The last fuels burn for the neutrinos

A star of twenty-five solar masses spends about seven million years burning hydrogen in its core, and most of a million burning helium. Then the sequence accelerates beyond anything the first two stages suggest. Carbon lasts a few centuries; neon, most of a year; oxygen, a few months; silicon, about a day. Then the core, now iron, collapses in less than a second.

The obvious explanation is that the later fuels release less energy. They do, but not by nearly enough: carbon burning releases about a twelfth as much energy per unit mass as hydrogen burning, and the stage lasts ten thousand times less. Something else is draining the core. It is not light — the star’s luminosity at its surface barely changes through all the late stages. It is neutrinos, produced in the core by the plasma itself at a rate that rises so steeply with temperature that from carbon burning onwards the core loses more energy to neutrinos than the whole star radiates as light, and everything about the end of a massive star follows from that.

Six stages, and the durations they last #

The durations are the output of full stellar-evolution calculations, which follow the star’s structure, its nuclear reactions and its losses from the beginning of hydrogen burning to the onset of core collapse.

The first two stages behave as stars are usually described. Hydrogen and helium burning are powered by fusion, and their energy is carried outward by radiation and convection and leaves the star as light. The rate of burning is set by the rate at which the star loses energy from its surface: a star in balance burns exactly as fast as it shines. Hydrogen fusion in the Sun runs astonishingly slowly for precisely that reason — the core produces only what the surface lets out — and helium burning, fed by a resonance that had to exist, lasts about a tenth as long because it releases about a tenth as much energy per unit mass and the star’s luminosity is somewhat higher.

After helium the rule changes. The core’s energy no longer has to find its way out through the star, because a new channel opens that bypasses the star entirely.

Neutrinos from a hot plasma #

At temperatures above about half a billion kelvin, the photons in a stellar core are energetic enough that some of them create electron–positron pairs, and at the same time some pairs annihilate. Almost always the annihilation produces two photons, which stay in the plasma. Very occasionally — about once in $10^{19}$ annihilations at these energies — the weak interaction lets an electron and a positron annihilate into a neutrino and an antineutrino instead. The neutrinos leave. They interact so weakly that the whole star is transparent to them, and the energy they carry is gone from the star within seconds.

The rate at which this happens rises extraordinarily steeply with temperature. The number of positrons in the plasma depends on the fraction of photons with energies above twice the electron’s rest energy, a number that grows exponentially with temperature near half a billion kelvin; the energy each annihilation carries away grows with it; the weak cross-section grows as the square of the energy. Together, over the range of the late burning stages, the energy loss rate from pair neutrinos rises roughly as the ninth power of the temperature. Double the core temperature and the neutrino losses rise five hundredfold.

Other processes produce neutrinos too — the decay of plasmons, collective oscillations of the plasma that behave like massive photons, dominates in dense, degenerate cores such as those of white dwarfs and red giants; photo-neutrinos and bremsstrahlung contribute at other densities — but in the non-degenerate cores of massive stars during carbon, neon, oxygen and silicon burning, pair annihilation is the leading term.

Burning to replace what the neutrinos take #

Once neutrino losses exceed the star’s light, the core’s burning rate is set by the neutrinos, not by the surface. The core burns fast enough to replace the energy the neutrinos carry off, and since the neutrino luminosity is enormous, it burns its fuel in a short time.

The estimate is crude: one temperature and one density per stage, a single core mass, a textbook formula for the neutrino emissivity. It gets carbon burning within a factor of seven and silicon burning within a factor of fifteen. What it gets right, unambiguously, is which loss sets the clock. Without neutrinos, a massive star’s late stages would last as long as its helium burning did, and its evolution would look like a slow, stately progression through heavier fuels. With them, the last three stages together last less than a year.

That changes the character of the whole end of the star’s life. In a neutrino-cooled core the burning regions are small and short-lived, the core contracts and heats to reach the next fuel as soon as the last is exhausted, and several burning shells can be active at once around an inert core — carbon burning in a shell outside neon burning outside oxygen burning — each shell racing through its own fuel on its own neutrino-driven clock. The cores of massive stars in their final months are not layered stable structures but a nest of flickering, convective, rapidly evolving shells, and three-dimensional simulations of the last hours find the oxygen-burning shell convecting violently enough to shake the core on the eve of collapse.

What each stage leaves behind #

The late stages are short, but they are where most of the elements between oxygen and iron are made, and the products of each are fixed by what can fuse at its temperature.

Carbon burning fuses pairs of carbon nuclei into neon, sodium and magnesium. Neon burning is not a fusion of neon with neon at all: at 1.5 billion kelvin the photons are energetic enough to knock a helium nucleus out of neon, and that helium is captured by other neon nuclei to make magnesium — a rearrangement driven by light. Oxygen burning fuses oxygen pairs into silicon, sulphur, argon and calcium. Silicon burning is again a rearrangement: photons break silicon and sulphur apart into protons, neutrons and helium nuclei, which are captured by other nuclei in a quasi-equilibrium that climbs towards the most tightly bound nuclei, the iron group, and stops there because nothing further releases energy.

The products are the elements that make up rocky planets — silicon, magnesium, sulphur, calcium and iron — and they are made in the last year of a massive star’s life, in a core racing its own neutrino losses. The biggest stars die first, and in dying they seed their surroundings with those elements within a few million years of forming; the ratio of these alpha elements to iron in old stars is a record of how quickly the first generations of massive stars exploded. The elements heavier than iron are not made by fusion at all, because fusion past iron costs energy; they are made by neutron capture, slowly in the envelopes of ageing stars or rapidly in explosive events, and the late stages of massive stars supply part of the neutrons for the first of those.

The surface never finds out #

The star’s envelope — the vast majority of its mass and all of its visible surface — responds to changes in the core on its thermal timescale, the time it takes to radiate away its own heat content, which for a red supergiant is thousands of years.

Carbon burning lasts a few centuries; everything after it, less than a year. The envelope has no time to respond to anything after carbon ignites, and the star’s appearance — its luminosity, its temperature, its position on the Hertzsprung–Russell diagram — is frozen at what the core was doing at the start of carbon burning. A red supergiant such as Betelgeuse looks the same whether its core is burning carbon with centuries to go or silicon with a day left, and nothing an astronomer can observe at the surface distinguishes the two. The prediction of when a nearby red supergiant will explode is therefore uncertain by the whole duration of carbon burning, a few hundred years to a few thousand, and when Betelgeuse dimmed dramatically in 2019 and 2020, the dimming — later traced to a cloud of dust ejected from its surface — said nothing about its core.

The one messenger that does carry news of the core in its last days is the neutrino itself. During oxygen and silicon burning the neutrino luminosity of a star at a few hundred parsecs would be large enough for the largest detectors to register an increase in their rate — a pre-supernova neutrino signal, days to hours before collapse. The energies are low, around a megaelectronvolt, and the detection depends on distinguishing the signal from background, but the next time a massive star within a kiloparsec or so is about to explode, the neutrinos that are running its clock may announce its death before its light changes at all.

Neutrinos that have been counted #

Neutrinos from stellar interiors are not hypothetical, and the ones that have been detected bracket the ones this essay is about.

The Sun’s core has been observed in neutrinos for more than half a century. They come from the fusion reactions themselves rather than from the plasma — a proton–proton chain releases neutrinos as part of turning protons into neutrons — and they carry about two per cent of the Sun’s energy output. They are the only thing that leaves the centre directly, and their flux has been measured precisely enough to be a thermometer for the core to a tenth of a per cent. In the Sun they are a minor loss; in a carbon-burning core the thermal neutrinos from the plasma are a thousand times the star’s light.

At the other end, on 23 February 1987, detectors in Japan and the United States registered two dozen neutrinos within about ten seconds from the core collapse of a blue supergiant in the Large Magellanic Cloud, Supernova 1987A. Those neutrinos carried off almost all of the gravitational energy of the collapsing core — about $3\times10^{46}$ joules, a hundred times the energy of the explosion that followed — and their arrival, three hours before the first light, confirmed that a collapsing core is cooled by neutrinos just as its last burning stages were. The last day of silicon burning sits between those two detections: a neutrino luminosity far above the Sun’s and far below a collapse, which the current detectors could just register from a star within a kiloparsec.

Stars as a particle-physics laboratory #

The same neutrino losses that shorten the late stages of massive stars cool other stars too, and because the losses are computed from particle physics, stellar observations constrain the particle physics.

In the degenerate helium core of a red giant, and in a young, hot white dwarf, neutrinos are produced mainly by plasmon decay, and they cool the object from inside. A white dwarf is a clock with no fuel in it, and for the first few million years after it forms, neutrinos rather than light dominate its cooling; the observed number of hot white dwarfs at each temperature in a star cluster matches the number the standard neutrino emission predicts. In a red giant, neutrino cooling delays the ignition of helium in the core until the core has grown to a particular mass, which sets the brightness of the brightest red giants in a population — the tip of the red-giant branch, a standard candle used to measure distances to nearby galaxies.

If neutrinos had a magnetic moment, or if other weakly interacting particles such as axions carried energy out of stellar cores, the cooling would be faster than the standard physics predicts, the helium flash would come later, and the tip of the red-giant branch would be brighter. The measured brightness of that tip in globular clusters sets some of the tightest limits there are on the neutrino’s magnetic moment and on the coupling of axions to electrons — limits obtained by treating a population of old stars as a detector whose sensitivity is the energy budget of their cores.

A brighter star, the same story #

The crossing point between light and neutrinos depends on how bright the star is, and a more luminous star crosses slightly later.

That insensitivity is the ninth power at work. The neutrino luminosity changes by a factor of three when the temperature changes by about thirteen per cent, so no plausible difference in the star’s light moves the crossing outside the gap between helium and carbon burning. Every star massive enough to burn carbon in its core burns it as a neutrino-cooled star, and the qualitative end of a massive star’s life — the accelerating sequence, the frozen surface, the short final stages — is the same across the whole range of masses that end in core collapse.

The contrast with hydrogen burning #

The steepness of the neutrino losses is worth setting beside the temperature dependence of the reactions themselves, which is also steep.

In hydrogen burning the steepness of the reaction rates acts as a thermostat. A core that heats slightly produces much more energy, expands, cools, and returns; the reaction rate’s sensitivity makes the star stable, and the burning rate is fixed by the surface’s demand. In the late stages two steep functions of temperature compete — the energy generation and the neutrino losses — and both are steeper than anything in hydrogen burning, with the carbon-burning rate rising as roughly the thirtieth power of temperature and silicon burning faster still. The core settles at the temperature where generation matches neutrino loss, and since both rise so steeply, that temperature is nearly fixed for each fuel: carbon burns near 0.8 billion kelvin, oxygen near two, silicon above three. The thermostat still works, but it is now regulated against a loss that leaves from the centre, not against the light leaving from the surface.

What the estimate leaves out #

The durations are rounded from one set of published models. Other groups’ models differ by tens of per cent in the late-stage durations, depending on the treatment of convection, of the carbon-burning reaction rate — itself uncertain by a factor of two — and of mass loss earlier in the star’s life, which sets the core mass.

The neutrino emissivity is a single fitting formula for pair annihilation in a non-degenerate plasma. At the densities of silicon burning the electrons are becoming degenerate, which suppresses pair creation, and plasmon and other processes contribute; the full treatment is a set of fitting formulae covering all of them.

Each stage is one temperature, one density and one core mass. In reality the temperature rises through each stage, the burning moves outward into shells, and the core mass that is burning changes. The crude estimate is a way of showing the energy budget, not a stellar model.

And the photon luminosity is held constant. It does rise somewhat through the late stages, but not by enough to matter against losses that exceed it by factors of thousands.

Still open: what the core looks like the moment before it falls #

The collapse of the iron core, and whether the resulting explosion succeeds, depends sensitively on the structure left by the last stages of burning — the size of the iron core, the density profile of the silicon and oxygen shells around it, and the violence of the convection in those shells. That structure is set during the last few hours, by neutrino-cooled shell burning that is three-dimensional, turbulent and still only beginning to be simulated realistically. One-dimensional stellar models produce a pre-collapse structure that varies erratically with the star’s initial mass — neighbouring masses giving very different cores — and it is not known whether that erraticism is real or an artefact of treating three-dimensional convection in one dimension. Since whether a massive star explodes or collapses quietly to a black hole seems to depend on exactly that structure, the last day of silicon burning, run on a neutrino clock nobody can observe directly, is where the fate of the star is decided.