A wall that leans towards the body that damps

Every hot Jupiter is in two tidal relationships at once. The star raises a tide on the planet — a bulge of the planet’s gas pointing at the star — and the planet raises a tide on the star. Both bulges lag behind the line between the two bodies because their material does not respond instantly, both lags produce torques, and both torques change the orbit. Either one, acting on an eccentric orbit, removes the eccentricity: the dissipation is largest at pericentre, where the tide is strongest, and the orbit is gradually rounded off.

The wall in the period–eccentricity plane — the period inside which hot Jupiters are circular — is usually attributed to the first of these, the tide in the planet, with the second set aside as smaller. The argument for setting it aside is that the star’s radius is a tenth of the planet’s distance from it at best, and the tide’s effect falls as the fifth power of that ratio. It is a reasonable argument, and it rests on a guess about how well the two bodies dissipate, because a star that dissipated a thousand times better than a planet would undo it.

There is a way to settle which tide does the work without knowing how well either body dissipates, and it uses something the wall has that neither tide has alone: a dependence on the planet’s mass, which runs in opposite directions for the two.

Two tides, two exponents of the mass #

The rate at which each tide damps the eccentricity, for a planet of mass $m$ on an orbit of semi-major axis $a$ and mean motion $n$ round a star of mass $M$, in the constant-$Q'$ approximation used throughout the hot-Jupiter literature, is

$$\frac{1}{\tau_p} = \frac{63}{4}\,\frac{n}{Q'_p}\,\frac{M}{m}\left(\frac{R_p}{a}\right)^5, \qquad \frac{1}{\tau_\star} = \frac{171}{16}\,\frac{n}{Q'_\star}\,\frac{m}{M}\left(\frac{R_\star}{a}\right)^5 .$$

Everything in the two expressions is shared except the bodies’ radii, their dissipation factors and one ratio of masses, which appears upside down in one relative to the other.

The reason is simple once stated. The tide in the planet is raised by the star, so its strength scales with the star’s mass, and it acts on the planet’s own orbital motion, so its effect on the orbit is divided by the planet’s mass: a heavier planet is harder for its own tide to move. The tide in the star is raised by the planet, so its strength scales with the planet’s mass, and its effect on the orbit — which is the relative orbit of the two — scales the same way. A heavier planet raises a bigger tide on its star and is circularised faster by it.

The two lines are mirror images in slope. Where they sit vertically depends on the two dissipation factors, which are the poorly known part — the planet’s is thought to be of order a million and the star’s larger, perhaps by a factor of ten to a thousand, but the evidence for both is indirect. With the values drawn, the planet’s tide is faster at every mass, by a factor of about three hundred at one Jupiter mass and three at ten.

The wall leans #

A population of hot Jupiters of a given age is circular inside the period at which the damping time equals that age, and eccentric outside it — if they arrived eccentric. The damping time rises steeply with period: at fixed mass, as $a^{13/2}$, or as the period to the thirteen-thirds. So the wall moves only slowly with anything else, but it moves, and the direction it moves with the planet’s mass is fixed by which tide is doing the damping.

The direction of the lean is a cleaner diagnostic than the position of the wall, for a reason the essay on the ratio gave in detail. The position depends on the age divided by the dissipation factor, and neither is known independently: a younger population with a better-dissipating planet and an older one with a worse-dissipating planet put the wall in the same place. The lean has no such degeneracy. A planet-dissipated wall falls with mass at any age and any dissipation factor; a star-dissipated wall rises. Changing the unknowns slides the lines up and down, and leaves their slopes alone.

The slopes are shallow — the mass enters to the power of three-thirteenths, so a hundredfold range in mass moves the wall by a factor of three. But the hot Jupiters span exactly that range, from a third of a Jupiter mass to more than ten, and the heavy end of it is well populated.

What the heavy planets say #

The observed planets in the figure are a selection chosen to make the comparison visible, and they include the planets that have been the subject of this argument since they were found.

The heavy and eccentric. HAT-P-2 b has nine Jupiter masses, a period of 5.6 days and an eccentricity of 0.52. XO-3 b has nearly twelve Jupiter masses, a period of 3.2 days and an eccentricity of 0.28. WASP-14 b has seven Jupiter masses, a period of 2.2 days and an eccentricity of 0.09. HAT-P-34 b, at three Jupiter masses and 5.5 days, has an eccentricity of 0.44.

The light and circular. HD 209458 b, HD 189733 b and WASP-12 b, between seven-tenths and a Jupiter mass and a half, at periods of one to three and a half days, are all circular to the precision of their measurements, as is nearly every planet below a couple of Jupiter masses inside about three days.

The heavy planets keep eccentricities at periods where the light ones have none. That is the lean the planet’s tide predicts and the opposite of the one the star’s tide predicts. Were the star doing the damping, the heavy planets would be the ones circularised first, out to longer periods than the light ones — and the most massive close-in planets would be the most reliably circular of all.

There are heavy circular planets too: WASP-18 b, ten Jupiter masses on a 0.94-day orbit, has an eccentricity of about a hundredth. At that period both tides are strong enough to circularise anything, and the planet sits well inside every version of the wall. It does not distinguish the two.

So the population, read for its lean rather than for the position of its edge, says the planet’s tide does most of the eccentricity damping across the hot-Jupiter mass range. That is what the estimate from the radii alone assumed, and it is now an inference from the planets rather than an assumption about them.

The wall a dissipative star would draw #

It is worth drawing the alternative rather than only describing it, because the difference is large enough to be read off a scatter plot.

In this world, the planets at the top right of the plot would be the first to be circular, and the eccentric planets would be the light ones at intermediate periods — a pattern that would look, to anyone plotting it, like a tendency for heavy planets to arrive on circular orbits. Nobody has suggested such a tendency, because the measured population shows the opposite. The figure exists to make clear that the opposite was not inevitable: it is a fact about how stars and planets dissipate, and not a consequence of the geometry of tides.

A star that dissipated that well would also drain hot Jupiters’ orbits quickly, since the tide in the star is the one that removes orbital energy once the planet’s rotation is locked. A hundred thousand for a Sun-like star would bring a ten-Jupiter-mass planet on a one-day orbit into its star in well under a million years, and planets like WASP-18 b would be rare to the point of absence. The survival of the heaviest, closest planets and the lean of the eccentricities point the same way, which is a consistency check the two arguments did not have to pass.

How heavy before the star takes over #

Because the two tides scale oppositely with mass, there is a mass at which they are equal — and above it the star’s tide dominates.

The crossover is independent of the period, which is a small surprise worth stating: both tides depend on the separation to the same power, so their ratio is a property of the two bodies and of nothing about the orbit. Setting the two rates equal gives

$$m_\times = M \left(\frac{63}{4}\cdot\frac{16}{171}\cdot\frac{Q'_\star}{Q'_p}\right)^{1/2}\left(\frac{R_p}{R_\star}\right)^{5/2},$$

and with a planet a tenth of the star’s radius the bracket is around a hundredth of the star’s mass for a ratio of ten in $Q'$ — fourteen Jupiter masses. For any ratio above about ten the heavy eccentric planets sit below the crossover, and their survival is consistent with the planet doing the work. For a ratio near one they would sit above it, the star’s tide would dominate, and they should have circularised first.

The observed lean therefore bounds the ratio as well as identifying the body. It says the star dissipates at least several times less well than the planet, which agrees with every theoretical expectation for a Sun-like star’s radiative interior and a giant planet’s convective envelope — and which was, until the population was large enough to show the lean, an expectation rather than a measurement.

A population of mixed ages #

The figures draw walls for one age. A real sample has host stars from a few hundred million to ten billion years old, and every star’s wall is at a different period.

The age spread smears each mass’s wall into a band rather than a line, but it does not change the lean. The light planets at a given period are circularised in a fraction of the time the heavy ones need, so at every age the light ones are rounder. A population of mixed ages shows a lean even more robustly than a single cluster would, because the lean is a ratio of damping times and the ratio is the same at every age.

This is the difference between the lean and the cut-off period that is an age in binary-star clusters. There, the wall’s position was the measurement, and it needed a cluster of stars with a common age to mean anything. Here the wall’s slope is the measurement, and it survives the mixing of ages that makes the position meaningless for field planets.

What else leans the same way #

The mass is not the only planetary property that enters the tide in the planet. The radius enters to the fifth power — larger planets have larger tidal bulges and dissipate more — and hot Jupiters’ radii vary by a factor of two, for reasons that are not fully understood.

That correlation is a complication for the argument and needs to be faced rather than avoided. If heavy planets were systematically small, they would be circularised more slowly for a reason that has nothing to do with the opposite mass scaling of the two tides — and the observed lean could be produced by the radius alone, even with the star doing the damping.

The size of the effect settles it. Across the hot-Jupiter range the radius varies by a factor of about one and a half between a typical light inflated planet and a typical heavy one, which changes the planet’s damping rate by a factor of seven and moves the wall by about sixty per cent. The mass varies by a factor of thirty, which changes the planet’s damping rate by thirty and the star’s by thirty in the other direction — a factor of nine hundred between the two tides’ relative importance. The radius effect is real, it points the same way, and it is too small to reverse the mass effect. A star-dissipated wall would still rise with mass after the radii had been allowed for.

Why the stars never showed this #

The circularisation walls of binary stars were measured decades before the planets’, in clusters whose ages are known, and no lean of this kind was ever read from them. The reason is the symmetry of the problem. In a binary of two similar stars, the tide each raises on the other has the same strength and dissipates in the same kind of material, so the two terms in the damping rate are equal and swapping which body does the work changes nothing. The masses enter both in the same way, and the mass dependence that remains is a weak one shared by both.

A hot Jupiter breaks the symmetry as completely as anything can: the two bodies differ by a factor of a thousand in mass, ten in radius, and an unknown factor in how they dissipate. That is why the question of which body does the damping can be asked of planets and could not usefully be asked of binaries — and why the competing damping mechanisms of the stellar case, which differ in their dependence on period, were separated by the slope of the wall against period rather than against mass.

The dissipation the lean attributes to the planet does not vanish. It heats the planet’s interior at a rate set by the eccentricity and by the same dissipation factor, as tides heat a moon kept eccentric by its neighbours, and for a hot Jupiter on an eccentric orbit that heating can be comparable to the starlight it absorbs. Tidal heating has been proposed as one of the explanations for inflated radii, which would close a loop: the planets whose eccentricity is being damped fastest are being heated most, and are swollen, and are therefore damped faster still.

Where the picture is shakier than it looks #

Constant $Q'$. Both rates assume a dissipation factor that is the same at every tidal frequency and for every planet and star. That is the assumption the essay on the ratio ended on, and it is the weakest part of the whole subject: dissipation in a convective envelope depends on the tidal frequency, and inertial waves in a rotating planet or star can make it depend on it sharply. The lean argument relies on the mass scaling of the two tides, which comes from where the masses enter the tidal potential rather than from the dissipation, and is therefore more robust to this than the wall’s position is — but if $Q'_p$ itself varied systematically with planet mass, the lean would carry that variation too.

How the planets arrived. The wall marks where eccentricity has been removed only if the planets arrived with it. A hot Jupiter that migrated inward through its disc arrives circular whatever its mass, and one that arrived by high-eccentricity migration and was circularised on the way in arrives circular at a period that depends on its pericentre history. If heavy planets arrived more often by the eccentric route, the observed lean would be partly a lean in delivery rather than in damping.

Eccentricity pumped from outside. A distant companion can keep a hot Jupiter’s eccentricity from decaying to zero by continually exciting it. Several of the heavy eccentric planets have long-period companions or trends in their velocities. Excitation by a companion does not depend on the hot Jupiter’s mass in the way damping does, so it would add scatter rather than lean, but it means a residual eccentricity is not always a sign that damping has been slow.

Small eccentricities are hard to measure. WASP-14 b’s 0.09 and WASP-18 b’s 0.01 are measured from radial velocities and the timing of secondary eclipses, and small eccentricities are biased upward by noise: a fit to a circular orbit’s velocities returns a positive eccentricity on average. The heavy planets’ large eccentricities, 0.28 to 0.52, are not affected; the boundary cases are.

The model here is a sketch. The walls are drawn for a single star mass and radius, a single age and constant radii within each curve, with dissipation factors chosen to be plausible rather than fitted. WASP-14 b sits inside even the planet-dissipated wall drawn here, which says either that its system is young, that its planet dissipates less well than the value drawn, or that something keeps its small eccentricity up. What the figures show is the geometry of the argument; a fit to the population needs each planet’s own mass, radius, host and age.

The surprising connection #

The opposite scaling of the two tides with the planet’s mass is the same physics that sets the direction of a moon’s tidal migration, seen through a different quantity. The Moon recedes from the Earth because the tide it raises on the Earth is carried ahead by the Earth’s rotation; the tide the Earth raises on the Moon has long since stopped mattering, because it locked the Moon’s rotation and now only damps the Moon’s small eccentricity. In each system, which body’s dissipation drives which change in the orbit is decided by which body is the lighter and which the smaller, and the answer for a hot Jupiter — the planet damps the eccentricity, the star drains the orbit — is the Earth–Moon answer with the masses reversed.

The heavy eccentric hot Jupiters are the systems where that division of labour is tested. They are heavy enough that the star’s tide on them is a hundred times stronger than on a Saturn, and still not strong enough to have rounded their orbits off. The planets keep their eccentricity because they are too heavy for their own tide to move, which is an odd thing for a planet to be.

Still open: whether the star’s tide is the one that drains the orbit #

The lean says the star dissipates poorly compared with the planet, which is what the eccentricity damping requires. The other tidal effect on a hot Jupiter — the slow shrinking of its orbit, which only the tide in the star can produce once the planet’s rotation is locked — depends on the star’s dissipation alone, and it has been measured directly in one system, where the orbital period of a planet a little over one day long is decreasing by about thirty milliseconds a year. The stellar dissipation factor that implies, a few hundred thousand, is lower than the lean seems to allow if planets dissipate at the rate drawn here: it would put the crossover mass at a few Jupiter masses, and the heavy eccentric planets above it. Whether that system is typical, with the star dissipating unusually well for a reason specific to its structure, or whether stellar dissipation is generally stronger than the eccentricities alone suggest — in which case the heavy eccentric planets are young, or are being kept eccentric from outside — is what timing more short-period planets for longer will decide.