A temperature that belongs to none of the gas

Most of the atomic gas in a galaxy has never been seen in visible light. It is too cold to glow, too thin to absorb starlight measurably, and spread through a volume so large that its total mass rivals that of the stars in some galaxies. What makes it observable at all is a single, almost absurdly weak transition: the flip of the electron’s spin relative to the proton’s in the ground state of hydrogen, which releases a photon of 21 centimetres wavelength, 1420.4 megahertz. An isolated atom waits about eleven million years to make it. There are so many atoms that the line is one of the brightest in the radio sky, and it is what the rotation curves of galaxies are traced in, far beyond the edge of their light.

The line can be seen two ways. Pointed at an empty patch of sky, a radio telescope sees the gas glowing: emission. Pointed at a bright distant radio source — a quasar, a radio galaxy, a supernova remnant — it sees the same gas cut a notch out of the source’s continuum: absorption. The two measurements look at the same atoms. They do not weight them the same way, and the difference between their weightings is the whole of what can be learned from them together. It is a temperature, and it is not the temperature of any of the gas.

Emission counts atoms, absorption counts them over their temperature #

The two levels of the transition are populated in a ratio fixed by a temperature, the spin temperature $T_s$, which in most of the interstellar medium is driven by collisions and by scattered Lyman-alpha photons to be close to the gas’s kinetic temperature. The energy of the transition corresponds to only 0.068 kelvin, so even the coldest interstellar gas has the two levels populated almost exactly in the ratio of their statistical weights, three to one. That single fact sets the behaviour of both measurements.

In emission, each atom in the upper level decays at the same tiny rate whatever the temperature, and since a fixed three-quarters of the atoms are always in the upper level, the light emitted per atom is the same at 50 kelvin as at 5,000. If the gas is transparent to its own line, the brightness integrated over the line is simply proportional to the number of atoms along the sight line:

$$N_{\rm HI} = 1.823\times10^{18}\int T_B\,dv \quad \mathrm{cm^{-2}},$$

with the brightness temperature $T_B$ in kelvin and the velocity in kilometres per second. No temperature appears. That is what makes the 21-centimetre line the standard way of weighing atomic gas.

In absorption the situation is different. An atom in the lower level absorbs; an atom in the upper level, stimulated by the passing photon, emits an identical one in the same direction. With the levels almost equally populated per state, the two processes almost cancel, and what survives is a net absorption proportional to the small excess of the lower level — which goes as $h\nu/kT_s$. The optical depth is

$$\tau = \frac{N_{\rm HI}}{1.823\times10^{18}\,T_s\,\Delta v},$$

per unit velocity width. Absorption counts atoms divided by their temperature. Cold gas absorbs a hundred times more strongly per atom than gas at a hundred times the temperature, and the warm gas that fills most of the volume barely absorbs at all.

The figure shows both spectra for one simple sight line, and the asymmetry is immediate. In emission the warm gas is a third of the signal; in absorption it is invisible. The cold cloud’s emission peak is capped well below its column’s worth because the cloud is close to opaque at line centre: its brightness temperature cannot exceed its own temperature, however many atoms are added. In absorption it is the dominant feature, with an optical depth just over one.

A temperature read off each channel #

Because the two measurements weight the gas differently, their ratio carries information neither has alone. For gas at a single temperature, radiative transfer gives the emission as $T_B = T_s(1 - e^{-\tau})$, and the absorption measures $\tau$ directly from the fraction of the background source’s light that gets through. The ratio $T_B/(1-e^{-\tau})$ is then the spin temperature, exactly, at every velocity.

That is the method of the emission–absorption surveys, which pair a spectrum towards a radio source with emission spectra taken just beside it on the sky, where the same gas is seen without the source. It needs bright background sources, of which there are a few hundred useful ones across the whole sky, and it needs care with the fact that the emission beside the source is not the emission in front of it. But in principle it is a thermometer that reaches gas no other instrument can.

Read channel by channel along a real sight line, the thermometer gives something stranger than a temperature. At line centre, where the cold cloud dominates both spectra, it reads close to the cloud’s own temperature — a little high, because warm gas in front of the cloud adds emission without adding absorption. In the line wings, where only warm gas contributes, it reads thousands of kelvin. In between it passes smoothly through every value, and no part of the gas has those intermediate temperatures. The channels are mixing the cold and warm gas in proportions that change with velocity.

The harmonic mean is what the line reports #

The number most often quoted is not channel by channel but integrated: the total emission divided by the total absorption. For a sight line of several components that are each transparent, that ratio is

$$\langle T_s \rangle = \frac{\sum_i N_i}{\sum_i N_i/T_i},$$

the column-weighted harmonic mean of the components’ temperatures. The structure follows directly from the two weightings: the numerator is what emission counts and the denominator is what absorption counts.

A harmonic mean is dominated by its smallest terms. If half the column is at 60 kelvin and half at 6,000, the arithmetic mean temperature of the atoms is about 3,000 kelvin; the harmonic mean is 119.

This is why the early surveys found “typical” spin temperatures of a hundred to a few hundred kelvin and why that number was for a long time read as the temperature of interstellar gas. It is the temperature of none of it. A measured integrated spin temperature of 250 kelvin says, in this two-phase picture, that about a fifth of the column is cold. It is a mixing fraction disguised as a thermometer reading, and the curve in the figure is the conversion between the two.

The conversion assumes the gas comes in two components, cold and warm, and knows their temperatures. That assumption is the content of the two-phase medium, in which thermal balance allows gas to sit stably near 50–100 kelvin or near 5,000–10,000 and drives anything in between into one or the other. The emission–absorption surveys are the most direct test of it, because decomposing the spectra into Gaussian components gives a temperature for each component, warm and cold separately. The result has been less clean than the theory. The cold components cluster where they should, at tens of kelvin. A substantial share of the warm gas — between a fifth and a half, depending on the survey and on how the decomposition is done — comes out at temperatures between 500 and 5,000 kelvin, in the range the thermal balance says should be unstable. Either that gas is being stirred out of equilibrium faster than it can cool, by the same turbulence that holds the gas layer open, or the decomposition is assigning widths that are partly turbulent to temperature. The thermometer that works is not quite sharp enough to tell.

Cold gas that hides from the emission #

The second consequence of the two weightings is more serious for anyone weighing gas. The optically thin formula counts atoms only if the gas is transparent. Cold gas, absorbing a hundred times more per atom, stops being transparent at modest columns.

When a cloud’s optical depth approaches one, its emission saturates. The brightness temperature of an opaque cloud is its own temperature and no more, whatever its column. A 60-kelvin cloud contributes at most 60 kelvin of brightness at line centre, and doubling its column then adds almost nothing to the line — the same saturation that makes a strong absorption line in a stellar spectrum stop growing in depth and grow only in its wings. A column read from the emission under the thin assumption then misses atoms, and the colder the gas, the earlier it starts.

The figure gives the size of the problem for a single cloud. At the columns typical of high Galactic latitude, a few times $10^{20}$ atoms per square centimetre, the correction is small. At $10^{21}$ and above, which describes much of the Galactic plane and every dense cloud complex, cold gas at 40 kelvin is undercounted by a factor of two. The correction depends on the temperature, which emission alone does not reveal; that is the trap. A spectrum that shows a flat-topped line is a hint, since saturation flattens the peak, but warm gas behind and in front fills in the profile and blurs the shape.

What a sight line cannot say about itself #

A real sight line crosses several clouds at several temperatures and velocities, embedded in warm gas whose column varies from place to place. The correction then depends on how the cold gas is divided — one thick cloud hides more than three thin ones of the same total — and that division is invisible in the emission.

Two things come out of the ensemble. The mean correction grows with the apparent column, as the single-cloud figure predicts, but more slowly, because many sight lines at high column are high because they contain more warm gas or several thin clouds rather than one opaque one. And the scatter at fixed apparent column is as large as the mean correction, so a survey that applies one correction factor to every pixel of a map is right on average and wrong almost everywhere. Only absorption against a background source, along that sight line, measures the correction directly. That is a few hundred measurements across the whole sky, against the millions of pixels in an emission map.

The surveys that have done this — the Arecibo Millennium Survey and its successors — find that over most of the sky at high Galactic latitude the thin column underestimates the true one by ten to thirty per cent, consistent with the moderate end of the figures here. The correction becomes larger, and more uncertain, towards dense regions. One analysis that used the dust’s far-infrared emission as an independent gauge of the total gas found corrections up to a factor of two over large areas of the sky, attributing them to optically thick cold atomic gas. The emission–absorption pairs, which measure the opacity directly, mostly do not see corrections that large, and the disagreement turns on what the dust is assumed to be doing — whether the dust per atom is constant, which is itself what the dust emission is being used to measure.

The gas that is in neither census #

The question matters because of a long-standing discrepancy. Comparing the gas mass inferred from 21-centimetre emission, plus the molecular gas traced by carbon monoxide, with the total gas implied by the dust or by the gamma rays produced when cosmic rays strike nuclei, there is more gas than the two tracers account for. The excess is concentrated around molecular clouds, in envelopes where the gas is transitioning from atomic to molecular. It has been called dark gas, and it amounts to tens of per cent of the atomic mass in the solar neighbourhood.

Two candidates for it make different predictions. One is optically thick atomic hydrogen of the kind this essay has been about: cold, dense atomic gas that the emission undercounts. The other is molecular hydrogen in the outer layers of clouds where the hydrogen has already turned molecular but carbon monoxide, which needs more shielding from ultraviolet light to survive, has not yet formed — molecular gas without the molecule normally used to see it. The first predicts deep, narrow 21-centimetre absorption towards the excess regions; the second predicts weak absorption and strong emission from ionised carbon, which survives where carbon monoxide does not. Where both tests have been made, the molecular explanation carries most of the weight, with optically thick atomic gas a minority contribution. Why the hydrogen turns molecular where it does — at a surface density that turns out to be set by the dust per atom — is the next question about this gas.

The same ratio, ten billion years ago #

The harmonic-mean argument has its cleanest application far outside the Milky Way, in gas no emission measurement can reach. A quasar’s spectrum crosses every galaxy along the line of sight, and the densest of those crossings show up as damped Lyman-alpha absorbers: columns above $2\times10^{20}$ atoms per square centimetre, measured from the damping wings of the hydrogen’s ultraviolet resonance line. That column counts every atom regardless of temperature, as 21-centimetre emission would. When the quasar is also a bright radio source, the same gas can be seen in 21-centimetre absorption against it, which counts atoms over temperature. The ratio is the integrated spin temperature of a galaxy’s gas billions of years ago, measured through the same two weightings.

The results are systematically higher than in the Milky Way. Local sight lines with comparable columns give harmonic-mean temperatures of a few hundred kelvin; many of the distant absorbers give a thousand or more, and several give no detectable 21-centimetre absorption at all. Read through the curve above, a spin temperature of a thousand kelvin means the cold phase holds only a few per cent of the column. The absorbers with the highest spin temperatures tend to be the ones with the fewest heavy elements, measured from their metal lines in the same spectrum.

That trend has a physical reading. The cold phase exists because gas can cool to tens of kelvin, and at those temperatures the cooling is done almost entirely by fine-structure lines of carbon and oxygen. With fewer metals the cooling is weaker, the pressure needed for a cold phase to exist is higher, and at a given pressure less of the gas can be cold. A galaxy poor in metals should therefore hold its atomic gas mostly warm, and its harmonic-mean temperature should be high. The 21-centimetre ratio is then not only a mixing fraction but, through the metals, a record of how far a galaxy’s chemistry had come. Two complications sit beside the trend: a quasar’s radio emission is often more extended than its ultraviolet point, so the two measurements may not sample the same gas, and the quasar itself can supply ultraviolet that warms the absorber. The trend has survived attempts to correct for both, though not with the precision the local surveys reach.

Why a weak line is so useful #

It is worth pausing on the physics that makes all of this possible, because it is unusual. In most spectral lines emission and absorption both depend strongly on temperature, and disentangling them requires a model of the atmosphere — the problem that runs through the formation of every line in a star. The 21-centimetre line escapes that because its energy is so small. The upper and lower levels are always populated in the ratio of their weights; the emission per atom is a constant of nature; and the temperature enters only through the tiny imbalance that sets the absorption. Each measurement depends on the temperature in the simplest possible way — not at all, and inversely — and that is why their combination is clean enough to be a thermometer, imperfect only because the sky does not keep its phases separate.

The same simplicity makes the 21-centimetre line the tool of choice for the early universe, where the spin temperature is set against the microwave background rather than against a radio source, and a cold gas casts a shadow on the background by exactly the absorption described here. There the issue is not a mixture of phases but the history of the temperature itself, and the line is being asked to measure it before any star had formed.

What the figures assume #

Every figure here treats each component of the gas as a single Gaussian at a single temperature and velocity, adds turbulent broadening as a fixed width, and assumes the spin temperature equals the kinetic temperature. Each simplification matters somewhere. The warm gas’s spin temperature can fall below its kinetic temperature at low density, where collisions are too rare to couple the levels and Lyman-alpha scattering is needed to do it, so the warm phase’s absorption is a little larger than the figures assume. Real cold clouds have internal temperature gradients, warmer at their edges, and are not single slabs. And the emission beside a radio source is not the emission in front of it, since the source’s beam and the telescope’s beam sample different gas — an uncertainty that dominates the error budget of the best surveys for the weakest, warmest components.

None of these changes the central structure. Emission counts atoms; absorption counts atoms over temperature; their ratio is a harmonic mean; and cold gas thick enough to matter is the gas the emission undercounts.

Still open: how much of the warm gas is really in between #

The emission–absorption surveys have become sensitive enough to detect the faint, broad absorption of the warm phase directly, by stacking many sight lines, and to fit its temperature rather than assume it. What they find is a warm phase cooler on average than thermal balance allows, with a substantial fraction in the range where gas should not linger. Whether that fraction is real — gas held out of equilibrium by turbulence, by heating from shocks, or by the intermittent passage of supernova remnants — or is a product of fitting Gaussians to profiles whose widths are partly turbulence rather than temperature, decides whether the two-phase picture is a description of the medium or only of its endpoints.

A cleaner answer needs the turbulent and thermal widths separated, which means comparing the 21-centimetre line with a line of a heavier atom formed in the same gas, whose thermal width is smaller by the square root of the mass ratio while its turbulent width is the same. Absorption lines of ionised metals in the ultraviolet do this towards bright stars, but they sample different gas, at different distances, through different apertures. The thermometer the 21-centimetre line supplies is the best the interstellar medium has, and the question it was built to answer — whether there are two temperatures and nothing in between — is still not settled by it.