Astronomy & Celestial Mechanics
the sky, computed
Generator

The fusion-rate generator

Energy generation against core temperature
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 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.

8 essays call fusion-rate. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

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. Stars

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.

 Inside a star that is holding itself up (polytrope n = 3). Temperature, density and pressure through a star, as fractions of their central values, against fractional radius. All three come from one numerical integration of the Lane–Emden equation at index 3, which is the hydrostatic balance written for a gas whose pressure is a power of its density. The inner half of the radius holds 90% of the mass, and the outer half is nearly weightless — which is why the load, and so the temperature, is concentrated where the burning is. Stars

A star is held up by its own weight

A star has a central temperature because it has a central pressure, and it has a central pressure because everything above is pressing down. The nuclear reactions do not set that temperature — they obey it.

 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. Stars

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.

 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. Stars

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.

 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. Stars

The explosion that never reaches the surface

When helium ignites in a degenerate core the thermostat every other burning stage runs on is disconnected. The runaway reaches ten billion solar luminosities for a few seconds, and the star's visible response is to get fainter.

 Two stars at the same point of every other diagram, 4.1 times apart in one. Above: the gravity-mode period spacing against the large frequency separation, for 90 shell-burning giants and 55 core-burning ones. These are the same stars in every other measurement. They have the same luminosity, the same temperature, the same colour, the same surface gravity and — inside the band drawn — the same Δν, which means the same mean density; the global scaling relations return the same mass and the same radius for both. What separates them is a quantity that comes from nowhere near the surface: ΔΠ₁ is set by the buoyancy frequency integrated across the core, and a core that has ignited helium is expanded and convective, so its integral is smaller and its period spacing larger. The two sequences do not touch — 61 seconds against 248 — and the gap sorts a catalogue of tens of thousands of giants into stars burning hydrogen in a shell and stars burning helium in a core, by a Fourier transform of a light curve. Below: what that looks like in the spectrum itself, drawn over two radial orders. The number of mixed ℓ = 1 modes between consecutive radial modes is Δν/(ΔΠ₁ν²), so the shell burner has about 12 of them and the core burner about 3: the star with the larger period spacing has the sparser spectrum. The picture cannot show what the same modes are also used for and cannot settle — the splitting of each mixed mode gives the rotation rate of the core separately from the envelope, and the cores come out spinning some ten times faster than the surface and a hundred times slower than any model of angular-momentum transport predicts. Stars

Two stars only a Fourier transform can tell apart

A giant burning hydrogen in a shell and one burning helium in its core sit at the same luminosity, the same temperature and the same mean density. Every scaling relation returns the same mass and radius for both. The gravity-mode period spacing is fifty seconds for one and three hundred for the other.

 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. Stars

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.

 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. Stars

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.

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