The physical seam · nuclear astrophysics, made operable

The Sun Is a Thousand Times Too Cold

Two protons have to touch for the Sun to shine, and to touch they must climb an electrical hill about 1.44 million electronvolts high. At the centre of the Sun a proton carries about 0.0013. Classical physics does not say the Sun shines slowly. It says the Sun does not shine. Start by watching exactly how badly it fails.

Drag the contact distance anywhere it could plausibly sit and the verdict does not soften. This is not a shortfall you fix with a hotter core or a bigger Sun. Under classical mechanics the expected number of protons in the entire Sun sitting above the barrier at any instant is a number with hundreds of zeros after the decimal point before the first digit. Counting attempts instead of occupancy does not rescue it either: the blinded re-derivation of this page put the total number of proton-proton encounters in the Sun's entire history at around 10⁸⁸, which still leaves about 10⁻³⁷⁸ successful classical crossings ever. That last figure is an order-of-magnitude estimate made offline, not recomputed in your browser, and it needs a core density and a collision cross section that nothing else on this page needs. The Sun is not dim. It is off.

What people believed before, and exactly how it failed

The nineteenth century had a good answer and it was wrong by two orders of magnitude. If the Sun shines by slowly contracting, gravity releases energy at a rate that sets a clock: divide the Sun's gravitational binding energy by its luminosity. Kelvin and Helmholtz did the sum and got tens of millions of years. Geologists, and later radiometric dating, kept handing back billions. Both cannot be right.

Eddington named the way out in 1920, before anyone could calculate it, and then wrote the sentence that this whole page is an argument with:

We do not argue with the critic who urges that the stars are not hot enough for this process; we tell him to go and find a hotter place. A. S. Eddington, The Internal Constitution of the Stars (1926), chapter XI, "The Source of Stellar Energy", p. 301. Checked word for word against a scan of the book (Internet Archive item internalconstitu0000ased, a Dover reprint): the running head on the page carrying this sentence reads "THE SOURCE OF STELLAR ENERGY 301", and the wording and punctuation above are as printed.

Eddington was right about the conclusion and wrong about the rhetoric. There is no hotter place. The stars really are not hot enough, by the factor Instrument 1 just computed. What rescues them is not temperature.

"Quantum tunnelling" is the right word and it is not yet an answer

Here is where the knowledgeable reader stops reading, because they already know the word. So let us spend the word immediately and see what it buys. Gamow's 1928 barrier-penetration factor for a bare Coulomb potential is exp(−2πη) = exp(−√(E_G/E)), and at the mean thermal energy of the solar core it is about 4.6 × 10⁻⁹. Nine orders of magnitude. The classical shortfall was four hundred and sixty-five. Tunnelling at the typical energy does not close the gap; it barely dents it.

The resolution is that the Sun does not burn at the typical energy. It burns in a window that neither factor predicts on its own, because one of them falls with energy and the other rises, and only their product has a maximum. That product is the thing almost nobody outside a nuclear-astrophysics course has ever actually looked at.

Two things in that panel are worth stopping on.

The first is the share of the rate inside the window. Around four fifths of the Sun's fusion comes from a band a few keV wide, centred several times above the mean thermal energy: a tail that classical physics says is empty and that quantum mechanics says is merely very thin. The integrand ratio tells you the price of being ordinary. A pair of protons at the Gamow peak reacts tens of times more readily, per unit of energy, than a pair at the mean thermal energy, and the peak sits where classical mechanics has already given up.

The second is that the textbook ΔE = 4√(E₀kT/3) Gaussian is an approximation and the panel shows you its error rather than hiding it. For the pp reaction in the Sun it is wrong by a few percent, because the true integrand is visibly skewed toward high energy while the Gaussian is symmetric. Switch to ¹⁴N + p and the error shrinks, because a taller barrier makes the peak sharper and the saddle-point expansion better. That is the approximation telling you exactly when to trust it.

Why the Sun burns hydrogen and a heavier star burns carbon

The exponent ν in that panel is the whole story of stellar structure in one number. It comes out of the same closed form for every reaction, and it moves only because the charges move. For p + p in the Sun it is near 4. For ¹⁴N + p, the slowest step of the CNO cycle and therefore its pacemaker, it is near 20. Bethe wrote down 4 and 24:

The proton-proton reaction (21), although it predicts the correct energy production in the sun, has a rather weak dependence on temperature. According to (19), (20), it behaves about as T⁴. H. A. Bethe, "Energy production in stars", Nobel Lecture, 11 December 1967.

Making reasonable assumptions of the reaction strength S(E), on the basis of general nuclear physics, I found in 1938 that the carbon-nitrogen cycle gives about the correct energy production in the sun. Since it involves nuclei of relatively high charge, it has a strong temperature dependence, as given in (19). The reaction with ¹⁴N is the slowest of the cycle and therefore determines the rate of energy production; it goes about as T²⁴ near solar temperature. Bethe, same lecture. The reactions of carbon and nitrogen with protons are the subject of his 1939 paper, Phys. Rev. 55, 434.

His first number is ours. His second is not, and we are not going to smooth that over: the same closed form gives 3.85 for p + p and 19.7 for ¹⁴N + p at a core temperature of 15.54 MK, and 24 is distinctly steeper than 19.7. What the formula does say is that ν climbs as the gas cools, because the Gamow peak retreats further into the tail. Select ¹⁴N + p on Instrument 3 and drag the temperature down: ν passes 24 near 8.7 MK. Whether Bethe was rounding, quoting a different reference temperature, or working from a rate calculation this two-term approximation does not capture, the lecture does not say.

Two rates with exponents 4 and 20 must cross somewhere. Where they cross is where a star stops being a proton-proton star and becomes a CNO star, and that crossing point is the reason the main sequence has the shape it has. It is also exquisitely sensitive to what you assume is in the gas, which is exactly how the field spent twenty years disagreeing about our own Sun.

Why the first step is so slow that we exist

Two protons that tunnel to contact still, overwhelmingly, do nothing. There is no bound state of two protons: a diproton is not a nucleus. For the collision to produce anything permanent, one of the two protons must turn into a neutron during the collision, which is a weak interaction, and the collision lasts of order 10⁻²¹ seconds. The weak force almost never manages it. That failure is our entire biosphere's budget.

Everything after that first step is fast, and it branches. Solar Fusion III's Figure 1 lays out the shares, and all of them are standard-solar-model outputs S model output rather than measurements: 99.8 % of first steps are p + p and 0.24 % are the pep reaction; of the ³He that results, 84.8 % is consumed by ³He + ³He, which closes the chain as pp-I, and 15.2 % by ³He + ⁴He, which opens pp-II and pp-III; only 0.016 % of the chain gets as far as ⁷Be + p, the branch that makes the ⁸B neutrinos that dominated forty years of solar neutrino experiments; and 0.000025 % is hep. The CNO cycle sits alongside all of it, worth about 1 % of the Sun's luminosity, and it is the subject of the last third of this page because that 1 % is the only direct assay of the core's composition anyone will ever have.

You can read the size of the bottleneck straight off one published table, without any theory at all: take the S-factor of the first step and the S-factor of the very next step, which is ordinary electromagnetism, and divide.

So how do we know any of this, rather than telling ourselves a consistent story?

Everything so far is a calculation about a place nobody can see. The Sun's centre is opaque: no photon made there reaches us as itself. There is exactly one direct probe, and it is the neutrinos, which leave the core essentially without interacting and arrive here eight minutes later carrying the reaction-by-reaction bookkeeping of the fusion that made them.

That gives a test with real teeth. If fusing hydrogen to helium powers the Sun, then a specific weighted sum of the neutrino fluxes must equal the Sun's measured brightness, because each neutrino species is emitted in a reaction that also released a definite amount of heat. Bahcall wrote the constraint out in general in 2002. It is arithmetic on nuclear masses, and it either closes or it does not.

The weight α on each flux is the heat the star actually kept from the reactions that made that neutrino, and it is a difference of atomic masses. For a pp-chain neutrino the bookkeeping is easy: every helium nucleus the chain builds releases the full Q and emits exactly two neutrinos, so α = Q/2 − <Eν> is exact. The CNO cycle is not so obliging. It emits its two neutrinos at very unequal points around the loop, one after a single proton capture and the other after three more, so halving Q is simply wrong there and gives a ¹³N coefficient nearly four times too large. Bahcall instead takes the mass difference across the arc of the cycle that ends in each neutrino, and so does the table below. The fourth column is his published coefficient, so you can see for yourself which of our eight the arithmetic reproduces and which it does not.

The neutrinos are also a thermometer, except when they are not

Because the pp-chain reaction rates go as steep powers of temperature, the fluxes read out the central temperature of a model. Bahcall and Ulmer measured those powers by running a thousand detailed solar models and fitting. Here are five published solar models built with the same evolution code, differing only in the assumed chemical composition, with their central temperatures and their fluxes. Fit the same power law yourself.

With all five models in the fit, which is where the instrument starts, the pp and ⁷Be exponents land inside Bahcall and Ulmer's error bars. The ⁸B exponent comes out at about 18.4 against their 24 ± 5, so it sits just below the lower edge of their bar, close enough to be a near miss rather than a contradiction. The three CNO exponents are not close to anything: they come out around 40, 44 and 59 against published values of 24.4, 27.1 and 27.8, with quoted errors of a few tenths.

That is not a broken fit; it is the physics announcing itself. A pp-chain neutrino flux depends on the core only through its temperature, so a single number T_c is very nearly a sufficient statistic for it. A CNO flux carries a direct factor of the catalyst abundance as well, and these five models move the abundance on purpose. No power of temperature can absorb that. Bahcall and Ulmer's exponents came from a thousand models in which many inputs varied at once; these five vary composition and nothing else, so the two experiments are asking different questions, and the CNO fluxes are where the difference between the questions becomes enormous.

Which is the entire premise of the CNO neutrino programme. A ¹³N or ¹⁵O neutrino is not another thermometer. It is a chemical assay of the solar core, and it is the only one there will ever be. Borexino performed it in 2020 and again, with a different method, in 2023.

Here, we report the direct observation, with a high statistical significance, of neutrinos produced in the CNO cycle in the Sun. BOREXINO Collaboration, Nature 587, 577 (2020). Their 2023 final analysis reports R(CNO) = 6.7 (+1.2/−0.8) counts per day per 100 tonnes and rejects the no-CNO hypothesis at greater than 5σ.

Detecting the CNO neutrinos is not the same as settling what the Sun is made of, and the difference is worth stating in the review's own words rather than ours:

Results show a ∼2σ tension with LZ metallicity determinations, while being in better agreement with HZ mixtures. While the error budget is presently dominated by the uncertainty of the Borexino CNO neutrino measurement, a significant contributor to the error (∼10%) is nuclear, due to uncertainties in S₁₁₄, S₃₄, and S₁₇. Acharya et al., "Solar fusion III", Rev. Mod. Phys. 97, 035002 (2025), Sec. II. LZ and HZ are the low- and high-metallicity abundance sets.

Two sigma is a lean, not a verdict, and about a tenth of the error is not astrophysics at all: it is three nuclear cross sections, one of which is the CNO bottleneck S₁₁₄ you can already drag on Instrument 4. The only direct assay of the solar core's chemistry currently points at the high-metallicity side, and it does not yet point hard.

Three places this is still genuinely open

What follows is not a summary of settled science. These are live disagreements, and each one is a control you can move.

One. The number that sets the Sun's clock has never been measured

The cross section of the driving reaction of the pp chain, ¹H(p, e⁺ν)²H, is too small to be measured and thus must be taken from theory. Acharya et al., "Solar fusion III", Rev. Mod. Phys. 97, 035002 (2025), Sec. I.C.

Two reactions in the chain have never been done on a laboratory floor, and this is the one that matters: the same review says "with the exception of the minor hep branch, all other reactions and decays have been studied in laboratories". So the unmeasured set is exactly p + p, which sets the Sun's clock, and ³He + p, which supplies about one solar neutrino in 10⁷. Neither has been done and neither probably can be. Worse, the published uncertainty on the first got larger between the 2011 evaluation and the 2025 one: Solar Fusion II recommended 4.01(1 ± 0.009) × 10⁻²⁵ MeV b, Solar Fusion III recommends 4.09(1 ± 0.015) × 10⁻²⁵ MeV b, "where we have added the correlated and uncorrelated errors linearly to be conservative".

Now, the trap. The obvious thing to do is perturb this number and watch the Sun's brightness move. That would be false. The Sun's luminosity is measured, and solar models are calibrated to reproduce it by adjusting the initial helium abundance and the mixing length. Change the pp cross section and the luminosity does not budge. What moves is the central temperature the model needs in order to deliver that fixed luminosity, and after it, the neutrino fluxes that depend steeply on temperature.

Two. Nobody knows what the Sun is made of, to the precision that matters

Solar models employing the LZ abundances fail to reproduce most helioseismic probes of solar properties. This disagreement constitutes the so-called solar composition problem ... that has defied a complete solution. All proposed modifications to physical processes in SSMs offer, at best, only partial improvements in some helioseismic probes. Acharya et al., "Solar fusion III", Rev. Mod. Phys. 97, 035002 (2025), Sec. II.

Redoing the solar photospheric spectrum with three-dimensional atmosphere models lowered the inferred metal abundance, and it broke the agreement between solar models and helioseismology that the older, higher abundances had enjoyed. Twenty years later it is not fixed. Put the two models on trial against the two cleanest seismic numbers and run the verdict yourself.

Three. Screening, which can counterfeit a solution

Electrons partially cancel the charge the two nuclei see, which lowers the barrier a little. In a laboratory it happens because of the atomic electrons of the target; in the Sun it happens because of the plasma. Both matter at the percent level, and neither is fully under control.

In the laboratory the effect is measured, and it comes out too big:

The values of U_e extracted from experiment using gaseous targets have often exceeded the adiabatic limit, defined in atomic physics as the difference between the electron binding energies of the separate atoms in the entrance channel and that of the composite atom ... This in turn has generated unease about the reliability of the values for S_b(E) extracted from laboratory measurements for use in astrophysics. Acharya et al., "Solar fusion III", Rev. Mod. Phys. 97, 035002 (2025), Sec. XII.

Since the astrophysical S-factors are extracted from laboratory data by removing a screening correction, an unexplained screening potential is an unexplained systematic sitting under every extrapolation to solar energies. Watch what it does.

And in the Sun, the same small correction is large enough to be dangerous, because it is the same size as the thing everyone is arguing about:

Nevertheless, we find that mild modification of the nuclear screening factors can re-match low-metallicity model predictions to observed fluxes, although it does not restore the agreement with the helioseismic frequency ratios. Salmon, Buldgen, Noels, Eggenberger, Scuflaire & Meynet, A&A 651, A106 (2021), abstract.

Read the second half of that sentence carefully, because the first half alone is the version that gets repeated. A screening tweak can buy you the neutrinos and still lose you the sound speed. That is what an unsolved problem looks like from the inside: several partial fixes, each of which purchases one agreement at the cost of another.