The heat that turns into matter

The share that is not half a kT ends with a warning about its own central figure. It traces a gas getting hot enough for its particles to become relativistic, so that their energy grows as their momentum rather than its square and the ratio of specific heats falls from five thirds to four thirds. But the drawing is for a gas of fixed particle number, and it notes that a gas hot enough for its electrons to be relativistic is hot enough to make electron–positron pairs, after which “the particle count is a function of temperature”. That is where this essay starts.

Every thermodynamics that does not include relativity treats the number of particles as a given. Heating a gas adds energy to the particles that are there. Relativity makes that an assumption rather than a law: energy and mass are the same quantity, a mass is an energy already, and nothing forbids a gas from spending some of its heat on making new particles, provided it makes them in combinations that conserve electric charge. An electron and a positron together cost 1.022 million electronvolts and carry no charge. At temperatures where the radiation in a gas carries photons of that energy in quantity, the gas will make them, and the number of particles becomes a thermodynamic variable — settled, like everything else in equilibrium, by what maximises the entropy.

How many pairs a hot gas makes #

The rule that fixes the number is the one the reaction that cannot go all the way uses for any chemical reaction: at equilibrium the chemical potentials on the two sides balance. The reaction here is light becoming a pair, $\gamma + \gamma \rightleftharpoons e^- + e^+$, and light’s chemical potential is zero, because the photons in a hot gas are not counted — walls and matter make and destroy them freely. So the electrons’ chemical potential and the positrons’ must add to zero. The second condition is charge. A gas of oxygen nuclei brings its own electrons, eight per nucleus, and the electrons must always outnumber the positrons by exactly that many.

The two conditions together fix everything, and the drawing solves them with the full relativistic Fermi–Dirac distribution for each species. The striking feature is how cool the gas can be when pairs appear. An electron’s rest energy corresponds to six billion kelvin, and yet at one billion — where the typical thermal energy is only a sixth of the cost of a pair — a thin gas already holds more positrons than it has electrons of its own. The reason is entropy. A pair has an enormous number of momentum states available to it, and although each one is suppressed by the Boltzmann factor for its rest energy, the sum over all of them wins. It is the same arithmetic that ionises hydrogen at temperatures far below what its binding energy suggests.

Density works against it. The electrons the oxygen supplies fill the lowest states, raising the electrons’ chemical potential, and since the positrons’ must be its negative, their numbers are pushed down. At a million grams per cubic centimetre there are two positrons per million electrons at a billion kelvin; at a thousand, more positrons than electrons. Pairs are a phenomenon of gas that is hot and thin.

Heat that goes into mass #

The pairs have to be paid for, and the drawing itemises the bill. Heat a gas that is making pairs and part of the heat goes into their rest mass. That part raises the gas’s energy without raising its temperature and without raising its pressure — rest mass exerts no pressure. Near two billion kelvin, more than a quarter of any heat added to a thin oxygen gas becomes matter.

That is a new term in the heat capacity, and it is large. Half a kT for every way of moving counts a gas’s heat capacity as the number of ways its particles can hold energy; a pair-producing gas has one more way, the creation of particles themselves, and near the threshold it is one of the most important. A gas whose heat capacity has a large component that does nothing for its pressure is a soft gas: squeeze it, and its temperature rises, but much of the work goes into making more pairs rather than into pushing back.

The stiffness that falls below four-thirds #

The stiffness that matters for a star is the adiabatic exponent $\Gamma_1$, the logarithmic slope of pressure against density when a gas is compressed too quickly for heat to escape. For an ordinary gas of atoms it is 5/3. For light alone it is 4/3. A star’s core is a mixture, and before pairs appear its exponent lies between the two — nearer 4/3 the more of its pressure is radiation, which in a very massive star is most of it.

Four thirds is the number on which a self-gravitating body’s stability turns. Squeeze a star uniformly and its gravity grows as the inverse of its radius, while its pressure, compressed adiabatically, grows as the density to the power $\Gamma_1$, which is the radius to the power $-3\Gamma_1$. The pressure force on a shell grows as $r^{2-3\Gamma_1}$ against gravity’s $r^{-2}$; the pressure wins on compression, and pushes the star back out, only if $\Gamma_1 > 4/3$. The mass no cold matter can hold up meets the same number from a different direction: a white dwarf whose electrons have become relativistic has an exponent of exactly 4/3, and at that point its radius has no preferred value and its mass has a ceiling.

The drawing shows pair production pulling a hot, thin core’s exponent below the line. As the temperature approaches a billion kelvin, pairs begin to form and every compression spends some of its work on making more of them, so the pressure rises less steeply than it would; the exponent drops. Well above the threshold, the pairs are themselves relativistic particles behaving much like radiation, and the exponent returns towards 4/3 from above. Between, for a range of temperature of about a factor of two, a thin enough core cannot hold itself up.

Why four-thirds is where the binding goes #

The same number appears in a star’s energy budget, and there it explains why a core below 4/3 does not merely wobble but runs away. The virial theorem, which weighs what cannot be put on a scale, ties a star’s internal energy to its gravitational energy. For a star whose gas has a single adiabatic exponent $\Gamma$, the total energy — internal plus gravitational — comes out as

$$E = \frac{3\Gamma - 4}{3(\Gamma - 1)}\, W,$$

where $W$ is the gravitational energy, which is negative. For an ordinary gas, with $\Gamma = 5/3$, the total is half the gravitational energy, and the star is firmly bound: it must lose energy to contract, and a ball of gas that loses energy heats up as it does so, which is the self-regulation that lets stars burn steadily for billions of years.

As $\Gamma$ falls towards 4/3 the factor in front falls to zero. The star’s total energy approaches zero — it is barely bound — and a small disturbance costs almost nothing. Below 4/3 the factor changes sign, and contraction releases energy rather than requiring it to be removed; there is then no restoring tendency at all, and a core that begins to shrink keeps shrinking, faster and faster, until something else — here, the ignition of its oxygen — intervenes. A pair-unstable core is not a star with a weak spot. It is a star whose binding energy has been spent on making electrons and positrons.

Where a core cannot stand #

The unstable region is a band in the plane of density and temperature, open towards low density and closed towards high. A star’s centre moves across this plane over its life, and for a star whose pressure is mostly radiation it moves roughly along lines of constant $T^3/\rho$ — lines of constant entropy per particle, because radiation’s entropy goes as the cube of the temperature. More massive stars have higher entropy and lie on lines further towards low density and high temperature. Stars of ordinary mass pass well to the right of the band, their cores too dense for pairs to matter. The most massive stars’ lines pass through it.

What happens to them was worked out in the 1960s by Gideon Rakavy, Giora Shaviv, Zalman Barkat and others, and the modern numbers come from Alexander Heger and Stanford Woosley’s calculations of 2002. After burning their helium, stars that end with oxygen cores of roughly 64 to 133 solar masses contract into the band. As their cores’ exponent falls below 4/3 they stop contracting slowly and begin to collapse. The collapse heats the oxygen, which begins to fuse explosively, and the thermonuclear energy released is more than the star’s entire gravitational binding. The star is blown apart completely — no neutron star, no black hole, nothing left — in a pair-instability supernova, which should be one of the brightest explosions in the universe, powered for months by the decay of several solar masses of freshly made radioactive nickel.

Around that range sit two others. Somewhat lighter cores enter the band only partly: they suffer a series of violent pulses, each ejecting a shell of material, before settling and collapsing normally — pulsational pair-instability. Heavier cores collapse so hard that the energy is absorbed by the breaking of nuclei into helium and neutrons before the oxygen can explode, and they fall directly into black holes. The instability therefore carves a hole in the masses of black holes that stars can make, which is where the physics of this essay meets the gravitational-wave detectors.

The gap in the black holes #

If single stars cannot leave black holes between about 50 and 130 solar masses, black holes in that range should be missing from nature — unless they are made some other way. The detectors that hear orbits shrinking as they radiate have been weighing black holes since 2015, and most lie below the gap. In 2019 they recorded GW190521, the merger of two black holes whose masses were estimated at about 85 and 66 solar masses; the heavier sits squarely in the gap, and the merger left a remnant of about 142 solar masses, the first clear black hole of intermediate mass.

The event does not refute the instability. It asks where such an object came from. The candidates are black holes that are themselves products of earlier mergers, grown in dense star clusters; stars that merged before collapsing; and uncertainties in the physics that sets the gap’s lower edge, the most important of which is the rate of one nuclear reaction — carbon-12 capturing a helium nucleus to make oxygen — which decides how much of a core ends as oxygen, and which is known experimentally to only about ten per cent at the energies that matter. Shifting that rate moves the gap’s lower edge by several solar masses, and the detectors’ growing catalogue is now being used to constrain the nuclear rate from the black holes rather than the other way round.

The heat the early universe got back #

The same pairs existed everywhere, once. In the first seconds after the Big Bang the universe was hotter than ten billion kelvin, and electrons, positrons and photons filled it in thermal equilibrium, with a pair gas very like the one in the drawings but with no net electrons to speak of — one extra electron for every billion pairs. The drawing counts the entropy of that mixture. Far above the threshold, light carries two units and the pairs, being fermions with four spin-and-charge states, carry seven eighths of four more: eleven halves in all. As the universe expanded and cooled through a few billion kelvin, pairs stopped being made faster than they annihilated, and the entropy they carried was handed over to the photons.

What makes this measurable is that one component was left out. Neutrinos had been in equilibrium with everything else, but they interact so weakly that they stopped exchanging energy at about ten billion kelvin, just before the pairs annihilated. The photons received the pairs’ entropy and the neutrinos did not, so afterwards the photons were hotter by the cube root of eleven fourths, a factor of 1.401. Today’s cosmic microwave background, at 2.7255 kelvin, therefore implies a cosmic neutrino background at 1.945 kelvin, with about 336 neutrinos in every cubic centimetre of space. They have never been detected directly — their energies are far too small — but their effect on the expansion rate at the time of the microwave background is measured, and matches the prediction for three species of neutrino to within a few per cent.

Particle number is not what relativity conserves #

The pair gas is a clean example of what changes in thermodynamics when relativity is taken seriously. In ordinary thermodynamics the number of each kind of atom is conserved, and a chemical potential is attached to each. In relativistic thermodynamics the conserved quantities are charges — electric charge, and numbers like baryon and lepton number that reactions cannot change — and chemical potentials attach to them, not to particles. Photons, which carry no charge, have no chemical potential at all; electrons and positrons have equal and opposite ones, because each carries one unit of charge of opposite sign.

The threshold is also a relativistic fact in a way that is easy to miss. Two photons can make a pair only if the invariant mass of the pair of them exceeds two electron masses — two photons of 511 keV meeting head-on just manage it, and two of a million electronvolts travelling side by side never do. In a thermal gas the photons arrive from all directions with a spread of energies, and the pairs are made by the energetic tail colliding at wide angles. The rate, as well as the equilibrium, is a question for relativistic kinematics.

And the rest-mass energy counted in the budget is exactly the energy that the box of light that weighs something says any bound system carries: a gas making pairs gains inertia and gravitational mass as it gains them, although none of its particles is moving faster.

Where the pair gas stops #

Equilibrium. Every figure assumes the pairs are made and destroyed fast enough to stay in equilibrium. In a stellar core, with densities of thousands of grams per cubic centimetre, they are, by an enormous margin; in the early universe they were, until the last stage of annihilation.

No neutrinos. A hot pair gas also loses energy: occasionally an electron and a positron annihilate into a neutrino and an antineutrino, which leave the star at once. In a massive star after helium burning this neutrino cooling is the dominant energy loss, and it is what makes the late stages of such a star race — carbon, neon, oxygen and silicon burning take centuries, years, months and days. None of that loss appears in an equilibrium equation of state.

An ideal oxygen gas. The ions are treated as a perfect gas and the electrons and positrons as non-interacting fermions. At the densities of the unstable region both are good approximations; the electrostatic interactions between particles would change the exponent in the third decimal place.

A single composition. The drawings are for pure oxygen. A real core has carbon, neon and magnesium mixed in, and its electrons per nucleon differ slightly, which shifts the region’s edges by a little.

What the pictures cannot show #

The pictures are equations of state: what a gas does if it is squeezed or heated a little. What they cannot show is what a star does when its core’s exponent falls below 4/3 — the collapse itself, the ignition of oxygen in a thin shell and then everywhere, the shock that runs outward through the star, and the light curve, months long, of an explosion that leaves nothing behind. All of that requires hydrodynamics and nuclear burning coupled together, and it is computed, not derived. The drawings say only that such a star has nowhere stable to stop.

Still open: has a pair-instability supernova been seen? #

The theory is half a century old, and the explosions it predicts have not been identified with certainty. They need stars far more massive than any that form today in galaxies like the Milky Way, where heavy elements drive mass away in stellar winds before the core can grow large enough; they should be commoner in the early universe, among the first generations of stars. A few unusually bright and slow supernovae — SN 2007bi is the most discussed — have been proposed as pair-instability explosions and disputed, because other mechanisms, a magnetar spinning down inside a normal supernova among them, can produce similar light curves. Whether any has been seen, and how common they were among the first stars, is being pursued with surveys that find thousands of supernovae a year and with searches for the chemical signature such explosions would leave in the oldest stars.

The habit worth carrying away is to ask which quantities a hot enough system still conserves. Once a gas can make particles from its own heat, particle number stops being a constraint and becomes a variable, and every response function — the heat capacity, the stiffness, the entropy — acquires a term for the cost of matter. In a star that term can decide whether the star survives, and in the early universe it decided, a few seconds in, the temperature of a background of neutrinos that still fills space.