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Resolving Star Cluster Formation in Galaxy Simulations with Cosmic Ray Feedback

This paper presents the first high-resolution simulations of star cluster formation incorporating dynamically coupled cosmic ray feedback, demonstrating that while cosmic rays primarily reduce star formation rates and cluster formation efficiency, they also produce more gravitationally bound clusters by lowering the turbulent energy budget of the interstellar medium.

Original authors: Brandon Sike, Mateusz Ruszkowski, Oleg Y. Gnedin, Yingtian Chen, Matthias Weber, Timon Thomas, Christoph Pfrommer

Published 2026-06-09

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CC BY 4.0

Original authors: Brandon Sike, Mateusz Ruszkowski, Oleg Y. Gnedin, Yingtian Chen, Matthias Weber, Timon Thomas, Christoph Pfrommer

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 galaxy as a giant, bustling city where stars are the citizens. In this city, stars don't just appear one by one; they are born in crowded neighborhoods called star clusters. These clusters are like massive apartment complexes where thousands of stars are packed together.

This paper is a report from a team of scientists who built a virtual city inside a supercomputer to understand how these star neighborhoods form and what happens to them. Specifically, they wanted to see how a mysterious force called Cosmic Rays (high-energy particles zooming through space) affects the construction of these star neighborhoods.

Here is the story of their findings, broken down into simple concepts:

1. The Virtual City and the "Ghost" Force

The scientists used a sophisticated computer program (a "moving-mesh" code) to simulate a patch of space similar to our own neighborhood in the Milky Way. They filled it with gas and let gravity do its work to form stars.

Usually, when stars are born, they explode as supernovae (giant stellar fireworks) that blow away the gas, stopping more stars from forming. This is the "feedback" loop. But the scientists added a new variable: Cosmic Rays. Think of Cosmic Rays as an invisible, high-pressure wind that pushes against the gas.

They ran three different versions of this simulation:

Version A (MHD): No cosmic rays. Just gas, gravity, and explosions.
Version B (CR-NL): Cosmic rays are everywhere and push on everything equally.
Version C (CR-NL-IN): The "realistic" version. Cosmic rays push on the gas, but in the cold, dense clouds where stars are born, they get "stuck" or dampened, so they don't push as hard locally.

2. The Main Discovery: The "Traffic Jam" Effect

The most important thing the scientists found is that Cosmic Rays act like a traffic jam for star formation.

In the version with Cosmic Rays (especially the realistic one), the total number of stars born dropped. Why? Because the Cosmic Rays pushed the gas apart, making it harder for the gas to collapse into dense clumps where stars are born.

The Analogy: Imagine trying to build a sandcastle. If a strong wind (Cosmic Rays) keeps blowing the sand away, you can't pile it up high enough to build a castle. You end up with fewer castles (stars).

3. What Happened to the Star Neighborhoods?

The scientists looked closely at the "apartment complexes" (star clusters) that did form. They compared the three versions to see if the Cosmic Rays changed the quality of the neighborhoods.

The Size and Shape: The clusters looked roughly the same size and shape in all three versions. The Cosmic Rays didn't change the blueprint of the buildings.
The "Tightness" (Virial Parameter): This is where it got interesting.
- In the version without Cosmic Rays (Version A), the star clusters were "loose." They had too much energy, moving around too fast, and were on the verge of falling apart (unbound).
- In the versions with Cosmic Rays (Versions B and C), the clusters were "tight." They were moving slower and were more likely to stay together as a group (bound).
- The Analogy: Think of a group of people at a party.
  - No Cosmic Rays: The party is wild. People are dancing too fast and jumping around. Eventually, the group will scatter and break up.
  - With Cosmic Rays: The party is calmer. People are moving slower, huddling closer together, and the group stays intact longer.

4. Why Did This Happen?

The scientists realized this wasn't because Cosmic Rays were gently holding the stars together. It was a side effect of the "traffic jam" mentioned earlier.

Cosmic Rays reduced the total number of stars being born.
Fewer stars meant fewer supernova explosions.
Fewer explosions meant less "turbulence" (chaotic shaking) in the gas.
Because the gas wasn't being shaken as violently by explosions, the new star clusters formed in a calmer environment.
Result: The clusters formed with less energy, so they were tighter and more stable.

5. The "Pre-Conditioning" Surprise

The scientists also checked where the supernova explosions happened. They wondered if Cosmic Rays would change where the explosions occurred (e.g., in dense gas vs. empty space).

The Result: It didn't matter. In all three versions, about 50% of the explosions happened in empty, diffuse gas, and 50% happened in dense gas. The Cosmic Rays changed the amount of gas, but they didn't change the pattern of where the explosions happened relative to the stars.

The Bottom Line

This paper is a breakthrough because it's the first time scientists have simulated star clusters forming while accounting for Cosmic Rays in a realistic way.

The takeaway: Cosmic Rays don't necessarily change the rules of how star clusters are built, but they act as a regulator. They slow down the rate at which stars are born. This slower pace creates a calmer environment, resulting in star clusters that are more stable and less likely to fly apart immediately.

The simulations showed that when you include these cosmic "winds," the results actually match what we see in the real universe very well, suggesting that Cosmic Rays are a key ingredient in the recipe for how galaxies evolve.

Technical Summary: Resolving Star Cluster Formation in Galaxy Simulations with Cosmic Ray Feedback

Problem Statement
Star clusters are the primary sites of massive star formation, which drives feedback processes shaping the interstellar medium (ISM) and regulating future star formation. A comprehensive understanding of galactic feedback requires resolving the parsec-scale formation of these clusters and the multiphase ISM. While cosmic rays (CRs) are known to contribute to ISM pressure and heating, their specific impact on star cluster formation rates (SFRs), cluster mass functions, and cluster boundedness remains uncertain. Previous studies have explored CR feedback in the context of galactic outflows or used subgrid models for clusters, but few have simultaneously resolved individual star particles, dynamically coupled CRs, and a feedback-regulated multiphase ISM. This work addresses the gap in understanding how CR transport models, specifically those based on self-confinement theory, influence the formation and properties of star clusters.

Methodology
The authors analyze high-resolution "tallbox" simulations performed using the moving-mesh code Arepo. The simulations employ the Crisp non-equilibrium thermochemistry and feedback model, which tracks cooling/heating processes for hydrogen and metals. The study utilizes three distinct physics cases to isolate the effects of CR transport:

MHD: Magnetohydrodynamics with CRs treated as a uniform heating/ionization term.
CR-NL: Two-moment CR magnetohydrodynamics (CRMHD) with nonlinear Landau damping (NLLD) as the sole damping mechanism, resulting in uniform CR-gas coupling.
CR-NL-IN (Fiducial): CRMHD with both NLLD and ion-neutral damping (IND). In this case, CRs decouple from cold neutral gas, reducing their local impact on star formation.

The simulations resolve gas down to a mass resolution of $10 M_\odot$ and a spatial scale of $\sim 0.5$ pc. Star formation is implemented via a Schmidt-type prescription where gas cells exceeding a density threshold ( $n_H = 10^3 \text{ cm}^{-3}$ ) are converted into collisionless star particles with a local efficiency of 100% per free-fall time. Stellar feedback is modeled exclusively through supernovae (SNe); radiative feedback and stellar winds are omitted to isolate the impact of CRs and SNe. SNe are injected stochastically for stars aged 3–38 Myr, with an additional 5% of SN energy injected as CRs in the CRMHD cases.

Star clusters are identified using a modified Friends-of-Friends (FoF) algorithm that incorporates a temporal coordinate, ensuring that identified groups consist of coeval stars within a spatial linking length of 2 pc. The analysis focuses on "young" clusters (ages $\sim 10$ Myr) to characterize their initial properties before significant tidal disruption.

Key Results

Cluster Mass Functions: All three simulation cases produce star cluster mass functions that follow a power law ( $dN/dM \propto M^{\alpha_{PL}}$ ) with slopes between $-2.0$ and $-2.4$ for masses $> 10^3 M_\odot$ . This matches observational trends where the slope correlates positively with the star formation rate surface density ( $\Sigma_{SFR}$ ). The inclusion of CRs shifts the data points along these observed relations but does not produce unphysical results.
Cluster Formation Efficiency (CFE): The CFE ( $\Gamma$ ) shows a positive correlation with $\Sigma_{SFR}$ when calculated for massive clusters ( $M > 5000 M_\odot$ ), consistent with theoretical predictions (e.g., Kruijssen 2012) and observations. However, for lower mass thresholds ( $M > 1000 M_\odot$ ), the CRMHD cases show a high, constant CFE, while the MHD case shows lower values. The authors suggest the high CFE in CRMHD cases may be an artifact of the high density threshold and efficiency used in the star formation prescription.
Supernova Clustering: Approximately 50% of SNe occur in diffuse, pre-conditioned gas ( $n_H < 10^{-2} \text{ cm}^{-3}$ ) across all three cases. This fraction is largely independent of the CR model, indicating that CR feedback does not significantly alter the local environment where the first few SNe explode, despite changing the global volume fraction of the diffuse ISM.
Cluster Properties and Virial Parameters:
- Radii and Velocity Dispersions: Star clusters in the MHD case exhibit larger half-mass radii and higher 1D velocity dispersions compared to the CRMHD cases.
- Boundedness: The MHD clusters are, on average, unbound ( $\alpha_{vir} > 1$ ), whereas the CRMHD clusters are, on average, bound ( $\alpha_{vir} < 1$ ).
- Mechanism: The authors attribute these differences to the turbulent energy budget of the ISM. The MHD case has the highest SFR, leading to a higher rate of SN energy injection and a more turbulent ISM. This higher ambient turbulence is inherited by the forming molecular clouds, resulting in star clusters with higher velocity dispersions that are less gravitationally bound. CR feedback reduces the SFR, thereby lowering the turbulent energy budget and producing more tightly bound clusters.

Significance and Claims
The paper claims to present the first simulations of star cluster formation that simultaneously include dynamically coupled CRs, a multiphase ISM, and individual star particles representative of massive stars. The primary significance lies in demonstrating that:

The Crisp feedback model can self-consistently reproduce the formation of star clusters with observed mass function slopes and environmental dependencies without fine-tuning, even in the absence of pre-SN radiative feedback.
The primary impact of CR feedback in this context is the reduction of the global SFR. This reduction lowers the turbulent energy budget of the ISM, which in turn produces star clusters that are more gravitationally bound (lower velocity dispersions) compared to simulations without effective CR feedback.
CRs do not appear to dynamically disrupt the collapse of individual dense molecular clouds or significantly alter the local preconditioning of SNe; their effect is mediated through the global regulation of the ISM state.

The authors conclude that while their idealized setup omits radiative feedback and stellar winds, the relative differences between the cases regarding CR impact on cluster boundedness and SFR are robust. The results provide new constraints for CR transport models and suggest that CR feedback acts to move simulation results along observed star cluster relations rather than creating new, unobserved regimes.