A soft centre the rest of the core outvotes

The usual account of the pair instability is a single sentence about a single place. When the centre of a very massive star passes about a billion kelvin, the photons there become energetic enough to turn into electron–positron pairs; making the pairs takes energy that was holding the star up; the core loses its support, collapses, and ignites its oxygen explosively. It is a correct account of what happens at the centre, and it leaves out the question that decides whether anything happens to the star.

Stability is not a property of a point. A star is unstable when a small compression of the whole body releases more energy than it costs, and that is a sum over every shell in it, weighted by how much of the pressure each one carries. A core can have gas at its centre that, taken alone, would not hold itself up, and still be perfectly stable, because the gas around the centre is stiff and there is more of it. The real threshold for the pair instability is where the soft gas wins that vote, and it lies at a core mass half again as large as the one at which the centre first goes soft.

A supermassive star is the case in which every shell has the same thin margin, so there is nothing to vote. A core that cannot hold up its envelope is a balance across a boundary. This one is a balance across the whole interior, and the argument is the arithmetic of that vote.

A gas that stops pushing back #

Compress an ordinary gas and the work done on it goes into the random motion of its particles: the temperature rises, the pressure rises with it, and for a monatomic gas the pressure goes up as the density to the power 5/3. That exponent, the adiabatic exponent $\Gamma_1$, measures a gas’s ability to push back. For a self-gravitating sphere the number to beat is 4/3. Squeezed homologously, the pressure a star needs to hold itself up rises as the radius to the power −4, and the pressure the gas supplies rises as the radius to the power $-3\Gamma_1$; the virial theorem makes the same statement about energy, and the two agree that 4/3 is the line.

A gas that is mostly radiation sits close to that line already, because radiation itself has $\Gamma_1 = 4/3$ exactly. That is the situation in the core of a very massive star after helium burning: a ball of carbon and oxygen, tens of solar masses, carrying more than half its pressure in light, with a margin over 4/3 of a few hundredths and falling as the mass rises.

Pairs take part of that margin away, and the thermodynamics of how they do it is a subject of its own; here it is an input. Above about a billion kelvin, light in equilibrium with the gas makes electron–positron pairs, each locking more than a million electronvolts into rest mass where it exerts no pressure, so compression goes partly into making mass instead of heat. Between roughly one and three billion kelvin, and at densities below about a million grams per cubic centimetre, where the electrons are not yet degenerate enough to block the states a new pair would need, $\Gamma_1$ falls below 4/3. Above that temperature the pairs are relativistic and push like more radiation, and $\Gamma_1$ climbs back. The figures here compute it from the full equation of state of ions, radiation and equilibrium pairs at every point, with no limiting form assumed.

What belongs to stellar structure is which gas in a core is in that range, and when. For an n = 3 star the central temperature and density are tied by $T_c \propto \beta\mu M^{2/3}\rho_c^{1/3}$, with β the gas share of the pressure from Eddington’s quartic, here at the mean molecular weight of ionised oxygen, 16/9. A contracting core’s centre climbs a line of slope three in the plane of log density and log temperature, and a heavier core’s line lies further towards low density, hotter at every density. A light core’s centre reaches a billion kelvin already too dense to make pairs and passes the soft region by; a heavy core’s centre goes through it. Away from the centre the gas is cooler and thinner by fixed ratios, $T = T_c\,\theta$ and $\rho = \rho_c\,\theta^3$ with θ the Lane–Emden function, so the whole core sweeps through the soft region as one, its centre leading.

The weight of the centre is small #

The criterion that decides whether a star is dynamically stable is an integral over the whole body:

$$\bar\Gamma_1 = \frac{\int \Gamma_1\,P\,dV}{\int P\,dV}, \qquad \text{unstable if } \bar\Gamma_1 < \tfrac43,$$

with a small relativistic correction that is negligible in these cores. Each shell’s vote is its pressure times its volume. The pressure is greatest at the centre, but the volume there is tiny, and the product $P\,4\pi r^2$ peaks well out from the middle. In an n = 3 polytrope it peaks at about a fifth of the radius and has fallen to half its peak by a third. The centre itself carries almost no weight at all.

This core is caught near its most dangerous moment, and the picture is not the one the single-sentence account suggests. The very centre has already been heated through the soft region and come out the other side; the soft gas is a shell, and it sits right on the peak of the weighting curve, so it is about as influential as soft gas can be. Two-thirds of the pressure-weighted volume is soft. And the core is stable, because beyond a third of the radius the gas is cooler, out of the pair-producing range, and stiffer by more than the soft gas is soft. The red area is deep but narrow; the green area is shallow but long. The integral of the two comes out positive by three thousandths, a margin as thin as a supermassive star’s, but a margin.

Which means that whether a core collapses cannot be read off its centre, or even off the size of its soft region. It has to be computed, at each moment of the contraction, as the balance of those two areas.

The vote, taken as the core heats #

A contracting core does not choose a temperature. It passes through all of them in turn, and the question is whether the weighted margin ever crosses zero on the way.

Every curve has the same shape. Below a billion kelvin the margin is set by the mixture of gas and radiation and changes slowly. Then the centre, followed by more and more of the core, enters the soft region, and the margin falls. It bottoms out between two and three billion kelvin at the centre, when the soft shell sits across the peak of the weighting, and recovers as the inner gas passes through the ceiling and the soft shell moves outward into regions that carry less weight.

The depth of the dip increases with mass for two reasons that work together. A heavier core starts closer to 4/3, because more of its pressure is radiation; and it is hotter at a given density, so more of it is inside the soft region at once. At fifteen solar masses of oxygen the margin never falls below about two hundredths. At twenty-five it falls to three thousandths and recovers. At forty it crosses zero and reaches minus one hundredth, and at eighty minus three hundredths.

A core whose margin stays positive contracts smoothly through the pair region and goes on to burn its oxygen in the ordinary way, quietly and in hydrostatic balance. A core whose margin crosses zero does not wait to reach the minimum. Once the weighted margin is negative a small compression feeds on itself, the core collapses on a free-fall time, and the collapse heats the oxygen until it ignites all at once. What happens after that depends on how much energy the explosion releases against how tightly the star is bound, which is a question about the burning rather than the instability.

Two thresholds, and the gap between them #

Reading the deepest point of each curve off as a function of mass gives the threshold, and doing the same for the centre alone gives the local criterion it is usually confused with.

The two curves cross zero at different masses, and the interval between them is the territory the single-sentence account gets wrong. A core of twenty-two solar masses of oxygen has gas at its centre that cannot hold itself up, by any local measure, and it does not collapse. The stiffer gas around it carries it through.

The whole-core threshold, about 27.5 solar masses in this construction, is a number with two kinds of uncertainty in it. One is the model: the core is an n = 3 polytrope of pure oxygen whose structure ignores the pairs, and a real core is a mixture of carbon and oxygen with a structure that the pairs do modify, in a direction that makes it slightly more unstable. The other is what the number is compared with. Evolutionary calculations that follow real stars find the first instability in stars whose helium cores exceed roughly thirty solar masses, which corresponds to somewhat lighter carbon–oxygen cores inside them. This construction lands in that neighbourhood rather than on a precise value, which is the most an n = 3 polytrope should be asked to do. What it does get exactly right is the structure of the answer: a local criterion that fails at one mass and a global criterion that fails at a larger one.

A majority is not enough #

The cleanest way to see what the global criterion demands is to ask how much of the core has to be soft before it gives way.

Below nineteen solar masses no gas in the core ever goes soft, because the centre’s track passes on the dense side of the soft region. Above that the soft share rises steeply, crosses a half at about twenty-three and a half solar masses, and at the threshold stands at two-thirds. For four solar masses of oxygen the soft gas is the majority of the vote and the core still holds.

The reason is that the soft gas is only slightly soft. At the threshold it falls short of 4/3 by about nine thousandths on average, weighted by pressure; the stiff gas around it exceeds 4/3 by about seventeen thousandths. The soft gas therefore needs nearly twice the weight of the stiff gas to cancel it, which is two-thirds of the total. That ratio is not a coincidence of these numbers. It is set by how far below 4/3 the pair-producing gas can go, which is never far, because pairs soften the gas only as long as they are a minor component and it stiffens again once they dominate; and by how far above 4/3 the rest of the core sits, which is set by its gas share β. The green curve is falling across the whole range, and that is the other half of why heavier cores go: as β falls, the stiff gas becomes less stiff, and its votes count for less.

Neutrinos hurry the core through and do not change the count #

A core near a billion kelvin is not contracting slowly. The same pairs that soften it also annihilate, and a small fraction of annihilations make a neutrino and an antineutrino instead of two photons. Those escape the star directly, and at these temperatures they carry off energy far faster than the surface radiates it: an oxygen core loses more to neutrinos than the whole star gives out as light, which is why the late burning stages last months and days rather than millions of years. The core is driven through the pair region on a timescale of years or less.

That changes how long the core spends near its most dangerous moment. It does not change whether the moment is dangerous. The vote above is about a dynamical instability, judged against the free-fall time, which for these cores is minutes; a neutrino loss that takes years to change the temperature appreciably is frozen on that timescale. $\Gamma_1$ is defined for adiabatic compression, the stability integral is evaluated with it, and the answer is the same whether the core arrived at a given state quickly or slowly. The losses decide the order in which the states are visited and nothing about each one.

The instability was found that way round. Fowler and Hoyle noticed in 1964 that pairs would lower $\Gamma_1$ in the cores of very massive stars, and in 1967 Rakavy and Shaviv, and separately Barkat, Rakavy and Sack, followed oxygen cores through the pair region with the stability integral rather than the central value, and found that the heavier ones collapsed and exploded. The local and global criteria were distinguished from the first calculation that took the question seriously; it is the retelling that has merged them.

Relativity takes a share of the vote #

The supermassive star is toppled by a term this essay has so far left out: in general relativity pressure gravitates, and a star needs a margin over 4/3 of $1.1245 \times 2GM/(Rc^2)$ at n = 3 just to stay where a Newtonian star would be neutral. In an ordinary star that term is a few millionths and nobody computes it. In an oxygen core at its most dangerous moment it is not so small. The core is then tens of solar masses inside a radius of under a hundred thousand kilometres, and its compactness $GM/Rc^2$ is around $5\times10^{-4}$ — the compactness of a white dwarf near its limiting mass.

The demand comes out at about a thousandth, nearly flat across the masses that matter, because a heavier core reaches its dangerous temperature at a proportionally larger radius. Against margins of a few hundredths it changes little, and against the margin near the threshold it changes the answer by about a solar mass: the purple curve crosses zero at 26.6 rather than 27.5. Measured against the pull the soft gas exerts on the vote — its pressure-weighted share times how far it falls below 4/3 — the relativistic term is a little under a fifth.

That is the right size for a correction and the wrong size for a cause. Nothing about the pair instability needs relativity, and every published threshold is computed without it to an accuracy that the composition of the core, not gravity, limits. But it is a reminder of what the supermassive star makes explicit. A criterion of the form “margin over 4/3 against something small” is only as safe as the margin is large, and in these cores the margin at the threshold is, by definition, zero, so every small term that adds to the demand moves the threshold in the same direction. The vote is close, and close votes are where the minor parties count.

What the collapse leads to #

A core whose weighted margin crosses zero collapses on a free-fall time, which for these cores is minutes, and the compression ignites its oxygen all at once. What follows is decided by the burning, not the instability: whether the energy released exceeds the core’s binding, and so whether the core pulses and settles, or is destroyed, or falls inward past the point where burning could stop it. The ranges of mass that lead to each are the subject of the thermonuclear calculations and of the black holes that are, and are not, found afterwards.

What the stability argument contributes to that story is where it starts. A threshold read off the centre would begin the violent outcomes a third lower in core mass than the integral does, and every consequence downstream — the masses at which pulses begin, the heaviest black hole a single star can leave — inherits the difference.

Still open: whether a real core counts the votes the same way #

Every number here comes from a core built as an n = 3 polytrope of pure oxygen, with the pairs ignored in its structure and included in its stability. Each of those choices moves the weighting, and the weighting is the whole of the argument.

A real core is not one polytrope. It is a carbon–oxygen ball with a composition gradient, burning shells at its edge, and a convective region whose extent changes as it contracts. The weight $P\,r^2$ peaks where the polytrope puts it, at about a fifth of the radius, only if the run of density is the polytrope’s; a core more centrally condensed than n = 3 moves the peak inward, onto the gas that goes soft first, and a less condensed one moves it out. The pairs themselves soften the core, which makes it more condensed, so including them in the structure should lower the threshold — by how much, an integral over a single model cannot say.

And the soft share at the threshold, two-thirds in this construction, is set by a ratio of two averages: how far below 4/3 the soft gas falls, and how far above it the stiff gas stands. Whether that ratio is a near-constant of the problem, so that “two-thirds of the pressure-weighted volume” is a rule a real core obeys, or an accident of this one equation of state and this one structure, is something only the full hydrodynamic calculations can test, and they report thresholds rather than votes. Reading the vote out of them — the soft share at the moment each modelled core begins to collapse — would turn a number from a polytrope into a measured property of massive stars.