Eddington Luminosity

What the limit is

Light carries momentum. When a photon scatters off a charged particle, it imparts a tiny push. Multiply that push by the enormous photon flux from a luminous object, and the cumulative force on the surrounding plasma can rival gravity. Above some critical luminosity, radiation pressure outweighs gravity and matter is driven outward.

That critical luminosity is the Eddington luminosity, L_Edd. It marks the edge of stable hydrostatic equilibrium for any luminous object surrounded by ionized matter. Below the limit, gravity wins; the surrounding matter falls or stays. Above the limit, radiation wins; the matter is blown away. For a star, exceeding the limit means losing mass through a radiation-driven wind. For an accreting compact object, exceeding the limit means halting the accretion that fuels its luminosity in the first place — a built-in self-regulation.

The Eddington limit was derived by Arthur Eddington in his 1926 monograph The Internal Constitution of the Stars. He used it to argue for an upper limit on stellar masses around 100 M☉. The same physics now appears in dozens of contexts unrelated to stellar interiors: nova outbursts, ultraluminous X-ray sources, quasar accretion, supermassive black hole growth in the early universe, and even gamma-ray burst photospheres.

Deriving the formula

Consider an object of mass M at the centre of a spherical envelope of fully ionized hydrogen. At distance r from the centre, take a proton-electron pair. Two forces act on it.

Gravity pulls inward with force

F_grav = G M (m_p + m_e) / r² ≈ G M m_p / r²

The proton dominates the mass since m_p ≈ 1836 m_e.

Radiation pressure pushes outward. Photons scatter mainly off electrons via Thomson scattering with cross-section σ_T = 6.65 × 10⁻²⁵ cm². Each photon transfers momentum p = E/c to the scattering electron. The radiation flux at distance r is L/(4πr²), so the force on a single electron is

F_rad,electron = (L / 4πr²) × (σ_T / c)

The electron transfers this momentum to the proton via Coulomb attraction (in the bulk plasma the proton-electron separation is locked tightly by the requirement of electrical neutrality). The effective force on the pair is therefore

F_rad = σ_T L / (4πr²c)

Setting F_grav = F_rad and cancelling 1/r²:

G M m_p / r² = σ_T L / (4πr²c)
L_Edd = 4π G M m_p c / σ_T

Plugging in constants (G = 6.674 × 10⁻⁸ cgs, m_p = 1.673 × 10⁻²⁴ g, c = 2.998 × 10¹⁰ cm/s, σ_T = 6.65 × 10⁻²⁵ cm², M_⊙ = 1.989 × 10³³ g):

L_Edd = 4π × 6.674e-8 × 1.989e33 × 1.673e-24 × 2.998e10 / 6.65e-25
      ≈ 1.26 × 10³¹ × (M / M_⊙) erg/s

Equivalently, in solar luminosity units (L_⊙ = 3.846 × 10³³ erg/s):

L_Edd ≈ 3.28 × 10⁴ × (M / M_⊙) L_⊙

So a 1 M☉ object has L_Edd ≈ 33 000 L_⊙ — far above the Sun's actual luminosity. A 10⁸ M☉ black hole has L_Edd ≈ 3.3 × 10¹² L_⊙ ≈ 1.26 × 10³⁹ erg/s, the regime occupied by bright quasars.

Opacity dependence: when Thomson is not the right cross-section

The formula above assumes ionized hydrogen with electron scattering as the only opacity source. In other regimes, different opacities take over and the Eddington limit changes accordingly. The general formula is

L_Edd = 4π G M c / κ

where κ is the relevant opacity (cross-section per unit mass). For ionized hydrogen, κ_T = σ_T / m_p ≈ 0.4 cm²/g. For ionized helium, κ ≈ 0.2 cm²/g (one electron per 2 m_p), so L_Edd is twice the hydrogen value. For neutral or partially ionized gas, line opacities at UV and optical wavelengths can be 10⁴–10⁶ times Thomson; the line-driven Eddington luminosity for hot massive stars is therefore far smaller than the canonical Thomson value at the same mass.

Opacity source	κ (cm²/g)	Conditions	Effective L_Edd	Where it matters
Thomson scattering (H⁺)	0.40	Ionized hydrogen	1.26 × 10³¹ M/M☉ erg/s	Stellar cores, AGN disks
Thomson scattering (He²⁺)	0.20	Ionized helium	2.52 × 10³¹ M/M☉ erg/s	Helium-burning stars
UV line opacity (CNO, Fe)	~10–10³	OB-star envelopes	~10⁻²–10⁻⁴ × Thomson	O-star and Wolf-Rayet winds
Bound-free (cool envelope)	~1–10	Partially ionized	~0.04–0.4 × Thomson	Cool LBV envelopes
H⁻ opacity	~10⁻³ × Thomson at peak	Sun-like photospheres	~10³ × Thomson	Cool main-sequence stars
Dust	~10–100	Cool, dusty (T < 1500 K)	~10⁻²–10⁻³ × Thomson	Red supergiants, AGB stars
Klein-Nishina (high E)	< σ_T at >100 keV	Hot relativistic plasmas	> Thomson	Compton-dominated coronae

This table makes one point clearly: the Thomson Eddington limit is the floor for opacity contributions, and any additional opacity reduces the effective L_Edd. Wolf-Rayet stars and luminous blue variables operate close to or above their effective Eddington limit precisely because UV line opacity from heavy elements multiplies the radiation force on the wind material.

What happens at and above the limit

Below L_Edd: stable hydrostatic equilibrium, mass can accrete in or stay confined to the surface gravitationally. This is normal stellar and accretion-disk behavior.

At L_Edd: marginal stability. Small perturbations launch outflows. The object's luminosity self-regulates.

Above L_Edd: radiation drives outflows. The system can persist briefly and locally above Eddington under several mechanisms:

• Radiation-driven mass loss. Material is shed as a wind, carrying momentum away with it. The remaining bound material relaxes back to or just below L_Edd. This is the steady-state mode for Wolf-Rayet stars and many massive O-stars.
• Inhomogeneous (porous) atmospheres. Density fluctuations let radiation escape preferentially through low-density channels while gravity holds the dense clumps. Effective opacity is reduced. LBV outbursts are thought to operate this way.
• Geometric beaming. Disks radiate preferentially perpendicular to their plane. Off-axis observers see lower flux; the locally observed luminosity along the polar direction can exceed L_Edd by orders of magnitude. Many ultraluminous X-ray sources are now thought to be beamed sub-Eddington flows.
• Slim accretion disks. When mass-accretion rate exceeds the critical Eddington rate, the disk thickens (becomes "slim" rather than thin), photons advect inward with the flow rather than escape, and luminosity saturates at log-Eddington while accretion proceeds rapidly.

Worked numerical example: a 10⁸ M☉ supermassive black hole

Take a supermassive black hole with M = 10⁸ M_⊙. Its Eddington luminosity is

L_Edd = 1.26 × 10³¹ × 10⁸ = 1.26 × 10³⁹ erg/s

For comparison, a typical galaxy of ~10¹¹ stars has total stellar luminosity ~10⁴⁴ erg/s. So this single black hole, if accreting at the Eddington limit, outshines about 10⁵ stars but still only ~0.001% of its host galaxy. However, that luminosity is concentrated in a region the size of a few solar systems — the black hole's accretion disk extends from ~3 R_S to ~1000 R_S, where R_S = 2GM/c² ≈ 3 × 10¹³ cm for 10⁸ M☉. Quasar surface brightness is therefore many orders of magnitude higher than any galaxy.

The corresponding mass-accretion rate at Eddington, with radiative efficiency η ≈ 0.1 (typical for thin disks):

L = η × Ṁ × c²
Ṁ = L_Edd / (η c²) = 1.26e39 / (0.1 × (3e10)²)
   ≈ 1.4 × 10¹⁸ g/s
   ≈ 22 M_⊙ per year

The Salpeter timescale — the e-folding time for black-hole growth at Eddington — is

t_Sal = M / Ṁ = (η c² σ_T) / (4π G m_p (1 - η))
      ≈ 4.5 × 10⁷ yr

(using η = 0.1 and the formula correcting for mass-energy lost as radiation rather than added to the BH). Doubling the BH mass requires ln(2) × t_Sal ≈ 31 Myr at full Eddington efficiency. Tripling requires ln(3) × 45 ≈ 50 Myr. So a 10¹⁰ M☉ supermassive black hole at z = 6 (when the universe was 1 Gyr old) growing from a 100-M☉ seed via continuous Eddington accretion needs ~18 Salpeter times = ~810 Myr — uncomfortably close to all of cosmic time available. This timing tension is the core puzzle of high-z quasar formation.

The brightest known quasars indeed sit near or just below their Eddington limits. SDSS J0100+2802 at z ≈ 6.3 has M_BH ≈ 1.2 × 10¹⁰ M☉ and bolometric luminosity ~4 × 10⁴⁷ erg/s, giving L/L_Edd ≈ 0.3. ULAS J1342+0928 at z ≈ 7.5 has L/L_Edd ≈ 1.5, suggesting brief super-Eddington phases or geometric beaming. These observations test the Eddington limit at its extreme.

Massive stars and the empirical Eddington edge

For stars without active accretion, the Eddington limit shows up in a different way: as the upper envelope of stellar luminosity at high mass. A main-sequence star's luminosity scales as L ∝ M^a with a ≈ 3.5 for low-mass stars, dropping to a ≈ 1.5–2 for the most massive stars. The drop is precisely because L approaches L_Edd at high mass and cannot grow proportionally with M anymore.

η Carinae, a luminous blue variable in the Carina Nebula, illustrates the dynamic edge of L_Edd. During its 1840s "Great Eruption," it briefly outshone every star in the sky except Sirius — peak luminosity ~5 × 10⁷ L_⊙, or ~5 × L_Edd for its current ~120 M☉ mass. That super-Eddington phase ejected ~10–20 M☉ of material, the famous Homunculus Nebula now seen in HST images. The eruption is the cleanest example of a massive star pushing past its Eddington limit and shedding mass in response.

Wolf-Rayet stars run continuously close to L_Edd. They have stripped their hydrogen envelopes via Eddington-driven winds and now expose hotter inner regions, where line-driven winds carry off ~10⁻⁵ M_⊙/yr at 1000–3000 km/s. The mass-loss rate scales with proximity to Eddington — empirically, Ṁ_WR ≈ 10⁻⁴ M_⊙/yr × (L/L_Edd)^2.4 — so the more luminous Wolf-Rayet stars lose mass faster, regulating themselves toward L_Edd.

The empirical upper mass limit for stable stars in the Milky Way is ~150–200 M☉ (Figer 2005; Crowther et al. 2010), set by the Eddington limit at solar metallicity. The most massive resolved stars in the LMC's R136 cluster reach 250–300 M☉ (R136a1, R136a2, R136a3, R136c) and are likely formed by stellar mergers at sub-solar metallicity, where line-driven winds are weaker because heavy-element opacity is reduced.

Quasars, black-hole growth, and the Magorrian relation

The Eddington limit places hard constraints on the early growth of supermassive black holes. The first quasars appear at z = 6–7, when the universe was 0.7–0.9 Gyr old, with masses of 10⁸–10¹⁰ M☉. To grow that large in such a short time at Eddington, a ~100 M☉ stellar seed would need to accrete continuously for ~700 Myr — leaving essentially no time for the seed to form. This is the "seeding problem" and has motivated theories of direct-collapse black-hole seeds (10⁴–10⁵ M☉ formed by gas collapse with no fragmentation) and super-Eddington accretion phases.

The Eddington limit also explains the Magorrian relation between black-hole mass and host-galaxy bulge mass: M_BH ≈ 0.001 M_bulge in nearby galaxies. Self-regulation through Eddington-limited accretion provides a natural feedback mechanism — when the BH approaches Eddington, its radiation drives gas out of the host galaxy, throttling its own growth. The asymptotic ratio depends on cooling timescales, gas dynamics, and feedback efficiency.

Variants and extensions

• Effective Eddington for line-driven winds. The Castor-Abbott-Klein (CAK) formulation parametrizes line opacity with a force multiplier M(t) ~ 10²–10³ at peak. The effective Eddington limit is L_Edd / (1 + M(t) Γ) and can be 100× lower than Thomson Eddington for hot O-stars, explaining wind structure and mass-loss rates.
• Photon trapping and slim disks. When accretion rate exceeds critical, photons advect with the flow rather than escaping. Luminosity saturates at L_Edd × (1 + ln(Ṁ / Ṁ_Edd)) — only logarithmically growing. Slim disks describe ULXs and tidal disruption events.
• Magnetic Eddington. Strong magnetic fields can confine plasma against radiation force, shifting the effective Eddington limit. Magnetar X-ray bursts radiate at apparent luminosities 10–1000× spherical Eddington because the field channels the radiation into beams.
• Compton Eddington at high temperatures. At T_e > m_e c² / k_B ≈ 10⁹ K, Klein-Nishina corrections reduce σ relative to Thomson, so L_Edd increases. Relevant for relativistic jets and gamma-ray burst photospheres where T can be ~10⁹–10¹⁰ K.
• Eddington in radiation-MHD simulations. Modern numerical work (Athena, RAMSES, FLASH) self-consistently treats radiation transport, magnetic fields, and gas dynamics, finding effective Eddington luminosities that depend on geometry, magnetic topology, and turbulence — generally allowing super-Eddington luminosities through inhomogeneous flows up to ~10× L_Edd.

Where the Eddington limit shows up

• Quasar luminosity function. The bright end of the quasar luminosity function cuts off around 10⁴⁷ erg/s for 10⁹-M☉ black holes, set directly by L_Edd. Surveys like SDSS, DES, eROSITA, and Euclid use this cutoff to characterize SMBH demographics across cosmic time.
• Wolf-Rayet wind mass-loss. The empirical relation Ṁ_WR ∝ (L/L_Edd)^2.4 (Hamann, Gräfener, Liermann 2006) parameterizes mass loss in Wolf-Rayet stars by their proximity to Eddington. This rate determines whether the star ends as a black hole or neutron star and is a major uncertainty in compact-binary merger rate predictions.
• Eta Carinae's Great Eruption (1837–1856). Brief super-Eddington phase that ejected ~10–20 M☉ in the Homunculus Nebula. Total radiated energy ~10⁴⁹–10⁵⁰ erg, roughly equivalent to a low-end supernova but the star survived. HST and JWST imagery resolves the bipolar nebula at distance ~7500 light-years.
• Ultraluminous X-ray sources (ULXs). Off-nuclear point sources in nearby galaxies with X-ray luminosities 10³⁹–10⁴¹ erg/s — far above Eddington for stellar-mass black holes. Most are now interpreted as super-Eddington stellar-mass BHs or neutron stars with beamed emission, after the 2014 detection of pulsations in M82 X-2 confirmed neutron-star ULXs.
• Tidal disruption events (TDEs). When a star is torn apart by a supermassive BH, the resulting accretion peaks at super-Eddington rates for weeks to months, then decays as t⁻⁵/³. The thermal X-ray and optical/UV flares (e.g., ASASSN-14li, AT2019qiz) trace Eddington-limited accretion onto SMBHs in real time.

Common pitfalls

• Forgetting the opacity in the formula. The 1.26 × 10³¹ erg/s/M☉ figure assumes pure ionized hydrogen with Thomson opacity. For helium-rich, metal-rich, or partially-ionized envelopes the effective L_Edd can differ by orders of magnitude.
• Treating super-Eddington as impossible. Super-Eddington flows occur. They simply cannot be steady, spherical, hydrostatic configurations. Time-dependent, anisotropic, or wind-launching solutions can sustain L > L_Edd briefly or locally.
• Confusing Eddington with "maximum stellar mass." The Eddington limit constrains the relationship between L and M, not M directly. Massive stars can exist above any naive mass cutoff if they reduce their effective luminosity (e.g., via low metallicity in early universe stars or through ongoing mass loss).
• Ignoring the radiative-efficiency factor for accretion. Mass-accretion rate at Eddington luminosity is Ṁ_Edd = L_Edd / (η c²), where η ranges from ~0.057 (Schwarzschild ISCO) to ~0.42 (extremal Kerr ISCO). Different spin assumptions give different Ṁ_Edd, important for SMBH growth models.
• Applying spherical Eddington to disks. Accretion disks are 2D, not spherical. The effective limit on luminosity differs from spherical L_Edd, especially when geometric beaming or photon trapping is important. Spherical L_Edd is a useful normalisation, but should not be taken as an exact upper bound for non-spherical flows.

Summary

The Eddington luminosity is the brightness ceiling at which radiation pressure on the surrounding plasma balances gravitational attraction. For ionized hydrogen, L_Edd = 1.26 × 10³¹ M/M☉ erg/s — a remarkably simple expression depending only on mass and fundamental constants. The limit governs the upper envelope of stellar luminosity (Wolf-Rayet, LBVs, η Carinae), the maximum sustained brightness of accreting compact objects (quasars, AGN, X-ray binaries, ULXs), and the e-folding time for supermassive black-hole growth (Salpeter time ~45 Myr at η = 0.1). It is one of the few astrophysical "constants of nature" that ties together stellar, accretion, and cosmological-feedback physics under a single derivation that takes one page and uses only Thomson scattering and Newtonian gravity.