A line darker than any temperature in the gas

The usual explanation of a dark absorption line is a statement about temperature. The light leaving a star at each wavelength comes from roughly the depth where the optical depth at that wavelength reaches two-thirds, and at the wavelength of a strong line the gas is so opaque that this depth lies much higher in the atmosphere than at neighbouring wavelengths. The temperature falls outward, so the line’s light comes from cooler gas and is fainter. A line is a thermometer reading of a higher layer.

That explanation assumes the gas at each depth emits like a black body at its local temperature — that its source function, the ratio of emission to absorption, equals the Planck function. The assumption is local thermodynamic equilibrium, and it holds when collisions between particles are frequent enough to set the populations of the atomic levels. For the strongest lines in a star’s spectrum, which form in its thin outer layers, it fails. Photons in those lines are absorbed and re-emitted many times before a collision can turn one into heat, and the radiation, not the temperature, decides how bright the gas is in the line.

The consequence is dramatic enough to state as a paradox. Take a gas at one temperature throughout. In equilibrium it can make no absorption line at all, since every depth it is seen from has the same brightness. Let the line photons scatter instead of thermalise, and the same gas makes a line almost black at its centre.

Scattering, destruction and escape #

The simplest model that shows the effect has two atomic levels. An atom in the lower level absorbs a photon at the line’s frequency and is excited. It can then do one of two things. It can re-emit a photon in the same line, in a random direction — the photon has been scattered, and it continues its random walk through the gas. Or a collision with an electron can de-excite it before it radiates, turning the photon’s energy into heat — the photon has been destroyed. The probability of destruction per absorption is ε, the ratio of the collisional de-excitation rate to the sum of that and the radiative rate.

Collisions also work the other way: an electron can excite an atom, which then radiates a photon in the line. That is how line photons are created, and the rate at which collisions create them, compared with the rate at which collisions destroy them, is set by the gas’s temperature. In equilibrium creation and destruction balance at the Planck function. A line source function can depart from it only because photons leave — and photons leave only from near the surface.

Deep in the atmosphere, a photon is destroyed long before it can random-walk to the surface, every photon created is destroyed nearby, and the source function is the Planck function. Near the surface, photons created there have a good chance of escaping instead of being destroyed, and the gas loses more line photons than collisions replace. The line’s source function falls below the Planck function. How far below depends on ε, and the answer is a single, exact result.

The square-root law #

The source function at the surface of an isothermal, semi-infinite atmosphere of two-level atoms is

$$S(0) = \sqrt{\varepsilon}\,B,$$

the Planck function times the square root of the destruction probability. The result is exact for complete redistribution of frequencies on scattering, and it is independent of the line’s profile, its strength and the details of the geometry. It is one of the few exact results in radiative transfer, and it says that a gas whose atoms destroy one line photon in a million absorptions has a line source function at its surface a thousand times below its Planck function.

The square root has a simple origin. A photon created at optical depth τ in a scattering medium random-walks, and to escape it must reach the surface, which takes of order $\tau^2$ scatterings. Each scattering risks destruction with probability ε, so it survives about $1/\varepsilon$ scatterings, and it can escape from depths up to about $1/\sqrt{\varepsilon}$. Below that thermalisation depth, the gas does not know there is a surface. Above it, the deficit of escaping photons grows towards the surface, and at the surface itself the balance between creation and loss settles at $\sqrt{\varepsilon}$ of the Planck value.

The two-stream treatment drawn here is an approximation, and for a line whose photons are redistributed over its Doppler profile on each scattering it underestimates the thermalisation depth: photons scattered into the line’s wings, where the gas is more transparent, travel much further in one step, and the true thermalisation depth for a Doppler profile is nearer $1/\varepsilon$ than $1/\sqrt{\varepsilon}$. The surface value, $\sqrt{\varepsilon}$, is the same either way.

A line in a gas at one temperature #

The emergent line follows from the source function by the same rule that made the line a thermometer: the light at each frequency comes from where the optical depth at that frequency is about two-thirds. In the far wings, where the line adds little opacity, that depth is in the continuum’s layers, the source function there is the Planck function, and the intensity is the continuum’s. At line centre the depth is far up, where the line’s optical depth is two-thirds and its source function has fallen to near $\sqrt{\varepsilon}$ of the Planck function. The line is dark not because the gas there is cool but because it is short of photons.

That is the opening figure, and the comparison with equilibrium is the point. A thermometer reading of an isothermal gas gives the same temperature at every depth, and an equilibrium spectrum of it is a featureless continuum. A scattering line in the same gas has a central intensity of a few hundredths of the continuum and a profile shaped by the line’s opacity, deepest where the line is most opaque and recovering to the continuum where the line’s opacity falls below the continuum’s.

The line’s strength relative to the continuum matters less than might be expected.

The central depth barely moves, because it is set by the source function where the line’s optical depth is two-thirds, and that depends on ε and not on how much stronger the line is than the continuum. What the line’s strength sets is how far into the wings the deficit extends — a stronger line stays opaque further from its centre and is broader. The depth and the width carry different information.

The core as a measure of collisions #

The central intensity therefore measures ε, and ε measures something other than temperature.

The destruction probability is the ratio of the collisional de-excitation rate to the total, and the collisional rate is proportional to the electron density. A line’s core depth is, to first approximation, a density measurement.

The numbers explain why the strongest lines of the Sun and similar stars cannot be read as thermometers. The resonance lines of sodium, calcium, magnesium and hydrogen form in the upper photosphere and the chromosphere, where electron densities are far too low for collisions to keep up with radiative decays, and their cores are set by the escape of photons. Their central intensities are a few per cent of the continuum — deeper than any temperature in the gas would allow in equilibrium — and the analyses that derive abundances from them must solve the transfer of radiation and the statistical equilibrium of the atomic levels together. That is what the phrase “non-LTE correction” in an abundance analysis means: a recomputation of the line with the populations set by the radiation as well as the collisions, which for the strongest lines can change the inferred abundance by a factor of two.

A second density measurement hiding in a spectrum #

A spectrum already contains one well-known density measurement. The ionisation balance of an element depends on the electron pressure through the Saha equation, so a luminosity class is a density measurement — a supergiant’s thin photosphere ionises its metals more readily than a dwarf’s at the same temperature — and the widths of pressure-broadened lines are the second thing a spectrum says, after composition. The scattering line’s core is a third route to the same quantity, and it works where the other two cannot.

The ionisation balance and the pressure broadening both read the density of the photosphere, where most of the continuum forms. The scattering core reads the density in the much thinner layers above, where the strongest lines form and where the ratio of collisions to radiative decays sets ε. That is why non-equilibrium line formation, once treated as a nuisance to be corrected for, became one of the principal tools for diagnosing the chromosphere: the depth and shape of a resonance line’s core, interpreted with the $\sqrt{\varepsilon}$ physics, give the electron density and temperature of a layer that emits almost nothing in the continuum.

The same physics extends to the magnetic field. The polarisation of light scattered in a strong line depends on how the scattering atoms are oriented, and a magnetic field reorients them; a line whose photons are scattered many times carries the imprint of the field on its scattering polarisation. Measuring a field from a line that will not visibly split relies on the same departure from equilibrium that darkens the line’s core.

How the square root was found #

The idea that a line’s source function need not be the Planck function is as old as the theory of stellar atmospheres. Arthur Schuster and Karl Schwarzschild in the first decade of the twentieth century modelled lines as pure scattering in a layer above the photosphere, and Edward Milne and Arthur Eddington in the 1920s wrote down the equations of a line formed by a mixture of scattering and true absorption, with a parameter equivalent to ε. What they lacked was a reason to take the scattering seriously, since most lines appeared to behave thermally.

The reason came from the Sun’s chromosphere, and from the realisation in the 1950s, associated above all with Richard Thomas, that the populations of the levels in the outer solar atmosphere are controlled by the radiation field rather than by collisions. The exact square-root result for the surface source function and the thermalisation depth for different line profiles were established in the 1960s by Eugene Avrett and David Hummer, by solving the transfer equation numerically for the two-level atom. Their results turned a qualitative idea into a set of numbers that every later treatment has reproduced, and the $\sqrt{\varepsilon}$ law is the one most often quoted.

Hydrogen as a test case #

The Balmer lines of hydrogen are an instructive intermediate case. They arise from the second level of hydrogen, whose population is a sensitive function of temperature through the Boltzmann factor, and in equilibrium that sensitivity makes the Balmer lines superb thermometers of hot stars. In cool stars like the Sun the Balmer line cores form high enough that the second level’s population is partly set by the Lyman-alpha radiation field — itself a scattering line, trapped and multiply scattered — and the cores depart from equilibrium while the broad wings, formed deeper, do not. A temperature derived from a Balmer line’s wings is therefore reliable and one derived from its core is not, which is the general pattern: the parts of a strong line formed deep are thermometers, and the parts formed high are not.

Where equilibrium is a good approximation #

Most of a star’s spectrum is not like this. The thousands of weak metal lines that dominate a cool star’s spectrum form deep in the photosphere, where the density is high, collisions are frequent and ε is close to one. For them local equilibrium is an excellent approximation, and the standard analysis that reads temperature and gravity from line strengths works. The departure grows for stronger lines, which form higher, and for lines of minority species whose populations are sensitive to the radiation field — iron’s neutral lines in metal-poor stars, whose thinner atmospheres let ultraviolet radiation over-ionise the iron, are the best-studied case.

The decisive quantity is always the same: the ratio of the collision rate to the radiative rate where the line forms. A line formed where that ratio is large is a thermometer. A line formed where it is small is a probe of the radiation field and the density, and reading it as a thermometer gives a temperature too low.

The same physics with a temperature rise #

The isothermal case isolates the effect, but real atmospheres have a temperature that falls outward through the photosphere and then rises again in the chromosphere. A scattering line formed through that structure carries both influences: where the line’s source function is still coupled to the temperature, it follows the temperature’s rise and makes the line’s core brighter; where it has decoupled, it falls towards the $\sqrt{\varepsilon}$ value regardless. The strongest lines in cool stars show exactly this — a central emission peak where the chromospheric rise is seen, with a dark reversal at the very centre where the topmost layers have decoupled.

That is the same reversal that makes the limb brighter than the middle in wavelengths formed above a temperature rise, and the same relation that ties a line’s appearance to the gradient of its source function rather than to the temperature at one depth. The emission cores of calcium’s H and K lines are the clearest case, and their width turns out to measure something unexpected about the star: its luminosity.

What the model leaves out #

The two-level atom is the simplest atom that scatters. Real resonance lines belong to atoms with many levels, and photons absorbed in one line can be re-emitted in another; the destruction probability then includes those channels as well as collisions, and it can be larger or smaller than the collisional value. Scattering in real lines is not completely redistributed in frequency: in the far wings of the strongest lines a photon is re-emitted close to the frequency it was absorbed at, which changes the wing profile substantially and is essential for the solar calcium and magnesium resonance lines. And real atmospheres are not plane-parallel and static: the chromosphere is a dynamic, structured layer, and a line’s core samples a range of conditions along each line of sight.

None of these changes the central fact, which is that a line whose photons scatter more often than they are destroyed is dark for reasons that are not thermal. The $\sqrt{\varepsilon}$ law is the clean statement of it, and it survives every refinement as the scale of the effect.

The general point: brightness is a balance, not a temperature #

A source function is a ratio of emission to absorption, and it equals the Planck function only when the processes that create and destroy photons are in balance with each other locally. Near a boundary that balance is broken, because photons escape through the boundary without being destroyed, and the deficit is felt to a depth set by how far a photon wanders before its destruction. The same reasoning governs the microwave background, whose spectrum became a perfect black body only because, at early times, photons were destroyed and created far faster than they could escape; once the creation processes became too slow, distortions could no longer be erased. A star’s strongest lines are the same physics in miniature: a gas that would be a perfect black body if its photons could not leave, and is not because they can.

Still open: how far equilibrium extends in the stars that matter most #

Modern abundance work for the Sun and nearby stars routinely includes departures from equilibrium for the elements where they are known to matter. For the oldest, most metal-poor stars — the ones whose compositions record the first generations of nucleosynthesis — the departures are largest, because their atmospheres are thinner, their electrons fewer and their ultraviolet radiation less blocked, and they are least well calibrated, because few laboratory collision rates exist for the atoms and conditions involved. Abundances in those stars change by factors of two depending on whether equilibrium is assumed. Whether the trends of composition with age that are drawn from them reflect the chemical history of the Galaxy, or partly the growing failure of equilibrium in thinner atmospheres, is a question that depends on collision cross-sections nobody has yet measured.