Introduction to Bondi accretion and spherical inflow
This Bondi accretion calculator estimates the classic spherical gas inflow rate onto a compact gravitating object such as a black hole, neutron star, white dwarf, or massive star. In the Bondi picture, the surrounding gas has very little angular momentum, so it does not need to form a disk before moving inward. Instead, gravity pulls gas radially inward while gas pressure pushes outward, and the balance between those effects determines whether the object can capture material efficiently.
Bondi accretion is often used as a first-pass astrophysics estimate because it reduces a complicated flow problem to a small set of measurable or assumed quantities. If you know the central mass M, the ambient gas density far from the object ρ∞, the gas sound speed cs, and the adiabatic index γ, you can get a physically meaningful order-of-magnitude inflow rate. That makes the model useful in lecture notes, quick research estimates, simulation sanity checks, and intuition building for black-hole growth or stellar feeding.
This page returns three outputs that are especially helpful when you want to interpret a Bondi problem quickly. First, it gives the Bondi radius rB, the approximate scale at which the object's gravity starts to dominate over thermal support. Second, it reports the mass accretion rate Ṁ in kg/s and in solar masses per year. Third, it compares that inflow to a built-in Eddington reference rate, which gives you a rough sense of whether the idealized spherical supply is tiny, moderate, or very large compared with a common luminosity-based benchmark.
How to use the Bondi accretion rate inputs
To use this Bondi accretion calculator well, think of each field as describing conditions far from the compact object, before the local infall strongly reshapes the gas. Start with the mass M in solar masses. A stellar-mass black hole might be around 10 M☉, while a supermassive black hole can range from millions to billions of M☉. Because the Bondi rate scales as M2, this input matters enormously.
Next enter the ambient density ρ∞ in kg/m³. This should represent the gas density well outside the capture region, not the compressed density close to the event horizon or stellar surface. Then enter the sound speed cs in km/s. The calculator converts that value to m/s automatically inside the script. Finally, provide the adiabatic index γ, which tells the model how compressible the gas is and which Bondi coefficient λ applies.
If your source gives number density instead of mass density, convert it before you calculate. A rough hydrogen-only conversion often used for quick estimates is 1 cm−3 ≈ 1.67×10−21 kg/m³ before any mean molecular weight correction. That shortcut is not exact, but it is enough for many back-of-the-envelope Bondi problems. Typical thermodynamic choices are γ = 5/3 for a monatomic ideal gas, γ = 1.4 for a diatomic gas, and values near 1 when the flow is better approximated as nearly isothermal.
- 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.
In practice, the easiest way to learn from the calculator is to change one variable at a time. Hold mass and density fixed, then try a hotter-gas case and a cooler-gas case to see how quickly the result changes. After that, vary density and notice how much more gently the output responds. This one-at-a-time approach mirrors how astrophysicists build intuition before moving on to more detailed disk, feedback, or numerical models.
The Bondi accretion formula behind the calculator
The Bondi accretion formula used on this page combines gravity, gas density, and sound speed into a steady spherical inflow estimate. The structure is simple but physically rich: denser gas provides more material to capture, a more massive central object deepens the gravitational potential, and a larger sound speed makes pressure support stronger and inflow harder. Among those inputs, the sound speed is often the most dramatic lever because it appears in the denominator to the third power.
Bondi accretion rate
The Bondi radius measures the scale where the gravitational pull of the accretor starts to compete strongly with the gas thermal energy implied by the sound speed. Outside that radius, the background medium behaves more like undisturbed ambient gas. Inside it, pressure has a harder time resisting infall, so the central object can influence the flow much more directly.
Bondi radius
The dimensionless Bondi coefficient λ depends on the adiabatic index, so the thermodynamics of the gas still matter even after you specify density and sound speed. This calculator keeps the standard expression below and also preserves special numerical handling near γ = 1 and γ = 5/3, where direct evaluation can become fragile in floating-point arithmetic.
Bondi coefficient
The script uses 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 built-in reference scaling ṀEdd = 1.4×1017(M/M☉) kg/s, so the displayed ratio is Ṁ/ṀEdd. That is a useful benchmark for context, but it is still only a benchmark: it does not replace a full radiation-hydrodynamic calculation with a chosen efficiency, opacity model, and flow geometry.
One more practical point matters when you experiment with the inputs. The usual analytic Bondi coefficient is most comfortable in the familiar range from nearly isothermal gas up to the monatomic ideal-gas value near 5/3. If you type a value of γ that makes the coefficient undefined in the current implementation, the calculator warns you instead of printing a misleading number. That safeguard is deliberate because a clean error message is better than a quietly nonsensical inflow rate.
How to interpret the Bondi radius, Ṁ, and Eddington-scaled result
The Bondi accretion result is easiest to read in three layers: capture scale first, inflow rate second, and Eddington comparison third. Start with rB. A larger Bondi radius means the central object can gravitationally influence gas from farther away, so a larger region of the background medium is part of the potential capture zone. If the radius is tiny compared with the structure you care about, the object has only a limited reach into that gas reservoir.
Next read the mass accretion rate Ṁ. That number tells you how much mass would flow inward per second in the idealized Bondi solution. The scaling laws help you see what is driving it. Because Ṁ ∝ M2, doubling the central mass raises the rate by a factor of four if the gas properties stay fixed. Because Ṁ ∝ ρ∞, doubling the density doubles the inflow. And because Ṁ ∝ cs−3, even a moderate decrease in sound speed can cause a very large jump in the predicted rate.
Finally, use Ṁ/ṀEdd as context rather than as a verdict. A tiny ratio often means the surrounding gas is too hot, too diffuse, or both to feed the object efficiently in the spherical approximation. A ratio near or above unity does not guarantee that a real source will accrete at that level. Instead, it signals that the idealized supply is large enough that omitted physics such as angular momentum, radiation pressure, jets, winds, turbulence, magnetic fields, or feedback could become the real bottleneck.
When the calculator seems to give a surprisingly large or surprisingly tiny answer, the first things to audit are the units and the sound speed. Density values are often copied from number-density discussions without converting to kg/m³, and sound speed is easy to misread if one source quotes km/s while another uses m/s. Those two unit slips can move the Bondi estimate by many orders of magnitude, so checking them usually explains the biggest surprises.
Worked example: a 10 M☉ compact object in diffuse gas
This Bondi accretion worked example shows how strongly the answer reacts to sound speed even when mass and density stay fixed. Suppose a 10 M☉ compact object sits in gas with ambient density 1×10−24 kg/m³, sound speed 10 km/s, and adiabatic index γ = 5/3. With those inputs, the calculator returns a Bondi radius of about 2.65×1013 m, or roughly 177 AU. The corresponding mass inflow rate is about 5.53×106 kg/s, which is approximately 8.77×10−17 M☉/yr. The Eddington-scaled ratio is about 3.95×10−12, so the idealized spherical inflow is extremely small compared with the built-in benchmark.
Now keep the same mass, density, and adiabatic index, but lower the sound speed from 10 km/s to 1 km/s. The Bondi radius rises to about 2.65×1015 m because rB scales as cs−2. The accretion rate rises to about 5.53×109 kg/s, or 8.77×10−14 M☉/yr, because Ṁ scales as cs−3. The Eddington ratio also grows by roughly a factor of 1000, reaching about 3.95×10−9. Nothing about the mass supply changed except the thermal state of the gas, yet the inflow estimate changed enormously.
That pair of numbers is the core lesson of Bondi accretion in one glance. Cooler gas is not just a little easier to capture; it can be dramatically easier to capture because pressure support weakens so quickly as the sound speed drops. In many real astrophysical environments, deciding whether the gas is hot and tenuous or cool and relatively quiescent matters more than fine-tuning the third decimal place of any one parameter.
Assumptions and limitations of the Bondi spherical inflow model
The Bondi accretion assumptions built into this calculator are the same ones that make the analytic solution elegant and restrictive. The model assumes steady, spherical, non-self-gravitating inflow with negligible angular momentum and a single thermodynamic description. Real accretion flows often violate one or more of those conditions, which means the Bondi answer should usually be read as a baseline estimate rather than as the last word on a source.
- Angular momentum: even a modest amount of rotation can divert gas into an accretion disk rather than letting it fall in radially.
- Magnetic fields and turbulence: magnetized or turbulent gas can redistribute support, change inflow paths, and launch outflows.
- Heating, cooling, and multiphase structure: a single fixed adiabatic index may not capture real gas that cools, heats, shocks, or fragments.
- Relative motion: if the object moves through the medium, Bondi–Hoyle–Lyttleton accretion may be a better approximation than static spherical Bondi flow.
- Numerical fragility near special γ values: the standard 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 full radiative transfer or feedback model.
These caveats do not make the calculator unhelpful. They explain what the calculator is best at: revealing the major scaling relations cleanly. If you need an exact prediction for a rotating, magnetized, feedback-dominated source, you need a richer model. If you need to understand how density, mass, and sound speed push an accretion estimate up or down, Bondi accretion is still one of the clearest starting points in astrophysics.
Background and physical intuition for Bondi capture
Bondi accretion remains popular because this specific spherical-capture problem turns messy gas dynamics into a memorable scaling law. Gas far from the accretor begins with pressure support set by temperature, composition, and microphysics. Gravity pulls inward. If the gas carries very little angular momentum, the flow can pass through a sonic point and continue inward as a transonic solution. That sonic transition is what elevates the Bondi rate beyond a mere dimensional guess: the smooth transonic condition selects a physically relevant accretion rate.
The Bondi radius is a compact way to picture where gravity starts to win over thermal motion. For a massive black hole in relatively cool interstellar gas, that radius can become very large in astronomical terms. For the same object in a hotter halo, the sound speed increases, thermal support strengthens, and the effective capture scale shrinks sharply. The exact object can therefore look well fed in one environment and nearly starved in another without changing anything about its own mass.
These scalings also appear in feedback arguments. If accretion energy heats the surrounding gas, then cs rises, the Bondi radius falls, and the Bondi rate drops steeply. That creates a natural self-regulation loop in many analytic discussions: energetic output can suppress the future inflow that powers it. Even when simulations later add boost factors or more elaborate sub-grid prescriptions, they often keep Bondi-like dependence at the core because the physical intuition is still valuable.
A simple mental picture helps. Imagine two gas reservoirs with the same density around the same compact object. In the colder reservoir, particle motions are slower, so gravity can pull material from farther out and focus more of it inward. In the hotter reservoir, faster random motions behave like stronger pressure support, which narrows the capture region and lowers the inward mass flux. That is why the sound-speed term dominates so many Bondi estimates and why careful unit handling matters so much when you use the calculator.
| M (M☉) | ρ∞ (kg/m³) | cs (km/s) | Ṁ (M☉/yr) | Ṁ/ṀEdd |
|---|---|---|---|---|
| 10 | 1e−24 | 10 | 8.8e−17 | 4.0e−12 |
| 10 | 1e−24 | 1 | 8.8e−14 | 4.0e−9 |
That comparison is not there to claim universal precision. It is there to make the Bondi dependencies visible at a glance. When you experiment with the form below, try changing only one field per run and compare your result against this table. You will quickly see whether the shift you produced came mainly from the mass-squared term, the linear density term, or the very steep inverse-cube sound-speed term.
Mini-Game: Bondi Radius Lock for sound-speed intuition
This Bondi Radius Lock mini-game turns the same spherical-accretion idea into a quick visual 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 output, 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 becomes more chaotic. If you can feel how the radius changes while you play, you will usually remember the formula more easily when you return to the input form.
A strong run feels a lot like reading the calculator output correctly. Staying on the cooler side enlarges the capture radius and makes cold packets 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 turbulent, echoing the astrophysical reality that heating and shock-like disturbances can make simple spherical accretion much harder to maintain.