A spectrum with no continuum left in it

An ionisation potential and a chemical bond are the same kind of number. Both are the energy needed to take a system apart into two pieces that were one, both are compared against $kT$ inside an exponential, and both sit in an equilibrium with a pressure in the denominator because one particle is becoming two.

The difference is the size. Hydrogen’s ionisation potential is 13.6 electronvolts and titanium oxide’s bond is 6.87, so the arithmetic that puts hydrogen’s Balmer maximum near ten thousand kelvin puts the survival of a molecule near three and a half. Below that temperature a stellar photosphere is a chemistry rather than a plasma, and its spectrum stops looking like a set of lines drawn on a background.

The equation is the same equation #

Written side by side the two look almost identical. For ionisation,

$$\frac{n_{i+1}n_e}{n_i} = \left(\frac{2\pi m_e kT}{h^2}\right)^{3/2}\frac{2Z_{i+1}}{Z_i}e^{-\chi/kT},$$

and for a molecule coming apart into two atoms,

$$\frac{n_A n_B}{n_{AB}} = \left(\frac{2\pi \mu kT}{h^2}\right)^{3/2}\frac{Z_A Z_B}{Z_{AB}}e^{-D/kT},$$

with $\mu$ the reduced mass of the pair and $D$ the dissociation energy. The free electron has been replaced by a free atom; that is the whole of the substitution. Both are statements of the same thermodynamics, and the second was derived within a few years of the first for the same reason — somebody wanted to know what a stellar spectrum was made of.

The exponential decides where the transition happens and the prefactor decides how sharp it is. Because $D$ appears inside $e^{-D/kT}$, a factor of two in the bond energy is not a factor of two in the temperature but very nearly one: a strongly bound molecule survives to a proportionally higher temperature, and the ordering of the bands in a stellar sequence is the ordering of the bond energies.

The carbon monoxide result is the single most consequential fact in cool-star chemistry, and it is worth drawing out because it explains a classification that otherwise looks arbitrary.

Carbon monoxide is so tightly bound that essentially all of the scarcer of carbon and oxygen ends up inside it, and whatever is left over is free to make everything else. In a star with more oxygen than carbon — which is nearly all of them — the surplus oxygen makes titanium oxide, vanadium oxide and water, and the star is an M type with oxide bands. In a star with more carbon than oxygen the surplus carbon makes diatomic carbon, cyanogen and silicon carbide, and the spectrum is completely different: a carbon star. The dividing line is not a ratio of a few to one. It is exactly one, because a molecule with an eleven-electronvolt bond consumes the minority species to near-completion, and the chemistry on either side of that point has nothing in common.

That is a threshold produced by an exponential in an equilibrium constant, and it sorts a whole population of stars into two families with no intermediate class worth the name.

Pressure protects a molecule #

The denominator in both equations carries a pressure, and the consequence is the same in both cases and reads as opposite in the spectrum.

More free electrons about means more recombinations, so a denser photosphere holds an atom neutral to a higher temperature — the effect that makes a luminosity class a density measurement. More free atoms about means more reassociations, so a denser photosphere holds a molecule together to a higher temperature. Both reactions are one particle becoming two and both are suppressed by crowding.

What differs is what the classifier sees. The atomic criterion is a ratio of ionised to neutral, and it falls as the gravity rises. The molecular criterion is a band strength, which rises with the amount of molecule, and so it rises as the gravity rises. The two run in opposite directions, and a classifier crossing from K into M has to reverse the sense of every rule.

The last observation in that caption is the same rule the atomic case obeys, and it is worth stating once for both. A ratio measured between a large population and a tiny one is measured badly, so the useful diagnostic at any temperature is whichever species has both of its forms present. That is why a classifier moves from calcium to strontium to iron as the temperature falls, and why below four thousand kelvin the criterion moves from an ionisation ratio to a band ratio: not because the physics changes, but because a different equilibrium is the one sitting near its midpoint.

Why the continuum disappears #

The real difficulty of a cool spectrum is not that the criteria change. It is that the thing every line depth is measured against stops existing.

An atomic line is a narrow feature. There are tens of thousands of them in a solar spectrum, they occupy a small fraction of the wavelength, and between them the spectrum reaches a level set by the continuous opacity. That level is the continuum, a depth rather than a surface, and every equivalent width in astronomy is an area measured against it.

A molecular band is not a narrow feature. A diatomic molecule’s electronic transition carries a vibrational structure, and each vibrational band carries a rotational structure of hundreds of lines, and at a few thousand kelvin many vibrational levels are populated at once. One electronic transition of titanium oxide therefore contributes thousands of lines spread over hundreds of ångströms, and there are several such transitions, and there is vanadium oxide, and water, and in the infrared the pressure-broadened wings of everything.

The result is that between four thousand kelvin and the bottom of the stellar sequence the opacity has no gaps left. What looks like a continuum in a low-resolution spectrum of an M dwarf is the tops of the strongest bands — a pseudo-continuum, whose level is set by the molecules rather than by the continuous opacity, and which moves when the molecular abundances move.

Everything downstream of that is affected. An equivalent width measured against a pseudo-continuum is not the quantity the curve-of-growth theory is about; a colour index is a ratio of band depths rather than of blackbody fluxes; and the true continuum has to be reconstructed from a model rather than observed, which turns every abundance below four thousand kelvin into a model-dependent statement.

A band carries an isotope ratio #

There is a payoff to all this that no atomic line can deliver, and it is the best reason to want molecular bands rather than merely to tolerate them.

A molecule’s rotational and vibrational energies depend on the reduced mass of its atoms, and the reduced mass depends on which isotope each atom is. Replace the carbon in carbon monoxide with carbon-13 and every line in every band shifts, by an amount that grows with the vibrational quantum number and is easily resolved in the infrared. The two isotopic species have their own band heads a few tens of wavenumbers apart, and their relative strengths are the ratio of the two isotopes directly.

An atomic line cannot do this. The isotope shift of an electronic transition is a hyperfine-scale effect, tiny for anything heavier than lithium, and swamped by every other broadening mechanism. The molecular shift is a mass effect on a mechanical oscillation and it is large.

What the ratio measures is worth stating because it is not a composition in the ordinary sense. Carbon-12 is made in helium burning and carbon-13 is made from it by proton capture in the CNO cycle, so a star born with the interstellar ratio near 89 shows a lower one once convection has dredged material from a region where the cycle has operated. Measured ratios in red giants run from about 25 down to 10, and each is a statement about how deep that star’s convective envelope reached and how long it stayed. A band head in the infrared is a probe of a region a thousand times deeper than anything the spectrum otherwise reaches, and it works because a bond is a spring whose frequency is set by a mass.

The same argument gives oxygen isotopes from carbon monoxide and from water, nitrogen from cyanogen, and magnesium from magnesium hydride, and every one of them is a constraint on a nuclear process rather than on an abundance.

What the hydrogen molecule does to the atmosphere #

The hydrogen case is worth a section because it is the one where the chemistry feeds back into the mechanics.

A gas of atomic hydrogen has a mean molecular weight near 1.3 once helium is counted; a gas of molecular hydrogen has one near 2.3. The pressure scale height goes as the inverse of that number, so an atmosphere that converts its hydrogen to molecules shrinks by nearly a factor of two in vertical extent at the same temperature and gravity.

There is also a thermodynamic consequence. Dissociating a molecule absorbs energy without raising the temperature, exactly as ionising an atom does, so the effective specific heat of the gas rises enormously in the dissociation zone and the adiabatic gradient falls. A region where hydrogen is coming apart is therefore convectively unstable almost regardless of its opacity, which is part of why the coolest stars are convective throughout and the mixing-length treatment their structure depends on has to be applied where it is least tested.

The same equilibrium that explains the spectrum sets the atmosphere’s scale height and its convective state, and in an atomic photosphere those three would be separate subjects.

What is actually measured, and what is assumed #

None of the fractions above is observable. What a plate records is a band’s depth, and turning that into a molecular abundance requires everything a line analysis requires and several things it does not.

The route runs through a model atmosphere. A temperature and pressure structure is assumed, the chemical equilibrium is solved at every depth for a few hundred species at once rather than for the single reaction drawn here, the radiative transfer is integrated through millions of molecular lines, and the resulting spectrum is compared with the observed one. The output is a temperature, a gravity and a set of abundances; the input includes every bond energy, every partition function and every line list.

Two of those inputs are the weak points and they are weak in different ways.

The partition functions cannot be approximated away. An atom’s partition function is close to its ground-state weight over a wide range of temperature, which is what lets the atomic figures in this collection carry that approximation honestly. A diatomic molecule at three thousand kelvin has of order a thousand rotational levels populated, so its partition function is in the hundreds and is a strong function of temperature. Setting it to one would move the dissociation temperature by more than the width of the whole M sequence, which is why the figures here carry rotation classically and vibration harmonically and say so. Real work carries them level by level from measured spectroscopy.

The line lists are the larger problem. Computing a spectrum dominated by molecular bands requires knowing the position and strength of every line in those bands, and for the important species that means of order $10^{9}$ transitions, most of them never measured. They come from quantum-chemical calculations anchored on the measured subset, and the accuracy of the anchoring is the dominant uncertainty in the atmospheres of M dwarfs and of the planets that orbit them. Successive revisions of a water line list have moved published temperatures of cool objects by hundreds of kelvin.

How the bands got their names #

The observational history is worth a paragraph because the bands were catalogued for decades before anybody knew what made them.

Secchi’s fourth spectral class, defined in the 1860s, was the one with broad dark bands sharp on one edge and fading on the other — the shape a band head has, though nobody described it that way. Those were the carbon stars. The M class as Cannon defined it was another set of bands, fluted the opposite way, and the classification was entirely a matter of matching appearances.

The identification came from the laboratory. A. Fowler produced the bands of titanium oxide in an arc in 1904 and pointed out the match to the M-star bands; the identification was argued about for nearly twenty years, because nobody could see why a metal oxide should exist in a star, and it was settled once the dissociation arithmetic above could be done. The same sequence happened with the carbon stars: the bands were matched to diatomic carbon and to cyanogen in the laboratory, and the reason they appeared together in one class and never with the oxides waited for the carbon monoxide argument.

That order — pattern, then identification, then explanation — is the same one the luminosity classes followed, and it recurs so often in this subject that its absence is the surprise. The classifiers were measuring a chemistry and calling it an appearance, and the measurement was good enough that the classes survived the explanation unchanged.

The subclasses of the M sequence are still defined that way. An M3 is not a temperature; it is a star whose titanium-oxide band depths at a stated pair of wavelengths match those of the standard star that defines M3. The temperature scale underneath is a calibration, and it is a hard one: the radii of a few dozen nearby M dwarfs have now been measured by interferometry, which with a bolometric flux gives an effective temperature with no atmosphere model in it at all, and those are what the rest of the scale is anchored on. There are fewer than fifty of them, and the scale for the whole population rests on that sample.

The sequence below the sequence #

The molecular argument runs out of stars before it runs out of temperature, and what happens past the end of the stellar sequence is where it now does most of its work.

Below about 2,400 K titanium oxide and vanadium oxide vanish from the spectrum — not because they dissociate, but because they condense. Titanium is removed into solid grains that settle out of the photosphere, so the bands that define the M sequence disappear over a narrow range of temperature and are replaced by the metal hydrides and alkali lines that define the L type. Below about 1,300 K methane appears, because the carbon monoxide equilibrium finally reverses in favour of $\mathrm{CH_4}$, and that is the T type.

Those two transitions define a classification of objects that are not stars, and both are chemistry rather than spectroscopy in the older sense. The spectral sequence stops being a temperature sequence of a gas and becomes a condensation sequence, which is the same physics that decides where a protoplanetary disc’s solids are applied to an atmosphere instead of to a disc.

The connection is not a metaphor. The relevant quantity in both cases is a partial pressure compared with a vapour pressure, and the reason both produce sharp transitions is the same: a vapour pressure is exponential in the inverse temperature, so a species goes from almost all gas to almost all condensate over a narrow band.

Where the argument stops #

Equilibrium is an assumption, and in the coolest atmospheres it fails. The chemistry above assumes every reaction has had time to reach its equilibrium at the local temperature and pressure. In an atmosphere where convection carries a parcel upward faster than a reaction can readjust, the abundance is frozen at the value it had deeper down. Carbon monoxide in the atmospheres of cool dwarfs and of giant planets is observed in quantities far above equilibrium for exactly this reason, and the excess is used as a measurement of the vertical mixing rate.

A single reaction is not a chemistry. The figures here solve one equilibrium at a time with the partners in excess. A real photosphere solves a few hundred simultaneously under mass conservation for every element, and the couplings matter: the titanium oxide abundance depends on how much oxygen carbon monoxide has already taken, which depends on the carbon abundance, which is itself being measured from the same spectrum.

The infrared is worse than the optical, and it is where the flux is. A cool star radiates most of its light beyond one micron, and that is where water’s bands sit — not a few hundred ångströms wide but continuous across whole photometric bands, with a structure that changes with temperature faster than anything in the optical. A magnitude measured in a near-infrared filter on an M dwarf is a measurement of how much water absorption fell inside the filter, which is why the colours of cool dwarfs are poor temperature indicators and why the useful ones are chosen to sit in the gaps between band systems rather than for any blackbody reason.

And condensation removes elements the gas-phase calculation still contains. Once grains form, the affected element is gone from the photosphere altogether, and no equilibrium among gas-phase species can say so. The transition at the bottom of the M sequence is invisible to everything in this essay.

Still open: which of the two descriptions is the approximation #

The habit worth carrying out of this is about which regime is treated as the special case.

The atomic photosphere is where the theory was built, and molecules arrive in it as a complication at the cool end. Counted by objects the arrangement is upside down: most stars are M dwarfs, most stars are therefore molecular, and the atomic photosphere is the exception that happens to include the Sun. The techniques that work superbly at six thousand kelvin — an equivalent width against a continuum, an abundance from a curve of growth, a temperature from a line ratio — are techniques for a minority of stars, and the majority has been analysed by a different method with much larger uncertainties for as long as anybody has tried.

What that costs is not yet settled. The abundances of M dwarfs matter for the planets they host, for the chemical history of the Galaxy’s most numerous population, and for the calibration of every survey that measures them in bulk; and the spread between independent analyses of the same star remains several times the quoted errors. Whether that closes with better line lists, with better model atmospheres, or with a measurement that avoids the continuum problem altogether is the question the field is currently answering, and the third possibility is the one nobody has a candidate for.