Astronomy & Celestial Mechanics
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The mass–luminosity relation — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Luminosity against mass, against a slope of 3.5. Main-sequence luminosity against mass, both in solar units, on logarithmic axes, over the range 0.079 to 63 solar masses. The measured curve comes from the eclipsing binaries and is the same in every drawing of it; what changes here is what it is compared against. The dashed line is a pure power law of exponent 3.5, and the curve crosses it rather than following it — the local slope runs from about 2.3 at the bottom of the range, where the interiors are convective, through nearly 4 near a solar mass where bound-free opacity dominates, to about 3 among the massive stars where electron scattering does. Quoting one exponent across the whole sequence is a convenience and the places it fails are the places the interior physics changes. Because the slope is between three and four across most of the range, a small spread in mass becomes an enormous spread in output: the 63-solar-mass end is 3.0e+9 times brighter than the 0.079-solar-mass end.

    Mass decides everything, by a power of three and a half

    Two stars of the same mass are almost the same star. Double the mass and the output multiplies by eleven — which is why a modest range of masses produces a colossal range of stars.

    part 1 · stars
  2. An exponent between 2.3 and 4, not a number. The local logarithmic slope of the mass–luminosity relation against mass — the exponent that "L ∝ M^3.5" stands in for — read off the same fit to eclipsing binaries that the relation itself is drawn from. It is a step function with two corners, and the steps are 2.3 below 0.43 M☉, 4 0.43 to 2 M☉, 3.5 2 to 55 M☉. The horizontal lines are what homology predicts: a star in radiative equilibrium has L ∝ μ⁴M³/κ̄, which gives an exponent of 3 when the opacity is electron scattering and a constant, 5.5 when it is Kramers and depends on density and temperature, and 1 where radiation pressure holds the star up and the luminosity is pinned to the Eddington value. The measured steps sit between those predictions rather than on them, which is what a real star does — no star is homologous, the low-mass ones are convective throughout and the heavy ones have convective cores, and the fitted exponents are what is left after all of that. The useful statement is not that the exponent is 3.5 but that it is between 2 and 5 everywhere and 4 in the middle, and that the reason it moves is a change of which opacity is carrying the energy out.

    An exponent that is a slope, not a law

    The phrase “L goes as M to the three and a half” stands in for a curve with three straight pieces and two corners. The exponent is 2.3 below half a solar mass, 4 in the middle and falls towards 1 at the top — and each change of exponent is a change in which opacity is carrying the energy out.

    part 2 · stars
  3. Brighter by mass, fainter by temperature. Main sequences from 0.5 to 1.3 solar masses in luminosity against effective temperature, hotter to the left, for solar composition, [Fe/H] = −1.5, from homology applied to the solar-composition relation. The dots join stars of the same mass: lowering the metals moves each one up and to the left, so a 0.8 solar-mass star goes from 0.41 to 0.66 solar luminosities and from 5049 to 5879 K. The main sequence is steep, so a move of that size to the left carries a star past solar stars much brighter than itself: at a fixed temperature of 5800 K the metal-poor sequences lie 0.59 magnitudes below the solar one. A metal-poor star is brighter than a solar one of the same mass and fainter than a solar one of the same temperature — which is why such stars were called subdwarfs.

    Brighter for its mass and fainter for its colour

    Two stars of the same mass are almost the same star only if they are made of the same material. Take most of the metals out and the opacity falls, the radiation leaks out faster and the star settles brighter and much hotter — so far hotter that, at a given colour, it sits below the ordinary main sequence. The relation between mass and light has composition inside it, and every mass, age and distance read off it inherits that.

    part 3 · stars
  4. The carbon a star makes, and the carbon it needs. For a 30 solar-mass star born with no carbon, two quantities against its central temperature, on logarithmic scales: the carbon fraction its core needs for the CNO cycle to carry its luminosity (falling steeply, because a hotter core needs less catalyst), and the carbon the triple-alpha process makes in the hundred thousand years the core takes to contract (rising even more steeply). They meet at 108 million kelvin and a carbon fraction of 1.7 × 10⁻¹⁰: 1.7 parts in ten billion by mass, made by the star itself, are enough to switch on the cycle that runs every massive star of the present day. The contraction stops there.

    A star that has to make its own catalyst

    Every massive star now forming burns hydrogen through a cycle of reactions that carbon, nitrogen and oxygen catalyse, and the first stars had none of those elements. A massive star made of hydrogen and helium alone contracts and heats until its core reaches a hundred million kelvin, where helium begins to fuse into carbon; a few atoms of carbon in ten billion are enough to start the cycle, and the star settles there — three times hotter at its centre than any star of the same mass today, and smaller, and bluer.

    part 4 · stars

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