A planet circularised at twice the distance that would destroy it

There are two ways to bring a giant planet from where it formed to a few days from its star. One is through the gas: the planet exchanges angular momentum with the disc it is embedded in and drifts inward, slowly and on a nearly circular orbit, stopping wherever the disc’s structure stops it. The other needs no gas at all. A distant companion — another planet, or a star — drives the giant planet’s eccentricity up, either slowly through a secular cycle or abruptly through a close encounter, until its closest approach to its star is a few hundredths of an astronomical unit. There the star raises tides in the planet, the tides dissipate energy, and the orbit shrinks.

The second route leaves a fingerprint that the first cannot. It is a statement about where the orbit ends, and it follows from one conservation law.

Energy leaves at periastron and angular momentum does not #

The tide a star raises on a planet is steep: its strength goes as the inverse cube of the separation, and the rate at which it dissipates energy goes as a much higher power. On an orbit of eccentricity 0.97 the planet spends almost all of its time far from the star, where nothing happens, and passes periastron in a few hours, where everything does. Each passage is a brief, hard squeeze.

A squeeze at periastron is nearly radial. It raises a bulge on the planet along the line to the star, and the bulge lags slightly because the planet’s interior is not perfectly elastic; the lag produces a small torque, but the dominant effect is a loss of energy with very little change of angular momentum. There is a deeper reason too. The only place the orbit’s angular momentum could go is into the planet’s spin, and a planet’s spin is a tiny reservoir: its moment of inertia is a quarter of its mass times its radius squared, against its mass times the orbit’s radius squared, a ratio of about one in a hundred thousand for a hot Jupiter. The spin fills up almost immediately — the planet is spun to roughly match its orbital speed at periastron — and after that the orbit’s angular momentum has nowhere to go.

So the orbital angular momentum is conserved while the energy drains away. For a Keplerian orbit the angular momentum per unit mass is $\sqrt{GMa(1-e^2)}$, which means the combination

$$a(1-e^2) = q(1+e)$$

is fixed throughout the circularisation, where $q = a(1-e)$ is the periastron distance. When the orbit has become a circle, $e = 0$ and the semi-major axis is that same quantity. The final radius is therefore $q(1 + e_0)$ — for any starting eccentricity near one, twice the closest approach.

That is worth stating as a sentence because it is so specific: a planet circularised by tides ends at twice the distance of its original periastron, and it gets there by following a track on which the periastron barely moves while the apastron comes in from tens of astronomical units. The tracks in the figure show it — near-vertical at the bottom, where the orbit is almost circular and the periastron and apastron are close together, and nearly horizontal at the top, where the orbit is almost parabolic and each passage takes energy without moving the closest approach.

The factor is not exactly two, and the amount by which it falls short is itself informative. A planet whose eccentricity was driven only to 0.9 before the tide took hold finishes at 1.9 times its periastron rather than 2; one delivered at 0.99 finishes at 1.99. A secular cycle raises the eccentricity slowly and hands the planet to the tide at the moment the periastron first becomes small enough to matter, which is when the eccentricity is highest — so it delivers close to the factor of two. A single violent scattering between two giant planets can deliver at a lower eccentricity with the periastron already small, and it leaves the planet slightly further in, relative to where it started. The difference is ten per cent at most, which is smaller than the uncertainty in any single planet’s Roche distance, and it cannot be used to tell the two apart for any one object. It would show only as a softening of the edge in a large enough sample.

A floor measured in the planet’s own size #

The periastron cannot be arbitrarily small. Inside a certain distance the difference between the star’s pull on the near and far sides of the planet exceeds the planet’s own gravity, and the planet overflows the region inside which its material is bound to it. That distance is the Roche distance, and for a planet it is

$$a_R \approx 2.16\,R_p\left(\frac{M_\star}{M_p}\right)^{1/3},$$

with a coefficient appropriate to a fluid body whose density rises inward as a giant planet’s does. A planet delivered with its periastron inside this distance loses its outer layers on the first passage and is progressively stripped or destroyed. Only planets with $q > a_R$ survive to circularise, and they circularise at $a_f = q(1 + e_0) > 2a_R$.

So the prediction is an inner edge at twice the Roche distance. What makes that prediction distinctive is its units. The Roche distance is proportional to the planet’s radius and to the inverse cube root of its mass, so the predicted edge is at a different orbital period for every planet. A light, puffy planet has its edge far out; a heavy, compact one can sit much closer to its star. Nothing about the star’s disc, its magnetic field or its spin enters.

The other route has an edge in different units entirely. A planet migrating through the disc stops where the disc stops — at the inner edge of the gas, which a young star’s magnetic field truncates near the radius where the gas orbits at the star’s own rotation rate. That is a period of a few days, and it is the same period for every planet regardless of its mass or radius.

That difference in units is the test. Given a sample of hot Jupiters with measured masses and radii, the inner boundary of their distribution can be drawn in period and in Roche distances, and the variable in which it is sharper says which route put them there. The calculation that first did this, in 2006, found the edge of the sample then known close to twice the Roche distance rather than at one — the value a simple picture of a planet spiralling in until it was torn apart would have given — and that result is one of the main reasons the eccentric route is taken seriously for hot Jupiters at all.

A ceiling as well as a floor #

The same steepness that makes the periastron the only place anything happens also puts an outer limit on the channel. The time to circularise rises very steeply with the closest approach — in the standard equilibrium-tide model, a planet delivered with its periastron at four Roche distances takes of order a hundred times longer to finish the journey than one delivered at two, because the tide’s grip weakens as a high power of the distance at which it acts. That is the same steep dependence that draws the circularisation wall in the period–eccentricity plane, where a factor of two in period separates orbits that have been rounded from orbits that have not.

For the high-eccentricity route it means that only planets delivered within a narrow range of periastra finish the journey within the age of their star. Too close and they are stripped; too far and they are still eccentric. The finished hot Jupiters therefore occupy a band a factor of two or so wide in Roche distances, starting at two — and the eccentric planets found with periastra just outside that band are the ones still on their way. Where exactly the band’s outer edge falls depends on the tidal quality factor of the planet, a number that cannot be computed from first principles and is inferred instead from the very populations it is used to explain; the two competing theories of tidal dissipation put it in different places by more than a factor of two in period.

That band is the one prediction of the channel that depends on the tide’s strength. The inner edge does not: it is set by conservation of angular momentum and by the Roche distance, and neither has a free parameter in it.

An edge that slopes with mass #

Translating the prediction into the variable a survey actually measures — the orbital period — shows its shape directly.

The observed hot Jupiters extend well inside the disc band, to periods of about a day, and the heaviest of them reach the shortest periods — a planet of ten Jupiter masses orbits its star in less than a day. Both facts are what the sloping floor predicts and neither is what the flat floor predicts. The shape of the upper edge of the short-period desert — the region where planets of intermediate mass are missing at periods under a few days — has also been fitted with exactly this dependence, a boundary tracking twice the Roche distance for planets of each mass.

That is not conclusive on its own, because a planet can arrive by the gas and then be moved further inward by the tide it raises on its star, which drains the orbit’s angular momentum into the star’s spin and shrinks a circular orbit directly. WASP-12 b’s orbit is measured to be decaying by exactly this process. Tidal decay after arrival blurs whichever edge the delivery drew, by an amount that depends on how dissipative the star is — a number known, for any given star, to about an order of magnitude.

Ten planets placed in their own units #

The test can be run on individual planets whose masses and radii are well measured, by placing each at its semi-major axis divided by its own Roche distance.

This is the kind of result that has to be stated carefully. The six at or above the line are consistent with the prediction. The four below it are not consistent with the simple version, and they are the most inflated planets in the sample — among them the hottest planet known, whose dayside is hotter than many stars. A giant planet’s radius is not fixed. Planets this close to their stars are observed to be larger than models of cooling giant planets allow, by up to eighty per cent in radius, and the proposed causes all act after arrival. A planet that circularised at twice its Roche distance when it was 1.2 Jupiter radii across and has since grown to 1.9 now sits at 1.26 Roche distances, without having moved at all.

So the test is cleanest for the planets that are least inflated, and among those the edge at two holds. For the inflated ones it becomes a test of a combined model — delivery, then inflation, then perhaps decay — in which each stage has its own uncertainty. Several of the planets below the line are also those whose orbits are measured or predicted to be decaying fastest, which moves them inward in the other variable. Neither effect is in the simple prediction; both push points below the line, and neither could push a planet above it.

A journey that costs many binding energies #

The energy that has to be removed along each track is enormous, and it has to go somewhere.

The energy is deposited inside the planet, at periastron, by the tide. If it were deposited quickly — faster than the planet could radiate it — the planet would be unbound many times over. That high-eccentricity migration works at all is therefore a statement about timescales: the circularisation has to take long enough that the planet radiates the heat as it arrives. For a giant planet with a closest approach of a few hundredths of an astronomical unit, the standard estimates put the circularisation time at hundreds of millions to billions of years, and a planet radiating sixteen binding energies over that time is a planet with a large internal heat source for as long as the journey lasts.

That makes a second, independent connection with the inflated radii. A giant planet with an internal heat source is larger than one cooling passively, and planets caught partway along a track — still eccentric and still being heated — should be inflated for that reason as well as for any other. The connection is real in direction and disputed in size, since the inflation seen in circular hot Jupiters long after circularisation needs a heat source that persists, and the tidal one does not.

The same numbers set a problem at the start of the journey. The first passages of a planet on an orbit of eccentricity 0.97 each take a large bite of energy, and it has been shown that the oscillations a periastron passage excites in the planet can grow from one passage to the next, chaotically, until they break and dissipate much faster than the smooth tide would. That shortens the early part of the track enormously and puts a large fraction of the energy into the planet over a short interval. Whether the planet survives the heating depends on where in its interior the oscillations break, and that is not settled.

Planets caught on the way #

If hot Jupiters were delivered this way, some should be caught in transit: planets on highly eccentric orbits with closest approaches of a few hundredths of an astronomical unit, moving along one of the tracks in the opening figure. They exist. One of the best-known has a period of 111 days and an eccentricity of 0.93, and at periastron it comes closer to its star than Mercury does to the Sun; its temperature has been measured to jump by hundreds of degrees within hours of each close passage.

Their numbers are a constraint on the whole channel. The time a planet spends on each part of a track is known from the tidal model, so a given rate of hot-Jupiter production by this route implies a given number of planets caught partway. Early estimates predicted that a transit survey the size of Kepler’s should have found several super-eccentric Jupiters on their way in; it found fewer than predicted, and the shortage has been used to argue that high-eccentricity migration cannot be the only route by which hot Jupiters arrive. The argument depends on the tidal model — a faster early phase, as the chaotic tides allow, would leave fewer planets to be caught — so it limits the channel’s share rather than excluding it.

What the tracks leave out #

The tracks are drawn for a planet whose spin has caught up with its orbit and whose tide is the smooth equilibrium tide. They leave out the planet’s structure, which decides how much energy each passage dissipates and therefore how long the journey takes; they leave out the companion, which is still there and still perturbing the orbit, so that the eccentricity is being driven up by one process while the tide drives it down, and the track is a compromise between the two; and they leave out the star’s own tide, which becomes important once the orbit is circular and acts on angular momentum rather than on energy.

They also say nothing about the angle between the orbit and the star’s equator. A planet delivered by a companion arrives on an orbit tilted however the companion’s perturbation left it, and the tilts of hot Jupiters’ orbits relative to their stars’ spins are observed to be widely distributed around hot stars and nearly aligned around cool ones. That is a separate fingerprint of the same channel, erased by a different tide on a different timescale, and it is the one most often quoted as evidence for this route.

And none of it identifies the companion. A secular cycle driven by a distant inclined companion, a close scattering encounter between two giant planets, and a slow chaotic diffusion of eccentricity among several planets all deliver the same kind of orbit to the start of a track. The circularisation erases which one it was.

Still open: what fraction arrived each way #

Everything here points to the same conclusion as the rest of the evidence on hot Jupiters: both routes operate. The edge at twice the Roche distance, the slope of the edge with mass, the misaligned orbits around hot stars and the planets caught on the way in all point to eccentric delivery for a substantial fraction of hot Jupiters. The resonant chains, the aligned orbits of the hot Jupiters that have nearby companions, and the shortage of super-eccentric Jupiters point to disc delivery for another substantial fraction. What is not known is the ratio, and every estimate of it depends on at least one quantity — the tidal dissipation in planets, the viscosity of discs, the inflation history of the planets themselves — that is uncertain by an order of magnitude. The cleanest remaining test is the one drawn here: a large sample of hot Jupiters with masses and radii measured well enough, and inflation modelled well enough, for the inner edge to be drawn in Roche distances and compared with the edge drawn in period.