Two methods, and one density

There is no measurement of what a planet is made of. There is a mass, there is a radius, and there is the ratio of one to the cube of the other — and everything said about rock, iron, water and hydrogen is an inference from that single number against a family of computed curves.

The number is worth the trouble, because it separates kinds of object that nothing else separates. And it exists only where two entirely different measurements have been made of the same body.

Each method delivers half of the answer #

A transit gives $R_p/R_\star$, and with a stellar radius, $R_p$. It says nothing whatever about mass: a hydrogen balloon and a ball of iron of the same size cast the same shadow.

A radial-velocity orbit gives $m_p \sin i$, and says nothing whatever about size: the star is pulled by a mass, and a mass is not a shape. Put them on the same object and something new appears. The transit fixes $\sin i \approx 1$, so the minimum mass becomes a mass; the mass and the radius give

$$\rho_p = \frac{3m_p}{4\pi R_p^3},$$

and the density is a statement about the material.

The Earth is 5.51 g/cm³; the Moon is 3.34; Jupiter is 1.33; Saturn is 0.69, less than water. Those four numbers separate iron-and-rock from rock-without-much-iron from hydrogen-and-helium without any further information, and they were the first things known about the planets that constrained their interiors at all.

The curves are computed, not drawn through the points #

The composition curves in the first figure come from integrating the equations of a self-gravitating sphere with a stated equation of state — the pressure at each depth supporting the weight above it, exactly the hydrostatic balance that holds a star up, with cold rock or iron in place of hot plasma.

Two features of the result are worth more than the curves themselves.

The exponent is not a third. A constant-density sphere has $R \propto M^{1/3}$. The computed curves go as $M^{1/3.7}$, because a heavier planet squeezes its own material into a denser state: the Earth’s core is compressed to about 13 g/cm³ against 7.9 for iron at the surface. Compression flattens the relation, and the flattening is a measurable prediction rather than a fitting parameter.

Above about ten Earth masses, a rocky planet stops being possible in practice. The curves continue mathematically, but a solid core that heavy accretes hydrogen from the disc faster than it can be lost, so nature does not build them. That threshold — the runaway accretion mass — is why the mass–radius plane is nearly empty between the rocky sequence and the giants.

At the other end, the hydrogen curve does something that ought to be startling and is instead familiar: it turns over. A cold hydrogen sphere’s radius rises as $M^{1/3}$ while the material behaves normally and falls as $M^{-1/3}$ once electron degeneracy is providing the pressure, with a maximum around three or four Jupiter masses. So Jupiter, a thirteen-Jupiter-mass brown dwarf and a 0.08-solar-mass star are all roughly one Jupiter radius across.

The gap in the middle, and the crowd on either side #

Between the rocky sequence and the giants there is a region of the plane that the solar system does not visit at all: bodies between two and four Earth radii, of which there are none between the Earth and Uranus. Around other stars they are the single most common kind of planet known.

The arithmetic of that last sentence is the reason the class is so awkward. Adding one per cent of a planet’s mass as hydrogen can add thirty per cent to its radius, because the gas sits on the outside where the gravity is weakest and the compression least. So a radius measurement is exquisitely sensitive to the envelope and nearly blind to the rock underneath — the opposite of the mass, which is nearly all rock. Two measurements that are each dominated by a different component is a good position to be in, and it is why this class can be characterised at all.

Kepler-51 b is the extreme case: about 3.7 Earth masses inside 7.1 Earth radii, a density of 0.06 g/cm³, roughly the density of the air in a room. Such a planet cannot be a scaled-down Jupiter; it is a small core inside an enormous, tenuous, and presumably temporary envelope.

What a density cannot say #

A density is one number, and a composition is a mixture — so the inference is degenerate, and severely.

A planet of 6 Earth masses and 2.5 Earth radii has a density of 2.1 g/cm³. That is consistent with a rocky core wrapped in a half-per-cent hydrogen envelope, and equally consistent with a body half water by mass with no envelope at all, and equally consistent with several intermediate mixtures. Three different worlds, one number.

Nothing about better measurement resolves this. It is not noise; it is that the map from composition to density is not injective. What resolves it, partially, is other information: the planet’s temperature and the mass of the star’s disc make some mixtures implausible, the host star’s own iron abundance is a weak prior on the planet’s, and — the only direct route — the composition of the atmosphere read from a transmission spectrum says what is on top of the envelope even when the density cannot say how much of it there is.

The honest form of the statement is therefore a family of allowed compositions rather than a composition, and the papers that report a single one are compressing an interval into a point.

What was actually measured #

HD 209458 b, 2000. Mass 0.69 Jupiter masses from the velocity curve, radius 1.35 Jupiter radii from the transit depth, and therefore a density of 0.37 g/cm³ — a third that of water, and about a quarter of what a cold hydrogen sphere of that mass should have. The planet is inflated, and twenty-five years later there is no agreed explanation. The candidates are all versions of depositing energy deep enough to slow the cooling: tidal heating, ohmic dissipation from winds moving through the planet’s magnetic field, downward transport of stellar heat. What is certain is the correlation — the more strongly irradiated a hot Jupiter is, the more inflated it tends to be, and below about 0.2 per cent of Jupiter’s insolation the effect vanishes.

CoRoT-7 b, 2009. The first transiting planet small enough to be rocky: 1.58 Earth radii, and a mass of about 4.8 Earth masses that took 106 radial-velocity measurements on an active star to establish. Density about 6.6 g/cm³ — an Earth-like composition, on an object with an orbital period of 20.5 hours and a day-side temperature above 2,000 K.

Kepler-36, 2012. Two planets in the same system with periods of 13.8 and 16.2 days — orbits separated by 10 per cent in distance — whose densities are 7.5 and 0.9 g/cm³. One is rock, the other is mostly envelope, and they formed within a hair’s breadth of each other. Whatever removed the atmosphere from one and left it on the other operated on a scale of a few per cent in orbital distance, which is the sharpest single piece of evidence for the process that carves the radius valley.

TRAPPIST-1, 2021. Seven planets whose masses come from their mutual perturbations rather than from any stellar wobble, and whose radii come from transits of a star small enough that the depths are half a per cent. The densities are all within a few per cent of each other and about 7 per cent below the Earth’s, which is a statement about a whole system’s building material rather than about one planet.

55 Cancri e, 2011–2016. Eight Earth masses inside 1.88 Earth radii, a density of 6.7 g/cm³, on an 18-hour orbit with a day-side temperature above 2,000 K. The density is consistent with an Earth-like interior and, when it was first measured with a larger radius, was consistent with a carbon-rich composition instead — which produced a widely reported “diamond planet” that a revised stellar radius removed. The episode is a clean demonstration of the point made twice above: the planet’s density is a stellar measurement wearing a planetary label, and it moves when the star does.

And the calibration nobody can repeat. Every one of these inferences is checked against exactly eight objects whose masses and radii are known independently and to absurd precision — the solar system’s planets, weighed by the spacecraft that flew past them. That is the entire calibration set. It contains no planet between 1 and 3.9 Earth radii, no planet hotter than 700 K, and nothing at all resembling the most common kind of planet in the galaxy. The curves are extrapolations away from the only points where they are anchored, and their agreement with the exoplanet data in the overlap region is the only reason to trust them outside it.

The same diagram read two other ways shows how much of it is the curves and how much is the points.

The mass from the clock instead of the star #

The density needs a mass, and the essay’s account gets it from the star’s reflex motion. There is a second route that uses no spectroscopy at all, and it works in exactly the systems where the first one struggles.

In a system with several transiting planets, the planets pull on each other. The consequence is that their transits do not arrive on a strictly periodic schedule: each is early or late by an amount that depends on the others’ masses, with a pattern whose period is set by how near the pair is to a resonance.

Measuring those transit timing variations and fitting a dynamical model returns the masses — from the same photometry that gave the radii, with no radial velocities involved.

The method’s reach is complementary rather than competing. Timing variations are largest for planets near a mean-motion resonance and for systems with low-mass stars, which is where the reflex velocity is smallest and hardest to measure; they are useless for a single planet, which perturbs nothing. So the two methods populate different parts of the same diagram.

They also disagreed, for a while, in a way that mattered. Masses from timing came out systematically lower than masses from velocities for planets of similar size — by tens of per cent — and the discrepancy was argued over for years. Part of it was a selection effect, since timing works best for the low-density planets near resonances, and part was a genuine difference in the populations the two methods reach.

A density derived from a timing mass and one derived from a velocity mass are therefore not interchangeable, and a catalogue that mixes them without saying which is which has an unlabelled systematic running through it.

The remedy adopted in practice is to keep the two populations separate in any statistical analysis and to prefer systems where both measurements exist, which is a few dozen planets out of several thousand.

Those few dozen are worth a disproportionate amount of telescope time for exactly that reason: they are the only objects that can calibrate one method against the other.

They are also the systems where a disagreement between the two would be visible at all, which is the more important half of the reason.

Where the picture stops #

A mass and a radius are not measured to the same accuracy. Radii from transits are typically good to a few per cent when the star is well characterised; masses from radial velocities are often good only to 20–30 per cent for small planets, because the signal is a few tens of centimetres per second against stellar jitter of a metre or two. Density goes as $m/R^3$, so the mass dominates the error budget for rocky planets and the radius dominates for giants.

Both depend on the star. $R_p = k R_\star$ and $m_p \propto M_\star^{2/3}$, so an error in the stellar parameters propagates into both. The densities in the Kepler catalogue shifted measurably when Gaia’s parallaxes revised the stellar radii, and some planets moved across composition boundaries as a result. Nothing about the planets changed.

A bulk density is an average. It says nothing about layering. Mercury’s density of 5.43 g/cm³ is nearly the Earth’s, and the interiors are not remotely alike: Mercury is 70 per cent iron by mass with a thin rock shell, the Earth is 32 per cent. Without a moment of inertia — which needs a spacecraft, or at least a resolvable figure — the layering is unconstrained.

Inflated giants break the curve entirely. A planet whose radius is set by an energy source rather than by its composition cannot be placed on a composition curve at all, and the hot Jupiters occupy a region of the plane where the curves have no predictive value. The honest treatment is to exclude them, which means the mass–radius relation is calibrated on the objects that are not doing anything interesting.

And the two numbers are not independent when one telescope measures both. A transit depth and a velocity amplitude come from different instruments, but both are scaled by properties of the same star, and those properties are usually derived from one spectrum. So the errors in $R_p$ and $m_p$ are correlated, and a density quoted with errors propagated as though they were independent is quoted too precisely. The correlation is not large for giants and is not negligible for the small planets, where it is the difference between excluding a composition and merely disfavouring it.

The generalisation #

Two measurements of the same object, neither of which means much alone, is the standard structure of astronomy rather than a special feature of this field.

A parallax and a brightness give a luminosity, and neither gives it alone. A colour and a luminosity place a star on the diagram that sorted the stars, from which a mass and an age follow — from two photometric numbers and a great deal of theory. An eclipsing double-lined binary gives absolute masses and radii, and is the only configuration that does; the entire calibration of stellar structure rests on the few hundred systems that happen to eclipse and show both spectra.

In each case the pattern is the same: one observable constrains a product or a ratio, another constrains a different combination, and the intersection is the quantity that was wanted. The awkward part is always the same too — the intersection is only as good as the assumption that both measurements refer to the same object.

That assumption fails more often than it sounds as though it could. A radial-velocity signal from a star and a transit signal from a fainter star two arcseconds away, blended in the same photometric aperture, will combine into a perfectly plausible density for a planet that does not exist. The routine defence is high-resolution imaging of every candidate host, which is why an exoplanet paper reporting a density usually contains an adaptive-optics image that appears to have nothing to do with the argument.

And two readings of the degeneracy the density leaves behind.

Where this goes next #

The mass–radius plane has structure in it that this essay has treated as background: the two populations of small planets, the near-emptiness between them, and the sharp upper edge of the rocky sequence. Those are not features of the measurement. They are features of how planets are made and unmade, and the histogram that shows them most clearly is a histogram of radius alone.

Later rungs on this anchor: interior models and their equations of state. The core mass fraction, and what a host star’s iron abundance predicts. Water worlds, and whether any have been identified. Inflated hot Jupiters and the candidate mechanisms. The moment of inertia, and why it needs a spacecraft. Ultra-short-period planets as stripped cores. Iron-rich planets and giant impacts. Density as a function of stellar irradiation. And the mass–radius relation of the solar system’s moons, which spans the same range with none of the ambiguity.