Eddington Luminosity Limit

The Eddington luminosity is the single most quoted brightness in high-energy astrophysics, and it wears two faces at once. Historically it is a stellar statement: Arthur Eddington asked how bright a star could be before its own radiation pressure lifted its outer layers off, and found a ceiling fixed by mass and opacity alone. Today it is an accretion statement: the maximum steady power a compact object can radiate while still pulling gas inward, the natural yardstick against which every quasar, X-ray binary, and growing black hole is measured. The two faces are one equation, $L_{\mathrm{Edd}}=4\pi G M m_p c/\sigma_T$ , read either as "radiation supporting an envelope" or "radiation expelling an accretion flow." The reason the limit is so clean is a coincidence of geometry that is not really a coincidence at all. Gravity from a point mass and photon flux from a point source both dilute as $1/r^2$ , so their ratio is a pure number set only by the accretor's mass and the microphysics of Thomson scattering — no reference to where the gas sits. Above the critical luminosity, radiation wins everywhere along a radial ray at once, and spherical accretion is choked off. This article derives $L_{\mathrm{Edd}}$ , converts it to a maximum accretion rate and an exponential growth clock (the Salpeter time), and then confronts the idealization with the real, genuinely super-Eddington universe of disks, beams, and pulsating ultraluminous X-ray sources.

Key takeaways

The Eddington luminosity $L_{\mathrm{Edd}}$ is the maximum steady luminosity at which outward radiation pressure on ionized gas still does not exceed inward gravity, for spherically symmetric accretion.
For pure ionized hydrogen it depends only on the accretor's mass: $L_{\mathrm{Edd}}=4\pi G M m_p c/\sigma_T\approx1.26\times10^{38}\,(M/M_\odot)\ \mathrm{erg\,s^{-1}}$ , or about $3\times10^4\,L_\odot$ per solar mass.
The limit is radius-independent because gravity and radiation force both scale as $1/r^2$ ; the balance is a pure ratio of gravity to Thomson scattering.
Divided by a radiative efficiency $\eta$ , the ceiling becomes an accretion-rate ceiling $\dot M_{\mathrm{Edd}}$ and an exponential growth clock: the Salpeter e-folding time $t_{\mathrm{Sal}}\sim5\times10^7\,\mathrm{yr}$ for $\eta=0.1$ .
It is a benchmark, not an absolute ceiling — non-spherical geometry, photon trapping, and magnetic channeling permit genuine super-Eddington accretion, as ULX pulsars radiating $\sim100\,L_{\mathrm{Edd}}$ prove.
Applications set lower bounds on quasar and AGN masses, cap the brightness of accreting neutron stars, and regulate black-hole growth through the $M$ – $\sigma$ relation.
Key idea: set the radiation force from Thomson scattering equal to gravity on an ionized hydrogen plasma; the $1/r^2$ dependence cancels, leaving a radius-independent maximum luminosity $L_{\mathrm{Edd}}=4\pi G M m_p c/\sigma_T$ fixed by the accretor mass alone.

The core insight

Set the radiation force from Thomson scattering equal to gravity on an ionized hydrogen plasma; the $1/r^2$ dependence cancels, giving a radius-independent maximum luminosity $L_{\mathrm{Edd}}=4\pi G M m_p c/\sigma_T$ that depends only on the accretor mass.

Where this lives today

The Eddington limit is the ruler laid against every accreting object now under study. The most urgent frontier is the high-redshift black-hole problem: JWST is finding surprisingly massive, often overmassive black holes — compact red active galaxies ("little red dots") and confirmed nuclei at $z>10$ — within the first $500$ million years of cosmic time, straining the Salpeter clock and pushing toward heavy direct-collapse seeds or sustained super-Eddington phases. In the local universe, the Event Horizon Telescope's images of M87 and Sagittarius A probe accretion in the strongly sub-Eddington, radiatively inefficient regime, testing the ADAF physics of the limit's low end. At the stellar scale, pulsating ultraluminous X-ray sources have overturned the assumption that $L_{\mathrm{Edd}}$ is a wall, demonstrating neutron stars radiating $\sim100\,L_{\mathrm{Edd}}$ through magnetically channeled columns. Tidal disruption events flare through and above the Eddington rate, letting us watch a black hole cross its own limit in real time. And in survey cosmology, Eddington ratios distilled from hundreds of thousands of SDSS and DESI quasar spectra anchor the $M$ – $\sigma$ relation that ties this compact-object microphysics to the assembly of galaxies.

The physical picture

Picture a single hydrogen atom's worth of plasma — one proton and one electron — hovering in the radiation field of an accreting object. Gravity pulls almost entirely on the proton, which carries essentially all the mass $m_p$ ; photons scatter almost entirely off the electron, through the Thomson cross section $\sigma_T$ . The two are locked together by electrostatics: if radiation pushed the light electron out and left the proton behind, the charge separation would build a field that instantly drags the proton along. Radiation therefore lifts the proton through the electron, and the plasma responds as one fluid. The luminosity at which that lift equals the proton's weight is the Eddington luminosity. The first figure makes the force balance — and its clean $1/r^2$ cancellation — one picture.

Figure

Force balance on one hydrogen unit at radius r: gravity GMm_p/r² pulls the proton inward while radiation Lσ_T/4πr²c pushes the electron outward. Both scale as 1/r², so the radius cancels and L_Edd = 4πGMm_p c/σ_T ≈ 1.26×10³⁸ (M/M_⊙) erg s⁻¹

The deep reason the ratio is so clean is that both a point mass's gravity and a point source's photon flux fall off as the inverse square of distance, so their quotient is independent of $r$ : if $L$ exceeds $L_{\mathrm{Edd}}$ anywhere along a radial line, it exceeds it everywhere, and the infalling gas is blown back. That single number is a conversion rate between mass and maximum steady power — roughly $1.3\times10^{38}\,\mathrm{erg\,s^{-1}}$ , or $3\times10^4\,L_\odot$ , per solar mass of accretor. The second figure plots it across nine decades of mass, from neutron stars to quasar engines.

Figure

The Eddington ladder: L_Edd = 1.26×10³⁸ (M/M_⊙) erg s⁻¹ spans nine decades — a 1.4 M_⊙ neutron star caps near 1.8×10³⁸, a 10 M_⊙ black hole near 1.3×10³⁹, and a 10⁸ M_⊙ quasar engine near 1.3×10⁴⁶ erg s⁻¹

The core equations

The central result is a force balance per hydrogen unit. Equate the radiation force on the electron, $L\sigma_T/(4\pi r^2 c)$ , to gravity on the proton, $GMm_p/r^2$ ; the $1/r^2$ cancels and the surviving luminosity, $L_{\mathrm{Edd}}$ , depends on nothing but the accretor mass and two microphysical constants.

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

where: $L_{\mathrm{Edd}}$ — maximum steady spherical-accretion luminosity; $M$ — mass of the accretor; $m_p$ — proton mass, $1.673\times10^{-24}\,\mathrm{g}$ ; $\sigma_T$ — Thomson cross section, $6.652\times10^{-25}\,\mathrm{cm^2}$ ; $c$ — speed of light; $G$ — gravitational constant. Reading it: the numerator is gravity's grip on the proton, the denominator radiation's grip on the electron. Evaluated with CODATA constants, the coefficient is worth memorizing — it converts accretor mass directly into a luminosity ceiling, equivalently about $3\times10^4$ solar luminosities per solar mass.

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

where: $M_\odot$ — solar mass, $1.989\times10^{33}\,\mathrm{g}$ . Reading it: a $1\,M_\odot$ accretor caps at $\approx1.3\times10^{38}\,\mathrm{erg\,s^{-1}}$ ; scale linearly for any other mass. The composition enters only through the number of electrons per unit mass. Writing the electron-scattering opacity per gram as $\kappa$ absorbs $\sigma_T$ and $m_p$ into one material coefficient, covering any ionized mixture.

L_{\mathrm{Edd}}=\dfrac{4\pi G M c}{\kappa} \qquad (3)

where: $\kappa$ — electron-scattering opacity (cross section per unit mass), $\mathrm{cm^2\,g^{-1}}$ . Reading it: pure hydrogen gives $\kappa=\sigma_T/m_p\approx0.40$ ; a solar mix gives $\kappa\approx0.34$ , because heavier elements carry fewer electrons per nucleon. Because sources are rarely exactly at the limit, the dimensionless Eddington ratio is the working diagnostic.

\Gamma_{\mathrm{Edd}}\equiv\dfrac{L}{L_{\mathrm{Edd}}} \qquad (4)

where: $\Gamma_{\mathrm{Edd}}$ — Eddington ratio; $\Gamma_{\mathrm{Edd}}=1$ marks the balance, $<1$ sub-Eddington, $>1$ super-Eddington. Reading it: quasars cluster around $\Gamma_{\mathrm{Edd}}\sim0.1$ – $1$ , running near capacity. The luminosity ceiling becomes a fuel-supply ceiling once we know the radiative efficiency $\eta$ — the fraction of accreted rest-mass energy emerging as light. Setting $L=\eta\dot M c^2$ equal to $L_{\mathrm{Edd}}$ gives the Eddington accretion rate.

\dot M_{\mathrm{Edd}}=\dfrac{L_{\mathrm{Edd}}}{\eta c^2}=\dfrac{4\pi G M m_p}{\eta\,\sigma_T c} \qquad (5)

where: $\dot M_{\mathrm{Edd}}$ — accretion rate delivering $L_{\mathrm{Edd}}$ ; $\eta$ — radiative efficiency, $\sim0.06$ – $0.4$ for black-hole accretion (canonically $0.1$ ). Reading it: for $M=10^8\,M_\odot$ and $\eta=0.1$ , $\dot M_{\mathrm{Edd}}\approx2\,M_\odot\,\mathrm{yr^{-1}}$ — a modest stellar diet fuels a quasar. Because $L_{\mathrm{Edd}}\propto M$ , an object that always accretes at its own limit grows exponentially. The e-folding time of that growth — the Salpeter time — depends only on $\eta$ and constants, not on the mass itself.

t_{\mathrm{Sal}}=\dfrac{\eta}{1-\eta}\,\dfrac{\sigma_T c}{4\pi G m_p}\approx5.0\times10^{7}\left(\dfrac{\eta/0.1}{1-\eta}\cdot 0.9\right)\mathrm{yr} \qquad (6)

where: $t_{\mathrm{Sal}}$ — e-folding time for Eddington-limited mass growth; the fixed group $\sigma_T c/(4\pi G m_p)\approx0.45\,\mathrm{Gyr}$ is the Eddington timescale. Reading it: for $\eta=0.1$ this is $\approx5\times10^7\,\mathrm{yr}$ ; dropping the small $(1-\eta)$ correction recovers the traditional quoted value $\approx45\,\mathrm{Myr}$ . Finally, in a sub-Eddington radiation field gravity is not switched off but reduced: the net inward acceleration is gravity diluted by the Eddington ratio — the form governing stellar envelopes and radiation-driven winds.

g_{\mathrm{eff}}=\dfrac{GM}{r^2}\left(1-\Gamma_{\mathrm{Edd}}\right) \qquad (7)

where: $g_{\mathrm{eff}}$ — net inward acceleration after radiation support. Reading it: as $\Gamma_{\mathrm{Edd}}\to1$ effective gravity vanishes and the atmosphere becomes unbound — Eddington's original stellar criterion.

Derivation ARadius-independence of the Eddington limit

The goal is to demonstrate from first principles that balancing radiation force against gravity on a fully ionized hydrogen plasma yields one critical luminosity that is the same at every radius, so that a source either satisfies the balance everywhere along a radial ray or nowhere. We treat isotropic radiation from a central point mass, compute the outward momentum deposited on electrons against the inward pull on protons, and watch the radial dependence cancel identically.

Step 1. At radius $r$ the energy flux of an isotropic source of luminosity $L$ is $S=L/(4\pi r^2)$ , and since photons carry momentum $E/c$ , the momentum flux is $S/c=L/(4\pi r^2 c)$ . Why: the inverse-square dilution of flux is the origin of the $1/r^2$ that will later cancel against gravity.

Step 2. An electron presents the Thomson cross section $\sigma_T$ to this momentum flux, so it feels an outward force $f_e=\sigma_T\,S/c=\sigma_T L/(4\pi r^2 c)$ . Why: Thomson scattering redirects photons that on average were moving radially outward, so their net momentum transfer to the electron is radially outward.

Step 3. Gravity acts on the proton, which carries essentially all the mass of the hydrogen unit, giving an inward force $f_g=GMm_p/r^2$ . Electrostatic coupling ties the electron and proton together, so it is legitimate to balance the force on the electron against the weight of its partner proton. Why: the plasma is quasi-neutral; any charge separation self-corrects, so electron and proton move as a single fluid element of mass $\approx m_p$ .

Step 4. Impose $f_e=f_g$ : $\sigma_T L/(4\pi r^2 c)=GMm_p/r^2$ . Both sides carry an identical factor $1/r^2$ , which divides out cleanly. Why: this is the physical heart of the result — the two competing forces share the same distance law, so their balance cannot depend on distance.

Step 5. Solve for the critical luminosity: $L=4\pi G M m_p c/\sigma_T\equiv L_{\mathrm{Edd}}$ . Because $r$ has vanished, this value holds simultaneously at all radii. Why: a source exceeding $L_{\mathrm{Edd}}$ overpowers gravity along the entire ray, choking spherical accretion rather than merely at one shell. *Dimensional check: $[GMm_p c/\sigma_T]=(\mathrm{cm^3\,g^{-1}\,s^{-2}})(\mathrm{g})(\mathrm{g})(\mathrm{cm\,s^{-1}})/\mathrm{cm^2}=\mathrm{g\,cm^2\,s^{-3}}=\mathrm{erg\,s^{-1}}$ , confirming the units are correct; the numerical value is $1.26\times10^{38}\,(M/M_\odot)\,\mathrm{erg\,s^{-1}}$ .*

Go deeper · Derivation B · 1 minFrom the luminosity ceiling to exponential black-hole growth

Derivation BFrom the luminosity ceiling to exponential black-hole growth

The aim is to convert the static luminosity limit into a dynamical statement about how fast an accreting black hole gains mass, and to extract the Salpeter e-folding time. We assume the hole always radiates at its own mass-dependent Eddington luminosity with fixed efficiency $\eta$ , track how much inflowing rest mass is retained versus radiated, and integrate the resulting linear equation — keeping the frequently dropped $(1-\eta)$ factor.

Step 1. Because $L_{\mathrm{Edd}}=(4\pi G m_p c/\sigma_T)\,M$ is linear in $M$ , an Eddington-limited hole obeys $L(t)=(4\pi G m_p c/\sigma_T)\,M(t)$ . Why: the luminosity is slaved to the instantaneous mass, so the growth equation will be self-referential and hence exponential.

Step 2. The rest-mass inflow needed to power $L$ is $\dot M_{\mathrm{acc}}=L/(\eta c^2)$ , but a fraction $\eta$ of that mass-energy is radiated away, so the hole actually gains $\dot M=(1-\eta)\dot M_{\mathrm{acc}}=(1-\eta)L/(\eta c^2)$ . Why: energy conservation demands that the radiated fraction not be counted toward the mass the hole keeps; ignoring this over-counts growth by a few percent.

Step 3. Substitute $L=L_{\mathrm{Edd}}(M)$ to get $\dot M=\dfrac{1-\eta}{\eta}\dfrac{4\pi G m_p}{\sigma_T c}\,M\equiv M/t_{\mathrm{Sal}}$ , a linear first-order equation with constant coefficient. Why: $\dot M\propto M$ is the signature of exponential growth, and the proportionality constant is the inverse e-folding time.

Step 4. Read off $t_{\mathrm{Sal}}=\dfrac{\eta}{1-\eta}\dfrac{\sigma_T c}{4\pi G m_p}$ and integrate to $M(t)=M_0\,e^{t/t_{\mathrm{Sal}}}$ . Why: the number of e-foldings available in cosmic time, $t_{\mathrm{age}}/t_{\mathrm{Sal}}$ , sets the largest mass a given seed can reach by a given epoch. *The fixed group $\sigma_T c/(4\pi G m_p)=0.45\,\mathrm{Gyr}$ ; with $\eta=0.1$ , $t_{\mathrm{Sal}}=(0.1/0.9)(0.45\,\mathrm{Gyr})\approx5.0\times10^7\,\mathrm{yr}$ . The often-quoted $\approx45\,\mathrm{Myr}$ drops the $(1-\eta)$ factor, i.e. sets $t_{\mathrm{Sal}}\approx\eta\times0.45\,\mathrm{Gyr}$ .*

Go deeper · Derivation C · 1 minThe Eddington factor and radiation-modified hydrostatic balance

Derivation CThe Eddington factor and radiation-modified hydrostatic balance

The purpose here is to recover Eddington's original stellar meaning of the limit — a ceiling on how luminous a star can be before radiation unbinds its envelope — by inserting radiation force into hydrostatic equilibrium rather than into an accretion balance, showing that radiation rescales gravity and that effective gravity vanishes at the same $L_{\mathrm{Edd}}$ derived above.

Step 1. Hydrostatic equilibrium reads $dP/dr=-\rho\,GM_r/r^2$ . In a radiation field the outward radiative acceleration per unit mass is $g_{\mathrm{rad}}=\kappa L/(4\pi r^2 c)$ — the momentum flux $L/(4\pi r^2 c)$ times the opacity per gram $\kappa$ . Why: opacity $\kappa$ is the cross section per unit mass, so $\kappa\times(\text{momentum flux})$ is a force per unit mass.

Step 2. Combine gravity and radiation into $g_{\mathrm{eff}}=\dfrac{GM}{r^2}-\dfrac{\kappa L}{4\pi r^2 c}=\dfrac{GM}{r^2}\left(1-\dfrac{\kappa L}{4\pi G M c}\right)$ . Why: both terms carry $1/r^2$ , so radiation rescales the local gravity by a radius-independent factor.

Step 3. Recognize the bracket as $1-L/L_{\mathrm{Edd}}=1-\Gamma_{\mathrm{Edd}}$ , using $L_{\mathrm{Edd}}=4\pi G M c/\kappa$ ; hence $g_{\mathrm{eff}}=(GM/r^2)(1-\Gamma_{\mathrm{Edd}})$ . Why: the accretion limit and the stellar limit are literally the same number.

Step 4. As $L\to L_{\mathrm{Edd}}$ , $g_{\mathrm{eff}}\to0$ : the atmosphere is marginally unbound and radiation drives a wind. Why: this ties the accretion yardstick back to Eddington's stellar-structure problem, closing the loop between the two faces of the limit. *As a check: setting $g_{\mathrm{eff}}=0$ gives $L=4\pi G M c/\kappa=L_{\mathrm{Edd}}$ , identical to equation (3); and dimensionally $[\kappa L/(4\pi r^2 c)]=(\mathrm{cm^2\,g^{-1}})(\mathrm{erg\,s^{-1}})/(\mathrm{cm^2}\cdot\mathrm{cm\,s^{-1}})=\mathrm{cm\,s^{-2}}$ , confirming the radiative term is indeed an acceleration.*

Limiting cases

How the limit and its consequences behave across physically distinct regimes:

Case	Result	Meaning
Pure ionized hydrogen	$L_{\mathrm{Edd}}=\dfrac{4\pi G M m_p c}{\sigma_T}$	The standard form — one electron per proton, opacity from Thomson scattering only; gives $1.26\times10^{38}\,(M/M_\odot)\,\mathrm{erg\,s^{-1}}$ .
General composition	$L_{\mathrm{Edd}}=\dfrac{4\pi G M c}{\kappa}$	With $\kappa\approx0.34\,\mathrm{cm^2\,g^{-1}}$ for a solar mix; heavier elements give fewer electrons per nucleon, so $L_{\mathrm{Edd}}$ drops modestly.
Super-Eddington disks	$L\gtrsim L_{\mathrm{Edd}}$	Non-spherical geometry and photon trapping in slim/advective disks sustain $L$ several times $L_{\mathrm{Edd}}$ ; radiation escapes along the poles while matter accretes near the plane.
Radiatively inefficient	$L\ll L_{\mathrm{Edd}}$	At very low $\dot M$ the flow becomes hot and geometrically thick (ADAF); most energy is advected into the hole rather than radiated, so $L\ll\eta\dot M c^2$ .
Marginal envelope	$\Gamma_{\mathrm{Edd}}\to1$	Effective gravity $g_{\mathrm{eff}}\to0$ : a stellar atmosphere becomes unbound and radiation pressure launches a wind — Eddington's original stellar criterion.

Common trapThe Eddington limit is a hard, unbreakable ceiling on luminosity

It is not. Equation (1) assumes spherical symmetry, steady state, isotropic radiation, and pure electron-scattering opacity — all of which real systems violate. In an accretion disk radiation escapes anisotropically (mostly along the poles) while gas accretes near the equatorial plane, so the two no longer compete along the same ray. Photon trapping in optically thick advective flows, relativistic beaming, and magnetic channeling onto small polar caps all permit genuinely super-Eddington luminosities. The clinching evidence is the pulsating ultraluminous X-ray sources: coherent pulsations prove they are neutron stars of $\sim1.4\,M_\odot$ , yet several radiate at $\sim100\,L_{\mathrm{Edd}}$ . Treat $L_{\mathrm{Edd}}$ as a yardstick and a stability threshold, never a law forbidding all $L>L_{\mathrm{Edd}}$ .

Worked examples

1 · The Eddington luminosity of a neutron star (fully worked)

A neutron star of mass $M=1.4\,M_\odot$ accretes fully ionized hydrogen. Compute its Eddington luminosity from constants, and say what the answer means for Type I X-ray bursts.

Step 1. Write the balance and gather cgs constants: $L_{\mathrm{Edd}}=4\pi G M m_p c/\sigma_T$ with $G=6.674\times10^{-8}$ , $m_p=1.673\times10^{-24}\,\mathrm{g}$ , $c=2.998\times10^{10}\,\mathrm{cm\,s^{-1}}$ , $\sigma_T=6.652\times10^{-25}\,\mathrm{cm^2}$ , and $M=1.4\times1.989\times10^{33}=2.78\times10^{33}\,\mathrm{g}$ .

Step 2. Numerator: $4\pi\times6.674\times10^{-8}\times2.78\times10^{33}\times1.673\times10^{-24}\times2.998\times10^{10}$ . Grouping, $4\pi\,G=8.39\times10^{-7}$ ; times $M$ gives $2.33\times10^{27}$ ; times $m_p$ gives $3.90\times10^{3}$ ; times $c$ gives $1.17\times10^{14}$ .

Step 3. Divide by $\sigma_T$ : $L_{\mathrm{Edd}}=1.17\times10^{14}/6.652\times10^{-25}=\mathbf{1.76\times10^{38}\,\mathrm{erg\,s^{-1}}}$ . Equivalently, use the shortcut $1.26\times10^{38}\times1.4=1.76\times10^{38}$ — the two agree, as they must.

Step 4. Interpretation: this is precisely the peak luminosity reached in photospheric-radius-expansion Type I X-ray bursts, where a thermonuclear flash on the neutron-star surface briefly drives the photosphere outward against gravity. Because the burst pins the luminosity at $L_{\mathrm{Edd}}$ , and $L_{\mathrm{Edd}}$ is a known function of mass, these bursts serve as standard candles for measuring distances to X-ray binaries. L_Edd ≈ 1.8×10³⁸ erg s⁻¹ for a 1.4 M_⊙ neutron star.

2 · How fast can a high-redshift quasar's black hole grow? (fully worked)

A quasar at redshift $z\approx7$ , when the universe was $\approx0.77\,\mathrm{Gyr}$ old, hosts a $10^9\,M_\odot$ black hole radiating near its Eddington limit. If it grew from a stellar seed by continuous Eddington-limited accretion with $\eta=0.1$ , is there enough time?

Step 1. The required luminosity is $L_{\mathrm{Edd}}=1.26\times10^{38}\times10^{9}=\mathbf{1.26\times10^{47}\,\mathrm{erg\,s^{-1}}}$ , among the most luminous objects known.

Step 2. Growth is exponential, $M(t)=M_0\,e^{t/t_{\mathrm{Sal}}}$ , with $t_{\mathrm{Sal}}=(0.1/0.9)(0.45\,\mathrm{Gyr})=\mathbf{5.0\times10^7\,\mathrm{yr}}$ .

Step 3. The e-foldings available by $z\approx7$ number $t_{\mathrm{age}}/t_{\mathrm{Sal}}\approx770/50\approx\mathbf{15}$ . Growing from a Population III seed of $M_0\sim100\,M_\odot$ to $10^9\,M_\odot$ needs $\ln(10^7)=\mathbf{16.1}$ e-foldings.

Step 4. Verdict: $16.1$ needed versus $\lesssim15$ available — a light stellar seed cannot quite make it with continuous Eddington-limited accretion. From a direct-collapse seed of $M_0\sim10^5\,M_\odot$ only $\ln(10^9/10^5)=\ln(10^4)=\mathbf{9.2}$ e-foldings are required, which fits comfortably. The high-z quasar problem: either heavy ( $\sim10^5\,M_\odot$ ) seeds or episodes of super-Eddington accretion are required.

3 · Minimum mass of an X-ray source

An X-ray source has bolometric luminosity $2.5\times10^{38}\,\mathrm{erg\,s^{-1}}$ . If it is Eddington-limited, what is the minimum accretor mass? Since $L\le L_{\mathrm{Edd}}=1.26\times10^{38}(M/M_\odot)$ , invert to $M\gtrsim2.5/1.26\approx\mathbf{2.0\,M_\odot}$ — a neutron star or low-mass black hole, but too massive for a white dwarf. An observed luminosity buys a lower bound on the mass of the engine behind it.

Physics of the portrait

The Eddington limit is where stellar structure and black-hole astrophysics turn out to be the same equation. It carries the name of the man who, in working out how radiation holds a star up, wrote down a luminosity ceiling that would later govern the brightest accreting objects in the universe.

Origins · 1 minArthur Eddington and how this was discovered

Arthur Eddington

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

Arthur Stanley Eddington was the leading astrophysicist of the early twentieth century and the architect of the theory of stellar interiors. In the 1910s and 1920s he established that a star's interior is in radiative equilibrium — energy diffuses outward as radiation, and radiation pressure is a major partner to gas pressure in supporting the star against gravity. Pushing that idea to its limit, he asked how luminous a star of given mass could be before radiation pressure alone exceeded gravity and blew the envelope apart, and arrived at the ceiling $L_{\mathrm{Edd}}\propto M$ that now bears his name; it explained why the most massive stars are rare and why they shed mass in powerful winds. His 1926 monograph The Internal Constitution of the Stars codified the mass–luminosity relation and the role of opacity. Eddington was also the observational champion of general relativity — his 1919 eclipse expedition measured the deflection of starlight by the Sun. Decades after his death, when Salpeter and Zel'dovich identified accretion onto compact objects as the quasar power source, his envelope criterion was reborn as the accretion limit and growth clock for black holes — the same $4\pi G M m_p c/\sigma_T$ reached from the opposite direction.

History

1916

— Eddington formulates radiative equilibrium in stellar interiors

1926

— The Internal Constitution of the Stars sets out the radiation-pressure luminosity limit

1964

— Salpeter and Zel'dovich propose accretion onto black holes as the quasar power source

1973

— Shakura and Sunyaev formulate the standard thin accretion-disk model

2014

— the ultraluminous X-ray source M82 X-2 is shown to be a neutron star radiating near 100 times its Eddington limit

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., §17.5 (accretion) and the stellar-structure chapters · Frank, King & Raine, Accretion Power in Astrophysics, 3rd ed., Ch. 1 · Rybicki & Lightman, Radiative Processes in Astrophysics (Thomson scattering and radiation force) · Shapiro & Teukolsky, Black Holes, White Dwarfs, and Neutron Stars, Ch. 13–14 · Binney & Tremaine, Galactic Dynamics, 2nd ed. (AGN and accretion) · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 14 (black-hole growth and feedback). The CODATA constants used throughout are $G=6.674\times10^{-8}$ cgs, $m_p=1.673\times10^{-24}$ g, $c=2.998\times10^{10}$ cm/s, $\sigma_T=6.652\times10^{-25}$ cm², $M_\odot=1.989\times10^{33}$ g; these give $L_{\mathrm{Edd}}=1.26\times10^{38}\,\mathrm{erg\,s^{-1}}$ per solar mass. The Eddington timescale group $\sigma_T c/(4\pi G m_p)=0.45\,\mathrm{Gyr}$ yields $t_{\mathrm{Sal}}=5.0\times10^7$ yr for $\eta=0.1$ (the traditional $\approx45$ Myr drops the $1-\eta$ factor). The pure-hydrogen opacity is $\kappa_{\mathrm{H}}=\sigma_T/m_p\approx0.40\,\mathrm{cm^2\,g^{-1}}$ and a solar mix gives $\kappa\approx0.34\,\mathrm{cm^2\,g^{-1}}$ .

Prerequisites. Thomson Scattering Cross Section · Newtonian Gravity and Hydrostatic Balance · Radiation Pressure and Momentum Flux · Neutron Stars. Leads to. Neutron Stars · White Dwarfs · Millisecond and Recycled Pulsars.

Ask the tutor: Ask me why a light stellar seed can't grow into a $z\approx7$ quasar, or how the same $L_{\mathrm{Edd}}$ governs both accreting black holes and the winds of the most massive stars.

Is the Eddington limit really an absolute maximum brightness?

No — it is a maximum only under its idealizing assumptions: spherical symmetry, steady state, isotropic radiation, and pure electron-scattering opacity. It is best understood as a stability threshold and an order-of-magnitude yardstick. Real accretion disks radiate anisotropically, so gas keeps falling in near the equatorial plane while photons escape along the poles; photon trapping, beaming, and magnetic funneling push effective luminosities well above $L_{\mathrm{Edd}}$ . The clearest proof is the pulsating ultraluminous X-ray sources, confirmed neutron stars radiating around one hundred times their own Eddington limit. So $L>L_{\mathrm{Edd}}$ is not forbidden; it just requires breaking spherical symmetry.

Why does gravity act on protons but radiation on electrons — and why can we balance them?

Gravity scales with mass, and the proton is about $1836$ times heavier than the electron, so essentially all the weight of a hydrogen unit is the proton's. Radiation scatters off charges via the Thomson cross section, which scales as the inverse square of particle mass, so photons couple almost entirely to the light electron. The two forces seem to act on different particles — but the plasma is quasi-neutral and electrostatically bound. If radiation nudges electrons outward relative to protons, the resulting electric field immediately pulls the protons along. Electron and proton move together as one fluid element of mass $\approx m_p$ , and balancing the radiation force per electron against gravity per proton is exactly correct.

What sets the numerical coefficient 1.26×10³⁸ erg s⁻¹?

It is $4\pi G m_p c/\sigma_T$ per solar mass, a pure combination of fundamental constants with the proton mass and Thomson cross section. Physically, $\sigma_T$ in the denominator is why the number is large: the cross section is small, so it takes an enormous luminosity for radiation to overcome gravity on a proton via a single electron. Numerically it is about $1.26\times10^{38}\,\mathrm{erg\,s^{-1}}$ per $M_\odot$ , equivalently $\sim3\times10^4\,L_\odot$ . Because it is linear in mass, you rarely recompute it — just multiply the coefficient by the mass in solar units.

What is the Salpeter time, and why is it sometimes quoted as 45 and sometimes 50 Myr?

The Salpeter time is the e-folding timescale on which an Eddington-limited black hole grows exponentially, $M(t)=M_0\,e^{t/t_{\mathrm{Sal}}}$ . It is the fixed group $\sigma_T c/(4\pi G m_p)=0.45\,\mathrm{Gyr}$ times $\eta/(1-\eta)$ . For $\eta=0.1$ the full expression gives $t_{\mathrm{Sal}}\approx50\,\mathrm{Myr}$ . The commonly cited $\approx45\,\mathrm{Myr}$ drops the small $(1-\eta)$ correction — neglecting the few percent of inflowing rest mass radiated away rather than added to the hole. Both are "the Salpeter time." Crucially, it is independent of the black-hole mass, which is what makes exponential growth so constraining.

If growth is exponential, why is there a high-redshift quasar problem?

Exponential growth is fast, but capped by the number of e-foldings the universe has had time to provide. By $z\approx7$ the universe is only $\approx0.77\,\mathrm{Gyr}$ old, giving at most $\sim15$ Salpeter times. Growing a $10^9\,M_\odot$ black hole from a light $\sim100\,M_\odot$ stellar seed needs about $16$ e-foldings of continuous accretion — just barely too many, assuming the hole never pauses. The observed existence of such holes points to either heavier "direct-collapse" seeds of $\sim10^4$ – $10^5\,M_\odot$ , needing only $\sim9$ e-foldings, or episodes of super-Eddington accretion that beat the Salpeter clock.

Does the Eddington limit also apply to ordinary stars?

Yes — that is where Eddington found it. Radiation rescales the effective gravity in a stellar envelope by $(1-\Gamma_{\mathrm{Edd}})$ , so as a star's luminosity approaches $L_{\mathrm{Edd}}$ its outer layers become marginally unbound and radiation pressure drives a wind. This is why the most massive, luminous stars — hot O stars, luminous blue variables, Wolf–Rayet stars — sit close to their Eddington limit and shed mass copiously, and why there is an effective upper limit to stellar masses. The accretion version and the stellar version are literally the same formula, $L_{\mathrm{Edd}}=4\pi G M c/\kappa$ , read as either "radiation expelling infall" or "radiation unbinding an envelope".