A luminosity class is a density measurement

The first rung of this anchor took the Saha equation and used it to explain why the Balmer lines peak at ten thousand kelvin: a competition between excitation, which needs heat, and ionisation, which removes the atoms. It held the electron pressure fixed at a solar value throughout.

That pressure is in the denominator, and it is not the same everywhere.

$$\frac{N_{i+1}}{N_i} \;=\; \frac{2kT}{P_e}\left(\frac{2\pi m_e kT}{h^2}\right)^{3/2}\frac{Z_{i+1}}{Z_i}\,e^{-\chi/kT}$$

More free electrons about means more recombinations, so a denser photosphere holds an element neutral to a higher temperature. And how dense a photosphere is depends on the star’s surface gravity, which is what separates a dwarf from a supergiant.

The consequence has been the second axis of stellar classification since the 1940s, and it is worth stating in the form it deserves. Two stars can have identical colours, identical temperatures, and spectra that no classifier would confuse — because one is a hundred times less dense than the other at the level where its light escapes. The information that separates them is a pressure, and the instrument that reads it is the same equation that gives the temperature.

A second consequence, less often noticed: molecules run the other way. A molecule of two atoms is two particles becoming one, so by the same principle a denser photosphere favours the molecule — which is why the titanium-oxide bands that define the M sequence are stronger in a dwarf than in a giant of the same temperature, and why the coolest luminosity criteria have the opposite sign to the ones drawn here.

The history is worth a paragraph because the two halves of the story arrived thirty years apart. Annie Jump Cannon’s classification, completed in the 1910s, was one-dimensional: a sequence of spectral types ordered by the appearance of the spectrum, later understood to be a temperature sequence. Antonia Maury had already noticed, in the 1890s, that some stars of a given type had unusually narrow and sharp lines — she gave them a separate designation and Hertzsprung realised those were the intrinsically luminous ones. It took until Morgan and Keenan’s atlas of 1943 for the second axis to be systematised, and until the Saha equation was applied quantitatively for anybody to say what it was.

The order of discovery is the usual one for this subject: the pattern was seen, then named, then explained, and the explanation turned out to be a single term in an equation that had been available since 1920.

Why the pressure follows a square root #

The relation between gravity and photospheric pressure is the one approximation in this essay and it is worth deriving rather than asserting.

The photosphere is the depth at which the optical depth is about two thirds. Integrating hydrostatic equilibrium down to that depth gives a gas pressure

$$P_{\rm gas} \;\approx\; \frac{g}{\kappa}$$

where κ is the opacity per unit mass. In a cool star that opacity is dominated by H⁻ — a hydrogen atom holding a second, loosely bound electron — and the abundance of H⁻ is proportional to the number of free electrons available to be captured. So κ is proportional to the electron pressure, and

$$P_e \;\propto\; P_{\rm gas} \;\propto\; \frac{g}{P_e} \qquad\Longrightarrow\qquad P_e \propto \sqrt{g}$$

A square root, from a self-consistency between the opacity and the thing the opacity is made of. It is worth noticing how peculiar that is: the pressure at a star’s surface depends on the opacity, the opacity depends on the free electrons, and the free electrons depend on the pressure. The photosphere finds its own level, and the exponent that comes out of the loop is not one that could be guessed from either half.

The relation is also the reason this is an approximation rather than an identity, and the figures say so. In a hotter star the opacity is not H⁻ but bound-free absorption from metals or, hotter still, electron scattering, whose opacity does not depend on the electron pressure at all — and then $P_e \propto g$ rather than $\sqrt{g}$. The square root is a cool-star result and the essay’s figures are cool-star figures. Over the range from a dwarf at log g = 4.4 to a supergiant at 1.2 that is a factor of forty in pressure, and by the Saha equation a factor of forty the other way in the ionisation ratio.

The size of the effect, in the units a classifier uses #

It is worth converting the numbers into the units the classification is actually written in, because a factor of forty in pressure sounds enormous and a spectral class is a narrow thing.

A luminosity class spans roughly one and a half in log g — from 4.4 for a dwarf to 3.0 for a giant, and 3.0 to about 1.2 from a giant to a supergiant. That is a factor of five in pressure per two classes, and by the Saha equation a factor of five in the ionisation ratio at fixed temperature.

A spectral subclass spans about two hundred kelvin in the middle of the main sequence, and by the first figure that is worth roughly the same factor in the ionisation ratio as one luminosity class. So the two axes are comparably sensitive, which is exactly the condition for a two-dimensional classification to be worth having: an axis far less sensitive than the other would be a decoration.

That balance is not designed. It is a coincidence of the numbers, and it is why the MK system’s two axes are both about ten steps long.

What a classifier actually measures #

The figures above are fractions, and a fraction is not observable: a line’s strength depends on how much of the element there is as well as on what state it is in.

What is observable is a ratio of two lines from two stages of one element, and that ratio divides the abundance out exactly.

Two things follow from that figure and both are historical as well as physical.

A spectrum carries two independent numbers. Which lines are present is mostly a temperature; the ratios between stages of one element are mostly a pressure. Neither is clean — the Saha equation contains both variables and so does every line ratio — but the two dependences have different shapes, an exponential and a first power, so a spectrum with lines of several ionisation potentials separates them. That is why a two-dimensional classification was possible in the 1940s, before anybody could compute either axis — the classifiers were reading a temperature and a density and calling them a type and a class.

A luminosity class is a gravity, and only then a luminosity. The name is a historical accident of the correlation: at a given temperature a low-gravity star is large, and a large star at that temperature is luminous. The spectrum knows nothing about the luminosity. It knows the pressure.

The distinction is not pedantry, and there is a case where it matters. A white dwarf has a surface gravity of log g near 8 — four orders of magnitude above a main-sequence star — and it is among the least luminous things in the sky. Its spectrum is unmistakable, and what makes it unmistakable is the pressure: Balmer lines hundreds of ångströms wide, and ionisation ratios that belong to nothing else. Classify it by its “luminosity class” and the class would mean the opposite of the name. The quantity being read is the density, and the naming survives because for ordinary stars the correlation holds.

There is a caveat about which lines, and it is not a small one. The criterion works best when the two stages have comparable populations, because a ratio of a large number to a tiny one is measured badly. For calcium at six thousand kelvin the neutral fraction is under one per cent, so the Ca I line is weak and the ratio is measured on its weakness — which is why the classical luminosity criteria in that temperature range use strontium and iron rather than calcium, chosen precisely because their ionisation potentials put both stages in play at the temperature concerned.

So a classifier uses different criteria at different temperatures, not because the physics changes but because the pair of lines whose ratio is well measured changes. That is a practical consequence of the same equation, and it is the reason the MK system’s criteria are a table rather than a formula.

Seeing it in the pressures themselves #

Setting the same element’s behaviour at the two extremes of pressure side by side makes the size of the effect plain.

Both of those figures are ionisation fractions, and no spectrum shows one. What a spectrum shows is a line’s strength, and a line counts the atoms in one particular level of one particular stage — so the excitation has to be put back on top of the ionisation before either figure can be read off a plate.

A third effect that is not ionisation #

The luminosity criteria a classifier uses are not all ionisation ratios, and it is worth naming the other one because it points the same way for a different reason.

A line’s width, away from its core, is set by pressure broadening: collisions with neighbouring particles interrupt the emitting atom and smear the transition’s energy. The broadening is proportional to the number density of perturbers, so a dwarf’s lines have broad damping wings and a supergiant’s do not. For hydrogen, where the mechanism is the linear Stark effect and the perturbers are charged, the effect is enormous — the Balmer wings of an A dwarf are tens of ångströms wide and those of an A supergiant are barely wider than the thermal core.

That is the sharpest luminosity criterion in the hot half of the diagram, and it is what Maury was seeing. It is independent of the ionisation argument and it depends on the same quantity, which is the reason a spectrum determines the gravity better than either criterion alone would.

What is actually measured #

None of the above is a fraction, and it is worth walking from the observation to the number.

A classifier compares the depths of two absorption lines on a photographic plate or a digital spectrum, and a depth is not a population — it is a point on a curve of growth whose shape depends on the broadening as much as on the number of absorbers. Those depths depend on the number of absorbers in the lower level of the transition, on the transition’s oscillator strength, on the amount of broadening, and on how the line is formed relative to the continuum — so the mapping from a depth ratio to a stage ratio is a curve of growth and a model atmosphere, not a division.

For classification, none of that is done. A classifier matches the spectrum against standards — a small set of stars whose types and classes were assigned by an earlier generation and are the definition rather than a measurement of anything. The physics in this essay explains why the matching works; it plays no part in the matching.

There is a subtlety in the standards worth naming, because it is where the physics leaks back in. The standard stars were chosen to be typical of their class, and “typical” was assessed at solar composition. A metal-poor star has fewer metals to donate electrons, so its photospheric electron pressure is lower at the same gravity, so its ionisation ratios look like a lower-gravity star’s — and a classifier matching it against solar-composition standards assigns it too luminous a class. That is a real and well-known bias, it is worth a class or more for the most metal-poor stars, and it is invisible to anybody who does not already suspect it.

For a quantitative analysis it is all done, and the output is a surface gravity with an uncertainty of about 0.15 in the logarithm, which is a factor of 1.4 in gravity and 1.2 in pressure. That is the accuracy with which a star’s density can be read off its light, and it is the weakest of the three fundamental parameters a spectrum yields — much worse than the temperature and worse than the composition read from what is missing.

The reason is visible in the equation. The temperature enters through an exponential and the pressure through a first power, so the same quality of spectrum constrains the first far better than the second — which is the same trade a temperature and a gravity that push against each other describes from the fitting side.

What the density says #

It is easy to lose track of what has been measured. A surface gravity is $GM/R^2$, so a spectrum that yields a gravity has said something about the star’s mass and radius jointly — and combined with a temperature and the Stefan–Boltzmann law it yields a luminosity, and with an apparent brightness a distance.

That chain is the spectroscopic parallax, and it works to about twenty-five per cent in distance for a single star — a rung of the ladder whose errors compound rather than a measurement in its own right. Twenty-five per cent is poor beside a trigonometric parallax and it has one enormous advantage: it does not degrade with distance. A spectrum of a supergiant at ten kiloparsecs gives the same fractional distance as a spectrum of one at ten parsecs, because the physics being read is the star’s own and not the geometry of the observation.

The whole of that rests on the pressure term. Remove it and a spectrum gives a temperature and nothing else, a star could be any size, and the only distances available would be the geometric ones.

The generalisation #

The structure worth extracting is that an equilibrium constant with a density in it turns a spectrum into a barometer.

Any reaction whose two sides have different numbers of particles shifts with pressure, by Le Chatelier’s principle, and ionisation is such a reaction: one atom becomes two particles. So the ionisation state of anything is a joint measurement of temperature and density, and separating them requires two observables — which a spectrum, with lines from many species at many ionisation potentials, has in abundance.

The same shape appears wherever this collection finds a two-parameter equilibrium. The ionisation front in a reionising universe is set by the same equation with the density supplied by cosmic expansion. Molecular equilibria in cool atmospheres are the same statement with dissociation in place of ionisation, and they are why the coolest stars are classified by molecular bands rather than by atomic lines. The ionisation that sets where a spectral line forms is the same equation again, read at a depth rather than at a surface. In every one of these cases the observable is a ratio of populations and the unknown is a pair of numbers, so one measurement is never enough and two of the right kind always are.

There is a second reading, about what makes a diagnostic clean. The line ratio in the third figure works because the abundance divides out exactly — the two lines come from the same atoms, so however many there are, the ratio does not care. That exactness is worth more than sensitivity: a diagnostic ten times more sensitive but carrying an abundance would be worse, because the abundance is not known. Look for the quantity that cancels before looking for the quantity that responds.

The corollary is a rule about classification schemes generally. When an empirical classification turns out to be two-dimensional, look for two independent variables in the physics — the classifiers were measuring something, and the number of axes they needed is a measurement of how many things.

Where the ladder goes next #

The next rung takes the same equation to the coolest stars, where atoms are not the story. Below about 4,000 kelvin the photosphere contains molecules — titanium oxide, water, carbon monoxide — whose dissociation equilibria obey the same arithmetic with a different energy, and whose bands dominate the spectrum so thoroughly that the continuum cannot be found. Classification there is a matter of band ratios, and the pressure dependence is stronger rather than weaker.

Further rungs on this anchor: the departure from local thermodynamic equilibrium, which breaks the Saha equation entirely in a thin enough atmosphere and does so worst in exactly the supergiants this essay’s extremes are; the partition functions, held at their ground-state weights throughout here, which matter at the tens-of-per-cent level and are themselves temperature-dependent sums; helium’s second ionisation, which is what sets the temperature scale of the hottest stars; and the ionisation of a stellar wind, where the density falls as the inverse square of the distance and every stage is present somewhere.