An element that was only empty space

On 29 August 1864, William Huggins pointed a telescope with a spectroscope attached at NGC 6543, a small bright planetary nebula in Draco. He expected the dark-line spectrum of a star, since many people believed nebulae were clusters of stars too distant to separate. What he saw was three bright lines on a dark background, which meant at once that this object was a thin luminous gas. One line was the hydrogen line that would later be called $\mathrm{H}\beta$, and the other two, green, near 5007 and 4959 ångströms, corresponded to nothing.

There was every reason for a confident reading. Spectroscopy had just shown a spectrum to be a chemical signature. In 1868, a line seen in the Sun’s chromosphere that matched nothing on Earth was attributed to an unknown element, which then turned up in a laboratory in 1895 as helium. By 1900 the same reasoning had been applied to these green lines, whose unknown source was given the name nebulium. Others were found: the coronal lines of an imagined element, coronium; an auroral green line from an unexplained source in the Earth’s own upper atmosphere.

Nebulium ran into trouble around 1914. Moseley’s X-ray work put every element in a numbered sequence, with only a few gaps. No gap left a place for a light element with so strong a line. The element had nowhere to go. In 1927 Ira Bowen, a physicist at Caltech, recalculated the energy levels of ionised oxygen, nitrogen and other common ions, and found that some of the low-lying transitions matched the nebular lines exactly. Those lines come from doubly ionised oxygen. They are absent from laboratory spectra not because they can’t be emitted but because a laboratory is too full.

An ion and its two ways down #

Consider an ion in a gas of electrons. An electron collision may lift it from its ground state to an excited level. The low levels of $\mathrm{O}^{2+}$ lie about two and a half electronvolts up, very convenient for a nebula at $10^4$ K with thermal electrons of about one electronvolt, enough to get into the tail of the distribution. Once in the excited level, there are two routes down. It may radiate. Or another electron may hit it and take the energy away.

For most atomic transitions, the first route wins by a huge margin. An ordinary, allowed transition has a lifetime of ten nanoseconds. For this level, the quantum selection rules that govern electric-dipole transitions do not allow it: the ion’s configuration of electrons is unchanged, and the parity is the same. The transition has to go by a higher-order process, and it is slower by a factor of some billions. Its radiative rate is about 0.02 per second. On average, the ion waits fifty seconds to emit.

That is the argument in one figure. The rate at which an ion is de-excited by collisions is proportional to the electron density and the radiative rate is constant, so the two cross at the critical density $n_c=A/q$. That is the density at which half of the excited ions will radiate. For a gas below it, every collision that excites the ion results in a photon, and the energy is lost from the gas. Above it the ion has its energy stolen before it can emit.

At the density of the best laboratory vacuum of Huggins’s day, around a millionth of an atmosphere, there are still about $10^{13}$ particles per cubic centimetre, nearly ten million times the critical density. And a discharge tube has walls: an excited ion in a tube at low pressure hits a wall in far less than fifty seconds. Nothing in the experimental physics of the 1860s or the 1920s could allow those lines to appear. A line that is common throughout the universe could not be produced by any physical means available on Earth.

The word “forbidden” has stayed in use, and it misleads. The transition has a definite rate that quantum mechanics calculates. It is a slow rate, and that is all. Where the line is not seen, it is because something else is faster. For this reason it would be better to say that the forbidden line is the natural behaviour of the atom, and a laboratory’s dense environment is what forbids it.

A line that sorts the gas #

Suppose the density is varied. The emission per unit volume per ion, for excitation by electrons and a chance of escape $1/(1+n/n_c)$, is

$$\varepsilon \propto \frac{n\,q_{12}\,h\nu}{1+n/n_c}.$$

At low density, every excitation gives a photon. The line’s strength per ion is proportional to the number of electrons and the total emission goes as $n^2$. At high density it saturates, with the populations of the levels at their equilibrium Boltzmann values. Then the emission per ion no longer depends on density.

That sorting of lines by density makes a spectrum a kind of density map. Each forbidden line turns off above its own critical density, and the critical densities differ by many orders of magnitude from one line to another, so a list of which lines are present says what range of densities is there. A quasar’s spectrum shows a broad region near the black hole with allowed and semi-forbidden lines, but no strong forbidden lines. Lines like [O III] are narrow only, and come from a separate, thin, much larger gas. That separation of the broad-line and narrow-line regions was one of the first findings about quasars, and it came from nothing more than this one fact: at the density of the broad-line gas, about $10^{10}$, forbidden lines are suppressed. So the absence of a broad [O III] line in those spectra is itself a density measurement.

A similar logic applies in the sense in which a stellar luminosity class measures a density. But stars are in or near thermal balance, and these lines come from a very different regime. An astrophysical spectrum, depending on its forbidden lines, says which kind of gas it comes from, in a way that needs only a look at a list of wavelengths.

A gauge for density #

Having two lines from one ion, with almost the same energy but different critical densities, gives something very useful. Temperature affects both equally, and the abundance of the ion cancels exactly. Their ratio depends on density alone.

This is one of the most used diagnostics in optical astronomy. Two lines separated by 15 ångströms, measured with a modest spectrograph, tell the density of a gas that is thousands of light years away. The same trick, with a density read from line ratios rather than from a stellar atmosphere, applies to the doublet of [O II] at 3726 and 3729 Å, and it works for distant galaxies at redshifts of several, using the spectrograph on the James Webb Space Telescope. It is remarkable that a physical measurement so direct can be made so far away, with no calibration at all, apart from the atomic physics. The atomic physics was largely calculated rather than measured, because the forbidden transitions cannot be measured in the laboratory.

Note the limitation. The diagnostic has a range. Below a density of about a hundred the ratio sits at its low-density limit and gives only an upper bound on the density. Above about ten thousand it has saturated at the other end, giving a lower bound. Only in between does it return a number. Different pairs of lines with different critical densities therefore cover different ranges of density, and a real survey would use several.

A thermometer from a higher level #

A ratio between levels of different energy gives a thermometer. For $\mathrm{O}^{2+}$, the strong 5007 and 4959 lines come down from an excited level about 2.5 electronvolts up, whereas the much weaker line at 4363 Å comes from a higher level, about 5.3 electronvolts up. Electrons must have the energy to reach it. In a gas at $10^4$ K the number with five electronvolts is small and strongly dependent on temperature.

The ratio can be read directly as an electron temperature, and with it one can then turn a line’s strength into an abundance. For that reason this temperature measurement has been at the core of chemical abundance studies of galaxies for half a century. It is what gives the oxygen abundance of an H II region, by what is called the direct method.

However, a curious problem arises. The weak line at 4363 Å is very faint, and it can only be detected when the gas is fairly hot, so the direct method tends to work only in metal-poor regions. Metal-rich regions, including those near the Sun’s abundance, are cool, and in them the line is too faint to detect at any great distance. To go further the method has been replaced by empirical calibrations of strong lines. Those disagree among themselves, at the level of up to a factor of several in oxygen abundance, and the question of what the absolute scale is, is not unlike other discrepancies in abundances. There is another wrinkle, in that fainter recombination lines of the same ion give an abundance higher by about a factor of two than the forbidden lines, in the same regions, with the discrepancy unresolved for several decades. It is one of the oldest standing puzzles in nebular astrophysics.

Why a nebula stays near ten thousand degrees #

A surprising consequence of this physics is that forbidden lines keep a nebula cool. Ionising photons deposit energy in the gas, at a rate related to the temperature of the star. Each photon’s excess over the ionisation threshold becomes kinetic energy in an electron. If there were no other process, this would heat the gas to the point where recombination radiation balanced, tens of thousands of kelvin.

Collisions with an ion then excite the low-lying levels, and each resulting line photon escapes the nebula without being reabsorbed, because the nebula is transparent to it. That’s a very efficient drain of heat, tuned to a thermal electron energy near an electronvolt.

Hence the near-universal temperature of nebulae. Whatever the input, in a solar-metallicity gas the balance falls near $10^4$ K, within a factor of less than two. Oxygen at a few parts in ten thousand can take away most of the energy from the gas, because the [O III] lines happen to be the ones whose energy lies just above the thermal energy of the electrons. At lower temperature few electrons have the energy to excite them. At higher temperature the cooling increases fast. The temperature is thus set by the atomic structure of oxygen, and the strength of the star’s radiation has a minor influence on it, which explains why it took decades for the first measurements to show any real variety.

It follows that the most prominent lines of a nebula are also its thermostat. The green colour of the Orion Nebula’s central region, through a telescope, is a direct view of the process keeping the gas at a fixed temperature. In a metal-poor nebula, as the plot shows, there is too little oxygen and the gas runs hotter, with stronger [O III] per unit oxygen, until at very low metal abundance hydrogen’s own Lyman-alpha cooling becomes the thermostat instead. And that sets an upper limit on the [O III] brightness of a galaxy that can be reached at any metallicity, a fact used to identify the extremely metal-poor galaxies found in the early universe.

What the two-level atom misses #

A real ion has more than two levels. The ground term of $\mathrm{O}^{2+}$ has three fine-structure levels, which themselves produce infrared forbidden lines at 52 and 88 micrometres, with much lower critical densities. Those contribute to the cooling as well. A full model needs at least five levels and collision strengths that have been computed, at great effort, by quantum scattering calculations. The sulphur ratio’s high-density limit, 0.57 in the figure against the true 0.44, shows the size of the error from the simple approach.

The gas is a single value. Nearly every nebula has a range of densities and temperatures along the line of sight. A ratio then gives some weighted mean, and the weight is different for each diagnostic. High-density clumps will be under-represented in lines that have low critical densities. The combination is enough to produce apparent discrepancies between diagnostics, and it may be part of the explanation for the difference with recombination lines.

The cooling curve is normalised. The chart gives the correct shape of the thermostat, and its absolute temperature has been normalised at the solar case, and a real calculation needs dozens of ions.

A spectrum alone doesn’t say it is nebular gas. Shocks also excite forbidden lines, at temperatures of a different kind. The ratio [O III] 5007 to $\mathrm{H}\beta$ alone cannot separate a photoionised region from a shock, though other lines can. The classic diagnostic diagrams exist for that purpose, sorting star-forming regions, active nuclei and shocks.

Coronium, geocoronium and the rest #

The same resolution applied, in time, to the other invented elements. The green aurora at 5577 Å was at first attributed to a “geocoronium,” which would have to be a light gas extending high above the atmosphere. McLennan and Shrum in Toronto showed in 1925 that it is neutral atomic oxygen, by producing it in a tube under carefully arranged conditions: a large vessel with helium and a little oxygen, where an excited oxygen atom could avoid collisions with walls and wait for the emission. That was the first laboratory measurement of such a line, and a nice piece of experimental ingenuity. Its lifetime is around a second, well below that of the nebular lines.

Coronium proved to be iron. Bengt Edlén and Walter Grotrian in 1939–1941 showed that the coronal green and red lines are forbidden transitions of very highly ionised iron, $\mathrm{Fe}^{13+}$ and $\mathrm{Fe}^{9+}$. That identification had a large consequence: an atom with thirteen electrons stripped needs a temperature of a million kelvin or so. It was the first proof that the solar corona is that hot, and a corona at that temperature was so far from expectation that it was doubted for a while. And so the second invented element led to one of the biggest problems of solar physics, which remains partly unsolved.

So each of these false elements was a real discovery of a state of matter. In each case the mistake was to assume that the new line required a new substance, when in fact it required a new condition. It is a pattern which the same chain of inference has often hit, and one that gives good reason to be careful about the first interpretation of an unknown line. A similar puzzle has outlasted them all. The diffuse interstellar bands, seen since 1919, have never been fully identified, with only a few assigned to a large carbon molecule.

A planetary nebula as a laboratory #

Planetary nebulae are the natural place in which to see all this at once, and that is why the first discovery was made in one. A star at the end of its life has blown off its outer layers, and its exposed hot core ionises them, in the kind of nested sphere that a count of photons produces. The density is a few thousand per cubic centimetre, well below the critical density of the strongest lines. Oxygen is at a few parts in ten thousand. Those conditions make [O III] by far the strongest visible line, typically several times the strength of hydrogen’s $\mathrm{H}\beta$, and it can carry as much as a tenth of the light of the star.

This has a useful consequence for distances. The brightest planetary nebulae in any galaxy have nearly the same [O III] luminosity, in old and young populations alike. So their luminosity function has a sharp cut-off, and observers, with a narrow filter at 5007 Å and another just off it, can find hundreds of them in galaxies many megaparsecs away, as points blinking on and off between the filters. No other emission gives such high contrast against the galaxy’s starlight. The reason that bright cut-off is so universal is still something of a puzzle, because the brightest nebulae are thought to come from fairly massive stars that old galaxies do not have. It remains a good distance measure that nobody fully understands.

So a transition with a lifetime of fifty seconds, in an ion that was once thought to be a new element, has become a tool for measuring the Hubble flow.

Still open: why the two ways of counting oxygen disagree #

Oxygen’s abundance in a nebula can be measured in two ways. The forbidden lines, as above, are bright and they depend exponentially on temperature. Oxygen also has much fainter recombination lines, which arise when an $\mathrm{O}^{3+}$ ion picks up an electron and cascades down. These depend only weakly on temperature. In principle they ought to agree, and in almost every nebula where both can be measured, they disagree by a factor of two or more.

The disagreement has been known for more than fifty years. Suggested explanations include temperature variations within the nebula, which would inflate the forbidden-line result’s weakness since it would be biased to the hot parts; cold, metal-rich clumps that emit recombination lines and no forbidden lines; and non-thermal electron energies in the gas, because a photoionised gas may not have had time to thermalise fully. Each has been argued and each has difficulties. The practical consequence is a factor of two uncertainty on the oxygen abundance scale of the universe, through a method that looks like a very direct measurement. Whatever the outcome, this seems again to be a question of what state the gas is in, and the atoms have the same message as they did in 1927: read what the conditions are, before concluding what the substance must be.