The continuum is made by one atom in ten thousand

Every measurement of a spectral line is a comparison. A line’s depth means nothing on its own; what is measured is how far the flux falls below the level it would have had with no line there, and that level is set by whatever absorbs light at every wavelength rather than at one.

For the Sun the answer to what does that absorbing was not settled until 1939, and it is strange enough that it was proposed before anybody had seen the ion involved in a laboratory. The Sun’s continuous opacity is dominated by $\mathrm{H}^-$: a hydrogen atom with a second electron attached, bound by 0.754 electronvolts, with exactly one bound state and no excited ones at all.

Two things follow immediately and neither is obvious. The abundance of that ion depends on how many free electrons there are to attach. And in a photosphere at five or six thousand kelvin, hydrogen is far too hard to ionise to supply them.

The electron pressure is not a free parameter #

The figures on the two essays before this one hold the electron pressure fixed and ask what it does to an element. That is the right way round for those arguments and it postpones a question: where does the electron pressure come from?

It is not an independent property of a gas. At a stated temperature and gas pressure it is whatever makes the ionisation of the gas produce exactly that many electrons, which is a fixed point rather than a choice:

$$P_e = P_{\rm gas}\,\frac{\varepsilon}{1+\varepsilon},\qquad \varepsilon = \sum_s A_s\,x_s(T, P_e),$$

with $A_s$ each species’ abundance and $x_s$ its ionised fraction, which itself depends on $P_e$ through Saha. The equation has the unknown on both sides, it is solved by iteration, and the iteration converges quickly because raising $P_e$ suppresses ionisation and therefore lowers $P_e$ again.

The answer it gives at solar values is a couple of newtons a square metre, and the structure of the answer is the interesting part. At the cool end the sum is dominated by a term with $A_s = 10^{-4}$, because that species is nearly fully ionised while hydrogen at $A_s = 1$ is ionised to a part in $10^{5}$. Multiply the two out and the trace species wins by a wide margin.

A photosphere’s free electrons are a property of its metals, and the metals are four orders of magnitude scarcer than the gas they are in. That is the arithmetic the rest of this essay is about.

What the ion is, and why it has one state #

$\mathrm{H}^-$ deserves a paragraph on its own terms because it is an unusual object.

A neutral hydrogen atom has no net charge and no permanent dipole, so a second electron approaching it feels nothing until it is close enough to polarise the atom. The resulting attraction is weak, and the ion it produces is bound by 0.754 electronvolts against hydrogen’s own 13.6 — a factor of eighteen. There is exactly one bound state. Every attempt to find an excited state of $\mathrm{H}^-$ finds a resonance that autoionises instead, so the ion has no line spectrum whatever.

That is what makes it a continuum absorber. With no bound-bound transitions, all of its absorption is bound-free — a photon above 0.754 eV detaches the extra electron, which is any photon shorter than about 1.64 microns — and free-free, where a passing photon is absorbed by an electron in the field of a neutral atom. The first gives a smooth opacity across the whole visible spectrum with an edge in the near infrared; the second takes over beyond the edge and rises with wavelength.

So the object responsible for the level every solar line is measured against contributes no lines of its own. Had $\mathrm{H}^-$ possessed a rich spectrum there would be no continuum in the Sun either, for the reason the coolest stars have none.

A star with fewer metals is more transparent #

The consequence is a direct one and it is measurable, which is unusual for a statement about an opacity.

Remove metals from a cool star and the electron supply falls, so the $\mathrm{H}^-$ abundance falls, so the continuous opacity falls. Light then escapes from deeper in — the optical surface is a depth rather than a place, and lowering the opacity moves that depth further down, into hotter and denser gas.

That does several things at once, and separating them is most of the work in analysing a metal-poor star.

The star is hotter at its optical surface than a metal-rich star of the same mass and age, which shifts its colours blueward and moves it on the colour–magnitude diagram by an amount that is not a change in its interior at all.

The lines are stronger than the abundance alone would give, because a line’s depth is a ratio of the line opacity to the continuous opacity, and the denominator has fallen along with the numerator. An iron line in a star with a tenth the iron is not a tenth as deep; some of the reduction is cancelled by a more transparent continuum. The cancellation is partial and it is temperature-dependent, and getting it wrong biases every abundance derived from the spectrum in the same direction.

And the atmosphere is more compact, because the pressure at a given optical depth is higher when the opacity is lower, which feeds back into the ionisation equilibria the first figure solves.

What is actually measured #

Nothing above is observed. What is observed is a flux, and the chain from one to the other is worth walking because each link carries a different kind of uncertainty.

The first link is the laboratory measurement of the ion itself, and it is the strongest. The photodetachment cross-section of $\mathrm{H}^-$ has been computed to high accuracy from first principles — it is a three-body Coulomb problem with two electrons, which quantum mechanics handles well — and measured in the laboratory since the 1950s. Of everything in this essay it is the part with the smallest error.

The second is the electron pressure, which is never measured and always solved for. It requires the abundances of all the electron donors, which is the thing being measured; the solution is therefore iterative in a second sense, with the abundances and the atmosphere converged together.

The third is the model atmosphere, which supplies the temperature and pressure as functions of depth. It is built by demanding that the total flux be the same at every level — radiative equilibrium — with the opacity computed from the same chemistry, so the opacity that the argument is about is also an input to the structure that produces it.

The fourth is the comparison, and here there is a genuine test. A model atmosphere predicts not only the line spectrum but the absolute shape of the continuum, including the $\mathrm{H}^-$ edge at 1.64 microns. That edge is observed in the solar spectrum, in the right place, with roughly the right depth. It is the single cleanest confirmation that the identification is correct, and it was not available when the identification was made.

There is a second and more surprising check. The solar limb darkening — how much fainter the disc is at its edge than at its centre — depends on how the source function varies with optical depth, which depends on how the opacity varies with wavelength. Measured limb darkening as a function of wavelength was one of the first tests the $\mathrm{H}^-$ hypothesis passed, and it is sensitive to the ratio of the bound-free to the free-free contribution rather than to their sum, which is exactly what an absolute flux measurement cannot separate.

The wavelength at which a star is most transparent #

The two absorption channels have opposite wavelength dependences, and where they cross is an observable that several parts of astronomy quietly depend on.

Bound-free detachment needs a photon above 0.754 electronvolts, so it switches off beyond 1.64 microns and its cross-section falls away toward that threshold from the short-wavelength side. Free-free absorption — a photon taken up by an electron passing a neutral atom — has no threshold and its cross-section rises roughly as the square of the wavelength. Add the two and the total opacity has a minimum, at about 1.6 microns, right where the bound-free contribution has nearly gone and the free-free has not yet grown.

That minimum is why the H photometric band exists where it does. A cool star is more transparent there than anywhere else in its spectrum, so light at 1.6 microns escapes from deeper, the emergent flux is higher than a smooth interpolation would give, and the spectral energy distribution has a bump in it. The bump is not a feature of the star’s temperature; it is a feature of the opacity of an ion.

Several techniques are built on the consequence. Surface-brightness relations for measuring stellar angular diameters are calibrated in the infrared partly because the opacity minimum makes the relation between colour and surface brightness tighter there. The near-infrared is where the extinction by dust is also lowest, so the two transparencies compound, and observations of the Galactic plane use the same band for both reasons at once. And in the atmospheres of brown dwarfs and giant planets the same minimum is what lets light from deep layers escape in narrow spectral windows between the molecular bands — the only route by which the deep temperature structure of such an object is observable at all.

One ion’s photodetachment threshold therefore sets which band a great deal of modern photometry is done in, and the connection runs from a three-quarter-electronvolt binding energy to the choice of a filter.

The negative ion is not alone, either, and the company it keeps says something about why it wins. Helium will also hold an extra electron transiently enough to absorb in the free-free channel, and so will molecular hydrogen; both contribute in the coolest atmospheres, where hydrogen is molecular and there is little else. The positive molecular ion $\mathrm{H}_2^+$ absorbs in the ultraviolet of cool stars. None of them competes with $\mathrm{H}^-$ across the visible in a star like the Sun, and the reason is the one that runs through this whole essay: the absorber has to be built out of the abundant species, and the only abundant species available is hydrogen. Everything else is a correction at the per-cent level, which is exactly where the modelling difficulty in the coolest objects now sits.

Why the identification took so long #

The history is instructive because the obstacle was not the measurement.

By the mid-1930s the solar continuous opacity was a known quantity: its magnitude followed from the observed limb darkening and the energy flux, and its wavelength dependence was measurable. What nobody could find was a source. Neutral hydrogen’s bound-free absorption requires an atom already excited to the second or third level, and at 6,000 K there are far too few of those. Metallic bound-free edges are in the wrong places and are too weak. Electron scattering is grey and a hundred times too small at these densities. The calculated opacity fell short of the measured one by more than an order of magnitude, and the discrepancy stood for years.

Rupert Wildt proposed $\mathrm{H}^-$ in 1939, on the basis of a cross-section that had been computed for an entirely different purpose and an existence that had been demonstrated theoretically by Bethe in 1929. The ion had never been observed. The proposal worked immediately, at the right magnitude and with the right wavelength dependence, and it was accepted within a few years.

The missing ingredient was not a measurement but a candidate, and the candidate was an object nobody had reason to think about because it had no spectrum to be catalogued by. That is a recurring shape: an absence in an inventory is hard to fill when the missing item is invisible by construction.

The same structure appears in the interstellar medium’s missing baryons and in every case where a budget fails to balance and the deficit is in something that does not emit.

The one part in ten thousand, stated carefully #

It is worth being precise about what the headline claim does and does not say, because it is easy to overstate.

It does not say the Sun’s opacity is made of metals. The absorbing particle is $\mathrm{H}^-$ and it is hydrogen. What the metals supply is the electron, and they supply it because their ionisation potentials are around 6 to 8 electronvolts rather than 13.6 — a difference in the exponent that a factor of $10^{4}$ in abundance cannot make up.

It does not say the dependence is proportional. The number of $\mathrm{H}^-$ ions goes as the electron pressure, and the electron pressure goes as the metal abundance only while the metals dominate the supply. Once hydrogen contributes, the dependence weakens and then vanishes — which the second figure shows happening over about a thousand kelvin.

And it does not apply to hot stars at all. Above about 8,000 K the continuous opacity is neutral hydrogen’s own bound-free and free-free absorption, and above about 15,000 K it is electron scattering, which knows nothing about composition whatever, so the mean opacity a stellar interior runs on changes character with depth. The metallicity dependence of a stellar continuum is a cool-star phenomenon with a sharp upper edge.

That last point is worth dwelling on. A single photosphere is not at one temperature. The Sun’s runs from about 4,400 K at the top of the region that matters to 9,000 K at the bottom of it, which straddles the handover in the first figure. So within one atmosphere the electron supply changes hands with depth, and the opacity’s sensitivity to composition is strong at the top and negligible at the bottom.

What a spectrum cannot separate #

There is a degeneracy here that limits abundance work and it is worth naming.

A line’s strength depends on the ratio of the line opacity to the continuous opacity. For an iron line in a cool star, both the numerator and the denominator contain the iron abundance — the numerator directly, the denominator through the electrons iron donates. The ratio is therefore much less sensitive to iron than a naive reading suggests, and the sensitivity itself depends on temperature and gravity.

There is a partial escape and it is the standard practice. Use lines of two ionisation stages of the same element and require them to give the same abundance. The neutral and ionised populations respond differently to the electron pressure, so demanding consistency between them constrains the gravity; demanding that lines of different excitation potential give the same abundance constrains the temperature — which is the composition read from what is missing. Both are the same trick: find a pair of observables whose dependence on the nuisance parameter has different shapes, and let the requirement of agreement fix it. That is the trade between a temperature and a gravity seen from the chemistry rather than from the fitting.

Where the picture stops #

One metal is not the metals. The figures here carry a single representative donor at 7.6 electronvolts, which is a fair stand-in for magnesium, silicon and iron together. It is not a stand-in for sodium, potassium and aluminium, whose potentials are near 5 and which are fully ionised further down the temperature scale than the representative is. Carrying them would move the cool end of the handover to lower temperature and change nothing about its shape; the published model atmospheres carry a hundred elements and thirty ionisation stages.

Local thermodynamic equilibrium is assumed throughout, and it is the assumption that fails first at the top. Every population in this essay is the one a Saha or Boltzmann equation gives at the local temperature, which requires collisions to dominate over radiative transitions. High in a photosphere the gas is thin, the radiation field is not the local blackbody, and the populations depart from equilibrium — worst in the most extended atmospheres, which are the supergiants the gravity figures use as their low-pressure limit.

And the whole treatment is one-dimensional and static. A real photosphere is a convective granulation pattern with rising hot columns and sinking cool lanes, and the ionisation equilibria are not linear in the temperature, so the average of the chemistry over the surface is not the chemistry of the average temperature. That non-linearity is exactly what the three-dimensional models corrected in the solar abundances, and it is where the argument below picks up.

Still open: the solar abundances, and an opacity that has to absorb the difference #

There is a live disagreement this essay’s arithmetic sits inside, and it is worth ending on because it is unresolved rather than tidy.

Spectroscopic determinations of the Sun’s oxygen and carbon abundances were revised downward by about a third in the 2000s, on the strength of three-dimensional hydrodynamic model atmospheres replacing one-dimensional static ones. The new values are widely regarded as better measurements. They also broke the agreement between solar models and helioseismology: with less oxygen the interior opacity falls, the convection zone’s base moves, and a prediction that had matched the seismic measurement to a part in a thousand stopped matching.

The proposed resolutions divide into two families. One says the abundances are still wrong and the three-dimensional atmospheres carry an error somewhere in exactly the chemistry described here — the electron donors, the continuum normalisation, the departures from local equilibrium in the lines used. The other says the abundances are right and the opacity inside the star is underestimated, which recent measurements of iron opacity at stellar-interior conditions have unexpectedly supported.

Both families are about an opacity nobody can measure directly in the object concerned, and both would be settled by a measurement that separates the photospheric determination from the interior model. There is no such measurement yet, and the discrepancy has stood for twenty years in the best-observed star there is.