Astronomy & Celestial Mechanics
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The only thing that leaves the centre

A photon made in the Sun's core takes a hundred thousand years to get out and arrives thermalised past recognition. A neutrino takes 2.3 seconds and arrives unchanged, so its flux is the fusion rate now — which is why a factor of three could not be absorbed by any adjustment to the Sun.

Assumes Fusion, Energy transport and Asteroseismology.

17 min read 8 figures Measuring the unreachableScales that defeat intuition

Everything this collection says about the inside of a star is an inference from its outside. A luminosity and a temperature at the surface, plus a set of equations, plus a composition assumed from somewhere else, give a run of pressure and temperature inwards — and the run is a solution, not a measurement. Nothing has ever been observed there.

The obstacle is not distance. It is opacity. A photon created in the core is absorbed and re-emitted so many times on the way out that its journey is a random walk of order a hundred thousand years, and by the time it emerges it carries the temperature of the surface it left from rather than the one it was born in. No photon that reaches the Earth has ever been near a fusion reaction.

One particle is different. A neutrino produced in the core crosses the entire Sun without interacting, leaves in 2.3 seconds, and arrives eight minutes later carrying the energy it was made with. Its flux is not a model output but the fusion rate itself, now.

Seven sources over twelve decades of flux, and one threshold below the one that matters. The Sun's neutrino spectrum at the Earth, with the continua drawn per unit energy and the two monoenergetic lines as spikes at their own energies. The pp reaction supplies 91 per cent of all of them and its endpoint is at 0.4233 MeV. The vertical lines are experimental thresholds, and they are the figure's argument: only gallium sits below that endpoint. Chlorine, which produced the deficit and held it for twenty years, could not see a single pp neutrino — it counted ⁷Be and ⁸B, which are a rare branch of a rare branch, together under a per cent of the total — and the water detectors that followed were higher still. So the discrepancy that eventually turned out to be a property of the neutrino was measured, for two decades, using the least representative one per cent of the flux available. The total drawn here is 6.54·10¹⁰ cm⁻² s⁻¹, and it is checkable without any stellar model at all: every completed chain turns four protons into helium, releases 26.73 MeV of which 0.59 leaves as neutrinos, and emits two neutrinos — so the solar constant of 1361 W m⁻² fixes the number at 6.5·10¹⁰ cm⁻² s⁻¹, within 0.6 per cent of the sum of the model's own branches. The dominant flux is a consequence of the Sun shining and of nothing else.
Fig. 1 The Sun’s neutrino spectrum at the Earth, over twelve decades of flux. The pp reaction supplies ninety per cent of all of them and its endpoint is 0.42 MeV; the vertical lines are the thresholds of the experiments that could see them, and the whole history of the subject is in their placement. Only gallium sits below that endpoint. Chlorine, which produced the deficit and held it for twenty years, could not see a single pp neutrino — it counted ⁷Be and ⁸B, which together are under a per cent of the total. The number the model predicts for the whole flux is checkable without any stellar model at all: two neutrinos per completed chain and 26.7 MeV released per chain means the solar constant fixes it.

The flux that follows from the luminosity

The pp flux is not really a prediction of a stellar model, and it is worth being clear about why, because it is the anchor everything else is measured against.

The Sun shines by turning four protons into one helium nucleus. That conversion releases 26.73 MeV, of which about 0.59 MeV leaves as neutrino kinetic energy and 26.14 MeV as everything else — heat, and ultimately light. It also produces exactly two neutrinos.

So the number of completed chains per second is the Sun’s luminosity divided by 26.14 MeV, and the neutrino flux at the Earth is twice that divided by the area of a sphere of one astronomical unit. Using the solar constant directly,

Φ  =  2×1361 W m−226.14 MeV  =  6.5×1014 m−2 s−1,\Phi \;=\; \frac{2 \times 1361\ \mathrm{W\,m^{-2}}}{26.14\ \mathrm{MeV}} \;=\; 6.5\times10^{14}\ \mathrm{m^{-2}\,s^{-1}},

or 6.5×10106.5\times10^{10} per square centimetre per second. Sixty-five billion neutrinos cross a thumbnail every second, and the number comes from the solar constant and a mass excess, with no opacity, no composition and no equation of state anywhere in it.

That is the constraint the deficit had to be explained inside. Whatever the Sun’s interior is doing, the total number of neutrinos is fixed to a per cent or two by the fact that it shines — the only escape being that the luminosity now and the fusion rate now differ, which requires a delay of a hundred thousand years and cannot change a long-term average.

Seven sources over twelve decades of flux, and one threshold below the one that matters. The Sun's neutrino spectrum at the Earth, with the continua drawn per unit energy and the two monoenergetic lines as spikes at their own energies. The pp reaction supplies 91 per cent of all of them and its endpoint is at 0.4233 MeV. The vertical lines are experimental thresholds, and they are the figure's argument: only gallium sits below that endpoint. Chlorine, which produced the deficit and held it for twenty years, could not see a single pp neutrino — it counted ⁷Be and ⁸B, which are a rare branch of a rare branch, together under a per cent of the total — and the water detectors that followed were higher still. So the discrepancy that eventually turned out to be a property of the neutrino was measured, for two decades, using the least representative one per cent of the flux available. The total drawn here is 6.54·10¹⁰ cm⁻² s⁻¹, and it is checkable without any stellar model at all: every completed chain turns four protons into helium, releases 26.73 MeV of which 0.59 leaves as neutrinos, and emits two neutrinos — so the solar constant of 1361 W m⁻² fixes the number at 6.5·10¹⁰ cm⁻² s⁻¹, within 0.6 per cent of the sum of the model's own branches. The dominant flux is a consequence of the Sun shining and of nothing else.
Fig. 2 The same spectrum with only the three branches the argument uses. Stripped of the rest, the shape of the problem is legible: pp supplies almost everything and stops at 0.42 MeV; ⁷Be is a line four decades down; ⁸B is a broad tail eight decades down and is the only part of it that reaches an energy a water Cherenkov detector can see. The luminosity constraint fixes the total, which is essentially the pp curve — and every experiment before gallium was measuring the two curves that together are under a per cent of it.

The branch that is a thermometer

The pp chain has side branches, and they are rare, and their rarity is what makes them useful.

Once a helium-3 nucleus exists it can react with another helium-3 — the main branch, which produces no further neutrinos — or with a helium-4 to make beryllium-7, which opens the two rarer ones. The beryllium can capture an electron, giving a monoenergetic ⁷Be neutrino, or capture a proton to make boron-8, which decays to give the ⁸B neutrinos — energetic, up to 15 MeV, and about one in ten thousand of the total.

Which branch a given reaction takes depends on the competition between rates with different temperature dependences, and the net result is that the ⁸B flux goes as roughly the twenty-fourth power of the core temperature.

Φ(8B)  ∝  Tc 24.\Phi(^8\mathrm{B}) \;\propto\; T_c^{\,24}.

A quantity that responds to the twenty-fourth power of another is an extraordinarily sensitive instrument. An eight per cent measurement of the ⁸B flux is a three-tenths of a per cent measurement of the temperature at the centre of the Sun — a number that no other observation of any star comes close to.

A flux that goes as the twenty-fourth power, and a core temperature known to a tenth of a per cent. Three neutrino fluxes against the temperature of the solar core, in units of the standard model's own value. They are drawn as power laws because that is what they are over this range, and the powers are the whole content: the pp flux goes as T^−1.1 — it falls slightly, because raising the temperature moves the chain toward its rarer branches — while ⁷Be goes as T¹⁰ and ⁸B as T²⁴. A quantity that responds to the twenty-fourth power of another is a thermometer, and a very good one: an eight-per-cent measurement of the ⁸B flux is a three-tenths of a per cent measurement of the temperature at the centre of the Sun. That is what made the deficit a problem rather than a discrepancy. Matching an observed ⁸B rate 34 per cent of the prediction needs a core 4.4 per cent cooler, marked here — and by the time the measurement was that good, helioseismology had fixed the sound speed through the same region to better than 0.1 per cent, the narrow band at the model's own temperature. The two constraints are 44 times apart. Either the Sun was not the Sun that five thousand measured oscillation frequencies described, or the neutrinos were not arriving as the neutrinos that left.
Fig. 3 The same three curves over a wider range of core temperature, which shows the lever at full length. Across twenty-two per cent in temperature the ⁸B flux moves by a factor of over a hundred and the pp flux by a few per cent — and it is that ratio between the two responses, rather than either one alone, that makes the branch a thermometer. A measurement of pp fixes the total and a measurement of ⁸B fixes the temperature, from the same star, at the same instant, by counting particles of the same kind.
A flux that goes as the twenty-fourth power, and a core temperature known to a tenth of a per cent. Three neutrino fluxes against the temperature of the solar core, in units of the standard model's own value. They are drawn as power laws because that is what they are over this range, and the powers are the whole content: the pp flux goes as T^−1.1 — it falls slightly, because raising the temperature moves the chain toward its rarer branches — while ⁷Be goes as T¹⁰ and ⁸B as T²⁴. A quantity that responds to the twenty-fourth power of another is a thermometer, and a very good one: an eight-per-cent measurement of the ⁸B flux is a three-tenths of a per cent measurement of the temperature at the centre of the Sun. That is what made the deficit a problem rather than a discrepancy. Matching an observed ⁸B rate 34 per cent of the prediction needs a core 4.4 per cent cooler, marked here — and by the time the measurement was that good, helioseismology had fixed the sound speed through the same region to better than 0.1 per cent, the narrow band at the model's own temperature. The two constraints are 44 times apart. Either the Sun was not the Sun that five thousand measured oscillation frequencies described, or the neutrinos were not arriving as the neutrinos that left.
Fig. 4 The lever, and the contradiction it created. Three fluxes against the core temperature, in units of the standard model’s own value. The pp flux falls slightly as the core heats, because raising the temperature moves the chain towards its rarer branches; ⁷Be goes as the tenth power and ⁸B as the twenty-fourth. Matching an observed ⁸B rate a third of the prediction needs a core four and a half per cent cooler — and by the time the measurement was that good, helioseismology had fixed the interior to a tenth of a per cent, the narrow band drawn at the model’s own value. The two constraints are forty-five times apart.

What was actually counted

Four generations of experiment, each measuring a different combination of the same flux.

Chlorine, from 1968. A tank of 615 tonnes of perchloroethylene a mile underground in South Dakota. A neutrino converts a chlorine-37 nucleus to argon-37, which is radioactive; the argon atoms — a few dozen, in a tank of 103010^{30} molecules — are flushed out with helium every few months and counted by their decays. The threshold is 0.814 MeV, above the pp endpoint, so the signal is ⁷Be and ⁸B. It measured about a third of the predicted rate, and did so for twenty years while nearly everyone assumed the model was wrong.

Kamiokande, from 1987. Water, and the Cherenkov light from an electron knocked forward by a neutrino. The threshold is high, several MeV, so it sees only ⁸B — but it sees the direction, and the events point away from the Sun. That was the first demonstration that the particles being counted came from where they were supposed to. It measured about half the prediction.

Gallium, from 1990. Two experiments, in Italy and Russia, using gallium-71 with a threshold of 0.233 MeV — below the pp endpoint, and therefore the only measurement ever made of the dominant branch. They measured about sixty per cent of the prediction.

Three experiments, three different combinations of branches, three different deficits. That pattern is itself informative: no single adjustment to the Sun reproduces all three ratios, because they weight the temperature-sensitive branches differently and a cooler core suppresses ⁸B far more than pp.

A flux that goes as the twenty-fourth power, and a core temperature known to a tenth of a per cent. Three neutrino fluxes against the temperature of the solar core, in units of the standard model's own value. They are drawn as power laws because that is what they are over this range, and the powers are the whole content: the pp flux goes as T^−1.1 — it falls slightly, because raising the temperature moves the chain toward its rarer branches — while ⁷Be goes as T¹⁰ and ⁸B as T²⁴. A quantity that responds to the twenty-fourth power of another is a thermometer, and a very good one: an eight-per-cent measurement of the ⁸B flux is a three-tenths of a per cent measurement of the temperature at the centre of the Sun. That is what made the deficit a problem rather than a discrepancy. Matching an observed ⁸B rate 60 per cent of the prediction needs a core 2.1 per cent cooler, marked here — and by the time the measurement was that good, helioseismology had fixed the sound speed through the same region to better than 0.1 per cent, the narrow band at the model's own temperature. The two constraints are 21 times apart. Either the Sun was not the Sun that five thousand measured oscillation frequencies described, or the neutrinos were not arriving as the neutrinos that left.
Fig. 5 The gallium deficit read as a temperature, which is the arithmetic that sentence rests on. Sixty per cent of the prediction requires a core about one per cent cooler; the chlorine result’s thirty-four per cent required four and a half. One Sun cannot be both, and no intermediate value satisfies either — a core cool enough to explain chlorine over-suppresses gallium, and one warm enough for gallium leaves chlorine unexplained. Two experiments with different thresholds are two equations in one unknown, and they are inconsistent.

The measurement that ended it

The Sudbury Neutrino Observatory used a thousand tonnes of heavy water, two kilometres underground in a nickel mine, and its advantage was that deuterium gives three different reactions with three different sensitivities to flavour.

Charged current — a neutrino converts the neutron in a deuteron to a proton, producing an electron. This requires the neutrino to be electron-flavour, so it counts ϕe\phi_e alone.

Neutral current — a neutrino breaks the deuteron apart without changing flavour. This happens equally for all three flavours, so it counts ϕe+ϕμτ\phi_e + \phi_{\mu\tau}.

Elastic scattering off an electron, which is sensitive to the other flavours at about 15 per cent of the electron rate.

Three measurements of two unknowns is one more than is needed, and the redundancy is the result.

Three reactions, three lines, one crossing: 1.76 electron-flavour out of 5.09. The Sudbury measurement, drawn the way it settled the question. The horizontal axis is the flux of electron neutrinos from ⁸B and the vertical axis is the flux of the other two flavours, both in units of 10⁶ cm⁻² s⁻¹, and each of the detector's three reactions constrains a different combination of the two. Charged current — a neutrino converting a neutron in a deuteron — happens only for electron flavour, so it is a vertical band at 1.76. Neutral current — a neutrino breaking the deuteron apart without changing flavour — counts all three equally, so it is a band of slope −1 at 5.09. Elastic scattering off an electron is sensitive to the other flavours at 15.5 per cent of the electron rate, so it is a third band of a third slope. Three independent measurements of two unknowns is one more measurement than is needed, and the crossing is the result: the total is 5.09, against a stellar model that had predicted 5.05 ± 0.9 for thirty years, and the electron-flavour part of it is 35 per cent. Nothing about the Sun was wrong. What every earlier experiment had measured was the fraction of neutrinos that were still electron-flavour when they arrived, and the missing two thirds had become something the detectors were nearly blind to on the way. The picture cannot show the mechanism — the flavour change is resonant, and it happens in the outer solar interior rather than in flight — but it does not need to: the sum is conserved and the parts are not.
Fig. 6 The answer, drawn the way it settled the question. Each reaction is a straight band in the plane of electron-flavour flux against the flux of the other two, and each band has a different slope because each reaction weights the flavours differently. They cross at one point. The total is 5.09 in units of 10610^6 cm⁻² s⁻¹, against a stellar model that had predicted 5.05±0.95.05 \pm 0.9 for thirty years; the electron-flavour part of it is 1.76, about a third. Nothing about the Sun was wrong. What every earlier experiment had measured was the fraction of neutrinos that were still electron-flavour on arrival.

That is the whole of it: a solar experiment measured a property of the neutrino. The particles do not arrive as they left, because they have mass — a fact the standard model of particle physics did not include — and the flavour composition changes on the way, resonantly, in the outer part of the Sun itself.

Three reactions, three lines, one crossing: 1.76 electron-flavour out of 5.09. The Sudbury measurement, drawn the way it settled the question. The horizontal axis is the flux of electron neutrinos from ⁸B and the vertical axis is the flux of the other two flavours, both in units of 10⁶ cm⁻² s⁻¹, and each of the detector's three reactions constrains a different combination of the two. Charged current — a neutrino converting a neutron in a deuteron — happens only for electron flavour, so it is a vertical band at 1.76. Neutral current — a neutrino breaking the deuteron apart without changing flavour — counts all three equally, so it is a band of slope −1 at 5.09. Elastic scattering off an electron is sensitive to the other flavours at 15.5 per cent of the electron rate, so it is a third band of a third slope. Three independent measurements of two unknowns is one more measurement than is needed, and the crossing is the result: the total is 5.09, against a stellar model that had predicted 5.05 ± 0.9 for thirty years, and the electron-flavour part of it is 35 per cent. Nothing about the Sun was wrong. What every earlier experiment had measured was the fraction of neutrinos that were still electron-flavour when they arrived, and the missing two thirds had become something the detectors were nearly blind to on the way. The picture cannot show the mechanism — the flavour change is resonant, and it happens in the outer solar interior rather than in flight — but it does not need to: the sum is conserved and the parts are not.
Fig. 7 The same three bands over a wider range of flux, which puts the crossing in context. Extending the axes changes nothing about where the lines meet, and it makes visible how far the intersection sits from the axes — the electron-flavour flux is about a third of the total, so the point is well inside the plane rather than near an edge. Three bands with three slopes meeting at one point is a two-parameter fit with one degree of freedom left over, and the leftover is what turns a determination into a test.

What the picture cannot show

The mechanism. The conversion is not a vacuum oscillation for the high-energy neutrinos; it is a resonant conversion driven by the electron density gradient the neutrinos pass through inside the Sun. That is why the survival probability depends on energy — about 55 per cent for the low-energy pp neutrinos and about 30 per cent for ⁸B — and the energy dependence is a separate measurement from the totals shown here.

The CNO cycle. About one per cent of the Sun’s energy comes from a catalytic cycle running on carbon, nitrogen and oxygen rather than the pp chain, and it produces its own neutrinos. Those were detected only in 2020, by Borexino, and their flux is a direct measurement of the metal content of the solar core — which bears on the composition discrepancy that helioseismology has been unable to resolve. The current value slightly favours the higher metallicity.

Seven sources over twelve decades of flux, and one threshold below the one that matters. The Sun's neutrino spectrum at the Earth, with the continua drawn per unit energy and the two monoenergetic lines as spikes at their own energies. The pp reaction supplies 91 per cent of all of them and its endpoint is at 0.4233 MeV. The vertical lines are experimental thresholds, and they are the figure's argument: only gallium sits below that endpoint. Chlorine, which produced the deficit and held it for twenty years, could not see a single pp neutrino — it counted ⁷Be and ⁸B, which are a rare branch of a rare branch, together under a per cent of the total — and the water detectors that followed were higher still. So the discrepancy that eventually turned out to be a property of the neutrino was measured, for two decades, using the least representative one per cent of the flux available. The total drawn here is 6.54·10¹⁰ cm⁻² s⁻¹, and it is checkable without any stellar model at all: every completed chain turns four protons into helium, releases 26.73 MeV of which 0.59 leaves as neutrinos, and emits two neutrinos — so the solar constant of 1361 W m⁻² fixes the number at 6.5·10¹⁰ cm⁻² s⁻¹, within 0.6 per cent of the sum of the model's own branches. The dominant flux is a consequence of the Sun shining and of nothing else.
Fig. 8 The full spectrum with the axis extended at both ends, which brings the CNO branches and the hep neutrinos onto the plot. The nitrogen-13 and oxygen-15 curves sit two decades below pp and overlap it in energy, which is the whole difficulty of the measurement: there is no window where CNO dominates, so detecting it means measuring a spectrum precisely enough to see a shoulder rather than counting events above a threshold. At the far end, hep is eight decades below ⁸B and has never been detected at all.

And the time. Everything measured is the fusion rate as of eight minutes ago. The luminosity is the fusion rate averaged over the photon diffusion time. Comparing them is the only test of whether the Sun is in a steady state that does not assume it, and the two agree — but at the several-per-cent level, which permits a good deal of unsteadiness.

What it cost to be sure

The subject is worth one paragraph of accounting, because the ratio of effort to information is unlike anything else in this collection.

Thirty-three years passed between the first chlorine result and the Sudbury announcement. In that time the deficit was reported, disbelieved, re-measured by three independent techniques, and explained a dozen ways — a cooler core, a mixed core, a lower metal abundance, a rotating core, an unknown energy-loss channel, and a change of the neutrino’s own nature. Every one of the astrophysical explanations was tested and every one failed against the oscillation frequencies, which arrived in usable form in the 1990s and shut the door.

The lesson that survives is about what a measurement is worth when its interpretation is not in dispute. The chlorine experiment’s result was correct from the beginning; what was wrong was the assumption every reading of it made, which was that a neutrino counted is a neutrino emitted. The deficit was never a measurement of the Sun and it was never wrong.

The detector that had to be cleaner than the signal

The experiments above counted a few dozen events a year against a background that had to be smaller still, and the engineering that made that possible is a subject of its own — one where the requirement is stated in a unit nobody else uses.

Borexino’s task was the hardest of them: to see the ⁷Be line at 0.86 MeV and eventually the pp neutrinos below 0.42 MeV, in real time, by the scintillation light from a recoiling electron. There is nothing distinctive about such an event. It is a low-energy electron, indistinguishable from the electron a natural radioactive decay produces, so the only way to see the signal is to remove essentially all of the radioactivity from the detector.

The target purity was around 10−910^{-9} becquerels per kilogram of uranium and thorium — some ten orders of magnitude below ordinary materials, and about a millionth of the radioactivity of a human being per unit mass. Achieving it meant distilling the scintillator, building the containment vessel out of nylon film manufactured in a low-radon environment, and surrounding the whole thing with concentric layers of progressively less pure liquid, each shielding the next.

One constraint could not be engineered away and is the most striking of them. The scintillator is an organic liquid made from petroleum, and organic material contains carbon-14, which beta-decays with an endpoint of 0.156 MeV — squarely inside the pp neutrino window. No purification removes it, because it is chemically identical to the carbon the scintillator is made of. What was done instead was to choose the feedstock: petroleum from a deep, old deposit, isolated from the atmosphere long enough that its cosmogenic ¹⁴C had decayed away. The ratio achieved was about 10−1810^{-18} relative to ¹²C, against 10−1210^{-12} in anything living.

That is the shape of the whole enterprise: the measurement of the centre of the Sun was limited by the age of the oil the detector was filled with.

The same instruments, pointed downwards

Once a detector exists that can count a handful of low-energy events a year against nothing, it can be aimed at something other than the Sun, and the nearest available target is directly underneath it.

The Earth’s interior contains uranium, thorium and potassium, whose decay chains emit antineutrinos. Those particles leave the planet as freely as solar neutrinos leave the Sun, and detecting them measures the radiogenic heat production of the Earth — a quantity as inaccessible to direct measurement as the solar core, and for the same reason.

KamLAND and Borexino have both detected them, at rates of a few events per year, and the inferred radiogenic contribution is around 20 terawatts against a total surface heat flow of about 47. The remainder is primordial heat left over from formation and from core differentiation, and the split between the two had been argued for decades on the basis of meteoritic compositions and nothing else.

The parallel with the solar case is close enough to be worth stating. In both, the interior is opaque to everything else; in both, the flux of a weakly interacting particle is a direct rate measurement rather than a model output; and in both, the difficulty is entirely in counting a handful of events against a background that must be engineered to nothing. A technique built to settle a thirty-year discrepancy about a star turned out to be the only direct measurement of how much heat the Earth makes, which is a fair example of what a genuinely new messenger buys.

There is one further target the same detectors can reach, and it is the one the field is waiting on. A core-collapse supernova in this galaxy would deliver a burst of some ten thousand events in a few seconds, carrying ninety-nine per cent of the energy of the explosion and arriving hours before any light — because the neutrinos leave at the moment of collapse while the shock takes hours to reach the surface. The 1987 event in the Large Magellanic Cloud produced twenty-four detected neutrinos across three detectors and confirmed the basic picture; a Galactic event would resolve the time structure of the collapse itself, which no other observation can reach. The detectors are watching continuously and have been for decades, which is an unusual thing for an instrument to do: sit ready for an event that happens perhaps twice a century, in the knowledge that the alternative is missing it.

The alert networks that connect the detectors exist for that reason. A burst is recognised within seconds by several instruments independently, cross-checked, and broadcast to optical observatories, so that telescopes can be pointed at the right part of the sky before the shock has surfaced. It is the one place in astronomy where a warning of a transient event can precede the event’s visible beginning, and the warning comes from particles that left the collapsing core while the star’s surface was still undisturbed.

Whether the next such burst is detected at all is therefore a question about maintenance schedules and duty cycles rather than about physics, which is an uncomfortable thing to say about the most informative observation the field could make.

The generalisation

The pattern is one of the most useful in observational physics: when the usual messenger is scrambled by the medium, find one that is not, even at enormous cost in rate.

The cost here is severe. A neutrino’s interaction cross-section is some twenty orders of magnitude below an atomic one, which is why a detector is a thousand tonnes underground and counts events in tens per year. What is bought is a signal that has passed through the entire body of a star without being touched.

The same trade appears elsewhere. Gravitational waves pass through everything and are detected at strains of 10−2110^{-21}. Cosmic rays arrive with their arrival directions scrambled by magnetic fields and their energies intact, so their spectrum is usable and their sky map is not. And within astronomy the same argument decides which wavelength to observe at: radio waves pass through dust that stops visible light, which is why the centre of the Galaxy is a radio and infrared object.

Neutrino astronomy is the extreme case of the trade, and outside the Sun it has succeeded exactly twice — twenty-four neutrinos from a supernova in 1987, and a diffuse high-energy flux of extragalactic origin. Two detections in forty years, from an instrument that measures the inside of the nearest star to a fraction of a per cent.

Where this ladder goes next

Later rungs on this anchor: the energy dependence of the survival probability, which is the direct signature of resonant conversion in matter and is measured by comparing the pp, ⁷Be and ⁸B results; the CNO flux as a measurement of the core’s metallicity, and the composition problem it bears on; the day–night asymmetry, a small effect from neutrinos passing through the Earth at night that constrains the mixing parameters independently; supernova neutrinos, where the burst carries ninety-nine per cent of the energy of the explosion and arrives hours before the light; and the diffuse supernova background, which is the sum of every core collapse in the history of the universe and has not yet been detected.

Where this goes next

Energy generation against core temperature. The proton–proton chain and the CNO cycle, in solar units, against core temperature on logarithmic axes. The CNO curve is far steeper, so the two cross at 18.8 million kelvin — above that temperature a star runs mostly on CNO, and below it mostly on pp. The furnace that runs cooler than a compost heap The Sun's core produces 276 watts per cubic metre. A human body produces more. The Sun is bright because it is enormous, and its fusion is astonishingly slow — which is exactly why it lasts. Which gradient is steeper, and where. The temperature gradient radiation would need in order to carry the whole luminosity, against the gradient a rising blob of gas follows, through an n = 3 polytrope with a Kramers opacity and an energy generation going as ρT^4. Wherever the first exceeds the second the layer is unstable and convects. This model puts the outer boundary at 89 per cent of the radius; helioseismology measures the base of the Sun's convection zone at 71.3 per cent, and the gap between the two numbers is the price of a polytrope with one opacity law rather than a solar model. The curve also rises above the adiabatic value inside 4 per cent of the radius, which is a convective core — it appears when the energy generation is concentrated enough, and it is the structure a star burning by the CNO cycle has. The part of a star that boils Energy leaves a star's interior either by radiation or by bulk motion, and which one happens is settled by comparing two temperature gradients. The comparison decides that the Sun's outer third churns and its core does not, and that a massive star does exactly the opposite. 14 radial orders of the Sun, at 135.1 μHz apart. The p-mode spectrum of the Sun — 1 solar mass in 1 solar radius — from the asymptotic relation with its second-order term, drawn as 14 radial orders of ℓ = 0, 1 and 2 under a Gaussian envelope centred on ν_max = 3,090 μHz. Two numbers are marked and they do very different work. The large separation, 135.1 μHz, is the spacing between consecutive ℓ = 0 modes and fixes the mean density. The small separation, 9.00 μHz, is 2.7 pixels on this axis — it fixes the age, and it is why the échelle diagram exists rather than being a convenience. The vertical axis is the measurement: each mode moves the surface by about 20.0 cm s⁻¹ at the peak, and brightens it by a few parts per million, which is why this was impossible before a decade-long velocity series. Each mode is one line: its true width is set by its lifetime and is far below a pixel here. The interior read from a comb of frequencies A star's surface moves by about twenty centimetres a second, in thousands of overlapping sound modes at once. Two numbers off that spectrum give a mass and a radius with almost no stellar model in the chain, and a third gives an age.

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Charged currentCore temperatureDetector thresholdFlavourHelioseismologyLuminosity constraintNeutral currentNeutrino fluxPp chainSolar neutrinoSolar neutrino problemStandard solar model