Astronomy & Celestial Mechanics
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Exoplanets

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.

Assumes Planet composition and Transits.

16 min read 9 figures Scales that defeat intuitionMeasuring the unreachable

There is a maximum size for a ball of hydrogen and helium. Add mass to a small one and it grows; keep adding and the growth slows, because the interior starts to become degenerate and degeneracy pressure resists compression without caring about temperature; past about the mass of Jupiter the radius turns over and begins to fall. The maximum is around 1.06 Jupiter radii, and it is remarkably insensitive to everything — to the exact mass, to the heavy-element content, and to the age past the first few hundred million years.

Hundreds of the planets that have been measured are much larger than that. Some are nearly twice as large. And the excess is not scattered randomly across the population: it appears above a threshold in the starlight the planet receives, and below that threshold it is absent.

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.
Fig. 1 Radius against incident stellar flux for a population of Jupiter-mass planets. The horizontal line is what a structural model gives for a cold, old ball of hydrogen and helium of that mass. Planets receiving less than about 2×10⁵ watts per square metre sit on it, with a median of 1.06 — which is the control the rest of the figure depends on, since it says the models are not wrong in general. Above the threshold the radii climb to a median of 1.75, and the onset is sharp enough to be called a threshold rather than a trend.

Why a cold planet has a maximum radius

The mass–radius relation for a self-gravitating sphere of hydrogen has two regimes and a turnover between them.

At low mass the material is essentially incompressible: the pressure needed to support the body is small compared with the pressure at which hydrogen’s own molecular structure resists, so the density is nearly constant and R∝M1/3R \propto M^{1/3}. Jupiter is near the top of that regime.

At high mass the electrons are degenerate, the pressure comes from the exclusion principle, and the relation reverses: R∝M−1/3R \propto M^{-1/3}, the same relation a white dwarf follows. More mass means a smaller object.

Between the two there is a maximum, and it falls near a Jupiter mass. That the solar system’s largest planet sits at almost exactly the peak of the curve is a coincidence in one sense and not in another: any body much more massive would be smaller, and any body much less massive would be smaller too, so the largest planets in any system are all about the same size.

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.
Fig. 2 The whole plane, with the turnover visible at the top right. The giant branch flattens and then bends over, and everything from a third of a Jupiter mass to the bottom of the stellar main sequence has a radius within about thirty per cent of Jupiter’s. That is why the inflated planets are conspicuous: they are not slightly above a trend, they are above a ceiling.

The insensitivity is worth dwelling on, because it is what makes the observation an argument. A cold hydrogen planet’s radius depends on its mass only weakly near the peak; on its heavy-element content weakly, because a twenty-Earth-mass core inside a Jupiter shrinks the planet by about five per cent; and on its age hardly at all after the first few hundred million years, because contraction is logarithmic in time. A model with all of those free still cannot produce 1.7 Jupiter radii at a Jupiter mass. There is no dial to turn.

That is unusual. Most discrepancies in this collection can be absorbed by an unmeasured parameter — an inclination, a composition, an opacity — and this one cannot, which is why it has been taken seriously since the first inflated planet was found in 2000 and has not been dismissed since.

Why irradiation is not enough

The obvious explanation is that the planets are hot because their stars are close. It does not work, and the reason is worth following because it is the whole argument.

A planet’s radius is set by the entropy of its interior — its deep adiabat — not by its surface temperature. The interior is convective, so it is on a single adiabat from the centre out to the radiative–convective boundary; where that adiabat sits decides the density profile and therefore the size. Cooling lowers the entropy and shrinks the planet; the planet’s whole evolution is a slow contraction as the interior entropy leaks away.

Starlight is absorbed near the top of the atmosphere and re-emitted from there. It heats the outer layers and it raises the temperature at the radiative–convective boundary, which pushes that boundary deeper and slows the rate at which the interior can cool. So irradiation delays contraction, and a strongly irradiated planet is larger at a given age than an isolated one.

But delay is not inflation. Slowing the cooling gets a planet to perhaps 1.2 Jupiter radii after a few billion years, not to 1.8. To hold an interior at high entropy for gigayears, energy has to be deposited below the radiative–convective boundary, at pressures of tens to hundreds of bars, where it can raise the adiabat rather than merely sitting on top of it.

That is what the threshold in the hero figure is evidence for. Above a certain flux, some fraction of the incident energy is finding its way to depth. There is a clean way to state the distinction. Delaying the cooling changes how far the entropy has fallen from its initial value; depositing heat at depth changes the value it is falling toward. The first has a floor set by the planet’s initial entropy and its age; the second does not. Since the initial entropy is set by formation and largely forgotten within a hundred million years, only the second can hold a five-gigayear-old planet at 1.8 Jupiter radii.

How much energy, and where

Working backwards from the observed radii, the required deposition is of order half a per cent of the incident stellar flux, delivered at pressures of tens of bars. That is a small fraction and a large absolute number — for a typical hot Jupiter, something like 101910^{19} watts.

Several mechanisms can in principle supply it.

Ohmic dissipation. A hot Jupiter’s atmosphere is hot enough to thermally ionise alkali metals, giving it a small conductivity. Winds of a kilometre a second crossing the planet’s magnetic field drive currents, and those currents dissipate resistively at depth. This is the leading candidate because it naturally produces a threshold: below about 1,500 kelvin there are not enough free electrons, and the mechanism switches off. The predicted threshold temperature converts to roughly the observed flux threshold.

Thermal tides. The asymmetric heating of a synchronously rotating planet produces a mass asymmetry that the star torques, and the work done appears as heat. The mechanism is attractive because it also explains why some hot Jupiters’ orbits are not fully circularised, and it is hard to calculate reliably.

Breaking waves. Gravity waves excited by the day–night temperature contrast propagate downward and break, depositing momentum and heat. The efficiency depends on the wave spectrum, which is a hydrodynamic simulation problem.

Tidal dissipation in the planet. If the orbit retains eccentricity, tides do work on the interior — the same mechanism that heats Io, scaled up enormously. It fails for the many hot Jupiters whose orbits are circular to the precision measurable.

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.
Fig. 3 Three planets on the other side of the same argument. Each is far less dense than any cold composition allows — Kepler-51b at a twentieth of water — and the explanations are the mirror images of the inflation this essay is about: a deep hydrogen envelope, a planet still contracting from formation, or a ring system inflating the apparent radius. A radius alone cannot separate them, which is why the inflation problem is stated as a correlation with irradiation rather than as a property of any individual planet.

It is worth noting how small half a per cent is as an efficiency, and how large as a requirement. Half a per cent of the incident flux is far less than the fraction of stellar energy a planetary atmosphere converts into winds, which is nearer ten per cent, so no energy budget is violated. What is demanded is not a large amount of power but a route for it: a way of moving energy from where it is absorbed, at millibar pressures, to where it can raise an adiabat, at tens of bars — four orders of magnitude in pressure, against a strongly stabilising temperature gradient. Every candidate mechanism is a proposal for that route rather than for the power.

Why a threshold is the strongest evidence

A correlation between radius and flux could mean many things. A threshold is much harder to explain away.

If the excess were an observational bias — if inflated planets were simply easier to detect — the bias would be smooth in flux, because detectability depends on transit depth and on period, both of which vary continuously. A sharp onset at a particular flux does not follow from any selection effect anybody has proposed.

If the excess were a delayed-cooling effect, it would also be smooth, because the delay is a continuous function of the irradiation.

A threshold points instead to a mechanism that switches on: something with an activation condition. Thermal ionisation of alkalis is exactly such a condition, since the electron density depends exponentially on temperature through the Saha equation, and a modest change in equilibrium temperature changes the conductivity by orders of magnitude. There is one further piece of evidence in the shape of the population, and it is less often quoted. The scatter in radius at a given flux is large — much larger than the measurement errors — and it does not shrink at high flux. Planets receiving nearly identical irradiation differ in radius by tens of per cent. So whatever is doing the inflating depends on something besides the flux: a magnetic field strength, a metallicity, a rotation rate, or a history. A mechanism driven by flux alone would produce a tight relation, and the relation is not tight.

What was actually measured

The radii are transit depths, and they inherit everything that goes with them: a radius ratio rather than a radius, so a stellar radius is required, and a limb-darkening treatment that biases the result at the per-cent level. For inflated hot Jupiters the ratios are large — a tenth or more — which makes the photometry easy and the limb-darkening systematic relatively less important than for small planets.

The masses come from radial velocities, and for these objects they are the easy part: a Jupiter-mass planet on a three-day orbit produces a reflex velocity of a hundred and fifty metres a second, which is enormous by modern standards.

The incident flux is computed rather than measured: it is the star’s luminosity divided by four times pi times the square of the semi-major axis, so it requires a stellar luminosity, which requires a distance. Before parallaxes at these distances, that was the largest uncertainty in placing a planet on the horizontal axis of the hero figure. It is now a minor term.

What is not measured at all is the deposition depth, the deposition rate, or the mechanism. Every statement in the previous section is an inference from a population’s radii, and the population’s radii are the only data.

Everything close in is circular, and nothing else has to be. Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed from τ_e = (2/21)(Q′/n)(M_p/M⋆)(a/R_p)⁵ for a Jupiter with Q′ = 1e+6 and an age of 3 billion years. It falls at 6.0 days, and the reason it is a wall rather than a slope is the fifth power: at half the period the timescale is 91 times shorter. Nothing inside it has a measurable eccentricity, and outside it eccentricities run to 0.95 — which is the number to hold on to, because a planet on a 0.93 orbit at 111 days comes within 0.030 AU of its star at periastron, closer than Mercury, and is being circularised as it is observed.
Fig. 4 A related population statistic, and a check on one of the mechanisms. Hot Jupiters at short periods are almost all circular, because tides circularise them quickly — which is what rules out ongoing tidal heating as the general explanation for inflation, since a circular orbit does no tidal work. The planets that are inflated and circular need something else, and that is most of them.
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.
Fig. 5 What a mass and a radius actually constrain, drawn as the space of compositions rather than as a point. For a five-Earth-mass planet measured to five per cent in radius, the allowed mixtures of iron, rock and water form a band across the ternary diagram rather than a point in it — three unknowns and two measurements. Every claim that a planet “is” rocky or “is” a water world is a claim about where in that band it sits, and the band is wide.

The other end of the same population

There is a mirror-image problem at small radii and it is worth mentioning because it constrains the same physics from the other side.

Some hot Jupiters are smaller than a pure hydrogen–helium planet of their mass should be, which requires heavy elements — a hundred Earth masses or more in the most extreme cases. Since the inflation mechanism acts on all of them, a planet’s observed radius is the sum of a heavy-element deficit and an inflation excess, and disentangling the two for any individual object is not possible.

That has a consequence for the previous rung’s argument. The heavy-element content of a giant planet is a formation diagnostic — it says how much solid material was accreted — and for hot Jupiters it cannot be measured without first knowing the inflation, which cannot be measured without first knowing the composition. The degeneracy that afflicts the small planets reappears here in a different form, with an unknown heating in place of an unknown water fraction.

The way around it is the same: use the planets where one term vanishes. Giant planets far enough from their stars to be un-inflated give clean heavy-element measurements, and those are what the metallicity–composition relation for giant planets is built on. The threshold and the degeneracy are the two halves of the essay, and each of them is worth reading at a second setting, because the first is a measurement and the second is a limit on measurement.

The inflation threshold at 4·10⁵ W m⁻², and the 0.66 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 4·10⁵ 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 1.72 — 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.
Fig. 6 The same population with the threshold placed at twice the flux. Fewer planets sit above it and those that do are still the inflated ones, which is the test: a threshold that sorted the population arbitrarily would not keep sorting it when it moved.
One mass and one radius, and every composition that gives them. A planet of 10 Earth masses and 2 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 29 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.
Fig. 7 The compositions allowed for ten Earth masses at two Earth radii. The allowed band runs across the whole diagram, so this object is consistent with a rock-and-iron interior under a thin atmosphere and with a water world under none — and no improvement in the radius alone resolves it.

The entropy a planet is born with

The argument above dismissed the initial entropy on the grounds that it is forgotten within a hundred million years. That is right for a five-gigayear-old hot Jupiter and it is the whole story for a young one, and the contrast is worth drawing because the same physics decides a completely different observation.

A giant planet forms hot, and how hot depends on how it formed. Gravitational instability — a fragment of a disc collapsing directly — retains most of the released gravitational energy and produces an object of high initial entropy: large, bright, and slow to cool. Core accretion, in which gas settles onto a solid core through an accretion shock, radiates much of that energy away as it arrives and produces a planet of low initial entropy: smaller, fainter, and starting further down the same cooling track.

The two are called hot and cold starts, and they differ in luminosity by orders of magnitude at ages of a few million years. They converge by a few hundred million, which is why nothing about a mature planet distinguishes them.

That makes young planets the diagnostic, and young planets are exactly what direct imaging finds. A planet imaged at ten million years is bright enough to detect precisely because it is still radiating its formation heat, and converting its measured luminosity into a mass requires knowing which track it is on — so a directly imaged companion’s mass is quoted as a range spanning the two assumptions, and the range is a factor of several.

The circularity is uncomfortable and is being broken from two directions. Dynamical masses for a few imaged companions, from the astrometric wobble they induce on their host, give a mass independent of any cooling model — and comparing that against the measured luminosity says which track the object is on. And the accretion shock itself has been observed in a couple of forming planets, through hydrogen emission produced as material falls in, which is a direct measurement of how much energy is being radiated away rather than retained.

A third route is opening and it belongs to this essay rather than to the imaging literature. A young planet that is still contracting has a radius larger than its mature value, so a transiting planet in a young cluster measures the cooling track directly — a radius and an age, with no luminosity model in between. A handful of such systems are now known, at ages of ten to a few hundred million years, and their radii are larger than the mature population’s by amounts consistent with ordinary contraction.

What makes them awkward is that they are also strongly irradiated, since a transiting planet in a young cluster is a close-in one, so the contraction being measured is the delayed contraction of this essay’s second section rather than the free one. Separating the two requires a young planet far enough out to be un-irradiated, which is precisely the object a transit survey cannot find.

A quantity that is forgotten is not the same as a quantity that never mattered, and the entropy this essay’s planets have lost is the one their younger relatives are measured by.

What would settle it

Three observations would distinguish the mechanisms and none of them is easy.

Re-inflation. If the heating mechanism operates continuously, a planet whose star brightens as it leaves the main sequence should grow — its interior entropy should rise again. Planets around evolved stars are the test, and a handful of inflated planets around subgiants have been found. The statistics are thin.

The mass dependence. Ohmic dissipation, tides and wave breaking predict different scalings of the inflation with planet mass and with magnetic field strength. Measuring the trend across the population needs many well-characterised planets over a wide mass range, which the wide-field surveys and the occurrence-rate work that turns a detection list into a population are now supplying.

The atmospheric composition. A mechanism that deposits heat at depth also mixes, and the resulting composition gradient differs between mechanisms. High-precision emission spectroscopy might see it. And the diagram itself with a different pair of objects picked out, because which planets are highlighted decides what the reader takes from it.

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.
Fig. 8 A hot super-Earth and a temperate one, on the same diagram with the solar-system bodies removed. They sit on nearly the same density curve and their surfaces differ by more than a thousand kelvin, which is the sharpest statement that a density constrains a bulk composition and says nothing whatever about a surface.

One more reading strips the model curves off the diagram the whole argument is made on.

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.
Fig. 9 The same population without the constant-density curves. What is left is a scatter of points and the solar-system bodies, and the inflated planets are still visibly above everything else — the one conclusion in this essay that does not need the curves to be drawn.

Where the ladder goes

The previous rung of this anchor was about a degeneracy — one density consistent with many compositions. This one is the opposite situation: a radius consistent with no composition, which is a much more useful kind of discrepancy, because it points at a missing physical process rather than at a missing measurement.

The thread that leads out of it goes toward the atmospheres. Everything in this essay depends on how energy moves between the outer layers and the deep interior of a strongly irradiated planet, and that is the same question as how a phase curve’s offset arises, how a day–night contrast is maintained, and why some planets’ spectra are flat and others’ are not. It is also, in the end, a question about where these planets came from: a hot Jupiter did not form where it is, and the entropy it arrived with depends on how it got there.

Where this goes next

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. Everything close in is circular, and nothing else has to be. Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed from τ_e = (2/21)(Q′/n)(M_p/M⋆)(a/R_p)⁵ for a Jupiter with Q′ = 1e+6 and an age of 3 billion years. It falls at 4.5 days, and the reason it is a wall rather than a slope is the fifth power: at half the period the timescale is 91 times shorter. Nothing inside it has a measurable eccentricity, and outside it eccentricities run to 0.95 — which is the number to hold on to, because a planet on a 0.93 orbit at 111 days comes within 0.030 AU of its star at periastron, closer than Mercury, and is being circularised as it is observed. A planet where one cannot form A Jupiter at four days orbits inside a region that was too hot to hold ice and too small to hold the material. It did not form there — and the distribution of eccentricities says which of two journeys brought it in. Lane–Emden solutions for n = 0, 1, 1.5, 3, 4.5, 5, and the one that has no surface. The dimensionless density θ against the dimensionless radius ξ, for polytropic indices 0, 1, 1.5, 3, 4.5, 5. Each curve is the whole structure of a star whose pressure is K times its density to the power 1 + 1/n: the equation of state and hydrostatic equilibrium leave one second-order differential equation, and this is its solution. Every curve starts at θ = 1 with zero slope, because the density is greatest at the centre and has no cusp there. What separates them is where they end. At n = 0 the density is uniform and the surface is at ξ₁ = 2.4495; by n = 3 it has moved out to 6.8968 and the central density is 54.2 times the mean. At n = 5 the curve reaches zero only at infinity — a configuration of infinite radius and, remarkably, finite mass — and every index above it has neither. The three curves that have closed forms, n = 0, 1 and 5, are drawn from the same numerical integration as the rest and agree with those forms to better than two parts in a million, which is what licenses reading the others off the picture. What the figure cannot show is the scale: ξ is radius divided by a length that depends on the central density and on K, so two stars of the same index and wildly different sizes have the same curve here. An equation of state is already a star Write down how a gas's pressure depends on its density, insist that the pressure hold the weight up, and everything else follows — the run of density, the fraction of the mass inside each radius, and, at one particular index, a mass that does not care what the radius is.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Degeneracy pressureEquilibrium temperatureHot jupiterInflated radiusInsolationInterior entropyMass radius relationOhmic dissipationPolytropeRadiative convective boundaryTidal dissipation