A spot a slow star hides inside its velocity

A star’s radial velocity is measured from the positions of its spectral lines, and the photons that carry the information are those on the sides of the lines, where a small shift changes the flux most. The photon limit that follows from that is now below a metre per second for a bright star, and the limit that matters in practice is the star itself: its surface is not uniform, and anything that makes one part of the rotating disc brighter or darker than another moves the lines’ centres as surely as a planet does.

A planet moves every line of the spectrum by the same velocity and leaves its shape alone. A spot does not: it removes light from the part of each line that its own patch of surface was contributing, and that leaves a dent in the line’s profile at a particular velocity. So in principle a spectrum can tell the two apart, by measuring not just where each line is but what shape it has. The question is how much of a spot’s signal is in its shape and how much in its apparent shift — because only the first part can be separated, and the second is indistinguishable from a planet by any measurement of the spectrum at one moment.

Why the shape is the only witness at one moment #

A shift in a line is a speedometer because the line’s rest wavelength is fixed by atomic physics, so any displacement is a velocity. That is what makes the method work at any distance, and it is also what makes a spot dangerous: the speedometer reads the line’s centre, and it has no way to ask why the centre moved. A planet pulls the star and leaves a mass that is only a lower bound; a spot leaves a velocity curve with no mass behind it at all, and the two velocity curves can be identical in period and amplitude.

Time gives one kind of witness. Planets have been withdrawn because their period was the star’s own rotation, or a harmonic of it, and the periodogram of the star’s brightness or of its magnetic indicators showed the same peak. That test needs many nights and a star whose rotation period is known, and it fails when the planet’s period happens to lie near the rotation’s. The shape of the line is the other witness, and the only one available in a single spectrum. What this essay asks is how much it can say — and the answer is a number that depends on two things, the star’s rotation and the instrument’s resolution, which the figures below take in turn.

What a spot does to a line #

A rotating star’s spectral line is the sum of the lines from every patch of its visible surface, each Doppler-shifted by that patch’s velocity towards or away from the observer. The approaching limb contributes at negative velocities, the receding limb at positive ones, and the centre of the disc at zero, each weighted by how bright the patch is. A star rotating at a few kilometres a second has lines a few kilometres a second wider than they would otherwise be, with a shape set by the rotation and by the darkening of the disc towards its edge.

A dark spot removes light from its patch. Where the spot sits on the receding side, it removes light that was contributing to the red half of the line, so the line’s red half is shallower than it should be and the line’s centre, weighted by the light, moves to the blue. The opening figure shows the dent: a bump in the line, positive where the missing light used to absorb, at the spot’s own rotational velocity, with a smaller adjustment elsewhere because the line is normalised to the total light.

Decomposed against the derivative of the line — the change a pure shift would make — that bump is mostly a shift. A shift of a profile is a positive bump on one side of the centre and a negative one on the other, and the spot’s dent, though centred on the spot’s velocity rather than symmetric about the line’s centre, overlaps that pattern closely when the spot’s velocity is within the line. What is left over is a smaller wiggle with no net shift, the part of the profile’s change that no velocity could make.

The velocity a spot fakes #

As the star turns, the spot crosses the disc, and its effect on the line’s centre follows it.

On the approaching limb the spot removes blue-shifted light and the line moves to the red; at the disc’s centre it removes light at zero velocity and the line does not move; on the receding limb the line moves to the blue. The signal has the period of the star’s rotation, and its amplitude is set by how fast the spot is moving across the line of sight, which is the rotation speed, times how much light it removes. For a spot of this size it is metres to tens of metres per second, and on a quiet star it can mimic a planet whose orbital period is close to the rotation period — or, as spots form and decay over weeks, a planet whose signal comes and goes.

The scaling is the one measured on real stars: the fake velocity grows about in proportion to the rotation speed and to the spot’s covering fraction. A spot covering a tenth of a per cent of a star rotating at one kilometre a second — roughly the Sun’s equatorial speed, and roughly the area of a large group of sunspots — fakes about two metres per second at its largest. The Earth’s pull on the Sun is nine centimetres per second. A planet like the Earth round a star like the Sun has to be found underneath spots whose signals are twenty times larger and come and go with the star’s activity.

The share that is not a shift #

The question the opening figure raised is how the separable remainder changes with the star and the instrument.

On a fast rotator the spot’s dent is a narrow feature on a broad line, far from a shift of the whole line, and half or more of its effect is in its shape. On a slow rotator the line is barely broader than its intrinsic width and the instrument’s resolution, the spot’s dent is as broad as the line itself, and after convolution with the instrument it is almost indistinguishable from a displacement of the whole profile. At half a kilometre a second, a third of a per cent of the signal is in the shape; at one kilometre a second, a little over one per cent.

The resolution matters less than might be expected. Doubling it from 100,000 to 200,000 barely changes the share for slow rotators, because the limit is not the instrument but the line: the intrinsic widths of the lines themselves, set by thermal motion and turbulence in the stellar atmosphere, are a few kilometres a second, and a spot’s dent on a star rotating more slowly than that is smoothed by the star’s own physics before the light leaves it.

This is the same reason the spots of fast rotators can be mapped. Doppler imaging reconstructs a star’s surface from the bumps that travel across its line profiles as it rotates, and it works only for stars rotating at tens of kilometres a second, where each bump is resolved into its position on the disc. For the slow, quiet stars that radial-velocity surveys target, the same bumps are there but they have merged into the line’s position.

What it takes to see the shape #

The remainder is small, but it is not zero, and with enough photons it can be measured.

A spectrum good enough to measure a metre per second has a signal-to-noise of a few hundred per pixel over thousands of lines. For this spot, half a per cent of the surface, on a star rotating at five kilometres a second, the shape is visible at a signal-to-noise of twenty. At one kilometre a second it needs three hundred — the same spectrum, just. A smaller spot needs more, in inverse proportion to its signal, and the spots that matter for an Earth-like planet’s search are smaller than this one. The shape is there to be measured, and on the slowest rotators it sits at the limit of what the photons allow.

That sets the strategy of the modern searches. They measure, alongside the velocity, quantities designed to track the line’s shape — the asymmetry of the averaged line, its width, its depth — and they model the velocity as a planet plus an activity signal correlated with those quantities. On fast rotators the correlation is strong and the activity can be removed. On slow ones the shape indicators are themselves noisy, measured from the same few per cent of the signal that the figure says is not a shift, and the activity term has to be inferred from its time behaviour instead: its period, its coherence over the weeks a spot lives, its relation to the star’s brightness and magnetic indicators.

Bright regions, which are worse #

Spots are not the dominant source of activity signals on slowly rotating Sun-like stars. Bright regions are.

The surface of a star like the Sun is covered in granules, rising hot gas and falling cool gas, and because the rising gas is brighter its blueshift dominates the lines: every line is shifted a few hundred metres per second to the blue by convection. In magnetic regions — the bright faculae that surround spots and cover several per cent of the Sun at activity maximum — the field suppresses the convection, and the lines there are less blueshifted. A facular region’s lines are shifted to the red relative to the rest of the disc, with no gap in the rotation profile needed.

That signal is a displacement of real lines. On a slow rotator it is closer to a pure shift than a spot’s signal is, half as much of it is in the shape at one kilometre a second, and it does not even need the region to be off-centre to produce a velocity: a facular region at the centre of the disc fakes a redshift with no rotation at all. For the Sun, the suppression of convective blueshift is the largest single contributor to the activity signal in its radial velocity over a magnetic cycle, a few metres per second, and it is the component least distinguishable from a planet by the shape of the lines.

On fast rotators the order reverses: the facular patch at the centre of a broad line is a narrow, displaced feature, more distinct than a spot’s dent. But on fast rotators the spots dominate anyway, and on slow ones the faculae do — so in each regime the dominant activity signal is the one the shape distinguishes worse.

Spots that the planet itself draws #

There is one dark region whose effect on the line is known in advance: a transiting planet. As a planet crosses its star’s disc it blocks light from the strip of surface behind it, at that strip’s rotational velocity, exactly as a spot does, and the lines shift by an amount and with a sign that depend on whether the planet crosses the approaching or the receding half. This is the Rossiter–McLaughlin effect, and it is used to measure the angle between a planet’s orbit and its star’s spin — the angle that turns out to be large around hot stars and one that only cool stars forget.

It is the spot problem with the spot’s path given. The transit’s timing and depth fix where the dark region is at each moment, and the velocity anomaly follows; its amplitude is a few metres to tens of metres per second for a giant planet on a star rotating at a few kilometres a second, comparable to the spots in the figures. On fast rotators the planet’s shadow appears as a travelling bump in the line profile, the same Doppler-imaging signature; on slow ones it appears only as a velocity anomaly, and the analysis treats it as a shift. The two cases bracket the one drawn here.

The velocity that is an average of lines #

The figures treat every line as identical, which real lines are not. Lines formed at different heights in the atmosphere sample convection differently; lines of different strength sample different depths; lines sensitive to magnetic fields broaden in active regions. A spot’s or facula’s effect therefore differs from line to line, and the velocity measured from many lines is an average of lines that disagree.

That disagreement is itself information. A planet moves every line by the same amount. Activity moves different lines by different amounts, so the velocity measured line by line, or in groups of lines selected by their formation depth or magnetic sensitivity, separates activity from a planet in a way the shape of an averaged line cannot. The arithmetic of the figures applies line by line — a slow rotator’s activity is mostly a shift in each line — but the shifts are not the same in every line, and the difference between them is the most promising route to seeing the activity of a slow, quiet star at all.

The photometric twin of the problem #

A spot dims the star as well as shifting its lines, and that is useful: a spot on the approaching limb darkens the star and reddens the lines, so the velocity and the brightness move together, the velocity crossing zero at the moment the star is faintest and swinging to either side as it brightens. A planet’s velocity has no brightness partner. Simultaneous photometry therefore predicts much of the spot signal — the so-called FF′ method writes the activity velocity as the flux times its time derivative, plus a term for the suppressed convective blueshift proportional to the flux deficit squared.

It predicts less of the faculae. A facular region near the disc’s centre is barely brighter than its surroundings in visible light, because faculae are bright mostly towards the limb, and yet its suppressed blueshift is at its largest there. So the component of activity that the line shape distinguishes worst is also the one the brightness tracks worst. It is the same pattern as the flicker of granulation, which sets a floor on what a brightness measurement can reach: the star’s own surface is the noise source, and the shortest timescales and the smallest structures carry their signal into the measurement without a separate tag.

The disc’s own darkening towards the edge enters here too. A spot near the limb is seen foreshortened and against a limb that is darker than the middle in visible light, so it removes less light than the same spot at the centre would, and its velocity signal peaks not at the limb, where the rotational velocity is largest, but partway in, where velocity times visible light is largest. The rotation curve in the second figure has that shape: its extremes sit an eighth of a rotation either side of the spot’s crossing for the slower star and a twelfth for the faster, well inside the quarter-rotation at which the spot would reach the limb.

What the model leaves out #

The star is a sphere seen equator-on with a single circular region of fixed contrast, every line has the same Gaussian shape and depth, and limb darkening is linear. Real spots have dark centres and lighter surrounds; real faculae are brighter towards the limb than at the centre; real lines have asymmetric shapes from convection, the asymmetry that changes in active regions. The star’s inclination, set here to ninety degrees, changes the projected rotation speed and the visibility of high-latitude spots. The spectrograph is a Gaussian of fixed resolution and the noise is photon noise only. And the activity signal on a real star is the sum of many regions forming, evolving and decaying, not one region going round.

None of these changes the central arithmetic: a region on a slowly rotating star changes the shape of a line by a few per cent of the amount it changes the line’s position, because the line is barely wider than the region’s own contribution to it.

Still open: whether the activity of a quiet star can be removed at ten centimetres a second #

The instruments now operating reach a few tens of centimetres per second in their stability, and the next generation is designed for ten. At that level, every Sun-like star’s activity is the dominant signal, and the question is not whether the activity can be detected but whether it can be removed to a precision better than an Earth’s reflex motion. The shape of the lines gives a few per cent of the needed information on a slow rotator; the line-by-line differences give more; the star’s brightness variations and magnetic indicators, measured at the same times, give more again. Whether together they reach ten centimetres per second is being tested now on the one star whose surface can be imaged while its velocity is measured — the Sun itself, observed as a star by small telescopes feeding the same spectrographs, with its spots and faculae mapped from space at the same moments.