Astronomy & Celestial Mechanics
the sky, computed
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Solar neutrinos — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    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.

    part 1 · stars
  2. A flux that measures a temperature to 0.12 per cent. Neutrino flux against central temperature, in units of the standard model's, for the three main solar channels. The exponents are not arbitrary: each reflects how far up the Gamow peak the reaction has to reach, so the channel with the largest Coulomb barrier is the steepest. The boron-8 flux goes as roughly the twenty-fourth power, which means a measurement good to 3 per cent constrains the Sun's central temperature to 0.12 per cent — better than any other technique by an order of magnitude. The same steepness is why the flux is useless as a check on anything else: a stellar model whose central temperature is uncertain at the half-per-cent level predicts this flux to within a factor, and the disagreement between two model families is far larger than the measurement.

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

    The boron-8 neutrino flux from the Sun's core rises as roughly the twenty-fourth power of the central temperature. That makes it the sharpest thermometer in astrophysics and simultaneously the most fragile prediction — a model uncertain in its central temperature by half a per cent predicts the flux to within a factor.

    part 2 · stars
  3. A survival probability with a step in the middle of the spectrum. The probability that an electron neutrino made in the core of the Sun is still an electron neutrino at the Earth, against its energy, from the resonant conversion it undergoes in the Sun's own electrons: mixing angle sin²θ₁₂ = 0.307, splitting Δm² = 7.4·10⁻⁵ eV², and an electron density at the point of production of 90 moles per cubic centimetre. At low energy the matter potential is small beside the vacuum term and the probability is the vacuum average, 0.54 at 0.3 MeV (the flat limit is 0.55). At high energy the matter term dominates, the neutrino is born almost entirely in the heavier mass state, and the probability falls to 0.32 at 10 MeV, close to sin²θ₁₂. The two terms are equal at 2.1 MeV, which falls in the gap between the ⁷Be line and the bulk of the ⁸B spectrum. The points are the measured survival probabilities for each source separately — pp, ⁷Be, pep and ⁸B from Borexino, and ⁸B from SNO — and every one of them lies within about two standard deviations of the curve: low-energy neutrinos survive about half the time, high-energy ones about a third.

    A survival that depends on the energy

    A low-energy neutrino from the Sun reaches the Earth still an electron neutrino about half the time, and a high-energy one about a third of the time. The difference is made by the Sun's own electrons, and the energy at which it switches over — near two million electronvolts — sits in the gap between the neutrinos that never meet a resonance and the ones that all do.

    part 3 · stars
  4. The rise in survival, and what two detectors see of it. The survival probability of ⁸B electron neutrinos against neutrino energy (grey), and what two detectors measure of it against the energy of the electron they record: the ratio of the counted spectrum to the one expected without conversion, for electron scattering in water (blue) and for charged-current capture on deuterium (orange), both smeared by a water detector's energy resolution. Between 3 and 10 MeV the survival probability rises by 0.122. Scattering shows a rise of 0.044 over the same recoil energies, 36 per cent of it; capture on deuterium shows 0.063, 52 per cent.

    A rise a detector shrinks to a tilt

    Between 10 and 3 MeV the chance that a solar neutrino arrives as it left should rise by a third of its value, and no detector has clearly seen it. Counting is not the obstacle: a large water detector has many times the exposure the rise needs. What hides it is that scattering shows only a third of the rise, spread over the spectrum as a tilt of a few per cent, and that half of the evidence for the tilt sits where the radioactive background is and the other half where an error in the energy scale looks the same.

    part 4 · stars

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