Eddington Luminosity Calculator
Calculate the Eddington luminosity limit for a stellar or compact object from its mass.
Used in stellar physics, black hole accretion, and X-ray binaries.
Eddington Luminosity
The Eddington luminosity (or Eddington limit) is the maximum luminosity a spherically-accreting object can have before its outward radiation pressure overcomes inward gravity, halting further accretion. At this point, gravity and radiation force on the surrounding gas cancel exactly.
Formula (electron-scattering opacity, hydrogen plasma)
L_Edd = 4π × G × M × m_p × c / σ_T
Numerical form:
L_Edd ≈ 1.26 × 10³¹ × (M / M_sun) W ≈ 3.27 × 10⁴ × (M / M_sun) L_sun
Where:
- G = gravitational constant
- M = object mass
- m_p = proton mass
- c = speed of light
- σ_T = Thomson scattering cross-section
- L_sun = 3.828 × 10²⁶ W
Worked Example — A 10 M_sun Black Hole
L_Edd = 3.27 × 10⁴ × 10 = 3.27 × 10⁵ L_sun ≈ 1.26 × 10³² W
Stellar-mass black holes in X-ray binaries that reach this luminosity show Eddington-limited accretion: any extra incoming matter gets blown back out as a wind.
Why It Matters
| Object | Typical L / L_Edd |
|---|---|
| Solar atmosphere | ~3 × 10⁻⁵ |
| O-type stars | 0.1–1 (close to limit) |
| Wolf-Rayet stars | Often near 1 |
| Active galactic nuclei | 0.01 – 1 |
| Quasars | Often near 1 |
| Ultraluminous X-ray sources | Apparently > 1 (geometrical beaming) |
The Eddington limit caps the brightness of the brightest stars and the rate at which black holes can grow by spherical accretion, with profound implications for galaxy formation.
Eddington Accretion Rate
The accretion rate corresponding to L_Edd, assuming radiative efficiency η:
Ṁ_Edd = L_Edd / (η × c²)
For a typical η = 0.1 black hole:
- Ṁ_Edd ≈ 2.2 × 10⁻⁸ × (M / M_sun) M_sun/year
So a 10 M_sun black hole at the Eddington limit gobbles ~2.2 × 10⁻⁷ M_sun/year — modest, but enough that early universe black holes hit a growth ceiling that demands seed masses larger than stellar.
Caveats
L_Edd assumes:
- Spherical, optically-thin accretion
- Pure ionized hydrogen (changes by ~2× for helium-rich plasma)
- Electron-scattering opacity dominates
Real systems can briefly exceed L_Edd via:
- Beamed (anisotropic) emission
- Radiation-pressure-driven outflows that deflect rather than reverse infall
- Non-spherical (disk + funnel) geometry near black holes
- High-spin, advection-dominated flows
These caveats are why ultraluminous X-ray sources (ULXs) appear to break the limit by factors of 10× or more in some directions, but their true 4π luminosity is closer to the limit.
Historical Note
Sir Arthur Eddington derived this limit in 1926 in his classic monograph The Internal Constitution of the Stars. A century later, it remains the single most useful upper bound on the brightness of any spherical, ionized object.