A fraction read in each channel's own units

A Jupiter four days from its star did not form there, and two journeys could have brought it. In one it migrated inward through the gas disc, carried by a torque that nearly cancels, and stopped where the disc ended, a few days’ orbit from the star. In the other a distant companion drove its eccentricity up until its closest approach skimmed the star, and tides then circularised the orbit at twice that closest approach — which, since a closer approach would have torn the planet apart, puts the eccentric channel’s inner edge at twice the planet’s Roche distance, a length set by the planet’s own radius and mass.

Both routes operate. The aligned orbits of hot Jupiters with nearby companions and the planets caught in resonant chains point to the disc; the misaligned orbits around hot stars and the planets caught mid-journey on eccentric orbits point to the companion. What has not been settled is the fraction, and the essay that drew the edge suggested how to settle it: measure each hot Jupiter’s distance from its star in units of its own Roche distance, and see how many sit at the eccentric channel’s edge. This essay does that with simulated populations whose fraction is known, and finds that the obvious way of doing it fails in an instructive way.

Why one channel has an edge and the other does not #

The eccentric channel’s edge is a consequence of two conservation statements, and it helps to have them in view. A planet whose eccentricity has been driven to near one by a distant companion’s slow torque loses orbital energy at each close passage, where the tide it raises on itself is strongest, and loses almost no angular momentum, because a torque applied at periastron for a short time has little lever arm. Angular momentum per unit mass is $\sqrt{GM_\star a(1-e^2)}$, so $a(1-e^2)$ stays fixed while $e$ falls to zero, and the final circle has radius $a(1-e^2) = q(1+e)$ — twice the periastron when $e$ starts near one.

The Roche distance, $a_R = 2.16\,R_p (M_\star/M_p)^{1/3}$, is where a fluid planet on a circular orbit fills its own Roche lobe. A planet whose first periastron fell inside it was stripped on that passage, and so the survivors circularise no closer than $2a_R$. For a Jupiter around the Sun that is about 0.021 astronomical units, a period of a little over a day; for a planet three times as massive it is closer, and for an inflated one further out.

The disc channel has none of this. A planet migrating through gas stops where the torque stops — at the inner edge of the disc, which a young star’s magnetic field holds open at a few stellar radii, a period of a few days — or where it meets a trap in the disc’s structure. The stopping point is a property of the disc and the star, and the planet’s own size and mass do not enter. That is why one channel’s natural unit is the planet’s Roche distance and the other’s is a period, and it is the whole reason the fitting in the second half of this essay has to be done the way it is.

Two shapes, not one edge #

The figure is the population the argument imagines. The eccentric channel begins sharply just inside twice the Roche distance — the few planets below it are the ones delivered on the least eccentric orbits, which circularise at $q(1+e_0)$ with $e_0$ nearer 0.9 than 1 — and piles up between two and five. The disc channel has no edge in these units at all. A disc stops a planet at a period, a few days, and a period is a different number of Roche distances for every planet: a light, puffy planet has a large Roche distance and sits at three; a heavy, compact one has a small Roche distance and sits at eight. The disc’s planets form a broad hump that overlaps most of the eccentric pile.

The second population in the figure is the one a survey actually has. Radii come from transit depths, which are only as good as the star’s radius, and masses from radial velocities, which are only as good as the star’s mass; ten per cent on each is typical for a well-studied hot Jupiter and better than most. The radius in the Roche distance should also be the radius the planet had when it circularised, which for a planet that has contracted or been inflated since is not the radius measured now. The edge at two loses its sharpness, and a few per cent of the eccentric planets spill inside it.

What the figure does not show is how much that blurring matters, and the first surprise is that it hardly does.

A few hundred planets, if the templates are right #

The fraction is fitted by treating each planet’s position as drawn from a mixture of the two channels’ distributions, each computed from a model of the channel with the same errors folded in, and asking which mixture makes the observed positions most probable.

Fifty planets say almost nothing: the fit is broad and this one happens to peak far from the truth. Two hundred fix the fraction to a tenth. Eight hundred fix it to a twentieth. The transit surveys and their radial-velocity follow-up have measured masses and radii for several hundred hot Jupiters, so the sample needed is roughly the sample that exists.

A template is a model, and that is the point at which the fit stops being statistics. The eccentric template here was built from a distribution of periastra and initial eccentricities, a rule that every survivor circularises fully, and a population of planets with masses and radii in stated ranges. The disc template was built from a range of stopping periods and the same planets. Change any of those assumptions and the template changes; the fit then reports, with full confidence and a narrow spread, the fraction that best reconciles the data with the wrong templates. The spreads in the figure above are the statistical floor, not the error.

The spread is set by the number of planets and hardly by how well each is measured. Errors of twenty per cent on every radius and mass, which would be a scandal for any individual planet, widen the fraction by a quarter. The reason is that the two channels differ in the whole shape of their distributions, not only at the edge. The eccentric pile is narrow and sits low; the disc’s hump is broad and sits high; even with the edge smeared into a ramp, a planet at 2.3 Roche distances is several times likelier to have come by the eccentric route than by the disc’s, and one at seven is likelier to have come through the disc. The edge is the most striking feature and not the one that carries the information.

That is the good news, and it is conditional on a clause the figures have so far assumed: that the templates — each channel’s predicted distribution — are right.

A history assumed wrong #

The eccentric channel’s edge is at twice the Roche distance the planet had when it circularised. A hot Jupiter’s radius is not a fixed quantity. Many are larger than any model of a cooling gas giant allows, inflated by some fraction of the starlight they absorb; their radii change as their stars brighten and as the planets themselves cool, and a planet that arrived a few hundred million years after its star formed was larger then than it is now.

A population that has contracted by ten per cent since circularising has its eccentric planets at 2.2 present Roche distances instead of two; read against a template that assumes no change, part of the eccentric pile looks like the disc’s hump and the fitted fraction falls. A population that has been re-inflated looks the other way. The shift for a true fraction of 0.7 is nearly half the range of possible answers.

This is the precision–accuracy split the previous figures hid. The fraction can be made precise with a few hundred planets measured to ten per cent; it is accurate only to the extent that the planets’ radius histories are known, and those are among the least settled questions about hot Jupiters. The inflation mechanism is not agreed on, and different mechanisms predict different histories — some keep a planet large from birth, others re-inflate it as its star brightens late in life. Each gives a different eccentric template and a different fraction from the same data.

The candidates for the inflation each leave a different history. If the extra heat is deposited deep in the planet by currents driven through its ionised atmosphere, the planet is held large for as long as it is irradiated, and its radius at arrival was close to its radius now. If the planet is large because it has not yet finished contracting from its formation, it was larger still when it arrived. If it is re-inflated as its star brightens on the main sequence, it was smaller then and has grown — and the planets around the oldest stars should show it most. A survey’s hot Jupiters span all three situations, and the bias in the fraction is the average of the three weighted by whichever mechanism operates, which nobody yet knows.

Each channel in its own units #

The obvious way to fit the fraction is in Roche distances, because that is where the eccentric channel’s edge is universal. It is also the way that fails worst when anything else about the population is wrong.

The failure is in the other channel. A template for the disc channel in Roche distances has to convert a period into a number of Roche distances for each planet, which depends on its mass and radius; build it for planets of one mass and apply it to heavier ones, and the disc’s hump is in the wrong place. Planets three times heavier than assumed have Roche distances smaller by the cube root of three, the disc’s planets sit at 1.44 times as many Roche distances as the template expects, and the fit hands most of them to the eccentric channel. A fit in periods alone has the mirror-image problem, smaller here: the disc’s template is universal in days, but the eccentric channel’s edge in days slides as the mass to the minus one half.

The repair is to stop forcing both channels onto one axis. Each planet has both a period and a Roche distance; for a given planet the two are tied by Kepler’s third law, $\log x = \tfrac23 \log P + \text{const}$, so a probability per unit of one converts into a probability per unit of the other by a factor of two-thirds. Compute the probability that a planet came by the eccentric route from where it sits in Roche distances, the probability that it came through the disc from where it sits in days, and compare them planet by planet. Neither template then depends on what the planets weigh, and the fit is flat across a factor of nine in mass. With the right masses it is also as precise as the best of the others, about four hundredths for four hundred planets.

The general habit is worth stating, because it applies well beyond hot Jupiters. A mixture of mechanisms should be fitted with each mechanism in the variable where it is simplest, not with all of them in whichever variable the most striking one suggests. The edge at twice the Roche distance is a fact about one channel. Treating it as the axis for both imports every uncertainty about the planets’ masses into the other channel’s template, where it does the most harm.

The same structure appears elsewhere in this subject, and the lesson transfers. A population of binary stars contains systems circularised by tides and systems born circular, and the period below which every orbit is round is a property of the tidal channel only; fitting the eccentricity distribution of all binaries in units of that cut-off would make the born-circular ones look like a tidal population of the wrong age. Wherever two mechanisms produce one population, each has a variable in which it is simple, and those variables are rarely the same one.

There is a practical cost to doing it properly. The planet-by-planet comparison needs each planet’s mass and radius, not just its period, because without them a planet’s position in Roche distances is unknown. Planets with periods but no masses — most of those found by transit surveys and never followed up with radial velocities — cannot enter the fit except through their periods, where they carry the eccentric template’s mass dependence with them. The fraction is a measurement on the well-characterised hot Jupiters, and extending it to the rest means modelling their masses, which reintroduces the assumption the method was designed to remove.

Two edges, and only one of them sloping #

The same physics is visible without any fitting, in the plane where hot Jupiters are usually plotted.

The colours are what a survey does not have. Without them the population is one cloud whose lower envelope is set by the eccentric planets at high mass — down to less than a day for the heaviest — and by both channels at low mass, where the eccentric edge rises into the disc’s band. The slope of that envelope is therefore a diagnostic in its own right: a population delivered entirely through the disc would have a flat lower envelope, one delivered entirely on eccentric orbits a lower envelope falling as the square root of the mass. The observed hot Jupiters do show an inner edge that falls with mass, which is one of the older arguments that a substantial fraction came by the eccentric route. But the envelope is a statistic of the few planets nearest the star, and it measures the slope better than it measures the fraction.

What else tells the routes apart #

The distribution of orbits is one witness, and the others measure different things.

The tilt between a planet’s orbit and its star’s equator is the most direct. A planet carried inward by a disc should orbit in the disc’s plane, aligned with the star; one delivered by a companion’s torque can arrive on any inclination. Hot Jupiters around cool stars are mostly aligned and those around hot stars often are not, which has been read as cool stars’ convective envelopes realigning their planets by tides — so the obliquity measures the delivery route only where tides have not erased it.

Companions are another. A Jupiter delivered on an eccentric orbit by a distant companion needs that companion still to be there, or at least to have been there; one that came through the disc may have close neighbours that an eccentric delivery would have scattered. Hot Jupiters with nearby small planets are rare, and the few known — several of them found as transits running late under a neighbour’s pull — favour the disc for those systems.

And there are the planets caught on the way: warm Jupiters on eccentric orbits whose periastra are close enough that tides will circularise them within the star’s lifetime. Their number, compared with the number already circularised, measures the rate of eccentric delivery directly — provided the mechanism that pumps the eccentricity is not switched off before they get there.

Each of these could be folded into the same mixture fit as an extra dimension, with each channel’s prediction for it, and each would reduce the dependence on the radius history. None removes it, because the orbits are where most of the planets’ information is.

What the simulation assumes #

The two channels are idealised. The eccentric channel delivers periastra drawn uniformly in the logarithm between 0.7 and 4 Roche distances, with eccentricities between 0.9 and 0.99, and assumes every planet that survives circularises fully; real delivery depends on the companion, the tidal dissipation and the time available, and a channel that delivered planets mostly near the disruption limit would make a sharper pile. The disc channel stops every planet uniformly between three and eight days, a stand-in for the inner edges of discs truncated by their stars’ magnetic fields; discs with different inner edges give a hump in a different place.

The planets’ masses and radii are drawn independently, which they are not: heavier hot Jupiters are on average smaller, and the most inflated planets are light. A correlated mass–radius relation moves both templates together and would change the numbers here, though not the conclusion that each channel should be fitted in its own units.

The radius history is a single fractional change, the same for every planet in a population or scattered about a mean. Real histories depend on the planet’s mass, the star’s age and type and the inflation mechanism, so the bias in the fraction is not a single shift but a different one at each mass.

And selection is ignored. Transit surveys find close-in planets more easily, so the observed population over-represents short periods, which in Roche distances favours the eccentric pile; a real fit has to fold the survey’s detection probability into both templates, as the projected spread of obliquities has to be unfolded before it says anything about the true one.

Still open: the radius each planet had when it arrived #

The fraction of hot Jupiters delivered on eccentric orbits can be measured from their orbits alone, with the samples that exist, to about a tenth — provided each channel is read in its own units and provided the planets’ radii at the time of their arrival are known. The first condition is a matter of method. The second is physics that has not been settled: why hot Jupiters are inflated, when the inflation happened, and how much of it a planet carried through the century or million years of its circularisation.

So the question of how hot Jupiters arrived turns, at the end, into a question about their interiors. A planet’s Roche distance is a statement about its density, and the density at delivery is what the edge was drawn in. The youngest transiting giants, measured around stars only tens of millions of years old, are the ones whose radii are closest to what they were at arrival. A sample of young hot Jupiters large enough to draw their edge on its own, in their own Roche distances, would measure the fraction without needing the history at all.