Bondi Accretion Rate Calculator

Introduction

The Bondi accretion model is a classic approximation for spherical gas inflow onto a gravitating object such as a black hole, neutron star, or star. It applies when the gas has low angular momentum (so it does not form a disk) and the flow can be treated as steady and radially symmetric. In this idealized limit, the accretion rate depends mainly on the central mass M, the ambient gas density at large distance ρ∞, the gas sound speed c_s, and the adiabatic index γ.

This page calculates three commonly used quantities from the Bondi solution: (1) the Bondi radius r_B, (2) the Bondi mass accretion rate Ṁ in both kg/s and M☉/yr, and (3) an Eddington ratio Ṁ/Ṁ_Edd using a simple reference scaling. The calculator is intended for quick estimates and for building intuition about how strongly Ṁ changes with temperature (via c_s) and mass.

How to use

1. Enter the central mass M in solar masses (M☉).
2. Enter the ambient density ρ∞ in kg/m³ (the density “far” from the accretor).
3. Enter the sound speed c_s in km/s. The calculator converts this to m/s internally.
4. Enter the adiabatic index γ (typical values: 5/3 for monatomic ideal gas, 7/5 = 1.4 for diatomic gas, and values near 1 for nearly isothermal behavior).
5. Select Compute Accretion to display r_B, Ṁ, and Ṁ/Ṁ_Edd.

Tip: If you are converting number density to mass density, a rough hydrogen conversion is 1 cm⁻³ ≈ 1.67×10⁻²¹ kg/m³ (multiply by mean molecular weight if needed). Many astrophysical papers quote density in particles per cm³, while this calculator expects kg/m³.

Formula (Bondi accretion)

For a steady, spherical flow with an adiabatic equation of state, the Bondi accretion rate can be written as:

Bondi accretion rate Ṁ = 4π λ ρ∞ (G M)² / c_s³

The Bondi radius is a characteristic scale where gravity begins to dominate over thermal pressure:

Bondi radius r_B = 2 G M / c_s²

The dimensionless coefficient λ depends on the adiabatic index γ. This calculator evaluates:

λ = (1/2)^{(γ+1)/(2(γ−1))} · ((5−3γ)/4)^{−(5−3γ)/(2(γ−1))}

Notes on units and constants used by the calculator: G = 6.6743×10⁻¹¹ m³·kg⁻¹·s⁻², M☉ = 1.98847×10³⁰ kg, and 1 year ≈ 3.154×10⁷ s. The sound speed input is in km/s and is converted to m/s before computing.

The page also reports an Eddington ratio using the reference scaling already implemented in the calculator: Ṁ_Edd = 1.4×10¹⁷(M/M☉) kg/s, and the ratio is Ṁ/Ṁ_Edd. This is a convenient benchmark, but it is not a full radiative-efficiency model.

Worked example

Suppose you want an order-of-magnitude estimate for a compact object embedded in diffuse gas. Use the following inputs:

• M = 10 M☉
• ρ∞ = 1×10⁻²⁴ kg/m³
• c_s = 10 km/s
• γ = 5/3 (monatomic ideal gas)

After you click Compute Accretion, the calculator will return a Bondi radius on the order of 10¹⁵ m (sub-parsec scale) and a very small Ṁ (typically far below 1 M☉/yr), along with an Eddington ratio that is usually ≪ 1 for such hot, tenuous gas. If you keep everything the same but reduce c_s by a factor of 10 (e.g., from 10 km/s to 1 km/s), Ṁ increases by roughly 10³ because Ṁ ∝ c_s⁻³.

Assumptions and limitations

Bondi accretion is intentionally simple. It is most useful as a baseline estimate, but it can be inaccurate when real systems violate the model assumptions. Common limitations include:

• Angular momentum: even modest rotation can form an accretion disk and reduce the spherical inflow rate.
• Magnetic fields and turbulence: these can change the effective pressure support and drive outflows.
• Heating/cooling and multiphase gas: the assumed equation of state (via γ) may not represent the true thermodynamics.
• Relative motion: if the accretor moves through the gas, Bondi–Hoyle–Lyttleton accretion may be more appropriate than pure Bondi.
• Breakdown near γ ≈ 1 or γ ≥ 5/3: the λ expression used here can become numerically fragile or non-physical for some γ values; interpret results cautiously and consider domain-specific references.
• Eddington comparison: Ṁ/Ṁ_Edd is a rough indicator only; radiative efficiency, opacity, and geometry matter in practice.

Despite these caveats, the Bondi rate remains widely used in analytic estimates and as a sub-grid prescription in simulations. It is especially helpful for understanding scaling: Ṁ increases with M² and ρ∞, and decreases sharply with increasing sound speed.

Background and interpretation (additional context)

The phenomenon of gas accretion onto a gravitating body permeates numerous astrophysical contexts. When a star, black hole, or other compact object moves slowly relative to the surrounding gas and the gas itself lacks significant angular momentum, the flow can be approximated as spherically symmetric. This idealized scenario, first explored by Hermann Bondi in 1952, yields a remarkably simple estimate of the mass accretion rate, capturing the interplay between gravity, pressure support, and the thermodynamic properties of the gas. The Bondi model has become a foundational tool for understanding black hole growth in galactic nuclei, star formation in molecular clouds, and the feeding of compact remnants in diverse environments.

At the heart of Bondi accretion lies the competition between the gravitational pull of the central mass M and the thermal motion of gas particles characterized by the sound speed c_s. Gas initially at rest at infinity with density ρ∞ experiences an inward gravitational force that accelerates it toward the accretor. As the gas falls inward, its density and velocity increase, while pressure gradients oppose the infall. The Bondi solution represents a steady-state flow in which these forces balance in a self-consistent manner, producing a characteristic mass flux through any spherical surface.

In the standard derivation, mass conservation implies a constant mass flux through spheres of radius r: 4π r² ρ v = Ṁ, where v is the radial velocity. Combining this with the momentum equation and an adiabatic equation of state leads to a critical (sonic) point where the flow transitions from subsonic to supersonic. Enforcing a smooth solution through that point fixes the accretion rate and yields the compact expression used by this calculator.

The Bondi radius provides a characteristic scale for the problem. Gas at distances r ≫ r_B feels only a weak gravitational tug and remains roughly static, while material within r_B can no longer resist infall. For supermassive black holes embedded in 10⁴ K interstellar gas with c_s ≈ 10 km/s, r_B can reach parsec scales, illustrating how large volumes of gas influence nuclear activity.

Despite its simplicity, Bondi accretion captures several key scaling relations. The cubic dependence on the inverse sound speed implies that small drops in gas temperature dramatically enhance the accretion rate. Likewise, the quadratic dependence on mass means heavier objects grow faster if sufficient fuel is available. These scalings underpin models of black hole feedback in galaxies: heating the surrounding medium can stifle further accretion, establishing self-regulating cycles between active galactic nuclei and their hosts.

To illustrate typical numbers, consider a 10 M☉ black hole embedded in a hot phase of the interstellar medium with density ρ∞ = 10⁻²⁴ kg/m³ (roughly 1 proton per cm³) and sound speed 10 km/s. For γ = 5/3, the resulting Ṁ is extremely small. By contrast, in colder molecular gas where c_s drops substantially, the rate rises sharply. The table below lists a few illustrative cases:

Beyond black holes, Bondi accretion informs star formation theory. During the collapse of molecular cloud cores, protostars accrete roughly spherical envelopes until rotation or magnetic braking reshapes the inflow. The Bondi rate sets a limiting timescale for how quickly a protostar can gain mass in the absence of disk processes. Similarly, neutron stars traversing the interstellar medium may accrete at the Bondi rate, producing weak X-ray emission detectable by sensitive observatories.

In modern numerical simulations, Bondi-like prescriptions are often used to estimate black hole growth based on local gas properties, sometimes with additional “boost factors” to mimic unresolved turbulence or multiphase structure. Comparing those prescriptions with observations helps test whether feedback, jets, or turbulence suppress or enhance inflow relative to the idealized Bondi prediction.