Astronomy & Celestial Mechanics
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Planet composition — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Mass against radius, and what lies between the curves. Planetary radius against mass on logarithmic axes, in Earth units, with composition curves computed from interior models rather than drawn through the points. The solid-planet curves are R ∝ M^(1/3.7) — flatter than a constant density because a heavier planet compresses itself — and the hydrogen curve turns over near three Jupiter masses, where degeneracy pressure takes over and adding mass makes the planet smaller. A mass alone or a radius alone places a planet on a line; only both together place it between two curves, and that is the whole argument for measuring a planet twice.

    Two methods, and one density

    A transit gives a radius. A wobble gives a mass. Neither says what a planet is made of, and the two together say it in one number — which is the only reason both are worth doing on the same object.

    part 1 · exoplanets
  2. One mass and one radius, and every composition that gives them. A planet of 5 Earth masses and 1.6 Earth radii, and the compositions consistent with it. The horizontal axis is the fraction of the planet's mass in an iron core and the vertical axis the fraction in a water layer outside the rock; the heavy curve is every pair that reproduces the measurement exactly, and the band around it is what the 0.05 Earth-radius uncertainty allows. The answer is a curve, not a point, and that is not a failure of precision. Two numbers cannot determine three components: a planet can be made denser by adding iron or lighter by adding water, and along this locus the two changes cancel exactly. The ends of it are not small variations on one planet. At the left is a body with no iron at all and 0 per cent of its mass in water; at the right, one with an iron core like Mercury's and 23 per cent water. Those have different formation histories, different interiors, different everything, and the same mass and radius to the precision anybody can measure them. Breaking the degeneracy needs an observation that is neither a mass nor a radius. The usual one is a transmission spectrum, which measures the atmosphere's scale height and so its mean molecular weight — a hydrogen envelope and a steam envelope differ by a factor of nine in that, and the corresponding factor in the size of the spectral features. What the picture assumes is that the planet is differentiated into clean layers, which is the standard assumption and is false in detail: water dissolves into silicate melt at these pressures, and a mixed interior sits at neither end of this curve.

    One density, and every planet that has it

    A mass and a radius are two numbers, and a differentiated planet has at least three components. The set of compositions matching a measurement is therefore a curve rather than a point — and its two ends are a body with no iron and half its mass in water, and a body with a Mercury-like core.

    part 2 · exoplanets
  3. The inflation threshold at 2·10⁵ W m⁻², and the 0.69 R_J above it. Radius against the starlight received, for a population of Jupiter-mass planets. The horizontal line is what a structural model gives for a cold, old, Jupiter-mass ball of hydrogen and helium: 1.06 Jupiter radii, and it hardly depends on mass at all in this range, because degeneracy is beginning to set in and the mass–radius relation is flattening toward its turnover. Planets receiving less than about 2·10⁵ watts per square metre sit on that line, with a median of 1.06, which is the control the rest of the figure depends on: the models are not wrong in general. Above the threshold the radii climb, reaching a median of 1.75 — half again the size a cold planet of the same mass can be — and the onset is sharp enough to be called a threshold rather than a trend. Starlight by itself will not do this. Irradiation is absorbed high in the atmosphere and re-emitted from there; it slows the escape of heat from below, which delays contraction, but it cannot deposit energy beneath the radiative–convective boundary, and it is the interior entropy that sets the radius. So the excess is evidence for a mechanism that carries roughly half a per cent of the incident flux down to pressures of tens of bars — ohmic dissipation of currents driven through a partly ionised atmosphere, breaking gravity waves, and tidally forced turbulence are the candidates, and the threshold is the number each of them has to reproduce. The points are a synthetic population from a seeded generator, not a catalogue; what is real is the threshold, the size of the excess, and the fact that the un-irradiated planets sit exactly where they should.

    A radius no cold planet is allowed

    A Jupiter-mass ball of hydrogen has a maximum size, and it is about 1.06 Jupiter radii however old or young it is. Hundreds of hot Jupiters are half again that, and the excess switches on sharply above a threshold in the starlight they receive — which means something is putting energy in deep.

    part 3 · exoplanets
  4. A gap 1 per cent deep that became 63. The distribution of planet radii, drawn three ways: the underlying distribution with a gap in it, the same distribution convolved with a 25 per cent stellar radius error, and convolved with a 5 per cent one. Every planet radius is the transit depth's square root multiplied by a stellar radius, so an error in the star is an error in the planet, and a population of planets inherits the population of stellar errors as a smearing. The gap is 1 per cent deep at the old precision and 63 at the new one, and its centre does not move — a symmetric smearing hides a feature without displacing it. That is what happened when parallaxes for a hundred thousand planet hosts arrived: no new planets were observed, and a feature that had been marginal became unambiguous.

    A planet radius is a stellar radius

    A transit measures a ratio and nothing else. Every planet radius ever published is that ratio multiplied by a stellar radius that came from somewhere entirely different, so a population of planets inherits the errors of a population of stars — and when the stars were measured better, a feature nobody could see became unmistakable.

    part 4 · exoplanets
  5. The radius a rocky planet would have, by core fraction: at 5 M⊕ an Earth-like and a Mercury-like planet differ by 8%. Radius against mass for rocky planets of one to eight Earth masses, from Zeng and collaborators' interior models, for core mass fractions of 0, 0.33, 0.5, 0.7, 1 — from a planet with no iron core to one that is all iron. The shaded band is the range of core fractions planets would have if they kept their host stars' iron, magnesium and silicon in the stars' own proportions, for stars whose iron-to-magnesium ratio is within 0.15 dex of the Sun's: 0.26 to 0.41. The Earth, at 0.325, sits in it. At five Earth masses an Earth-like planet and one with a core fraction of 0.7, like Mercury, differ in radius by 8 per cent — a difference that transit and radial-velocity measurements can now resolve.

    More iron than the star could give

    A rocky planet is made of what its star was made of, minus the gases: iron, magnesium and silicon in the star's proportions, which for the Sun predict an iron core of a third of the planet's mass — and the Earth's core is 32.5 per cent. A handful of small exoplanets are so dense that their cores must hold seven-tenths of their mass, more iron than any star known to host planets could have supplied. Either something removed most of their rock after they formed, or the rock and the metal were separated before they did.

    part 5 · exoplanets
  6. A second peak, seen through three precisions. The distribution of measured core fractions for a population in which 90 per cent of planets sit at their stars' prediction (0.32, with an intrinsic scatter of 0.03) and 10 per cent at 0.65 (scatter 0.08), each planet compared with its own star, observed with measurement errors of 0.04, 0.08, 0.15 in core fraction. At 0.04 the dense group is a separate hump; at 0.08 it is a shoulder; at 0.15 it has become a long upper tail on a single peak, indistinguishable by eye from a population that is simply broad. The current measurements sit between the second and third curves.

    A second peak the errors have to let through

    Whether the densest small planets are a population of their own or the far edge of the ordinary one is not a question about any single planet. It is a question about the shape of a histogram seen through measurement errors as wide as the gap it is looking for, and the number of planets it needs rises fifteen-fold for every factor of three in lost precision — which puts the answer a few dozen planets beyond the samples that first suggested it.

    part 6 · exoplanets
  7. Rise, steady state and decline. The mass of magnesium (solid) and iron (dashed) in a white dwarf's mixing layer, each as a fraction of its steady-state value for magnesium, during an accretion episode that delivers both at equal rates for 5 magnesium sinking times and then stops; time in units of the magnesium sinking time. Iron sinks 1.6 times faster. While they rise, both grow together and their ratio is the accreted ratio; once accretion has run for a few sinking times each element saturates at its rate times its own sinking time, so the layer holds 0.63 times the accreted iron-to-magnesium ratio; after accretion stops iron vanishes first, and 7 sinking times later the layer holds 0.009 of the accreted ratio. The same body seen at three moments reads as three compositions.

    A planet read in the ashes of its star

    A white dwarf's gravity drags every heavy element out of its visible surface, so a white dwarf showing magnesium, iron and silicon is swallowing rock now. Its spectrum is the only place where the bulk composition of a rocky body round another star can be read element by element. But each element sinks at its own rate, so the ratio seen depends on when in the meal the spectrum was taken — and only the hottest hydrogen white dwarfs say.

    part 7 · exoplanets

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