A width that knows how bright the star is

The calcium K line is the deepest feature in the visible spectrum of a cool star: a broad, dark trough several nanometres wide near 393 nanometres, formed by singly ionised calcium, an ion abundant in the photosphere and extremely opaque at that wavelength. At its centre, in stars like the Sun and cooler, there is something unexpected — a small bright emission feature, a fraction of a nanometre wide, sometimes split by a dark central reversal. The bright core is made in the chromosphere, the thin layer above the photosphere where the temperature, after falling outward, begins to rise again.

Olin Wilson spent decades measuring those emission cores at Mount Wilson, because their brightness tracks a star’s magnetic activity and varies through its activity cycle. In 1957, with Vainu Bappu, he reported something about their widths that had nothing to do with activity. The width of the emission core, measured in velocity units, was tightly correlated with the star’s absolute magnitude — its true luminosity — and the correlation held over the whole range of cool stars, from dwarfs fainter than the Sun to supergiants a hundred thousand times brighter, with a scatter of about half a magnitude.

The relation is remarkable for what it ignores. The emission’s strength varies by large factors between stars of the same luminosity, and in a single star through its cycle; the width does not. The star’s temperature changes the whole spectrum; the width hardly cares. A measurement made on a narrow piece of one line in a star’s spectrum gives its luminosity to a factor of 1.6, and its distance to about a quarter.

A bright core in a dark line #

The core is bright because of where it forms. A line forms at the depth where it becomes opaque, and in a line as strong as calcium K that depth varies enormously across the profile: the far wings form deep in the photosphere, the core forms at the top of the chromosphere, and every frequency in between samples a different height. Reading the profile from the wings inward is reading the atmosphere’s source function from the bottom upward.

In the photosphere the temperature falls outward, and so does the source function, so moving from the wings towards the core the intensity falls: the broad dark trough. Above the temperature minimum, a few hundred kilometres up in the Sun, the temperature rises again through the chromosphere, and where the line’s source function is still coupled to the temperature it rises with it. Frequencies whose light comes from those layers are brighter than those from the temperature minimum below, and the profile shows emission peaks. Above that, in the uppermost layers, the line’s photons scatter many times before being destroyed and the source function decouples from the temperature, falling towards the square root of the destruction probability times the Planck function: the dark central reversal.

The labels for the features go back to the early spectroscopists: K1 for the dark minima on either side of the emission, K2 for the emission peaks, K3 for the central reversal. The width Wilson and Bappu measured is essentially the separation of the outer edges of the emission, close to the K1 minima, and those minima are the key to the physics.

Why the width grows with the column #

The K1 minima are the frequencies whose light comes from the temperature minimum. On the far side of each — further from line centre — the light comes from deeper, hotter photospheric gas; on the near side, from the rising chromosphere. The minima’s position in the profile is therefore set by the frequency at which the line reaches an optical depth of about one at the depth of the temperature minimum.

At that frequency the line’s opacity comes from its damping wings, which fall off as the inverse square of the distance from line centre. The optical depth at a distance x from line centre down to the temperature minimum is the column of calcium above the minimum times the wing opacity, which goes as $1/x^2$. Setting it to one gives

$$x_{\rm K1} \propto \sqrt{m_{\rm min}},$$

the distance of the minima from line centre grows as the square root of the column of gas above the temperature minimum. A star with ten times more gas above its temperature minimum has an emission core about three times wider. The schematic profile shows exactly that scaling, and it is the physical content of the width: the core’s width is a measurement of how much chromosphere a star has above its photosphere, in grams per square centimetre.

The law has a lower edge, and the figure starts above it on purpose. The argument needs the K1 minima to sit in the damping wing, where the opacity falls as the inverse square of the distance from line centre. For a thin enough column the temperature minimum is reached at a frequency still inside the line’s Doppler core, where the opacity falls as a Gaussian instead, and there the width stops tracking the column and is set by the thermal and turbulent velocities of the gas. In this model that happens below a line-centre optical depth of about $10^{4.5}$ at the temperature minimum. Where a real dwarf’s chromosphere sits against that edge is a question for a model atmosphere rather than this schematic one, but the relation’s straightness down to the faintest dwarfs it was calibrated on suggests they lie above it, in the regime where the width measures the column.

How a column becomes a luminosity #

A column of gas becomes a luminosity through the surface gravity. A giant star has a much lower surface gravity than a dwarf of the same mass, because its radius is larger, and lower gravity means an extended atmosphere: the pressure at the temperature minimum is set by the weight of the gas above it, and for a given pressure there is more gas above when gravity is weaker. In Thomas Ayres’s analysis of 1979, the column above the temperature minimum scales roughly as the inverse square root of gravity, which makes the width scale as the gravity to the minus one-quarter.

At a fixed mass and temperature, a star’s luminosity is proportional to the square of its radius and so to one over its gravity. A width scaling as $g^{-1/4}$ is then a width scaling as $L^{1/4}$, and the luminosity–width relation that follows has a slope of ten magnitudes per factor of ten in width.

The theory gets the essentials right. It explains why the width depends on luminosity, through gravity; why it hardly depends on temperature, because the temperature dependence of the column partly cancels; and why it is insensitive to activity, because the activity heats the chromosphere and brightens the emission without much changing how much gas lies above the temperature minimum. It does not get the slope. The stars spread over fifteen magnitudes while their widths spread over a factor of ten, and the simplest scaling would need a factor of thirty.

Several things could bend the slope. Giants and supergiants are not more luminous dwarfs of the same mass; along the relation, mass and temperature change too, and both enter the scaling. The chromospheres of giants are not simple scaled-up solar chromospheres: their temperature minima sit at different column densities, and their turbulent velocities, which broaden the line, are larger. Each refinement moves the predicted slope towards the observed one; none has reproduced it from first principles cleanly enough to replace the empirical calibration. The relation remains better established than its explanation.

Measuring a width that has no sharp edges #

The emission core does not have edges; it rises from the K1 minima to the K2 peaks and falls again, and a width has to be defined before it can be measured. Wilson and Bappu used the separation of the outer edges of the emission at roughly half its height above the K1 minima, measured by eye on photographic plates, and the practice since has been to fit the profile and take a width defined in the same spirit. The definition matters at the level of the relation’s scatter: two observers measuring the same stars with different conventions can differ by several per cent in width, which is a few tenths of a magnitude.

The instrument contributes too. The emission core of a dwarf is about thirty kilometres a second wide, a few hundredths of a nanometre at 393 nanometres, and a spectrograph must resolve it — a resolving power of at least twenty thousand, preferably more — and its own instrumental width must be subtracted, in quadrature or by fitting. Several broadening mechanisms widen a line in the same way, and the instrumental profile is one more; at the resolution of the original photographic work it was a significant fraction of the dwarfs’ widths, and part of the original relation’s scatter at the faint end was instrumental.

Three routes to a luminosity #

The Wilson–Bappu effect is one of three ways a spectrum gives a star’s luminosity, and they read different layers. The luminosity classes of the spectral classification are set by ionisation balances and line ratios that depend on the photosphere’s electron pressure; a supergiant’s thin photosphere ionises its metals more readily. The widths of pressure-broadened lines read the same photospheric pressure through collisions. Both are, at bottom, measurements of the photosphere’s density, which is set by the surface gravity.

The K-line core reads the chromosphere instead — the column of gas above the temperature minimum, a few hundred kilometres higher. It is set by the same gravity, through the weight of the overlying gas, but by a different physical route and in a different layer, so its errors are independent of the photospheric indicators’. That independence is what made it valuable as a check, and it is why the three methods were combined in the classification of cool giants long before parallaxes could adjudicate.

All three, ultimately, place a star on the diagram that sorted the stars by luminosity at a given temperature. The Hertzsprung–Russell diagram was first drawn with luminosities from parallaxes and from spectroscopic indicators of this kind, and its vertical axis owes as much to the widths and ratios of lines as to geometry.

A relation that survived the parallaxes #

Wilson and Bappu calibrated the relation with the few stars whose distances were known in 1957 — nearby dwarfs with trigonometric parallaxes, and giants in clusters at known distances. The calibration rested on a handful of points and a long extrapolation. When the Hipparcos satellite measured parallaxes for tens of thousands of stars in the 1990s — distances that are not simply one over the parallax when the parallax is small — the relation was recalibrated against precise distances for hundreds of stars with measured K-line widths, and it survived almost unchanged: the slope and zero point moved by small amounts, and the scatter stayed near half a magnitude.

That survival is the strongest evidence that the relation reflects physics rather than a selection of the original sample. A relation found in a few dozen stars and confirmed in hundreds, measured with an independent method of a different kind, is unlikely to be an accident. And the scatter that remains has structure: it correlates with metallicity, with rotation and with how the width is measured, which suggests that a more complete theory would reduce it.

A width as a distance #

The practical use of the relation was as a distance indicator for cool stars too far away for parallaxes. A spectrum of moderate resolution, enough to resolve an emission core a few tenths of an ångström wide, gives the width; the relation gives the absolute magnitude; the apparent magnitude gives the distance.

The method is imprecise. Half a magnitude of intrinsic scatter is a quarter in distance even with a perfect measurement, and the steep slope amplifies every error in the width: a width wrong by ten per cent is a magnitude wrong by two-thirds. Parallaxes from space have made it unnecessary for the stars of the solar neighbourhood. But it requires only a spectrum, works for any cool star with a visible chromospheric core, and is independent of the star’s colour and reddening, which spoil photometric distances. It remains useful for stars in crowded or dusty fields where colours are unreliable, and as a check on luminosities derived in other ways.

The width and the activity cycle #

The emission core’s strength was what Wilson set out to measure, and the Mount Wilson program that he began in 1966 followed the calcium emission of about a hundred nearby stars for decades. It found that many cool stars have activity cycles like the Sun’s — the emission rising and falling over years as magnetic regions come and go — and that the cycle’s character depends on the star’s rotation and age. That record is the foundation of what is known about how stellar activity behaves through a cycle in stars other than the Sun.

The width is remarkably indifferent to all of it. As the emission strengthens through a star’s cycle the core’s width changes by a few per cent at most, much less than its brightness. The indifference follows from the physics: the activity heats the chromosphere and raises its source function, brightening the emission, but the K1 minima are fixed by the column above the temperature minimum, which the activity barely changes. Two measurements of the same line — its brightness and its width — record two unrelated properties of a star, its magnetism and its gravity.

The same structure in other lines #

Calcium K is not the only line with a chromospheric emission core. The magnesium II h and k lines in the ultraviolet, accessible only from space, have deeper and broader emission, and their widths show a Wilson–Bappu relation of their own, with a similar slope. Hydrogen’s Lyman-alpha line, formed in the transition region above the chromosphere, has a width that also correlates with luminosity. The existence of the same relation in different lines, formed at different heights with different atomic physics, is further evidence that it is a statement about the star’s gravity and the structure of its outer atmosphere rather than a property of one ion.

The temperature reversal that makes the emission is the same one that makes a limb brighter than the centre at wavelengths formed above the temperature minimum, and that turns dark lines into emission at the limb during an eclipse. The chromosphere was discovered in exactly that way, as a bright pink rim of emission lines seen for a few seconds at the start and end of totality, and the K-line emission core is the same rim seen against the disc of a star too far away to eclipse.

Where the relation stops #

The relation applies only to stars that have a chromosphere visible in calcium, which means stars cooler than about early F type. Hotter stars have no convective envelope to drive the magnetic and acoustic heating that makes a chromosphere, and their K lines show no emission core at all. At the cool end, the coolest M dwarfs have cores so faint that the width is hard to measure against noise, and the relation’s scatter grows.

Within its range, three things widen the scatter. Composition: at fixed gravity a metal-poor star has a different column of calcium above its temperature minimum, and the width shifts accordingly, so a calibration made on stars near the Sun’s composition misplaces the metal-poor ones by a few tenths of a magnitude. Binarity: a spectrum that combines a giant and a dwarf shows two emission cores superposed, and a single width fitted to the blend means neither star. And rapid rotation, which smears the core beyond its intrinsic width in the few cool stars that spin fast — mostly young stars and close binaries — and makes them look more luminous than they are.

None of these is a failure of the physics described above; each is a second parameter the simple gravity scaling leaves out. The relation is a one-parameter description of stars that vary in at least three, and its half-magnitude scatter is about what the other two parameters would produce.

Two astronomers and a plate collection #

Olin Wilson spent most of his career at Mount Wilson on the calcium emission of cool stars, and the photographic plates he and his collaborators accumulated from the 1930s onward were the raw material of the relation. Vainu Bappu, then a young astronomer from India working at Mount Wilson and Palomar, worked with him on the measurements. Bappu went on to build observational astronomy in India, directing the Kodaikanal observatory and founding the observatory at Kavalur whose large telescope now bears his name. The relation carries both their names, and it is one of the few results in stellar spectroscopy found entirely by measurement and still used through the measured calibration rather than through any theory.

Still open: the slope from first principles #

The Wilson–Bappu relation is used through an empirical calibration because no model of stellar chromospheres reproduces it from first principles across the full range of luminosity. Modern chromospheric models are built star by star, fitted to many lines at once, and they reproduce individual stars’ K-line profiles well; what they have not done is predict, from the physics of how chromospheres are heated, why a supergiant’s chromosphere has the column it does relative to a dwarf’s. Chromospheric heating — by acoustic waves, by magnetic dissipation, in proportions that vary from star to star — is itself not settled, and the Wilson–Bappu slope is one of the cleanest quantitative tests any theory of it has to pass. A theory that heats chromospheres correctly should produce a slope of fifteen magnitudes per decade of width. None yet does without adjustment.