A flux that is a thermometer to a tenth of a per cent

Neutrinos are the only thing that leaves the Sun’s centre directly. Everything else — every photon, every measurement of the surface — arrives after a random walk of tens of thousands of years, thoroughly reprocessed. A neutrino flux is a measurement of the core as it is now.

What it measures, above all, is a temperature. The nuclear reactions that produce the neutrinos have rates that depend on temperature with enormous exponents, and the steepest of them is the sharpest thermometer in the subject.

The only thing that leaves the centre is therefore also the thing whose interpretation is most sensitive to what the centre is doing, and the two properties are the same property seen from two sides.

The two properties together make the neutrino fluxes an unusual instrument. They are the most precise measurement of a stellar interior that exists, and they are almost useless for checking a stellar model, and both statements follow from the same exponent.

Where the exponents come from #

The steepness is not a coincidence and it is not the same for every reaction. It comes from the shape of the Gamow peak.

Two nuclei fuse only by tunnelling through their mutual Coulomb barrier, and the tunnelling probability rises very steeply with energy. The number of nuclei available at a given energy falls exponentially, from the Maxwell–Boltzmann tail. The product of a steeply rising and a steeply falling function is a narrow peak — the Gamow peak — sitting far above the mean thermal energy.

Raising the temperature moves that peak, and because it is narrow, a small move changes the reaction rate a great deal. How much depends on how far up the tail the peak sits, which depends on the Coulomb barrier, which depends on the charges of the nuclei involved.

So the proton–proton reaction, between two singly charged nuclei, has a modest exponent of about 4. The reaction producing beryllium-7 involves a helium nucleus and has an exponent near 11. And the one producing boron-8 involves beryllium and a proton at a higher barrier still, giving roughly 24.

That progression is a useful thing to carry, because it explains a fact about the Sun that is otherwise mysterious: the furnace runs cooler than a compost heap in power per unit volume, and yet its reaction rates are ferociously sensitive to temperature. Both follow from the same narrow Gamow peak. The peak is far out on the tail, so very few particles are in it — hence the feeble volumetric rate — and moving the tail moves the count enormously — hence the exponent. A slow reaction and a steep one are the same statement.

There is a further consequence of the narrowness that is worth stating because it explains why the extrapolation problem below is so severe. The Gamow peak for the boron-8-producing reaction sits at about 18 keV, which is far below the lowest energy any accelerator experiment can reach with a usable counting rate — the Coulomb barrier suppresses the rate so effectively that a laboratory measurement at solar energies would take longer than the age of the Earth. Every cross-section used in a solar model is therefore measured at tens or hundreds of keV and extrapolated downward by a theoretical function. The steepness that makes the flux a good thermometer is the same steepness that makes its cross-section unmeasurable at the temperature it matters at.

The same steepness, read two ways #

An exponent of 24 does two opposite things, and which of them matters depends on which direction the inference runs.

As a measurement, it is superb. A flux measured to three per cent constrains the temperature to 0.12 per cent. Nothing else comes close: helioseismology, which measures the sound speed profile to a part in a thousand throughout most of the Sun, constrains the central temperature to a few tenths of a per cent because the innermost region is the one the acoustic modes sample worst.

As a prediction, it is hopeless. A standard solar model computes the central temperature from the composition, the opacity, the equation of state and the age. Each of those carries an uncertainty, and their combined effect on the central temperature is of order half a per cent. Raised to the twenty-fourth power, that is a factor of 1.13 — thirteen per cent, which is four times the measurement error.

So the flux is a precise measurement of a quantity the model predicts imprecisely, and the comparison between them is dominated by the model.

That asymmetry is the most useful thing about the whole business. It means the fluxes are not a test of the model in the ordinary sense; they are a measurement that the model must accommodate, and the accommodation constrains whatever in the model is least well known.

There is a second asymmetry that follows and is worth stating separately. Because the different channels have different exponents, measuring two of them measures two different combinations of the model’s inputs, and their ratio is a much sharper diagnostic than either alone. The ratio of the boron-8 to the beryllium-7 flux goes as the difference of the exponents — about 13 — and it is nearly independent of the overall normalisation, so it cancels several of the shared systematics. That is the same manoeuvre as taking the ratio of two spectral diagnostics with different sensitivities, and it works for the same reason.

What the fluxes arbitrate #

The quantity they have most usefully constrained is the Sun’s own composition, in an argument that has been running for twenty years.

The solar photospheric abundances of carbon, nitrogen and oxygen were revised downward by about a quarter around 2005, when three-dimensional model atmospheres replaced one-dimensional ones. The revision improved the spectroscopy — a better measurement — and it broke the solar model, because those elements are a substantial source of opacity in the radiative interior.

Lower opacity means a shallower temperature gradient, a smaller convection zone and a lower central temperature. The revised models disagree with helioseismology on the depth of the convection zone, on the surface helium abundance and on the sound speed profile, by many times the measurement errors.

The neutrino fluxes are a third opinion. The boron-8 flux depends on the central temperature and therefore on the opacity; the CNO fluxes depend directly on the abundance of carbon and nitrogen in the core. Measuring both separates the two dependences.

The measurement of the CNO flux in 2020 was aimed exactly at this, and its answer is currently mildly in favour of the higher abundances — which is to say, mildly against the improved spectroscopy. The uncertainty is still large enough that the question is open.

It is worth being explicit about how the three measurements are combined, because the logic is not a simple comparison. The helioseismology constrains the sound speed throughout the radiative interior, which is a strong constraint on the temperature and composition profile everywhere except the innermost few per cent. The neutrinos constrain the innermost few per cent, where seismology is weakest. And the spectroscopy constrains the surface composition, which is the boundary condition for both. The three are complementary in exactly the way a well-designed experiment would arrange, and they were not designed — they are three unrelated techniques that happen to have blind spots in different places, which is the most fortunate configuration a difficult measurement can be in. Two instruments blind in opposite directions is the deliberate version of the same thing.

What was actually measured #

Four fluxes have been measured and each required a different technique.

Boron-8, from the highest-energy neutrinos, measured by water Cherenkov detectors and by heavy-water detectors sensitive to all three neutrino flavours. The total flux agrees with the standard model to about three per cent, which is the measurement in the first figure.

Beryllium-7, a monoenergetic line at 0.86 MeV, measured by a liquid scintillator detector deep underground. Its flux is a much less steep function of temperature, so it constrains a different combination of the model’s inputs.

The proton–proton flux, the dominant one and by far the hardest, measured in 2014. Its agreement with the model at the one per cent level is a direct confirmation that the Sun’s luminosity now equals its nuclear energy generation now — a statement that could not previously be checked, because the photons take tens of thousands of years to emerge and the neutrinos take eight minutes.

The CNO fluxes, detected in 2020 at about the twenty per cent level, and the first direct evidence that the cycle operates in the Sun at all.

One more fact about the CNO measurement deserves stating because it is a rare case of an astronomical result that is also a statement about the universe’s history. The CNO cycle requires carbon and nitrogen to be present as catalysts, so it cannot operate in a star made of pure hydrogen and helium. Detecting it in the Sun confirms that the cycle runs in stars of the Sun’s mass at the Sun’s metallicity, which fixes the transition mass above which it dominates — and that transition mass decides which stars have convective cores, which is the subject of the previous rung. A neutrino detector a kilometre underground measures a rate that determines whether a star of a given mass has a mixed core, which is a chain of inference worth admiring.

Where the picture stops #

Three, and the first is the largest term in the error budget.

The cross-sections are laboratory measurements at the wrong energy. Reaction rates at solar temperatures correspond to energies far below what an accelerator can reach with useful counting rates, so the measured cross-sections are extrapolated down using a theoretical energy dependence. The extrapolation for the boron-8-producing reaction is the single largest uncertainty in the predicted flux, and it is larger than the measurement.

The comparison is against a model, not against a theory. A standard solar model is a specific calculation with specific inputs — an opacity table, an equation of state, a diffusion prescription, an assumed initial composition. Agreement between a flux and a model tests the whole assembly, and a disagreement does not say which input is at fault.

And the Sun is one star. Every constraint here is a measurement of one object’s core, and the inferences drawn from it — about opacities, about abundances, about the reaction rates — are then applied to every star in every model. That is a reasonable thing to do and it is worth being aware of how narrow the base is.

And a fourth, and it is about time. Neutrinos leave the core in eight minutes and photons take tens of thousands of years, so a neutrino flux measures the Sun now and a luminosity measures it then. That is usually described as an advantage and it is also a complication: the two are being compared as though they described the same epoch, and the assumption that the Sun’s output has been constant over a photon diffusion time is an assumption rather than a measurement. It is almost certainly right — nothing in a main-sequence star changes on that timescale — and the neutrino measurement is the only thing that could ever check it.

Why a steep exponent is a mixed blessing #

The underlying point deserves stating on its own, because the same structure appears wherever a measurement depends steeply on a parameter.

A steep dependence makes the observable a precise measurement of the parameter and a poor prediction from it. That is a single mathematical fact with two faces: the derivative that amplifies the parameter into the observable also divides the observable’s error by the same factor when the inference runs backwards. Whether the steepness is a gift or a curse depends entirely on which quantity is measured and which is computed.

The subject is full of examples on both sides. The seismic scaling relations amplify a small error into a large one, because the wanted quantity is downstream of the steepness. Here the wanted quantity is upstream, and the same exponent works in the observer’s favour.

The practical rule that follows is worth stating: when a relation is steep, arrange to measure the quantity that appears with the large exponent and infer the one that appears with the small one. A great deal of experimental design is that sentence applied.

For the Sun, that means the fluxes are best used to measure the central temperature, and then to constrain whatever the model needs in order to reproduce it — rather than as a check on a prediction. That reframing is what turned solar neutrinos from a thirty-year anomaly into an instrument.

A last observation about what makes this a good instrument rather than merely a sensitive one. A thermometer with a huge exponent is only useful if the other things the observable depends on are known. The boron-8 flux depends on the temperature to the twenty-fourth power and on the abundance of helium-3, on the cross-sections, and on the flavour conversion — and each of those had to be pinned down independently before the temperature could be extracted. The measurement became an instrument in 2002, when the total flux across all flavours was measured and the particle physics was separated from the astrophysics; before that the same number had been available for thirty years and meant nothing about the Sun at all. A mean opacity dominated by where the gaps are is the remaining input on the same list, and it is where the current disagreement lives.

End on what the episode demonstrates about anomalies. The solar neutrino problem — a measured flux a third of the predicted one — stood for thirty years, and for most of that time the leading explanation was that the solar model was wrong. It was not. The Sun’s central temperature is now known to a tenth of a per cent and agrees with the model; the missing neutrinos had changed flavour on the way. That is worth remembering whenever an astronomical measurement disagrees with an astrophysical prediction: the disagreement is a joint statement about the object, the model and the physics in between, and the third of those is the one nobody looks at first. The Pioneer residual went the other way — a decade of proposals to modify gravity, resolved by the spacecraft’s own waste heat — and the two together are a fair sample of how such things end.

One more property of a steep power law deserves recording, because it decides how the comparison between measurement and model should be reported. Inverting a twenty-fourth power turns a factor-of-two disagreement in flux into a three per cent disagreement in temperature, so a result that sounds catastrophic in one currency is unremarkable in the other. Both statements are true and they invite opposite reactions, which is why the choice of variable is not cosmetic. The convention that has settled is to quote the inferred central temperature rather than the flux ratio, precisely because it does not overstate the discrepancy — and the same convention makes an agreement look less impressive than it is, since matching a flux to ten per cent is matching a temperature to four parts in a thousand. A steep exponent compresses on one side of the inversion and expands on the other, and reporting only one side is a way of choosing how surprised the reader should be.

Where the ladder goes next #

The immediate next rung is the cross-section extrapolation: how a reaction rate measured at accelerator energies is extended down to the Gamow peak, what the theoretical dependence assumes, and why it is the leading uncertainty. Further up sits the abundance problem itself — the disagreement between spectroscopy, helioseismology and neutrinos about what the Sun is made of, and what would resolve it.