Eddington Luminosity

The Eddington luminosity is the point where a star, or a parcel of gas falling onto a compact object, runs out of room between two opposing forces. Gravity pulls matter inward with an acceleration $GM/r^2$ . Radiation streaming outward from the interior deposits momentum on that same matter — every photon scattered or absorbed hands over its momentum $E/c$ — and the resulting push scales as $\kappa L/(4\pi r^2 c)$ . When the outward push grows to equal the inward pull, the net force vanishes: hydrostatic equilibrium can no longer be sustained, and any further luminosity drives the outer layers off as a wind. The luminosity at that crossover is $L_{\mathrm{Edd}}$ . What makes the result so clean, and so useful, is a cancellation. Both accelerations fall off as $1/r^2$ , so the radius drops out entirely when they are set equal. What survives is a critical luminosity — not a luminosity-at-some-radius — fixed by only two things: the enclosed mass $M$ and the opacity $\kappa$ . For the hot, fully ionized plasma inside a massive star or an accretion flow, the dominant continuum opacity is Thomson scattering off free electrons, and $L_{\mathrm{Edd}}$ takes its canonical value of roughly $3\times10^{4}$ solar luminosities per solar mass. This one relation reaches across astrophysics. It caps how bright and how massive a star can be, powers the eruptive mass loss of luminous blue variables such as η Carinae, and through the Eddington accretion rate it sets the fastest clock on which supermassive black holes can grow. Real systems radiate below the idealized limit — line and bound-free opacities raise the effective coupling, and rotation and asymmetry intrude — but every correction is measured against the same benchmark Eddington wrote down a century ago.

Key takeaways

The Eddington luminosity $L_{\mathrm{Edd}}$ is where the outward radiative force on matter exactly balances the inward gravity on the same matter.
Setting $\kappa L/(4\pi r^2 c) = GM/r^2$ , the $r^{-2}$ cancels and leaves $L_{\mathrm{Edd}} = 4\pi G M c/\kappa$ — fixed by mass and opacity alone, independent of radius.
For ionized hydrogen with electron-scattering opacity, $L_{\mathrm{Edd}} = 4\pi G M m_{\mathrm{p}} c/\sigma_{\mathrm{T}} \approx 1.26\times10^{31}\,(M/M_\odot)\ \mathrm{W} \approx 3.3\times10^{4}\,(M/M_\odot)\,L_\odot$ .
Above $L_{\mathrm{Edd}}$ the effective gravity $g_{\mathrm{eff}} = (GM/r^2)(1-\Gamma)$ goes negative: static support fails and radiation blows matter away.
The same physics caps accretion at $\dot{M}_{\mathrm{Edd}} = L_{\mathrm{Edd}}/\eta c^2$ and sets the Salpeter e-folding time $t_{\mathrm{Edd}} \approx 4.5\times10^{7}\,(\eta/0.1)$ yr for black-hole growth.
Higher opacity lowers the limit: metals, lines, and ionization bumps make the effective $L_{\mathrm{Edd}}$ much smaller than the electron-scattering value.
Key idea: set the radiative acceleration $\kappa L/(4\pi r^2 c)$ equal to gravity $GM/r^2$ ; the $r^{-2}$ cancels, giving a maximum luminosity $L_{\mathrm{Edd}} = 4\pi G M c/\kappa$ set only by mass and opacity.

The core insight

Set the radiative force per unit mass, $\kappa F/c = \kappa L/(4\pi r^2 c)$ , equal to the gravitational acceleration $GM/r^2$ ; the $r^{-2}$ dependence cancels identically, yielding a maximum luminosity $L_{\mathrm{Edd}} = 4\pi G M c/\kappa$ that depends only on the enclosed mass and the opacity.

Where this lives today

The Eddington limit is a working tool across modern astrophysics, not a museum piece. In stellar physics it underlies the upper end of the initial mass function: radiation-hydrodynamics simulations show radiation pressure on dust reversing infall onto forming massive stars, and the absence of ordinary stars much above $\sim150$ – $300\,M_\odot$ is read as an Eddington-related cap. It governs the eruptions of luminous blue variables — η Carinae shed roughly ten solar masses in its 1840s "Great Eruption" while briefly exceeding its limit — and the winds that strip Wolf–Rayet stars bare. Among compact objects the Eddington rate is the reference against which every accretor is measured: X-ray binaries switch spectral states at characteristic fractions of $\dot{M}_{\mathrm{Edd}}$ , and ultraluminous X-ray sources radiating $10$ – $100\times$ their spherical limit forced the field to take super-Eddington, beamed accretion seriously. At the largest scales, the billion-solar-mass quasars found at $z>7$ press the Salpeter clock so hard they are among the strongest arguments for heavy direct-collapse seeds — so the relation Eddington wrote for stars now constrains cosmic reionization.

The physical picture

Picture one electron and its proton floating in the radiation pouring out of a star. Photons scatter off the electron — the proton is far too massive to scatter appreciably — and each event transfers momentum outward, so the electron feels a steady outward force proportional to the photon flux. It cannot run off and leave the proton behind: the tiny charge separation sets up an electric field that drags the proton along, and the pair moves as a unit. The outward push acts through the electron's cross-section $\sigma_{\mathrm{T}}$ ; the inward gravity acts on the pair's mass, essentially $m_{\mathrm{p}}$ . The Eddington luminosity is where these two cancel.

Figure

A proton-electron parcel in the radiation field of a star: radiation scattering off the electron pushes the pair outward with acceleration κL/(4πr²c) while gravity pulls it inward with GM/r². Both scale as 1/r², so at twice the radius each force is one quarter as strong yet the balance is identical — the radius cancels and L_Edd = 4πGMc/κ depends only on M and κ

The cancellation is what turns a local statement into a single number: if the parcel is balanced at one radius it is balanced at every radius, because moving it out by a factor of two weakens both forces by the same factor of four. So one luminosity characterizes the whole configuration, and the identical formula governs a $100\,M_\odot$ star and gas spiralling onto a black hole a million times smaller. The electron-scattering form makes $L_{\mathrm{Edd}}$ a straight line in mass, about $3.3\times10^{4}\,L_\odot$ per solar mass; overplot the main-sequence luminosity, which climbs far more steeply as $L\propto M^{3.5}$ , and the two lines cross where stars begin to press against their own light.

Figure

The Eddington luminosity L_Edd = 3.3×10⁴ (M/M⊙) L⊙ rises linearly with mass (slope 1 in log-log) while the main-sequence luminosity L ≈ (M/M⊙)^3.5 L⊙ climbs far more steeply; the curves cross near 60–70 M⊙, above which a main-sequence star approaches its own Eddington limit and becomes radiation-driven and unstable

The core equations

The general form holds for any opacity source. Set the radiative acceleration equal to gravity on the same matter; since the flux is $L/4\pi r^2$ and the momentum deposited per unit mass is $\kappa/c$ times the flux, the radius cancels and a single critical luminosity survives.

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

where: $L_{\mathrm{Edd}}$ — Eddington (critical) luminosity; $G$ — gravitational constant; $M$ — mass interior to the layer; $c$ — speed of light; $\kappa$ — opacity (cross-section per unit mass), in $\mathrm{m^2\,kg^{-1}}$ . Larger opacity means matter couples more tightly to the radiation, so less luminosity is needed to unbind it — the limit drops. For the hot, fully ionized plasma of massive-star envelopes and accretion flows the opacity is Thomson scattering, $\kappa_{\mathrm{es}} = \sigma_{\mathrm{T}}/m_{\mathrm{p}}$ for pure hydrogen. Substituting gives the canonical Eddington luminosity, in which the electron cross-section and the proton mass appear together.

L_{\mathrm{Edd}} = \frac{4\pi G M m_{\mathrm{p}} c}{\sigma_{\mathrm{T}}} \qquad (2)

where: $m_{\mathrm{p}} = 1.673\times10^{-27}\ \mathrm{kg}$ — proton mass; $\sigma_{\mathrm{T}} = 6.652\times10^{-29}\ \mathrm{m^2}$ — Thomson cross-section. The pairing is physical: electrons do the scattering (through $\sigma_{\mathrm{T}}$ ) but protons carry the weight (through $m_{\mathrm{p}}$ ), and electrostatic coupling locks them together. Assembling the constants collapses everything into a benchmark worth memorizing: a fixed number of watts, or of solar luminosities, per solar mass.

L_{\mathrm{Edd}} \approx 1.26\times10^{31}\left(\frac{M}{M_\odot}\right)\ \mathrm{W} \approx 3.3\times10^{4}\left(\frac{M}{M_\odot}\right)L_\odot \qquad (3)

where: $M_\odot = 1.989\times10^{30}\ \mathrm{kg}$ — solar mass; $L_\odot = 3.828\times10^{26}\ \mathrm{W}$ — solar luminosity. The Sun runs about thirty thousand times below its own Eddington limit. Composition enters through the electron count: only free electrons scatter, and a mixture has fewer per unit mass than pure hydrogen because heavy elements carry roughly two nucleons per electron. The opacity therefore depends on the hydrogen mass fraction $X$ .

\kappa_{\mathrm{es}} = \frac{\sigma_{\mathrm{T}}}{m_{\mathrm{p}}}\,\frac{1+X}{2} = 0.02\,(1+X)\ \mathrm{m^2\,kg^{-1}} \qquad (4)

where: $X$ — hydrogen mass fraction. Pure hydrogen ( $X=1$ ) gives $\kappa_{\mathrm{es}} = 0.04\ \mathrm{m^2\,kg^{-1}}$ ; a solar mix ( $X\approx0.70$ ) gives $\approx0.034$ , raising the limit slightly. How close a system runs to its limit is measured by a single dimensionless ratio, the Eddington ratio.

\Gamma \equiv \frac{L}{L_{\mathrm{Edd}}} = \frac{\kappa L}{4\pi G M c} \qquad (5)

where: $\Gamma$ — Eddington ratio; $\Gamma<1$ is sub-Eddington, $\Gamma=1$ critical, $\Gamma>1$ super-Eddington. Because $\kappa$ can spike in specific layers, $\Gamma$ is a local quantity — a shell can reach $\Gamma=1$ while the star as a whole stays below its limit. Mechanically, radiation does not so much oppose gravity as dilute it: subtracting the radiative acceleration from the gravitational one leaves an effective gravity reduced by $(1-\Gamma)$ .

g_{\mathrm{eff}} = \frac{GM}{r^2}\,(1-\Gamma) \qquad (6)

where: $g_{\mathrm{eff}}$ — net inward acceleration the gas feels. As $\Gamma\to1$ the star is nearly weightless; for $\Gamma>1$ , $g_{\mathrm{eff}}<0$ and matter accelerates outward — no hydrostatic solution exists. The identical balance limits accretion. Gas releasing gravitational energy converts a fraction $\eta$ of its rest mass to light, $L=\eta\dot{M}c^2$ ; requiring $L\le L_{\mathrm{Edd}}$ caps the steady spherical accretion rate.

\dot{M}_{\mathrm{Edd}} = \frac{L_{\mathrm{Edd}}}{\eta c^2} = \frac{4\pi G M m_{\mathrm{p}}}{\eta\,\sigma_{\mathrm{T}} c} \qquad (7)

where: $\dot{M}_{\mathrm{Edd}}$ — maximum steady accretion rate; $\eta$ — radiative efficiency ( $\approx0.1$ for a black hole, up to $\approx0.4$ for maximal spin). Above this rate the flow's own radiation exceeds $L_{\mathrm{Edd}}$ and chokes the inflow. Since $\dot{M}_{\mathrm{Edd}}\propto M$ , a black hole accreting at its limit grows exponentially, $M\propto e^{t/t_{\mathrm{Edd}}}$ , with a mass-independent e-folding time — the Salpeter time.

t_{\mathrm{Edd}} = \frac{\eta\,\sigma_{\mathrm{T}} c}{4\pi G m_{\mathrm{p}}} \approx 4.5\times10^{7}\left(\frac{\eta}{0.1}\right)\ \mathrm{yr} \qquad (8)

where: $t_{\mathrm{Edd}}$ — Eddington (Salpeter) e-folding time. The characteristic time $\sigma_{\mathrm{T}} c/(4\pi G m_{\mathrm{p}}) \approx 4.5\times10^{8}\ \mathrm{yr}$ times $\eta$ gives $\approx45\ \mathrm{Myr}$ ; reaching $10^9\,M_\odot$ takes about twenty e-foldings.

Derivation AThe critical luminosity from hydrostatic equilibrium

The goal is to obtain $L_{\mathrm{Edd}} = 4\pi G M c/\kappa$ not from a heuristic balance on a floating particle but from the momentum equation of the gas itself, so the meaning is unambiguous: the Eddington luminosity is the largest luminosity for which a static pressure structure can still hold matter up against gravity. We start from hydrostatic equilibrium, isolate the radiative part of the pressure gradient, and ask when it alone would exceed gravity.

Step 1. Write hydrostatic equilibrium for a spherical star: the total pressure gradient supports the weight of the overlying gas, $\mathrm{d}P/\mathrm{d}r = -GM\rho/r^2$ , where $\rho$ is the local density and $M$ the enclosed mass. Why: this is the statement that in a static star every layer is held up by the pressure difference across it; if that balance fails, the layer accelerates.

Step 2. Identify the radiative part. Radiative transfer gives the force per unit volume from the flux $F=L/4\pi r^2$ as $\kappa\rho F/c$ , which is minus the radiation-pressure gradient: $\mathrm{d}P_{\mathrm{rad}}/\mathrm{d}r = -\kappa\rho L/(4\pi r^2 c)$ . Why: each unit mass intercepts momentum at rate $\kappa F/c$ ; multiplying by $\rho$ turns it into a force per volume, which is a pressure gradient.

Step 3. Impose the ceiling. Radiation cannot push a static layer outward harder than gravity holds it, so $\kappa\rho L/(4\pi r^2 c) \le GM\rho/r^2$ , with equality the maximum. Why: if the radiative gradient alone exceeded what balances gravity, gas pressure would have to be negative to compensate — impossible — so no static solution exists past equality.

Step 4. Cancel and solve. The density $\rho$ and $r^{-2}$ divide out on both sides, leaving $\kappa L/(4\pi c) = GM$ , hence $L_{\mathrm{Edd}} = 4\pi G M c/\kappa$ . Why: cancelling $\rho$ and $r^{-2}$ makes the answer a single luminosity rather than a shell-by-shell condition — the limit is global in form though enforced locally. *Numerically, $4\pi G c/\kappa_{\mathrm{es}} \times M_\odot = 1.26\times10^{31}\ \mathrm{W}$ with $\kappa_{\mathrm{es}}=0.0398\ \mathrm{m^2\,kg^{-1}}$ for pure hydrogen.*

Go deeper · Derivation B · 1 minFrom opacity to the canonical number

Derivation BFrom opacity to the canonical number

The goal is to turn $L_{\mathrm{Edd}} = 4\pi G M m_{\mathrm{p}} c/\sigma_{\mathrm{T}}$ into the benchmark $3.3\times10^{4}\,L_\odot$ per solar mass by first justifying the electron-scattering opacity and then inserting CODATA constants, so the famous number is traceable rather than quoted — and along the way it becomes clear why the proton mass enters a formula about electron scattering, and how composition shifts the answer.

Step 1. Count the scatterers. Only free electrons scatter through $\sigma_{\mathrm{T}}$ ; ionized hydrogen has one electron per proton, so $n_{\mathrm{e}}/\rho = 1/m_{\mathrm{p}}$ and $\kappa_{\mathrm{es}} = \sigma_{\mathrm{T}}\,n_{\mathrm{e}}/\rho = \sigma_{\mathrm{T}}/m_{\mathrm{p}}$ . Why: opacity is an absorption area per gram — literally the electron cross-section times how many electrons a gram of plasma holds.

Step 2. Handle a mixture. Heavy elements give about one electron per two nucleons, so $n_{\mathrm{e}}/\rho \approx (1+X)/(2m_{\mathrm{p}})$ and $\kappa_{\mathrm{es}} = 0.02(1+X)\ \mathrm{m^2\,kg^{-1}}$ . Why: swapping hydrogen for helium or metals halves the electrons per gram, so a helium-rich envelope's limit is roughly twice the pure-hydrogen value.

Step 3. Substitute into equation (1). With $\kappa_{\mathrm{es}} = \sigma_{\mathrm{T}}/m_{\mathrm{p}}$ , $L_{\mathrm{Edd}} = 4\pi G M c\,(m_{\mathrm{p}}/\sigma_{\mathrm{T}})$ — the proton mass migrates into the numerator though it played no part in the scattering. Why: gravity acts on mass, which lives in the protons; the electrons only set how tightly radiation grips, so both particle properties survive.

Step 4. Insert numbers ( $G=6.674\times10^{-11}$ , $c=2.998\times10^{8}$ , $m_{\mathrm{p}}=1.673\times10^{-27}$ , $\sigma_{\mathrm{T}}=6.652\times10^{-29}$ , SI): the coefficient $4\pi G c\,m_{\mathrm{p}}/\sigma_{\mathrm{T}} = 6.32\ \mathrm{W\,kg^{-1}}$ , times $M_\odot$ gives $1.26\times10^{31}\ \mathrm{W}$ , and over $L_\odot$ gives $3.3\times10^{4}$ . Why: reducing the formula to watts per kilogram makes the limit a one-line estimate for any object once its mass is known. *As a check: $6.32\ \mathrm{W\,kg^{-1}}\times1.989\times10^{30}\ \mathrm{kg} = 1.26\times10^{31}\ \mathrm{W}$ , and dividing by $L_\odot = 3.828\times10^{26}\ \mathrm{W}$ gives $3.29\times10^{4}\,L_\odot$ per $M_\odot$ — the standard $\sim3\times10^{4}\,L_\odot$ per $M_\odot$ benchmark.*

Go deeper · Derivation C · 1 minThe Eddington accretion rate and the Salpeter time

Derivation CThe Eddington accretion rate and the Salpeter time

The goal is to show that the same luminosity ceiling, applied to gas falling onto a compact object, produces both a maximum accretion rate and a fixed exponential growth clock for black holes — the Salpeter time — whose independence of black-hole mass is what turns it into a hard constraint on early quasars. We convert the luminosity limit into a mass-flow limit through the radiative efficiency, then integrate the growth law.

Step 1. Relate luminosity to accreted mass. A flow $\dot{M}$ onto a compact object of efficiency $\eta$ radiates $L=\eta\dot{M}c^2$ ; demanding $L\le L_{\mathrm{Edd}}$ gives $\dot{M}_{\mathrm{Edd}} = L_{\mathrm{Edd}}/\eta c^2 = 4\pi G M m_{\mathrm{p}}/(\eta\sigma_{\mathrm{T}} c)$ . Why: accretion luminosity and rate are the same physics seen two ways, tied by how efficiently gravitational energy becomes light before the gas disappears.

Step 2. Note $\dot{M}_{\mathrm{Edd}}\propto M$ : a larger hole accepts proportionally more gas, so $\dot{M} = M/t_{\mathrm{Edd}}$ with $t_{\mathrm{Edd}} = \eta\sigma_{\mathrm{T}} c/(4\pi G m_{\mathrm{p}})$ , constant in $M$ . Why: proportional growth is exponential growth — the hallmark of Eddington-limited accretion is that the hole doubles on a fixed timescale however big it already is.

Step 3. Integrate to $M(t) = M_0\,e^{t/t_{\mathrm{Edd}}}$ and evaluate: $\sigma_{\mathrm{T}} c/(4\pi G m_{\mathrm{p}}) = 4.5\times10^{8}\ \mathrm{yr}$ , so $t_{\mathrm{Edd}} = 4.5\times10^{7}\,(\eta/0.1)\ \mathrm{yr}$ ; the retained fraction $(1-\eta)$ lengthens it to $\approx5\times10^{7}\ \mathrm{yr}$ . Why: this ~45–50 Myr e-folding is the fastest a hole can grow at its limit — about twenty e-foldings, roughly $0.8$ Gyr, to reach a billion solar masses from a modest seed. *As a check: $\eta\sigma_{\mathrm{T}} c/(4\pi G m_{\mathrm{p}}) = 4.50\times10^{7}\ \mathrm{yr}$ for $\eta=0.1$ . To put the scale in perspective, twenty e-foldings ( $e^{20}\approx4.9\times10^{8}$ ) would grow a $10^{3}\,M_\odot$ seed to $\sim5\times10^{11}\,M_\odot$ ; a $10^{9}\,M_\odot$ hole therefore needs only $\sim14$ e-foldings, corresponding to $\sim0.65$ Gyr.*

Limiting cases

How the limit shifts as the opacity and the setting change:

Case	Result	Meaning
Electron scattering, hot ionized H	$L_{\mathrm{Edd}} = \dfrac{4\pi G M m_{\mathrm{p}} c}{\sigma_{\mathrm{T}}}$	Canonical form; $\kappa_{\mathrm{es}} \approx 0.04\ \mathrm{m^2\,kg^{-1}}$ (pure H) gives $3.3\times10^{4}\,L_\odot$ per $M_\odot$ .
General composition	$\kappa_{\mathrm{es}} = 0.02\,(1+X)\ \mathrm{m^2\,kg^{-1}}$	Solar $X\approx0.70$ : $\kappa_{\mathrm{es}}\approx0.034$ , raising $L_{\mathrm{Edd}}$ $\sim18\%$ ; helium-rich ( $X\to0$ ) doubles it.
Line-driven / opacity bumps	$L_{\mathrm{Edd}}^{\mathrm{eff}} = \dfrac{4\pi G M c}{\kappa_{\mathrm{tot}}} \ll \dfrac{4\pi G M c}{\kappa_{\mathrm{es}}}$	Iron-bump and He/H ionization make $\kappa_{\mathrm{tot}}\gg\kappa_{\mathrm{es}}$ , so specific layers go super-Eddington first.
Accretion onto a compact object	$\dot{M}_{\mathrm{Edd}} = \dfrac{L_{\mathrm{Edd}}}{\eta c^2}$	With $\eta\approx0.1$ the infall's light caps the steady inflow and fixes the Salpeter clock.
Super-Eddington, non-spherical	$\Gamma>1$ locally, funnels and outflows	Photon trapping and geometry let disc + jet systems exceed $L_{\mathrm{Edd}}$ globally (ULXs) yet stay stable.

Common trapThe Eddington luminosity is a hard, universal ceiling on how bright anything can be.

It is neither hard nor universal without qualification. First, it is a ceiling, not a typical value — the Sun runs at $\Gamma\approx3\times10^{-5}$ , and quoting $L_{\mathrm{Edd}}$ as an object's actual luminosity is simply wrong. Second, the famous $3\times10^{4}\,L_\odot$ per $M_\odot$ assumes pure electron scattering; opacity from lines, bound-free and free-free absorption, and ionization bumps can be orders of magnitude larger, so the effective limit that drives winds is much lower — envelopes reach $\Gamma=1$ locally while the global luminosity stays sub-Eddington. Third, it is a spherical, steady balance; break the symmetry — a disc with evacuated polar funnels — and the apparent luminosity can exceed $L_{\mathrm{Edd}}$ by large factors, as ultraluminous X-ray sources show. The Eddington limit is a benchmark and an instability threshold, not a law forbidding brighter light.

Worked examples

1 · How close to the edge is a massive O star (confidence builder)

A hot main-sequence O star has mass $M=30\,M_\odot$ and luminosity $L\approx1.5\times10^{5}\,L_\odot$ . Find its Eddington luminosity and Eddington ratio, and say what they imply.

Step 1. Eddington luminosity from the benchmark (3): $L_{\mathrm{Edd}} = 3.3\times10^{4}\times30\ L_\odot = \mathbf{9.9\times10^{5}\,L_\odot}$ , about $3.8\times10^{32}\ \mathrm{W}$ (pure electron scattering; a solar mix would raise it $\sim18\%$ ).

Step 2. Eddington ratio: $\Gamma = L/L_{\mathrm{Edd}} = 1.5\times10^{5}/9.9\times10^{5} = \mathbf{0.15}$ — about one-seventh of the continuum limit, sub-Eddington globally but nowhere near the Sun's margin.

Step 3. Interpretation: a global $\Gamma=0.15$ is already close enough that local opacity enhancements matter. Near the iron opacity bump at $T\sim2\times10^{5}\ \mathrm{K}$ the opacity climbs several-fold, pushing those layers to $\Gamma\gtrsim1$ and launching a radiation-driven wind — which is why O stars lose mass at $10^{-6}$ – $10^{-5}\,M_\odot\,\mathrm{yr^{-1}}$ despite a sub-limit average. Γ ≈ 0.15 globally, but the iron bump makes the envelope locally super-Eddington — hence the wind.

2 · Can a billion-solar-mass quasar exist at z ≈ 7

Quasars of $M\approx10^{9}\,M_\odot$ are observed only $\sim0.75$ Gyr after the Big Bang. Starting from a $M_0=100\,M_\odot$ stellar-collapse seed accreting continuously at the Eddington limit with $\eta=0.1$ , is there time to grow one?

Step 1. Number of e-foldings required: $N = \ln(M/M_0) = \ln(10^{9}/10^{2}) = \ln(10^{7}) = \mathbf{16.1}$ . Each e-folding takes the Salpeter time.

Step 2. Growth time with $t_{\mathrm{Edd}} = 4.5\times10^{7}\ \mathrm{yr}$ (equation 8, $\eta=0.1$ ): $t = N\,t_{\mathrm{Edd}} = 16.1\times4.5\times10^{7} = \mathbf{7.3\times10^{8}\ yr} \approx 0.73\ \mathrm{Gyr}$ ; the $(1-\eta)$ retained-mass factor stretches this to $\approx8\times10^{8}\ \mathrm{yr}$ .

Step 3. Compare with the clock: the universe is only $\sim0.75$ Gyr old at $z\approx7$ . The growth time essentially equals the age available — it works only if accretion stays at the Eddington limit for the entire time, from a fairly massive seed, uninterrupted. Continuous Eddington growth from a 100 M⊙ seed just barely reaches 10⁹ M⊙ in the available time — the tension that motivates heavier seeds (10³–10⁵ M⊙ direct-collapse holes) or super-Eddington episodes. A third setting closes the loop between the limit and observation: a type-I X-ray burst on a neutron star that reaches photospheric radius expansion is pinned at $L_{\mathrm{Edd}}$ , so with $L_{\mathrm{Edd}}$ fixed by fundamental constants and the well-constrained $\sim1.4\,M_\odot$ mass, the burst becomes a standard candle. Measuring its peak flux $F$ and inverting $F = L_{\mathrm{Edd}}/4\pi d^2$ yields the distance — a $6\times10^{-8}\ \mathrm{W\,m^{-2}}$ peak from a helium-photosphere ( $X=0$ , $L_{\mathrm{Edd}}\approx3.5\times10^{31}\ \mathrm{W}$ ) burster sits at $d\approx220\ \mathrm{pc}$ , with photospheric composition the dominant systematic.

Physics of the portrait

The limit carries the name of the man who, more than anyone, turned the interior of a star from speculation into calculation — and who first saw radiation pressure as a structural force rather than a curiosity.

Origins · 1 minArthur Eddington and how this was discovered

Arthur Eddington

1882–1944 · radiative equilibrium and the internal structure of stars

Arthur Stanley Eddington was the pre-eminent theoretical astrophysicist of the early twentieth century, and the architect of the idea that a star's interior is governed by the balance between gravity and radiation. In papers from 1916 onward, culminating in his 1926 masterwork The Internal Constitution of the Stars, he treated the interior as gas in radiative equilibrium, with the outward photon flux providing a real pressure that helps hold the star up. From this he derived the mass–luminosity relation and its sharp consequence: since radiation pressure grows with luminosity, there is a maximum beyond which radiation would overwhelm gravity and blow the star apart. That ceiling — set by the electron-scattering opacity he could already estimate — is the Eddington luminosity, and it explained why the most massive stars are so rare and restless. Eddington also led the 1919 solar-eclipse expedition that confirmed the bending of starlight predicted by general relativity. His limit outlived its original context entirely: when quasars and accreting black holes were discovered decades later, the same balance he wrote for stars became the natural yardstick for the brightest objects in the universe.

History

1916

— Eddington develops the theory of radiative equilibrium in stellar interiors

1921

— the balance of radiation pressure against gravity sets a ceiling on stellar luminosity

1926

— Eddington's The Internal Constitution of the Stars codifies the mass-luminosity relation and the limit

1964

— Salpeter and Zel'dovich apply the limit to accretion onto massive black holes

1973

— Shakura and Sunyaev build the Eddington rate into the standard thin accretion-disc model

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., §10.6 and §§15–16 · Kippenhahn, Weigert & Weiss, Stellar Structure and Evolution, 2nd ed., §5 · Frank, King & Raine, Accretion Power in Astrophysics, 3rd ed., Ch. 1 · Binney & Tremaine, Galactic Dynamics, 2nd ed. · Mo, van den Bosch & White, Galaxy Formation and Evolution, §14 (black-hole growth and the Salpeter time). The key numbers from CODATA: the coefficient $4\pi G c\,m_{\mathrm{p}}/\sigma_{\mathrm{T}} = 6.32\ \mathrm{W\,kg^{-1}}$ gives $L_{\mathrm{Edd}}(M_\odot) = 1.26\times10^{31}\ \mathrm{W} = 3.3\times10^{4}\,L_\odot$ ; $\kappa_{\mathrm{es}} = \sigma_{\mathrm{T}}/m_{\mathrm{p}} = 0.0398\ \mathrm{m^2\,kg^{-1}}$ (pure H), $0.02(1+X)$ for a mixture; the Salpeter time $t_{\mathrm{Edd}} = \eta\sigma_{\mathrm{T}} c/(4\pi G m_{\mathrm{p}}) = 4.5\times10^{7}\,(\eta/0.1)\ \mathrm{yr}$ — the bare characteristic time $\sigma_{\mathrm{T}} c/(4\pi G m_{\mathrm{p}}) = 4.5\times10^{8}\ \mathrm{yr}$ is multiplied by $\eta$ . Representative figures: $\Gamma\approx0.15$ for a $30\,M_\odot$ , $1.5\times10^{5}\,L_\odot$ star; $\sim16$ e-foldings ( $\sim0.73$ Gyr) to grow $10^{2}\to10^{9}\,M_\odot$ ; radiative efficiency $\eta\approx0.06$ – $0.42$ (Schwarzschild to maximal Kerr).

Prerequisites. Hydrostatic Equilibrium in Stars · Radiation Pressure in Stellar Interiors · Opacity and Radiative Transfer in Stars. Leads to. Radiative Temperature Gradient · Convective Energy Transport · Equations of Stellar Structure.

Ask the tutor: Ask me to work through the Eddington ratio of any star or the Salpeter growth time of any black-hole seed, and we will chase the numbers together.

Is the Eddington luminosity a hard limit that nothing can exceed?

No — it is a threshold for one idealized case: steady, spherically symmetric accretion or support with pure electron-scattering opacity. Break any of those and objects routinely shine brighter. Ultraluminous X-ray sources radiate ten to a hundred times their spherical limit because the accretion is disc-like: radiation escapes through evacuated polar funnels while matter falls in along the plane, and photon trapping in the dense flow advects energy inward rather than halting it. What $L_{\mathrm{Edd}}$ really marks is where the simplest static, spherical solution stops existing. Past it the system does not vanish — it finds a more violent one: a wind, an outflow, or a beamed super-Eddington disc.

Why does the proton mass appear when it is the electrons that scatter the light?

Because two particles play two roles. Radiation grips the matter by scattering off electrons, so the outward force is set by $\sigma_{\mathrm{T}}$ . But gravity acts on mass, and essentially all the mass of an ionized hydrogen plasma is in the protons — the electron is $1836$ times lighter. The two cannot separate: if radiation pushed the electrons ahead, the charge imbalance would set up an electric field that drags the protons along. So the plasma responds as a unit — the electron "feels" the light, the proton provides the "weight" — and $L_{\mathrm{Edd}}=4\pi G M m_{\mathrm{p}} c/\sigma_{\mathrm{T}}$ encodes exactly this division of labor.

How can a whole star stay below the limit while parts of it exceed it?

The Eddington ratio $\Gamma = \kappa L/(4\pi G M c)$ is local because $\kappa$ is local. The luminosity through a shell is roughly constant across the envelope, but $\kappa$ is not: near $2\times10^{5}\ \mathrm{K}$ the "iron bump" from partially ionized iron makes it spike several-fold, and the hydrogen and helium ionization zones do the same when cooler. There $\Gamma$ can reach one even though the global luminosity is a modest fraction of the electron-scattering $L_{\mathrm{Edd}}$ . Such a layer cannot be held up statically — it inflates, turns convective, and drives mass loss. This local super-Eddington physics is the modern explanation for the winds and eruptions of luminous blue variables.

Does the Eddington limit set the maximum mass of a star?

It contributes, but is not the whole story. On the main sequence luminosity rises steeply, roughly $L\propto M^{3.5}$ , while $L_{\mathrm{Edd}}\propto M$ rises only linearly, so the curves converge and a massive enough star approaches its own limit. Combined with opacity bumps that push envelopes locally super-Eddington, this makes very massive stars unstable to pulsations and radiation-driven mass loss, capping how massive a star can be and stay bound. Observationally the cap sits around $150$ – $300\,M_\odot$ , but the exact value also depends on how stars form and on metallicity. The Eddington limit sets the scale of the cap without fixing a single sharp number.

What is the Salpeter time and why does it constrain early quasars?

The Salpeter time $t_{\mathrm{Edd}} = \eta\sigma_{\mathrm{T}} c/(4\pi G m_{\mathrm{p}}) \approx 45\ \mathrm{Myr}$ (for $\eta=0.1$ ) is the e-folding time for a black hole growing at its Eddington rate. Because that rate is proportional to mass, growth is exponential with this fixed timescale, so building a hole a factor $f$ larger takes $\ln f$ Salpeter times regardless of the seed mass. From $100\,M_\odot$ to a $10^{9}\,M_\odot$ quasar is $\ln(10^{7})\approx16$ e-foldings, about $0.7$ – $0.8$ Gyr of uninterrupted feeding. Since we observe such quasars less than a billion years after the Big Bang, the timing is uncomfortably tight — motivating heavier seeds ( $10^{4}$ – $10^{5}\,M_\odot$ direct-collapse holes) or brief super-Eddington episodes.

Does composition or metallicity change the Eddington luminosity?

Yes, through the opacity. For pure electron scattering $\kappa_{\mathrm{es}}=0.02(1+X)\ \mathrm{m^2\,kg^{-1}}$ , so a hydrogen-rich gas ( $X=0.7$ ) has a slightly lower limit than helium-rich gas ( $X=0$ ), which has half the electrons per gram and hence twice the limit. The larger effect comes from bound-free, free-free, and line opacities, which grow strongly with metallicity: a metal-rich envelope has a much higher effective opacity than electron scattering alone, so its effective Eddington luminosity is lower and it drives stronger winds. This is why metal-rich massive stars lose mass more readily than metal-poor ones — which in turn leaves metal-poor stars heavier remnants, shaping the masses of the black holes they form.