Why hydrogen's lines are strongest where hydrogen is not

The hydrogen lines in a stellar spectrum are strongest in A stars, at about 10,000 K. They are weak in the Sun, weaker in cool red stars, and weak again in the hottest blue ones.

Read that as a statement about composition and the conclusion follows immediately: A stars are made of hydrogen and other stars are not. Read it as a statement about temperature and the conclusion is the opposite — every one of these stars is overwhelmingly hydrogen, and what varies is how much of it is in a state that can absorb a Balmer photon.

The second reading is the correct one, and getting from the first to the second took until 1925.

Two exponentials pulling opposite ways #

A line’s strength depends on how many absorbers are available, and “absorber” means an atom in one specific state of one specific ionisation stage.

The Balmer lines — Hα at 656 nm, Hβ at 486, and the rest — are transitions upward from $n = 2$. An atom in the ground state $n = 1$ cannot absorb them; nor can a proton. So the population that matters is

$$\frac{N(n{=}2)}{N(\mathrm{H})} = \frac{N(n{=}2)}{N(\mathrm{H\,I})} \times \frac{N(\mathrm{H\,I})}{N(\mathrm{H})},$$

and the two factors move in opposite directions.

The first is Boltzmann’s. The energy gap from $n = 1$ to $n = 2$ is 10.2 eV, and the ratio of populations is $(g_2/g_1)e^{-\Delta E/kT}$ — which at 6,000 K is $10^{-8}$, at 10,000 K is $3\times10^{-5}$, at 20,000 K is $10^{-2}$. Rising, steeply.

The second is Saha’s. Ionisation costs 13.6 eV, and the balance between neutral and ionised depends on temperature the same way plus a factor for the density of free electrons to recombine with:

$$\frac{N_{\mathrm{II}}}{N_{\mathrm{I}}} = \frac{2kT}{P_e}\left(\frac{2\pi m_e kT}{h^2}\right)^{3/2}\frac{Z_{\mathrm{II}}}{Z_{\mathrm{I}}}\,e^{-\chi/kT}.$$

Falling, steeply, once past about 9,000 K.

Their product has a maximum. At 9,870 K, with a photospheric electron pressure of 20 N/m². That maximum is what the spectral sequence is sorted by.

The strongest line in the solar spectrum is not hydrogen’s #

The single deepest absorption feature in the visible solar spectrum is Ca II K, at 393.4 nm. Next to it is Ca II H at 396.8. Fraunhofer labelled them H and K in 1814, before anybody knew what they were, and they are the most prominent things in the catalogue.

Calcium is $2.2\times10^{-6}$ of the Sun’s atoms. Hydrogen is 92 per cent of them. The strongest line in the spectrum belongs to an element present at two parts in a million, and it is stronger than the lines of an element four hundred and fifty thousand times more abundant.

Every part of that is explained by the two figures above. At 5,800 K:

• Calcium is essentially all Ca II, and the Ca II K transition comes up from Ca II’s ground state, so nearly every calcium atom in the photosphere is available to absorb it.
• Hydrogen is essentially all neutral, and the Balmer transitions come up from $n = 2$, which holds about one hydrogen atom in $10^8$.

Multiply through: the available calcium absorbers outnumber the available hydrogen absorbers by a factor of a few hundred. The line strengths follow.

What was actually measured #

The observation is an equivalent width: the area of a line, expressed as the width of a completely black rectangle absorbing the same energy. It is measured by integrating the depth of the line across its profile relative to the continuum either side, and it is one number per line per star, in units of milliångström.

Getting from an equivalent width to a number of absorbers is not a matter of reading the depth, and it is the step where most of the difficulty lives. So an abundance determination needs weak lines, which are hard to measure; a model atmosphere, to supply the temperature and pressure at each depth; a partition function for each stage of each element; and an oscillator strength for each transition, measured in a laboratory. The chain from a dark line to “this star is 92 per cent hydrogen by number” runs through all four, and each of them was a research programme.

The measurement Payne actually made was of relative line strengths across the spectral sequence. She had Harvard’s collection of objective-prism plates — the largest body of stellar spectra in existence, classified by Annie Jump Cannon — and she measured how the strength of each line varied from class to class. The variation was smooth and continuous in temperature, and it fitted the Saha equation, which Meghnad Saha had published in 1920 — an equation about ionisation in a gas, arriving in astronomy five years later and settling what stars are made of and which had not yet been applied at this scale. The classification sequence O–B–A–F–G–K–M turned out to be a temperature sequence and nothing else.

The result that had to be withdrawn #

Payne’s 1925 thesis concluded that hydrogen and helium are enormously more abundant in stellar atmospheres than any other element — hydrogen by about a million times relative to the metals, which is roughly right.

Henry Norris Russell, who refereed it, did not believe it. The prevailing view, which he held and had contributed to, was that the Sun’s composition resembled the Earth’s: dominated by iron, silicon, magnesium and oxygen. It was not an unreasonable position. Meteorites and the Earth’s crust are the only material anybody had ever analysed, and the solar spectrum is full of iron lines, each one a dark gap where something was removed — thousands of them, far more than of anything else.

The thesis as published contains the sentence that the hydrogen and helium abundances are “almost certainly not real”. Payne added it at Russell’s urging. Four years later Russell reached the same conclusion by a different route and published it, and the result is often attributed to him.

Why the iron lines are so numerous is the same argument again, and it is worth finishing because it removes the last support from the old view. Iron is $3\times10^{-5}$ of the Sun’s atoms — and it is the element whose nuclear binding energy is the highest of all, which is a different fact with a different cause and is often run together with this one. It appears in thousands of lines because a neutral iron atom has 26 electrons in a complex configuration with an enormous number of low-lying energy levels, many of them populated at 5,800 K, and each populated level is the lower state of many transitions. Hydrogen has one electron and a handful of lines. The number of lines an element shows measures the complexity of its atom, not how much of it there is.

The same argument, one step colder #

Below the temperatures where the essay’s competition plays out, a second equilibrium takes over and behaves the same way with different actors.

Atoms can bind into molecules, and whether they do is decided by a balance between the binding energy and the thermal energy — the same shape of expression as the ionisation balance, with a dissociation energy in place of an ionisation potential. Below about four thousand kelvin the balance tips, and molecules survive in a stellar photosphere.

The consequence for a spectrum is dramatic rather than gradual. A molecule has vibrational and rotational levels on top of its electronic ones, so instead of a line it produces a band — hundreds of closely spaced transitions covering tens of nanometres. Titanium oxide is the conspicuous case: it appears in the coolest stars and its bands are so strong and so broad that they depress the continuum over whole regions of the visible spectrum.

That is why the coolest spectral classes are defined by molecular bands rather than by atomic lines, and why the sequence’s appearance changes character at that point rather than continuing smoothly.

The steepness is worth noticing. Titanium is a trace element — about one atom in ten thousand million — and its oxide dominates the visible appearance of the most common stars in the Galaxy. A species can be negligible by abundance and decisive by opacity, which is the same lesson the essay draws about hydrogen, arrived at from the opposite direction.

Both curves move with the electron pressure, and it is worth seeing how far, because the electron pressure is set by the surface gravity and is therefore the second axis of every spectral classification.

Where the equilibrium does not hold #

The whole argument assumes the gas is in thermal equilibrium, with the ionisation set by collisions at the local temperature. That is true in a stellar photosphere, where the density is high enough that collisions dominate, and it is false in most of the volume of the universe.

In an interstellar cloud around a hot star, the density is so low that collisions are rare and the ionisation is set by an entirely different balance: photons from the star ionise atoms, and free electrons recombine with ions at a rate proportional to the density. The equilibrium is between photoionisation and recombination, not between collisional ionisation and recombination, and the resulting ionisation fraction depends on the star’s ultraviolet output and the gas density rather than on the gas temperature.

The difference is stark enough to be worth a number. A stellar photosphere at ten thousand kelvin has hydrogen only slightly ionised; a nebula at the same temperature is ionised essentially completely, because the ionising photons keep arriving regardless of how cool the gas is. The gas is cool and ionised at once, which the essay’s equation cannot produce at any density.

So the Saha equation is a statement about a particular regime, and applying it outside that regime gives answers wrong by orders of magnitude. The regime is precisely the one where the essay’s subject lives — dense, collisional, and in equilibrium with its own radiation — which is a stellar interior and its photosphere, and very little else.

Why abundances are quoted relative to the Sun #

The balance in this essay sits underneath every abundance measurement ever made, and it is the reason those measurements are almost never quoted as absolute numbers.

To turn a line’s strength into an abundance, one needs the fraction of the element in the ionisation stage the line belongs to and the fraction of those in the lower level of the transition. Both come from the equations above, and both depend steeply on the temperature and on the electron pressure — so an error of a hundred kelvin in an assumed temperature can move a derived abundance by tens of per cent, and different lines of the same element move by different amounts.

The remedy is to stop asking for the absolute number. Measure the same lines in the star and in the Sun, analyse both the same way with the same model atmospheres and the same atomic data, and report the difference. The systematic errors — in the transition probabilities, in the model atmosphere’s structure, in the treatment of the line formation — are then largely common to the two analyses and cancel.

That is why stellar compositions appear in the literature as logarithmic ratios relative to solar, and why a paper reporting an iron abundance to 0.03 dex relative to the Sun may carry an absolute uncertainty ten times larger. The precise quantity and the accurate one are different numbers, and the field has settled on quoting the first because it is the one that supports the comparisons anybody wants to make.

The arrangement has a weakness worth naming. It makes the Sun’s own composition the zero point of everything, so a revision of the solar abundances shifts the entire scale — which is what happened when the solar oxygen abundance was revised downward in the 2000s, and why that revision propagated into subjects with no obvious connection to it.

It is an unusual arrangement and it works, provided everybody using the scale remembers that its zero is a measurement rather than a definition.

Where the model stops #

Local thermodynamic equilibrium. The Saha and Boltzmann equations describe a gas in equilibrium at one temperature. A stellar atmosphere is not: it is precisely the region where photons are escaping, so the radiation field is not the local Planck function and the level populations need not follow Boltzmann’s law. For most lines in most stars the approximation is good to tens of per cent. For strong lines formed high in the atmosphere, for hot stars where radiative rates dominate collisional ones, and for the resonance lines of the light elements, the departures are factors — and a modern abundance is quoted with a “non-LTE correction” that is itself a computation.

Partition functions. The ground-state weights used in these figures are the first term of a sum over all levels, and the sum formally diverges: an atom has infinitely many bound states converging on the ionisation limit. It is cut off by the fact that a real atom in a plasma is perturbed by its neighbours and loses its highest levels, which makes the cut-off a function of the density. For hydrogen at photospheric densities the correction is negligible; for iron it is tens of per cent.

One temperature and one pressure. An atmosphere has a temperature gradient — that is what makes it an atmosphere — and different lines form at different depths. A curve computed at a single $T$ and $P_e$ is a caricature of a stratified structure, and the real calculation integrates through it.

And the electron pressure is not independent. $P_e$ appears in the Saha equation, and in a hydrogen-dominated atmosphere the free electrons come mostly from ionised hydrogen — so the equation’s input depends on its own output and has to be solved self-consistently. The curves here take $P_e$ as given, which is a step the real computation does not get to take.

What the picture cannot show #

The depth. Every curve is drawn at one temperature at a time, and a real line is formed over a range of depths spanning thousands of kelvin. The observed strength is an integral through that range weighted by where the photons escape.

The abundance. The two curves in the first figure are each normalised to their own maximum, which is the only way to draw them on one axis, and it deliberately throws away the factor of 450,000 between hydrogen and calcium. That factor is the whole reason the comparison is interesting, and it is in the caption because it cannot be in the drawing.

Helium. The second most abundant element shows no lines at all in the Sun’s visible spectrum, because its first excited state is 19.8 eV up and the Boltzmann factor at 5,800 K is $10^{-17}$. It was discovered in the chromosphere during an eclipse, where the conditions are entirely different, and named for the Sun twenty-seven years before it was found on Earth. A figure of solar line strengths has an invisible 8 per cent of the atoms.

And the same two quantities read over the whole range of stellar temperatures rather than the part where the maximum sits.

Where the ladder goes next #

Later rungs on this anchor: the Saha equation derived, from the partition function of a free electron. The negative hydrogen ion, H⁻, which supplies most of the continuous opacity in the solar photosphere and exists only because there are enough free electrons from the metals — so the continuum against which every line is measured is itself a product of ionisation. Non-LTE line formation, as a subject. The spectral classification of the coolest objects, where molecules rather than atoms carry the diagnosis and TiO bands take over. Metallicity as a measurement, and the [Fe/H] scale that every stellar population study runs on. And the ionisation structure of a nebula, where the same equation is solved with a radiation field instead of a temperature and produces Strömgren spheres — the same balance that decides how much of a galaxy’s gas can form stars.

Payne’s thesis was called by Otto Struve “undoubtedly the most brilliant PhD thesis ever written in astronomy”. The measurement in it was of line strengths on photographic plates. The result was what the universe is made of, and the only thing standing between the two was an equation about which atoms are in which state.