A star tipped over at a thousand times its horizon

A white dwarf near the Chandrasekhar mass and a neutron star are the two objects in which general relativity is expected to matter for stellar structure, and for an obvious reason: they are compact. The ratio $GM/Rc^2$, which measures how far a body is from being a black hole, is a few times $10^{-4}$ for a white dwarf and a few tenths for a neutron star. The Sun manages $2\times10^{-6}$, and the corrections relativity makes to its structure are smaller than the uncertainty in its opacity.

There is a third object for which relativity is decisive, and it is not compact at all. A star of a million solar masses, if one ever forms, becomes unstable while its radius is nearly a thousand times its Schwarzschild radius and its mean density is below that of the air at sea level. Nothing about it looks relativistic. The ratio $GM/Rc^2$ is about $5\times10^{-4}$ at that moment, comparable to a white dwarf’s, and in a white dwarf that number is a footnote.

The difference is not in the size of the correction. It is in the size of what the correction is compared with. A star held up almost entirely by radiation is balanced on a margin so thin that a correction of a few parts in ten thousand is enough to remove it, and the whole of this essay is the arithmetic of that margin: how thin it is, why, and where the thin margin and the small correction cross.

A star at 4/3 resists compression by nothing at all #

The standard argument for stability begins with a homologous squeeze. Take a star in hydrostatic equilibrium, shrink every radius inside it by the same small factor, and ask whether the pressure that results is more or less than what is needed to hold the compressed star up. If more, the star springs back. If less, the compression continues by itself.

The weight to be held up scales one way. The gravitational pull on a shell grows as the inverse square of its radius, and the area over which the pressure must support it shrinks as the square, so the pressure needed to hold the star up rises as $R^{-4}$. The pressure available scales another way. Compressing a gas adiabatically raises its pressure as the density to the power $\Gamma_1$, the adiabatic exponent, and density goes as $R^{-3}$, so the pressure rises as $R^{-3\Gamma_1}$. The two exponents are equal when $\Gamma_1 = 4/3$, and that is the threshold: above it the available pressure outruns the demand on compression and the star is stable; below it the star cannot stop.

The real criterion weights $\Gamma_1$ by where the pressure is, $\bar\Gamma_1 = \int \Gamma_1 P\,dV / \int P\,dV$, because a star is not one gas. But the threshold is the same number, and it is the number the virial theorem makes exact: at $\bar\Gamma_1 = 4/3$ a star’s total energy is zero whatever its radius, so it can expand or contract with no cost and nothing to restore it.

An ordinary monatomic gas has $\Gamma_1 = 5/3$, a third of a unit clear. Radiation has $\Gamma_1 = 4/3$ exactly, because its pressure is a third of its energy density and a photon gas compressed adiabatically keeps that ratio. A star whose pressure is part gas and part radiation sits in between, and how far in between is fixed by one number, the share β of the pressure that the gas carries.

The margin falls as the square root of the mass #

The adiabatic exponent of a gas–radiation mixture was worked out by Chandrasekhar in 1939, and it has a simple limit:

$$\Gamma_1 = \beta + \frac{(4-3\beta)^2(\gamma-1)}{\beta + 12(\gamma-1)(1-\beta)} \;\approx\; \frac43 + \frac{\beta}{6}, \qquad \gamma = \tfrac53,$$

the approximation holding when the gas carries only a small part of the pressure. The share of the pressure carried by gas is not free. If the star is an n = 3 polytrope with β the same throughout — Eddington’s standard model, which fixes the index from the ratio of radiation to gas rather than from any assumption about the run of density — then β is set by the mass alone, through Eddington’s quartic, $1-\beta = 0.003\,(M/M_\odot)^2\mu^4\beta^4$. For a heavy star β is small, the quartic reduces to $\beta \approx 4.2\,\mu^{-1}(M/M_\odot)^{-1/2}$, and the margin above 4/3 is a sixth of that.

The curve begins at the Sun with a margin of a third, which is a star that resists compression as strongly as any ordinary gas can. By a hundred solar masses it has fallen to about a tenth; radiation is now carrying more than half the pressure and the star is radiating at a large fraction of its Eddington limit, which is the same fact read off the surface rather than the interior. Beyond a few hundred solar masses the curve is a straight line on these axes, a slope of minus one half, and at a million solar masses the margin is $1.2\times10^{-3}$.

Nothing in the Newtonian theory makes that small number dangerous. The margin is positive however small it is, and a positive margin means a stable star. A Newtonian star of a hundred million solar masses would contract slowly on its radiative timescale, its margin unchanged, until something else happened to it. The margin does not shrink as the star shrinks, because β depends only on the mass. It is a property of the star, not of its state.

That is the first half of the argument, and the second half is that there is something which does shrink as the star shrinks: its distance from being a black hole.

Pressure has weight, and relativity charges for it #

In Newtonian gravity mass is the source of the field and pressure is not. In general relativity both are, along with the energy of everything in the star. The pressure that holds a star up therefore also adds to the gravity it is holding up against, and the internal energy of the gas adds to it too. The effect is small in any star that is far from compact, because the pressure’s contribution enters divided by $c^2$, but it has a definite sign and a definite scaling with radius.

Run the homologous squeeze again with that included. The demand for pressure now rises slightly faster than $R^{-4}$, because the extra gravity of the pressure itself grows as the star compresses: the correction to the binding energy goes as $(GM/Rc^2)$ times the Newtonian binding energy, so as $R^{-2}$ against the Newtonian $R^{-1}$. A star that was exactly neutral at $\bar\Gamma_1 = 4/3$ is now unstable at that value, and it needs a margin to stay up. Chandrasekhar worked out the size of that margin for a polytrope in 1964, and Feynman reached the same result independently at about the same time:

$$\bar\Gamma_1 - \frac43 \;<\; K\,\frac{2GM}{Rc^2} \quad\Longrightarrow\quad \text{unstable}, \qquad K = 1.1245 \text{ for } n = 3.$$

The relativistic demand is proportional to the compactness. At the Sun it is about $5\times10^{-6}$, against a margin of a third. At a white dwarf it is about $5\times10^{-4}$, against a margin that only becomes small as the electrons go relativistic and the star nears its limiting mass — which is why relativity shaves the Chandrasekhar mass of a carbon white dwarf by a fraction of a per cent rather than determining it. In both cases the correction is the kind of term that precesses a perihelion by forty-three arcseconds a century: real, measurable, and dwarfed by everything else.

For a supermassive star “everything else” is the margin of the previous section, and it is not large.

Where the margin and the demand cross #

Measured in units of its own Schwarzschild radius $R_s = 2GM/c^2$, the relativistic demand is $1.1245\,R_s/R$, the same diagonal line for every star whatever its mass. The margin, in the same units, is a horizontal line whose height is set by the mass. A contracting star moves leftward along its horizontal line, and goes unstable where that line meets the diagonal.

The crossing radius is just $K$ divided by the margin, and with the small-β form of the margin and the quartic’s form of β it reduces to a single expression:

$$\frac{R_\text{crit}}{R_s} = \frac{1.1245}{\bar\Gamma_1 - 4/3} \;\approx\; \frac{6.75}{\beta} \;\approx\; 1.6\,\mu\left(\frac{M}{M_\odot}\right)^{1/2}.$$

The heavier the star, the further out it goes. That runs against every intuition about relativity, in which more mass means more compactness and more mass means relativity matters more, and the inversion is the point. A heavier supermassive star is more radiation-dominated, so its margin is thinner, so a smaller relativistic correction suffices to remove it, and a smaller correction is reached at a larger radius. At a hundred million solar masses the star becomes unstable at nearly ten thousand times the radius of its own horizon, where the compactness $GM/Rc^2$ is about $5\times10^{-5}$ — a quarter of a white dwarf’s, where the same correction barely registers.

At the light end, the crossing comes in tight. A star of a thousand solar masses would have to shrink to thirty Schwarzschild radii, about ninety thousand kilometres or an eighth of the Sun’s radius, before relativity tipped it over, and nothing of a thousand solar masses ever gets there by contraction: it ignites hydrogen at a radius hundreds of times larger and settles down to burn. Relativity is not what limits the masses of ordinary massive stars. It becomes the limit only where the margin is thin enough for the crossing to come before anything else does.

A star that goes unstable at the density of the upper atmosphere #

Converting the crossing back into ordinary units gives the size at which a supermassive star goes, and the density it has when it does.

A star of a million solar masses becomes unstable at a radius of 19 AU — larger than the orbit of Uranus — and at that moment its mean density is $2\times10^{-5}$ grams per cubic centimetre, about a sixtieth of the air in a room. At ten million solar masses its radius is 600 AU and its mean density is seven parts in a billion of a gram per cubic centimetre, about the density of the atmosphere eighty-five kilometres up, near where meteors burn. The collapse that follows takes it inside its horizon in roughly a free-fall time, which for such a diffuse object is days to months.

The orange curve is steep because two effects compound. The critical radius grows as $M^{3/2}$, since it is a multiple $\propto M^{1/2}$ of a Schwarzschild radius that is itself $\propto M$, and the mean density is the mass over the cube of that radius: $M / M^{9/2} = M^{-7/2}$. Ten times the mass gives a star three thousand times thinner at the moment it goes, which is the drop between the last two orange points.

The same scaling fixes the central temperature at the critical point, which falls as the inverse of the mass. For an n = 3 polytrope the central temperature is $T_c \approx 0.85\,\beta\mu m_u GM/(kR)$, and with $GM/R$ fixed at the crossing by the margin, $T_c \propto \beta^2 \propto M^{-1}$: about two billion kelvin at ten thousand solar masses, two hundred million at a hundred thousand, twenty million at a million. The last is the temperature at the centre of the Sun. A star of a million solar masses reaches the radius at which it collapses with its centre no hotter than a star that is burning hydrogen slowly — and that sets up the race that decides whether such stars ever burn at all.

Burning loses the race above a few hundred thousand suns #

A contracting star stops contracting when nuclear burning at its centre supplies the energy it radiates. For a supermassive star that energy is the Eddington luminosity, since a radiation-dominated star radiates at very nearly its limit, and the fuel is hydrogen burned by the CNO cycle, the only hydrogen-burning chain fast enough to matter at these temperatures. Whether the star ever burns depends on whether its burning power reaches the Eddington luminosity before its radius reaches the critical one. Since the central temperature only rises as the star contracts, the burning power is greatest at the critical point, and the comparison need be made there alone.

Below about a hundred thousand solar masses the critical point is so hot that CNO burning, if it had waited that long, would exceed the Eddington luminosity by many orders of magnitude, and so it does not wait: the star ignites at a larger radius, stops contracting, and burns hydrogen as an enormous, short-lived, radiation-dominated main-sequence star. Above a few hundred thousand solar masses the critical point is cool, the burning there falls short of the Eddington luminosity by a margin that grows steeply with mass, and the star reaches its critical radius and collapses with its hydrogen untouched.

The crossings are surprisingly insensitive to composition. A hundred thousand times less carbon, nitrogen and oxygen lowers the limit by only a factor of about two. The reason is that at the temperatures in question the CNO cycle runs at its saturated rate wherever it runs at all: above roughly a hundred million kelvin each catalyst nucleus captures protons faster than the cycle’s two positron decays can process them, so the rate depends on how much catalyst there is and on nothing else, and is enormous even for a trace. What pushes the limit down is the steep fall of central temperature with mass, which drags the burning region below the saturation threshold. A star made of the primordial mixture, with no carbon at all until it makes some by burning helium, is at the extreme of this, and detailed models of such stars find the same qualitative answer — collapse without ignition above roughly a hundred thousand solar masses.

What this construction leaves out is as important as what it includes. It treats the whole star as one polytrope with β fixed, ignores the energy the burning would release in the moments before collapse, and asks only whether burning could balance the radiated luminosity. Calculations that follow the collapse itself find that in a narrow range of masses, and with enough catalyst, the burning ignited by the compression is explosive and can reverse the collapse. That possibility sits near the crossings on this figure, not far from them, and it is an argument about dynamics that a static criterion cannot make.

Ten thousand years from a cloud to the brink #

The last number is how long any of this takes. A supermassive star contracting towards its critical radius is powered only by the energy it releases by contracting, and for a star of mixed gas and radiation the total energy is not the Newtonian binding energy but a small part of it:

$$E = \frac{\beta}{2}\,\Omega = -\frac{3\beta}{4}\,\frac{GM^2}{R}.$$

The radiation’s energy almost exactly cancels the gravitational energy, because radiation is the gas with $\Gamma_1 = 4/3$, and a star made of it alone would have zero total energy. The fraction that is left, proportional to β, is all the star has to radiate, and it radiates it at the Eddington luminosity.

The energy to be lost at the critical radius scales as $\beta GM^2/R_\text{crit}$, which with $R_\text{crit} \propto M/\beta$ is $\propto \beta^2 M$. The Eddington luminosity is $\propto M$. The ratio, the contraction time, is $\propto \beta^2 \propto 1/M$, and it is short: ten thousand years at a hundred thousand solar masses, a thousand at a million. That is the Kelvin–Helmholtz time of a body that heats as it loses energy whose stored energy has been almost cancelled by its own light, and on any cosmological or even stellar timescale it is an instant. A supermassive star above the burning limit does not linger as a star. It assembles, contracts, and collapses, and the interval between the gas arriving and the hole forming is shorter than the time the gas took to arrive.

Why anyone needs a star that does this #

No supermassive star has been observed, and the argument above does not require one to exist. It became urgent for a reason it was not written for. Quasars are seen when the universe was well under a billion years old, powered by black holes of a billion solar masses and more. A black hole growing by accretion is limited by the Eddington rate to e-folding its mass in roughly fifty million years, and starting from the remnant of an ordinary massive star, a few tens of solar masses, there is barely time to reach a billion even at the limit throughout.

A seed of a hundred thousand or a million solar masses shortens the problem by a factor of ten thousand. And the supermassive star above the burning limit is the most direct way to make one: gas that collects fast enough in the centre of a young halo, and is kept from fragmenting into ordinary stars, would assemble into such a star and collapse to a black hole of nearly its own mass within ten thousand years, without ever burning. The instability that seemed like a pathology of an idealised polytrope is a route to the black holes at the centres of galaxies, and whether nature used it is one of the open questions of early galaxy formation.

The same arithmetic constrains that route. A seed formed this way must have been more massive than a few hundred thousand solar masses to avoid burning first, or it would spend millions of years as a star, shedding mass in a radiation-driven wind and reaching core collapse lighter than it began. And it must have been assembled in less than its contraction time, or the outer layers would still be arriving when the centre went.

What the polytrope assumes and what survives #

Three simplifications carry this argument, and each deserves stating.

The first is that the star is a single n = 3 polytrope with β constant throughout. A real radiation-dominated star is close to this — it is convective almost everywhere, and a convective, radiation-dominated gas does have nearly constant β — but not exactly, and the coefficient $K$ is specific to the index. The structure of the answer, a margin of order β against a demand proportional to compactness, does not depend on the details.

The second is that the star is not rotating. Rotation adds a support that does not depend on $\Gamma_1$, and a rotating supermassive star can be stabilised to considerably smaller radii than a static one. But it cannot be stabilised indefinitely, because the rotational support of a contracting star grows only until the star reaches the rate at which its equator sheds mass. A uniformly rotating supermassive star spun to that limit is found to go unstable at about 430 times $GM/c^2$ — some two hundred Schwarzschild radii, whatever its mass — which for a star of a million solar masses moves the crossing inward by a factor of about five without changing the shape of the problem.

The third is that the instability is judged on the star as a whole, by a pressure-weighted average of $\Gamma_1$. That is the correct criterion, and it is also the one that makes a radiation-dominated star so exposed: β is nearly the same everywhere, so there is no stiff region anywhere in the star that could compensate. A star in which only part of the gas went soft would be judged differently, by how much of its pressure lay in the soft part, which is exactly the situation the pair instability creates in the oxygen cores of much smaller stars.

Still open: whether nature ever built one #

The instability is secure; the objects are not. Every observable consequence of a supermassive star is indirect, and most of them are shared with other routes to the same black holes.

The cleanest test would be the collapse itself. A star of a million solar masses going inside its horizon in a matter of days would radiate a burst of gravitational waves at frequencies near a millihertz, which is the band a space-based interferometer is designed for, and the waveform of a collapsing radiation-dominated star differs from that of two black holes merging. Whether such a burst is loud enough to be detected from the redshifts at which these stars would have formed depends on how much of the collapse is asymmetric, which in turn depends on how fast the star was rotating — the same quantity that moves the critical radius.

The other test is statistical. If the first massive black holes formed this way, they began at a hundred thousand solar masses or more, and the population of black holes in small galaxies today should carry a floor near that mass rather than a continuous tail down to the stellar remnants. Black holes in dwarf galaxies are hard to weigh, and the handful measured so far neither establish such a floor nor exclude one. The question of whether the universe ever made a star that relativity toppled at a thousand times its horizon is, for now, a question about the bottom end of a mass function that has barely been sampled.