One density, and every planet that has it

A transit gives a radius and a wobble gives a mass, and the two together give a density — which is the reason both are worth doing on the same object. That essay ended by noting that a density still does not determine a composition uniquely. This one is about what the ambiguity actually looks like, because it is much larger than the phrase suggests.

A differentiated rocky planet has, at minimum, an iron core, a silicate mantle and possibly a water layer. That is three components. A mass and a radius are two numbers. The system is underdetermined by construction, and no improvement in precision changes that.

Why the trade is exact #

Iron is dense and water is not, so adding iron shrinks a planet of fixed mass and adding water swells it. Along the locus in the figure the two changes cancel exactly, which is why the curve is a curve rather than a short segment.

The reason the cancellation is so nearly perfect over such a wide range is that the density contrast between the components is large. Iron at these pressures is about eight times denser than water ice, so a small change in the iron fraction is compensated by a modest change in the water fraction, and the locus is shallow. If the components were closer in density the locus would be steeper and shorter, and the degeneracy would be less severe.

There is a second, independent degeneracy hiding in the same figure. Water and hydrogen are both light, so a planet can be made large either by adding a thick water layer or by adding a thin hydrogen envelope — and a hydrogen envelope of one per cent of the planet’s mass swells it about as much as thirty per cent water. Since a per cent of hydrogen and a third of the planet’s mass in water are enormously different in what they say about formation, that is the more consequential ambiguity of the two.

How wide the ambiguity is, in the quantities people care about #

It is worth converting the locus into the statements a reader would want to make about the planet.

At the left-hand end, the planet has no metallic core. Its interior is silicate rock under a water layer nearly a thousand kilometres deep, and there is nothing in it to run a dynamo. At the right-hand end, the planet is nearly half iron by mass — an extreme even by Mercury’s standards — under a water layer half again as deep. Between those, every intermediate.

Those two bodies would have formed differently, would have different magnetic fields, different surface conditions, different outgassing histories, and different prospects for anything living on them. The measurement that cannot distinguish them is the best measurement anybody has, and it is quoted to three significant figures.

The honest way to present such a result is as a joint posterior rather than as a best-fit composition, and the better papers do. The habit of quoting “a planet of 5 M⊕ and 1.6 R⊕ has a 20 per cent core” — the best fit under an assumed prior — is common and misleading, because the prior is doing most of the work.

What breaks it, and what it costs #

Only a measurement that is neither a mass nor a radius can help, and there are three of them.

A transmission spectrum. During a transit, some starlight passes through the planet’s atmosphere, and molecules absorb at their own wavelengths. The size of the resulting features depends on the atmosphere’s scale height, which is proportional to temperature over mean molecular weight. A hydrogen atmosphere, with a mean molecular weight around 2.3, has a scale height nine times that of a steam atmosphere at 18 — so its spectral features are nine times larger, and a flat spectrum where a hydrogen envelope would give a large signal is evidence against the hydrogen solution.

The stellar composition. A planet’s iron-to-silicon ratio is expected to reflect the disc it formed in, which reflects the star. Measuring the star’s iron and magnesium and silicon abundances gives a prior on the planet’s core fraction — not a measurement, and one that fails for planets whose formation was violent, but a considerable narrowing of the locus.

Population statistics. If a large sample of planets is measured, the distribution of masses and radii constrains the population’s compositions even when no individual planet is determined. The radius valley — a deficit of planets at about 1.8 Earth radii — is the standard example, and it exists because planets with thin hydrogen envelopes lose them and planets without never had them.

None of the three is free. A transmission spectrum needs a bright star and many transits, and a flat one is ambiguous between a heavy atmosphere and a cloudy light one. A stellar abundance is a prior rather than a measurement of the planet, and it fails for exactly the planets whose formation was unusual — which are the interesting ones. And a population argument says nothing about any individual object.

There is a fourth route that works only rarely and is worth mentioning because it is clean. If a planet is in a multi-planet system with strong mutual perturbations, the timing variations of its transits give a mass independently of the star’s reflex motion — and in a few systems both methods have been applied to the same planet and disagreed, which is a check nobody enjoys having.

What the models themselves rest on #

The composition curves are not empirical. They are the output of interior structure calculations, and those calculations require the density of iron, of silicate rock and of water ice at pressures of megabars and temperatures of thousands of kelvin.

Those conditions are reached in the laboratory, and the measurements are not easy. Iron’s equation of state above about two megabars comes from shock compression experiments, and the different experimental techniques — gas guns, laser drive, diamond anvils with dynamic ramp loading — do not entirely agree. Water’s high-pressure phase diagram contains a superionic phase whose existence was disputed for two decades and whose density is materially different from what earlier models assumed.

A second assumption inside the same curves is easier to state and just as consequential: that the core is pure iron. The Earth’s is not. Its density is some ten per cent below pure iron at the same pressure, which is attributed to a few per cent by mass of a light element — sulfur, silicon, oxygen or carbon, and the argument over which has run for sixty years. Ten per cent in the core’s density moves the core mass fraction inferred from a given mass and radius by several per cent absolutely, in the same direction for every planet, and no measurement of an exoplanet can detect it. So the composition curves carry a systematic set by a debate about the Earth’s own interior, and a paper quoting a core fraction to a per cent is quoting a precision the underlying equation of state does not have.

The consequence is that a published core mass fraction carries an error bar from a laboratory measurement, and that error bar is usually not propagated. Two groups using different equations of state and the same mass and radius routinely publish core fractions differing by more than either quotes.

Where the degeneracy actually bites #

It matters most for the planets that are most interesting, which is unfortunate.

Sub-Neptunes, between about 1.7 and 4 Earth radii, are the commonest kind of planet in the Galaxy and have no analogue in the solar system. Whether a given one is a rocky core with a per cent of hydrogen, or a water-rich body with a steam envelope, is a question the density cannot answer, and the two possibilities imply completely different formation locations — inside or beyond the line where ice counts as rock — and therefore different migration histories.

Potentially habitable planets. Whether a temperate planet around an M dwarf is a rocky world with an ocean or a mini-Neptune with a hundred bars of hydrogen is the whole question, and their densities can be identical. Where the zone itself lies is a separate argument that this one has to be settled before.

The most massive rocky planets. Above about eight Earth masses, the composition curves converge: compression flattens the mass–radius relation and a given radius is consistent with a wider range of compositions. The degeneracy gets worse with mass, not better.

An argument that is not about exoplanets #

The degeneracy is not a peculiarity of distant systems. It applies to every body whose interior has been inferred from a mass and a radius, and the solar system’s own cases are instructive because there the answer is sometimes known independently.

Mercury’s density is high, and for a century that was read as a large iron core — correctly, as it turns out, but the reading was not forced by the density alone. What settled it was a moment of inertia measured from the planet’s spin state, which is a third number and breaks the degeneracy exactly as a transmission spectrum does for an exoplanet.

Ganymede and Callisto have nearly the same mass and nearly the same radius, and therefore nearly the same density. Their moments of inertia are 0.3115 and 0.3549, and those two numbers say that one is fully differentiated with an iron core and the other is a barely-sorted mixture of rock and ice. Two bodies indistinguishable by density, distinguished decisively by a flyby.

The lesson transfers directly. What every exoplanet composition is missing is the third measurement, and the solar system shows what that third measurement is worth: not a refinement, but the difference between two qualitatively different objects.

Everything here is a ratio to a star #

There is a systematic running under every number in this essay that is easy to lose, and it is worth stating in its strongest form: nothing about a planet is measured; every quantity is a ratio to its star, and the star is measured separately.

A transit gives the ratio of the planet’s radius to the star’s. A radial velocity gives a quantity proportional to the planet’s mass divided by the star’s mass to the two-thirds power. So a planet’s radius carries the star’s radius error in full, and its mass carries two-thirds of the star’s mass error.

The density is worse than either, because it is a mass over a radius cubed, and the two stellar errors do not cancel. A ten per cent error in the stellar radius is a ten per cent error in the planetary radius and a thirty per cent error in the planetary density — which on the composition locus above is most of the width of the ambiguity.

The consequence is that published planet densities have moved substantially without any planetary measurement being repeated. Two things did it. Parallaxes from an astrometric satellite gave distances to the host stars, which fixed their luminosities, which fixed their radii — and a great many stars turned out to be larger than catalogued, because they were subgiants misclassified as dwarfs. And asteroseismology gave direct densities for the brightest hosts, which are the best stellar radii available anywhere.

Both revisions moved planetary radii upward for the misclassified hosts, by tens of per cent in the worst cases, which moved those planets across the composition curves and in several cases across category boundaries.

A planet’s composition is a statement about its star’s radius before it is a statement about the planet, and the largest improvements in exoplanet interiors over the last decade came from stellar astrophysics rather than from anything pointed at a planet.

The star as a prior, and where the prior fails #

The suggestion above that a star’s composition constrains its planets’ deserves expanding, because it is the cheapest of the three degeneracy-breakers and its failures are informative.

The reasoning is that a planet and its star form from the same material, so the ratios of the refractory elements — iron, magnesium, silicon — should be similar in both. Measuring those ratios in the star’s spectrum then predicts the planet’s core mass fraction, because the core is essentially the iron and the mantle is essentially the magnesium silicates.

Applied to the Sun and the Earth it works: the Earth’s core fraction inferred from solar abundances is about a third, and the measured value is close to that. Applied statistically to exoplanet hosts it also works, in the sense that the population’s densities track the population’s stellar iron abundances.

Where it fails is on individual objects, and it fails in one direction. There is a small population of planets far denser than any stellar composition allows — bodies that appear to be almost entirely iron. No formation process starting from a disc of the observed composition makes such an object.

The standard explanation is a giant impact: a collision energetic enough to strip most of the silicate mantle away, leaving the core. That is also the leading explanation for Mercury, whose density is anomalous in the same direction and by a comparable factor.

So the prior is right for the population and wrong for exactly the objects whose densities are most striking — which is the same shape as every other statistical argument in this essay. A prior derived from a formation model cannot constrain a planet whose history was not the one the model describes, and the planets worth arguing about are disproportionately those.

What was actually measured #

Neither the mass nor the radius in the hero figure is an observation.

A radius is a fitted parameter of a transit model, and it carries a stellar-atmosphere systematic through the limb-darkening coefficients. It is also a ratio: what the transit gives is $R_p/R_\star$, so the planet’s radius is only as good as the star’s, and stellar radii come from a combination of parallax, photometry, spectroscopy and — for the best cases — asteroseismology.

A mass comes from the star’s reflex velocity, which is measured against the star’s own surface motion: granulation, oscillation and magnetic activity all produce apparent velocity shifts of the same size as a small planet’s signal. The masses of the smallest planets are therefore the noisiest quantities in the whole chain, and a twenty per cent mass error is a twenty per cent density error.

So the point in the mass–radius plane has an error ellipse, and the composition locus has to be intersected with it. In practice the published answer is a posterior distribution over compositions, and it is broad — not because the measurements are poor, but because the mapping is not invertible.

What the picture does not show #

The hero figure draws a clean two-component locus and every part of that cleanliness is an idealisation.

It assumes the planet is differentiated into distinct layers. Water dissolves into silicate melt at these pressures in quantities of tens of per cent, and a mixed mantle has a density between the layered alternatives — so the real locus is not a curve in a two-dimensional composition space but a surface in a larger one.

It assumes the planet is cold. A young, hot planet is larger at the same composition, by tens of per cent in the first few hundred million years, and the thermal state depends on the formation history and on the tidal heating. Applying a cold curve to a young planet returns too much water.

And it assumes there is no atmosphere. A hydrogen envelope of a tenth of a per cent of the mass adds several per cent to the radius, which on this locus is worth a great deal of water — and a tenth of a per cent of hydrogen is a quantity no measurement of the interior can exclude.

The degeneracy has one clean escape and it is worth ending on, because it is the only place in this essay where a radius says something a composition cannot fake.

What makes that argument different from everything before it is that no composition can produce those radii. A Jupiter-mass planet made of pure hydrogen, cold and old, has a radius near 1.06 Jupiter radii, and that is an upper bound rather than a modelling choice — nothing lighter than hydrogen exists to swell it further. A planet at 1.7 radii is therefore not a planet of unknown composition; it is a planet with an energy source, and the degeneracy that dominates the rest of the diagram simply does not apply.

That is the general shape of every escape from a degeneracy in this subject. The ambiguity is bounded by the extremes of the allowed range, so an observation that falls outside the range is informative no matter how badly the interior of the range is constrained. Most measured planets sit inside it, which is why most of them are ambiguous — and why the inflated ones, which are the least typical objects on the diagram, are the ones whose radii are actually saying something. The lesson generalises past exoplanets: a measurement that cannot distinguish between models inside their common range can still be decisive against the range itself, so the useful objects in any degenerate problem are the ones near or beyond an edge. Finding those is a different observational programme from characterising the typical case, and the two are often confused. A survey optimised to measure the typical planet well will find few of the informative outliers, and a survey built to find outliers will say little about the population — which is why the field runs both and why the tension between them is a standing argument about telescope time.

Where the ladder goes #

The first rung of this anchor established the density and flagged the ambiguity. This one has drawn it. The next is about a class of planet where the ambiguity is not the problem — where the radius is larger than any composition allows, so that something other than composition has to be supplying it.

There is also a rung about what happens when the third measurement arrives. Transmission spectroscopy at the precision now available does distinguish hydrogen from steam for the best targets, and the answers have been surprising: several sub-Neptunes have flat spectra where hydrogen was expected, and at least one has molecular features consistent with a hydrogen envelope over a deep water layer. The degeneracy is being broken one object at a time, and each break is a formation history rather than a composition.