The second thing a spectrum says

A spectral type is a temperature. That is the first thing a spectrum says, and it is what the sequence O B A F G K M encodes: which lines are present, and in what proportion, is decided almost entirely by how hot the gas is, because temperature decides which atoms are ionised and which levels are populated.

It is also, on its own, not enough to tell a star’s distance — because two stars with identical spectra in that sense can be a hundred thousand times apart in luminosity. Something else in the spectrum has to separate them, and something else does. It is not which lines are there. It is how wide they are.

Why the pressure follows the gravity #

The chain from a star’s size to its line widths has three links and each is short.

Pressure at the photosphere. In a grey atmosphere, the gas pressure where the optical depth reaches about two-thirds is $P \approx \tfrac23 g/\kappa$ — the weight of the column above, divided by how opaque that column is. So at fixed temperature and composition, $P \propto g$ directly. A red giant with $\log g = 1$ has a photospheric pressure ten thousand times lower than a dwarf with $\log g = 5$.

Damping rate. A radiating atom is interrupted by collisions with its neighbours, and the interruption rate is proportional to their number density. A shorter uninterrupted train of waves is a broader line, by the uncertainty relation applied to a duration — this is one of the three mechanisms that set a line’s width, and it is the only one of the three that depends on the pressure.

Wing shape. A collision-broadened line is Lorentzian in its wings, falling as $\Delta\lambda^{-2}$ rather than as an exponential, so the wings extend enormously further than the thermal core. The point at which the line reaches a given depth therefore moves as $\sqrt{\Gamma}$, and $\Gamma \propto P \propto g$, giving the square-root law the figure measures rather than assumes.

The classes are contours #

The Yerkes or MK system labels a star with a Roman numeral — I for supergiants, III for giants, V for the main sequence — beside its spectral type. The numerals look like a taxonomy and are something better. That agreement is the whole argument, and it is worth saying why it is not circular. The gravity on the axis is computed from a mass, a luminosity and a temperature: geometry and the Stefan–Boltzmann law, with no spectrum in it. The gravity a spectroscopist quotes is fitted to line profiles, with no luminosity in it. They are two independent determinations of one number, and they agree to a couple of tenths of a dex.

A spectrum becomes a distance #

Once the class is known, so is the absolute magnitude — a G8 III sits near $M_V = +0.8$, a G8 V near $+5.6$ — and a brightness becomes a distance the moment the intrinsic brightness is known.

The name for this is spectroscopic parallax, and it is a bad name: no parallax is involved, nothing is triangulated, and the accuracy is a matter of how tightly the class predicts the magnitude rather than of any geometry. What it has over an actual parallax is reach. The trigonometric triangle runs out where the annual shift falls below the astrometric error; a spectroscopic distance depends only on being able to take a spectrum, and a spectrum of a supergiant can be taken across the Galaxy — which is why supergiants were the first tracers of the structure measured from inside it.

What was actually measured #

The classification was built before its physical meaning was established, and the sequence of events is worth having.

Antonia Maury, working on Draper Catalogue spectra at Harvard in the 1890s, noticed that some stars of a given type had unusually sharp lines and marked them with a lowercase “c” in an additional column. She had no theory for it. Ejnar Hertzsprung realised in 1905 that the c-stars were the ones with small proper motions — which, at a given apparent brightness, means they are distant, which means they are intrinsically luminous. The sharpness of a line was a luminosity indicator eighteen years before anyone knew why, and Maury’s column is the direct ancestor of the MK luminosity class.

The physical explanation followed from Saha’s ionisation equation in 1920 and Cecilia Payne’s application of it in 1925 — the work that showed why hydrogen’s lines are strongest where hydrogen is not. The MK system in its modern form — Morgan, Keenan and Kellman, 1943 — is defined not by physics at all but by a set of standard stars: a spectrum is classified by comparison against a reference spectrum taken with the same instrument, which makes the system reproducible independently of whatever theory is current.

Which collisions, and it matters which #

“Pressure broadening” is a single phrase covering three physically distinct interactions, and which one dominates decides the exponent in front of everything above.

Van der Waals broadening is perturbation by neutral hydrogen atoms, and it is what broadens the metal lines of cool stars. The interaction energy falls as $R^{-6}$, and the resulting damping constant goes as the neutral density — that is the case the opening figure is drawn for.

Stark broadening is perturbation by charged particles, and it is what broadens the hydrogen lines. Hydrogen is unusual in having a linear Stark effect — its levels are degenerate in a way that most atoms’ are not — so the interaction falls only as $R^{-2}$ and the line width goes as the electron density to the two-thirds power. The Balmer wings of an A star are the classic luminosity indicator for exactly this reason: they are enormous, they are easy to measure, and they respond to gravity more steeply than a metal line does.

Resonance broadening is perturbation by atoms of the same species, which matters only for hydrogen in cool stars and for the strongest resonance lines.

The practical consequence is that a classifier uses different features in different temperature regimes, and the switch is not arbitrary — it follows which perturber is abundant. For the hot stars it is the Balmer lines and electrons; for the cool ones it is Ca II, Mg I and neutral hydrogen. The luminosity class is one number obtained by three different pieces of physics, joined by a system of standard stars that does not care which.

What it reaches, and what has overtaken it #

For seventy years the spectroscopic parallax was the workhorse distance for anything beyond a few hundred parsecs. Gaia has changed that arithmetic completely: trigonometric parallaxes are now available for over a billion stars, at accuracies of tens of microarcseconds, which reaches most of the Galactic disc and is better than a spectroscopic distance almost everywhere the two overlap.

What survives is the part Gaia cannot do. A trigonometric parallax needs the star to be seen, at an astrometric precision that degrades with brightness and with crowding — and the most luminous stars, which are the ones visible at the greatest distances, are also the rarest and the most likely to sit in obscured regions. For a supergiant in a distant spiral arm behind five magnitudes of dust, a spectrum is still what there is.

More importantly, the relationship has inverted. Gaia now calibrates the classification rather than the other way round: with a geometric distance in hand, the absolute magnitude of every class at every type can be measured rather than assumed, and the residual scatter — which is what the method’s accuracy has always been limited by — is finally an observable instead of an estimate.

Getting the gravity from the lines without the wings #

The wings of the strong lines are the direct route to a surface gravity and they are not the route most survey spectra can take, because a wing needs resolution and signal-to-noise that a million-star survey cannot spend on each object. Two indirect routes are used instead, and both are worth knowing because they fail in different ways.

Ionisation balance. An element present in two ionisation stages must give the same abundance whichever stage it is measured from. Iron is the usual choice, because a stellar spectrum carries hundreds of Fe I lines and dozens of Fe II lines. The ratio of the two populations is set by the Saha equation, which depends on the electron pressure and therefore on the gravity — so the gravity is adjusted until the two sets of lines agree.

That is elegant and it is entangled. The same adjustment also depends on the temperature, on the abundance itself and on the model atmosphere’s structure, so what is really being done is a simultaneous fit of four quantities to one spectrum, and the gravity is the least well constrained of them. Systematic offsets of 0.2 to 0.3 in $\log g$ between different groups analysing the same spectra are routine.

Asteroseismology. A star with a convective envelope oscillates, and the frequency at which its oscillation power peaks scales as

$$\nu_{\max} \propto \frac{g}{\sqrt{T_{\rm eff}}},$$

because that frequency is set by the acoustic cut-off at the surface, which is a property of the gravity and the sound speed there.

So a photometric time series of sufficient length gives $\log g$ to about 0.01 dex — an order of magnitude better than any spectroscopic method — from a measurement that involves no lines, no model atmosphere and no abundances. The catch is that it requires weeks of continuous photometry, which exists for the few hundred thousand stars that space photometry missions have watched and for nobody else.

The two are therefore used together: the seismic gravities are the calibration and the spectroscopic ones are the survey. Fixing $\log g$ at the seismic value and refitting the spectra removes the degeneracy that made the spectroscopic gravity poor, and improves the temperatures and abundances at the same time — which is why the overlap between a seismic mission’s field and a spectroscopic survey’s footprint is one of the more valuable pieces of sky there is.

A third opinion, from the transit #

There is one more route to a stellar surface gravity, available for a small set of stars and independent of everything above.

A transiting planet’s light curve gives the ratio of the orbital semi-major axis to the stellar radius, from the transit’s duration and shape. Kepler’s third law then converts that ratio, together with the period, into the star’s mean density:

$$\rho_\star = \frac{3\pi}{GP^2}\left(\frac{a}{R_\star}\right)^3.$$

No spectrum, no model atmosphere, no distance — a density from a period and a shape.

Density is not gravity, but the two are related through the radius, so a density plus a mass gives a gravity, and a density plus a spectroscopic temperature places the star on an evolutionary track much more tightly than a spectroscopic gravity does. In practice the transit density is often used in place of $\log g$ altogether, and the spectroscopic gravity is discarded.

The three routes have almost nothing in common, which is what makes their agreement worth something: one reads the pressure at the line-forming layer, one reads an acoustic frequency at the surface, and one reads the shape of a shadow crossing the disc.

The wings are the observable and the gravity is the quantity, so both are worth drawing at a second temperature — because the relation between them depends on which atoms are doing the colliding.

The case where the gravity is the whole measurement #

There is one class of star where the surface gravity is not a nuisance parameter or a distance indicator but the primary observable, and it shows what the method can do when the effect is large.

A white dwarf has the mass of a star inside the volume of a planet, so its surface gravity is around $10^6$ times the Sun’s — $\log g$ near 8 against the Sun’s 4.44. The pressure at the line-forming layer is correspondingly enormous, and the hydrogen lines are broadened by collisions with charged particles into features hundreds of ångströms wide.

Those profiles are the measurement. Fitting the shapes of the Balmer lines against model atmospheres gives the temperature and the gravity simultaneously, and because the effect is so large the gravity comes out to about 0.05 dex — far better than for any ordinary star, from the same physics that gives 0.3 dex for a main-sequence dwarf.

And a gravity is a mass, because a white dwarf’s radius is not a free quantity: the degenerate mass–radius relation fixes $R$ given $M$, so $g = GM/R^2$ can be inverted for the mass alone. That is how the mass distribution of tens of thousands of white dwarfs is known — not from orbits, which exist for a handful, but from the widths of four hydrogen lines apiece.

The chain is worth admiring for how short it is. A spectrum gives a line profile; a line profile gives a gravity; a gravity plus one piece of theory gives a mass. No distance, no orbit, no companion, no assumption about the star’s history. It is the same measurement as the luminosity classes of this essay, applied where the effect is six orders of magnitude larger, and it returns a quantity that for ordinary stars requires two of them in orbit.

It is also the measurement that turns a spectrum into a cooling age, since a white dwarf’s radius follows from its mass and its luminosity then follows from its temperature — which places the star on a cooling track whose elapsed time is read off directly.

The white dwarf case also inverts the usual difficulty. For an ordinary star the pressure broadening has to be disentangled from thermal and rotational widths of comparable size; here it exceeds them by orders of magnitude, so the profile is a nearly pure record of one effect and the modelling burden shifts entirely onto the line-broadening theory itself.

Where the model stops #

The method has three well-known failure modes and each of them was found the hard way.

The mass enters. $\log g = \log(GM/R^2)$, so converting a measured gravity into a luminosity requires a mass, and the mass is usually assumed from the spectral type. A factor of two in mass is 0.3 dex in $\log g$, which is a factor of two in luminosity and 40% in distance. That is the floor on a spectroscopic parallax and it cannot be improved by better spectra.

Metallicity is a third dimension. Fewer metals means less opacity, which raises the pressure at optical depth two-thirds and broadens the lines — so a metal-poor dwarf can mimic a metal-rich star of higher gravity. The MK system is calibrated on nearby stars of roughly solar composition, and applying it to halo stars produces systematic errors that were not understood until the 1950s.

Reddening. The apparent magnitude has to be corrected for dust before the modulus can be taken, and an uncorrected extinction makes a star look further away. The correction is itself estimated from colours, and a star’s colours are what the temperature was read from, so the errors are correlated. And the profile fitted with the two velocity fields the gravity measurement has to be separated from, since both of them broaden a line and neither of them is a pressure.

Where this ladder goes next #

The rung below reads composition from which lines are missing; this rung reads pressure from how wide they are. The next reads motion.

A line’s shape carries a rotation as well — a star’s projected rotation broadens every line by the same velocity, with a characteristic flat-topped profile that is quite unlike a Lorentzian and can be separated from it. Beyond that lies magnetic splitting, which duplicates lines rather than broadening them, and the asymmetries that convection puts into line bisectors, which are how granulation on another star is detected.

Each of those is one more number pulled out of a single feature, and they are the reason the phrase “the spectrum of a star” understates what is being recovered: a temperature, a gravity, a composition, a rotation, a magnetic field, a radial velocity and a turbulence, from a single trace of intensity against wavelength.