Eddington Luminosity and Accretion Limits

The Eddington luminosity is the most useful number in the physics of accretion, and it earns that place by being almost embarrassingly simple. It is the luminosity at which the outward radiation force on ionized gas exactly cancels the inward pull of gravity from a central mass: below it gravity wins and gas can fall inward or stay bound; above it radiation wins and the gas is blown away as a wind. What makes the balance clean is that both forces obey the same inverse-square law — radiation flux dilutes as $1/r^2$ , and so does gravity — so the radius cancels identically and a single critical luminosity, not a luminosity-at-a-radius, marks the tipping point. For ionized hydrogen that luminosity depends on nothing but the central mass: $L_{\mathrm{Edd}}\approx1.26\times10^{38}\,(M/M_\odot)\,\mathrm{erg\,s^{-1}}$ . The mechanism is a division of labour between the two species of a plasma. Radiation couples overwhelmingly to electrons through Thomson scattering, because the cross section scales as the inverse square of the particle mass; gravity couples to protons, which carry essentially all the mass. Electrostatic attraction chains the two together, so radiation must lift, and gravity must hold, the combined pair — and the relevant opacity is $\kappa_{\mathrm{es}}=\sigma_T/m_p$ . From this one scale grow the Eddington accretion rate, the radiative-efficiency budget of black holes, the Salpeter time governing how fast holes grow, and the Eddington ratio $\lambda=L/L_{\mathrm{Edd}}$ that sorts accretion states from thin disks to starved advection-dominated flows. It is also a limit made to be broken: disk geometry, photon trapping, and beaming let real sources shine above it.

Key takeaways

The Eddington luminosity $L_{\mathrm{Edd}}$ is where the outward radiation force on ionized gas exactly balances gravity, a natural ceiling on steady spherical accretion.
For ionized hydrogen it depends on mass alone, $L_{\mathrm{Edd}}=4\pi GMm_pc/\sigma_T\approx1.26\times10^{38}\,(M/M_\odot)\,\mathrm{erg\,s^{-1}}$ , because radiation and gravity both fall as $1/r^2$ and the radius cancels.
Radiation pushes electrons (Thomson scattering), gravity pulls protons, and Coulomb attraction locks the pair together — so the electron-scattering opacity $\kappa_{\mathrm{es}}=\sigma_T/m_p\approx0.40\,\mathrm{cm^2\,g^{-1}}$ sets the limit.
The Eddington rate $\dot M_{\mathrm{Edd}}=L_{\mathrm{Edd}}/(\eta c^2)$ and the Salpeter e-folding time $t_{\mathrm{Sal}}\approx50\,\mathrm{Myr}$ (for $\eta=0.1$ ) govern exponential black-hole growth.
Real sources routinely exceed $L_{\mathrm{Edd}}$ when the geometry is non-spherical or radiation escapes anisotropically — the limit is an ordering scale, not an absolute ceiling.
Key idea: set the radiation force per electron $\sigma_T L/(4\pi r^2c)$ equal to the gravity per electron–proton pair $GMm_p/r^2$ ; the radius cancels and leaves $L_{\mathrm{Edd}}=4\pi GMm_pc/\sigma_T$ , a function of mass alone.

The core insight

Set the outward radiation force on an electron, $F_{\mathrm{rad}}=\sigma_T L/(4\pi r^2c)$ , equal to the gravitational force on its bound proton, $F_{\mathrm{grav}}=GMm_p/r^2$ ; both scale as $1/r^2$ , so the radius cancels identically and leaves the mass-only critical luminosity $L_{\mathrm{Edd}}=4\pi GMm_pc/\sigma_T$ .

Where this lives today

The Eddington scale is the working currency of high-energy astrophysics, and its two most vivid modern arenas both sit away from $\lambda=1$ . At the low end, the Event Horizon Telescope resolved the shadows of Sgr A and M87, both accreting at Eddington ratios of $10^{-8}$ to $10^{-5}$ — radiatively inefficient, advection-dominated flows whose faintness is a direct consequence of being far sub-Eddington. At the high end, JWST has uncovered "little red dots" and $z>7$ quasars that appear to host $10^{7}$ – $10^{9}\,M_\odot$ black holes when the Universe was under a gigayear old, sharpening exactly the seed-and-growth tension of Worked Example 2 and reviving interest in super-Eddington accretion. Pulsating ultraluminous X-ray sources such as M82 X-2 — a neutron star radiating at $\sim100$ times its spherical Eddington limit — are the cleanest laboratories for how beaming and magnetically reduced opacity let real sources break the naive bound. The Eddington ratio also remains the primary axis along which X-ray binaries switch between hard and soft states, and radiation-pressure feedback near $L_{\mathrm{Edd}}$ is the leading mechanism invoked for the $M$ – $\sigma$ relation.

The physical picture

Picture a single electron–proton pair in the ionized gas around a bright central mass. Two forces act on it, through different particles. Gravity pulls almost entirely on the proton, which carries $1836$ times the electron's mass. Radiation pushes almost entirely on the electron: the Thomson cross section scales as $1/m^2$ , so the electron intercepts photon momentum roughly $3\times10^6$ times more effectively than the proton. Were the two free, the light would sweep the electrons out and leave the protons behind — but that charge separation would raise an enormous electrostatic field almost instantly, chaining each electron to a proton. The plasma responds as neutral pairs: radiation lifts the pair through its electron, gravity holds it through its proton, and the competition is between $\sigma_T L/(4\pi r^2c)$ and $GMm_p/r^2$ .

Figure

Force balance on one electron–proton pair at radius r from a luminous mass M: radiation drives the electron outward with F_rad = σ_T L/(4πr²c), gravity pulls the proton inward with F_grav = GMm_p/r², both fall as 1/r², so the ratio is fixed and the balance L = L_Edd holds at every radius

Because both forces carry the same $1/r^2$ factor, their ratio is a pure number that does not care how far the gas sits from the source. That is the deep reason the Eddington limit is a single luminosity rather than one that depends on where you look: if radiation overwhelms gravity at one radius, it overwhelms it at every radius, and the whole envelope lifts at once. Think of $L_{\mathrm{Edd}}$ as the weight the star's own light can just barely lift — a more massive object has stronger gravity and can shine brighter before its radiation blows its atmosphere off, hence the strict proportionality $L_{\mathrm{Edd}}\propto M$ . Notice what does not appear: the photon energies. The limit is a statement about momentum, $L/c$ , not about energy, which is why $c$ and the scattering cross section carry the physics while the spectral shape does not. That mass proportionality spans an astonishing range — from neutron stars to the most luminous quasars, one straight line on a log–log plot.

Figure

The Eddington luminosity is strictly linear in mass, L_Edd ≈ 1.26×10³⁸ (M/M_⊙) erg s⁻¹, one line across ten decades: a 1.4 M_⊙ neutron star at 1.8×10³⁸, a 10 M_⊙ stellar black hole at 1.3×10³⁹, Sgr A* near 5×10⁴⁴, and a 10⁹ M_⊙ quasar at 1.3×10⁴⁷ erg s⁻¹

The core equations

The definition is a force balance. Writing the radiation force per electron as the momentum flux $L/(4\pi r^2c)$ over the Thomson cross section $\sigma_T$ , and the gravity per pair as $GMm_p/r^2$ , the $1/r^2$ cancels and leaves a luminosity fixed by mass and constants alone.

L_{\mathrm{Edd}}=\frac{4\pi G M m_p c}{\sigma_T} \qquad (1)

where: $L_{\mathrm{Edd}}$ — critical (Eddington) luminosity; $G$ — gravitational constant; $M$ — central mass; $m_p$ — proton mass; $c$ — speed of light; $\sigma_T=6.65\times10^{-25}\,\mathrm{cm^2}$ — Thomson cross section. The content is a ratio of constants times $M$ : no density, no radius, no temperature. Plugging in CODATA values gives the number every accretion astronomer carries in their head, worth memorizing in both unit systems: $1.26\times10^{38}\,\mathrm{erg\,s^{-1}}$ or $1.26\times10^{31}\,\mathrm{W}$ per solar mass.

L_{\mathrm{Edd}}\approx1.26\times10^{38}\left(\frac{M}{M_\odot}\right)\mathrm{erg\,s^{-1}} \qquad (2)

where: $M_\odot=1.989\times10^{33}\,\mathrm{g}$ — solar mass. A $10^8\,M_\odot$ hole caps near $10^{46}\,\mathrm{erg\,s^{-1}}$ , a $10\,M_\odot$ hole near $10^{39}\,\mathrm{erg\,s^{-1}}$ . It is often cleaner to package the microphysics into an opacity per unit mass. For pure ionized hydrogen the electron-scattering opacity is $\kappa_{\mathrm{es}}=\sigma_T/m_p$ ; anything raising the opacity above this floor lowers the effective limit.

L_{\mathrm{Edd}}=\frac{4\pi G M c}{\kappa},\qquad \kappa_{\mathrm{es}}=\frac{\sigma_T}{m_p}\approx0.40\ \mathrm{cm^2\,g^{-1}} \qquad (3)

where: $\kappa$ — mass opacity; $\kappa_{\mathrm{es}}$ — its Thomson value for pure H ( $\approx0.34\,\mathrm{cm^2\,g^{-1}}$ for cosmic composition, $\kappa_{\mathrm{es}}=0.20(1+X)$ ). Larger $\kappa$ grips the gas more tightly, so a lower luminosity already unbinds it. A luminosity becomes an accretion rate through $L=\eta\dot Mc^2$ , where $\eta$ is the fraction of infalling rest-mass energy radiated. Setting $L=L_{\mathrm{Edd}}$ defines the Eddington accretion rate — the maximum steady feeding rate of a spherical accretor.

\dot M_{\mathrm{Edd}}=\frac{L_{\mathrm{Edd}}}{\eta c^2}=\frac{4\pi G M m_p}{\eta\,\sigma_T c} \qquad (4)

where: $\dot M_{\mathrm{Edd}}$ — Eddington accretion rate; $\eta$ — radiative efficiency, $\approx0.057$ (Schwarzschild) to $0.42$ (maximal Kerr), canonically $0.1$ ; note $\dot M_{\mathrm{Edd}}\propto M$ . The dimensionless Eddington ratio $\lambda$ (also written $\Gamma$ ) places a source on the accretion ladder and enters the equation of motion directly: radiation pressure reduces the effective gravity by $(1-\lambda)$ .

\lambda\equiv\frac{L}{L_{\mathrm{Edd}}},\qquad g_{\mathrm{eff}}=\frac{GM}{r^2}\,(1-\lambda) \qquad (5)

where: $\lambda$ — Eddington ratio; $g_{\mathrm{eff}}$ — net inward acceleration. At $\lambda=1$ effective gravity vanishes; at $\lambda>1$ it reverses and the atmosphere is driven off as a wind. Finally, because $L_{\mathrm{Edd}}\propto M$ , a hole feeding at fixed Eddington ratio grows at a rate proportional to its own mass — exponential growth. The e-folding time is the Salpeter time, set by fundamental constants and the efficiency.

M(t)=M_0\,e^{t/t_{\mathrm{Sal}}},\qquad t_{\mathrm{Sal}}=\frac{\eta}{1-\eta}\,\frac{\sigma_T c}{4\pi G m_p}\approx50\ \mathrm{Myr}\ (\eta=0.1) \qquad (6)

where: $t_{\mathrm{Sal}}$ — Salpeter e-folding time; $M_0$ — seed mass. The prefactor $\sigma_T c/(4\pi Gm_p)=Mc^2/L_{\mathrm{Edd}}\approx0.45\,\mathrm{Gyr}$ is the Eddington timescale; $\eta/(1-\eta)$ accounts for mass radiated rather than swallowed.

Derivation AThe Eddington luminosity from momentum balance

The goal is to derive the mass-only critical luminosity from first principles by balancing, at an arbitrary radius, the outward force that scattered radiation exerts on the electrons of a fully ionized hydrogen plasma against the inward gravity the central mass exerts on the protons, and to make explicit why the result is independent of radius, density, and spectral shape. We work in spherical symmetry, steady state, and with Thomson scattering as the only opacity — exactly the assumptions whose later violation explains super-Eddington sources.

Step 1. Start with the energy flux at radius $r$ from an isotropic source, $F=L/(4\pi r^2)$ . Divide by $c$ to get the momentum flux, $F/c=L/(4\pi r^2c)$ , since a photon of energy $E$ carries momentum $E/c$ . Why: the Eddington limit is a competition of forces, and force is a flow of momentum — the factor of $c$ that converts $L$ into a momentum current is why $c$ appears in the final formula while photon energies do not.

Step 2. An electron in this flux presents cross section $\sigma_T$ and re-radiates the intercepted photons isotropically, so it absorbs the net forward momentum and feels $F_{\mathrm{rad}}=\sigma_T\,L/(4\pi r^2c)$ . The proton, with cross section smaller by $(m_e/m_p)^2\sim3\times10^{-7}$ , feels essentially none of it. Why: the $1/m^2$ scaling assigns the radiation force to the electrons and lets us treat the proton as radiatively inert.

Step 3. Gravity acts on mass, so the force on the pair is $F_{\mathrm{grav}}=GM(m_p+m_e)/r^2\approx GMm_p/r^2$ , carried almost entirely by the proton. The electron and proton cannot separate: any charge imbalance raises a restoring electrostatic field, so the plasma moves as neutral pairs and the two forces act on one object. Why: without the Coulomb lock the light would simply expel the electrons and the argument would collapse — it is what makes "radiation lifts the pair, gravity holds the pair" correct.

Step 4. Set the two forces equal at the critical luminosity, $\sigma_T L_{\mathrm{Edd}}/(4\pi r^2c)=GMm_p/r^2$ . The $1/r^2$ appears on both sides and cancels identically, leaving $L_{\mathrm{Edd}}=4\pi GMm_pc/\sigma_T$ . Inserting $G=6.67\times10^{-8}$ , $m_p=1.673\times10^{-24}\,\mathrm{g}$ , $c=3.00\times10^{10}\,\mathrm{cm\,s^{-1}}$ , $\sigma_T=6.65\times10^{-25}\,\mathrm{cm^2}$ gives $1.26\times10^{38}(M/M_\odot)\,\mathrm{erg\,s^{-1}}$ . Why: the cancellation of $r$ is the whole reason a single luminosity — not a run of luminosities — defines the limit, and it is exact because radiation dilution and gravity share the inverse-square law of a point source.

Go deeper · Derivation B · 1 minThe Eddington rate, the Salpeter time, and exponential growth

Derivation BThe Eddington rate, the Salpeter time, and exponential growth

The aim here is to convert the luminosity limit into a limit on how fast a black hole can grow, and to show that Eddington-limited feeding produces strict exponential growth with an e-folding time built only from constants. The chain is: accretion luminosity fixes an accretion rate through the efficiency $\eta$ ; only the un-radiated fraction $(1-\eta)$ actually adds to the hole; and because $L_{\mathrm{Edd}}\propto M$ , the growth rate is proportional to the mass, which integrates to an exponential. This is the calculation that sets the clock on quasar assembly in the early Universe.

Step 1. Relate luminosity to accretion rate through the efficiency: $L=\eta\dot M_{\mathrm{acc}}c^2$ , so feeding at the Eddington luminosity requires $\dot M_{\mathrm{acc}}=L_{\mathrm{Edd}}/(\eta c^2)=4\pi GMm_p/(\eta\sigma_T c)$ , proportional to $M$ . Why: $\eta$ is where the inner-disk physics and black-hole spin enter — it sets how much luminosity a given rest-mass flow produces, and hence how much accretion the Eddington ceiling permits.

Step 2. Distinguish what falls in from what stays. Of the rest-mass energy reaching the hole, a fraction $\eta$ leaves as radiation, so the mass grows only at $\dot M_\bullet=(1-\eta)\dot M_{\mathrm{acc}}=\frac{1-\eta}{\eta}\frac{4\pi Gm_p}{\sigma_T c}M$ . Why: neglecting the $(1-\eta)$ correction is the common slip that turns the honest $50\,\mathrm{Myr}$ e-folding time into the often-quoted $45\,\mathrm{Myr}$ — it is the difference between rest-mass accreted and rest-mass retained.

Step 3. Recognize $\dot M_\bullet=M/t_{\mathrm{Sal}}$ with $t_{\mathrm{Sal}}=\frac{\eta}{1-\eta}\frac{\sigma_T c}{4\pi Gm_p}$ . Being linear in $M$ , it integrates to $M(t)=M_0\,e^{t/t_{\mathrm{Sal}}}$ . The prefactor $\sigma_T c/(4\pi Gm_p)=Mc^2/L_{\mathrm{Edd}}=0.45\,\mathrm{Gyr}$ is the Eddington timescale; for $\eta=0.1$ , $t_{\mathrm{Sal}}=(0.1/0.9)(0.45\,\mathrm{Gyr})=50\,\mathrm{Myr}$ . Why: exponential growth follows directly from $L_{\mathrm{Edd}}\propto M$ — a bigger hole eats proportionally faster, so growth compounds.

Step 4. Read off the clock. Growing a $100\,M_\odot$ stellar seed to a $10^9\,M_\odot$ quasar is seven decades in mass, or $\ln(10^7)=16.1$ e-foldings, taking $16.1\times50\,\mathrm{Myr}\approx0.81\,\mathrm{Gyr}$ of uninterrupted Eddington accretion. Why: the Universe is only $\sim0.7\,\mathrm{Gyr}$ old at $z\simeq7.5$ , so this is the crux of the early-quasar problem — continuous Eddington growth from a stellar seed barely fails, forcing heavier seeds or super-Eddington episodes.

Go deeper · Derivation C · 1 minThe generalized limit and the effective Eddington factor

Derivation CThe generalized limit and the effective Eddington factor

The purpose of this final derivation is to generalize away from pure Thomson scattering and show how the same balance, rewritten per unit mass with a general opacity, both recovers the standard result and predicts where the naive limit fails: when the opacity is enhanced by lines, dust, or metals the effective Eddington luminosity drops, and near the limit the effective gravity is scaled by $(1-\Gamma)$ so the atmosphere becomes marginally unbound. This reframing connects the textbook limit to line-driven stellar winds and to radiation-pressure feedback in galactic nuclei.

Step 1. Recast the balance per gram of plasma. The radiation force per unit mass is $\kappa F/c=\kappa L/(4\pi r^2c)$ , where $\kappa$ is the opacity (cross section per unit mass); gravity per unit mass is $GM/r^2$ . Equating gives $L_{\mathrm{Edd}}=4\pi GMc/\kappa$ , and with $\kappa=\sigma_T/m_p$ this reproduces equation (1). Why: writing the limit as $4\pi GMc/\kappa$ isolates all the microphysics into a single number $\kappa$ , so any process that changes the opacity changes the limit readably.

Step 2. Introduce the Eddington factor $\Gamma\equiv L/L_{\mathrm{Edd}}=\kappa L/(4\pi GMc)$ and write the net inward acceleration as $g_{\mathrm{eff}}=(GM/r^2)(1-\Gamma)$ . For $\Gamma<1$ the gas is bound with reduced gravity; at $\Gamma=1$ it floats; for $\Gamma>1$ the net force is outward. Why: expressing the result as a modification of gravity, not a hard wall, makes the limit dynamical — it explains winds and inflated envelopes rather than an on/off switch.

Step 3. Let the opacity exceed the electron-scattering floor. UV resonance lines in a partially ionized wind, or dust in a cool torus, raise the effective opacity to $\kappa_{\mathrm{eff}}=\mathcal{M}\,\kappa_{\mathrm{es}}$ with a force multiplier $\mathcal{M}$ of tens to thousands, so $L_{\mathrm{Edd}}^{\mathrm{eff}}=4\pi GMc/\kappa_{\mathrm{eff}}$ falls well below the Thomson value. Why: this is why hot stars and AGN drive powerful line- and dust-driven winds far under the classical limit — the gas they push is far more opaque than bare electrons, so it unbinds long before $L$ reaches $L_{\mathrm{Edd}}$ .

Limiting cases

How the Eddington scale behaves in the regimes that matter:

Case	Result	Meaning
Pure ionized hydrogen	$L_{\mathrm{Edd}}=\dfrac{4\pi GMm_pc}{\sigma_T}\approx1.26\times10^{38}\dfrac{M}{M_\odot}\,\mathrm{erg\,s^{-1}}$	The canonical case: opacity is Thomson scattering per proton mass, giving the mass-only scaling.
Cosmic composition	$\kappa_{\mathrm{es}}=0.20(1+X)\approx0.34\,\mathrm{cm^2\,g^{-1}}$	Helium adds mass without free electrons, so opacity per gram drops and $L_{\mathrm{Edd}}$ rises $\sim20\%$ .
Enhanced opacity (lines, dust)	$L_{\mathrm{Edd}}^{\mathrm{eff}}=\dfrac{4\pi GMc}{\kappa_{\mathrm{eff}}},\ \kappa_{\mathrm{eff}}\gg\kappa_{\mathrm{es}}$	Force multipliers of $10$ – $10^3$ lower the limit — the engine of line-driven and dusty winds.
Super-Eddington accretion	$L\gtrsim L_{\mathrm{Edd}},\ \dot M\gg\dot M_{\mathrm{Edd}}$	Disk geometry and photon trapping let matter accrete through the plane while excess light escapes along the poles.
Radiatively inefficient	$\lambda=L/L_{\mathrm{Edd}}\ll1$	Energy is advected inward rather than radiated (ADAFs) — hot, geometrically thick, and dim, as in Sgr A*.

Common trapThe Eddington luminosity is an absolute ceiling on observed brightness

It is not. The derivation bakes in spherical symmetry, steady state, isotropic radiation, and pure electron-scattering opacity — and each can be broken. In a thin or slim disk, radiation escapes preferentially along the rotation axis while matter spirals in through the dense midplane, so the apparent luminosity along the poles can exceed $L_{\mathrm{Edd}}$ without violating any force balance in the gas. At very high rates photons are trapped in the inflow and advected across the horizon before they can push back, and in magnetized neutron stars the cross section is suppressed below $\sigma_T$ so the local limit rises. Pulsating ultraluminous X-ray sources shine at $100$ – $500$ times their naive spherical limit for exactly these reasons. Treat $L_{\mathrm{Edd}}$ as the natural ordering scale, not a hard wall.

Worked examples

1 · A supermassive black hole at the limit (fully worked)

The luminous quasar 3C 273 has a bolometric luminosity of roughly $L\approx4\times10^{46}\,\mathrm{erg\,s^{-1}}$ and a black-hole mass estimated near $M\approx4\times10^8\,M_\odot$ . Find its Eddington luminosity, its Eddington ratio, and the accretion rate needed to power it at radiative efficiency $\eta=0.1$ .

Step 1. Eddington luminosity from equation (2): $L_{\mathrm{Edd}}=1.26\times10^{38}\times(4\times10^8)=\mathbf{5.0\times10^{46}\,\mathrm{erg\,s^{-1}}}$ — the ceiling under the spherical assumptions.

Step 2. Eddington ratio: $\lambda=L/L_{\mathrm{Edd}}=(4\times10^{46})/(5.0\times10^{46})=\mathbf{0.8}$ . The quasar radiates at $80\%$ of its Eddington limit — a radiatively efficient, thin-disk state near the top of the ladder, with no need for super-Eddington physics.

Step 3. Accretion rate from equation (4): $\dot M=L/(\eta c^2)=(4\times10^{46})/[0.1\times(3\times10^{10})^2]=4.4\times10^{26}\,\mathrm{g\,s^{-1}}$ . Converting, $4.4\times10^{26}\times(3.16\times10^7)/(1.99\times10^{33})=\mathbf{7.0\,M_\odot\,yr^{-1}}$ . So 3C 273 runs at λ ≈ 0.8 and must swallow about 7 solar masses of gas per year.

2 · Can a stellar seed become a quasar in time? (fully worked)

A Population III remnant of $M_0=100\,M_\odot$ accretes continuously at the Eddington limit with $\eta=0.1$ . How long to reach $10^9\,M_\odot$ , and is that compatible with the quasar J1342+0928 observed at redshift $z=7.54$ , when the Universe was $\approx0.69\,\mathrm{Gyr}$ old?

Step 1. Count e-foldings: the mass grows by $10^9/10^2=10^7$ , so $t/t_{\mathrm{Sal}}=\ln(10^7)=7\ln10=\mathbf{16.1}$ e-folds.

Step 2. Insert the Salpeter time. From equation (6) with $\eta=0.1$ , $t_{\mathrm{Sal}}=\frac{0.1}{0.9}\times0.45\,\mathrm{Gyr}=50\,\mathrm{Myr}$ , so $t=16.1\times50\,\mathrm{Myr}=\mathbf{0.81\,Gyr}$ . (Dropping the $(1-\eta)$ factor gives $t_{\mathrm{Sal}}=45\,\mathrm{Myr}$ and $t=0.72\,\mathrm{Gyr}$ — a $10\%$ shift that matters at this margin.)

Step 3. Compare to the clock. The required $0.81\,\mathrm{Gyr}$ exceeds the $0.69\,\mathrm{Gyr}$ age of the Universe at $z=7.54$ , so uninterrupted Eddington growth from a $100\,M_\odot$ seed does not make it. The tension eases with a heavier seed: a $10^4\,M_\odot$ direct-collapse black hole needs only $\ln(10^5)=11.5$ e-folds, or $0.58\,\mathrm{Gyr}$ , which fits. The earliest quasars therefore favour heavy seeds, brief super-Eddington bursts, or both — a live research problem.

Physics of the portrait

The Eddington luminosity carries the name of the man who first understood that radiation pressure is a load-bearing part of a star, and that a star cannot shine arbitrarily brightly without tearing itself apart.

Origins · 1 minArthur Eddington and how this was discovered

Arthur Eddington

1882–1944 · stellar structure, radiation pressure, and the maximum-luminosity limit

Arthur Stanley Eddington was the dominant astrophysicist of the early twentieth century and the architect of the theory of stellar interiors. Working in the 1910s and 1920s, he showed that a luminous star is held up not by gas pressure alone but by radiation pressure, and that the balance between gravity and the outward push of escaping photons governs its whole structure. In The Internal Constitution of the Stars (1926) he laid out the mass–luminosity relation and, from the same reasoning, the argument that there is a maximum luminosity beyond which radiation pressure exceeds gravity and the outer layers are blown off — the limit now bearing his name, originally derived for massive stars rather than for accretion. Eddington was also a central figure in relativity, leading the 1919 solar-eclipse expedition whose measurement of light bending confirmed general relativity. He clashed famously with the young Chandrasekhar over whether a white dwarf could have a maximum mass, insisting — wrongly — that "some law of Nature" must prevent complete gravitational collapse. The irony is deep: the same radiation-pressure limit he championed for stars became, decades later, the master scale for the accreting black holes whose existence he had refused to countenance.

History

1916

— Eddington's radiative-equilibrium model puts radiation pressure at the centre of stellar structure

1926

— Eddington's The Internal Constitution of the Stars states the maximum-luminosity argument

1964

— Salpeter and, independently, Zel'dovich propose accretion onto massive black holes as the quasar engine

1969

— Lynden-Bell models quasars as supermassive accreting black holes in galactic nuclei

1973

— Shakura and Sunyaev formulate the thin-disk model, embedding the Eddington rate in accretion theory

Going further

Texts. Frank, King & Raine, Accretion Power in Astrophysics, 3rd ed., Ch. 1 (the Eddington limit and accretion basics) · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed. (stellar structure and the radiation-pressure limit) · Rybicki & Lightman, Radiative Processes in Astrophysics, Ch. 1 (Thomson scattering and momentum flux) · Mo, van den Bosch & White, Galaxy Formation and Evolution, §14 (black-hole growth, Salpeter time, feedback) · Binney & Tremaine, Galactic Dynamics, 2nd ed. (galactic-nucleus context). Inserting CODATA constants ( $\sigma_T=6.65\times10^{-25}\,\mathrm{cm^2}$ , $m_p=1.673\times10^{-24}\,\mathrm{g}$ ) into $L_{\mathrm{Edd}}=4\pi GMm_pc/\sigma_T$ returns $1.26\times10^{38}(M/M_\odot)\,\mathrm{erg\,s^{-1}}$ , and $\kappa_{\mathrm{es}}=\sigma_T/m_p\approx0.40\,\mathrm{cm^2\,g^{-1}}$ . The Eddington timescale is $\sigma_T c/(4\pi Gm_p)=Mc^2/L_{\mathrm{Edd}}\approx0.45\,\mathrm{Gyr}$ , giving $t_{\mathrm{Sal}}=\frac{\eta}{1-\eta}(0.45\,\mathrm{Gyr})\approx50\,\mathrm{Myr}$ for $\eta=0.1$ ; the frequently quoted $45\,\mathrm{Myr}$ drops the $1-\eta$ factor.

Builds on: Radiation pressure and radiative transfer · Thomson scattering · Newtonian gravity of a point mass. Leads to: Thin Accretion Disks and the Shakura–Sunyaev Model · Advection-Dominated Accretion Flows · Supermassive Black Holes · Quasars and Quasar Evolution · AGN Feedback.

Ask the tutor: Ask me why the earliest $10^9\,M_\odot$ quasars strain Eddington-limited growth, or to redo any estimate here for a mass or efficiency you choose.

Is the Eddington luminosity a hard upper limit on how bright a source can be?

No — it is a limit only under its own assumptions: spherical symmetry, steady state, isotropic radiation, and pure electron-scattering opacity. Break any of them and a source can shine brighter. A disk lets radiation leak out along the poles while matter keeps accreting through the plane; at very high rates photons are trapped and dragged inward before they can push back; magnetized neutron stars suppress the cross section and raise the local limit. Pulsating ultraluminous X-ray sources at $100$ -plus times their spherical limit are real. Treat $L_{\mathrm{Edd}}$ as the scale that orders accretion phenomenology, not an inviolable ceiling.

Why does the proton mass appear when radiation only pushes on the electrons?

Because gravity and radiation act on different members of the same bound pair. Radiation pressure couples to the electron through its large Thomson cross section, but gravity couples to mass, and the mass of the pair is essentially the proton mass. Electrostatic attraction prevents the electron from being blown away alone, so radiation must lift the whole pair while gravity holds it. Setting the force on the electron equal to the gravity on the proton is what puts $m_p$ into the numerator of $L_{\mathrm{Edd}}=4\pi GMm_pc/\sigma_T$ . Worked per unit mass, the same physics reappears as the opacity $\kappa_{\mathrm{es}}=\sigma_T/m_p$ .

Why is the Eddington luminosity independent of radius and density?

Both competing forces share the same $1/r^2$ falloff of a point source, so when you set them equal the radius cancels exactly. Density never enters because the balance is written per particle (or per unit mass) — a denser gas has more particles feeling the force, but each obeys the same balance, so the critical luminosity is unchanged. This radius- and density-independence is precisely why a single luminosity marks the limit rather than a different threshold at every location.

What sets the radiative efficiency η, and why does it matter so much?

The efficiency is the fraction of infalling rest-mass energy that escapes as radiation, $L=\eta\dot Mc^2$ , fixed by how deep the gas can radiate before crossing the horizon. For a non-spinning (Schwarzschild) hole the innermost stable circular orbit gives $\eta\approx0.057$ ; a maximally spinning (Kerr) hole reaches $\eta\approx0.42$ ; the canonical $0.1$ is a convenient middle. It matters because it sets both the accretion rate for a given luminosity ( $\dot M_{\mathrm{Edd}}\propto1/\eta$ ) and the growth clock ( $t_{\mathrm{Sal}}\propto\eta/(1-\eta)$ ): a high-spin hole grows more slowly at fixed Eddington luminosity because more of what falls in leaves as light.

How do the earliest quasars grow so fast if the Salpeter time is 50 Myr?

This is one of the sharpest open problems in the field. Growing a $100\,M_\odot$ stellar-remnant seed to $10^9\,M_\odot$ takes about $16$ e-folds, $\sim0.8\,\mathrm{Gyr}$ of continuous Eddington accretion — slightly longer than the age of the Universe at $z\simeq7.5$ . The escape routes: a heavier seed (a $10^4$ – $10^5\,M_\odot$ direct-collapse hole needs far fewer e-folds), brief super-Eddington bursts, or a duty cycle near unity so growth is never interrupted. JWST's discovery of overmassive black holes and "little red dots" at high redshift has made this tension more acute.

Does the Eddington limit apply to stars as well as accreting black holes?

Yes — that is where Eddington first derived it. The most luminous massive stars sit close to their Eddington limit, which is thought to cap stellar masses near $\sim100$ – $300\,M_\odot$ and to drive the eruptions of luminous blue variables like Eta Carinae. In stars the opacity is enhanced by iron-group lines in the outer layers, so the effective limit is reached well below the pure-Thomson value, producing "porous" super-Eddington atmospheres. The identical force balance governs both a $200\,M_\odot$ star shedding its envelope and a $10^9\,M_\odot$ black hole regulating a galaxy.