Astronomy & Celestial Mechanics
the sky, computed
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Energy transport — the series

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

    part 1 · stars
  2. A free parameter worth 88 kelvin across its plausible range. The effective temperature a stellar model predicts, against mass, for three values of the mixing-length parameter. The parameter has no derivation: it is the distance a convective blob is supposed to travel before dissolving, in units of the local pressure scale height, and it is fixed by requiring that a model of the Sun reproduce the Sun. The three curves span 88 kelvin, which at fixed luminosity is a radius difference of 1.5 per cent — comparable to the precision with which radii are now measured by interferometry and by eclipsing binaries. Every stellar age, every isochrone and every mass inferred from a position in the temperature–luminosity plane depends on the value chosen, and there is no reason beyond convenience to expect the solar value to apply to a red giant or to a metal-poor dwarf.

    A length nobody derived, fitted to one star

    Convection in a star is turbulent, three-dimensional and impossible to compute inside an evolution code. What is used instead is one number — how far a blob of gas travels before dissolving — fixed by requiring that a model of the Sun come out with the Sun's radius, and then applied to every star ever modelled.

    part 2 · stars
  3. Larger and cooler, along a line of constant light. The fractional change in effective temperature against the fractional change in radius for a star whose luminosity is unchanged, and for luminosities −10% and +10%, from L = 4πR²σT⁴. Holding the luminosity, a radius 10 per cent larger needs a temperature 4.7 per cent lower: about half the radius change, with the sign reversed. The shaded box is the range reported for red dwarfs in eclipsing binaries against standard models — radii 5 to 15 per cent too large, temperatures 3 to 5 per cent too low — and it lies along the line of constant luminosity.

    Too large and too cool by just enough to keep the light

    Red dwarfs in eclipsing binaries, whose radii are measured directly, come out several per cent larger than stellar models predict for their masses and a few per cent cooler. They are not brighter or fainter. The light is decided deep inside, by mass and opacity, and whatever disturbs the convecting outer layers can change only the radius and temperature through which that light leaves.

    part 3 · stars
  4. A reservoir so large that it does not notice. The fractional increase in the heat of the Sun's convective envelope if a fraction 0.001, 0.003, 0.01 of the surface is held under spots, against time. The envelope holds about 7×10⁴⁵ erg, 58 thousand years of the Sun's light. Spots covering 0.3% of the surface for an eleven-year cycle add 4.2e-7 of that, a temperature change of 0.4 parts per million at most, spread over the whole envelope. So the blocked heat simply remains in the star, and the surface flux shows the deficit. It levels off only after the envelope's thermal time, when the star has adjusted.

    A spot whose missing heat goes nowhere

    A sunspot blocks most of the energy trying to come up beneath it, and a large group visibly dims the whole Sun by a thousandth. Energy is conserved, so the blocked flux ought to come out somewhere else, and for a century observers looked for the bright ring around every spot. There is none to speak of. The heat has gone into the rest of the convection zone, spread so fast and so widely that the surface cannot notice, and it will stay there for longer than human history. For the time being the Sun simply is fainter, without any contradiction.

    part 4 · stars

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