A neighbour an eccentric arrival cannot leave

Hot Jupiters arrive by two routes, and the orbits alone can say in what proportion — provided each route is read in its own units and the planets’ radii at the time they arrived are known. One route is the disc: a giant planet embedded in the gas it formed in is carried inward by the small residual of two nearly cancelling torques, and parks where the disc’s inner edge stops it. The other is violence: a distant companion, or a scattering between planets, throws the giant onto an orbit so eccentric that its closest approach is a few hundredths of an astronomical unit, and tides at each close passage drain its energy until it settles on a circle at twice the distance that would have torn it apart.

The two routes leave different orbits behind, and the distribution of orbits is where most of the evidence lies. But they also leave different neighbourhoods. The disc is a calm environment; planets migrating through it can migrate together, trap one another in resonance, and arrive in company. The violent route is a demolition. A Jupiter whose orbit stretches from a few hundredths of an astronomical unit to several astronomical units passes through everything in between, twice an orbit, for thousands of orbits.

Every distance between, twice an orbit #

The figure follows one eccentric delivery from the moment its periastron reaches the star’s vicinity. Tides raised on the planet at each close passage take energy out of the orbit and very little angular momentum, so the orbit shrinks while its periastron stays put, and the quantity $a(1-e^2)$ is conserved throughout. The apoastron, which begins near ten astronomical units, falls toward the periastron, and the orbit ends as a circle at twice its periastron distance — a five-day hot Jupiter, in this example.

Between the two curves lies every distance the planet visits. For most of the delivery, that is everything from three hundredths of an astronomical unit to beyond one. A small planet orbiting anywhere in that range, on a nearly circular orbit, is crossed by a body three hundred times its mass on every one of the giant’s orbits. The giant’s gravity at closest approach changes the small planet’s orbit by an amount comparable to its own, and after a few hundred crossings the small planet has been thrown out of the system, onto the star, or into the giant itself. The delivery takes thousands to millions of orbits, depending on how efficiently the planet dissipates tides. Nothing survives it.

The strip at the bottom of the figure is where a neighbour would have to be for it to be found the way most hot Jupiter companions are found: transiting the star along with the hot Jupiter, at a period of less than about fifty days. The migrator’s orbit covers that strip completely until its semi-major axis has shrunk to fifteen hundredths of an astronomical unit, and its periastron passes through the strip’s inner edge for the rest of the delivery. A hot Jupiter that arrived this way has an empty neighbourhood out to well beyond the periods a transit survey can find.

That is why the companions are evidence. An eccentric arrival cannot leave a close neighbour, so every hot Jupiter that has one came through the disc. The converse does not hold — a disc delivery need not leave a neighbour either — and that asymmetry decides what the count can say.

The first one, and why it was a surprise #

For the first fifteen years of hot Jupiters, none had a neighbour. Radial-velocity surveys found them alone, ground-based transit surveys found them alone, and the first space survey to stare at a single field for four years found its hot Jupiters alone as well, while finding nearby companions for a large share of the warm Jupiters further out. The absence fitted the violent account so well that it was read as evidence for it: hot Jupiters were lonely because they had cleared their surroundings on the way in.

Then, in 2015, the space telescope’s second mission found that WASP-47 b, a four-day hot Jupiter known from the ground, had a planet of about twice the Earth’s radius on a nineteen-hour orbit inside it and a Neptune on a nine-day orbit outside it. Both transit, both are close enough to the giant to be strongly perturbed by it, and the timing of the transits weighs all three. No eccentric delivery could have left that arrangement, so WASP-47 b came through the disc. Since then a handful more have been found, most by the small, periodic shifts that a neighbour imposes on a hot Jupiter’s transit times rather than by a second transit, and the loneliness of hot Jupiters has become a rate rather than a rule.

The rate is what this essay is about. One counterexample proved that the disc route exists; the share of hot Jupiters with neighbours is what says how often it was taken.

A floor and not a value #

Suppose a fraction $r$ of hot Jupiters are found with nearby companions. All of those were disc deliveries. The disc’s share $f_d$ of all hot Jupiters must then be at least $r$, and is larger by however many disc deliveries ended up with no visible companion. If a disc delivery is seen to have a companion with probability $k$, the expected companion rate is $r = k f_d$, and the share is $f_d = r/k$.

The figure is three straight lines, and the whole difficulty is in their slopes. The observed companion rate is measured: transit surveys and the timing variations that an unseen neighbour imposes on a hot Jupiter’s transits put it at a few to about ten per cent, depending on the sample and how thoroughly each system was searched. Seven per cent is used here as an illustration. What converts it into a share is $k$, and $k$ is the product of three separate chances. A disc delivery must have left a neighbour in the first place. The neighbour must line up so that it also transits. And it must be large enough for its transit to be detected. Each of those can be estimated; only one of them can be computed.

The neighbour has to line up #

A hot Jupiter that transits its star tells the observer that its orbit is seen almost edge on — tilted from the line of sight by less than the star’s angular radius as seen from the planet. A neighbour further out subtends a smaller star, so its band of transiting inclinations is narrower. Even if the two orbits lie in exactly the same plane, the outer one transits only if that common plane is tilted by less than its narrower band.

The surprise is that flatness hardly matters. With the orbits exactly coplanar, the chance that a neighbour at twenty days transits along with a three-day hot Jupiter is the ratio of their orbital radii, 28 per cent, and that is the ceiling — no spread of inclinations can raise it. A spread of half a degree leaves it unchanged, because the hot Jupiter’s own transit band is already wider than that. Systems built in a disc are flat to a degree or two, as the multi-planet systems found by transit surveys show, and for such systems the geometry rather than the tilt hides most neighbours. Only a spread of several degrees, larger than the transiting band itself, begins to cut the yield further.

So a neighbour at a week is caught most of the time and a neighbour at fifty days less than a fifth of the time. The neighbours a disc leaves are not uniformly distributed in period — resonant chains put them at specific period ratios, two to one and three to two being common — and a hot Jupiter at three days with a companion in the two-to-one resonance at six days would be caught most of the time. The ones further out are mostly missed.

And it has to be large enough #

The second factor is detection. A transit’s depth is the square of the ratio of the planet’s radius to the star’s, and its significance grows with the square root of the time spent in transit summed over every transit observed. A neighbour of two Earth radii crossing a Sun-like star dims it by a third of a thousandth.

The detection curves are steep. Above two and a half Earth radii, a companion is found at any period up to forty days; below one and a half it is found only at the shortest periods, and an Earth-sized neighbour not at all. The companions known beside hot Jupiters are mostly two to four Earth radii, which is what these curves would let through, and it means that a population of smaller neighbours, of the kind that are common around ordinary stars, could exist beside hot Jupiters and not have been counted.

The calculation assumes a bright star watched for close to three years by a space telescope — the most favourable case. Surveys from the ground, or space surveys that spend less time on each field, push the curves to the right, and the hot Jupiters observed that way have searches for small neighbours that are close to blind.

What can be computed #

Putting the two computable factors together gives the chance that a neighbour which survived the disc delivery is actually counted.

For a neighbour of typical size, spread evenly in the logarithm of its period between four and a hundred days, the chance of being counted is about a third with good photometry and a sixth with poor. Better photometry than three hundred parts per million hardly helps, because at that level the geometry is the limit; worse photometry halves the yield, because detection becomes the limit. Neither number depends on how the hot Jupiter arrived. It is the denominator a companion survey has to divide by before its rate means anything.

Timing helps here, and it changes the factors. A neighbour that does not transit can still be found if it perturbs the hot Jupiter’s transit times, and the perturbation is largest near resonance, where the disc’s company is most likely to sit. A neighbour in the two-to-one resonance with a three-day hot Jupiter, even one too small or too tilted to transit, shifts the giant’s transits by tens of seconds over a cycle of months — measurable with the precision space photometry gives a hot Jupiter’s deep transit. Timing searches raise the counted fraction for resonant neighbours toward the fraction that exist, and leave the non-resonant ones, further out, to the transit geometry.

That leaves the third factor, which cannot be computed from first principles: the probability that a disc delivery leaves a neighbour at all. Warm Jupiters — giants at ten to a few hundred days, which most accounts say arrived through the disc — have nearby small companions about half the time. If hot Jupiters delivered by the disc kept neighbours as often, and a third of those were counted, $k$ would be about a sixth and a seven per cent companion rate would imply that at least forty per cent of hot Jupiters came through the disc. If hot Jupiters delivered by the disc keep neighbours less often — because the disc’s inner edge, where they stop, is a crowded place where companions are pushed into the star or excited by the giant’s resonances — $k$ is smaller and the implied share larger.

A sharper count of an assumed scale #

The companion rate can be measured to any precision a large enough sample allows. The figure shows what that precision buys.

The statistics behave as statistics do: the width of the likelihood falls as the square root of the sample, and eight hundred hot Jupiters, about the number transit surveys have now found, pin the share to a few per cent. But the horizontal axis is in units of $k$. Halve the probability that a disc delivery shows a companion, and every curve slides to twice the share with the same relative width. The sample buys precision; the scale is assumed. A share of fourteen per cent known to three per cent is the answer for $k = 0.5$, and a share of twenty-eight per cent known to six per cent is the answer for $k = 0.25$, from the same data.

The same structure appeared in counting planets that were not seen: an occurrence rate is a count divided by a completeness, and the completeness carries every assumption the count does not. Here the completeness has a geometric part that can be computed exactly, a photometric part that can be computed from the noise, and a physical part — whether the disc left a neighbour — that is a property of the delivery itself.

Two routes that can be told apart once #

There is one situation in which the floor becomes a value. If some hot Jupiters are known on independent grounds to have come through the disc — because their orbits are aligned with their stars’ equators in systems too hot for tides to have realigned them, or because they sit in a resonant chain with their neighbours that no violent delivery could have assembled — then the probability that a disc delivery leaves a visible neighbour can be measured on them directly. The resonant hot Jupiter systems found so far, a handful, are exactly this sample, and they are too few to fix $k$ to better than a factor of two.

The orbits offer the other route. The fit of the mixture in each channel’s units gives the disc’s share from the shape of the distribution of orbital distances, with no reference to neighbours, and its weakness — the dependence on the planets’ radii when they arrived — has nothing in common with the neighbours’ weakness. If the orbits give a share of forty per cent and the companions a floor of fourteen with $k = 0.5$, the two agree, and the comparison says that $k$ cannot be much larger than a sixth. A combined analysis would measure $k$ as a by-product.

What the sweep does not show #

The figure of the sweep treats the eccentric delivery as a clean, isolated process: a giant on a fixed periastron, losing energy steadily, with nothing else in the system. Two complications soften the conclusion that it leaves nothing behind.

A neighbour can be acquired afterwards. If small planets form or migrate into the region after the giant has circularised — from material left in the disc, if the disc had not yet gone — a hot Jupiter delivered eccentrically could still end up with a companion. High-eccentricity migration is usually slower than a disc’s lifetime, which makes this unlikely, but a fast delivery by planet–planet scattering early in the disc’s life is not ruled out.

And a neighbour can be lost after a disc delivery. A hot Jupiter carried in by the disc may still have its system destabilised later, by an outer giant’s secular forcing or by the star’s own tides acting on the inner planets, and the neighbours of a disc delivery may be removed long after it. The probability that a disc delivery shows a companion today, $k$, then includes how long the system has had to lose it.

Still open: how often the disc leaves company #

The neighbour test is clean in one direction. A hot Jupiter with a close companion came through the disc, and nothing about its radius, its mass or the tidal physics of its envelope changes that. In the other direction it is a floor, and the floor’s height is the observed companion rate divided by a probability the observations do not supply.

Two of that probability’s three factors are computed here — the geometry of co-transit, which caps a neighbour at twenty days at about a quarter even in a perfectly flat system, and the detection of shallow transits, which halves the count for the smaller neighbours. The third, whether disc migration typically arrives in company, is a question for simulations of migrating systems in discs and for the young systems where the disc has only just gone. A hot Jupiter found around a star of a few tens of millions of years with a neighbour in resonance would say that company is common; a dozen found alone around young stars would say the floor is close to the answer, and that most hot Jupiters came the violent way.