A model of the star that can carry the planet

Two of the most celebrated nearby planets were withdrawn because their signals turned out to be their stars rotating. A dark spot carried across the disc removes light from the approaching limb and then from the receding one, the line centroid moves, and at the rotation period a few metres a second appear from no planet at all. The lesson drawn from those withdrawals was that the star has to be modelled, not merely checked afterwards.

The model that became standard treats activity as correlated noise. It does not try to draw the spots. It says only that the activity signal at two times is correlated in a particular way — strongly for times a rotation period apart, weakly for times far apart because spots evolve, and not at all for times separated by an odd half-rotation — and lets the data choose the actual curve. That is a Gaussian process, and fitting it alongside the planet is now the routine way of announcing one.

It works. It also has a property its users know about and its readers often do not. A noise model flexible enough to describe a star is flexible enough to describe a planet, and the question of which one a given signal belongs to does not go away because the model is good. It moves somewhere less visible.

Noise with a shape #

A Gaussian process is a statement about covariance. It says that the velocity measured at time $t_1$ and the velocity measured at $t_2$ are jointly Gaussian, with a covariance that depends only on the interval between them, $k(t_1 - t_2)$. Everything about the model is in the function $k$, which is called the kernel.

The kernel used for starspots is quasi-periodic:

$$k(\tau) = A^2 \exp\!\left(-\frac{\tau^2}{2\lambda^2}\right)\exp\!\left(-\frac{\sin^2(\pi\tau/P_{\rm rot})}{2w^2}\right).$$

Each factor has a physical reading. $A$ is the amplitude of the activity signal. The periodic factor says that a spot pattern repeats after one rotation, $P_{\rm rot}$, and $w$ controls how much harmonic structure the repetition can have — a small $w$ allows sharp, many-harmonic curves of the kind a spot produces as it crosses the limb. The decaying factor says that the pattern forgets itself over a time $\lambda$, because spots are born and die over weeks.

Fitting the model means choosing those four numbers, and then the data determine the activity curve itself: the most probable curve given the covariance and the points. The planet is added as a sinusoid — or a Keplerian — with its own amplitude and phase, and everything is fitted together.

The appeal is obvious. The kernel encodes what is known about spots without pretending to know where any spot is, and its parameters have names that can be checked against photometry, which measures the rotation period and the spot lifetime directly. A star whose brightness varies with a 25-day period and a 40-day coherence time should have a velocity kernel with the same numbers, and in practice that is how $P_{\rm rot}$ and $\lambda$ are usually constrained.

What the model costs before any data arrive #

A planet on a circular orbit of known period is linear in two numbers, the amplitudes of the sine and cosine terms. For a given kernel the best fit is a generalised least-squares solution with an exact covariance, and that covariance depends only on the times of observation and on the kernel — not on what was observed. How much the activity model costs a planet can therefore be computed before a single velocity is taken.

The answer has three features, and all three are about overlap in frequency.

The cost peaks at the rotation period. A planet whose period equals the star’s is a sinusoid at exactly the frequency the kernel puts most of its power at. For sixty velocities over two hundred days with three metres a second of activity and a forty-day spot lifetime, the error on the planet’s amplitude is nearly seven times what it would be against white noise. The kernel is saying, correctly, that a signal at that frequency is exactly what the star is expected to produce.

It peaks again at half the rotation period. A spot is visible for half a rotation and invisible for the other half, so its signal is not a sinusoid and carries a strong second harmonic. The kernel’s small $w$ allows for that, and a planet at half the rotation period sits in the harmonic.

It rises towards long periods. Spot patterns that evolve over a hundred days or more look, over a single season, like slow trends — and so does a planet on a long orbit. The longer the spot lifetime, the further this region extends.

And the width of every peak is set by the spot lifetime. A pattern that decays in fifteen days is quasi-periodic only loosely, so its power is spread across a broad band of frequencies around the rotation, and planets anywhere within that band are expensive. A pattern that persists for a hundred and fifty days is nearly periodic, its power is concentrated, and the planets it obstructs are confined to a narrow window. The stars with the most coherent spots are, counter-intuitively, the easiest to search between harmonics.

Harmonics, aliases and the eccentric planet #

The forbidden periods multiply once sampling enters. A star observed from one site can be observed only at night, so every series has a gap every day, and every periodic signal has aliases at frequencies offset by one cycle per day. The aliases of a real period are peaks in the periodogram that are not there, and the aliases of the rotation period and its harmonic are frequencies at which activity has power in the sampled data even though the star does not rotate at them. A planet near any of them pays part of the price a planet at the rotation period pays.

The harmonic at half the rotation period interacts with planet shapes in a particular way. An eccentric planet’s velocity curve is not a sinusoid either; to first order in the eccentricity it is a fundamental plus a second harmonic of exactly the same kind that two circular planets at a two-to-one period ratio would produce. A planet whose period is the rotation period’s double has its own harmonic at the rotation period, so a modest eccentricity puts part of the planet’s signal at the very frequency the kernel claims for the star. The fitted eccentricity of such a planet, and therefore its inferred dynamical history, depends on how the activity was modelled.

None of this is exotic. Rotation periods of Sun-like stars run from a few days for young ones to a month for old ones, and of M dwarfs to a hundred days and more. The habitable zones of M dwarfs lie at orbital periods of weeks to a few months — directly on top of those rotation periods and their harmonics. The stars around which an Earth-sized planet in the habitable zone produces the largest velocity signal are exactly the stars whose rotation sits where the planet would be.

Two models through the same points #

The opening figure shows what this means for one data set. A planet with a period 1.08 times the rotation period is put into simulated activity, and two models are fitted: activity plus planet, and activity alone, each with its own amplitude and lifetime chosen to fit best. Both pass through every point. The likelihood ratio is about 35 in favour of the planet — which sounds large and is not, by the standards a planet claim needs once the extra parameters are paid for, as the Bayesian treatment of planet detection makes explicit.

The activity-only model is not wrong in any way a residual could show. It has used its freedom — a slightly larger amplitude, a slightly longer coherence — to put the planet’s cycles into the spot pattern, and a spot pattern evolving over forty days can produce a 27-day signal that drifts slowly out of phase with a 25-day rotation as easily as a planet can. What would separate the two is time: given several seasons, the spot pattern would have been reborn many times with random phases, while the planet would have kept its own.

That is the central point, and it is worth stating plainly. At the rotation period, a planet and a star are distinguished not by the shape of the signal but by its persistence. Anything that looks only at a season’s worth of data is looking at a question that season cannot answer.

The same experiment run with the planet somewhere else shows what the model is good for. Move the planet to 1.6 times the rotation period, keep the same star, the same spots and the same sixty nights, and the activity-only model can no longer follow it.

A spot pattern drifting slowly out of phase can imitate a period a few per cent from the rotation; it cannot imitate one sixty per cent away, because that would require the pattern to slip by more than half a cycle every rotation, which a forty-day coherence does not allow. The activity-only fit goes through the activity and leaves the planet’s cycles in the residuals, and the likelihood ratio, a factor of some sixty million, is decisive. The model has done exactly what it was built to do: it has removed the star and left the planet standing, clearer than any analysis that treated the activity as white noise could have left it.

So the Gaussian process is not a planet-eating machine. It is an honest statement of which frequencies the star occupies, and it returns planets that sit elsewhere with their evidence largely intact. What it cannot do is invent a separation where the physics has none, and near the rotation period the physics has none within a season.

Fitting the noise drains the evidence #

The figure on costs held the kernel’s parameters at their true values. No analysis knows them. The rotation period can be fixed from photometry, but the activity’s velocity amplitude and its coherence time in velocity have to be fitted along with the planet — and fitting them changes the answer in a way that is easy to miss.

For a planet well away from the rotation period — at 1.6 or 2.5 times it — the data prefer a planet by about thirty units of log-likelihood when the activity model is fixed at the truth, and by about twelve when its amplitude and lifetime are fitted. The planet is still found in every data set at those periods. But more than half of the evidence for it has gone, and at four times the rotation period, where the long-period rise begins, the number of data sets clearing a strong threshold falls from eleven of fourteen to five.

The mechanism is not that the noise model fits the planet’s curve. It is that the alternative hypothesis, no planet, is given more room. Without a planet in the model, the fit raises the activity amplitude to account for the extra variance and lengthens the coherence time to make the extra signal look like slowly evolving spots, and the no-planet model gets much better at describing data that contain a planet. The planet model, by comparison, gains nothing from the same freedom. The difference between them — the evidence — shrinks.

The amplitude of a planet that is found comes back without bias, within a few per cent of the truth on average. That is what makes the effect easy to miss. Every individual announcement looks right: the planet is there, its mass is right — as right as a velocity’s mass ever is — and its error bar is honest. What has changed is the number of planets that cross the threshold, and that number is the input to every occurrence rate built from velocity surveys. A survey analysed with a flexible activity model and one analysed without it see different populations from the same stars, and the difference is largest exactly where the small, temperate planets are.

Persistence is the only separator at the rotation period #

At the rotation period itself, the evidence for a two-metre-per-second planet in sixty velocities over a season collapses to a log-likelihood gain of about two, whether the activity model is fixed or fitted. Almost no data set detects it. The question is whether more data can rescue it, and the answer depends on what kind of more.

Against white noise the error on the planet’s amplitude falls as one over the square root of the number of velocities, without limit. Against modelled activity at the rotation period it falls for the first few dozen points — which are needed to resolve the spot pattern at all — and then stops: about 1.5 metres a second for a hundred-day span, 1.2 for two hundred days and 0.9 for four hundred, whether the span holds forty-five velocities or two hundred and sixty.

The floor is set by how many independent spot patterns the span contains. Within one spot lifetime, the activity at the rotation period is a coherent sinusoid indistinguishable from a planet; over several lifetimes it is a sinusoid whose phase wanders, and a planet’s phase does not. More points within a lifetime describe the same wandering more finely and add nothing new. More lifetimes add new wanderings to average over. The floor falls roughly as the square root of the span in spot lifetimes, which is a statement about the calendar rather than the telescope.

The practical consequence is a scheduling rule that runs against instinct. A survey with a fixed budget of nights, aimed at a planet near the rotation period of a spotted star, should spread them over years rather than concentrating them in a season, even though concentrating them makes each season’s activity curve far better determined. The determined curve is not what separates the planet from the star.

What was actually measured #

The technique has a short history, and its successes and failures both run through these figures.

Successes. Fitting activity as a quasi-periodic process alongside planets turned several contested velocity signals into secure detections, and removed others. Planets around active M dwarfs and young stars — where the activity is tens of metres a second and the planets are a few — would be unreachable without it. In the most careful analyses the kernel’s parameters are fitted jointly to the velocities and to an activity indicator measured from the same spectra, such as the strength of the calcium emission lines or the width of the cross-correlation profile, or to simultaneous photometry. Those indicators respond to the spots but not to the planet, so they constrain the kernel from data the planet cannot have contaminated. The most powerful of them come from the spectrum itself: lines that form at different depths in the atmosphere respond differently to the convection a spot suppresses, so the spread of velocities across lines is an activity signal with no planet in it, measured simultaneously and with the same photons.

Failures. Several planets announced with Gaussian-process activity models were later disputed when more seasons of data arrived or when the indicators were modelled jointly. Most sat near the rotation period or one of its harmonics. In some, the planet’s significance fell as the noise model was made more flexible; in others it rose. A planet whose existence depends on the noise model’s flexibility is a planet whose existence depends on a prior — the same structural problem as the prior on its amplitude, arriving through the covariance instead.

What is reported. A modern detection paper states the kernel, the priors on its parameters, and usually a comparison of models with and without the planet, scored by an evidence that includes the activity model’s flexibility. The figure on drained evidence is a reason to read that comparison carefully: it is a test against the most flexible description of the star the analysts allowed, and a more flexible one would have drained more.

Where the simulations stop #

One kernel, used for both simulating and fitting. Every figure here generates activity from a quasi-periodic process and fits it with the same family. Real spots are not a Gaussian process — they are discrete, they appear at preferred latitudes, they migrate — and a real fit is always to some extent misspecified. Misspecification usually makes things worse, not better: structure the kernel cannot describe is left in the residuals, where a periodogram can mistake it for a planet.

A fixed grid for the noise model’s parameters. The fitted versions choose the activity amplitude and lifetime from a grid of thirty-six combinations by maximum likelihood, with the rotation period held at its true value. A full analysis marginalises over continuous parameters, which smooths the evidence but does not change its direction.

Circular planets, single seasons, white instrumental noise. Each simplification would change the numbers. None would restore the evidence the flexible model takes, because that loss comes from the competing hypothesis’s freedom, which is a property of the model and not of the data’s details.

Simulated data. The numbers are for sixty velocities and three metres a second of activity. A quiet star with half a metre a second of activity has every curve moved down by a factor of six, and the same shapes.

The cost of an honest null hypothesis #

The general lesson is about what a model comparison actually compares. A planet is detected when a model containing it describes the data better than a model without it, and how much better depends as much on the model without it as on the model with it. Making the null hypothesis more flexible is the honest thing to do — a star with spots is a star with spots, and a null that pretends otherwise will find planets in them — and it is not free. Every unit of freedom given to the alternative is a unit of evidence taken from the claim.

A flexible noise model is part of the hypothesis being tested, not part of the measurement. Its parameters are as much a claim about the star as the planet’s are a claim about the system, and the honest report of a detection is the evidence with the noise model’s freedom stated, not a number with the noise model hidden inside the error bar.

The other reading is about time. Between a signal and its imitator, what separates them is often not anything in a snapshot but something in a sequence: a periodic signal that persists against one that is reborn. The exposure that averages away a star’s oscillations is chosen by the star’s own frequencies; the observing span that separates a planet from its star’s rotation is chosen by the star’s spot lifetimes. Both are scheduling decisions set by the star, and neither can be bought with a bigger telescope.

Still open: a separator that is not time #

Persistence works, slowly, and a survey cannot wait a decade for every candidate. The other separator is colour. A planet moves every spectral line by the same velocity, because a Doppler shift is a property of the whole star. A spot does not: it is cooler than the photosphere, not black, and its contrast — and with it the velocity it produces — depends on the wavelength it is observed at. Whether velocities measured in two colours at once can tell a planet from a spot in a single season, and how far towards the infrared the advantage holds before the star’s magnetic field takes it back, is the question the next generation of spectrographs, covering the optical and the near-infrared simultaneously, is built to answer.