A peak that neither survey can see alone

The first giant planets found round other stars were in the wrong place. 51 Pegasi b orbits its star in four days, twenty times closer than Mercury orbits the Sun, and for several years afterwards most of the giants discovered were similar: hot Jupiters, and giants on eccentric orbits of a few months. The Solar System’s own giants, on orbits of twelve and thirty years, looked like a special case.

They were not. They were the hardest case to find. A radial-velocity survey measures the wobble of the star, and a wobble can be recognised only after most of a cycle has been watched; a planet on a twelve-year orbit takes twelve years to reveal. As the long surveys accumulated decades of data, the giants turned up further and further out, and when their occurrence was corrected for what each survey could not see, the distribution was the opposite of the early impression. Giant planets become commoner with distance from their star, rising by a factor of several between a tenth of an astronomical unit and a few, and the hot Jupiters that were found first are a rare population at the bottom of that rise.

Then, somewhere past the orbits the surveys can follow, the rise has to stop.

The rise, and why it is believed #

The inner part of the curve is the best-measured statement about giant planets there is. The surveys that established it — at Lick and Keck, at the European HARPS spectrograph, at the Anglo-Australian Telescope — each followed a few hundred to a thousand Sun-like stars for between one and three decades, and their completeness can be computed star by star: for each star, and each possible planet mass and period, whether a planet of that kind would have produced a detectable signal in the velocities actually taken on the nights actually observed. Dividing the detections by that completeness gives the rate, and three independent groups with three independent instruments find the same rise.

The rise is steep. Per logarithmic interval of distance, a star is several times more likely to have a giant at two or three astronomical units than at a tenth of one. The occurrence of hot Jupiters is about half a per cent to one per cent; the occurrence of giants between one and five astronomical units is of order ten times that. The first giants found were found first because they were close in, not because they were typical, which is the plainest example in the whole census of a survey drawing its own sky.

The rise is also what the formation of giant planets predicts. A giant core has to reach about ten Earth masses and start capturing gas before the disc disperses, and cores grow fastest where there is most solid material to build them. Inside the snow line, at a few astronomical units for a Sun-like star, the solids are rock and metal; outside it, water ice condenses too and the surface density of solids jumps by a factor of two or more. The rise towards a few astronomical units is the curve climbing towards the place where cores are easiest to build.

The bottom of the rise has a bump of its own #

The rise is not perfectly smooth at its inner end, and the exception is instructive. Hot Jupiters are not simply the low tail of a power law; they pile up at orbital periods of about three days, with a sharp inner edge near two and a distinct deficit of giants between about ten days and a hundred — the so-called period valley. The pile-up and the valley are the fingerprints of how giants arrive close in.

A giant cannot form where the hot Jupiters are: there is too little solid material and too much heat so close to the star. The hot Jupiters formed further out and were moved inward — either by migrating through the gas disc, or by being thrown onto a highly eccentric orbit by a companion and then circularised by tides at the pericentre. The second route delivers planets to about twice the tidal-disruption distance, which for a Jupiter round a Sun-like star is a period of a few days, and that is where the pile-up is. The period valley is the gap between the planets that were moved all the way in and the ones that stayed near where they formed.

So the curve drawn here, a single rise, is a smoothing of two populations: a small, relocated one close in, and a large, roughly in-place one that rises towards the snow line. The relocated population holds perhaps a tenth of the giants, and its details do not affect the peak. But it is a reminder that a smooth fitted curve is a summary of whatever shaped the population, and that a distribution drawn from its statistics has several histories folded into it.

Where the radial velocities run out #

A radial-velocity survey can report a planet with confidence once it has watched a substantial part of an orbit, and with some confidence from a partial orbit if the curvature of the velocity trend is measured well. For a survey with twenty years of data the practical limit is an orbital period of a few decades — about seven to ten astronomical units round a Sun-like star.

Near that limit the survey’s completeness falls, and the planets it does detect there have poorly measured periods and masses. The rate in the outermost bins is therefore built from few detections, each with a large correction, and the curve’s shape there is uncertain in exactly the way that matters: whether it is still rising, has flattened, or has begun to fall. The long surveys found a flattening or a turnover in their outermost bins, at two to three astronomical units in most analyses — but at a significance that depends on those bins, and with a slope beyond the peak that the radial velocities alone could not pin down.

What a partial orbit does give is a curvature. A velocity series covering a third of a twenty-year orbit shows a smooth arc, and an arc can be fitted by a range of periods and masses that trade against each other: a longer period with a heavier planet produces nearly the same arc as a shorter one with a lighter planet, as long as the part of the orbit observed is the same. One eccentric orbit can even imitate two circular ones when the coverage is poor. The long-period planets at the edge of the window therefore enter the rate as probability distributions over period and mass, not as points, and the outermost bins are populated by smeared-out contributions from planets whose periods are uncertain by factors of two. The shape of the curve near the peak is partly the shape of those smears, which is one reason different analyses of overlapping data put the peak anywhere from two to four astronomical units.

The other window #

Direct imaging finds giant planets in the opposite regime. It separates a planet’s own light from its star’s, which demands a contrast of a million or more at a fraction of an arcsecond, and it can only be done for planets far from their stars — tens of astronomical units for the nearest young stars — and for planets that are young and still hot from their formation, which glow in the infrared at a contrast the instruments can reach. Its window is at large separations and high masses, and its sensitivity declines towards small separations for the same reason the radial-velocity sensitivity declines towards large ones.

The imaging surveys of the 2010s observed several hundred young stars each, with instruments on eight-metre telescopes designed for the purpose. What they found, for Sun-like stars, was very little: planets of five to thirteen Jupiter masses between ten and a hundred astronomical units round about one star in a hundred or fewer — the largest survey reported 0.8 per cent for stars up to about a solar mass. Round more massive stars the rate was several times higher, and the planets found there, the four giants of HR 8799 among them, are what most images of exoplanets are pictures of. Round Sun-like stars wide-orbit giants are rare.

Joining them with an extrapolation #

The imaging result becomes a measurement of the turnover only when it is set against what the radial-velocity distribution would predict at the same separations. That needs two extrapolations — in distance, from a few astronomical units out to tens, and in mass, from the whole range of giants down to only the heaviest, which is all the imaging can see — and it is worth doing both explicitly.

The factor of nineteen is the part of this that is not in doubt. If giant planets continued to become commoner with distance as they do inside three astronomical units, imaging surveys would be finding wide-orbit giants round one Sun-like star in seven. They find them round one in a hundred or fewer. Something has to bring the curve down between the edge of one window and the start of the other, and the peak is where it turns.

The factor of two and a half against the flat continuation is a weaker statement, and it depends on the extrapolation in mass. The mass function of giants — how many there are per logarithmic interval of mass — is measured by the radial-velocity surveys, mostly at masses below a few Jupiters, and it falls gently. The imaging surveys see only the top of it, above five Jupiter masses, so the predicted imaging rate is proportional to the share of the mass function that lies up there, and that share is an extrapolation of a slope measured mostly at lower masses.

That is a fair picture of what is known. The existence of a turnover is secure: no reasonable mass function rescues a continued rise. Its steepness is not, because the imaging window sees only the heaviest giants and the radial-velocity window sees mostly the lighter ones, and the two have to be joined through a mass function that neither measures across the whole range. The peak is established by two surveys that each see one side of it, through a join that requires knowing how planets are distributed in mass as well as in distance.

Counting all of them #

The number most often quoted about giant planets is the fraction of stars that have one. It turns out to be a reading of a curve at a chosen distance, and the curve beyond the peak is the part least measured.

The spread from twelve to twenty per cent is not large as astronomical uncertainties go, but it changes a qualitative statement. At the low end most Sun-like stars have no giant planet anywhere and the Solar System’s pair of gas giants makes it a system of perhaps one in ten. At the high end one star in five has at least one, and the Solar System is ordinary in having giants, if unusual in having two. Whether Jupiter is a common or an uncommon planet is decided by the slope beyond the peak, and the slope beyond the peak is decided by the scarcity of planets nobody has seen.

The Solar System, sitting at the peak #

It is worth noticing where the Solar System’s giants fall on the curve. Jupiter is at 5.2 astronomical units and Saturn at 9.5 — just beyond the peak, where the curve has turned over and is falling again but has not yet fallen far. The Sun’s snow line in its formation era was at roughly 2.7 astronomical units, a little inside Jupiter’s orbit, and the peak of the distribution for Sun-like stars sits in the same neighbourhood.

That coincidence is the most direct connection between the planetary census and the theory of how giants form. The rise inside the peak is the region where cores can form but less readily, because the solids are rock alone; the peak is at and just beyond the snow line, where ice doubles the solids and cores grow fastest; and the fall beyond is where orbits are slow and discs thin, so cores grow too slowly to catch the gas before it disperses. The distribution of giant planets in distance is a map of where the conditions for building them were best, drawn from the statistics of planets that formed in discs long since gone.

The same reasoning applies round other stars, with the snow line moving with the star’s luminosity. For a star of half a solar mass the snow line was nearer one astronomical unit, and the peak of that star’s giant planets should move in with it — the microlensing surveys, which find giants round small stars at a few astronomical units, see a distribution of cold giants round M dwarfs consistent with a peak nearer the star. For stars of two solar masses and above, the imaging surveys see more wide-orbit giants, and the radial-velocity surveys of evolved intermediate-mass stars see their giant occurrence peaking at a few astronomical units and falling sharply beyond. A peak that tracks the snow line from star to star would be the census confirming the theory’s central idea directly.

Two methods joined across a gap #

The structure of this argument — a quantity measured by one method over one range, by another method over another, and joined through a region where neither is comfortable — is common enough in astronomy to have a characteristic weakness.

Every distance in astronomy is measured with the one before it: a method calibrated on nearby objects is applied to farther ones, and the calibration is transferred through the overlap between them. The weakness is always in the overlap, where each method is near its limit and the objects are few. The giant-planet census has the same shape with a gap instead of an overlap. The radial-velocity surveys reach to perhaps seven astronomical units with poor completeness, the imaging surveys start at perhaps ten with poor completeness, and the join is made through two models — a mass function and a functional form — rather than through objects both methods see.

The gap is being closed by a third method. Astrometry measures a star’s wobble across the sky rather than along the line of sight, and its sensitivity rises with the planet’s distance from its star — the opposite of radial velocity’s — so it is most sensitive in exactly the gap. The Gaia spacecraft has measured the positions of a billion stars to tens of microarcseconds over about a decade, and its full data release is expected to reveal thousands of giant planets between one and a few astronomical units round nearby stars, with their true masses rather than the minimum masses the radial velocities give.

What the extrapolations assume #

The curve is a broken power law. A real distribution has no reason to turn over at a single point with two constant slopes on either side; it is the simplest shape consistent with a rise and a fall, and the published fits that use smoother forms find the peak in the same place but disagree on its sharpness.

The mass function is assumed independent of distance. It is drawn from the radial-velocity surveys inside a few astronomical units and applied at tens. If the heaviest giants are preferentially at large distances — which some formation routes predict, gravitational instability in a massive outer disc among them — the imaging window would contain more of them than the extrapolation allows, and the turnover would be steeper than it looks.

Imaging rates depend on how bright a young giant is. The masses of directly imaged planets are inferred from their brightness and their age through models of how giants cool, and the models disagree, most for the youngest and most massive planets. A “five Jupiter-mass” planet from one model is sometimes three or eight from another, and the imaging rate for a fixed mass range moves with that.

And the star samples differ. The imaging surveys observe young stars, because only young planets are bright enough; the radial-velocity surveys observe old, quiet ones, because young stars are too active for precise velocities. Joining them assumes that the planets round a young star at forty million years are the same population as those round an old star at five billion, which, for giants at tens of astronomical units, has not been tested.

Still open: whether the peak has the same shape everywhere #

The existence of a peak is secure; its position, its sharpness and its dependence on the star are not. If the peak moves with the snow line, stars of different mass and luminosity should show it at distances scaling roughly with the square root of their luminosity, and stars of different metallicity should show it at the same place but with different heights — since the iron in a star governs how many giants form rather than where the ice condenses. The astrometric census will measure the peak’s position round hundreds of stars at once, with masses rather than lower limits, and for the first time within a single method. Whether it finds a peak that tracks luminosity, or one fixed in astronomical units whatever the star — which would point to something other than the snow line setting where giants are built — is the question the occurrence rates have been approaching from both sides without being able to reach.