A planet that was the star's own rotation

The rung below this one turned a periodic Doppler shift into a mass, with $\sin i$ as its single acknowledged unknown. The inference is sound and the arithmetic is exact. What it assumes is that the periodic shift is a shift — that the whole spectrum has moved bodily because the star’s centre of mass is moving.

There is a second way to make a spectral line’s centroid move, and it does not involve the star going anywhere.

A star is a rotating disc, and light from its approaching limb is blueshifted while light from the receding limb is redshifted; the line profile is the sum, symmetric about the star’s true velocity. Remove some light from one side of that disc — put a dark spot there — and the sum is no longer symmetric. The centroid moves. As the star turns, the spot crosses from the blue side to the red side, and the centroid moves back.

The size of the effect #

Two mechanisms contribute and they scale differently.

The flux deficit. A spot covering a fraction $f$ of the disc at projected position $x$ (in units of the stellar radius) removes light that was carrying velocity $x\,v\sin i$. The centroid shifts by approximately $f\,x\,v\sin i$ weighted by the spot’s projected area, which itself goes as the cosine of its longitude. Putting those together gives a term proportional to $\tfrac{1}{2}f\,v\sin i\,\sin 2\theta$ — and only for the half rotation the spot is visible.

For $f = 0.004$ and $v\sin i = 3$ km s⁻¹ that is about 5 metres a second. A spot four times larger, on a star rotating twice as fast, gives forty.

The suppression of convective blueshift. The photosphere is granular: hot rising columns cover most of the area and are brighter, cool sinking lanes cover less and are darker, so the disc-integrated spectrum is blueshifted relative to the true velocity by a few hundred metres a second, which is the granular flow of the convection zone seen in aggregate. A magnetic region suppresses the convection beneath it, removing that blueshift locally and producing an apparent redshift proportional to the covered area alone.

For the Sun this second term is the larger one, because the Sun rotates slowly. It scales with the area rather than with $v\sin i$, and it is the reason a quiet, slowly rotating star still has a metre-per-second floor.

Two ways it is caught #

The shape of the signal in time. A Keplerian orbit produces a curve whose harmonic content is fixed by its eccentricity: a circular orbit is a pure sinusoid with no power at half the period at all. A spot signal is truncated by visibility, so it always has substantial harmonic content, and the harmonics are at exactly $P/2$ and $P/3$ rather than at arbitrary frequencies.

The practical consequence is that an activity signal usually appears in a periodogram as a family — a peak at $P_{\rm rot}$ and further peaks at its harmonics — while a planet appears as one peak plus whatever the sampling adds.

Two things can put a peak where no signal is, and they are worth separating. The first is the sampling itself.

The second is what that window does to a real peak, which is not to hide it but to copy it.

The correlation with an activity indicator. Several quantities measured from the same spectra track magnetic activity and cannot respond to a planet: the emission cores of the calcium H and K lines, the width and asymmetry of the cross-correlation function, the depth of the Hα line, and the bisector. If the velocity correlates with any of them at the candidate period, the candidate is a rotation.

If it correlates with none of them, that is weaker evidence than it appears, because the indicators are noisy and a null correlation at a few metres a second is a null result at low significance.

What was actually withdrawn #

α Centauri B b, announced in 2012 at a period of 3.24 days and a semi-amplitude of 51 centimetres a second — an Earth-mass planet around the nearest sunlike star, at a distance fixed by parallax to four figures, from four years of spectra and a signal below the level anyone had previously claimed.

The withdrawal, in 2015, was not about spots. It was about the analysis. The velocities had been filtered to remove a much larger activity signal at the star’s 37-day rotation period, and the filtering was done in a way that created power near 3.2 days: the observing window, convolved with the removed signal, put a spurious peak exactly where the planet was claimed. Injecting a synthetic signal of the same amplitude at a different period into the same pipeline and recovering it at 3.2 days settled it.

Barnard’s star b, announced in 2018 at a period of 233 days and 1.2 metres a second, a super-Earth around the second-nearest system, from 771 measurements over two decades.

The withdrawal, in 2021, was about rotation. Barnard’s star turns in about 140 days, and the 233-day signal turned out to correlate with activity indicators once a longer and better-sampled data set existed — and to change amplitude between observing seasons, which a planet cannot do. What had been missing in 2018 was not a diagnostic but time: the signal was real in the data and was not coherent over the longer baseline.

Neither retraction was a failure of the measurement. In both cases the velocities were correct to the precision claimed. What was wrong was the step from a periodic signal to a planet, and in both cases the step was taken at an amplitude where the star’s own contribution is comparable with the signal.

What is done about it now #

The response has been to model the star rather than to filter it.

The standard approach treats the activity contribution as a Gaussian process — a correlated noise model with a covariance function containing the rotation period and a decay timescale for how long a spot pattern persists — fitted simultaneously with the Keplerian parameters rather than removed beforehand. That the model has physical parameters is the point: the fitted rotation period must agree with the one seen photometrically, and the fitted decay time must be a few rotations, which is the same discipline a physically parameterised noise model imposes anywhere.

Two further tactics have proved more useful than any statistic.

Observe where the star is quieter. Spot contrast falls towards the infrared, because a spot is cooler rather than dark and the contrast between two blackbodies shrinks at long wavelengths. A signal that persists from the optical to the near-infrared with the same amplitude is achromatic and therefore a Doppler shift; one that shrinks is a spot.

Observe more often. Activity is correlated on the rotation timescale, so measurements spaced by hours are not independent, while a planet’s signal at a different period is. Dense sampling within a night does nothing for a spot’s contribution to the nightly mean and everything for the ability to separate timescales.

The period that is not one number #

The account above treats the rotation period as a single frequency to be identified and avoided. A star does not have one.

The Sun’s equator turns in about twenty-five days and its poles in about thirty-four, and the difference is not a subtlety — it is a spread of a third. A spot at one latitude therefore produces a signal at a different period from a spot at another, and since spots migrate in latitude over a magnetic cycle, the period a star appears to have drifts with time.

Three consequences follow for anybody trying to separate an activity signal from a planet.

The activity power is not at a frequency; it is in a band, whose width is the differential rotation and whose position within the band depends on which latitudes are currently spotted. A periodogram peak from activity is therefore broader than a planet’s, and comparing peak widths is a genuine if weak diagnostic.

A noise model that assumes a single rotation period is misspecified. The standard covariance function used for these fits contains one period, and fitting it to a star with substantial differential rotation returns some average with a decay timescale shorter than the physical spot lifetime — the model absorbing the latitude spread into the wrong parameter.

And a planet whose period falls inside the band is very hard to establish. The band for a Sun-like star spans a third of the rotation period, and a candidate inside it has to be distinguished from activity by coherence over many years rather than by frequency.

A quantity that is quoted as a number and is really a distribution is the most reliable way to be wrong slowly, and the rotation period of an active star is such a quantity. The photometric period measured from a light curve is itself an average weighted by where the spots were during the observation, so even the independent check the previous section relied on is measuring something that moves.

Where the model stops #

A spot is not the only kind of active region. Bright faculae cover more area than spots on a quiet star and have the opposite photometric sign but the same convective-blueshift signature, so on a Sun-like star the two contributions partially cancel — and the cancellation is not exact and varies over the magnetic cycle.

Granulation and oscillation are separate problems. Convective cells evolve in minutes and p-mode oscillations in five, and both contribute velocity noise at the metre-per-second level. Both are beaten by averaging over the right interval, and the right interval is a property of the star that has to be known first.

And the cycle itself is a long-period signal. A magnetic cycle changes the mean convective blueshift over years, producing a velocity trend with a period of a decade — indistinguishable in a short data set from a distant giant planet. The Sun’s own cycle would appear as a several-metre-per-second signal at eleven years to an observer watching it as a star.

The picture cannot show the surface. Everything above is inferred from disc-integrated light. Which latitude a spot is at, how many there are, and whether the star’s rotation axis is even near the plane of the sky are not measured for any of these targets, and the models are parameterisations rather than descriptions.

How a claim is made now #

The two withdrawals changed the practice of the field more than they changed its physics, and the procedures that resulted are worth recording because they are unusually explicit for observational astronomy.

Injection and recovery, on the real data. Before a signal is claimed, synthetic Keplerian signals of the same amplitude are injected into the actual velocity time series at a grid of periods, and the pipeline is run on each. The question asked is not whether the candidate is significant but how often a signal of that amplitude at that period is recovered, and how often the pipeline produces one where none was injected. That converts a significance into a measured false-alarm rate for the specific data set.

Splitting the data. A planet is coherent, so the signal should be present with the same period and phase in the first half of the campaign and in the second. An activity signal is not, and the two withdrawn cases both failed this test once the baselines were long enough to apply it. The test costs nothing except patience and it is now routine.

Blind analysis. In some collaborations the velocities are analysed with an unknown offset added, or the candidate period is hidden from the person tuning the noise model, precisely because the tuning has enough freedom to produce or suppress a marginal signal. That is the same reasoning that led time-delay cosmography to blind its fits, and for the same reason: the systematic is a modelling choice made by a person who knows what answer is wanted.

And publishing the velocities. The single most effective change has been that the data are released. Both withdrawals were made by groups other than the discoverers, using the published measurements, which is only possible where the measurements are published.

None of that raises the precision. What it does is make the step from a periodic signal to a planet auditable, which is the step both retractions were about — and the field’s detection threshold has moved upward as a result, which is an unusual direction for a threshold to move and the right one.

It is worth noticing what the same discipline implies for the non-detections, which are never published and which carry information. A survey that observes two hundred stars for a decade and reports three planets has also established, for the other hundred and ninety-seven, upper limits on what could be present — and those limits are the occurrence rate, which is the quantity the field actually wants. Injection and recovery on the real data produces exactly those limits as a by-product, star by star, at no extra observational cost.

The reason they are so rarely quoted is that a limit is only as good as the noise model behind it, and the noise model is the star. A limit computed under a white-noise assumption is far too tight, and one computed under a correlated model depends on parameters fitted to the same data. So the field’s headline statistic — how common Earth-like planets are around Sun-like stars — inherits the whole of this essay’s difficulty, and is quoted with error bars that are honest about the counting and silent about the modelling.

What it costs the field #

The arithmetic of the limitation is worth stating plainly, because it decides what is reachable.

The Earth’s pull on the Sun is a semi-amplitude of 9 centimetres a second over a year. A modern spectrograph’s photon-limited precision on a bright star is a few tens of centimetres a second per exposure, so with enough exposures the instrument is not the obstacle: a thousand measurements at 30 cm s⁻¹ average to one. The obstacle is that the star’s own contribution does not average down, because it is correlated on the rotation timescale and modulated on the magnetic-cycle timescale, and both of those are comparable with or longer than the orbital period being searched for.

That is the whole reason an Earth analogue around a Sun analogue has not been found by this method after thirty years of trying. It is also why the two withdrawn planets above were both around very nearby, very well observed stars — the targets with the most data are the ones where a marginal signal accumulates enough significance to be published, and they are the same targets whose activity has been characterised well enough for someone to notice later.

The same arithmetic decides what a marginal detection looks like on the page, and the honest answer is that it looks like nothing in particular.

What that picture makes concrete is why the field moved from thresholds to models. A periodogram peak is a statistic computed under the assumption that everything not the signal is independent from one measurement to the next, and stellar activity is the opposite of independent. So the false-alarm probability attached to a peak is not merely optimistic; it is computed under a hypothesis nobody believes, and its being small says nothing about whether the peak is a planet.

The replacement is to model the correlated part explicitly — a Gaussian process whose covariance is quasi-periodic on the rotation timescale — and to ask whether a Keplerian is still required once that is in the model. It is a much weaker test, in the sense that far fewer signals survive it, and that is the point. The signals that do survive are the ones whose amplitude cannot be absorbed by any plausible description of the star, and the price of that confidence is that the ninety-centimetre-a-second regime is no longer approached at all. That is a real cost and it is worth naming as one. The method’s theoretical precision has improved by two orders of magnitude in thirty years and its practical reach has improved by rather less, because what was gained in the instrument was spent on being honest about the star. The next factor will not come from a better spectrograph; it will come from a better model of a stellar surface, which is a problem in convection rather than in optics. The instruments capable of the measurement already exist and have existed for a decade; what does not exist is a way to tell a stellar surface’s motion from a planet’s, and that is where the effort now goes.

The generalisation #

The structure of the problem is the one that recurs whenever an instrument measures a summary statistic of a resolved object it cannot resolve.

A radial velocity is one number extracted from a line profile, and the extraction throws away everything about the profile’s shape. Any process that changes the shape without moving the star therefore enters the number and cannot be distinguished from a motion — unless a second statistic, sensitive to shape and not to motion, is kept as well.

That is exactly the argument behind reading a magnetic field off a line’s circular polarisation rather than off its width, behind using a transit’s shape rather than its depth to reject a blended eclipsing binary, and behind checking the colour of a transit to reject one. The general rule: when the measurement is a projection, keep a second projection that responds differently. A single number, however precise, cannot distinguish two causes that produce the same number.

Where this ladder goes next #

Later rungs on this anchor: Gaussian-process models in detail, and the danger that a flexible noise model absorbs the planet as well as the star; line-by-line velocity extraction, in which lines with different sensitivities to temperature and to magnetic fields are used as separate instruments on the same spectrum; the infrared, where the spot contrast is small and the technique’s floor for M dwarfs is set; the solar telescope observations that measure this contribution directly by watching the Sun as a star while resolving its surface; and the ultimate limit, which is the granulation of a photosphere that cannot be modelled away and sets the precision at which an Earth analogue around a Sun analogue becomes detectable.