A floor the star sets for itself

The precision with which a star’s brightness can be measured has a sequence of limits, and each has been the subject of its own account. The first is counting: a brightness is a number of photons, and its error is the square root of the number. For faint stars the sky in the same aperture is brighter than the star, and sets the noise instead. The aperture’s size trades the star’s light against the sky’s. And comparison stars on the same frame cancel what the atmosphere and the instrument do to all of them at once, at the price of their own noise.

Every one of those is a property of the measurement. Go to space, use a large enough aperture, observe a bright enough star, and they can all be made small. What is left is the star.

A Sun-like star’s surface is not uniformly bright. It is covered with convection cells — granules — each a column of hot gas rising from below, cooling at the surface and sinking at its edges, each a few hundred kelvin hotter or cooler than its neighbours and living for about ten minutes. The disc of the star is the sum of millions of them, and the sum fluctuates. That fluctuation is not instrumental, does not cancel against comparison stars, and does not shrink with a larger telescope. It is the floor the star sets for itself.

A surface that boils #

The granules on the Sun were first resolved in the nineteenth century as a texture like rice grains, and their nature was settled in the twentieth: each is the top of a convective plume. Hot gas rises in the bright centre at one or two kilometres a second, radiates at the surface, cools by a few hundred kelvin, and sinks in the dark lanes between granules. In visible light a granule’s centre is about fifteen per cent brighter than its lanes. A granule lives for eight to ten minutes before it fades, splits or is replaced, and at any moment the visible disc of the Sun carries about two million of them.

They are as wide as they are because convection organises itself on the depth over which the pressure changes appreciably. Below the surface the pressure scale height is a few hundred kilometres; a rising column cannot stay coherent over many scale heights without breaking up, and the cells that reach the surface are roughly ten scale heights across. In a star whose gravity is weaker, the atmosphere is more extended, the scale height longer and the granules larger, in proportion. That one relation, a granule’s size following the scale height, is what makes the flicker a function of the star’s gravity and not just of its temperature.

How much a star flickers #

The flicker of a disc made of many granules depends on how many there are. If each granule is brighter or darker than average by a fixed fraction, independently of the others, the fluctuation of their sum falls as one over the square root of their number. The number is the star’s surface area divided by a granule’s area, and a granule’s size is set by the depth over which the pressure in the star’s atmosphere falls by a factor of $e$ — the pressure scale height, $H_p = kT/\mu m_H g$ — because convective cells cannot be much wider than they are deep. Granules are about ten scale heights across; on the Sun that is about a thousand kilometres.

So the flicker goes as the granule size divided by the star’s radius:

$$\sigma \propto \frac{H_p}{R} \propto \frac{T}{g R} \propto \frac{T R}{M}.$$

A star with lower surface gravity has a larger scale height, larger granules, fewer of them, and less averaging.

Normalised to the Sun’s measured granulation flicker of about fifty parts per million, the scaling gives forty for a K dwarf of three quarters of a solar mass, eighty-eight for a subgiant of twice the Sun’s radius, and over three hundred for a red giant. The red giant’s granules are so large that only a few thousand cover its disc, against a couple of million on the Sun, and their flicker is plain in a light curve. Granulation in red giants is measured routinely, and its amplitude and timescale are one of the ways the oscillations that reveal a star’s interior are separated from the background they sit on.

How it averages over a transit #

A transit search does not look at the brightness in single exposures. It looks for a dip lasting hours, so what matters is the noise averaged over the transit’s duration. Granulation flicker is correlated: a granule lasts minutes, so successive exposures within a few minutes see the same granules. Over windows longer than the correlation time the fluctuations average down, as one over the square root of the number of independent intervals in the window.

The correlation time scales as a granule’s size over its speed, which also grows as the gravity falls: about four minutes on the Sun, fourteen on a subgiant of twice its radius. Over the thirteen hours an Earth-like planet takes to cross a Sun-like star, the Sun’s granulation averages down to about five parts per million. The subgiant’s averages to seventeen: it starts larger and averages less, because its granules live longer.

Five parts per million is small against an Earth’s transit depth across the Sun, eighty-four. Seventeen is not small against the transit depth across the subgiant, which is four times smaller, because the depth goes as the inverse square of the star’s radius: twenty-one parts per million.

Oscillations average away first #

Granulation is not the only motion of a Sun-like star’s surface on timescales of minutes. The whole star rings, in acoustic oscillations driven by the same convection, with periods around five minutes on the Sun; their frequencies measure the star’s interior. In brightness they are smaller than granulation — a few parts per million on the Sun — and they are periodic, with many modes beating together, so over a window of hours they average away almost completely. In a transit search they are invisible.

Granulation is the faster of the two noises that do not average away: random, correlated over minutes, and so beaten down as the square root of the window. Supergranulation, correlated over a day, is beaten down far less over a thirteen-hour transit, and magnetic activity, correlated over the star’s rotation, hardly at all. The noise that matters for a transit is set by the slowest of a star’s surface motions that is still faster than the rotation — and granulation, being the fastest of those, is the floor under the floor.

Where the star overtakes the photons #

The comparison that decides whether a search is photon-limited or star-limited is between the two noises in the same window.

The telescope in the figure is Kepler-like: seventeen parts per million of photon noise in six and a half hours on a twelfth-magnitude star, falling by a factor of ten for every five magnitudes of brightness. Over a thirteen-hour transit window the photon noise and a Sun-like star’s granulation are equal at magnitude 10.1. For fainter stars, which were most of Kepler’s targets, the photons dominate; for brighter ones, the star does. A single Earth transit across a twelfth-magnitude Sun is detected at 6.4 standard deviations; across the brightest stars, however bright, at no better than about seventeen, because the star’s own flicker does not shrink with brightness.

That floor is the most optimistic one, because granulation is the best-behaved of a star’s variations. Beneath the granules are larger convective structures — mesogranulation and supergranulation, cells tens of thousands of kilometres across that live for hours to a day — whose brightness contrast is smaller but whose timescales match a transit’s, so they average less. And on longer timescales still, magnetic regions rotating across the disc add brightness variations of their own. Each lies above the granulation floor drawn here.

An Earth twin around a slightly larger Sun #

The scalings combine into a strong dependence on the star’s size, and the stars a survey watches are not all the Sun’s size.

Hold the star’s mass at the Sun’s and let its radius grow, as a Sun-like star’s does near the end of its life on the main sequence and afterwards. The transit depth falls as the square of the radius. The granulation flicker grows in proportion to it. The transit lasts longer, which averages more noise away, but not enough to compensate. Four transits of an Earth around an eleventh-magnitude Sun are detected at eighteen standard deviations; around the same star at 1.5 solar radii, at seven. A survey requiring 7.1 standard deviations, as Kepler’s pipeline did, crossing a threshold that is itself a ramp, loses its Earth twins around stars larger than about one and a half solar radii.

That matters more than it seems because of which stars a survey observes. A survey selects targets by brightness, and at fixed brightness it reaches farther for intrinsically brighter stars. Slightly evolved stars, larger and more luminous than the Sun at the same temperature, are overrepresented among the bright targets for exactly the reason a magnitude-limited sample is brighter than the population it came from. They look like Suns in their colours and are not Suns in their radii, and many of them are hosts around which an Earth cannot be found.

Hotter and cooler than the Sun #

The scaling runs in temperature as well as gravity. Hotter dwarfs, of spectral type F, have higher temperatures and slightly lower gravities than the Sun, so their scale heights are longer, their granules larger and their flicker stronger; their granulation contrast is also higher, which the simple scaling leaves out. A Sun-sized planet’s transit across an F star is shallower, because the star is larger, and the noise is larger: F stars are harder hosts for small planets than their brightness suggests.

Cooler dwarfs go the other way. A K or M dwarf has a higher gravity, a shorter scale height, smaller granules and more of them, and its granulation flicker is smaller than the Sun’s; a transit across it is also deeper, because the star is smaller. For granulation alone, the smallest stars are the best hosts for finding small planets. Their variability is dominated instead by magnetic activity — spots that persist for many rotations and flares that brighten them by per cents in minutes — which a granulation floor does not capture and which can produce periodic signals mistaken for planets in velocity as in brightness. The star’s floor changes character down the main sequence: convective flicker for the Sun and hotter, magnetic noise for the cool dwarfs.

What Kepler expected and what it found #

Kepler was designed to find Earth-sized planets in Earth-like orbits around Sun-like stars, and its design assumed those stars would be as quiet as the Sun. Satellite radiometers that have monitored the Sun’s total brightness since 1978 put the Sun’s variability on a six-and-a-half-hour timescale at about ten parts per million, at solar minimum and maximum alike.

The stars were twice as noisy. Ronald Gilliland and his colleagues measured the intrinsic variability of Kepler’s quietest Sun-like stars in the mission’s first months and found about twenty parts per million on six and a half hours, against the Sun’s ten — a difference that turned the four-year mission’s target of finding Earth twins from comfortable to marginal and was part of the case for extending it. Granulation alone, in this model, gives seven parts per million on that timescale, and the Sun’s whole variability ten; the rest of the stars’ excess must come from the larger and slower structures and from magnetic activity.

Why the stars were noisier than the Sun is partly the selection above — many of them are older and larger than the Sun, or younger and more active — and partly a hint that the Sun may be quieter than typical for its kind. Either way the result changed the arithmetic of the search: the floor was set by the stars, and the stars were not the Sun.

A floor that is also a measurement #

The granulation that limits a transit search is the same physics that limits radial-velocity searches, seen in brightness instead of velocity. In both, the answer is partly an observing strategy: in velocity, an exposure length tuned to the star’s oscillations averages them away, and in photometry a transit’s long duration averages the granulation. And in both, what is left after the averaging is the slowest of the star’s surface motions, which no strategy can remove.

The flicker is also information. Its amplitude and its timescale both scale with surface gravity, so a light curve’s background noise measures the star’s gravity — the method called flicker, which gives a surface gravity to about a tenth of a dex from a few months of photometry, for stars too faint or too quiet for their oscillations to be resolved. The noise that hides a planet tells how big the star it orbits is, and so what the planet’s radius would have been.

Many transits add; one star does not average with itself #

A floor set by the star is not absolute, because transits repeat. Granulation during one transit is independent of granulation during the next a year later, so the star’s noise in the folded light curve falls as the square root of the number of transits, exactly as the photon noise does. Four transits halve both. That is why the signal-to-noise figure above counts four, and why a mission’s length matters as much as its aperture for the planets with the longest periods: Kepler’s four years gave at most four transits of an Earth twin, its extension would have given more, and the failure of its pointing wheels in 2013 ended the extension.

But the stars’ slower noise does not average the same way. Magnetic activity on a star like the Sun waxes and wanes over a cycle of years, and the variability at transit timescales is larger near activity maximum than near minimum. Transits observed during an active phase are each noisier, and adding them improves the signal-to-noise more slowly than the square root of their number would suggest. The floor for a single transit is set by the star’s surface at that moment; the floor for a planet is set by the star’s surface over the years it takes to watch the planet transit three or four times, which for an Earth twin is a good fraction of a stellar activity cycle.

There is a corresponding limit on what more telescope can buy. A larger aperture lowers the photon noise; it does nothing to the star. For the brightest Sun-like stars, a bigger telescope observing one transit does no better than a smaller one, and the only way to reach an Earth is to observe more transits or more stars.

What the model leaves out #

The flicker is scaled from the Sun with a single exponent, ignoring the dependence of the granules’ brightness contrast on temperature, which makes hotter stars’ granulation stronger than the scaling gives. The correlation is a single exponential, where real granulation has a broader spectrum. Only granulation is modelled; supergranulation, faculae and spots, which dominate on the hours-to-days timescales of transits, are named but not drawn. The photon noise is for one telescope at one bandpass, and the transit durations assume central transits of an Earth at one astronomical unit regardless of the star’s luminosity, which makes the evolved stars’ transits somewhat too short.

What survives is the structure: a floor set by the star, independent of the telescope, rising as the star’s gravity falls, and reached for the bright stars every transit search most wants.

Still open: how quiet a Sun-like star can be #

The next space photometry missions aim at bright Sun-like stars, specifically to reach the regime where the photon noise is below the stars’ own. For them the stellar floor is the whole problem. Whether that floor can be modelled and subtracted — granulation as a known spectrum fitted alongside the transit, supergranulation and activity as correlated noise learned from the light curve itself — or whether it has to be beaten by observing more transits, depends on how much of the stars’ variability is structured enough to predict. The Sun’s own record, measured to a part per million by radiometers for more than four decades, is the template; the stars that will host the first Earth twins found by transit will be compared with it, and the question of how quiet a Sun-like star can be is the question of how many of them have to be watched to find one quiet enough.