A planet has one velocity in every colour

A planet pulls its star towards it and away again, and the star’s spectrum shifts as a whole. Every absorption line moves by the same fraction of its wavelength, so every line moves by the same velocity: a hydrogen line in the blue, an iron line in the yellow and a carbon monoxide band in the infrared all report the identical number. That is what a Doppler shift is. It does not care which line it is measured from, and it does not care what colour.

The signals that imitate planets are not Doppler shifts of the star. A dark spot rotating across the disc does not move the star at all. It removes light from one side of the rotating surface and then from the other, which makes each line’s profile lopsided and its measured centre move, at the rotation period, by metres a second. That movement is produced by an absence of light, and the amount of light a spot fails to emit depends strongly on the colour it is observed in.

So the cleanest test of whether a velocity signal is a planet is to measure it in two colours at once. A planet gives the same answer in both. A spot gives a smaller answer in the red. The test needs no extra seasons, no model of the star and no assumption about how spots evolve. What it needs is a spot that acts by removing light, and a star whose other magnetic effects stay out of the way — and neither condition can be taken for granted.

A spot is cooler, not black #

A sunspot looks black in a photograph because it is next to the photosphere. On its own it is a blazing surface at about four thousand kelvin, brighter per square metre than an incandescent lamp. Starspots on other stars are similar: regions several hundred to fifteen hundred kelvin cooler than their surroundings, where a concentrated magnetic field suppresses the convection that would otherwise carry heat to the surface.

The deficit of light from a spot is therefore the difference between two Planck curves, one at the photosphere’s temperature and one at the spot’s. Its contrast — the fraction of the photosphere’s light it fails to emit — is $1 - B_\lambda(T_{\rm spot})/B_\lambda(T_\star)$, and that fraction is far from constant across the spectrum.

The reason is the shape of the Planck curve. On its short-wavelength side, a blackbody’s brightness depends exponentially on temperature: a few per cent cooler is tens of per cent fainter. On its long-wavelength side, the Rayleigh–Jeans tail, brightness is simply proportional to temperature: a few per cent cooler is a few per cent fainter. Moving from the visible to the infrared moves the observation from the first regime towards the second, and the same physics that makes colour a thermometer makes a spot’s contrast fall steadily with wavelength.

For a spot 600 kelvin cooler than a Sun-like photosphere, the contrast is about 0.4 at 0.55 micrometres and 0.17 at 2.2. For one 1500 kelvin cooler it falls from 0.8 to 0.4. For a spot 300 kelvin cooler on a 3300-kelvin M dwarf it falls from 0.55 to 0.2. In each case the infrared sees less than half the spot the visible does.

Two ways a spot moves a line #

The velocity a spot produces is not only a matter of missing light, and the second mechanism behaves differently with colour.

The flux deficit is the geometric effect already described. A rotating star’s spectral lines are broadened by its rotation, the approaching limb contributing blueshifted light and the receding one redshifted. A spot on the approaching side removes some blueshifted light, the line’s centroid moves red, and half a rotation later the reverse. The size of the effect is the spot’s contrast times its area times the rotation speed, so it inherits the contrast’s wavelength dependence directly.

The suppression of convective blueshift is subtler and, on a quiet Sun-like star, larger. A normal photosphere is covered in granules: bright, rising columns of hot gas separated by dark, sinking lanes. Because the rising gas is brighter, the light-weighted average velocity of the surface is a net approach of a few hundred metres a second — a blueshift built into every line. A magnetic region, spot or bright facula, suppresses the convection beneath it, and the blueshift there vanishes. As the region rotates into view, the star’s lines move red by an amount set by the region’s area and by how much blueshift it removed, and the size of that shift varies from line to line with the depth at which each line forms. Its dependence on the colour of the line, though, is weak, and it is treated here as none.

So the colour test is strong for one kind of activity and weak for the other. A young, fast-rotating star covered in large cool spots produces velocity signals dominated by the flux deficit, and its spot velocity falls by more than half from the visible to the infrared. An old, slowly rotating, Sun-like star produces much smaller signals, and on the Sun itself the larger part comes from suppressed blueshift in faculae and plage — bright magnetic regions, not dark spots — whose velocity changes with colour far less. The quiet stars around which Earth twins are sought are therefore exactly the stars on which the colour test is weakest.

Two channels at once #

The test works only if both colours are measured at the same moment. Activity evolves: spots are born and decay over weeks, and a velocity measured in the blue one night and the red the next is a comparison between two different stars. Modern spectrographs record a wide range of wavelengths simultaneously across dozens of separate orders, and the velocity can be measured in each order or each group of orders independently, from the same photons and the same instant.

The opening figure plots the result for two simulated stars. A star with a planet gives points along the diagonal: whatever the velocity in the blue channel, the red channel reports the same number, and the fitted slope is one within the noise. A star with an evolving spot and no planet gives points along a shallower line, with a slope of about 0.64 for the spot drawn — which is just the ratio of the spot’s contrasts at the two wavelengths. Each point is one night. The separation needs no time series longer than a rotation and no model of how the spot evolves; a few dozen nights in one season suffice when the signal is several metres a second.

In practice the comparison is made across the whole spectrum at once. The velocity measured in each order is plotted against the logarithm of that order’s central wavelength, and the slope of that line on each night — a chromatic index — is an activity indicator measured from the same data as the velocity itself. A planet produces a chromatic index of zero on every night. A flux-deficit spot produces an index that varies with the rotation and correlates with the velocity, and a velocity signal that correlates with its own chromatic index is a signal that is not a planet.

The limit is noise. The chromatic signature is the difference between two velocities, and the difference is a fraction of the signal: one minus the contrast ratio, or about a third for the spot drawn here. Measuring it to the same significance as the signal itself needs roughly three times the precision, and each channel carries only part of the photons. For a star with a two-metre-per-second spot signal and a precision of a metre per channel, the slope is barely measured. The test is strongest exactly where it is least needed, on very active stars with large signals, and weakest on the quiet stars where a planet of a metre a second is being sought.

The slope is a thermometer #

The chromatic slope does more than reject a planet. It measures the spot.

The ratio of a spot’s velocity amplitudes in two channels is the ratio of its contrasts there, and the ratio of two contrasts at two known wavelengths depends on only one unknown, the spot’s temperature, once the photosphere’s temperature is known from the spectrum. A slope of 0.64 between 0.45 and 0.8 micrometres on a 5100-kelvin star corresponds to a spot about nine hundred kelvin cooler, and a shallower slope means a cooler spot. Size does not enter: a larger spot makes both velocities larger by the same factor and leaves their ratio unchanged.

That separates two things a single-colour velocity confuses. The amplitude of a spot signal is the product of the spot’s area and its contrast, so a small, very cool spot and a large, mildly cool one give the same velocity in one colour. In two colours they do not. The colour ratio fixes the contrast, the amplitude then fixes the area, and the velocity series becomes a measurement of the spot’s size and temperature — the same two quantities photometric light curves try to extract from the depth and colour of the dimming, and which they confuse in the same way.

Bright regions behave in the opposite sense. A facula is hotter than its surroundings, at least near the limb, so its contrast is negative, it adds light rather than removing it, and its colour dependence is reversed in sign: brighter in the blue. A star whose velocity signal comes from a mixture of dark spots and bright faculae has a chromatic slope that is a weighted mixture of the two, and can be close to one even when neither component is. On the Sun, observed as a star by small telescopes feeding the same spectrographs that search for planets, the velocity variations over a rotation are dominated by faculae and by their suppression of convective blueshift, and the chromatic signal is small. The Sun is the star a planet search most wants to be able to read, and it is not a star that shows its activity in colour.

The infrared and the M dwarfs #

The colour of a velocity matters for a second reason, and it points the same way. The smallest stars are the most favourable targets for finding small planets — they host more of them, and a given planet pulls a light star harder — and they emit very little visible light.

A 3300-kelvin M dwarf emits most of its photons near 1.5 micrometres, and a 2800-kelvin one near 1.8. An optical spectrograph collects the faint blue tail of such a star, and the photon-limited precision of a velocity falls as one over the square root of the photons collected. Moving to the near-infrared gains several times the photons and, for the coolest stars, a large factor in precision, even though the information a velocity is measured from lives in the slopes of spectral lines rather than in the photons themselves, and the infrared spectra of M dwarfs are carved by molecular bands whose lines are often blended.

The case for the infrared therefore combined two arguments: more photons, and fainter spots. Several spectrographs were built on it, some covering the visible and the near-infrared simultaneously so that the chromatic test could be applied to every star they observed. The early results were instructive, because the second argument turned out to be weaker than the contrast curves had suggested.

The field takes back part of the gain #

A starspot is a magnetic structure, and a magnetic field splits spectral lines. The Zeeman effect separates each line into components displaced by an amount proportional to the field strength, to the line’s magnetic sensitivity, and to the square of its wavelength. Expressed as a velocity — a displacement divided by the wavelength — the splitting grows in proportion to the wavelength itself. A kilogauss field splits a typical infrared line by more than a kilometre a second, and the splitting is a measurement of the field in its own right.

Spread across a rotating star’s line profile, the split light from a magnetic spot distorts the profile asymmetrically as the spot rotates, just as the missing light does, and the distortion moves the measured velocity. Its size depends on the field, the spot’s area, the lines used and how the velocity is extracted, and it grows towards the red while the flux-deficit term shrinks.

The sum has a minimum, and where it sits depends on the field. With a weak Zeeman term the spot signal falls all the way into the infrared, and observing at two micrometres reduces it by more than a factor of two. With a Zeeman term comparable to the flux term at one micrometre, the minimum moves back into the red end of the visible, and the infrared is no quieter than the optical. M dwarfs are the most magnetic of all main-sequence stars, with surface fields of several kilogauss covering much of their surfaces, and they are the stars for which the infrared was supposed to help most.

The relative size of the two terms is not known in general, and that is why three are drawn. It depends on the geometry of the spots, the field’s strength inside and outside them, and which lines are used. Simulations of spotted M dwarfs found the infrared advantage reduced or removed for strong fields, and spectrographs covering both ranges have found active M dwarfs whose infrared velocity scatter is smaller than their optical scatter and others whose is not.

What was actually measured #

The test has one celebrated success, and it came before the instruments were designed for it.

In 2008 a planet of about ten Jupiter masses was announced around TW Hydrae, a young star still surrounded by the disc it formed from. The evidence was a velocity signal of about two hundred metres a second at a period of 3.56 days, measured in the visible. A planet so young and so close to a star still building its planets would have been a remarkable constraint on how fast giants form.

Before the year was out the star was observed with an infrared spectrograph at about 1.5 micrometres. If the signal had been a planet, it would have had the same two hundred metres a second there. It had about thirty-five. The visible signal had the period of the star’s rotation, which photometry also showed, and its colour dependence was that of cool spots on a young star. The planet was withdrawn, on evidence collected in a few nights.

Young stars have remained the hardest case since, because they are heavily spotted, rotate in days, and are the stars around which the youngest planets would be most informative. The test has since been folded into routine analysis as the chromatic index, computed for every spectrum a visible-and-infrared instrument takes, alongside the line-shape and activity indicators that do the same job for the convective part of the signal.

Where the simple picture stops #

Blackbody spots. The contrast curves treat the spot and the photosphere as blackbodies. Real spots have molecular absorption the photosphere lacks, umbrae and penumbrae with different temperatures, and they are seen against a limb-darkened disc whose own colour changes towards the edge. The wavelength dependence survives; the precise numbers do not.

An achromatic blueshift. The suppressed convective blueshift is taken to be the same at every wavelength. It is not quite: lines of different strengths and formation heights carry different blueshifts, and the mix of lines changes across the spectrum. On the Sun the residual chromaticity is small.

A Zeeman term of unknown size. The field’s contribution is drawn proportional to wavelength with three arbitrary amplitudes, because its true size depends on the star. The figure’s content is the shape of the trade, not the position of the minimum for any particular star.

One spot. The simulated stars have a single evolving spot. A star with many spots at different longitudes has a smaller, more complex signal; its chromatic slope is the same, because every spot has the same temperature contrast, but its amplitude is smaller and the slope harder to measure.

A shift that belongs to the whole star #

The general statement is about what a Doppler shift is and what it is not. A velocity is a property of a body as a whole, and every part of its light carries it equally. Any signal that depends on which part of the light it is measured from — which wavelength, which line, which depth of formation — is not a velocity of the body, whatever else it may be.

A real motion is the same in every channel that can measure it; an imitation usually is not. The planet is the one signal in the spectrum guaranteed to be achromatic, line-independent and shape-preserving, and every activity indicator is a way of checking one of those guarantees. The colour test checks the first, the line-by-line spread checks the second, and the line bisector checks the third.

The other reading is that the checks are not independent of the star. The colour test works on spotted stars and fails on faculae-dominated ones; it works in the infrared until the field takes over. Each diagnostic has a domain where the star cooperates, and the quiet, Sun-like stars around which the smallest planets are sought are the ones where the stars cooperate least with every diagnostic, because their activity is mostly the kind that changes nothing but the velocity.

Still open: signals too slow for any of this #

Every test so far — the Gaussian process, the colour, the line shapes — is aimed at signals with periods of days to weeks, where a star’s rotation lives and where a season of data covers many cycles. A velocity series also contains signals far slower: a steady drift over years that may be a companion on a decades-long orbit, or the star’s magnetic cycle, or both. A drift is a single number, a slope, and a slope is consistent with an enormous range of companions. What a slope does and does not say, and what other measurements must be added to turn it into a mass, is the problem of the longest periods — where the companion’s orbit is longer than the data and the only evidence is that the star is accelerating.