The furnace that runs cooler than a compost heap

The Sun’s core is at fifteen million kelvin, and the phrase invites a picture of something violent. The number that goes with it does not fit the picture at all.

Averaged over the energy-generating region, the Sun produces about 276 watts per cubic metre. A compost heap manages several hundred. A human body, per unit volume, produces about 1,300. By any measure of intensity, the centre of the Sun is a feeble source.

The Sun is bright because it is very large. Multiply a feeble volumetric rate by $10^{27}$ cubic metres and the result is $3.8 \times 10^{26}$ watts — the luminosity that the mass sets. The luminosity comes from the size, not from the ferocity, and the slowness of the reaction is the reason the Sun has lasted four and a half billion years rather than four and a half thousand.

Why anything fuses at all #

Two protons repel. Bringing them close enough for the strong force to bind them means pushing through a Coulomb barrier, and the arithmetic of that barrier says the Sun should not work.

The barrier’s height is about 550 keV for two protons at the range where the strong force takes over. The mean thermal energy at fifteen million kelvin is 1.3 keV — four hundred times too small. Even the fast tail of the Maxwell–Boltzmann distribution does not help enough: the fraction of protons with 550 keV at that temperature is of order $e^{-400}$, which is zero for every practical purpose. Classically, the Sun cannot burn.

It burns because the barrier does not have to be climbed. Quantum mechanics allows a particle to appear on the far side of a barrier it could not surmount, with a probability that falls exponentially with the barrier’s width and the particle’s mass. Tunnelling was proposed by George Gamow in 1928 to explain alpha decay, and applied to stellar fusion by Atkinson and Houtermans the following year — before anyone knew what the reactions were.

The combination of two exponentials running opposite ways produces the characteristic behaviour of every nuclear reaction rate in a star. The number of particles with a given energy falls exponentially as energy rises; the tunnelling probability rises exponentially as energy rises. Their product is sharply peaked at an energy well above the thermal mean and well below the barrier — the Gamow peak — and essentially all fusion happens in that narrow window. For the Sun, the peak sits near 6 keV, which is about five times the mean thermal energy and about a ninetieth of the barrier.

That is why the rate is so steep a function of temperature. Nudging the temperature moves the whole distribution slightly and shifts a great deal of area into or out of a very narrow window.

The bottleneck, which is a weak interaction #

The Sun’s dominant reaction is the proton–proton chain, and its first step is the slowest reaction of astrophysical importance anywhere.

Two protons collide and must become a deuteron. But a deuteron is a proton plus a neutron, so one of the protons has to convert — emitting a positron and a neutrino — during the collision itself. That is a weak-interaction process, and the weak interaction is called weak for a reason: it is some twenty orders of magnitude less probable than the strong or electromagnetic processes that would otherwise dominate.

The consequence is a number with no intuitive scale. A given proton in the Sun’s core waits, on average, about nine billion years before it fuses. Every second, $10^{38}$ of them succeed — but there are $10^{56}$ protons available, so the per-proton rate is vanishingly small.

Nothing else in the chain is slow. Once a deuteron exists it captures another proton within a second or so, making helium-3; two helium-3 nuclei then combine to give helium-4 and release two protons. The entire rate of the Sun’s energy production is set by a weak-interaction step at the very beginning, and everything after it waits.

There is something worth pausing on there. If the weak interaction were somewhat stronger, the Sun would burn through its hydrogen in a fraction of the time; if somewhat weaker, it would not ignite at all. The lifetime of a star is set by a coupling constant from particle physics, and the fact that it lands within a factor of a few of the timescale on which planets become geologically interesting is not explained by anything in astronomy.

The other chain, and the temperature that switches it on #

There is a second route to the same end product, and above a threshold temperature it takes over completely.

The CNO cycle uses carbon, nitrogen and oxygen as catalysts. A carbon-12 nucleus captures four protons in sequence, passing through nitrogen and oxygen isotopes, and eventually spits out a helium-4 and returns to carbon-12 unchanged. The net reaction is identical — four protons become one helium nucleus — but the intermediate steps involve nuclei of charge 6, 7 and 8 rather than 1.

Higher charge means a higher Coulomb barrier, so the cycle requires a hotter core to run at all. Once it does run, the same higher barrier makes it far more sensitive to temperature: the pp chain goes roughly as $T^4$, the CNO cycle as $T^{18}$.

Two exponentially different curves cross once, and the crossing is what the first figure computes: 18.8 million kelvin. Below that a star runs mostly on the pp chain; above it, mostly on CNO. The Sun at 15.7 million kelvin generates about 99% of its energy by the pp chain. A star of 1.3 solar masses is hot enough at the centre for CNO to dominate, and everything more massive runs on it almost exclusively.

The consequence for stellar structure is large. The CNO cycle’s extreme temperature sensitivity concentrates the energy generation in a very small central region, and a large flux through a small region cannot be carried by radiation alone. Massive stars therefore have convective cores, stirring fresh hydrogen inward and extending their lives; low-mass stars have radiative cores and burn only what is already there. One exponent decides the internal architecture.

What the curve permits, and where it stops #

Fusion releases energy because of a single curve, and the curve has a maximum.

The energy released by fusing hydrogen to helium is the difference in binding energy per nucleon between the two — from zero to 7.07 MeV — which corresponds to converting 0.7% of the rest mass into energy. That figure is the one every stellar lifetime calculation uses.

Going further up the curve pays progressively less. Helium to carbon gains 0.6 MeV per nucleon; carbon to oxygen gains 0.3; silicon to iron gains 0.04. A massive star that burns through the whole sequence spends millions of years on hydrogen, hundreds of thousands on helium, and — at the end — about a day on silicon, because the energy yield has collapsed while the luminosity has not.

The peak is worth being precise about, because the usual statement is slightly wrong. Iron-56 is nearly always named as the most tightly bound nucleus. It is not: nickel-62 is, at 8.795 MeV per nucleon against iron-56’s 8.790. The reason iron-56 dominates in nature is a separate fact about how stellar cores actually assemble nuclei under conditions of nuclear statistical equilibrium, where nickel-56 is produced and then decays to iron-56. The energetic maximum and the observed abundance peak are two different things that happen to sit next to each other.

Beyond the peak, fusion costs energy rather than releasing it. A star that reaches iron in its core has run out of fuel in the only sense that matters, and what follows is collapse — which is where every element heavier than iron comes from, since making them requires energy to be put in rather than taken out — which is why the heavy elements exist because of the short-lived stars.

What was actually measured #

Everything above is a claim about the interior of a star, which no light escapes from. The photons made in the core take of order $10^{5}$ years to random-walk out, thoroughly reprocessed into a thermal continuum; nothing about the reactions survives the journey.

Neutrinos do escape. They interact only weakly — the same weakness that makes the first reaction slow makes the neutrinos it produces nearly untouchable — so they leave the core in about two seconds and arrive at the Earth eight minutes later, carrying direct information about reactions happening now.

The measurement was attempted from 1968, by Raymond Davis, using 380 cubic metres of dry-cleaning fluid in a South Dakota gold mine. Chlorine-37 captures a neutrino and becomes argon-37; the argon atoms are flushed out and counted individually. The expected rate was a few atoms per day, in a tank containing $10^{30}$ chlorine atoms.

He found about a third of the predicted number, and kept finding a third for thirty years. The solar neutrino problem was one of the longest-running discrepancies in physics, and the two candidate explanations were that the solar model was wrong or that something happened to neutrinos in transit.

The solar model was right. Neutrinos come in three flavours and oscillate between them, and the chlorine experiment was sensitive only to electron neutrinos. The Sudbury Neutrino Observatory settled it in 2001 by measuring both the electron-neutrino flux and the total flux of all flavours: the electron flux was a third of the prediction and the total flux matched it exactly. Neutrinos have mass, which the Standard Model had not required, and the discovery came out of an experiment about the Sun.

The measurement is now precise enough to see individual branches. Borexino has detected the neutrinos from the first pp reaction directly, at the predicted rate, and in 2020 detected the CNO cycle’s neutrinos in the Sun — confirming that about 1% of the Sun’s energy comes from a cycle whose dominance had been inferred for eighty years from theory alone.

The crossing between the two chains moves with the amount of catalyst available, and it is worth reading at two more compositions, because the composition is the one thing in the comparison that is not a property of the star’s mass.

The thermostat, which is the real reason it is stable #

The most consequential property of stellar fusion is one that follows from the steepness rather than from any particular reaction.

A rate going as $T^{18}$ is a violently unstable thing to have inside a star, and it is stabilised by hydrostatic equilibrium. Suppose the core produced slightly too much energy. It heats, so it expands; expanding gas cools; a cooler core, on a $T^{18}$ curve, produces dramatically less. Suppose it produced too little: it contracts, heats, and the rate climbs back. The steepness that looks dangerous is what makes the regulation sharp.

That is why the luminosity is set by the envelope rather than by the core. The core does not decide how much energy to make; it settles at whatever temperature makes exactly as much as the envelope is leaking. Fusion is the follower in that relationship.

The regulation depends on one thing: that the gas expands when heated. A degenerate gas does not — its pressure is set by density alone — so ignition in degenerate material has no thermostat and runs away. That is precisely what happens in a helium flash, and in a type Ia supernova, where the runaway consumes the whole star in seconds.

Where the model stops #

Two chains. Above the hydrogen-burning stages there are several more — triple-alpha for helium, then carbon, neon, oxygen and silicon burning — each with its own threshold and its own drastically shorter timescale.

Spherical, non-rotating, non-magnetic. Rotation mixes material into the burning region and extends a star’s life, and it is measurable from the width of a spectral line; the effect is large for massive stars and is one of the main uncertainties in their models.

A static composition. The core’s hydrogen fraction falls as it burns, which changes the mean molecular weight, which changes the pressure, which changes the temperature the thermostat settles at. A star brightens as it ages for this reason — the Sun is about 30% more luminous now than when it formed.

Reaction rates from the laboratory. The cross-sections are measured at energies far above the Gamow peak, because at stellar energies the rates are too low to measure at all, and then extrapolated downward. That extrapolation is the largest nuclear uncertainty in stellar models, and the reaction that limits the carbon-to-oxygen ratio in the universe is still known to only about 20% — an uncertainty that propagates into every abundance measured in a stellar spectrum.

The first figure has a limitation worth stating. It plots the rates against temperature, which suggests that a star chooses a temperature and gets a rate. The causation runs the other way: the star’s luminosity is imposed from outside, and the core’s temperature is whatever produces it. The correct reading of the crossing point is not “above 18.8 MK, more energy is released” but “a star whose envelope demands enough luminosity will have settled at a temperature above 18.8 MK, and there the CNO cycle is what supplies it”.

The comparison in the title is worth one qualification, because it is easy to over-read. The Sun’s core is feeble per unit volume and it is not feeble per unit mass relative to anything terrestrial over a long enough span: the total energy released by a kilogram of the Sun’s hydrogen over the star’s life exceeds anything chemistry can do by a factor of about ten million. What the low volumetric rate measures is the pace, and the pace is what determines whether a star lasts long enough for anything to happen around it.

The reaction nobody has measured #

Every rate in this essay comes from a cross-section, and cross-sections are measured in laboratories. One of them is not, and it is the one the Sun’s entire output depends on.

The first step of the proton–proton chain — two protons becoming a deuteron, with a positron and a neutrino emitted — has never been observed in any experiment. Its cross-section at the energies relevant to the Sun is of order $10^{-47}$ square centimetres, which is some twenty orders of magnitude below what an accelerator can detect, and the shortfall is not one that better equipment closes.

The reason is the same weakness that makes the reaction slow. A proton beam fired at a hydrogen target produces elastic scattering at an enormous rate and fusion at essentially none, and no counter can distinguish one event in $10^{20}$.

So the rate is computed. The weak interaction’s coupling is measured elsewhere — in neutron decay, in muon decay, in nuclear beta decays — and the matrix element for two protons becoming a deuteron is calculated from that coupling and from the deuteron’s known structure. The calculation is a theoretical one and is quoted with an uncertainty of about half a per cent.

That is an uncomfortable position for the most consequential rate in stellar astrophysics, and the discomfort is mitigated in two ways.

The first is that the same theory predicts related reactions that can be measured, and it gets them right. The second is that the Sun itself is the experiment: the measured neutrino flux from the first reaction is a direct count of how often it happens, and it agrees with the computed rate.

The Sun’s luminosity is checked against a calculation of a reaction nobody can perform, and the check is made by counting neutrinos from a star.

And two more, at the composition of the first stars and at the reaction that closes the chain.

The ladder from here #

Later rungs on this anchor: the Coulomb barrier and the Gamow peak derived. Tunnelling probability, and why the rate is so steep. The pp chain’s three branches. The CNO cycle in detail, and the bi-cycle. The triple-alpha process, and the Hoyle resonance — predicted from the existence of carbon before it was found. Advanced burning stages and their timescales. Nuclear statistical equilibrium and the iron peak. The s- and r-processes for the elements past iron. The solar neutrino problem and its resolution. Borexino’s detection of the CNO neutrinos. And the first stars, which had no CNO catalysts and had to reach far higher core temperatures to burn at all.

Eddington argued in 1920 that the Sun ran on subatomic energy, against a chorus saying the interior was not hot enough. His reply was that the critics should “go and find a hotter place” — and he added that if subatomic energy really was being used in stars, it brought the prospect of controlling that power measurably closer. The remark was made twenty-six years before the first reactor and a century before any fusion experiment returned more energy than it consumed.