A super-Earth's centre placed beyond the anvils by two numbers

The pressure at the centre of a body in equilibrium has a floor that assumes nothing — no composition, no temperature, no equation of state. Hydrostatic balance and the fact that density cannot be negative are enough to show that

$$P_c \ \ge\ \frac{GM^2}{8\pi R^4},$$

so a mass and a radius put a lower limit on how hard the centre is squeezed. For the Sun the limit is true and nearly useless: the Sun’s real central pressure is about five hundred times higher, because its density rises by a factor of a hundred towards the middle and the bound knows nothing about that concentration. The essay that introduced the bound left a practical question for planets, and it has a sharper answer than the Sun’s case suggests. For a planet made of rock and iron, the gap between the floor and the truth is not a factor of five hundred. It is a factor between three and seven, and it cannot be much larger, because rock and iron simply do not differ in density by enough.

That turns a bound into an estimate. A planet whose mass and radius have been measured has a central pressure known, from those two numbers alone, to within about a factor of two — enough to say which kind of laboratory experiment its interior corresponds to, before anything is assumed about what it is made of. For the super-Earths, the answer is that their centres lie beyond the reach of every static experiment, and for the heavier ones beyond the reach of most dynamic ones.

How far above the floor real bodies sit #

The six bodies with well-modelled interiors show the pattern. The Moon, nearly uniform in density with a core of only a few per cent of its mass — its moment of inertia, 0.393 of $MR^2$, says so — sits at 3.2 times its floor. Mercury and Mars, with substantial iron cores, sit near 5. The Earth, the most compressed and most differentiated of the four, sits at 6.3: its floor is 58 gigapascals and its centre is at 364. Jupiter, whose hydrogen is far more compressible than rock, is at about 11. The Sun is at more than five hundred.

The spread among the rocky bodies is a factor of two. The spread across all six is a factor of nearly two hundred. The bound is weak in exactly the cases where density varies a great deal through the body, and the only bodies in which it varies a great deal are the ones made of gas.

The floor itself comes from an elegant piece of reasoning. The pressure at the centre is the integral of the weight of each layer from the surface inward, and if the mass is regarded as the variable of integration rather than the radius, the integral can be bounded below using only the facts that the radius of each shell is less than the planet’s radius and that the enclosed mass grows from zero to $M$. What the derivation throws away is how the mass is distributed, and the size of the gap is a measure of how much that distribution matters.

Why even a uniform body sits at three #

The factor of three for a uniform body is worth understanding, because it is the price of making no assumption, and every number in the figure above is that price multiplied by something. In the derivation, every shell is treated as if it sat at the planet’s surface: the weight of a shell of mass $dm$ carrying the mass $m$ inside it is $Gm\,dm/4\pi r^4$, and replacing $r$ by the full radius $R$ makes each term as small as it could possibly be. For a uniform body the shells are spread through the volume, the inner ones much closer to the centre than $R$, and their true weights are larger. Integrating the true weights gives $3GM^2/8\pi R^4$ — exactly three times the floor.

That makes three the cost of not knowing where the mass is when the mass is spread as evenly as it can be. Nothing in a self-gravitating body in equilibrium spreads it more evenly than uniformly: a density that rose outward would be top-heavy and would overturn. So three is a second inequality, as firm as the first for any body that is not unstable: the central pressure is at least three times the floor whenever the density does not increase outward. The Moon, at 3.2, is as close to that limit as any natural body in the Solar System. Its rock is almost the same density from the crust to the small core, and its central pressure of about five gigapascals — the pressure a few hundred kilometres down in the Earth’s mantle — is within a few per cent of what a uniform sphere of its mass and size would have.

Everything above three is information about structure. For the Sun it is a factor of almost two hundred more, the signature of a core a hundred times denser than the mean. For the rocky planets it is at most a factor of two or so, which says that rock-and-iron bodies have little structure to hide. That is the difference the rest of this essay turns into a measurement.

A floor a dense core can raise only so far #

The reason rocky planets cluster so tightly becomes clear in the simplest model with any structure at all: a core of one uniform density inside a mantle of another. The ratio of central pressure to floor then depends on just two numbers, the core’s share of the mass and the ratio of the two densities. With no core, or a core of the same density, the body is uniform and the ratio is exactly three. A dense core raises it. But it raises it by a bounded amount, because a core that holds most of the mass is itself nearly a uniform body, and the ratio falls back towards three at both ends. Each curve has a maximum somewhere in the middle, near a core of a third of the mass.

The height of the maximum depends on the density contrast. For a contrast of 1.5 it is under four. For 2.5 — roughly what compressed iron against compressed silicate gives at the pressures of the Earth’s interior — it is 5.8. Even a contrast of four, more than any plausible pair of planetary materials, leaves the ratio below nine.

The real rocky planets sit where this predicts. Mars, with a core of a quarter of its mass and a density contrast near two, sits at 4.8 on its two-layer curve; the Earth, with a core of a third and a contrast of 2.4, at 5.6; Mercury, whose core is more than two-thirds of its mass, at 4.2. The Earth’s true value, 6.3, is a little higher than its two-layer value because each layer is also compressed by the weight above it, so density increases with depth within the mantle and within the core as well as across the boundary. That adds a modest amount, not a factor of several.

So for anything built of rock and iron, whatever the proportions, the central pressure lies between three and about seven times the floor. The lower limit is a theorem for any body whose density does not rise outward — the uniform body is the minimum — and the upper limit comes from the bounded density contrast of the materials. It is not a theorem, but there are no rocky materials that break it.

Which laboratory a planet’s centre belongs to #

The floor depends on the square of the mass and the inverse fourth power of the radius. Rocky planets grow in radius only slowly with mass — about as the 0.27 power, because their material is compressed by its own weight and grows denser as the planet grows — so the floor rises nearly in proportion to the mass. A planet of ten Earth masses has a floor of about half a terapascal, eight times the Earth’s.

What makes this useful is the laboratory scale it can be laid against. Static compression in a diamond-anvil cell holds a sample at a fixed pressure for as long as needed, measures its structure with X-rays, and heats it with lasers. It reaches the Earth’s central pressure routinely and about half a terapascal with effort; specialised designs have gone higher, but every result there is hard won. Beyond that, the only way to compress rock or iron is dynamically: a laser pulse shaped to ramp the pressure up over a few nanoseconds without a shock, during which the sample’s density and sometimes its crystal structure can be measured before it disassembles. Ramp compression has reached several terapascals in iron, magnesium silicate and diamond. Shock compression goes higher still, but along a path that heats the material far above any planet’s interior.

The floor, multiplied by three to seven, crosses the static limit between about 1.3 and 3.3 Earth masses. Any rocky planet heavier than that has a centre at pressures no anvil has held. The floor alone reaches the static limit only at about eleven Earth masses, and that is the claim requiring no assumption at all: a rocky planet of eleven Earth masses has a centre beyond the static limit whatever it is made of.

The measured planets show the same picture from the data. GJ 367 b, under two-thirds of an Earth mass but so small that it must be mostly iron, already has a floor of 0.1 terapascals, nearly double the Earth’s. Kepler-10 b, HD 219134 b and K2-229 b have floors of 0.17 to 0.21; 55 Cancri e has 0.29. Their centres, at three to seven times those values, run from under one terapascal to about two. None is within reach of a static experiment, and the heaviest is inside the range only ramp compression has explored.

How much of a super-Earth is beyond the anvils #

The centre is one point, and the question that matters for building a model is how much of the planet lies in regimes that have not been measured. Taking an Earth-like structure — a core of a third of the mass, a density contrast of 2.4 — and scaling its pressures with the floor as the planet grows, the fraction of the mass above any threshold follows. At one Earth mass, nothing exceeds half a terapascal. At five Earth masses, a third of the planet does, and a tenth is above one terapascal. At ten Earth masses more than half the planet is beyond the static limit and a third is beyond one terapascal.

That is the sense in which one density is consistent with many compositions for the larger super-Earths. The mass–radius curves for iron, rock and their mixtures used to infer what a planet is made of are computed from equations of state that, over much of the planet, have never been measured at the relevant pressure and are extrapolated from lower ones. Magnesium silicate is predicted to change its crystal structure, and perhaps to break down into its oxides, somewhere near a terapascal; iron’s melting curve, which decides whether a super-Earth’s core is liquid and so whether it can sustain a magnetic field, is measured at the Earth’s core and extrapolated beyond. The floor tells a modeller, before the first iteration, how much of the answer will rest on extrapolation.

How well two numbers give the floor #

A bound built from two measurements inherits their errors, and with unequal weights. The mass enters squared, the radius to the fourth power, so each per cent of radius uncertainty is worth two of mass. For a transiting planet the radius is the depth of the transit times the radius of the star, and the star’s radius is usually the larger part of the error: three per cent is now achievable for well-studied bright stars, and six per cent is common. The mass comes from the star’s reflex velocity and is often known to ten per cent or worse.

With a three-per-cent radius and a ten-per-cent mass, the floor is known to about a quarter of its value. That is better than it sounds, because the conclusion it supports is about factors rather than per cents. The floor times three to seven spans a factor of two and a half, the gap between the static and the dynamic regimes is a factor of ten, and a quarter is small against either. A planet’s interior is placed in its laboratory regime reliably from data that already exist.

The floor lifts off once there is gas #

The argument works only because rock and iron are similar in density, and it fails the moment the planet holds gas. A sub-Neptune is a rocky core inside an envelope of hydrogen and helium that may be only a few per cent of the mass but doubles the radius. The floor, computed from the total mass and the total radius, then falls as the inverse fourth power of a radius that the light envelope has inflated, while the central pressure is still set mostly by the rocky core beneath. The ratio climbs without limit.

With one per cent of the mass in an envelope a hundred times less dense than rock, the central pressure is already sixteen times the floor; with an envelope a thousand times less dense — closer to a hot, puffy sub-Neptune — it is more than a hundred. The floor is still true, as it is for the Sun. It has stopped saying anything.

That failure is informative in its own right. A planet whose floor is a good estimate is a planet without a substantial envelope, and the condition for the floor to be a good estimate is the same condition that puts it below the radius gap that separates rocky planets from those with gas. The bound is useful exactly for the planets on the rocky side of the gap, which are the planets whose interiors are in the extreme regime it describes.

The same reasoning one step further #

The two-layer argument is the planetary version of a statement that runs through the whole family of these bounds. Equilibrium gives inequalities without any knowledge of the material; the virial theorem gives a floor under a star’s mean temperature; relativity puts a ceiling on how compact any star can be. Each bound is loose in proportion to how much the body’s structure departs from the extreme case, and for rocky planets the departure is small because the materials are similar. A rocky planet is the one kind of body in astronomy for which the equation of state is nearly irrelevant to the order of magnitude of its interior, because rock is nearly incompressible on the scale of the bound, and that makes it the case in which the bound comes closest to being a measurement.

It is also the reason a planet’s size can be written in the constants of nature at the low-mass end: a body of atoms held up by electrostatic forces has a density close to that of its atoms, nearly independent of its mass, and the same near-constant density that fixes its size keeps its central pressure within a factor of a few of the floor.

What the figures leave out #

The two-layer bodies have uniform layers, which understates the central pressure by about ten per cent for the Earth and somewhat more for heavier planets, where self-compression is stronger; the mass fractions drawn above the static limit are therefore lower limits. The scaling of radius with mass as the 0.27 power is an approximation to computed mass–radius relations for Earth-like composition over one to twenty Earth masses. The central pressures of the Solar System bodies are taken from published interior models: the Earth’s is measured seismically to within a few per cent, Mars’s and Mercury’s are constrained by their moments of inertia and tidal responses to perhaps twenty per cent, and Jupiter’s depends on a core whose size and diluteness are still argued over. The laboratory limits are rough guides rather than sharp lines; static experiments above half a terapascal exist but are rare, and the pressures quoted for ramp compression are peak values held for nanoseconds. The exoplanet masses and radii are published central values, rounded, and several carry uncertainties large enough to move their floors by a third.

Still open: whether the extrapolated rock is the rock inside #

The floor says how much of a super-Earth lies beyond the measured equations of state; it does not say what the rock does there. That is the question ramp-compression experiments are now pressing on: magnesium silicate taken to a terapascal and beyond, with its crystal structure read by X-ray diffraction in the nanoseconds before it flies apart, iron’s melting point measured at pressures three times the Earth’s centre. Each such point replaces a stretch of extrapolation in the mass–radius curves with a measurement, and so changes what a measured density says about a planet’s iron fraction.

The interior also has a temperature, which the bound ignores entirely and which decides whether a super-Earth’s mantle convects, whether its core is liquid, and whether it can hold a magnetic field. A floor on pressure has no counterpart for heat. Whether the heaviest rocky planets are geologically alive or frozen stiff by pressures that raise their melting points beyond their internal temperatures is a question their mass and radius cannot answer, and the bound’s own success at placing them in a laboratory regime is a reminder of how much of a planet two numbers leave undecided.