Astronomy & Celestial Mechanics
the sky, computed
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Fusion — the series

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

    part 1 · stars
  2. A furnace with the thermostat taken out. Left, the two pressures in a red giant's helium core at 10⁶ g/cm³. The ideal-gas pressure rises with temperature, as the whole regulation of a star depends on it doing; the degenerate electron pressure is a horizontal line, because the exclusion principle does not know the temperature. At the ignition point near 10⁸ K the degenerate term is 5.1 times the gas term, so a temperature rise there raises the total pressure by almost nothing and the core does not expand. Right, what that costs. Both curves start at the same temperature with the same triple-alpha rate, whose logarithmic slope is 41 — the famous T⁴⁰, differentiated here rather than quoted. The regulated core settles; the degenerate one, with nowhere to put the extra energy, goes vertical. The flash reaches 10¹⁰ solar luminosities for a few seconds and not one photon of it is seen: every erg goes into lifting the degeneracy, and the star's visible response is to become fainter and settle on the horizontal branch.

    The resonance that had to exist

    Three alpha particles cannot meet at once, and the two-body intermediate falls apart in 8×10⁻¹⁷ seconds. That is a hundred thousand times longer than a crossing time, which is just enough — provided a nuclear level sits at 7.65 MeV. Hoyle argued from the existence of carbon to the existence of the level, and it was found where he said.

    part 2 · stars
  3. The core's neutrino luminosity against its temperature, and the star's light. The luminosity carried off by neutrinos from pair annihilation in a burning core of 3 solar masses, in erg per second on a logarithmic axis, against the core's temperature in billions of kelvin, along a rough track on which the density rises as the cube of the temperature, with each burning stage marked at its own published temperature, density and core mass. The horizontal line is the star's light, about 1.5 × 10⁵ times the Sun's, nearly constant through all the late stages. The neutrino luminosity rises roughly as the ninth power of the temperature and overtakes the light at about 0.60 billion kelvin — between helium and carbon burning. From carbon burning on, carbon 76, neon 3·10⁴, oxygen 10⁵, silicon 7·10⁵ times the light is lost as neutrinos. The core is no longer cooled by the surface it shines through; it is cooled directly, from inside, by particles that leave without interacting.

    The last fuels burn for the neutrinos

    A massive star burns hydrogen for millions of years and silicon for about a day. The collapse in duration is not because the late fuels are poorer; it is because, from carbon burning on, the core loses more energy to neutrinos than the whole star gives out as light, and the neutrino losses rise as the ninth power of the temperature.

    part 3 · stars
  4. The abundances the slow neutron-capture process leaves behind. The abundance of each mass number produced by the s-process model at a mean neutron exposure of 0.3 per millibarn, relative to the iron seed, on a logarithmic axis. Dividing σN by the cross section turns the steps into peaks: nuclei at the closed shells capture so reluctantly that material piles up in them. The model's peaks sit at mass numbers 88, 138, 208 — strontium and zirconium, barium, and lead — the three s-process peaks of the solar system's abundances. The peak at barium stands 7 times above the abundances just below it. The abundance pattern of the heavy elements is a record of nuclear structure: the magic numbers of the shell model are written into the composition of every star, by a process that happened in other stars before the Sun formed.

    A flow that piles up at the magic numbers

    Half the elements heavier than iron were built one neutron at a time, in ageing stars, over thousands of years. The flow of matter up the chart of nuclei settles into a balance in which each nucleus is as abundant as it is reluctant to capture — and it jams at the closed neutron shells, leaving the peaks at strontium, barium and lead that every star's spectrum records.

    part 4 · stars

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