Eddington Luminosity Calculator

Find the maximum luminosity a compact object can sustain before radiation pressure overcomes gravity, and the critical accretion rate at that limit.

💡 Eddington Luminosity Calculator
Compact object mass10.00 M☉
M☉
1 M☉100 M☉
Observed luminosity
L☉
Eddington Luminosity
—
In Watts
—
Max Accretion Rate (η = 10%)
—
Critical Mass
—
In Kilograms
—

💡 What is the Eddington Luminosity?

The Eddington luminosity (also called the Eddington limit) is the maximum luminosity a gravitationally bound object can emit through steady accretion before radiation pressure overcomes gravity. At this critical value, the outward force of photon pressure on ionised hydrogen exactly balances the inward gravitational pull. The formula was derived by Sir Arthur Eddington in 1916 using the equilibrium condition for stellar structure.

The Eddington limit applies most usefully to compact accreting objects: neutron stars in X-ray binaries, stellar-mass black holes, and supermassive black holes at the centres of galaxies. When an X-ray binary shines near 10 to the 38 watts, it is close to its Eddington limit. When quasars outshine entire galaxies, they are accreting close to the Eddington rate for hundred-million solar mass black holes.

A common misconception is that the Eddington limit is absolute. In practice, sources can briefly exceed it through geometric beaming (where emission is not isotropic), super-Eddington winds, or radiation-dominated columns above neutron star surfaces. Ultraluminous X-ray sources (ULXs) are thought to be stellar-mass objects in super-Eddington accretion states.

The accretion rate at the Eddington limit, called the Eddington accretion rate, depends on the assumed radiative efficiency. This calculator uses the standard thin-disk value of 10 percent. Higher efficiency (relevant for rapidly spinning black holes) reduces the required accretion rate; lower efficiency (slim disks, advection-dominated flows) increases it. The Eddington accretion rate is a key parameter for modelling black hole growth over cosmic time.

📐 Formula

LEdd  =  4πGM mpc ÷ σT  ≈  1.26 × 1031 × (M / M☉) W
G = gravitational constant = 6.674 × 10−11 m3 kg−1 s−2
M = mass of the compact object (kg)
mp = proton mass = 1.673 × 10−27 kg
c = speed of light = 2.998 × 108 m/s
σT = Thomson cross-section = 6.652 × 10−29 m2
Eddington accretion rate: &Ṁ;Edd = LEdd / (η c2), where η ≈ 0.1 (10% efficiency)
Example: A 10 M☉ black hole has LEdd = 1.26 × 1032 W ≈ 329,000 L☉

📖 How to Use This Calculator

Steps

1
Select calculation mode — choose Find L_Edd to compute the luminosity limit from a mass, or Find Mass to find the mass implied by an observed luminosity.
2
Enter the mass or luminosity — in Find L_Edd mode, type a mass in solar masses or drag the slider (range 1 to 100 M☉, or type larger values). In Find Mass mode, enter a luminosity in solar luminosities.
3
Read the results — the calculator shows the Eddington luminosity in solar luminosities and watts, plus the maximum accretion rate in solar masses per year at 10 percent radiative efficiency.

💡 Example Calculations

Example 1 — Stellar-Mass Black Hole (10 M☉)

Cygnus X-1 type black hole: mass = 10 solar masses

1
Apply LEdd = 1.26 × 1031 W × 10 = 1.260 × 1032 W.
2
Convert to solar luminosities: 1.260 × 1032 / 3.828 × 1026 = 329,154 L☉.
3
Eddington accretion rate (η = 0.1): &Ṁ; = 1.260 × 1032 / (0.1 × (3 × 108)2) = 1.40 × 1016 kg/s = 2.224 × 10−7 M☉/yr.
LEdd = 329.154 thousand L☉ (1.260 × 1032 W), max accretion = 2.224 × 10−7 M☉/yr
Try this example →

Example 2 — Sgr A* (Milky Way Central Black Hole, 4 × 106 M☉)

Sagittarius A*: mass = 4,000,000 solar masses

1
LEdd = 1.26 × 1031 × 4 × 106 = 5.040 × 1037 W.
2
In solar luminosities: 5.040 × 1037 / 3.828 × 1026 = 131.661 billion L☉.
3
Eddington accretion rate: 0.0890 M☉/yr. Sgr A* today accretes a tiny fraction of this and is one of the faintest known galactic nuclei.
LEdd = 131.661 billion L☉ (5.040 × 1037 W), max accretion = 0.0890 M☉/yr
Try this example →

Example 3 — Finding Black Hole Mass from Quasar Luminosity

A quasar shines at 1013 solar luminosities (near Eddington limit)

1
Switch to Find Mass mode. Enter luminosity = 1 × 1013 L☉.
2
M = LEdd / (1.26 × 1031) × M☉ = (1013 × 3.828 × 1026) / 1.26 × 1031 = 3.038 × 108 M☉.
Critical mass = 3.038 × 108 M☉ (6.043 × 1038 kg)
Try this example →

❓ Frequently Asked Questions

What is the Eddington luminosity formula?+
The Eddington luminosity is L_Edd = 4πGMm_p*c/σ_T, which simplifies to approximately 1.26 × 10^31 watts per solar mass. The formula assumes fully ionised hydrogen and Thomson electron scattering opacity. For a 1 M☉ object, L_Edd ≈ 1.26 × 10^31 W ≈ 32,900 solar luminosities.
Why does the Eddington limit depend only on mass?+
Because both radiation pressure and gravitational force scale with distance in the same way (both go as 1/r^2), the distance cancels out. The equilibrium condition reduces to a balance between luminosity and mass multiplied by fundamental constants. The object's size, temperature, and composition do not appear in the final expression for fully ionised hydrogen.
How does the Eddington limit explain quasar luminosities?+
Quasars with luminosities of 10^13 to 10^14 solar luminosities must be powered by black holes of 100 million to 1 billion solar masses accreting near their Eddington limits. This was one of the key arguments for supermassive black holes long before they could be observed directly.
What is a super-Eddington accretor?+
A super-Eddington accretor is a compact object receiving mass at a rate above the Eddington limit. Instead of being disrupted, the excess energy is carried away by radiation-driven winds or collimated jets. Ultraluminous X-ray sources (ULXs) are thought to be neutron stars or stellar-mass black holes in super-Eddington accretion states, sometimes outshining small galaxies.
What is the Eddington accretion rate for a stellar black hole?+
For a 10 solar mass black hole with 10 percent radiative efficiency, the Eddington accretion rate is about 2.22 × 10^−7 solar masses per year, or about 1.4 × 10^16 kg per second. Real X-ray binaries are believed to accrete at fractions of this rate most of the time.
Does the Eddington limit apply to normal stars?+
Yes. Very massive stars above roughly 100 solar masses approach their Eddington limits in their interiors, driving extreme stellar winds and mass loss. The most luminous observed stars (such as Eta Carinae) are near or occasionally above their Eddington limits, leading to instability and outbursts. The Eddington limit therefore sets an approximate upper bound on stellar masses.
How does radiative efficiency affect the Eddington accretion rate?+
The Eddington accretion rate is &Ṁ;_Edd = L_Edd / (η c^2). Higher efficiency η means less mass is needed to generate the same luminosity, so &Ṁ;_Edd is smaller. Thin accretion disks achieve η ≈ 6 to 42 percent depending on black hole spin. Radiatively inefficient flows can have η well below 1 percent, requiring much higher mass inflow rates.
Is the Eddington luminosity the same as the maximum luminosity of a black hole?+
The Eddington limit is a limit on steady-state accretion-powered luminosity. Black holes can also produce brief bursts far exceeding this value through tidal disruption events, jet emission, or gravitational-wave powered mergers. For sustained luminosity from accreted gas, however, the Eddington limit is the standard theoretical ceiling.
What is the Eddington luminosity of Sgr A*, the Milky Way's central black hole?+
Sgr A* has a mass of about 4 million solar masses, giving an Eddington luminosity of roughly 131 billion solar luminosities (5 × 10^37 W). Its actual luminosity today is about 10^27 W, more than 10 orders of magnitude below the Eddington limit, making it extraordinarily quiet for a galactic nucleus.
How is the Eddington luminosity used in cosmology?+
The Eddington limit constrains how fast black holes can grow. A black hole accreting continuously at its Eddington limit doubles its mass on a timescale of about 45 million years. Growing from stellar-mass seeds to 10^9 solar mass quasars in under 1 billion years requires sustained near-Eddington accretion, which is an active topic of research in high-redshift galaxy formation.
Tags: Eddington Luminosity Eddington Limit Radiation Pressure Accretion Rate Calculator Formula Free Online Astrophysics

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What is the Eddington luminosity for a 10 solar mass black hole?

For a 10 solar mass black hole, the Eddington luminosity is approximately 329,000 solar luminosities, or 1.26e32 watts. This is the maximum sustained luminosity before radiation pressure disrupts the infalling material.

Why do neutron stars and black holes have Eddington limits?

The Eddington limit arises because photons carry momentum. When outward radiation pressure on infalling ionised hydrogen equals the inward gravitational force, further accretion is halted. The limit depends only on the mass and fundamental constants.

Can objects exceed the Eddington luminosity?

Yes, temporarily. Gamma-ray bursts, ultraluminous X-ray sources, and some novae exceed the Eddington limit through geometric beaming, super-Eddington winds, or short-lived outbursts where radiation pressure has not yet had time to act.

What is the Eddington accretion rate?

The Eddington accretion rate is the mass inflow rate that produces the Eddington luminosity, given a radiative efficiency. For 10 percent efficiency, it equals L_Edd divided by 0.1 times c squared.

How does the Eddington limit apply to quasars?

Quasars are supermassive black holes accreting near their Eddington limits. A quasar with luminosity 10 to the 13 solar luminosities implies a black hole mass of roughly 300 million solar masses.

Does the Eddington limit depend on the type of accreting material?

The standard formula assumes fully ionised hydrogen, where opacity is dominated by Thomson electron scattering. For helium-rich or metal-rich plasmas the limit differs slightly, but the hydrogen approximation is standard for most applications.

What is the Eddington luminosity for Sgr A*?

Sgr A* has a mass of about 4 million solar masses, giving an Eddington luminosity of roughly 131 billion solar luminosities. Today it accretes far below this limit and is very dim compared to active galactic nuclei.

Is the Eddington limit the same for all types of compact objects?

The formula is the same for black holes, neutron stars, and white dwarfs. However, the physical consequence differs: for white dwarfs, exceeding the limit can trigger a Type Ia supernova rather than disrupting an accretion disk.

📌 Quick Tips

💡The Eddington luminosity scales linearly with mass, so doubling the mass doubles the limit.
💡Real sources like X-ray binaries can briefly exceed the Eddington limit via geometric beaming or anisotropic emission.
💡The accretion rate shown assumes 10 percent radiative efficiency, typical for thin accretion disks around black holes.
💡Sgr A*, the Milky Way's central black hole at 4 million solar masses, radiates far below its Eddington luminosity today.
💡Quasars accreting near their Eddington limit are the most luminous persistent objects in the universe.