The lines a synthesis keeps pay for the ones it lacks

The classical way to measure how much of an element a star contains is to find its absorption lines one at a time, measure each one’s equivalent width — the area it removes from the spectrum — and turn that area into an abundance through the curve of growth. The method is transparent and it has two weaknesses. It needs clean lines, isolated from their neighbours, and in a cool star’s crowded spectrum there are few; and it needs a continuum to measure the area against, which in the same crowded spectrum is never seen, only inferred from the highest points, which are themselves pulled down by weak lines.

Spectrum synthesis answers both. Instead of measuring lines, it computes the whole spectrum of a region — every line in it, blended as they are blended in the star, from a model atmosphere and a list of lines with their wavelengths, excitation energies and oscillator strengths — and adjusts the abundances until the computed spectrum matches the observed one pixel by pixel. Blends are no longer a problem, because the synthesis contains them. The continuum can be fitted along with everything else. And every pixel carries information, so a region of a few tens of ångströms with a hundred lines gives an abundance whose random error is far smaller than any single line’s. Nearly every large spectroscopic survey now measures its stars this way.

The synthesis is only as complete as its line list, and line lists are not complete. What happens to an abundance when the list lacks some of the lines in the star is not a random error that the many pixels average away. It is a systematic one, with a sign, and its sign depends on a choice the analyst makes about the continuum.

A forest with its weak lines missing #

The figures use a simulated metal-line forest: a ten-ångström window near 5000 Å holding 160 lines at random positions, with strengths drawn so that most are weak and a few are strong, as in real spectra of cool stars, where the number of lines rises steeply towards the weak end. Each line is given the same Doppler width, the spectrum is the exponential of their summed optical depths — a single absorbing layer rather than a model atmosphere, which is enough to give the right amount of blending — and it is observed through an instrument of a given resolving power, with noise. The synthesis is given a line list that contains only the lines above some strength, as real lists effectively do: strong lines of common elements are measured in the laboratory, while weak lines, lines of rare species and lines between highly excited levels are often missing, or present with wavelengths and strengths that are guesses. It fits one abundance, scaling every listed line together, and either holds the continuum at its true level or fits it too.

With the continuum held at its true level, the fit returns an abundance 0.19 dex too high — half as much again as the true value. The reason is visible in the spectrum. Wherever a missing line is blended with a listed one, the listed line appears deeper than the list says it should, and the only way the synthesis can deepen it is to raise the abundance. Wherever a missing line stands alone, the synthesis cannot reproduce it at all, and the residual shows a dip that no parameter fills. The fit is a compromise between those two: it makes the listed lines stronger to absorb as much of the missing absorption as it can, and leaves the rest as residuals.

How much the missing lines cost #

The error rises with the share of the absorption that the missing lines carry, and the rise depends strongly on the resolving power. At a resolving power of a hundred thousand the lines are resolved, most of the missing ones stand clear of the listed ones, and even with half the absorption missing the error is only 0.06 dex: the fit leaves the missing lines as residuals rather than attributing them. At five thousand every line is blended with every other within an ångström, the missing absorption is inseparable from the listed lines’ wings, and the same incompleteness produces an error of more than 0.7 dex — a factor of five in the abundance.

A share of a third or a half of the absorption in missing lines sounds extreme, and in a well-studied star in the visible part of the spectrum it is. It is not extreme in the blue and ultraviolet, where the density of lines rises and laboratory data thin out; in the spectra of cool stars dominated by molecules, whose millions of lines are known largely from calculation; or in the near-infrared, where the spectroscopic surveys of the last decade have worked and where many lines were identified only once those surveys had observed them. The line lists for those regions have been built partly by fitting the very spectra they are used to analyse.

Resolution decides where the absorption goes #

The two treatments of the continuum give errors of opposite sign. With the continuum held, the missing absorption goes into the listed lines, and the abundance is too high, by an amount that falls steadily as the resolving power rises and the missing lines separate out. With the continuum fitted, the synthesis has a second way to account for absorption it cannot explain line by line: it lowers the continuum. A forest of weak missing lines depresses the spectrum broadly, and a lower continuum mimics that depression cheaply. The fit lowers it by more than the missing lines justify, because a lower continuum also makes every listed line shallower relative to it, and then compensates by lowering the abundance. The result is an error of the opposite sign — the abundance too low — and smaller, but no less systematic.

That reversal connects the synthesis to the older method directly. The pseudo-continuum that makes equivalent widths of isolated lines come out too small is the same depression, and a synthesis that fits its own continuum inherits it. What the synthesis adds is the choice: a synthesis whose continuum is set by the model atmosphere and a flux calibration, rather than by the data, pushes the error the other way. Neither choice removes it; each chooses which parameter absorbs it.

Errors that average away, and one that does not #

The synthesis’s great advantage over line-by-line analysis is statistical. Every line’s oscillator strength — the atomic quantity that says how strongly it absorbs — is known only to some precision, often a tenth of a dex for lines measured in the laboratory and much worse for lines computed from theory. A single line’s abundance inherits its oscillator strength’s error in full. A synthesis of a hundred lines inherits the average of a hundred errors, which is ten times smaller if they are independent.

The figure separates the two kinds of error. Oscillator strengths wrong at random by a tenth of a dex give a single window an abundance error of under two hundredths of a dex — the synthesis of many lines has already averaged most of the atomic uncertainty away — and fitting many windows together reduces it further. Missing weak lines give an error of 0.16 dex in one window and the same 0.16 dex in thirty-two. Every window is missing the same kind of lines for the same reason, so every window is biased in the same direction, and averaging gives the bias to more decimal places. The more lines a synthesis uses, the smaller its random error becomes and the more its total error is dominated by what it does not know.

Why the missing lines are the weak ones #

The number of lines of each strength in a stellar spectrum rises steeply towards the weak end: for every line deep enough to measure by eye there are ten shallower ones and a hundred shallower still. Most lines are weak, and a line list is incomplete almost entirely at that end, for a reason that has nothing to do with astronomy. Laboratory measurements of oscillator strengths need a line strong enough to measure in a laboratory source, and the weak lines in a star are weak because they come from rare species, from levels high above the ground state, or from transitions that are nearly forbidden — all of which are also weak in the laboratory. Theoretical calculations fill some of the gap, with errors that grow as the lines get weaker.

What matters for an abundance is not how many lines are missing but how much absorption they carry, and that is where the steep count bites. A single weak line removes little light, but there are so many of them that together they can remove as much as the few strong lines do. In the simulated forest the 137 weakest lines — six in seven of the lines — carry a third of the absorption. In the blue and ultraviolet spectrum of a cool star, where the lines are denser still, the share is higher; the continuum itself is set there by very few absorbers, and the forest of weak metal lines on top of it is what makes that part of the spectrum so hard to model.

A famous blend #

The clearest demonstration that a missing contribution in a synthesis can move a fundamental number is the oxygen abundance of the Sun. For decades one of its principal indicators was a single forbidden line of neutral oxygen at 6300 Å — weak, formed in conditions close to equilibrium, and therefore thought to be one of the most reliable. The line turned out to be blended with a weaker line of nickel almost exactly at the same wavelength, whose contribution had been underestimated. Allowing for it properly, together with a three-dimensional model of the solar surface in place of a one-dimensional one, lowered the oxygen abundance derived from that line by a substantial fraction, and the revision — along with others made for similar reasons — lowered the Sun’s total abundance of heavy elements.

The consequence has not gone away. A Sun with less oxygen and fewer heavy elements has a lower opacity in its interior, and models of the solar interior built with the revised abundances no longer match the sound speeds measured by helioseismology as well as the old ones did. Either the abundances are still wrong, or the opacities computed for the interior are, or something else is missing from the models. It is a problem that started with a line in a synthesis being charged for absorption that belonged to another line.

Tuning the list against the Sun #

The standard remedy for incompleteness has an instructive side effect. A line list can be calibrated against a star whose spectrum and composition are well known — the Sun, or a well-studied giant for surveys of giants — by adjusting each line’s oscillator strength until the synthesis of that star matches its observed spectrum. Such astrophysical oscillator strengths fix the wrong laboratory values and, in doing so, also absorb the missing lines: a listed line blended with a missing one acquires an oscillator strength strong enough to account for both. For stars like the calibrator, the result is excellent, because the missing lines are present with the same strengths, and the absorbed error cancels. That is the logic of differential analysis, applied at the level of the line list.

For stars unlike the calibrator it can be worse than no calibration. A missing line of a molecule or of a highly excited level changes strength with temperature quite differently from the listed line whose oscillator strength absorbed it. In a star a thousand degrees cooler than the Sun, the missing molecular line may be ten times stronger, while the listed atomic line has been given the Sun’s version of the blend. The error that cancelled at the Sun’s temperature reappears elsewhere, with a sign and size set by how the missing species’ lines scale — and it appears as a trend of abundance with temperature, which is one of the persistent, poorly explained systematics in large survey catalogues.

The same bookkeeping in a velocity #

Line lists are also used to measure velocities: a spectrum is cross-correlated with a mask of line positions, and the shift that maximises the correlation is the star’s velocity. A missing line matters much less there, because a velocity is measured from where lines are rather than how deep they are, and a weak line missing from the mask costs only its own small contribution to the precision. A blend matters more. A listed line blended with an unlisted one has its apparent centre pulled towards its neighbour, and if the neighbour’s strength changes — with the star’s activity, with its temperature across a spot — the pull changes, and the measured velocity moves with no motion of the star. A velocity is an average of lines that disagree, and the disagreement is partly the unmodelled blends; the mask designers for the most precise velocity instruments select lines partly by how isolated they are, and discard those whose neighbours are unknown.

The residuals as a measurement #

The same incompleteness that biases the abundance also leaves a record. At high resolution and high signal, the missing lines that stand clear of listed ones appear in the residuals as dips the synthesis cannot fill, and their number grows with the signal-to-noise ratio: at fifty, a fifth of the missing lines are visible; at sixteen hundred, four-fifths. Residuals that recur at the same wavelengths in the spectra of many stars mark lines nobody has identified, and how their strengths change with the stars’ temperatures and gravities says which element and ionisation stage produce them. A great deal of the progress in line lists over the last two decades has come from exactly this: large surveys producing thousands of high-quality spectra, whose synthesis residuals were stacked to find, identify and calibrate the missing lines.

That turns the incompleteness into a measurable quantity for every spectrum. The residuals of a synthesis, summed over a region, measure how much absorption the list fails to account for; if they are large, the abundance from that region is suspect by an amount the figures above can estimate. A synthesis that reports its residuals alongside its abundances is reporting its own systematic error, and one that does not is hiding it in its continuum or its abundance.

What the simulation leaves out #

The forest is a single layer with one Doppler width, not a model atmosphere; real lines form at different depths, have wings shaped by collisions, and respond to temperature and pressure through the model. None of that changes the accounting drawn here — missing absorption must be put somewhere — but it changes the numbers, because a strong line’s wings interact with nearby weak lines differently from a Gaussian core. The missing lines are the weakest ones, which is the common case; lines missing because their species is absent from the list, or because their wavelengths are wrong by more than a line width, bias things differently, and a line with the right strength at the wrong wavelength is in some ways worse than a missing one, because it removes flux where none should be removed and leaves the true position unexplained.

The simulation also fits a single abundance for all listed lines. A real synthesis fits several elements, each from its own lines, and the missing absorption is then shared among them in proportion to how much each element’s lines overlap the missing ones — which means that elements with few lines, measured from a crowded region, carry the largest share of the error. And it takes the model atmosphere to be right. A synthesis with an incomplete list is compared with an observed spectrum through a model whose own opacity is computed from the same incomplete list, so the model’s temperature structure is subtly wrong too — the opacity that the missing lines should have added is missing from the atmosphere as well as from the spectrum.

Still open: how complete a list has to be #

The question the synthesis method has never fully answered is how complete a line list must be for a given precision. For abundances to a hundredth of a dex — the precision now claimed for differential analyses of stars like the Sun, and needed to see the chemical signatures of planet formation in their atmospheres — the figures suggest that the missing absorption must be below a few per cent of the total in the regions used, at high resolution. Whether the line lists in use meet that, region by region and star type by star type, is not known in general, because the only way to know is to find the missing lines, which requires the high-resolution, high-signal spectra that most surveys do not take. The lists are being completed star by star, residual by residual, and the precision of every abundance measured by synthesis rests on how far that work has got in the wavelengths it used.