A brightness that would blow the star apart

A star is held up by its own weight — a pressure gradient balancing gravity at every depth — and the pressure is usually taken to be the gas pressure. It is not the only one. Radiation carries momentum, and a flux of photons passing outward through matter exerts an outward force on it.

That force is proportional to the luminosity. The weight it has to overcome is proportional to the mass. So there is a luminosity at which the two are equal, and above it the outer layers of a star are not held.

Four lines of algebra, and nothing else in it #

The derivation is short enough to give completely, which is the point of the quantity.

A free electron in a radiation field of flux $F$ scatters photons and absorbs their momentum. The force on it is $\sigma_T F/c$, with $\sigma_T$ the Thomson cross-section. The gravitational force on the same electron is negligible, but the electron cannot leave on its own: pulling it away from the protons sets up an electrostatic field that drags them with it. So the radiation force on one electron is opposed by the gravitational force on one proton-plus-electron pair, essentially $GMm_p/r^2$.

Setting $F = L/4\pi r^2$ and equating:

$$\frac{\sigma_T}{c}\frac{L}{4\pi r^2} = \frac{GMm_p}{r^2} \quad\Longrightarrow\quad L_{\rm Edd} = \frac{4\pi GMc\,m_p}{\sigma_T} = \frac{4\pi GMc}{\kappa}.$$

The radius cancels. Both sides fall as $1/r^2$, so the balance is the same at every depth and the limit is a property of the object rather than of a surface. Numerically it comes to $3.8\times10^4$ solar luminosities per solar mass — so the Sun’s ceiling is about 38,000 times its actual output.

The only material property in it is the opacity $\kappa$, and the value used above is pure electron scattering in a hydrogen-rich gas, $0.034\ \mathrm{m^2\,kg^{-1}}$. That is the smallest opacity a stellar plasma can have, so the limit computed with it is the most generous possible one. Every other source of opacity — bound-free, bound-bound, the millions of iron lines that matter enormously in a hot star’s envelope — makes the real limit lower, and in the line-driven case by a large factor.

There is a case in which that understatement becomes enormous, and it decides which objects the limit binds at all. The derivation coupled the radiation to the electrons and the gravity to the protons, and what holds the two together is an electrostatic field. Dust grains are held to the gas by collisions instead, and a grain’s cross-section for absorbing starlight, per unit of its own mass, exceeds electron scattering per unit mass by two or three orders of magnitude. A dusty gas is therefore unbound at a luminosity a thousand times below the electron-scattering ceiling — provided the grains drag the gas along with them, which they do, for the same reason an electron cannot leave without its protons.

So one object has two Eddington limits, three decades apart, and which of them applies depends on the temperature. Inside the radius at which grains sublimate, near 1,500 kelvin, there is no dust and the limit is the one derived above; outside it the effective ceiling is far lower, and a modest luminosity drives an outflow. That boundary is why the winds of cool luminous giants are dust-driven and start at a particular radius, why an active nucleus can sweep gas out of its host while radiating at a per cent of its nominal limit, and why the same nucleus cannot clear the gas inside its own sublimation radius.

The general point survives the arithmetic: the limit is a statement about a ratio of opacity to gravity, and every number quoted for it is a statement about which absorber was assumed.

The high-mass main sequence is parallel to its own ceiling #

The hero figure’s most informative feature is one the drawing was not built to show, and it survived a check rather than being asserted.

The mass–luminosity relation is usually quoted as $L \propto M^{3.5}$, and it is, over the middle of the range. Above about 55 solar masses it flattens to $L \propto M$ — the same power as the Eddington limit. The two lines never cross on the plot; they run parallel, separated by a constant factor, at 83 per cent.

That is not a coincidence and it is not a fit. A star that massive is supported largely by radiation pressure rather than gas pressure, and a radiation-pressure-supported star is by construction near its Eddington limit — that is what the statement means. The mass–luminosity relation flattens because the star cannot be brighter than its own ceiling, so the ceiling’s slope becomes the relation’s slope. The relation stops being about nuclear physics and starts being about the limit.

The observational consequence is the upper end of the stellar mass function. Stars above about 150 solar masses are rare to the point of controversy; the record holders in R136, the cluster at the heart of the Tarantula Nebula, are quoted around 200 to 300, and every such claim is contested because the objects may be unresolved multiples. Whatever the exact number, the mechanism setting it is the one above: a more massive star is closer to the fraction of its limit at which its envelope is no longer bound.

What happens when it is exceeded #

A star that briefly exceeds its limit does not explode. It sheds.

Eta Carinae is the standing example. In the 1840s it brightened to become the second-brightest star in the sky, radiated for roughly two decades at a luminosity now estimated at several times its Eddington limit, and ejected between ten and twenty solar masses of material — which is visible today as the Homunculus Nebula. It survived. The event is called the Great Eruption, and no accepted model explains what triggered it.

The steady version of the same physics is the radiation-driven wind. An O star does not exceed its continuum Eddington limit, but the millions of metal lines in its ultraviolet spectrum are individually far more opaque than electron scattering, and a line at the right Doppler-shifted wavelength intercepts photons enormously more efficiently. The result is a wind driven off the surface at two or three thousand kilometres a second, carrying $10^{-6}$ to $10^{-5}$ solar masses a year.

That rate sounds negligible and is not. Over a three-million-year lifetime it removes tens of solar masses, so a massive star’s lifetime ends with a substantially smaller star than it began with, and the mass that arrives at the supernova depends on how much the wind took. Since the wind is line-driven it depends on metallicity, so the same initial mass leaves a different remnant in a metal-poor galaxy than in a metal-rich one — which is one of the reasons the black holes LIGO detects are heavier than the ones in the Milky Way.

The same limit, applied to something that is not a star #

An accreting object radiates because infalling matter releases gravitational energy, and the radiation pushes back on the matter that is still falling. So the limit becomes a limit on the accretion rate.

This is where it does its most consequential work.

The arithmetic on that figure is the origin of a real and unresolved problem. Growing from a 100-solar-mass seed to $10^9$ takes $\ln(10^7) = 16$ e-foldings, or about 800 million years at the Eddington rate with no interruptions — and the most distant quasars are seen at ages under 700 million years. The available time is not comfortably longer than the required time; it is shorter.

The proposed resolutions are all uncomfortable in their own way, and each is testable. Heavier seeds — the direct collapse of a $10^4$ to $10^5$ solar mass gas cloud without fragmenting into stars — remove four of the sixteen e-foldings. Lower radiative efficiency shortens the Salpeter time, at the price of a hole that spins slowly. Or the limit is genuinely exceeded for extended periods.

What is actually measured #

The Eddington limit is never measured directly. What is measured is the Eddington ratio, $\Gamma = L/L_{\rm Edd}$, and getting it requires two independent quantities.

The luminosity comes from the observed flux and a distance. The mass, for an active galactic nucleus, comes from the width of its broad emission lines combined with the size of the broad-line region — the size being measured by reverberation mapping, which times the delay between a change in the continuum and the response of the lines, and converts that delay into a light-travel distance. It is a mass obtained from a light-crossing time and a velocity, and it rests on the same virial argument that weighs a galaxy by its velocity dispersion.

The distribution that results peaks well below one — most accreting black holes at any time are radiating at a few per cent of their limit — with a tail approaching and occasionally exceeding it.

The limit is also a statement about the interior #

Everything above treats the Eddington limit as a condition at a star’s surface. The same inequality applies at every depth, and there it has a name of its own.

The ratio of radiation pressure to total pressure inside a star, usually written $1 - \beta$, is exactly the ratio of the local luminosity to the local Eddington luminosity. So a star radiating at a per cent of its ceiling is a star whose interior is supported at the per cent level by radiation, and a star at 83 per cent of its ceiling is supported almost entirely by it.

That matters because the two kinds of pressure behave differently under compression. Gas pressure goes as $\rho^{5/3}$ for an adiabatic monatomic gas; radiation pressure goes as $\rho^{4/3}$. A star supported by radiation is therefore a star whose effective adiabatic index is close to 4/3, and 4/3 is the threshold at which a self-gravitating sphere becomes marginally unstable — a small compression releases exactly as much energy as it costs.

So the same number that limits a star’s luminosity limits its stability, through a completely different argument. This is Eddington’s standard model, and it is the reason the upper mass limit is sometimes derived from pulsational instability rather than from the wind: both are the statement that $1-\beta$ has approached one. The degenerate case is the mirror image — a white dwarf approaching the Chandrasekhar mass also approaches an index of 4/3, from the other direction and for a quite different reason, and becomes unstable at the same threshold.

Where the limit is beaten, and how #

The derivation assumed spherical symmetry, and that assumption is the weakest thing in it.

Matter falling onto a compact object almost always has angular momentum and forms a disc. A disc is flat: the radiation escapes preferentially perpendicular to it, while the matter arrives along it. The photons and the infalling gas are then not in each other’s way, and accretion rates well above the spherical Eddington value become possible.

The observational proof arrived in 2014. Ultraluminous X-ray sources — off-nuclear point sources in nearby galaxies with apparent luminosities of $10^{39}$ to $10^{41}$ erg per second — had been assumed to be intermediate-mass black holes, on the grounds that nothing lighter could be that bright. Then NuSTAR found pulsations in M82 X-2, at 1.37 seconds. A pulsation means a rotating magnetised surface, which means a neutron star, which means a mass of about 1.4 solar masses and an Eddington luminosity a hundred times below what the object is emitting.

That single measurement converted the Eddington limit from a hard ceiling into a statement about a particular geometry. Several such pulsating sources are now known. The limit still holds for spherical accretion; almost nothing in the universe accretes spherically.

A ceiling used as a candle #

There is one setting in which the limit is reached so reliably that it serves as a measuring instrument rather than as a constraint.

A neutron star accreting hydrogen from a companion piles it up on the surface until the base of the layer ignites, and the burning is a runaway that consumes the accumulated fuel in seconds. These are Type I X-ray bursts, and the brightest of them reach the neutron star’s Eddington luminosity exactly rather than approximately — because a luminosity above it is not held, and the excess goes into lifting the photosphere instead of into the flux.

The signature is unmistakable, which is what makes the identification trustworthy. During such a burst the inferred emitting radius grows by a factor of ten or more while the colour temperature falls, and then the photosphere settles back onto the surface at a moment called touchdown, when the radius returns to the star’s own. Throughout the expansion the bolometric luminosity is very nearly constant, because it is pinned; at touchdown it is a known luminosity belonging to a known mass.

A known luminosity and a measured flux are a distance. The burst therefore supplies exactly the quantity that an absolute magnitude never is — a standard candle whose standard value comes from a piece of physics rather than from calibration against a lower rung — and distances to several dozen bursters in this galaxy are obtained this way.

The catch is the one the derivation has been carrying all along. The touchdown luminosity is $4\pi GMc/\kappa$, and $\kappa$ depends on the hydrogen fraction of the accreted layer, which is a property of the donor star rather than of the burst. A helium-rich donor gives an opacity smaller by nearly a factor of two, a ceiling larger by the same factor, and a distance wrong by its square root if the composition is guessed. So the cleanest product of these bursts is not the distance at all but the ratio of the neutron star’s mass to its radius, which the same measurement constrains and which no assumption about the donor touches.

What the picture cannot show #

A time dependence. Both figures are steady-state. A star near its limit is not steady — it pulses, it ejects shells, and the luminous blue variables change their apparent temperature by thousands of kelvin on decade timescales while their bolometric luminosity stays roughly constant.

The right opacity. Every number here uses electron scattering. In the iron opacity peak, at temperatures around 200,000 kelvin in a massive star’s envelope, the true opacity is several times higher, and the envelope is locally super-Eddington even when the star as a whole is not. That local inflation is the leading explanation for why massive stars have such extended, unstable outer layers, and none of it is in a figure that uses one opacity.

And the duty cycle. The growth curves assume continuous accretion. Real black holes accrete in episodes, and the fraction of the time a hole spends actively accreting — the duty cycle — is what converts an e-folding time into an actual growth history. It is estimated from the fraction of galaxies hosting an active nucleus, which is a few per cent, and inserting that honestly makes the early-quasar problem far worse than the figure suggests.

And the mass in the formula is the mass inside. For a star radiating from its surface the enclosed mass is the whole star, and the limit is unambiguous. For an accreting disc it is the compact object’s mass, and for a wind launched from a point well outside a stellar core it is the mass interior to that radius — which for a red supergiant’s extended envelope is not the same number as the star’s total mass. The cancellation of the radius that made this quantity so clean holds only where both forces are exerted on the same material by the same enclosed mass, and the interesting cases are the ones where they are not.

One more opacity closes the comparison the essay opened with.

Where the ladder goes next #

Two rungs. The first is the wind itself as a subject: how a line-driven outflow is computed, why the mass-loss rate scales with metallicity, and what that does to the mass function of black-hole remnants. The second is the super-Eddington regime taken seriously — advection-dominated flows, photon trapping, and the beamed geometries that let an object radiate a hundred times a limit derived on the assumption that it could not.