What this calculator does
Bondi accretion is the classic back-of-the-envelope model for spherical gas inflow onto a gravitating object. It is most often used for black holes, neutron stars, white dwarfs, and stars sitting inside gas that is not rotating much. In that idealized limit, the gas does not settle into a disk first. Instead, pressure and gravity compete in a nearly radial flow, and the inflow rate can be estimated from just a few inputs: the central mass M, the ambient gas density far away ρ∞, the gas sound speed cs, and the adiabatic index γ.
This page turns those inputs into three quantities that astronomers often want immediately: the Bondi radius rB, the mass accretion rate Ṁ in both SI and solar-mass units, and a quick Eddington-scaled comparison using the reference relation already built into the calculator. The output is useful for order-of-magnitude thinking because the Bondi formula makes the parameter dependence very clear: denser gas feeds the object faster, heavier objects accrete faster, and hotter gas with a larger sound speed is much harder to pull inward.
How to enter the inputs
Start with the mass M in solar masses. A stellar black hole might be around 10 M☉, while a supermassive black hole could be millions or billions of M☉. Then enter the ambient density ρ∞ in kg/m³. This is the density well outside the capture region, where the gas still reflects its background environment rather than the immediate pull of the compact object. The sound speed cs belongs in km/s; the script converts it to m/s automatically. Finally, supply the adiabatic index γ, which represents the gas thermodynamics.
If your source gives number density rather than mass density, convert it first. A commonly used rough hydrogen conversion is 1 cm−3 ≈ 1.67×10−21 kg/m³, before any mean molecular weight correction. That is only an estimate, but it is often enough for a quick Bondi calculation. Typical γ values are 5/3 for a monatomic ideal gas, 7/5 = 1.4 for a diatomic gas, and values close to 1 when nearly isothermal behavior is a better approximation.
- Enter the central mass M in M☉.
- Enter the ambient density ρ∞ in kg/m³.
- Enter the sound speed cs in km/s.
- Enter the adiabatic index γ.
- Click Compute Accretion to calculate rB, Ṁ, and Ṁ/ṀEdd.
The formula behind the calculator
In the steady, spherical Bondi solution, the accretion rate is written as a dimensional prefactor times a thermodynamic correction factor λ. The overall structure is simple: gas density sets how much fuel is available, the central mass sets how strong gravity is, and the sound speed sets how much thermal pressure resists infall. The result is especially sensitive to cs, which enters as an inverse cube.
Bondi accretion rate
The Bondi radius is the length scale where the gravitational potential energy per unit mass becomes comparable to the gas thermal energy scale represented by the sound speed. Outside that radius the gas is only weakly disturbed. Inside it, pressure can no longer easily hold the material up against infall.
Bondi radius
The coefficient λ depends on the adiabatic index. This calculator uses the standard expression below, with numerical safeguards for the common limit cases near γ = 1 and γ = 5/3.
Bondi coefficient
The constants in the script are G = 6.6743×10−11 m³·kg−1·s−2, M☉ = 1.98847×1030 kg, and 1 year ≈ 3.154×107 s. The Eddington comparison follows the implemented reference scaling ṀEdd = 1.4×1017(M/M☉) kg/s, so the reported ratio is simply Ṁ/ṀEdd. That ratio is helpful as a benchmark, but it should not be mistaken for a full radiation-hydrodynamic model with a chosen radiative efficiency and geometry.
How to interpret the result
The first output, rB, tells you the approximate capture scale. A larger Bondi radius means the accretor influences gas farther out. The second output, Ṁ, gives the estimated spherical mass inflow rate. Because the Bondi rate scales as M2, doubling the mass does much more than double the inflow; it multiplies it by four if the surrounding gas properties stay fixed. By contrast, doubling the ambient density only doubles the accretion rate. The most dramatic sensitivity usually comes from the sound speed: if cs drops by a factor of 10, Ṁ rises by a factor of about 1000.
The Eddington ratio gives context. If Ṁ/ṀEdd is tiny, the Bondi inflow is far below a common reference scale associated with luminous, radiation-limited accretion. That often happens in hot, diffuse gas. If the ratio comes out much larger in your estimate, that does not automatically mean a real object must accrete at that rate. Instead, it means the idealized spherical supply is large enough that additional physics, such as angular momentum, radiation pressure, outflows, or feedback, would likely become important.
Worked example
Suppose a 10 M☉ compact object sits inside dilute gas with ambient density 1×10−24 kg/m³ and sound speed 10 km/s. If you take γ = 5/3, the calculator returns a Bondi radius on the order of 1015 m and a very small mass accretion rate, typically far below 1 M☉/yr and far below the Eddington benchmark. That is the normal outcome for hot, tenuous gas: pressure support wins over gravity until the gas gets very close.
Now keep the same mass and density but lower the sound speed from 10 km/s to 1 km/s. The Bondi radius grows by a factor of 100 because rB ∝ cs−2, and the accretion rate rises by about 1000 because Ṁ ∝ cs−3. That single change is enough to build intuition for why cooler gas can feed compact objects so much more efficiently than hotter gas.
Assumptions and limitations
Bondi accretion is deliberately simple, which is exactly why it is useful and why it must be interpreted carefully. The model assumes steady, spherical, non-self-gravitating inflow with negligible angular momentum and a reasonably well-described equation of state. In real astrophysical systems, each of those assumptions can fail.
- Angular momentum: even a modest amount can produce an accretion disk instead of purely radial inflow.
- Magnetic fields and turbulence: these alter the effective support of the gas and can trigger outflows or anisotropic inflow.
- Heating, cooling, and multiphase structure: one fixed adiabatic index may not capture the real thermodynamics.
- Relative motion: if the object moves through the gas, Bondi–Hoyle–Lyttleton accretion can be a better description.
- Numerical fragility near special γ values: the usual λ expression needs care near γ = 1 and γ = 5/3, so this page applies special-case limits there.
- Eddington comparison: the displayed ratio is a reference scaling, not a complete radiation model.
Even with those caveats, Bondi accretion remains a standard teaching and estimation tool because it isolates the core scaling relations so cleanly. It shows, in one line, why dense cold gas is easy to accrete and hot rarefied gas is not.
Background and physical intuition
The reason the Bondi problem keeps appearing in astrophysics is that it compresses a complicated dynamical story into a surprisingly transparent estimate. Gas far from the accretor begins with some pressure support set by its temperature and composition. Gravity tries to pull that gas inward. If the gas has very little angular momentum, the flow can pass through a sonic point and continue inward in a roughly spherical way. That sonic transition is what makes the Bondi solution more than a dimensional guess: the requirement of a smooth transonic flow selects the physically relevant accretion rate.
The Bondi radius is a convenient shorthand for where the contest between thermal support and gravity turns decisively in gravity's favor. In galactic nuclei, that radius can reach parsec scales for massive black holes embedded in relatively cool interstellar gas. In diffuse hot halos, the same black hole may have a much smaller effective capture region because the sound speed is larger. For stellar remnants moving slowly through the interstellar medium, the numbers are smaller, but the same logic applies.
These scalings matter in feedback arguments. If a black hole heats its surroundings, then cs rises and the Bondi rate falls steeply. That creates a natural tendency toward self-regulation: energetic output can reduce the supply that powers future accretion. Many analytic models and simulation sub-grid prescriptions are built around exactly that idea, even when they later modify the basic Bondi expression with boost factors or efficiency corrections.
For another concrete mental picture, imagine two gas reservoirs with the same density and the same central mass. In the colder reservoir, the random motions are weaker, so gravity can influence material from farther away and focus more of it inward. In the hotter reservoir, fast thermal motion behaves like stronger pressure support, shrinking the capture radius and reducing the inward flux. That is why the sound-speed term is so dominant in practice and why it is worth checking units carefully before interpreting a result.
| M (M☉) | ρ∞ (kg/m³) | cs (km/s) | Ṁ (M☉/yr) | Ṁ/ṀEdd |
|---|---|---|---|---|
| 10 | 1e−24 | 10 | 6e−14 | 4e−12 |
| 10 | 1e−24 | 1 | 6e−11 | 4e−9 |
That comparison is not meant to provide exact predictions for every environment. Instead, it is there to build intuition. The colder case differs from the hotter case almost entirely because of the sound-speed dependence. If you are estimating growth rates for compact objects, testing several plausible values of cs is often more revealing than fussing over very small changes in other parameters.
Mini-Game: Bondi Radius Lock
If you want a fast intuition builder, try the optional mini-game below. It turns the Bondi idea into a small skill challenge: you tune the gas sound speed left and right, which expands or shrinks the Bondi radius ring around the central object. Cold blue packets prefer a larger capture radius, warmer packets want a middle radius, and hotter violet packets need a tighter inner ring. Dense packets are worth extra points because higher ρ∞ boosts Ṁ directly, while red shock fronts are dangerous because hot feedback can suppress inflow.
The game is separate from the calculator result, so it will not change the math above. Its only purpose is to make the scaling memorable through action: larger rings correspond to lower sound speed, density surges pay out more, and later phases become harder as the environment gets more turbulent.
A good run usually feels like the calculator itself. When you stay on the cooler side, the capture radius grows and cold packets are easier to collect. During density surges, the same alignment earns more because the available fuel has increased. When the late feedback storm arrives, the game becomes more chaotic, echoing the astrophysical reality that heating and turbulence can make clean spherical accretion much harder.