An exposure time chosen by the star

An Earth-mass planet a year from a Sun-like star moves that star by nine centimetres a second. The star’s own surface moves by ten times that on timescales of minutes, and by several times that on timescales of hours and days, with no planet present at all. The spectrograph is no longer the limit — the best instruments are stable to about ten centimetres a second over years — and neither, for a bright star, is the photon noise. What stands between the current generation of velocity surveys and an Earth twin is the star itself.

The star’s surface contributes in three ways that differ in their timescales, and the timescale is everything. Magnetic spots and bright plage rotate with the star over days to weeks, and they imitate planets outright. Faster than that, the surface boils: convective cells rise and sink on timescales from ten minutes to a day. Faster still, the whole star rings, with acoustic oscillations at periods around five minutes. The first can only be modelled. The other two can in principle be averaged away, and how well depends on matching the averaging to the star.

Why an Earth is nine centimetres a second #

The number that sets the whole problem follows from the reflex motion. A planet of mass $m$ on a circular orbit of period $P$ around a star of mass $M$ moves the star with a semi-amplitude

$$K = 28.4\ \mathrm{m\,s^{-1}}\left(\frac{m\sin i}{M_\mathrm{J}}\right)\left(\frac{P}{1\ \mathrm{yr}}\right)^{-1/3}\left(\frac{M}{M_\odot}\right)^{-2/3},$$

which for Jupiter at Jupiter’s distance gives the Sun’s 12.5 metres a second, for a hot Jupiter at three days gives about 140, and for the Earth gives 0.089. An Earth is 318 times less massive than Jupiter and its period is twelve times shorter, and the cube root of twelve buys back only a factor of 2.3.

The same formula says where the problem is easiest. A star of a fifth of the Sun’s mass is about two hundred times less luminous, so its temperate zone sits at a period of two or three weeks rather than a year; the shorter period and the lighter star both raise K, and an Earth-mass planet there moves its star by about a metre a second. That is why the first temperate Earth-mass planets found by velocities were found around red dwarfs, and why an Earth twin around a Sun-like star — the case this essay is about — is the hardest version of the measurement by an order of magnitude.

A star that rings at five minutes #

A star like the Sun is a resonant cavity for sound. Convection near the surface excites acoustic waves continuously, and the waves that are trapped between the surface and a turning point deep inside set up standing modes — thousands of them, at frequencies spaced by a nearly constant interval, whose pattern is how the interior is read. Each mode moves the surface up and down, and the whole disc’s light carries a velocity signal that is the sum of all the modes visible in integrated light.

For the Sun the modes cluster around a frequency of 3.1 millihertz, a period of five and a half minutes, in an envelope about a millihertz wide. Individual modes have velocity amplitudes of a few tens of centimetres a second, and their sum in a short exposure is a velocity that wanders by tens of centimetres a second over minutes. That is small against a hot Jupiter and large against an Earth.

The frequency at the centre of the envelope, $\nu_{\max}$, is set by the star’s surface gravity and temperature: it scales as the gravity divided by the square root of the temperature, which is why it is one of the two numbers asteroseismology calibrates against the Sun. A smaller, denser star rings faster and more quietly. A star that has begun to swell towards the subgiant branch rings more slowly and with larger amplitude, because the oscillation amplitude grows roughly with the ratio of the star’s luminosity to its mass.

Why one oscillation vanishes and a band does not #

Averaging a sinusoid over a window of length T multiplies its amplitude by a factor $\sin(\pi\nu T)/(\pi\nu T)$. That factor is exactly zero whenever the window contains a whole number of periods, and between those zeros it falls as one over the window’s length. A single oscillation at five and a half minutes, averaged over exactly five and a half minutes, disappears completely.

The star does not ring at one frequency. It rings across a band a millihertz wide, and no single exposure length is a whole number of periods of every mode at once. The best that can be done is to put the first null near the middle of the band, where most of the power is, and accept that the modes on either side are only partly suppressed. For the envelope drawn, the first minimum is at 5.8 minutes and fifteen per cent of the rms survives it; beyond that, the residual falls only as one over the exposure length, so going from ten minutes to twenty buys a factor of two.

This is where the star’s noise and the counting noise trade against each other. The error bar that comes from counting photons falls as the square root of the exposure, so for a bright star a short exposure already reaches centimetres a second of photon noise and a long one is spent on precision that is not needed. The oscillations reverse that logic: the exposure is set not by how many photons are wanted but by how long the star takes to ring through its own cycle.

The practical consequence was worked out from the Sun and applied to stars: an exposure of fifteen minutes or so leaves the oscillation residual well under ten centimetres a second for a solar-type star, and an exposure of a minute or two — which a bright star can easily afford, since it is not short of photons — leaves most of the oscillation in place. A bright star observed quickly can therefore be noisier than a fainter one observed slowly. The photon noise goes down with the shorter exposure’s greater efficiency; the stellar noise goes up, and for a bright star it is the larger of the two.

An exposure set by the star #

Since $\nu_{\max}$ differs from star to star, so does the exposure that suppresses the oscillations best.

This leads to a recommendation that was made explicitly in 2019: tune each star’s exposure time to its own $\nu_{\max}$, which is known in advance from its spectroscopic gravity and temperature. A fixed exposure for every target is the wrong choice for almost every target. For the dwarfs that dominate planet searches the difference is a few minutes and a few centimetres a second. For the evolved stars that bright, nearby samples inevitably contain, it is the difference between an oscillation residual of ten centimetres a second and one of fifty.

There is a limit to how far tuning can go. The optimum depends on the envelope’s width as well as its centre, and the width is not known in advance as well as $\nu_{\max}$ is; a star whose modes are unusually spread out has a shallower minimum. And the oscillations are stochastic: each mode is continuously excited and damped, so its amplitude and phase change over a few days, and the suppression achieved on one night is not guaranteed on the next. Tuning makes the oscillations a small term in the budget. It does not remove them from it.

A night on a quiet star #

The oscillations are the fast part of the surface’s motion, and averaging deals with them. What is left after averaging is slower.

The slower wander is granulation. The visible surface of a Sun-like star is tiled with about a million convective cells at any moment, each a thousand kilometres across. The centre of each cell is hot gas rising at about a kilometre a second, bright and blueshifted; the lanes between cells are cooler gas sinking, darker and redshifted. Because the rising gas is brighter, it dominates the integrated light, and every spectral line is shifted to the blue by a few hundred metres a second — the convective blueshift. That shift is not a problem while it is constant. It is not constant: the number of cells, their brightness contrast and their velocities fluctuate, and with a million cells on the disc the fluctuation in the average is of order a thousandth of the shift. That is tens of centimetres a second, changing on the lifetime of a granule — about ten minutes.

Three kinds of boiling, three correlation times #

Granulation is not the only convective pattern. Beneath and between the granules there is a larger, longer-lived structure called mesogranulation, with cells several thousand kilometres across that persist for hours, and beneath that the supergranulation, with cells thirty thousand kilometres across, horizontal flows of a few hundred metres a second, and lifetimes of about a day. Each pattern contributes its own velocity fluctuation, and each fluctuation is correlated in time over roughly the pattern’s lifetime.

The mathematics of averaging a correlated signal is simple and decisive. Averaged over a window much shorter than its correlation time, a correlated fluctuation is not reduced at all: the window sees one value. Averaged over a window much longer than its correlation time, it falls as the inverse square root of the number of independent correlation times in the window. The transition is at the correlation time itself, and there is no way to do better than the inverse square root once the window is long.

The consequence for strategy is that an exposure longer than about fifteen minutes gains little on a single visit, because it is already past the oscillations and still far short of the mesogranulation’s correlation time. What helps is to visit the star several times in a night, separated by hours, so that the visits sample independent realisations of the mesogranulation — a strategy that was proposed from solar simulations in 2011 and is now the standard design for Earth-twin searches. But nothing within a night helps with the supergranulation. A night’s worth of observations sees one realisation of it. Only averaging across many nights reduces it, and between nights everything else about the star, the instrument and the atmosphere also changes.

The amplitudes drawn are representative values for a quiet Sun-like star. Published estimates of each vary by factors of about two, from different simulations and from different ways of analysing the Sun’s own disc-integrated velocities, and the supergranulation’s contribution is the least well known of the three because it is the slowest and needs the longest continuous records to measure.

Nine centimetres a second, priced in nights #

Once the fast noise has been averaged within each night, what is left per night behaves roughly as independent noise from night to night, and the arithmetic of measuring a small sinusoid in independent noise is a square law.

The semi-amplitude of a sinusoid fitted to N points of noise σ has an uncertainty of σ times the square root of two over N. Requiring the amplitude to be seven times its own uncertainty — a reasonable standard for a claim that will be scrutinised, though no pipeline’s threshold is as sharp as the number quoted for it — gives $N = 2(7\sigma/K)^2$. The square is what makes the numbers brutal. Halving the nightly noise cuts the required nights by four; doubling it multiplies them by four. For an Earth twin, with K = 0.09 metres a second, a nightly noise of one metre a second requires twelve thousand nights, which is thirty years of observing a single star every clear night.

A dedicated programme can afford a few hundred nights on each of a few dozen stars over a decade. That arithmetic works only if the residual noise per night, after oscillations and granulation have been averaged and activity has been modelled, is below about sixteen centimetres a second. The supergranulation alone, on the figures above, contributes more than that.

The floor has to be modelled, not averaged #

That conclusion — that averaging alone cannot reach an Earth twin — is why the frontier of the subject has moved from instruments to the star.

The most direct response is to observe the Sun. The Sun can be observed as a star, its light integrated over the whole disc and fed into the same spectrographs used for planet searches, every day for years, while at the same time its surface is imaged at high resolution. Every velocity excursion in the integrated light can then be matched to the surface features that caused it. Several solar telescopes now do exactly this, and they have measured the Sun’s own convective and magnetic velocity noise with no planet to confuse it — which is the calibration every model of stellar noise was missing.

The second response uses the spectrum itself. Granulation does not shift every line equally, because lines form at different heights and the convective flows change with height; a line formed deep in the photosphere is blueshifted more than one formed higher up. A velocity measured from thousands of lines is an average of lines that disagree, and the disagreement is information. Measuring the velocity separately for groups of lines with different formation heights, and looking for the differential signature that convection produces and a planet does not, is the line-by-line approach, and it is currently the most promising way to reduce the floor below what averaging reaches.

The third is the observing strategy itself. Multiple visits per night sample the fast noise; long, uninterrupted seasons sample the slow; and simultaneous photometry measures the brightness changes that accompany some of the velocity changes. None of these removes the floor. Each converts part of it from noise into a modelled signal — and a modelled signal has to be modelled correctly, since a model flexible enough to absorb the star’s slow wander is also flexible enough to absorb a planet with a period of a year. The same care that keeps a single eccentric orbit from being split into two planets is needed to keep a planet from being folded into the noise model.

What the figures leave out #

Every figure here uses a simplified description of the star. The oscillations are drawn as a smooth Gaussian envelope when they are a comb of discrete modes with finite lifetimes. The convective patterns are drawn as fluctuations with a single exponential correlation, when their real spectra are broader and their amplitudes change with the magnetic cycle. And the most dangerous contribution of all, the rotational modulation of spots and plage, is absent — not because it is small but because it is periodic, and averaging does nothing to a periodic signal whose period is days to weeks. The figures describe the part of the problem that averaging can address, and the conclusion is that even that part is not solved by averaging alone.

They also describe a quiet star. A younger or more active star has larger convective fluctuations, a stronger rotation signal and a noise floor several times higher. The Earth-twin search is feasible, if at all, only for the quietest nearby Sun-like stars, and the selection of those stars — by their activity, their rotation and their oscillations — is itself a measurement that precedes the search.

Still open: whether the Sun’s floor is every star’s #

The whole calculation rests on amplitudes and timescales measured or simulated for the Sun. Other Sun-like stars of the same temperature and gravity should have similar granulation, and their oscillations follow the scaling relations well. But the supergranulation, which sets the floor, has been measured in any detail only on the Sun, and its dependence on temperature, rotation and metallicity is not known. A star with weaker supergranulation than the Sun would be a much better target; one with stronger would be hopeless. The decade-long, many-visits-per-night programmes now starting on a few dozen of the nearest quiet stars will measure it as a by-product — the noise floor of each target is the first thing their data will reveal — and only then will it be clear how many of those stars an Earth twin could be found around at all.