A planet's size written in the constants of nature

A white dwarf is held up by a rule about counting: electrons cannot share a quantum state, so squeezing them into a smaller volume forces them into states of higher momentum, and the pressure that results needs no heat at all. The same rule makes a white dwarf smaller as it gets heavier, and it sets a mass beyond which no cold star can be held up. Both results come from balancing the pressure of degenerate electrons against gravity, and both can be written in terms of fundamental constants alone.

The same rule applies to any body cold enough and dense enough for its electrons to be degenerate, and that includes giant planets and brown dwarfs. Jupiter’s interior is at millions of atmospheres, and its electrons are degenerate over most of its volume. What distinguishes a planet from a white dwarf is not the physics of its pressure but the size of the competing effect. In a white dwarf degeneracy is everything. In a small planet or a moon the electrons are bound in atoms, and the atoms resist being pushed together by their own electrostatic structure. A body of hydrogen passes from one regime to the other as its mass rises, and the passage happens at a definite mass.

This essay derives that mass, finds it to be about the mass of Jupiter, and shows that it is fixed by the same constants that fix the white dwarf limit — with one number between them, the strength of electromagnetism. The largest planets are as large as they are because the fine-structure constant is about a hundred and thirty-seventh.

A radius that stops growing #

The relation between mass and radius for cold bodies of hydrogen and helium was first worked out in full by Zapolsky and Salpeter in 1969, using equations of state for matter from the density of ice to the density of white dwarfs. Its shape is simple.

At small masses the radius rises as the cube root of the mass: adding material adds volume at constant density, as it does for a pile of rock. At large masses it falls, as the inverse cube root. In between there is a maximum, and it is broad. The curve drawn here, an interpolation between the two limits normalised to Jupiter, peaks at three or four Jupiter masses and a little above Jupiter’s radius, and between a third of a Jupiter mass and the smallest hydrogen-burning stars — nearly three hundred times heavier — the radius never strays outside a factor of two.

Real objects follow it where they are cold. Saturn and Jupiter lie close to the curve. The transiting brown dwarfs CoRoT-3 b and KELT-1 b, twenty and twenty-seven times Jupiter’s mass, are about Jupiter’s size, a little above the cold curve because they are young and still warm; LHS 6343 C, sixty-three Jupiter masses and old, is smaller than Jupiter, close to the curve. The two red dwarfs, TRAPPIST-1 and Proxima Centauri, sit well above it, because hydrogen fusion in their cores supplies heat and the heat supplies pressure. A star is a cold body that has been inflated by its own fire.

Two laws that meet at the top #

The two limits come from two different sources of pressure, and the turnover is where they trade places.

The first law needs only the size of atoms. Matter that is held up by its atomic structure has a density of order one proton mass per atomic volume, $\rho \sim m_p/a_0^3$, where $a_0$ is the Bohr radius; gravity in a small body cannot squeeze it much harder, so the density is roughly fixed and $R \sim a_0 (M/m_p)^{1/3}$.

The second law needs only the exclusion principle. Electrons packed to a number density $n$ have momenta of order $\hbar n^{1/3}$, and their pressure is of order $\hbar^2 n^{5/3}/m_e$. Hydrostatic balance requires the pressure at the centre to be of order $GM^2/R^4$, and with $n \sim M/(m_p R^3)$ that gives

$$R \sim \frac{\hbar^2}{G m_e m_p^{5/3}}\, M^{-1/3},$$

the relation that makes a white dwarf smaller as it gains mass.

The crossover, from the constants #

Setting the two radii equal gives the mass at which degeneracy takes over. The Bohr radius is $\hbar^2/(m_e e^2)$, so the factor $\hbar^2/m_e$ in the degenerate radius is $e^2 a_0$, and most of the constants cancel:

$$M_\times \sim \frac{(e^2/G)^{3/2}}{m_p^{2}} = m_p\left(\frac{\alpha}{\alpha_G}\right)^{3/2}, \qquad R_\times \sim a_0 \left(\frac{\alpha}{\alpha_G}\right)^{1/2}.$$

Here $\alpha = e^2/\hbar c \approx 1/137$ is the fine-structure constant, which measures the strength of electromagnetism, and $\alpha_G = G m_p^2/\hbar c \approx 5.9\times10^{-39}$ is its gravitational counterpart, the strength of gravity between two protons. Their ratio, about $1.2 \times 10^{36}$, is the number by which electricity outweighs gravity between two particles; the crossover mass is the proton’s mass times that number to the power three-halves.

Evaluated, the crossover is 1.2 Jupiter masses and 0.82 Jupiter radii. The construction knows nothing about the solar system — no planet formation, no disc, no history — and it lands on Jupiter. Order-of-magnitude arguments like this one leave out numerical factors of a few, and the detailed calculation puts the actual maximum radius at three or four Jupiter masses rather than one; but the scale is right to within those factors, and the scale is the point. A planet is the size it is because a hydrogen atom is the size it is and because gravity is $10^{36}$ times weaker than electricity.

The same construction, one factor changed #

The mass beyond which no cold star can be held up comes from the same balance pushed further. As a degenerate body grows heavier its electrons are squeezed until their momenta approach $m_e c$, the pressure law softens to $\hbar c\, n^{4/3}$, and the radius drops out of the balance altogether. What is left is a single mass,

$$M_{\rm Ch} \sim \frac{(\hbar c/G)^{3/2}}{m_p^2} = m_p\, \alpha_G^{-3/2},$$

which evaluates to 1.85 solar masses before the numerical factor and the electrons-per-nucleon correction bring it to the familiar 1.4.

The two masses have the same form. One is the proton mass times $(\alpha/\alpha_G)^{3/2}$, the other the proton mass times $(1/\alpha_G)^{3/2}$. Their ratio is $\alpha^{3/2}$, about one part in sixteen hundred. The largest cold planet and the largest cold star are the same construction, with the strength of electromagnetism entering once in the planet — because atoms are what hold a planet up — and not at all in the star, where the electrons have been stripped from their atoms and only the exclusion principle and relativity remain. Planets and white dwarfs sit on either side of the first crossover; white dwarfs and neutron stars on either side of the second.

The sequence continues upward. A body above the Chandrasekhar mass has no cold equilibrium as a white dwarf; if its matter is dense enough for neutrons to become degenerate, the same reasoning with the neutron mass in place of the electron’s gives neutron stars, and their maximum mass is set by the stiffness of matter at nuclear density, which is not a combination of fundamental constants anyone can yet compute. At the low end the sequence continues downward too: below the planetary crossover, bodies are held up by the strength of their materials rather than by atoms in bulk, and the scale at which a body becomes round under its own gravity is set by the strength of rock or ice — again a combination of electromagnetic and gravitational constants, with chemistry in between.

Why the smallest star is eighty Jupiters #

The cold curve also explains a third mass scale that sits between the planetary crossover and the white dwarf limit: the lowest mass at which a ball of hydrogen can become a star. A contracting body heats as it shrinks, because gravity does work on it, and if nothing stopped the contraction its centre would eventually reach the ten million degrees or so at which hydrogen fuses fast enough to replace the heat it radiates. Degeneracy stops it. As the electrons become degenerate, further contraction goes into raising their Fermi energy rather than the temperature of the gas, and the central temperature reaches a maximum and then falls as the body settles towards its cold radius.

The maximum temperature rises with mass, and mechanics alone sets it, before fusion has any say: the gas’s thermal energy per particle at the centre cannot exceed something of order $GMm_p/R$, and the radius at which degeneracy halts the contraction is the cold radius of the figures. For a body of a few tens of Jupiter masses the peak central temperature is a few million degrees, enough to burn the small amount of deuterium present but not ordinary hydrogen. Only at about seventy-five to eighty Jupiter masses does the peak reach the temperature at which hydrogen burning balances the losses, and that is the boundary between brown dwarfs and stars.

The boundary’s position follows from the constants too: it is where the gravitational energy of a body at its degenerate radius corresponds to a temperature at which protons can tunnel through their mutual repulsion at a useful rate, which brings in the fine-structure constant through the height of the Coulomb barrier. The three masses — the largest planet near a Jupiter mass, the smallest star near eighty, and the largest white dwarf near fifteen hundred — are three crossings of the same few physical effects, and their ratios are powers of $\alpha$ and of the ratio of the electron and proton masses, with numerical factors that the detailed calculations supply.

Planet, brown dwarf, star: one curve, three names #

The flatness of the curve has also made the classification of these objects a matter of argument rather than measurement. A planet and a brown dwarf of the same radius orbiting the same star differ in mass, in formation history and in whether they ever burned deuterium, and none of those is visible in a radius. The working definition used by the International Astronomical Union draws the line at about thirteen Jupiter masses, where deuterium burning switches on — a line set by the same peak-temperature argument at a lower threshold. Others prefer a definition by formation: a planet forms in a disc round a star, a brown dwarf forms like a star from a collapsing cloud. The two definitions overlap, and objects of fifteen Jupiter masses that formed in discs and objects of ten that formed alone are both known.

The cold curve does not care about either definition. It describes a single family of bodies held up by the same two pressures, from Saturn to the smallest red dwarfs, and its only feature is the broad maximum where they trade places. On the mass–radius diagram the classes are not separated by any gap or kink; they are separated by mass, which the radius cannot show. What separates them in the census is the “brown dwarf desert” — the scarcity of companions between about ten and eighty Jupiter masses on close orbits round Sun-like stars — and that is a fact about how such companions form, not about how they are held up.

The same size, a hundred times the density #

The flatness of the radius has an immediate consequence. If the size hardly changes while the mass changes by a factor of three hundred, the density changes by nearly that factor.

Saturn would float on water. Jupiter is a little denser than water. A brown dwarf of sixty Jupiter masses is about the size of Jupiter and more than a hundred times as dense, its mean density above that of lead and its central density of order a thousand times that of water. The density is the cleanest discriminant between the classes, and it requires both a mass and a radius.

A transit cannot tell, and a wobble can #

That has a practical consequence for how planets and brown dwarfs are found. A transit survey measures a radius — strictly, a ratio of the planet’s radius to its star’s — and almost nothing else.

A one-per-cent transit across a Sun-like star is equally well explained by a warm Saturn, by Jupiter, by a young brown dwarf and by an old one near the hydrogen-burning limit. It can even be a low-mass star in an eclipsing binary, which is a common contaminant of transit surveys: a red dwarf of a tenth of a solar mass has a radius only about fifteen per cent larger than Jupiter’s. The transit says the object is of planetary size. It does not say it is a planet.

The star’s wobble does the separation that the transit cannot. Its amplitude is proportional to the companion’s mass, so it rises by the full factor of three hundred where the radius rises by none. A Jupiter on a five-day orbit moves its star by about a hundred metres a second; a brown dwarf of seventy Jupiter masses moves it by eight kilometres a second, enough to be measured with a modest spectrograph in a single night. That is why every transiting planet candidate is followed by a measurement of the star’s motion, and why the transiting brown dwarfs, rare as they are, have masses and radii measured as well as any objects in the sky: the transit gives the radius and the inclination, and the inclination turns the wobble’s lower bound into a mass.

Where the curve is not the answer #

The cold curve is a floor, not a prediction. A body is larger than the curve says if it is hot, and every real object is warmer than absolute zero. Giant planets and brown dwarfs are born hot from the energy of their collapse and contract as they cool, over billions of years for the most massive; a young brown dwarf can be twice the size of an old one of the same mass. Hot Jupiters on close orbits are often half as large again as the curve allows, because something deposits energy deep in their interiors that the starlight alone cannot account for. And stars are much larger than the curve near the hydrogen-burning limit, because fusion holds them up.

Composition moves the curve too. A planet with a heavy core is smaller than a pure hydrogen–helium one of the same mass, and the more heavy elements it holds the smaller it is; Saturn sits above the cold curve partly because the curve is for solar composition and partly because of residual heat. The maximum shifts to lower mass and smaller radius for heavier compositions, as the constants argument predicts: in the derivation, a heavier nucleus per electron changes the mass per atomic volume.

What the derivation leaves out #

The crossover is an order-of-magnitude estimate, and the factors it omits are not small. The atomic regime is treated as if matter had the density of a single hydrogen atom per Bohr volume; real hydrogen at a few thousand atmospheres is molecular, compressible and denser than that estimate by a factor of a few. The degenerate regime neglects the electrostatic interaction between electrons and nuclei, which lowers the pressure by a significant fraction at planetary densities — the very effect that, included properly, produces the rounded peak at three or four Jupiter masses rather than a sharp corner near one.

The interpolation drawn in the figures, $R \propto M^{1/3}/(1+(M/M_c)^{2/3})$, has the right limits and the right shape but is a fit, not a derivation; detailed equations of state for hydrogen at megabar pressures, now constrained by shock experiments and by the interiors of Jupiter and Saturn measured by spacecraft gravity fields, are what the published mass–radius relations use. The measured points are taken from published values with their own uncertainties of a few per cent, which do not affect the argument.

Still open: a curve tested at its peak #

The cold mass–radius relation has been tested at Jupiter and Saturn, where interior models have spacecraft gravity data to work with, and at the white-dwarf end, where degenerate electrons have been checked against measured masses and radii to a few per cent. In between, at the peak itself, it rests on few objects. Transiting brown dwarfs are rare — the “desert” of companions between about ten and eighty Jupiter masses on close orbits is one of the cleanest features of the exoplanet census — and most of those known are young enough that their radii are set by their remaining heat rather than by the cold equation of state.

The peak is where the two pressures are comparable and where the equation of state is least certain: hydrogen there is partially degenerate, strongly coupled and, at the centre, close to becoming metallic. Old, cold brown dwarfs with masses and radii measured to a few per cent would test the curve where theory is weakest. The space photometry surveys are finding them one at a time around faint red stars, where the transit is deep, and each one is a measurement of how matter behaves halfway between an atom and a star — at the place where, by the constants of nature, the largest planets have to be.