How much of a line is not in its depth

The colour of a star is a thermometer and its lines are an inventory. A stellar spectrum’s dark lines say which elements are present. Turning that into how much of each element is present is a different problem, and the honest version of it starts with an inconvenient fact: a line’s depth is bounded and its cause is not.

A line can be at most completely black. Once its centre has reached zero, adding a hundred times more absorbing atoms cannot make the centre darker, because there is nothing left to remove. What does keep changing is the line’s width — and therefore its area.

The measurable is an area #

The quantity spectroscopy uses is the equivalent width: the width of a perfectly black rectangle that removes as much light as the line does. It is defined by an integral over the line profile,

$$W_\lambda = \int \left(1 - \frac{F_\lambda}{F_{\rm cont}}\right) d\lambda,$$

and its virtue is that it is independent of the instrument. A spectrograph with poor resolution smears a line out, making it shallower and wider, and leaves the area untouched. Depths are instrumental; areas are not. Equivalent widths are quoted in milliångströms, and a good measurement of a weak solar line reaches a few per cent.

Three regimes, and why the middle one is nearly blind #

The shape of the curve follows from the shape of the line profile, which has two parts with entirely different mathematics.

The core is a Gaussian, because atoms are moving. Thermal motion at a few thousand kelvin gives each atom a small Doppler shift, the shifts are distributed as a Maxwellian, and the resulting absorption profile is $e^{-u^2}$ in units of the Doppler width. Gaussians fall off very fast.

The wings are Lorentzian, because the atomic transition has a finite lifetime. Natural broadening from the uncertainty principle, and collisional broadening from perturbations by neighbouring particles, both produce a profile falling as $1/\Delta\nu^2$ — slowly, so that far from the core the Lorentzian is enormously larger than the Gaussian even though the Gaussian dominates near the centre.

The convolution of the two is the Voigt profile, and the ratio of the Lorentzian width to the Doppler width is a single parameter, usually called $a$. For a solar line it is of order $10^{-3}$ to $10^{-2}$.

Now the three regimes fall out.

When the line is weak, no photon is absorbed twice, and the area is simply the number of absorbers times the strength of the transition. $W \propto N$. The figure’s fitted slope on this stretch is 0.97, and the small departure from one is the beginning of the next regime.

When the core saturates, the centre is black and further absorbers can only widen the black region. But the Gaussian falls as $e^{-u^2}$, so pushing the black region outward by one Doppler width requires $e$ times as many atoms as the last one. The width grows as $\sqrt{\ln N}$, which is barely growth at all. The figure finds the shallowest stretch of its own curve and measures a slope of 0.14 there.

When the wings take over, the Lorentzian’s $1/\Delta\nu^2$ tail becomes the dominant absorber, and integrating it gives $W \propto \sqrt{a N}$: a slope of exactly one half, which the figure measures as 0.50.

The middle regime is the problem. A line on the flat part of the curve is very nearly useless as an abundance indicator. Its equivalent width can be measured to one per cent and the abundance inferred from it will still be uncertain by a factor of several, because the inverse of a nearly flat function is a nearly vertical one. Most of the strong, obvious, easily measured lines in a stellar spectrum are on the flat part.

What has to be known before an abundance comes out #

The curve of growth converts an equivalent width into a column of absorbers, and getting from that column to an abundance requires four things that are not observations of the star.

The oscillator strength. How strongly a given transition absorbs is an atomic property, measured in a laboratory or computed, and for many transitions it is the largest uncertainty in the whole chain. Abundances quoted to 0.05 dex frequently rest on $\log gf$ values good to 0.1.

The excitation and ionisation state. Only the atoms in the lower level of the transition absorb, and what fraction of the element is in that level depends on temperature through the Boltzmann distribution and on the ionisation state through the Saha equation. So an abundance requires a temperature, which requires a model.

The model atmosphere. The whole treatment above imagines a slab of absorbers in front of a continuum source. A real atmosphere has no such division: the line and the continuum form at different depths, over a range of depths, at different temperatures. The one-dimensional models used for decades are hydrostatic, plane-parallel and in local thermodynamic equilibrium, and all three are approximations of known inadequacy.

And the broadening. The Doppler width computed from the temperature alone does not fit real lines. It is systematically too small.

Microturbulence, which is a fitted parameter with a story attached #

That last point produced one of the more candid inventions in astrophysics.

If lines of the same species and the same excitation but different strengths are measured, they should all give the same abundance. Using thermal broadening alone, they do not: strong lines give systematically higher abundances than weak ones. The discrepancy is exactly what would happen if the lines were less saturated than the model says, which is what would happen if there were extra small-scale motion widening the core.

So an extra broadening term is added, called microturbulence, and its value — around 1 kilometre a second for the Sun — is chosen for each star by requiring that the abundance come out independent of line strength.

It is worth being clear about what that is. It is a free parameter fitted to remove a trend that the model cannot otherwise remove. There is a physical process behind it — the granulation of a convective atmosphere really does produce velocity fields of about that size, and the convection that produces them is computable — but the number used is not measured from the granulation; it is fitted from the spectrum. Three-dimensional hydrodynamic model atmospheres, which resolve the convection, reproduce the line profiles without any microturbulence parameter at all, which is the strongest evidence that the fudge was standing in for something real.

The one place the saturated regime is escaped #

The damping wings are usually a nuisance. In one important case they are the entire measurement, and it is worth seeing because it inverts everything above.

A quasar’s light passes through intervening gas clouds, each of which imprints an absorption line of neutral hydrogen at its own redshift. Where the column density is very high — above about $2\times10^{20}$ atoms per square centimetre — the Lyman-$\alpha$ line is deep into the damping regime, and its wings are broad enough to be resolved and fitted directly.

Because $W \propto \sqrt{N}$ there, the column density follows from the width of the wings with no dependence on temperature, no dependence on turbulence, and no dependence on how the gas is distributed along the sight line. These are the damped Lyman-$\alpha$ systems, and the neutral-hydrogen columns measured from them are good to a few per cent — better than almost any other column density in astronomy. The regime that destroys precision in a stellar atmosphere restores it here, because it is the regime in which the profile’s shape depends on one thing only.

Which lines to choose, which is the whole craft #

Everything above turns into one practical instruction, and it is the instruction that separates a usable abundance from a decorative one: measure weak lines.

A line on the linear part of the curve converts a one per cent error in equivalent width into a one per cent error in abundance. A line on the flat part converts the same one per cent into something between ten and fifty. So an analysis works with lines whose equivalent widths are small — for the Sun, below about 100 milliångströms, and preferably below 50 — and the difficulty is that weak lines are hard to measure at all: they are shallow, they sit close to the noise, and there are usually other lines blended with them.

That tension sets the design of every high-resolution spectrograph used for abundances. What is needed is not depth of exposure on bright objects but resolution — enough to separate a 30-milliångström line from its neighbours — combined with a very high signal-to-noise ratio, because a two per cent line has to be measured against a continuum defined to better than one. Instruments like HARPS and ESPRESSO reach resolving powers above 100,000 for exactly this reason, and the same instruments do double duty finding planets, because a reflex velocity is measured from the same lines by a shift rather than by an area.

There is a second instruction hidden in the shape of the curve, and it is the one that makes the whole method self-checking. Because the three regimes have different slopes, a set of lines of the same species spanning a range of strengths must all lie on one curve — and fitting them all at once determines both the abundance and the broadening, rather than assuming one to get the other. A single line gives an abundance with an assumption attached; twenty lines of one species give an abundance and a test.

What was actually measured, and what it cost #

The curve of growth was developed by Marcel Minnaert, Albrecht Unsöld and others in the 1930s, and it made quantitative stellar abundances possible for the first time. Cecilia Payne’s 1925 conclusion that stars are overwhelmingly hydrogen came from the Saha equation applied to line strengths, before this machinery existed; the curve of growth is what turned that qualitative result into a table of numbers.

The most instructive modern episode is the solar oxygen abundance. For decades the accepted value came from one-dimensional model atmospheres and sat near $\log \epsilon_{\rm O} = 8.9$. Three-dimensional hydrodynamic models, which resolve the granulation and remove the need for microturbulence, revised it downward to about 8.69 — a reduction of some 40 per cent in the amount of oxygen in the Sun.

That should have been a straightforward improvement, and it caused a decade-long problem. The Sun’s interior structure is known independently, from the sound speeds helioseismology measures, and the old abundances reproduced it beautifully. The new ones do not: they change the opacity in the radiative interior, which moves the base of the convection zone away from its measured depth. Two of the best measurements in solar physics disagree, and the disagreement is a spectroscopic revision that everyone believes is more correct.

What the picture cannot show #

A depth of formation. Every figure here treats the absorbers as a slab in front of a continuum. In a real atmosphere the line and the continuum form over overlapping ranges of depth, and a strong line’s core forms much higher than its wings — so a single line samples a range of temperatures, and the “temperature” used in the analysis is a weighted average whose weights depend on the line.

Departures from equilibrium. The Boltzmann and Saha distributions assume the gas is in local thermodynamic equilibrium. In the thin outer layers where strong lines form it is not, and the corrections — non-LTE effects — can reach several tenths of a dex for some species. They are computed rather than measured, from atomic data that is itself uncertain.

And the continuum. The equivalent width is defined relative to the continuum level, which in a crowded spectrum is not observable: there is no wavelength where no line absorbs. What is used is a fitted upper envelope, and in a metal-rich star or a fast rotator that envelope can sit several per cent below the true continuum, which biases every equivalent width in the same direction.

The spectrum that determines its own parameters #

The list of things that have to be known before an abundance comes out included a temperature, a surface gravity and a microturbulence — none of them observations of the star in the ordinary sense. In practice they are not taken from elsewhere. They are extracted from the same spectrum, by three requirements that have to hold simultaneously.

The first fixes the temperature. Lines of one species arising from different lower energy levels have populations set by the Boltzmann distribution, so getting the temperature wrong makes the abundances derived from high-excitation and low-excitation lines disagree systematically. Requiring no trend of abundance with excitation potential determines the temperature.

The second fixes the surface gravity. The Saha equation partitions an element between its neutral and singly ionised states according to the electron pressure, which depends on the gravity. So requiring that lines of the neutral species and lines of the ionised species of the same element give the same abundance determines the gravity — usually done with iron, which has hundreds of usable lines in both stages.

The third fixes the microturbulence, and it is the requirement already described: no trend of abundance with line strength.

Three conditions, three parameters, and they are solved together by iteration, because each one shifts the others. The result is a set of spectroscopic parameters that are internal to the analysis, and their virtue is exactly that: they are consistent with the lines being used, which is what an abundance needs.

Their weakness is the same thing. A systematic error in the model atmosphere shifts all three together in whatever combination keeps the three conditions satisfied, and nothing internal to the spectrum reveals it. Comparing spectroscopic temperatures against those from an angular diameter and a flux is the external check, and the two agree well for solar-type stars and diverge for giants and for metal-poor stars — which is where the model assumptions are worst.

Cancelling the atomic physics #

There is one arrangement that removes the largest uncertainty in the whole chain, and it is the reason a handful of stellar abundances are quoted to a hundredth of a dex while most are quoted to a tenth.

The oscillator strengths are the problem: they are laboratory or theoretical quantities, they carry errors of tens of per cent, and they enter every abundance directly. But they are properties of the atom rather than of the star, so they are identical for every star observed in the same line.

A differential analysis exploits that. Instead of computing an abundance from each line, compute the difference between the equivalent width in the target star and in a reference star — usually the Sun — line by line, using exactly the same line list, the same model atmosphere code and the same analysis. The oscillator strength cancels out of the difference, and so does a good deal of the model atmosphere’s systematic error, provided the two stars are similar enough that their atmospheres resemble each other.

The precision this reaches is remarkable: differences of 0.01 dex, one part in forty in an abundance ratio, for stars closely resembling the Sun.

What it has been used for is the detection of very small chemical anomalies. Solar twins — stars matching the Sun in temperature, gravity and metallicity — show abundance patterns that vary at the 0.02 dex level in ways correlated with the condensation temperature of the element, which has been interpreted as a signature of rocky material having been removed from, or added to, the star’s convective envelope.

The price is that the answer is a difference and not a value. A differential analysis says a star has 0.03 dex more iron than the Sun and says nothing about how much iron the Sun has, which is the quantity the absolute analyses supply badly.

The curve’s shape depends on the damping parameter and its three regimes are visible in the profiles, so both are worth reading at a second value.

The curve of growth is one of the few places in this subject where the difficulty is not the measurement but the interpretation: the equivalent width is easy and the abundance behind it requires everything else about the atmosphere to be known first.

Where the ladder goes next #

Two rungs. The first is the profile as a diagnostic of velocity rather than of number — rotational broadening, macroturbulence and the asymmetries granulation imprints on line bisectors, which are how a stellar surface’s motions are measured without resolving it. The second is spectral synthesis, which abandons the curve of growth entirely: instead of reducing a line to one number and inverting a curve, compute the whole spectrum from a model atmosphere and adjust the abundances until it matches, which is what every modern analysis does and which the curve of growth remains the way of understanding.