The photons on the sides of the lines

A shift in a spectral line is a speedometer, and the best spectrographs now read it to better than a metre a second — a shift of three parts in a thousand million in the wavelength of a line, a hundredth of a pixel on the detector. They do it by averaging thousands of lines, each of which disagrees slightly with the others about the star’s motion for reasons of its own. That is the systematic part of the problem. This essay is about the other part, which sets a floor no amount of care can go below: how precisely a spectrum can measure a velocity when the only noise is the counting of photons.

The answer is not simply “more photons, better precision”, though that is true. It depends on where in the spectrum the photons fall, and most of them fall where they are useless. A velocity is measured only by the photons on the sides of the absorption lines, where the spectrum changes steeply with wavelength and a small shift changes the count in a pixel measurably. The continuum between the lines, which holds most of the light, measures nothing about velocity at all, because shifting a flat spectrum leaves it unchanged.

Information in a slope #

The calculation, due in this form to Pierre Connes in 1985 and made the standard for spectrograph design by François Bouchy, Francesco Pepe and Didier Queloz in 2001, is short.

A pixel holds $A$ photons, counted with the Poisson uncertainty $\sqrt{A}$ that counting always carries. If the star’s velocity changes by $\delta v$, the spectrum shifts, and the count in the pixel changes by $(dA/dv)\,\delta v$. The shift is detectable in that pixel when the change exceeds the noise, so the pixel alone measures the velocity to $\sqrt{A}/|dA/dv|$. Independent pixels combine in inverse quadrature, which gives the whole spectrum’s precision:

$$\sigma_v = \left[\sum_i \frac{(dA_i/dv)^2}{A_i}\right]^{-1/2}.$$

Each pixel contributes its squared slope divided by its count. In the continuum the slope is zero, and the contribution is zero however many photons the pixel holds. At the bottom of a line the slope is also zero. On the flanks the slope is large, and it is largest where the line is deepest and narrowest.

It is conventional to fold everything except the photon count into one number. Writing $\sigma_v = c/(Q\sqrt{N})$, with $N$ the total number of photons, defines the quality factor $Q$ — a property of the spectrum’s shape alone, independent of how bright the star is or how long it was observed. A cool star with many deep lines has a high $Q$; a hot star with few lines, or a fast rotator with broad ones, a low $Q$. Two stars observed to the same signal-to-noise can differ in achievable precision by more than an order of magnitude, entirely through $Q$.

Depth, not strength #

The information calculation cares about a line’s shape in a way that its usual measure does not. A line is conventionally described by its equivalent width, the amount of light it removes, and a line’s equivalent width is mostly not in its depth: as a line strengthens it saturates, its core stops deepening, and it grows by spreading its wings.

For velocity, the core is what matters. A saturated line with a flat, dark core and broad wings has most of its equivalent width in the wings, where the slope is gentle, and its information is concentrated in the two steep walls of the core. A weak line, unsaturated, is a shallow dip with gentle sides and little information. The most valuable lines are the moderately strong ones — deep enough that their walls are steep, not so strong that their extra absorption has moved into the wings. Two lines of equal equivalent width can differ in velocity information by a large factor, and a list of lines chosen for chemical analysis, which values lines for their sensitivity to abundance, is not the list a velocity measurement would choose.

That difference matters for how line lists are built. The masks used for cross-correlation were long constructed by selecting lines by depth in a spectrum of a reference star, which approximates the information weighting; the optimal method weights every pixel by its measured slope and needs no list at all.

Rotation spreads the information thin #

A rotating star’s surface moves towards the observer on one side and away on the other, and every line is broadened by the spread of velocities across the disc. The broadening conserves the line’s equivalent width — the amount of light it removes — so a broader line is also a shallower one.

The three-halves law is the reason planet searches choose their stars so carefully. The Sun rotates at about two kilometres a second at its equator; stars much hotter than about 6,500 kelvin generally rotate at tens of kilometres a second, because they lack the deep convective envelopes whose magnetic winds brake the rotation of cooler stars. An F star rotating at 20 kilometres a second has lines about five times broader than the Sun’s and gives velocities about ten times less precise from the same photons, and an A star rotating at 100 kilometres a second is essentially unusable. The stars round which planets have been found by their wobble are overwhelmingly the slow rotators, and that is a property of the instrument’s information budget as much as of the planets.

Cool stars have more to say #

The number of lines matters as much as their width, and it depends strongly on temperature.

The lines of a stellar spectrum come from atoms, ions and molecules absorbing at their own wavelengths, and which absorbers are present depends on the temperature. A hot A star’s atmosphere is mostly ionised hydrogen, and its visible spectrum is dominated by a handful of broad hydrogen lines. A Sun-like star adds thousands of lines from neutral and singly ionised metals. A K dwarf adds more still, and in an M dwarf molecules — titanium oxide, water, iron hydride — form and fill the spectrum with bands of tens of thousands of lines. What a spectrum says about composition is carried by the same lines that carry the velocity.

So Q rises steeply towards cooler stars, and a K dwarf observed to the same signal-to-noise as a G dwarf gives a better velocity. The M dwarfs are richer still but faint in the visible, where most spectrographs work; their lines are densest in the red and near infrared, where they emit most of their light, and the newer spectrographs built for M-dwarf planet searches work there for that reason.

The trade between the two halves of the spectrum is itself an information calculation. In the blue a Sun-like star has the most lines per nanometre and the fewest photons; in the red, fewer lines and more photons. For a Sun-like star the product of the two favours the blue and green, where the planet-search spectrographs were first built. For an M dwarf, which emits ten or a hundred times more light in the near infrared than in the blue and has deep molecular bands there, the product favours the red, and instruments such as CARMENES, SPIRou and NIRPS extend the velocity measurement to one or two microns. Where a spectrograph should look is decided by where the slope-weighted photons of its target stars are, and for the commonest stars in the Galaxy that is not where the eye would look.

What resolution buys #

A spectrograph’s resolving power sets how finely the spectrum is sampled, and it is tempting to assume that finer is always better. At a fixed number of photons it is not.

The flattening is where the stars’ own lines take over from the instrument. A star’s lines have an intrinsic width of a few kilometres a second from thermal motion and turbulence in its atmosphere, and once the spectrograph resolves that width, there is no steeper slope to find. The planet-search spectrographs — HARPS, ESPRESSO, the instruments that followed — work at resolving powers between about 100,000 and 140,000, which is the point at which a slowly rotating Sun-like star’s lines are just resolved. Higher resolution costs throughput, detector area and stability, and buys almost nothing.

What a mask throws away #

Practical velocity measurements do not use the optimal weighting directly. The traditional method cross-correlates the spectrum with a mask — a template of line positions, each an open slot — and measures the velocity from the peak of the resulting correlation function.

A binary mask gives every line equal weight, or weights set by an approximate depth, and it samples each line through a slot of fixed width rather than weighting each pixel by its slope. That wastes information: the pixels at the bottom of a line, which carry none, are counted as heavily as the flanks, and lines of very different information content are treated alike. Measured against the photon limit, a simple mask typically reaches within a few tens of per cent of the optimal precision — close, but not at it. The modern alternative uses the star’s own spectrum as the template: all the observations of a star are co-added into a high signal-to-noise model of its spectrum, and each new observation is fitted to that model pixel by pixel, which weights every pixel by its actual slope and reaches the photon limit. The same approach can fit each line separately, and that is what makes it possible to see the lines disagreeing with one another.

The same arithmetic, measuring other things #

The calculation is not special to velocities. It is the general rule for how precisely a shift of any known shape can be measured in the presence of counting noise, and the same form appears wherever astronomy measures a position, a time or a displacement.

The position of a star on a detector is measured by the shift of its image, and the precision is set by the slope of the image’s profile: a sharp image gives a precise position, a blurred one a poor one, and the continuum of sky around it contributes noise but no information. The Gaia spacecraft’s positions, which reach tens of microarcseconds, are this calculation applied to the one-dimensional profile of each star as it crosses the focal plane. The arrival time of a pulse is measured by the shift of its profile in time, and a millisecond pulsar is a good clock because its pulse is narrow and bright — the same steep-slope argument, applied to a profile in time rather than in wavelength. And the time of a transit is measured by the ingress and egress, the only parts of the light curve where the brightness is changing; the flat bottom and the flat out-of-transit baseline set the depth and contribute nothing to the timing.

In every case the information is in the slopes, and in every case the design of the measurement is a matter of putting the photons where the slopes are — narrow pulses, sharp images, deep lines, short ingresses — rather than of collecting more photons where the signal is flat. The general statement is the Fisher information of a counting measurement, and its bound on the achievable error, the Cramér–Rao bound, is the photon limit in all of them.

The photons an Earth twin needs #

The photon limit converts directly into a number of photons, and for the hardest target in the field the number is large.

The reflex motion that an Earth-mass planet on a one-year orbit induces in a Sun-like star is nine centimetres a second, and detecting it at several standard deviations over many epochs needs per-epoch precision of a few tens of centimetres a second at worst. A Sun-like star observed across the visible spectrum has a quality factor of order ten to twenty thousand, and $\sigma_v = c/(Q\sqrt{N})$ then requires of order $10^{10}$ to $10^{11}$ detected photons per epoch to reach ten centimetres a second. That is achievable only for the brightest Sun-like stars, on telescopes of eight metres or larger, with exposures of tens of minutes.

The photon budget is therefore not the binding constraint for the nearest stars, and has not been for a decade; for them the limit is the star’s own surface. For stars a few magnitudes fainter it is, and the number of stars around which an Earth twin can be sought by its wobble is set by how many are bright enough to supply the photons, which is a few dozen. The mass the wobble gives is a lower bound even then, and every one of those few dozen stars has to be slowly rotating and cool enough to have the lines — the same selection the curves above describe.

The photon limit and the real limit #

The photon limit is a floor, not a forecast. For a bright Sun-like star observed for fifteen minutes with a four-metre telescope, it is a few tens of centimetres a second; the velocities actually achieved are around a metre a second.

The gap between the photon limit and the achieved precision is filled by three kinds of error. The instrument’s wavelength calibration must be stable to a few centimetres a second over years, which requires spectrographs in vacuum tanks at constant temperature, fed by optical fibres that scramble the starlight, and calibrated against laser frequency combs whose lines are spaced by an exact radio frequency. The Earth’s atmosphere imprints its own absorption lines and must be modelled away. And the star itself moves its lines without moving: convection, spots and oscillations shift and distort the lines by amounts of order a metre a second, and those shifts, unlike photon noise, do not average down with more light.

The last of these is now the binding limit for Earth-mass planets round Sun-like stars, whose reflex velocity is nine centimetres a second. It is a limit set by the star’s own surface, and the photon calculation says something useful about it too: the lines that carry the most velocity information are the deep, narrow ones, and the deep lines form highest in the atmosphere, where convection has died away, while the weak lines form deeper, among the moving granules. A velocity weighted towards the information is weighted towards the lines least disturbed by the star’s surface — a coincidence of physics that the optimal weighting exploits for free.

What the synthetic spectra simplify #

The lines are Gaussian and placed at random. Real lines have Voigt profiles with damping wings, their depths follow a curve of growth rather than a uniform distribution, and their positions cluster by element and molecule. The precisions drawn are representative of the scaling, and within a factor of about two of the published values for real spectral types.

Rotation is a Gaussian broadening that conserves equivalent width. The true rotational kernel has sharper edges than a Gaussian, and its effect on the slopes is somewhat different in detail; the three-halves law holds for both.

The noise is pure photon noise. Detector read noise, background light and the noise of the reference spectrum all add to it, and for faint stars they dominate; the figures describe bright stars.

And the spectrum spans 100 nanometres at 550. Real spectrographs cover 300 to 400 nanometres or more, and the precision scales as the square root of the coverage if the information density is uniform, which it is not: the blue carries more lines per nanometre and fewer photons, the red fewer lines and more photons, and where the balance lies depends on the star.

Still open: whether the information in a spectrum can separate the star from the planet #

The photon limit treats every shift of the spectrum as a velocity. The star’s surface produces changes that are not pure shifts — lines that deepen, narrow, or develop asymmetries as spots and faculae rotate across the disc — and a pure Doppler shift moves every line by the same amount in velocity while a surface effect does not. In principle, the same information calculation that sets the photon limit can be extended to separate the two: to ask how precisely a spectrum can measure a pure shift in the presence of a distortion of known form, and which lines and which pixels carry the most information about each. That turns the problem of stellar activity from one of averaging into one of estimation, and the methods now being developed fit each observed spectrum as a shifted template plus a small set of distortion shapes learned from the star’s own history. How much of the gap between a metre a second and nine centimetres they can close depends on how many distinct ways a stellar surface can change its spectrum — which is a question about the star, asked through its photons.