Composition, read from what is missing

Auguste Comte wrote in 1835 that of the stars, humanity would never know their chemical composition. The statement is usually quoted to make him look foolish, and it deserves better than that: it was a reasonable inference from everything then known. Composition is a property of matter, matter cannot be brought back, and light was not thought to carry that kind of information.

Twenty-four years later Kirchhoff and Bunsen showed that it does — and that the information is carried by the wavelengths where light fails to arrive.

Why the lines are dark #

An atom absorbs at exactly the wavelengths whose photon energies match a difference between two of its energy levels, and those level differences are fixed by the atom’s structure. The set of wavelengths is a signature, and no two elements share one.

In a star, the continuum comes from the dense layers below, where the gas is opaque and radiates thermally. Above them sits a cooler, thinner atmosphere. A photon of the right wavelength for some transition in that atmosphere is absorbed and re-emitted in a random direction, so most of the ones headed toward the observer are scattered out of the beam. The result is a dip at that wavelength: not an absence of photons but a deficit.

Two facts follow, and neither is obvious.

The gas must be cooler than the source behind it. Kirchhoff’s laws state the arrangement: a hot dense body gives a continuum, a hot thin gas gives bright emission lines, and a cool thin gas in front of a continuum gives dark absorption lines. The same gas produces bright lines or dark ones depending only on what is behind it, which is why a nebula seen against empty sky shows emission and a stellar atmosphere shows absorption.

The lines are not black. A strong line removes most but never all of the light at its centre, because the absorbing layer has a finite depth and the wings of the line profile come from progressively deeper, hotter material. The residual intensity at line centre is itself a measurement — of the temperature at the height where that line forms.

The trap: line strength is not abundance #

The obvious reading of a spectrum is that a strong line means a lot of the element. It is wrong, and getting it wrong held the subject up for two decades.

A line at a given wavelength is produced by a transition from a particular energy level of a particular ionisation state. For hydrogen’s visible Balmer lines, the transition starts from the first excited level — an electron already in $n = 2$. In a cool star almost every hydrogen atom is in the ground state, so almost none can absorb at those wavelengths and the lines are weak. In a hot star the hydrogen is ionised, so there is no bound electron at all and the lines are weak again. Only in between, around 9,500 K, is a useful fraction of the hydrogen both excited and un-ionised.

Now hold the composition fixed and move the temperature in each direction, which is the experiment this essay is about. Nothing below changes what the gas is made of; every difference between the three spectra is a difference in which states its electrons occupy.

The same inversion happens again at the other end, and for the opposite reason. Where a cool star’s metals absorb because their electrons are still bound, a hot star’s have lost the electrons that would do the absorbing.

Those three figures have identical compositions. Everything that differs between them is temperature, acting through which states the atoms are in.

Payne’s resolution #

The consequence for classification is the central episode in the history of the subject.

Stellar spectra were sorted at Harvard in the 1890s and 1900s, mostly by Annie Jump Cannon, who classified something over 350,000 of them by eye. The classes were originally lettered A, B, C… by the strength of the hydrogen lines, and had to be reordered and pruned when it became clear the alphabetical sequence was not physical. What emerged was the sequence O B A F G K M, which is continuous, one-dimensional, and was of unknown meaning.

The prevailing view was that it was a composition sequence: A stars had more hydrogen, K stars more metals. It was a natural reading and it was wrong.

In 1925 Cecilia Payne, in her doctoral thesis, applied Meghnad Saha’s ionisation equation — which gives the fraction of a gas in each ionisation state as a function of temperature and pressure — to the Harvard spectra. The result was unambiguous. The whole sequence was a temperature sequence, running from about 40,000 K at O to about 3,000 K at M, and the enormous variation in line strengths was almost entirely a variation in excitation and ionisation. It is the same axis colour supplies from two filters, arrived at from completely different data.

The same calculation gave a second result. Working backwards from the line strengths to the actual abundances, she found that hydrogen and helium were more abundant than everything else by four orders of magnitude — that stars are, essentially, hydrogen.

That conclusion was rejected. Henry Norris Russell, reviewing the thesis, told her it was “clearly impossible”, and the published version contains a sentence saying the hydrogen and helium abundances are “almost certainly not real”. They were real. Russell reached the same conclusion himself four years later by a different route and is often credited with it. Payne’s thesis has been called the most brilliant ever written in astronomy, and the retraction sits in the middle of it.

The second axis, and why one was not enough #

The spectral sequence is one-dimensional, and one dimension is not enough to describe a star. The correction is worth having because it is the point at which spectroscopy and the diagram that sorted the stars become the same subject.

Two stars can share a temperature and be nothing alike. A red giant and a red dwarf both sit near 3,500 K, both show the same species, and both are class M — but one is a hundred times the other’s radius and a hundred thousand times its luminosity. The spectral class cannot tell them apart, because the class is a temperature and they have the same temperature.

What does tell them apart is the pressure in the atmosphere. A giant’s surface gravity is a thousandth of a dwarf’s, so its outer layers are enormously less compressed. Collisions between atoms are correspondingly rarer, and since collisions perturb an atom’s energy levels and smear its transitions, the lines in a giant are narrower than the same lines in a dwarf at the same temperature. The effect is pressure broadening, it is largest in the strong hydrogen lines, and it is measurable from the shape of a line rather than its position or depth.

That gives the second axis: the luminosity class, from I for supergiants through V for main-sequence dwarfs, assigned by measuring the widths of a few lines. The Sun is G2V; Betelgeuse is M2I; Sirius B is a white dwarf and sits off the scheme entirely. Two letters and a numeral place a star on both axes of the diagram — and neither requires knowing its distance, which is the whole reason the classification survives.

It is worth noticing what has happened. A spectrum contains at least four independent measurements, carried by four different properties of the same lines: their positions give velocity, their identities give composition, their depths give temperature, and their widths give pressure. Nothing else in astronomy delivers four unrelated quantities from one observation.

What was actually measured #

Turning a line’s depth into an abundance is a chain with several links, and each has to be stated because published abundances are quoted to two decimal places and the reader should know what supports them.

The measured quantity is the equivalent width: the width, in wavelength, of a completely black rectangle removing the same total amount of light as the line does. It is a robust number — insensitive to the instrument’s resolution, which smears a line without changing its area — and it is what appears in any abundance analysis.

Converting it into a number of atoms requires a model. Specifically, it requires: the star’s effective temperature and surface gravity, obtained from its colour and from the pressure-broadened wings of other lines; a model atmosphere giving temperature and pressure as functions of depth; the transition’s oscillator strength, a quantum-mechanical quantity measured in laboratories or computed; and the relation between abundance and equivalent width, which is not linear.

That last point is the curve of growth, and it has three regimes. A weak line grows in proportion to abundance. A saturated line — one whose centre has absorbed nearly everything available — barely grows at all as more atoms are added, so its equivalent width is a poor abundance indicator. A very strong line grows again, as the square root, because the damping wings take over. Choosing lines on the linear part of the curve is most of the craft of abundance analysis.

The precisions that result are worth being frank about. A differential analysis — comparing a star against the Sun, using the same lines and the same method — reaches about 0.02 dex, five percent. An absolute analysis, which needs the oscillator strengths to be right in an absolute sense, is good to perhaps 0.1 dex, twenty-five percent. The gap between those two numbers is the reason nearly every abundance in the literature is quoted relative to the Sun.

The convention itself encodes the difficulty. Metallicity is written $[\text{Fe/H}]$, a logarithm of the iron-to-hydrogen ratio relative to the Sun’s, so that the Sun is 0 by definition and a star with a tenth the Sun’s iron is $-1$. Everything is anchored to one star because the anchor cancels most of the systematics.

The generalisation: the same lines, everywhere else #

The technique was invented for the Sun and applies without modification to anything that emits light through gas, which turns out to be almost everything.

Interstellar gas. A star seen through a cloud shows the cloud’s absorption lines superimposed on its own, distinguishable because they are far narrower — the cloud is cold, so thermal broadening is small — and at a different Doppler shift. A single line of sight can show a dozen separate clouds at a dozen velocities, and the composition and temperature of each is measured separately.

Planetary atmospheres. A transiting exoplanet’s atmosphere absorbs a little extra light at its own wavelengths during transit, so subtracting the in-transit spectrum from the out-of-transit one gives the atmosphere’s spectrum. The effect is a few parts in $10^{4}$, and water, carbon dioxide, methane and sodium have all been detected this way.

The intergalactic medium. A distant quasar’s spectrum shows a dense forest of hydrogen absorption lines, each from a separate cloud at a separate redshift along the line of sight. The Lyman-alpha forest is a core sample of the universe between here and the quasar, and the statistics of the lines constrain how matter is distributed on large scales.

The first stars. The most metal-poor stars known have $[\text{Fe/H}]$ below $-7$ — ten million times less iron than the Sun. Their compositions are thought to record the yield of a single supernova from the first generation of stars, so a spectrum taken now is a measurement of an object that existed thirteen billion years ago and left nothing else behind — the yield of a star that lived a few million years. Two further consequences of that arrangement are worth recording. The blends make the derived abundance of an element depend on which lines were chosen, so two analyses of the same star using different line lists can differ by more than either quotes as its uncertainty. And in the most metal-rich stars the forest closes over entirely: there is no true continuum anywhere in the visible, the fitted level is a pseudo-continuum defined by convention, and every equivalent width measured against it inherits that convention.

An element found in the Sun before it was found on Earth #

The technique’s most persuasive demonstration is the one case where a spectrum identified something nobody had a sample of.

During the total eclipse of August 1868, observers looking at the chromosphere — the thin layer visible for a few seconds at the edges of totality, where the gas is seen against dark sky rather than against the photosphere, so its lines are bright rather than dark — recorded a yellow emission line near 587.6 nanometres. It sat close to the pair of sodium lines Fraunhofer had labelled D, and it was not either of them.

No terrestrial substance produced a line at that wavelength. Norman Lockyer proposed that it belonged to an element not known on Earth, and the name helium was coined for it. The proposal was not universally accepted, and for a quarter of a century the evidence for the element consisted of one line in one spectrum with nothing to compare it against.

It was isolated in 1895, from a uranium mineral, and its spectrum matched.

Two things about the episode are worth carrying. The first is that the identification was made on a wavelength alone: the argument was that the line’s position did not coincide with any known element’s, which is a purely negative statement and was strong enough to name a new element. That is only possible because the wavelengths are sharp and the catalogues were already good.

The second is that the same reasoning has failed. A set of lines in the corona was attributed in the same period to another new element, provisionally called coronium, and turned out to be iron — ionised thirteen times over, in gas at a million kelvin, in a state no nineteenth-century laboratory could produce. The lines were real and unmatched, and the conclusion drawn from that was wrong, because the catalogue of known spectra was a catalogue of what could be made in a discharge tube.

An unmatched line means the identification is not in the catalogue, and the catalogue’s limits are set by what a laboratory can reach. That is the standing caution on every argument from absence in this subject.

Where the model stops #

Local thermodynamic equilibrium. The standard analysis assumes the level populations are set by the local temperature. In thin, strongly irradiated atmospheres they are not, and the corrections — “non-LTE” — can reach several tenths of a dex for some elements, in a direction that is not always known.

One-dimensional atmospheres. Real stellar surfaces are convecting, with hot rising cells and cool sinking lanes. Averaging them into a single depth-dependent model biases derived abundances; three-dimensional hydrodynamic models changed the accepted solar oxygen abundance by nearly a factor of two in the 2000s, which propagated into models of the Sun’s interior and produced a discrepancy with helioseismology that is still open.

Photospheres only. A spectrum measures the composition of the layers the light escapes from. For most elements that is representative of the whole star, and for some it is not — a star that has dredged up processed material from its own core shows a surface composition it was not born with, which is one of the ways the mass–luminosity relation acquires scatter.

Detectable species only. An element with no transitions at accessible wavelengths, or one entirely ionised in the conditions present, is invisible. Helium is nearly undetectable in the spectra of cool stars, and its abundance in them is inferred rather than measured.

The figures have a limitation shared with every drawn spectrum. Real absorption lines number in the tens of thousands across the visible range, blended into a near-continuous forest in which the true continuum level is itself a matter of judgement — for a cool or metal-rich star there may be no wavelength anywhere that is genuinely unabsorbed. These figures show eight lines cleanly separated on a clean continuum, which is what a spectrum looks like only for the hottest, cleanest stars. The problem the analysis actually faces is deciding where the top of the curve is.

Between the two extremes the essay has drawn there is a temperature at which the hydrogen lines are at their strongest, and it is worth putting a spectrum there.

The reason the hydrogen sequence peaks rather than rising is worth stating once more in its own terms, because it is the single point Payne’s thesis turned on. Absorbing at Balmer wavelengths requires an atom already in the first excited state, which needs 10.2 electron volts to reach and which almost no atom occupies at 4,000 kelvin. Raising the temperature populates that state exponentially fast. Raising it further ionises the atom entirely, and an ion has no Balmer lines at all. The strength is a product of a rising factor and a falling one, and a product of that shape has a maximum.

What makes the argument decisive rather than merely plausible is that the maximum’s position is calculable. The Boltzmann factor is set by 10.2 electron volts and the Saha factor by 13.6, both measured in the laboratory, and the temperature at which their product peaks comes out near 9,500 kelvin with no adjustable parameter anywhere. The observed A-type stars have the strongest hydrogen lines in the sequence and their temperatures are near 9,500 kelvin.

That is the whole of the resolution, and it is the reason the composition of the Sun is 71% hydrogen rather than the 40% iron the line strengths appear to say.

The ladder from here #

The same absorption, read through a planet’s atmosphere rather than a star’s, is a radius that depends on the colour it is measured in.

Later rungs on this anchor: Kirchhoff’s three laws, and the conditions each requires. The Saha equation, and ionisation as a function of temperature and pressure. The Boltzmann distribution over excitation levels. The spectral sequence and its luminosity classes. Equivalent width and the curve of growth. Oscillator strengths, and where they come from. Model atmospheres, one-dimensional and three. Non-LTE corrections. Metallicity, and the chemical evolution of the galaxy. Interstellar absorption and the diffuse interstellar bands, whose carriers are still unidentified after a century. Transmission spectroscopy of exoplanets. And the Lyman-alpha forest.

Fraunhofer catalogued 574 dark lines in the solar spectrum in 1814 and labelled the strongest with letters. He had no idea what they were, but he measured their wavelengths carefully enough that his letters are still in use, and his D line is still where sodium is.