More small planets round smaller stars

Three stars in every four in the solar neighbourhood are red dwarfs — M dwarfs, between about a tenth and a half of the Sun’s mass. Most stars are small, and most of the light is not: a census by number is a census of M dwarfs, and a census by brightness a census of the few stars like the Sun and heavier. The same split runs through the planets. The planet-hunting surveys began with Sun-like stars because they are bright and quiet, and a large share of what is known about planets is a statement about a minority of hosts.

When the Kepler survey’s small-planet catalogue was divided by the mass of the host star, and each part corrected for everything the survey could not see, two results came out that run in opposite directions. Small planets — between one and four times the Earth’s radius, on orbits shorter than about two hundred days — are about twice as common round M dwarfs as round Sun-like stars, at around two and a half per star against one. Giant planets are several times rarer: about one M dwarf in thirty has a Jupiter within a few astronomical units, against about one Sun-like star in fifteen.

The two trends are not two halves of one statement about “planets”. They are two different statements about two populations that respond to the host in opposite senses, and the contrast between them is the thing to explain.

The giant half is the easier one #

The rate of giant planets rises steeply with the iron in the host star, and the reading that makes sense of the slope is a race: a core has to reach about ten Earth masses and start pulling in gas before the disc’s gas disperses in a few million years, and the rate at which it grows depends on how much solid material is available near it. Double the solids and the core finishes sooner, and the fraction of discs in which it finishes in time rises far more than twofold, because winning is rare.

A star’s mass enters the race the same way its metals do, through the solids. A disc round a more massive star is, on average, more massive — the millimetre-wave surveys of young discs in nearby star-forming regions find their dust masses rising roughly as the 1.3 to 1.9 power of the star’s mass. Less material, slower cores and a gas disc that disperses on much the same few-million-year clock: the giant rate falls with stellar mass for the reason it falls with metallicity. Round an M dwarf there is also less time than it looks, because orbital periods at a given distance are longer round a less massive star and a core’s growth is counted in orbits.

That part of the story is the one the core-accretion picture predicted before it was measured, and the measured slope — close to linear in stellar mass at fixed metallicity, from the radial-velocity surveys of stars between a third of a solar mass and two — is comfortably inside what the models give. The surprise is the other curve.

Small stars are easier hosts to search, and that is not the explanation #

Before the small-planet trend can be read as a fact about planets, it has to survive the obvious objection: every small planet round a small star is easier to find, so of course more of them turn up.

Each of the three curves is a separate advantage. A planet transits with probability equal to the star’s radius over the orbit’s, and a temperate orbit round a small star is so much closer in that the probability triples even though the star is smaller. The year is so short that dozens of transits fit into a survey instead of three or four, so a shallow signal can be built up out of many repetitions. And the transit depth — the fraction of the star’s light the planet blocks — rises as the square of the planet-to-star radius ratio, which is ten times larger for an Earth crossing a 0.3-solar-radius star than for one crossing the Sun. The reflex velocity is five times larger too.

None of this is the source of the trend, because none of it survives the correction that an occurrence rate is. A rate is a count divided by the probability of having detected what was counted, and every one of these advantages is inside that probability: the transit probability goes into the denominator directly, the number of transits and the depth go in through the survey’s measured completeness. What remains after the division is the rate the survey would have found with perfect sensitivity. The M dwarfs’ two and a half small planets per star are after that division, and the doubling relative to Sun-like stars is a doubling in the rate, not in the detections.

What the selection does do is fill the catalogue of temperate small planets almost entirely with M-dwarf systems — Proxima Centauri b, the seven planets of TRAPPIST-1, LHS 1140 b — and that is a separate fact about the catalogue from the fact about the rate. A reader of the list of known temperate planets sees a sample in which small stars are overrepresented many times over, and the part of that overrepresentation that is real is the factor of about two in the curve at the top.

What the doubling is measured against #

The M-dwarf rate rests on a much smaller sample than the Sun-like one, and on stellar parameters that were, for most of the survey’s life, markedly worse.

The Kepler field held about 150,000 target stars, of which fewer than four thousand were M dwarfs, and the small-planet rate round them is built from a few hundred candidates. That is enough for a rate with an uncertainty of ten or fifteen per cent in the bins that matter, and not enough to divide further — by metallicity, by age, by multiplicity — without the numbers in each bin falling to a handful.

The larger problem was the stars themselves. A planet’s radius is measured as a ratio to its star’s, so every error in a stellar radius is an error of the same fraction in every planet round that star. The M dwarfs in the original input catalogue had radii estimated from colours calibrated on hotter stars, and they came out too large by typically ten to twenty per cent for the smallest of them. When the radii were redetermined — first from spectra, then from Gaia parallaxes, which turn a measured brightness and temperature directly into a radius — the planets shrank with their stars, and planets moved across the one-Earth-radius and four-Earth-radius boundaries that define the bins. The rate of small planets round M dwarfs survived the revision with its doubling intact; the rate of Earth-size ones in particular did not, and moved by more than its quoted error. A rate per star is only as good as the ruler used on the star, and for small stars that ruler was, until recently, the least certain number in the calculation.

The same rock from less material #

The small-planet curve becomes striking when it is converted from planets to mass.

A small planet round an M dwarf is on average smaller than one round a Sun-like star — the radius valley that divides rocky planets from those with gas envelopes sits lower round smaller stars, and the planets on either side of it are scaled down too. Multiplying the planets per star by a typical mass per planet gives the mass of solid material locked up in close-in planets, and that product turns out to depend on the host’s mass hardly at all.

A small star ends up with roughly the same inventory of close-in rock as the Sun’s kind of star, from a disc that held about a sixth as much solid material. There are three ways that can be true, and the occurrence rates alone cannot choose among them.

The first is efficiency: planet formation round small stars converts a larger share of the available solids into planets. That would need the processes that lose solids — dust grains dragged inward by the gas and swallowed by the star, planetesimals ejected or ground back into dust — to be weaker in low-mass discs, and there is no strong reason to expect them to be.

The second is delivery. The solids in a disc do not stay where they formed. Pebbles of millimetres to centimetres drift inward through the gas, and a planet built close to the star can be made largely of material that condensed much further out and arrived by drift. If the drift is efficient, the close-in inventory measures the solids of the whole disc funnelled inward, not the solids that happened to be close in — and a small disc, whose outer parts are not far away, could deliver a larger fraction of what it had.

The third is that the discs measured are not the discs that made the planets. The dust masses come from discs a few million years old, when a large share of their solids may already be locked in bodies too large to radiate at millimetre wavelengths. If planet formation runs faster round small stars, their discs would look poorer at that age for the very reason that they have already made their planets.

A clock counted in orbits #

There is one further difference between small stars and large ones that none of the three readings above uses, and it pushes in the same direction as all of them.

The growth of a planet from smaller bodies is counted in orbits, not in years: the rate at which a protoplanet sweeps up material is proportional to how often it passes through that material, which is to say to its orbital frequency. At a fixed temperature in the disc — the relevant comparison, since the temperature decides what can condense — the orbit round a small star is much tighter and much faster. At the distance where a 0.3-solar-mass star gives the Earth’s insolation the year is 28 days, so a protoplanet there completes thirteen orbits for every one the Earth completes at the same temperature round the Sun.

A disc’s gas disperses on a clock measured in years, and it is not very different between small stars and large ones — a few million years for both. So a planet-forming region round a small star gets more than ten times as many orbits in which to grow before the gas goes, and any process whose outcome depends on the number of orbits available — the accretion of pebbles, the assembly of cores from planetesimals, the migration of what has formed — runs further to completion there. That argues for the efficiency reading and against it at once: small planets finish growing round small stars, which is consistent with more of them; and giant cores, which have to reach ten Earth masses at the snow line where the orbits are slow round every star, gain nothing, because the snow line round a dim star is close in but its orbital period is still long compared with the close-in region.

How much the conclusion depends on the slope assumed #

The disc scaling is the least certain number in that figure, and the conclusion should be checked against the other end of its range.

At the steep end the efficiency explanation becomes very difficult: a disc that turned more than half its solids into planets inside half an astronomical unit would leave almost nothing for the rest of the system, and the M dwarfs with outer planets detected by microlensing show that some material does end up further out. The steepness of the disc scaling therefore pushes towards the delivery reading, or towards discs whose measured masses understate what they began with. That is a real constraint, obtained from nothing more than multiplying three power laws together, and it is where the occurrence rates stop being a census and start being evidence about how planets form.

The same data under the other published slopes #

The small-planet and giant exponents also have ranges, and a slate of plausible values from the published fits should be drawn before anything is built on the factor of seven.

That is a useful property and worth noticing for its own sake. Many of the systematic errors in these surveys push both curves the same way — an error in the stellar masses of M dwarfs, for instance, which are harder to measure than those of Sun-like stars, slides both curves along the axis together. The ratio of the two populations cancels a good share of that, and the statement that small stars favour small planets over giants by a large factor is more robust than either of the statements it is made from.

Where the giants round small stars went #

The giant curve drawn is for giants within 2.5 astronomical units, the range the radial-velocity surveys covered, and it is not the only measurement of giants round small stars.

Microlensing finds planets round whatever star happens to pass in front of a background star, and since most stars are M dwarfs, most microlensing hosts are too. It is most sensitive to planets at a few astronomical units — near the Einstein radius of a lens of a third of a solar mass halfway to the galactic bulge — which is beyond the snow line of such a star. What it finds there is that planets of Neptune’s mass ratio and above are common round M dwarfs, and giants of Jupiter’s mass are not rare: the rate of cold giants round M dwarfs from microlensing is a few per cent to ten per cent, depending on the mass range, which is larger than the close-in radial-velocity rate for the same stars.

So the giants round small stars are not absent so much as further out, beyond where the radial-velocity surveys could follow them for enough of an orbit, and at the distances where the snow line of a small star sits. That is what the core-accretion picture would predict: the cores form where solids are most plentiful, at and beyond the snow line, and round a small star they have less material and fewer orbits to work with, so fewer of them finish and those that do are less likely to migrate inward. The two methods are measuring different parts of one distribution, and joining them needs a model of how planets are spread in distance — which is where the census of giant planets has to be joined across methods.

Most planets orbit red dwarfs #

Put the occurrence together with the census of stars and the typical planet in the galaxy comes into focus.

Three quarters of stars are M dwarfs, and each has about twice as many small planets as a Sun-like star. So something like five in every six small close-in planets in the galaxy orbit an M dwarf, and the systems the early surveys were built to find — a Jupiter-like giant round a Sun-like star — are a small minority of planetary systems by number. The Solar System, with no close-in small planets at all and a giant at five astronomical units round a comparatively heavy star, is unusual on both counts.

That changes what a question like “how common are Earth-like planets” is actually about. The rate of Earth-size planets in the habitable zone is higher per star round M dwarfs, the M dwarfs are more numerous, and their temperate planets are the easiest to detect and to study. The nearest temperate Earth-size planet to the Sun is almost certainly round an M dwarf — Proxima Centauri b, at 1.3 parsecs, may be it. Whether such a planet is habitable is a separate matter: an M dwarf spends its first billion years far brighter in the ultraviolet and X-rays relative to its bolometric output than the Sun did, and a planet in its habitable zone sits ten times closer to that radiation. Whether a small planet keeps its atmosphere round such a star is being tested now, planet by planet, with the transmission and emission spectra the transit geometry makes possible.

What the curves do not settle #

The curves are power laws standing in for fits. The published small-planet trend differs between analyses by a factor of about two in its exponent, depending on the period range, the size range and whether the planets are counted per star or by the fraction of stars with at least one. The giant trend is measured over a narrower range of stellar masses than the curve is drawn across, and below about a third of a solar mass it rests on a handful of detections.

The mass per planet is a guess. It converts a radius distribution into a mass through a mass–radius relation that is itself uncertain by a factor of two for the sub-Neptunes, which carry most of the mass; the inventory figures would move with it, though the direction of the conclusion would not.

Disc masses are dust masses. They assume a dust opacity and a temperature, both of which may scale with stellar mass, and they are measured at a single age. A trend in the conversion from millimetre flux to mass would move the upper curve without moving anything real.

And a rate per star hides multiplicity. Two and a half small planets per M dwarf could be most M dwarfs with two or three planets, or a minority with seven and most with none. The number of planets per system is harder to measure than the rate, and the answer changes what “a typical M dwarf system” looks like far more than it changes the curve.

Still open: whether the planets remember their host in its own units #

A planet’s mass is naturally measured in Earth masses, and the curves above count planets that way. A disc’s physics is naturally measured relative to its star: the disc mass scales with the star’s, the snow line moves with the star’s luminosity, and the timescales scale with the orbital period at a given insolation. That suggests counting planets by the ratio of their mass to their star’s rather than by their mass — and when the microlensing surveys, which find planets round M dwarfs at a few astronomical units, and the transit and radial-velocity surveys, which find them closer in round every kind of star, are both expressed that way, their distributions of mass ratio show a turnover at similar values of order $10^{-4}$. Whether that similarity is a real universality — the planet-forming process scaling with its star so completely that the star’s mass drops out — or an accident of comparing different distances with different methods, would decide whether “more small planets round smaller stars” is a fact about small stars or a fact about the units the question was asked in.