said, "Okay, you are now lighter. Let's look up the life story of a star that was born at this lighter weight." It erased the star's history.

This paper presents a new, upgraded map. It gives stars a memory. It tracks how their internal "engine" (the core) changes even when they lose or gain weight. This leads to much more accurate predictions about how big their black holes will be and how they interact with their partners.

The Core Problem: The "Engine" vs. The "Body"

Think of a massive star like a car.

The Body: The outer layers of gas (the total mass).
The Engine: The central core where nuclear fusion happens (the convective core).

The Old Way (The "Hurley" Model):
If you stripped the paint off a car (mass loss), the old map assumed the engine instantly became smaller to match the new, lighter car. It didn't matter that the engine was actually built for a heavy truck; the map just said, "Now you are a small car, so you have a small engine."

The Reality (The "Shikauchi" & "Brcek" Model):
In reality, if a massive star loses its outer layers late in its life, its engine is still huge and full of "burned fuel" (helium). The engine doesn't shrink just because the body got lighter. In fact, if the star gains weight from a partner, the engine can actually grow and mix in fresh fuel, making the star feel "younger" (a process called rejuvenation).

The old map missed this. It would underestimate how big the engine (helium core) really was.

What This New Framework Does

The authors built a new set of rules for the COMPAS code (specifically a mode they call BRCEK) that fixes these issues:

1. Tracking the "Engine" (Convective Core)

Instead of guessing the engine size based only on the star's current weight, this new system calculates how the engine grows and shrinks step-by-step.

Analogy: Imagine a baker kneading dough. If you take a piece of dough away (mass loss), the remaining dough doesn't instantly become a different recipe; it just has less volume. The new model tracks exactly how much "dough" is left and how the ingredients inside have mixed.

2. The "Rejuvenation" Effect

When a star steals mass from a partner, it doesn't just get heavier; it gets a fresh supply of hydrogen fuel.

Analogy: Think of a tired runner who suddenly gets a caffeine shot and a fresh pair of shoes. They don't just run faster; they effectively "reset" their fatigue. The new model calculates exactly how much the star's "age" resets when it steals fuel, preventing the star from dying too early in the simulation.

3. Shrinking the "Body" (Radius)

The old map thought that when a star lost its outer layers, it would stay puffy and large.

Analogy: Imagine a balloon. If you let the air out, it shrinks. The old map kept the balloon inflated even after the air was gone. The new model realizes that once the heavy outer layers are stripped away, the star becomes much more compact and dense, like a deflated balloon that has been squeezed tight.

What Changed in the Results?

By using this new "memory" system, the authors found several surprising differences compared to the old way:

Bigger Engines: Stars end their main life with much larger helium cores than we thought.
Heavier Black Holes: Because the engines are bigger, when these stars explode and collapse, they leave behind heavier black holes.
Different Shapes: Stripped stars are much smaller and denser than previously predicted.
New Binary Systems: The new model can create specific types of binary star systems (like a black hole orbiting a massive star) that the old model simply couldn't produce. One example they found looks very similar to a real system in our galaxy called Cygnus X-1.

Why Does This Matter?

The authors aren't just fixing a math problem; they are fixing the foundation of how we understand the universe's "graveyard" of dead stars.

Gravitational Waves: When black holes collide, they create ripples in space-time. The size of these ripples depends on how heavy the black holes are. Since this new model predicts heavier black holes, it changes the predictions for what our detectors (like LIGO) should hear.
Population Synthesis: This is the science of simulating millions of stars at once to see what the universe looks like. With this new map, the simulation is more realistic, helping astronomers understand why we see the stars and black holes we do.

Summary

In short, this paper says: "Stars remember their past." By giving the computer code a way to track a star's internal history—whether it lost weight, gained weight, or had its engine mixed up—the authors have created a more realistic simulation of the universe. This leads to bigger black holes, smaller stripped stars, and a better understanding of the cosmic dance between binary stars.

1. Problem Statement

Rapid binary population synthesis (BPS) codes, such as COMPAS, are essential for modeling large populations of binary stars to predict gravitational wave sources. However, these codes typically rely on simplified analytical fits (e.g., Hurley et al. 2000) that treat main-sequence (MS) stars as having a fixed internal structure determined solely by their current total mass and age.

Key Limitations of Existing Models:

Loss of Evolutionary History: When a star loses mass (via winds or Roche-lobe overflow), standard models simply switch to a new evolutionary track corresponding to the new mass, effectively erasing the star's history.
Underestimation of Core Mass: If a massive star loses a significant portion of its envelope late in the MS, standard models assign it a helium core mass based only on its current low mass. This ignores the fact that the core had already grown significantly while the star was more massive. This leads to a systematic underestimation of the terminal-age main sequence (TAMS) helium core mass.
Inadequate Treatment of Accretion: Standard models lack a physical treatment for "rejuvenation," where an accreting star mixes fresh hydrogen into its core, effectively resetting its evolutionary clock.
Radius Discrepancies: Existing prescriptions often overestimate the radii of stripped MS stars, leading to inaccurate predictions for mass transfer onset.

2. Methodology and Framework

The authors present a new MS evolution framework, implemented in COMPAS as the BRCEK prescription, which explicitly tracks the evolution of the convective core mass and central chemical composition.

Theoretical Basis:

Core Evolution: The framework builds on the semi-analytical model of Shikauchi et al. (2025), which solves coupled differential equations for the central helium fraction ( $Y_c$ $Y_{c}$ ) and convective core mass ( $M_c$ $M_{c}$ ) under arbitrary mass-loss histories.
- Equation (1) tracks $Y_c$ evolution based on luminosity and core mass.
- Equation (2) tracks $M_c$ evolution, accounting for natural shrinkage and mass-loss-induced shrinkage via a function $\delta$ representing core "inertia."
Extension to Mass Gain (Rejuvenation): The authors extend the Shikauchi model to handle mass accretion. They performed new MESA simulations of accreting stars to derive a new fitting function for $\delta$ $δ$ during mass gain.
- Rejuvenation Logic: If accretion causes the convective core to grow ( $\Delta M_c > 0$ ), the model recalculates the central helium fraction ( $Y_c$ ) by mixing fresh hydrogen from the envelope into the core. This lowers $Y_c$ , effectively making the star appear younger.
- Surface Composition: The framework tracks the surface helium abundance ( $Y_s$ ). When the star is stripped below its CNO-processed core mass, $Y_s$ increases, which is used to modify the stellar radius.

Implementation in COMPAS:

New Options: A new command-line option --main-sequence-core-mass-prescription was added with modes: HURLEY (original), MANDEL (qualitative workaround), and BRCEK (new framework).
Radius Scaling: Since a full analytical radius prescription for mass-changing stars is unavailable, the authors retain the Hurley et al. (2000) radius formula but scale it based on surface helium abundance (Eq. 19). This ensures stripped stars remain more compact, consistent with detailed models.
Luminosity: For $M_{ZAMS} \geq 15 M_\odot$ , luminosity is calculated based on $M_c$ and $Y_c$ (Shikauchi et al. 2025). For lower masses, the original Hurley prescription is used.
MS Mergers & CHE: A new fitting function (Eq. 21) was derived from MESA models to determine the initial convective core mass of merger products and chemically homogeneous evolution (CHE) stars based on their total mass and helium abundance.

3. Key Contributions

Physical Core Tracking: The first rapid BPS implementation that dynamically tracks convective core mass and composition evolution under arbitrary mass-loss and mass-gain histories.
Stellar Rejuvenation Model: A physically motivated treatment for MS accretors that accounts for core growth and hydrogen mixing, preventing the artificial "freezing" of core evolution during mass transfer.
Improved Radius Prescription: A scaling law for stripped MS stars that reduces their radii as surface helium increases, correcting the overestimation of radii in previous models.
Merger/CHE Initialization: A fitting function to accurately initialize the core mass of merger products and spun-down CHE stars, improving the modeling of these rare but critical evolutionary pathways.

4. Results

The authors compared the BRCEK framework against the standard HURLEY and MANDEL prescriptions using single-star and binary-star simulations.

Single Star Evolution:

Higher Core Masses: BRCEK predicts significantly larger helium core masses at TAMS for stars that experience mass loss, particularly for $M_{ZAMS} > 50 M_\odot$ .
Compact Radii: Stripped MS stars in the BRCEK framework are more compact, aligning better with detailed MESA-based models (e.g., Posydon).
Remnant Masses: Due to larger pre-collapse cores, BRCEK predicts systematically higher black hole (BH) masses for high-mass stars across various metallicities.

Binary Star Evolution:

Case A Mass Transfer: In binary systems undergoing Case A mass transfer (donor on MS), the donor retains more mass in the BRCEK framework because the reduced radius (due to higher surface helium) allows it to disengage from Roche-lobe overflow earlier.
Rejuvenation Effects: Accretors in BRCEK show realistic core growth and rejuvenation, whereas standard models often fail to capture the internal structural changes.
Cygnus X-1 Analogue: The framework successfully reproduced a high-mass X-ray binary (HMXB) configuration similar to Cygnus X-1 (a $\sim 21 M_\odot$ BH and a $\sim 40 M_\odot$ MS star), a system that could not be formed under the HURLEY formalism.
Blue Stragglers (BSSs): In population synthesis, BRCEK produces a distinct separation between merger products (hotter, more luminous) and accretors in the HR diagram, whereas HURLEY produces a uniform distribution.

5. Significance

This work represents a major step forward in the physical realism of rapid binary population synthesis. By moving beyond simple mass-age tracks to a core-tracking framework, the authors:

Correct Systematic Biases: They eliminate the underestimation of core masses in stripped stars, which directly impacts predictions for the mass distribution of compact objects (neutron stars and black holes).
Improve Gravitational Wave Predictions: The resulting shift toward more massive black holes and altered binary configurations will significantly impact the predicted chirp mass distribution and merger rates detectable by LIGO/Virgo/KAGRA.
Enable New Physics: The ability to track surface abundances and model rejuvenation allows for the study of Wolf-Rayet progenitors, HMXBs, and Blue Stragglers with unprecedented detail in a rapid code.

The authors note that while the framework is robust, future work will focus on refining the radius prescription for all mass ranges and exploring the sensitivity of results to different convective boundary mixing parameters.