The brightness a mass cannot exceed

Light carries momentum and exerts a pressure, and for a dust grain the balance against gravity depends on the grain’s size. This essay is about the case where the size cancels too, and the answer is a property of the source alone.

Fig. 1 The Eddington luminosity against mass, with the main sequence beside it. The Sun sits at 2.6 × 10⁻⁵ of its own limit, a ten-solar-mass star at 1.2 × 10⁻², and a hundred-solar-mass star at 0.83 — which is why the two lines converge at the top and why stars of a few thousand solar masses do not exist.

The cancellation

Consider a parcel of ionised gas at radius $r$ from a source of luminosity $L$ and mass $M$ .

The radiation flux there is $L/4\pi r^2$ . It pushes on the free electrons by Thomson scattering, each electron presenting a cross-section $\sigma_T$ , so the outward force per unit mass of gas is $\kappa L/4\pi r^2 c$ with $\kappa = \sigma_T/m_p$ the opacity.

Gravity pulls inward with $GM/r^2$ per unit mass. Setting the two equal:

\frac{\kappa L}{4\pi r^2 c} = \frac{GM}{r^2} \quad\Longrightarrow\quad L_{\text{Edd}} = \frac{4\pi GMc}{\kappa}

The radius has gone. That is the whole of the argument and it is worth dwelling on: because both forces obey the same inverse-square law, their ratio is the same everywhere, and a source is either above the limit at every radius or below it at every radius.

The cancellation works because both forces fall the same way, and the reason is pure geometry: each is a fixed quantity spread over a sphere whose area grows as the square of the radius. Gravity spreads a flux of field lines and radiation spreads a flux of photons. The identical geometry is what makes the comparison between them independent of where it is made — the ratio of the two is the same at the surface of the star and at the edge of the galaxy, which is why the Eddington limit is a statement about a luminosity rather than about a distance.

Numerically, with the electron-scattering opacity for solar composition, $L_{\text{Edd}} = 1.47\times10^{31}$ watts per solar mass — about 38,000 times the Sun’s luminosity for each solar mass of material.

Two species, one force each

There is a subtlety in that derivation which is usually passed over and which decides the value of the constant.

The radiation pushes on electrons: Thomson scattering has a cross-section inversely proportional to the square of the particle’s mass, so a proton’s is a factor of $(m_p/m_e)^2 \approx 3.4\times10^6$ smaller and can be ignored entirely. Gravity pulls on protons, which carry essentially all the mass.

The blow-out band has two edges, not one. The ratio of radiation force to gravity against grain radius, with the radiation-pressure efficiency included: a grain much smaller than the wavelength of the light barely interacts with it — the efficiency falls as the fourth power of the size, which is Rayleigh's law — so the ratio stops rising and turns over. The dashed line is the same ratio with the efficiency taken as one, which is the usual drawing and is right only to the right of the turnover at 115 nm. The consequence is that a grain can be too small to be blown out as well as too large. Taking the threshold at a half — the value at which a grain released from a circular orbit is unbound — the band runs from 48.2 nm to 574 nm, and the largest ratio any grain of this material reaches is 1.87. Everything outside that band stays, and what stays does not stay put: it spirals. — Fig. 2 Which particle feels the push. A charge shaken by a passing wave re-radiates, and the momentum it removes from the beam is the pressure that beam exerts — but the rate depends on the charge’s acceleration, so it goes as one over the mass squared. The radiation therefore pushes on electrons and effectively not on protons at all, while gravity pulls almost entirely on the protons. The two species are held together by the electric field between them, which is why the limit applies to the plasma as a whole rather than to either component.

So the two forces act on different particles, and the reason the gas moves as one is electrostatic: any tendency for the electrons to be pushed out ahead of the protons separates charge, and the resulting field — over a screening length — drags the protons along. The Eddington limit is therefore a statement about a plasma held together by its own internal field, and it does not apply to neutral gas, where the opacity is a different and usually much larger number.

The number that carries all of this is a single cross-section. Thomson scattering has $\sigma = 6.65\times10^{-29}$ square metres, and it enters the limit through the opacity $\kappa = \sigma/m_p$ — one scattering cross-section divided by one particle mass. That is the whole of the microphysics: everything else in the Eddington limit is $G$ , $c$ and geometry.

What it does to stars

A star’s luminosity rises steeply with mass — roughly as $M^{3.5}$ on the upper main sequence — while its Eddington limit rises only linearly. The two lines therefore converge, and the convergence is fast.

At one solar mass the star is at a few parts in a hundred thousand of its limit. At ten solar masses it is at one per cent. At a hundred it is at 0.83, and the outer layers are being held down by a small residual difference between two large forces.

Three things follow, and all of them are observed. Massive stars have strong radiation-driven winds, losing a substantial fraction of their mass over their lifetimes. They are unstable: the luminous blue variables sit near their limits and undergo eruptions in which they shed solar masses at a time. And there is an upper limit to stellar mass in the region of a few hundred solar masses — not because heavier ones cannot be assembled in principle, but because the radiation of the assembling star drives away the gas that would have made it heavier.

The other end of the same story is the smallest mass that can collapse. Which disturbances grow rather than travel is decided by gravity against pressure; what can be assembled at all is decided by gravity against radiation. The two limits together bracket the range of stellar masses, from about a hundredth of a solar mass at the bottom to a few hundred at the top, and neither bound comes from anything about nuclear burning.

The number in the middle of a massive star

Inside a star the same ratio appears as a term in the structure equations rather than as a boundary condition, and it changes what the star is.

Hydrostatic equilibrium says the pressure gradient balances gravity. In a star near its Eddington limit, a large part of that pressure is radiation pressure, $aT^4/3$ , rather than gas pressure — and the two behave differently under compression. Gas pressure rises as the four-thirds power of density at constant entropy; radiation pressure rises as the four-thirds power too, but with the temperature tied to the density in a way that makes the effective adiabatic index approach 4/3 exactly.

An index of 4/3 is the boundary of stability. A star whose pressure is dominated by radiation is therefore only marginally bound, in the same sense and by the same arithmetic that makes the Chandrasekhar mass a limit: the energy of a small compression is neither restored nor released, and any additional effect decides the outcome.

A radiation-dominated star sits at a marginal place. The restoring force against a change of size has nearly cancelled — gravity pulling in, radiation pushing out, the two scaling together — so the star responds to small disturbances by pulsating, by shedding mass, or by both. That is not a failure of the model; it is what being near a limit looks like from inside, and the most massive stars known are observed doing exactly it.

That is why the upper end of the stellar mass range is populated by objects that are visibly unstable rather than by ordinary bright stars, and why the eruptions of luminous blue variables are usually discussed with the Eddington ratio in the first sentence.

What it does to black holes

For an accreting object the limit binds the other way. A black hole’s luminosity comes from the matter falling onto it, $L = \eta\dot{M}c^2$ with $\eta$ around 0.1, so a cap on the luminosity is a cap on the rate at which mass can be taken on.

How fast a black hole is allowed to grow. The mass of a black hole accreting at its own Eddington limit, against time, for four starting masses. Radiating a fraction 0.1 of the rest mass it swallows, the hole's mass grows exponentially with an e-folding time of 38.5 million years — the Salpeter time, which contains no mass at all, because both the luminosity cap and the mass supply scale together. Growth is therefore a matter of how many e-foldings are available rather than of how much gas there is. Reaching a billion solar masses takes 709 Myr from 10 M☉, 621 Myr from 100 M☉, 532 Myr from 1000 M☉, 355 Myr from 10⁵ M☉. That arithmetic is the whole of the early-quasar problem: billion-solar-mass black holes are observed less than 700 million years after the Big Bang, which allows about fifteen e-foldings, and a stellar-mass seed needs eighteen. Something has to give — a heavier seed, a period of accretion above the limit, or a lower radiative efficiency — and which of the three it is remains open. The limit is not a law of nature in the way the speed of light is: it assumes spherical accretion of ionised hydrogen, and both assumptions can be broken. — Fig. 3 The mass of a black hole accreting at its own limit, against time, for four seeds. The growth is exponential with an e-folding time of 38.6 million years — which contains no mass, because the cap and the supply scale together — so reaching a billion solar masses takes 709 Myr from a stellar-mass seed and 355 from a hundred-thousand-solar-mass one.

Because both the cap and the mass scale with $M$ , the growth is exponential and its e-folding time — the Salpeter time — is the same for every mass:

t_S = \frac{\eta\kappa c}{4\pi G} \approx 39\ \text{million years}

So growth is measured in e-foldings rather than in gas supply, and the arithmetic becomes a constraint on cosmology. Quasars of a billion solar masses are observed at redshift 7, less than 700 million years after the Big Bang — about fifteen e-foldings’ worth of time, against the eighteen a stellar-mass seed would need.

That gap is a real and open problem. The candidate resolutions are a heavier seed — direct collapse of a pristine gas cloud into a black hole of 10⁴ to 10⁵ solar masses — or a period of accretion faster than the limit, or a lower radiative efficiency during the growth phase. All three are being pursued, and the interest of the Eddington argument here is that it turns an observation of distant quasars into a statement about physics at much earlier times.

The efficiency in the growth time comes from a radius. Accretion luminosity is released from matter falling to a depth set by the hole’s size, so the fraction of rest mass converted is of order ten per cent rather than the nuclear 0.7 per cent — and that factor of fifteen appears directly in how fast a hole can grow, because a brighter process at fixed luminosity swallows less mass.

Where the limit is exceeded, and how

The derivation makes two assumptions that are stated rarely and broken often.

It assumes spherical symmetry. Matter arriving in a disc and radiation leaving along the poles are not in each other’s way, and such a flow can carry many times the Eddington rate while remaining below the limit in every direction that matters. Ultraluminous X-ray sources exceed the spherical limit for their inferred masses by factors of tens to hundreds, and geometry is the leading explanation.

It assumes the radiation escapes. In a sufficiently dense inflow the photons are trapped and carried inward with the gas rather than pushing outward against it — the flow is optically thick and the photon diffusion time exceeds the infall time. That regime is called hyper-Eddington accretion, and it is one of the routes by which an early black hole might grow fast enough.

And the opacity is not always electron scattering. For neutral or dusty gas the opacity is orders of magnitude larger, so the effective limit is orders of magnitude lower — which matters a great deal for how a forming star clears its envelope, and is why dust is the agent that stops accretion in many models of massive star formation.

Which grains the light wins. The radiation force on a spherical grain divided by the gravitational force on it, against the grain's radius, on logarithmic axes. Both forces fall as the inverse square of the distance, so the ratio does not depend on how far away the grain is — only on how big it is. Light acts on the cross-section and gravity on the volume, so the ratio goes as 1/a, and the two are equal at 287 nm for material of density 2000 kg/m³. Anything smaller than that is expelled; anything larger stays. — Fig. 4 The grain-sized version of the same comparison, where the ratio depends on the size of the grain because the cross-section and the mass scale differently. In the Eddington case the cross-section per unit mass is a property of the plasma and not of any particle’s size, which is precisely why a single number can be quoted for a whole star.

The wind, which is where the limit is actually met

A star below its Eddington limit for electron scattering can still be above it for particular wavelengths, and that is how a radiation-driven wind works.

The opacity in the derivation is the electron-scattering one, which is grey — the same at every wavelength. Real gas also has lines, and at the wavelength of a strong line the opacity is thousands of times larger. A parcel of gas sitting in the outer layers of a hot star is therefore above the local Eddington limit for the light in that narrow line, even while the total flux is comfortably below the grey limit.

That would remove very little momentum on its own, because the line absorbs a narrow slice of the spectrum and quickly saturates. What makes it work is the Doppler shift: as the parcel accelerates outward it shifts into fresh, unabsorbed continuum, so a single line keeps finding new photons to take momentum from. The wind accelerates itself by running away from its own shadow.

The wind is where the limit is actually met, and it is met from below rather than exceeded. A star approaching its Eddington luminosity does not explode; it drives a wind, shedding the outer layers that the radiation can no longer hold down. The limit therefore enforces itself continuously, by removing mass until the luminosity and the mass are back in the permitted ratio.

The resulting mass-loss rates for the most luminous stars reach 10⁻⁵ solar masses a year, which over a few million years is a substantial fraction of the star. So the Eddington limit does not merely cap what can exist: through the wind it actively removes what is near it, which is one reason the observed mass distribution falls off where it does.

When the photons cannot get out

The second escape route deserves a criterion, because “the radiation is trapped” is a statement with a radius attached.

Photons in an inflow do not travel freely; they scatter, and their outward progress is a random walk. The time to diffuse out from a radius $r$ therefore grows as the square of the optical depth rather than linearly, while the gas falling in takes a time set by the free-fall speed. Where the first exceeds the second, a photon produced at that radius is carried inward with the gas faster than it can climb out, and it never delivers its momentum to anything on the way.

Setting the two equal gives a trapping radius proportional to the accretion rate times the opacity over the speed of light. Below the Eddington rate that radius is inside the innermost stable orbit and nothing is trapped; well above it, the trapping radius moves out and a growing fraction of the radiation is swallowed rather than radiated.

Which changes the character of the limit entirely. Below it, radiation is the thing that stops accretion; above it, radiation stops escaping, so the feedback that was supposed to enforce the limit is disabled by the very conditions that violated it. The flow becomes optically thick and radiatively inefficient, its luminosity rises only logarithmically with the accretion rate, and there is no obvious ceiling on the rate at all.

That is the mechanism the fast-growth proposals for early black holes rest on, and it is testable in principle by what such a flow looks like from outside: much more mass arriving than the luminosity suggests, and an outflow driven from the trapping region rather than from the surface. Whether it happens at the rates required is a question about the gas supply rather than about the physics.

The other end of the scale

The limit is a ceiling and it is worth noticing that there is no corresponding floor, because the most studied black hole in the sky sits nine orders of magnitude below it.

The four-million-solar-mass black hole at the centre of the Galaxy has an Eddington luminosity of about $6\times10^{37}$ watts. It radiates something like $10^{29}$ — a ratio of order $10^{-9}$ , which is not a small departure from the limit but a total absence of relevance to it.

The gas supply is not the explanation. There is measurably enough material flowing toward it to power a luminosity a thousand times higher if that material fell in and radiated with the usual efficiency. What happens instead is that at very low accretion rates the inflowing gas becomes too tenuous to radiate efficiently: the ions and electrons stop exchanging energy with each other quickly enough, the ions get most of the heat and cannot radiate it, and the energy is carried across the horizon or blown out in a wind rather than emitted.

So the efficiency $\eta$ that appears in every growth calculation above is not a constant. It is around a tenth for a flow near the Eddington rate and can be $10^{-3}$ or lower for a flow far below it, which means the same accretion rate produces wildly different luminosities depending on the regime.

The consequence for the growth argument is worth stating, because it cuts the other way from the trapping one. A low efficiency means less radiation for a given inflow, which relaxes the feedback and lets mass accumulate faster — so the same physics that makes a starved black hole faint also makes it easier to feed. Both of the escapes from the limit in this section work by making the radiation less effective, and neither of them requires anything about gravity to change.

Eddington’s own use of it

The limit is now quoted mostly in connection with accreting black holes, which did not exist as objects of study when it was derived, and the original argument is worth recording.

Eddington was building the first quantitative theory of stellar structure in the years around 1920, and the term he needed was the contribution of radiation pressure to holding a star up. For the Sun it is negligible; for a star of tens of solar masses it is not, and the ratio of radiation pressure to total pressure became a parameter of his standard model.

The limit falls out of that as the case where the ratio reaches one, and Eddington’s interest in it was as an explanation for why the observed range of stellar masses is so narrow. Stars span a factor of ten million in luminosity and only a few hundred in mass, and the compression at the top of that range is the convergence this essay’s first figure draws.

It is a fair example of a result outliving its purpose. The stellar-structure question it was invented for is now a small part of a much larger computational subject; the ratio it produced turned out, half a century later, to be the organising number for accretion onto compact objects, a phenomenon nobody had conceived of. What made it portable is that the derivation contains no property of a star — only a mass, a luminosity and an opacity, and any object with those three has an Eddington limit whether or not it is a star.

Where else the same cancellation appears

In the Chandrasekhar limit, structurally. That limit exists because for a relativistic degenerate gas the pressure and gravity scale the same way with radius, so the radius cancels and what is left is a mass. The mass no cold matter can hold up and the brightness no mass can exceed are the same kind of argument with different quantities in it.

In the Toomre and Jeans criteria, where a comparison of two rates yields a length rather than a strength.

And in the classification of forces generally. Whenever two effects share a scaling, their ratio is a pure number and becomes a property of the system rather than of the measurement — which is why so much of astrophysics is a list of dimensionless ratios with names attached.

The same cancellation appears wherever two effects share a geometry. A quantity that scales with mass set against another that also scales with mass leaves a pure number, and pure numbers are where the physics is. The virial relation between a bound system’s kinetic and potential energy is the same manoeuvre in a different subject, and it is worth recognising the shape: when a comparison turns out to be independent of size, some geometry has cancelled.

How the limit is used as a measurement

A ratio that is hard to exceed is also a way of estimating things that cannot be measured directly, and this one is used that way constantly.

An active galactic nucleus has an observable luminosity and an unobservable mass. Assuming it radiates at some fraction of its Eddington limit converts one into the other, and the assumption is defensible because the distribution of measured Eddington ratios for such objects is fairly narrow — mostly between a hundredth and one. Black hole masses in distant quasars are quoted on that basis more often than on any other.

The same logic runs backwards for X-ray binaries: a source whose luminosity exceeds the Eddington limit of a neutron star is evidence that the accretor is a black hole, and for years that was one of the standard arguments for identifying one. The discovery of pulsating ultraluminous X-ray sources — neutron stars, unambiguously, radiating at hundreds of times their spherical limit — is what forced the geometry question into the open.

The limit is used as a measurement, which is the most practical thing about it. If a source is radiating at its Eddington luminosity then its brightness fixes its mass — so an observed luminosity becomes a lower bound on the mass of whatever is producing it, with no need to resolve anything. That is how the masses of the first quasars were estimated, and the answers were large enough to make how they grew so quickly an open question.

What the pictures cannot show

The opacity used is a single number. Electron scattering at solar composition gives 0.034 m² per kilogram, and the limit quoted elsewhere as 1.26 × 10³¹ watts per solar mass uses the pure-hydrogen value instead. Real opacities depend on temperature and composition and are tabulated rather than computed, and near the surface of a massive star they are dominated by iron lines rather than by electron scattering.

The main sequence drawn is an empirical fit. Its three-piece power law is a summary of measurements and not a derivation, and the top end is uncertain because very massive stars are rare and their masses are inferred rather than weighed.

The limit says nothing about time. A star above its Eddington limit does not explode; it drives a wind, loses mass, and moves down toward the limit. The dynamics of that adjustment is the subject and none of it is in the ratio.

And the growth calculation assumes continuous accretion. Real black holes accrete in episodes with long gaps, so the observed duty cycle enters, and every quoted growth time is a lower bound in a way the exponential curve does not show.

The ladder from here

Later rungs on this anchor: the Eddington limit for a non-spherical flow, and what “super-Eddington” actually means in a disc; photon trapping and the hyper-Eddington regime; the Eddington factor as a term in stellar structure rather than as a boundary condition; and radiation-driven winds, where the opacity is line-dominated and the limit becomes wavelength-dependent.

The neighbouring ladders are light has a pressure, which is the quantity being compared with gravity, the size the light cannot blow away, which is the same comparison where the radius does not cancel, and the mass no cold matter can hold up, which is the same shape of argument with degeneracy pressure in place of radiation.