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Accretion-modified Stars in Accretion Disks of Active Galactic Nuclei: Contribution to AGN disk viscosity

This study demonstrates that accretion-modified stars (AMSs) embedded in active galactic nucleus (AGN) disks drive powerful outflows that generate turbulence and provide an efficient mechanism for angular momentum transport, effectively contributing to the disk's viscosity with scaling relations dependent on the central black hole mass and accretion rate.

Original authors: Jun-Rong Liu, Jian-Min Wang, Hua Feng

Published 2026-02-27

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CC BY 4.0

Original authors: Jun-Rong Liu, Jian-Min Wang, Hua Feng

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, swirling whirlpool of gas and dust surrounding a supermassive black hole at the center of a galaxy. This is an Active Galactic Nucleus (AGN). For decades, astronomers have been puzzled by a specific problem: How does this gas actually fall into the black hole?

To fall in, the gas needs to lose its "spin" (angular momentum) and move inward. In a normal whirlpool, friction (viscosity) helps slow things down. But in these giant cosmic disks, the gas is so hot and thin that normal friction is too weak to do the job. It's like trying to stop a spinning ice skater by blowing on them; it just doesn't work fast enough.

This paper proposes a new, powerful mechanism to solve this mystery: The "Cosmic Jet Engine" effect created by smaller black holes living inside the disk.

Here is the story of how the authors, Liu, Wang, and Feng, explain this:

1. The Unwanted Guests: Stellar Black Holes

Deep inside this giant gas disk, there are hundreds or thousands of smaller black holes (stellar-mass black holes). Think of them as tiny, hungry sharks swimming in a giant ocean of gas.

These sharks are constantly eating the gas around them.
Because the gas is so dense, they eat way too much, far faster than physics usually allows.

2. The "Bondi Explosion" (Scenario A)

When a shark eats too fast, it gets a stomach ache. In physics terms, the gas piles up, gets super hot, and creates a massive outflow or wind that blasts outward.

The Analogy: Imagine a person trying to drink a milkshake through a tiny straw. If they suck too hard, the straw bursts, and the milkshake sprays everywhere.
These "bursts" happen in cycles. The black hole gorges itself, explodes outward with a shockwave, clears a hole in the gas, and then has to wait for the gas to refill the hole before it can eat again.
This cycle is called a "Bondi Explosion."

3. The Steady Wind (Scenario B)

In some cases, the black holes don't explode in bursts. Instead, they maintain a steady, powerful wind, constantly pushing gas away while eating at a steady, maximum rate.

The Analogy: Think of a garden hose running at full blast, constantly pushing water away from the nozzle.

4. The Magic: How This Moves the Galaxy

Here is the brilliant part of the paper. When these black holes blast out gas (either in explosions or steady winds), they crash into the surrounding disk gas.

The Analogy: Imagine a crowded dance floor (the galaxy disk). If one dancer suddenly spins wildly and pushes everyone else, the whole crowd starts to swirl and mix.
These blasts create turbulence (chaos). This turbulence acts like a giant, cosmic mixer. It shoves the inner gas outward and pulls the outer gas inward.
This process transfers angular momentum (the spin) from the inside to the outside, allowing the inner gas to finally fall into the central supermassive black hole.

5. The Results: Solving the Viscosity Puzzle

The authors did the math to see how effective this "black hole mixer" is.

They found that these outflows create enough turbulence to act as a super-viscosity.
In the outer regions of the galaxy disk, this mechanism is incredibly efficient. It provides the exact amount of "friction" needed to make the galaxy work as we see it.
They calculated that depending on how many black holes there are and how they are distributed, this mechanism could be responsible for 1% to 100% of the viscosity needed to run the galaxy.

Why This Matters

It solves a 50-year-old mystery: It explains how gas actually gets to the center of galaxies to feed the supermassive black holes.
It connects the small and the big: It shows how tiny black holes (the size of a city) can control the behavior of a galaxy (the size of a solar system).
It predicts what we might see: If this theory is right, we should see specific patterns in how galaxies spin and how they emit light (and gravitational waves) from these "exploding" black holes.

In a nutshell: The galaxy isn't just a smooth, quiet whirlpool. It's a chaotic, turbulent place where thousands of "hungry" black holes are constantly blowing bubbles and creating shockwaves. These shockwaves act like a giant cosmic blender, mixing the gas and allowing the galaxy to function. Without these "accretion-modified stars," the galaxy would likely be stuck, unable to feed its central monster.

1. Problem Statement

Active Galactic Nuclei (AGN) are powered by gas accretion onto central supermassive black holes (SMBHs). A fundamental challenge in AGN physics is explaining the origin of viscosity in accretion disks, which is required to transport angular momentum outward and allow matter to accrete inward. While mechanisms like Magnetorotational Instability (MRI) and Gravitational Instability (GI) are well-studied, they often struggle to explain viscosity levels, particularly in the outer regions of AGN disks.

Recent observations (e.g., gravitational waves from heavy binary black hole mergers) and theoretical models suggest that stellar-mass black holes (sMBHs) can exist and grow within AGN disks. These objects, termed "Accretion-Modified Stars" (AMS), accrete gas from the dense disk environment, often at hyper-Eddington rates. The authors investigate whether the mechanical feedback (outflows) from these AMSs can generate sufficient turbulence to drive angular momentum transport, thereby acting as a significant source of disk viscosity.

2. Methodology

The authors construct a theoretical framework to calculate the effective viscosity parameter ( $\alpha_{AMS}$ ) contributed by a population of sMBHs embedded in a standard Shakura-Sunyaev accretion disk.

Physical Scenarios: Two distinct accretion regimes for AMSs are modeled:
- Scenario A (Episodic Bondi Explosion): sMBHs undergo hyper-Eddington accretion via Bondi accretion. This leads to the formation of a "Bondi explosion," where powerful outflows create a low-density cavity. The process is episodic, alternating between an active outflow phase (Type I) and a quiescent "rejuvenation" phase (Type II) where the cavity refills.
- Scenario B (Steady Eddington Accretion): sMBHs accrete at the Eddington limit, launching steady, optically thick winds.
Angular Momentum Transfer Mechanism: The study posits that outflows from AMSs collide with the surrounding disk gas, generating shocks. These shocks fragment into small-scale turbulent vortices. The interaction redistributes angular momentum: fluid elements facing the inner shock lose angular momentum, while those facing the outer shock gain it, resulting in a net outward transport.
Parameter Space Exploration:
- SMBH Properties: Mass ( $M_p$ ) ranging from $10^7$ to $10^9 M_\odot$ and dimensionless accretion rates ( $\dot{M}_p$ ) from 0.1 to 10.
- sMBH Population: The study varies the sMBH mass function ( $dN/dM_s \propto M_s^{-\beta}$ with $\beta = 1, 2, 3$ ) and their radial spatial distribution ( $dN/dR \propto R^\gamma$ with $\gamma = -1, 0, 1, 2$ ).
- Total Mass Fraction: The total mass of sMBHs is parameterized as a fraction $\zeta$ of the disk gas mass (reference $\zeta = 0.01$ ).
Calculation: The authors derive the time-averaged angular momentum transfer rate ( $\dot{J}$ ) per sMBH and integrate over the population to calculate the global effective viscosity $\alpha_{AMS}$ by equating the migration timescale driven by AMS feedback with the standard viscous timescale.

3. Key Contributions

Novel Viscosity Mechanism: The paper proposes that AMS feedback is a critical, previously under-quantified source of viscosity in AGN disks, particularly in the outer regions where MRI and GI may be less effective.
Scaling Relations: The study derives explicit scaling relations for the effective viscosity based on SMBH mass and accretion rate for two distinct physical regimes (episodic vs. steady).
Population Constraints: It quantifies how the distribution of sMBHs (mass function and spatial concentration) regulates the overall disk viscosity, providing a link between gravitational wave observables (sMBH populations) and electromagnetic disk properties.
Self-Consistency Check: The authors demonstrate that for certain high-mass/high-accretion scenarios, the AMS contribution can exceed standard viscosity limits ( $\alpha > 0.1$ ), suggesting a need for self-consistent "AMS-AGN disk" models in future work.

4. Key Results

Viscosity Magnitude:
- Scenario A (Episodic): $\alpha_{AMS}$ ranges from $\sim 3 \times 10^{-4}$ to $0.01$ (for $\zeta=0.01$ ).
- Scenario B (Steady): $\alpha_{AMS}$ ranges from $\sim 3 \times 10^{-3}$ to $0.04$.
- In both scenarios, viscosity is significantly higher in the outer disk ( $R/R_g > 10^3$ ) compared to the inner disk, where AMS feedback dominates the transport mechanism.
Scaling Laws:
- Scenario A: $\alpha_{AMS} \propto \zeta M_p^2 \dot{M}_p$ . The strong dependence on $M_p^2$ is driven by the increase in disk height ( $H$ ) and Hill radius ( $R_{Hill}$ ) with mass, which enhances the energy release of Bondi explosions.
- Scenario B: $\alpha_{AMS} \propto \zeta M_p^{1.5} \dot{M}_p^{0.1}$ . The steady wind scenario is less sensitive to the accretion rate but still scales strongly with SMBH mass.
Impact of Distribution:
- A steep mass function (more low-mass sMBHs, $\beta=3$ ) combined with an outer-disk concentration ( $\gamma=2$ ) maximizes viscosity, potentially reaching $\alpha \gtrsim 0.1$ in the outer disk.
- Conversely, flat mass functions and inner-disk concentrations yield lower viscosity values.
AMS Evolution: In Scenario A, the duty cycle ( $\delta$ ) is very low ( $\sim 0.004 - 0.016$ ), meaning sMBHs spend most of their time in the low-activity rejuvenation phase. This episodic nature prevents runaway growth of sMBHs while still providing significant cumulative feedback.

5. Significance

Solving the Low-Viscosity Problem: The results suggest that AMS feedback provides an efficient mechanism to resolve the long-standing "low-viscosity problem" in AGN disks, particularly in the outer regions where standard mechanisms struggle.
Multi-Messenger Connection: The study bridges the gap between gravitational wave astronomy (sMBH mergers, EMRIs) and AGN disk physics. The derived viscosity depends on the sMBH population parameters ( $N_0, \beta, \gamma$ ), which can be constrained by future space-based GW detectors (LISA, Taiji, Tianqin).
Disk Structure and Stability: By providing a mechanism for angular momentum transport and heating, AMS feedback may help maintain gravitational stability in the outer Shakura-Sunyaev disk and alter the disk's density and temperature profiles.
Future Observational Signatures: The model predicts that collective thermal emission from AMS populations could influence the Spectral Energy Distribution (SED) of AGNs, potentially explaining shallow spectral shapes in quasar composites, and that AMS-driven outflows could produce detectable $\gamma$ -rays in radio-quiet AGNs.

In conclusion, this work establishes that the mechanical feedback from accretion-modified stars is a viable and potentially dominant source of viscosity in AGN disks, offering a unified explanation for disk evolution, sMBH growth, and multi-messenger signatures.