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Radiating Bondi Flows I: Dimensionless Framework and Constant Opacity Solutions

This paper extends the classic Bondi accretion model to include radiative feedback in gas-pressure-dominated environments, demonstrating through numerical and analytic solutions that radiative heating and cooling primarily suppress accretion rates in a manner dependent on optical depth, luminosity, and cooling time, with significant implications for planet formation.

Original authors: Avery Bailey, Andrew Youdin, Kaitlin Kratter

Published 2026-03-24

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CC BY 4.0

Original authors: Avery Bailey, Andrew Youdin, Kaitlin Kratter

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 you are trying to fill a giant bucket with water using a hose. In the classic version of this problem (the "Bondi accretion" model from 1952), the water flows smoothly into the bucket. The speed of the flow depends only on how big the bucket is and how fast the water is moving in the hose. It's a simple, predictable system.

But in the real universe, things are rarely that simple. This paper, "Radiating Bondi Flows I," asks a new question: What happens if the water in the hose is so hot that it starts glowing and shooting out its own light?

Here is the story of the paper, broken down into everyday concepts.

1. The Setup: The "Hot" Bucket

In space, objects like planets, stars, or black holes pull in gas from their surroundings. Usually, we assume this gas is just cold, invisible stuff falling in. But in reality, as gas falls toward a massive object, it gets squeezed and heated up.

When gas gets hot enough, it doesn't just sit there; it starts to radiate energy (like a glowing lightbulb). This paper looks at what happens when that glowing gas pushes back against the gravity trying to pull it in.

The Analogy: Imagine trying to run toward a fan that is blowing air at you. The harder you run (gravity pulling in), the more the fan pushes back (radiation pressure). If the fan is too strong, you can't get close to it.

2. The Main Discovery: The "Feedback Loop"

The authors found that this glowing gas acts like a brake.

In the old, simple model, gas falls in at a steady rate. In this new model, the gas heats up, glows, and that light pushes the incoming gas away.

The Result: The more the gas glows, the less gas actually gets inside.
The Metaphor: It's like trying to eat a bowl of soup that is steaming hot. The steam (radiation) rises up and hits your face, making it hard to get the spoon (the gas) into your mouth. The hotter the soup, the less you can eat.

3. The Four "Knobs" of the Universe

The authors realized that to understand this braking effect, you only need to turn four "knobs" (parameters):

How "thick" the gas is (Optical Depth): Is the gas like clear glass (thin) or like a thick fog (thick)? If it's thick fog, the heat gets trapped, and the braking is stronger.
How bright the object is (Luminosity): How much light is being shot out? Brighter means more braking.
How fast it cools down (Cooling Time): If the gas can cool down instantly, it stops glowing and falls in easily. If it holds onto its heat (slow cooling), it keeps glowing and pushing back.
The "Gasiness" of the gas: A technical number describing how the gas behaves when squeezed.

4. The Three Regimes (The "Traffic Lights" of Space)

The paper maps out what happens when you turn these knobs to different settings. They found three main zones:

The Green Light (Easy Flow): If the gas is thin, dim, and cools down fast, it falls in almost as fast as the old models predicted. It's like a gentle breeze.
The Yellow Light (The "Goldilocks" Zone): If the gas is getting hot but not too hot, the flow slows down. The amount of gas falling in drops dramatically based on how bright the object is.
The Red Light (The "Stop" Sign): If the gas is thick, very bright, and holds its heat, the braking is so strong that the flow almost stops. The gas forms a hot, glowing shell around the object and refuses to fall in.

5. Why This Matters: Building Planets

You might ask, "Why do we care about gas falling into a black hole?" The authors point out that this is crucial for building planets.

When a giant planet (like Jupiter) is forming, it pulls in gas from the disk around a young star.

The Old View: We thought the planet just sucked up gas at a steady, predictable rate.
The New View: As the planet grows, it gets hot. That heat pushes back on the gas. This might explain why some planets stop growing at a certain size, or why they take longer to form than we thought.

6. The "Convection" Twist

The paper also checks if the hot gas might start churning like boiling water (convection).

The Metaphor: Imagine a pot of soup. If the bottom is hot, the soup bubbles up. The authors checked if this bubbling would help the gas get past the "brakes."
The Finding: Mostly, no. The gas cools down too fast or the flow is too smooth for the bubbling to help much. The "brakes" (radiation) still win.

Summary

This paper is a user manual for the universe's "braking system."

It tells us that when gas falls into space objects, it doesn't just fall silently. It heats up, glows, and pushes back. If the object is too bright or the gas is too thick, the gas gets stuck in a hot, glowing halo, and the object stops growing as fast as we used to think.

In a nutshell: Gravity tries to pull gas in, but the gas's own light pushes it out. The balance between these two forces determines how fast planets and stars can grow.

1. Problem Statement

The paper addresses the limitations of the classical Bondi accretion model (H. Bondi, 1952), which assumes adiabatic, spherically symmetric flow onto a gravitating body. While the original model is foundational, it neglects radiative feedback (heating and cooling), which is critical in many astrophysical environments.

Previous extensions of Bondi accretion have largely focused on radiation-pressure-dominated regimes (e.g., black hole accretion) or optically thin cases. However, there is a gap in understanding gas-pressure-dominated regimes where radiative heating significantly alters the flow but does not dominate the pressure (e.g., protoplanetary disks, giant planet formation). The authors aim to construct steady-state, spherically symmetric accretion solutions that include radiative energy transport to determine how radiative feedback modifies the accretion rate ( $\dot{M}$ ) relative to the adiabatic rate ( $\dot{M}_{ad}$ ).

2. Methodology

2.1. Governing Equations

The authors extend the standard fluid equations (mass and momentum conservation) by adding an energy equation that accounts for entropy changes due to radiative heating and cooling.

Variables: The system is solved for density ( $\rho$ ), Mach number ( $M$ ), specific entropy ( $s$ ), luminosity ( $L$ ), and radiation energy density ( $E_r$ ).
Radiative Transfer: They employ a frequency-integrated approximation using the Planck mean opacity ( $\kappa_P$ ) for emission and the Rosseland mean opacity ( $\kappa_R$ ) for absorption. The system is closed using a flux-limited diffusion approximation (via a closure relation for the radiation pressure tensor).
Equation of State: An ideal gas equation of state is used, assuming gas pressure dominates over radiation pressure.

2.2. Dimensionless Framework

To generalize the results, the authors non-dimensionalize the equations using the Bondi radius ( $r_B = GM/2c_{s,\infty}^2$ ) and a characteristic luminosity scale ( $L_B$ ). The problem is reduced to dependence on four dimensionless parameters:

$\gamma$ : Adiabatic index (fixed at $7/5$ for diatomic gas in most results).
$\tau_B$ : Optical depth through the Bondi radius ( $\kappa \rho_\infty r_B$ ).
$\tilde{L}_\infty$ : Dimensionless luminosity at infinity ( $L_\infty / L_B$ ).
$\beta$ : Dimensionless cooling time ( $t_{cool} c_{s,\infty} / r_B$ ).

2.3. Numerical Solution

Sonic Point Challenge: Unlike the adiabatic case, the sonic point location is not known a priori in radiative flows. The authors use a shooting method, integrating inward from a large outer radius ( $r \approx 10^{12} r_B$ ) for various trial accretion rates ( $\dot{M}$ ).
Convergence: They identify the unique transonic solution where the flow passes smoothly through the sonic point (neither undershooting nor overshooting).
Radiative Equilibrium Switching: To handle numerical stiffness at large radii where the flow is near radiative equilibrium, they integrate a simplified equilibrium system first, then switch to the full disequilibrium system closer to the accretor.

3. Key Contributions

Dimensionless Parameter Space: The paper provides the first comprehensive mapping of radiating Bondi flows in the gas-pressure-dominated regime across a wide parameter space ( $\tau_B, \tilde{L}_\infty, \beta \in [10^{-3}, 10^3]$ ).
Analytic Scaling Laws: The authors derive analytic scaling laws for the accretion efficiency ( $f_{acc} \equiv \dot{M}/\dot{M}_{ad}$ ) in different regimes, confirming numerical results.
Convection Analysis: They analyze the stability of these solutions against convection (Schwarzschild criterion), determining when radiative suppression might be mitigated by convective transport.
Analytic Formula: An approximate analytic formula for $f_{acc}$ is derived (Appendix B) to facilitate use in population synthesis models without running full simulations.

4. Key Results

4.1. Radiative Suppression of Accretion

Contrary to radiation-pressure-dominated environments where radiation can drive outflows, in gas-pressure-dominated regimes, radiative feedback primarily suppresses accretion.

Mechanism: Radiative heating increases the temperature and sound speed of the inflowing gas, pushing the sonic point inward and reducing the density at the sonic point, thereby lowering $\dot{M}$ .
Magnitude: Suppression can be orders of magnitude, particularly at high optical depths, high luminosities, or long cooling times.

4.2. Regimes and Scaling Laws

The authors identify distinct regimes based on the interplay of $\tau_B$ , $\tilde{L}_\infty$ , and $\beta$ :

Isothermal Regime (Rapid Cooling):
- Occurs when cooling is very fast ( $\beta$ is small) and luminosity is low.
- $f_{acc} \approx 2.97$ (approaching the isothermal limit).
No-Solution Regime:
- A region where no transonic solution exists (typically low $\tilde{L}_\infty$ relative to $\tau_B$ and $\beta$ ) because the luminosity would become negative before reaching the sonic point. Astrophysical flows naturally avoid this region.
Optically Thin Suppression ( $\tau_B \lesssim \tilde{L}_\infty$ ):
- High luminosity heats the gas significantly.
- Scaling: $f_{acc} \sim \tilde{L}_\infty^{-5/4}$ .
- Independent of opacity and cooling time.
Optically Thick Suppression ( $\tau_B \gtrsim \tilde{L}_\infty$ ):
- High Luminosity Case: The flow is heated by diffusion.
  - Scaling: $f_{acc} \sim (\tilde{L}_\infty \tau_B)^{-5/8}$ .
- Low Luminosity Case: Luminosity varies radially; accretion is suppressed by the cooling time.
  - Scaling: $f_{acc} \sim \tau_B^{-10/11} \beta^{-5/11}$ .
  - Note: The authors note that low-luminosity, radially varying solutions may not admit self-consistent interior solutions and are likely less astrophysically relevant.

4.3. Convection and Stability

Radiative suppression creates negative entropy gradients (heating from the outside in), which are Schwarzschild-unstable and prone to convection.
However, the authors find that for typical parameters (especially in planet formation), the radiative cooling time is often shorter than the convective growth time.
Consequently, convection is often inefficient at transporting heat away, and the purely radiative solutions (with suppressed accretion) remain valid. Convective corrections are only significant if the radiative-convective boundary (RCB) lies very close to the sonic point.

5. Significance and Applications

Planet Formation: This work is directly applicable to the runaway accretion phase of giant planet formation. In protoplanetary disks, gas is cold (gas-pressure dominated), but young planets have significant intrinsic or accretion luminosity. Previous models often assumed adiabatic accretion; this paper shows that radiative feedback can significantly reduce the accretion rate, potentially altering the timescales for gas giant formation.
AGN Disks & Stellar Binaries: The models offer a framework for accretion in Active Galactic Nucleus (AGN) disks (where gas pressure may dominate in certain regions) and wind accretion in binary systems where the accretor's intrinsic luminosity affects the capture rate.
Theoretical Foundation: By providing a dimensionless framework and analytic scalings, this paper serves as a "Paper I" for a series, allowing future work (Paper II) to incorporate realistic, non-constant opacities and specific hydrodynamic simulations for these applications.

In summary, the paper establishes that radiative feedback is a potent suppressor of Bondi accretion in gas-pressure-dominated environments, providing the necessary theoretical tools to quantify this effect for modern astrophysical modeling.