A peak that should move with the star

Giant planets round Sun-like stars are not spread evenly in orbital distance. Their occurrence rises outward from the hottest orbits, reaches a maximum somewhere between two and four astronomical units, and falls beyond it. The rise is measured by two decades of radial-velocity monitoring; the fall is inferred by joining those surveys to direct imaging across a gap neither method covers, and the existence of a peak between them is now secure.

The peak sits close to where water ice first condensed in the Sun’s disc, the line beyond which ice counts as rock. That is the obvious explanation for it: beyond the snow line the solid material available to build a core roughly doubles, cores grow fast enough to capture gas before the disc disperses, and giant planets form. If that is why the peak is there, the peak should move when the snow line moves, and the snow line moves with the star.

This essay works out where the peak should sit round other stars under each of the hypotheses that have been proposed for it, and asks which measurement can tell them apart. The answer turns out to be organised around a single fact that is easy to miss: every hypothesis is calibrated on the Sun, so every hypothesis agrees about stars like the Sun, and the surveys that found the peak were made of stars like the Sun.

Where the ice begins round a smaller star #

The snow line is a temperature, about 150 kelvin for water at the low pressures of a protoplanetary disc. In a disc warmed only by its star’s light, the temperature at a distance r falls as $L^{1/4} r^{-1/2}$, so the radius where it reaches 150 kelvin scales as the square root of the luminosity. Round the young Sun it sat at about three astronomical units.

The question is which luminosity. A main-sequence star’s luminosity rises steeply with its mass, roughly as $M^4$ near the Sun’s mass, and a snow line set by it would move as $M^2$. But planets form in the first few million years, when the star is still contracting towards the main sequence and is brighter than it will later be, and the luminosity of a star a million years old rises much more gently with mass — closer to $M^2$, which puts the snow line at roughly $a \propto M$. The heating of the disc by its own accretion adds a further dependence that weakens the stellar one. Both exponents are rough, and the young-star case is the more physical; but they bracket the range.

Two other hypotheses say the peak has nothing to do with ice. If giants migrate inward through their discs and stop where something halts them — the inner edge of a dead zone, a pressure trap — the peak could sit at a fixed orbital period, or at a fixed distance, independent of the star. Each is a single exponent: the peak’s distance goes as $M^q$, with q = 2 or 1 for the two snow lines, 1/3 for a fixed period and 0 for a fixed distance.

Which ice #

The water snow line is not the only one. A disc has a staircase of condensation fronts, one for each volatile — water at about 150 kelvin, carbon dioxide near 70, carbon monoxide near 20 — and each sits further out than the last. Round the young Sun, the carbon monoxide line was at some tens of astronomical units. A peak in the giants’ occurrence at three astronomical units is the water line’s or nobody’s; no other ice condenses near there.

Water is also the ice that matters most for building cores. It is the most abundant condensable after rock, and its condensation roughly doubles the surface density of solids, where carbon dioxide and carbon monoxide add smaller increments further out where the orbital periods are long and growth is slow. The water line has a second effect that may matter as much: just inside it, water vapour diffusing inward and re-freezing as it crosses the line piles up solid material in a narrow ring, and pebbles drifting inward from beyond it are slowed there. Either effect makes the region near the water line the likeliest place for a core to grow large quickly.

Those are reasons to expect the peak to follow the water line. They are not evidence that it does, and the same arguments were made, with the same confidence, before it was known whether the peak existed at all. The way to find out is to move the water line and watch.

The peak drawn round four stars #

Under the young-star snow line, the peak moves in proportion to the star’s mass.

The curves are the shape measured round Sun-like stars — rising as $a^{0.8}$ and falling as $a^{-1}$ — with the height scaled by the star’s mass, since giants are rarer round smaller stars in rough proportion. A star of a third of the Sun’s mass would have its giants piled up at 0.9 astronomical units, inside the orbit of the Earth; one of one and a half solar masses at four and a half, a little inside Jupiter’s.

The same figure drawn under the fixed-distance hypothesis has every peak at three astronomical units, and differs only in height.

The difference between the two figures is the whole of the test. In one, small stars have their giants close in; in the other, at the same distance as the Sun’s. Nothing else in either figure distinguishes them, and both are drawn through the same curve for the Sun.

Four hypotheses that agree about one star #

Plotted against the star’s mass, the four hypotheses are four straight lines on logarithmic axes, all passing through the Sun.

They fan out from a single point, and the fan opens on both sides — but much faster towards small stars, because the smallest stars that host giants are a factor of three or four below the Sun in mass, while the largest are only a factor of one and a half or two above it. Round a star of 0.3 solar masses, the predictions span a factor of eleven in distance. Round a star of 1.5 solar masses, a factor of two.

The point where they meet is where the surveys that discovered the peak looked. Radial-velocity programmes were designed around Sun-like stars, which are bright, quiet and rich in sharp spectral lines, and most of the giants with well-measured orbits beyond an astronomical unit orbit stars between 0.8 and 1.2 solar masses. Across that range the four hypotheses differ by less than the width of the peak itself. No number of additional Sun-like stars can separate them. The test has to be made where the lines have diverged, and that means round small stars — the commonest in the Galaxy and, for giant planets, the hardest.

The same hypotheses, read in years #

A telescope does not measure orbital distances. It measures periods, and Kepler’s third law reshuffles the hypotheses.

The period at a given distance is longer round a smaller star, because the star’s weaker gravity drives a slower orbit. A peak fixed at three astronomical units round a star of 0.3 solar masses has a period of nine and a half years — almost twice the Sun’s peak period, and at the edge of what any present survey can have followed through a full orbit. A peak following the young-star snow line round the same star has a period of a year and a half, well inside the reach of every method.

That asymmetry matters for how the test can fail. A survey of small stars that finds giants piling up at periods of a year or two has found the snow-line peak. A survey that finds the giants’ occurrence still rising at its longest periods is consistent with a fixed-distance peak but does not prove it, because the rise could continue to a peak just beyond the baseline. The fixed-distance hypothesis is the one that is hardest to confirm and easiest to mistake for an incomplete survey — the same trap that the planets that were not seen lay for every occurrence rate near the edge of a survey’s sensitivity.

A wobble the same for every star #

The census that can reach small stars at these periods in large numbers is astrometry: the space mission measuring the positions of a billion stars repeatedly over a decade sees each star’s wobble about the centre of mass it shares with its planets. The wobble’s angular size is the planet-to-star mass ratio times the orbit’s size divided by the distance, $\alpha = (m/M)(a/d)$.

Here is the surprising thing the arithmetic does. A smaller star wobbles more for the same planet, as $1/M$. A peak that moves inward with the star puts the planet on a smaller orbit, as $M^q$. If the peak follows the young-star snow line, q = 1, and the two effects cancel exactly: a Jupiter at the peak wobbles its star by the same angle whatever the star’s mass, about a tenth of a milliarcsecond at thirty parsecs. If the peak is fixed in distance, the small stars wobble three times more. If it follows the main-sequence snow line, three times less.

So the snow-line hypothesis predicts that an astrometric survey’s sensitivity to the peak’s giants is the same round every star at a given distance — and since small stars are far commoner than Sun-like ones, a volume-limited astrometric sample reaches many more of them. The fixed-distance hypothesis predicts that small stars’ giants are the easiest of all to see astrometrically, but at periods so long they are seen as accelerations rather than orbits — the regime in which an astrometric catalogue yields a floor under the unseen companion’s mass rather than a mass. Each hypothesis says not only where the peak is but how the survey that looks for it will be biased, and the two predictions point in opposite directions.

Iron raises the peak and leaves it in place #

A star’s mass and luminosity set where its snow line was. Its iron sets how much solid material was available to build cores, and the giants’ rate depends on it steeply — roughly as the square of the iron abundance.

The condensation temperature of water does not depend on how much iron is in the disc, so in the snow-line picture metallicity should change the peak’s height and nothing else. That separation is useful. The hypotheses about position can be tested in a sample sorted by stellar mass, and the hypothesis about height in a sample sorted by iron, and neither needs the other’s answer. If the peak moved with metallicity, it would mean something other than ice was setting its position — a migration that depends on the disc’s solid content, for example, or cores that grow fast enough to reach runaway accretion only inside a radius that depends on how much material there is.

There is a reason to expect a little movement even so. A metal-rich disc may be more opaque and therefore cooler at a given distance, pushing the snow line slightly inward, and a richer supply of solids lets cores reach the critical mass at greater distances where growth is slower. Both effects are small compared with the factor-of-three shifts between the mass hypotheses, which is what makes the stellar-mass test the cleaner one.

How many giants the test needs #

The peak is broad. Its position, estimated from a sample of detected giants, carries an uncertainty that falls only as the square root of their number.

For the broken power law drawn here, the peak’s position from N planets is known to about $1.1/\sqrt{N}$ in the natural logarithm of distance. The young-star snow line and the fixed distance differ, round a star of 0.3 solar masses, by $\ln(1/0.3) = 1.2$. Separating them at three standard deviations then needs about eight detected giants — and since only about three per cent of such stars host a giant within thirty astronomical units, a complete search of some two or three hundred stars. Separating the snow line from a fixed period, a subtler difference, takes about twenty giants. Near half a solar mass every requirement has roughly tripled, and by 0.8 solar masses they are in the hundreds.

These are lower limits. The calculation assumes the shape of the peak is known and only its position is fitted, that every giant’s orbit is measured, and that the sample is complete across the whole decade either side of the peak. A real survey fits the shape too, has incomplete sensitivity at the long-period end exactly where the fixed-distance hypothesis puts the peak, and measures some orbits only as accelerations. But the order of magnitude is the point: a few dozen giants round the smallest stars would answer a question that a thousand round Sun-like stars cannot.

What the small stars already say #

The giants of small stars are rare and have been studied mostly by other methods. Radial-velocity surveys of red dwarfs, which host more small planets than Sun-like stars do, find giants within a few astronomical units round a few per cent of them, a rate several times lower than round the Sun, and too few to locate a peak. Gravitational microlensing is most sensitive to planets near the Einstein radius of the lensing star, which for a typical half-solar-mass lens in the Galactic bulge is two to three astronomical units — just beyond where the snow line of a star of that mass would have been. Microlensing has found cold giants round such stars at rates that are not small, which is evidence that giants form beyond the snow line round small stars too; but because a microlensing planet is seen once and its host’s mass is usually inferred rather than measured, the method samples the occurrence in a band rather than mapping a peak.

The hypotheses have also been argued about from the theory side for twenty years. Planet-formation models that grow cores by accreting planetesimals predict that giants become very rare round small stars, because their discs are less massive and their orbital timescales longer at the snow line, so cores grow too slowly to capture gas before the disc disperses. Models that grow cores by accreting pebbles are less sensitive to that and predict more giants round small stars, preferentially near the snow line — the same growth that, in the Sun’s disc, may have let Jupiter’s core shut off the supply of pebbles to everything inside it. Where the peak sits round a red dwarf is therefore a test of formation as well as of where the ice was: a peak at the young-star snow line with a height not far below the Sun’s would favour the pebbles.

What the hypotheses leave out #

The snow line moves while the planets form. As a disc’s accretion rate falls and its star contracts, the snow line moves inward over a few million years. A peak set by the snow line records where it was while cores were growing, which could differ from the simple scalings by a factor that itself depends on the star’s mass.

Migration blurs the peak. Giants that formed at the snow line move afterwards — inward through the disc, or outward or inward by scattering with each other — and the observed peak is the formation peak convolved with that redistribution. If migration is stronger round small stars, the peak would be broader and displaced there, and the measured position would not be the snow line’s.

The peak’s shape is assumed. The figures treat it as a sharp break in a power law. The real distribution may be smoother, which makes the position harder to measure and the counts in the last figure larger.

Companions truncate discs. About a quarter of red dwarfs have a stellar companion, and a companion within a few tens of astronomical units truncates the disc at a fraction of the separation, removing the outer part where a fixed-distance peak would sit while leaving the snow line untouched. A sample that includes such binaries would see giants suppressed beyond a few astronomical units for a reason unrelated to either hypothesis, and the test has to be made on single stars or on binaries wide enough not to matter — a selection that has to be made from the same astrometric data, before the planets are counted.

The height’s scaling is assumed. Giants are drawn in proportion to the star’s mass, as broad surveys suggest; the exact dependence is uncertain, and since it multiplies the number of stars a survey needs, it matters to the feasibility of the test without changing which hypothesis it would pick.

Still open: whether the ice put the giants there #

The position of the giants’ peak round the Sun’s kind of star is now known to within a factor of about two, and its proximity to the snow line is the most natural explanation for it. The explanation predicts how the peak should move round other stars, and three alternatives predict differently. All four agree at one solar mass, which is where the measurement was made.

The test is a few dozen giants round stars of a third to a half of the Sun’s mass, with orbits measured rather than inferred, and the astrometric census of the coming data releases is the first survey large enough to provide them. If those giants pile up at one or two years round the smallest stars, the peak is the snow line’s and the giants were built where water froze. If their occurrence is still climbing at a decade, the peak is something fixed in the geometry of every disc — and the ice that seemed to explain the Sun’s giants will have been a coincidence of the one star where every explanation agrees.