The metre per second that is not the star

The first rung of this ladder made a line shift into a speedometer: a wavelength displaced by a part in ten thousand is a velocity of thirty kilometres a second, and the arithmetic is one line long.

Reading that speedometer to a metre a second is a different undertaking, and almost none of it is about the Doppler effect. A metre a second is a shift of one part in $3\times10^8$ — a thousandth of a pixel on a large spectrograph, and a ten-thousandth of the width of the line being measured. Nothing about the measurement is direct, most of what is measured is not the star, and the thing that finally limits it is the star’s own surface.

What has to be subtracted #

The observer is on a rotating planet, orbiting a barycentre, in a solar system whose own centre of mass moves as Jupiter goes round. Each of those contributes.

The Earth’s orbit gives up to 29.79 km/s along a line of sight in the ecliptic, and the component along any particular line of sight is $V_\oplus\cos\beta$ times a sinusoid in the season, with the eccentricity adding a second harmonic. This is the term the figure shows, and it must be removed to about a centimetre a second — one part in three million — for a planet at the Earth’s own mass to be recoverable.

The Earth’s rotation adds up to 0.46 km/s, depending on latitude and hour angle. It changes over an exposure, so what is removed is the exposure-weighted mean, and computing that weight requires knowing how the flux arrived through the exposure — which is why precision spectrographs carry an exposure meter recording photon arrival rate through the integration.

The Earth’s own motion about the Earth–Moon barycentre contributes 12 m/s. That is smaller than most planetary signals and larger than many, and it is not optional.

The solar system barycentre is the reference the whole correction targets, and it moves relative to the Sun by up to 12 m/s as the giant planets orbit. Referring velocities to it rather than to the Sun is what makes a measurement in 1995 comparable with one in 2025. Beyond those sit corrections that would be absurd anywhere else in astronomy and are routine here. Light travel time across the Earth’s orbit, which shifts the epoch by up to 8.3 minutes and matters because the star’s own velocity is changing. The relativistic transformation between the observatory’s frame and the barycentric one, at the level of $v^2/c^2 \sim 10^{-8}$, or three metres a second. And the target’s own proper motion, which changes the direction of the line of sight and therefore which component of the Earth’s velocity projects onto it — a secular acceleration of a few centimetres a second per year for a nearby high-proper-motion star, small and cumulative.

Measuring a thousandth of a pixel #

Nothing above helps if the spectrograph cannot deliver the raw shift. Two problems have to be solved: knowing the wavelength scale, and beating the fact that a pixel is enormous.

The second is easier than it sounds and worth stating because it is often mistaken for a limitation. A spectral line spread over ten pixels, with a thousand photons in each, has a centroid determined far better than a pixel: the precision of a centroid goes as the line width divided by the signal-to-noise, times the square root of the number of lines used. A spectrum with 10,000 usable lines at signal-to-noise 200 gives a formal precision below a metre a second. The information is there in the photons; the difficulty is entirely in the calibration. The calibration problem is the hard one. A spectrograph’s wavelength scale drifts with temperature and air pressure — a millikelvin changes the refractive index of the air in the beam enough to move the spectrum by metres per second — and the drift has to be tracked continuously, not measured once.

Three answers have been used. An iodine cell placed in the beam superimposes thousands of narrow molecular lines on the star’s own spectrum, so that both go through the same optics and suffer the same drift; the disadvantages are that it absorbs half the light and only works over a hundred nanometres. A thorium–argon lamp fed through a second fibre gives a simultaneous reference, which is what HARPS used to reach a metre a second; the disadvantage is that the lamp’s lines are unevenly distributed, of wildly varying brightness, and age. A laser frequency comb produces a spectrum of evenly spaced lines whose frequencies are tied to an atomic clock, which is the current answer and is good to a few centimetres a second — the calibration is no longer the limit.

What is left is the star #

When the barycentric correction is right and the calibration is at centimetres per second, the residual scatter for a quiet solar-type star is around one metre a second. That is not instrumental. It is the star.

A star’s surface is convecting. Granules rise, are hotter and brighter than the intergranular lanes, and are moving upwards at a kilometre a second; the lanes are cooler, darker and descending. The surface-averaged velocity is therefore not zero — the rising material is brighter, so it is over-weighted — and the resulting convective blueshift is of order 300 m/s for the Sun. It is not a problem while it is constant, and it is not constant: it varies with magnetic activity, because magnetic fields suppress convection.

Three timescales of variability follow, and each is dealt with differently.

Oscillations, at five minutes for a solar-type star, with an amplitude of tens of centimetres a second in velocity. Averaged away by exposing for longer than the period, which is why precision velocity exposures are fifteen minutes rather than three.

Granulation, on timescales of minutes to hours, at a metre a second or so. Partly averaged by long exposures and by observing more than once a night.

Rotational modulation by spots and plage, at the rotation period, with an amplitude that reaches tens of metres a second for an active star. Not averageable at all, because it repeats — and that is the dangerous one, because a periodic signal at a period of days to weeks is precisely what a planet looks like.

The test that separates them #

The two are separable, and the test is a good one because it does not depend on any model of the star.

A planet moves the whole star. Every photon leaving every part of the surface is shifted by the same amount, so the line profile translates rigidly: its shape is untouched, and its shape includes the position of its own midpoint at every depth.

A spot changes the surface. It removes flux from one velocity, which distorts the profile — and a distorted profile has a moved centroid and a changed shape.

That is why a radial-velocity detection is confirmed by a bisector that did not move rather than by a period that repeated. The supporting checks are of the same kind: the signal should be achromatic, since a Doppler shift is a fractional change in wavelength and a spot’s effect is not; it should not correlate with a chromospheric activity index such as the core emission in the calcium H and K lines; and its period should not be the star’s rotation period or a simple fraction of it.

Several announced planets have been withdrawn on exactly these grounds. The most instructive is not a withdrawal but a reinterpretation: a signal at the rotation period that persists for a decade is not obviously a spot, because spots do not, and separating a long-lived active region from a planet in a spin–orbit resonance is a live problem rather than a solved one.

What the numbers actually are #

It is worth setting the scales side by side, because the ratios are what make the subject difficult and they are not intuitive.

The Earth’s orbital velocity along a typical line of sight: 29,000 m/s. The Earth’s rotation: 460. A hot Jupiter’s reflex on a solar-type star: 50. Jupiter’s reflex on the Sun: 12.5. A star’s granulation: 1. Instrumental stability with a frequency comb: 0.1. The Earth’s reflex on the Sun: 0.09.

The signal being sought is smaller than the correction being applied by five and a half orders of magnitude, and smaller than the star’s own surface motion by a factor of ten. The measurement is not photon-limited, not instrument-limited, and not calibration-limited; it is limited by the object.

That is an unusual position for an astronomical measurement to be in, and it changes what progress looks like. A larger telescope does not help. A better spectrograph does not help. What helps is either observing quieter stars, or modelling the activity well enough to subtract it — which means understanding stellar surfaces at a level nobody needed before.

The single line that is made out of thousands #

Nothing above says how the shift is actually extracted, and the answer explains both why the precision is achievable and where a whole class of systematic error enters.

A spectrum of a solar-type star contains tens of thousands of absorption lines, most of them blended with their neighbours, many of them weak enough to be invisible individually. Fitting them one at a time throws away most of the information and most of the signal-to-noise. What is done instead is a cross-correlation: the observed spectrum is compared against a mask — a list of line positions with weights, essentially a comb — at a range of trial velocities, and the correlation as a function of trial velocity is computed.

The result is a single function with a well-defined minimum, and it behaves like an average line profile with the combined depth of everything that went into it. Fitting a Gaussian to it returns a velocity, and the precision of that fit is the precision of ten thousand lines measured at once. The same function returns three further numbers as a free by-product: its depth, its width, and its bisector — which is where the activity test of the previous section is performed.

The mask is the weak point, and it is a weak point of a specific and unavoidable kind. A mask is built for a spectral type, so a star observed with a mask made for a hotter star is being correlated against lines whose relative strengths are wrong. The resulting cross-correlation function is subtly asymmetric, its minimum is displaced, and the displacement is of order tens of metres a second.

That does not matter for a planet search, because it is constant and every velocity is shifted by the same amount. It matters enormously the moment two instruments or two masks are compared, and it is the reason a radial velocity survey’s internal precision is a hundred times better than its agreement with anybody else’s numbers. Precision and accuracy are separated here by a factor that most of the field never has to confront, because the science is done entirely in differences.

The measurement’s two failure modes sit at opposite ends of the velocity scale, and both are worth drawing at the sizes they actually occur at.

The velocity a star does not have #

That separation has a limit that is worth following to its end, because it says something about what a radial velocity even is.

Suppose every instrumental and analysis difficulty were removed and the true position of every line were known to a centimetre a second. The velocity extracted would still not be the star’s motion, for two reasons that have nothing to do with motion at all.

Gravitational redshift. Light climbing out of the Sun’s potential well is redshifted by $GM/Rc^2$, which is 636 metres a second. Every line in the solar spectrum is displaced by that amount, in exactly the way a recession would displace them, and no measurement made on the spectrum alone can tell the difference.

Convective blueshift. The granulation asymmetry described above weights the rising bright material more heavily than the sinking dark material, displacing the lines the other way by a few hundred metres a second — and by a different few hundred for every line, since lines forming at different depths sample different convective velocities.

So the Sun’s apparent radial velocity, measured from its spectrum, is a few hundred metres a second and its actual motion towards or away from the observer is whatever it is. The two are related by a correction that must be modelled, star by star, and that is currently uncertain at the tens of metres a second level for anything other than the Sun.

The consequence is that the absolute radial velocity of a star is among the least well determined of its basic quantities, while the change in that velocity is among the best determined quantities in astronomy. A field that measures a difference to ten centimetres a second and the value itself to three hundred metres a second is unusual, and the asymmetry is entirely because the star’s surface contributes a constant that the orbit does not.

A practical consequence follows for anybody comparing published catalogues. A velocity quoted for a star by two surveys can differ by a hundred metres a second with neither being in error, because each is reporting the position of its own cross-correlation function against its own mask. What is comparable between them is the slope of a time series, never its offset, and a catalogue that combines archival velocities from different instruments has to fit an offset per instrument before it can fit anything physical.

The convention that has grown up around this is to publish velocities on an instrument’s own scale and to treat the zero point as a nuisance parameter, marginalised over rather than measured. It works, and it means the archive holds a great many velocities that are internally superb and mutually incomparable.

Where the model stops #

Three limits.

The barycentric correction is only as good as the ephemeris, the observatory position and the timing. Modern planetary ephemerides are good to centimetres; a station position to millimetres; a timestamp to a millisecond. Each is comfortably sufficient, which is worth saying because it locates the difficulty: the correction is not the limiting term and has not been for two decades.

The target’s coordinates enter the correction, and an error in them projects a fraction of the Earth’s velocity into the answer at the annual period. An arcsecond of coordinate error puts 0.14 m/s of spurious annual signal into a target near the ecliptic — comparable to an Earth-mass planet, and at a period that would be extremely hard to argue away.

And the bisector test fails at low signal-to-noise and for slow rotators. Measuring a bisector requires resolving the line profile, so it needs high resolution and high signal; and a star rotating at 2 km/s has a line barely wider than the instrumental profile, so a spot’s distortion is buried. The stars that are quietest, and therefore best for finding small planets, are exactly the stars on which the discriminating test is weakest.

And the diagnostic that separates a genuine shift from a surface effect, at a velocity where both are plausible.

Where this ladder goes next #

This rung establishes what a radial velocity measurement consists of and where its floor comes from.

Above it lies the modelling of stellar activity: Gaussian-process regression against activity indicators, line-by-line velocity analysis that exploits the different sensitivities of different lines to convection, and the attempt to reach ten centimetres a second on a star that moves at one metre.

Beside it lies the same measurement made differently. Astrometry measures the transverse component of the same reflex orbit, and the two together give the inclination and therefore the true mass rather than a lower bound; a transit gives the inclination directly when the geometry permits.

And below it, as a discipline rather than a technique, lies the habit this rung is really about: when a measurement’s limiting error is the object rather than the apparatus, the way forward is to find a second observable that responds differently to the signal and to the noise. The bisector is that observable here, and the pattern recurs wherever a systematic mimics a discovery.