Two counts of the same oxygen, and the gas each one sees

The bright green light of a planetary nebula was once taken for an element nobody had found. It turned out to be doubly ionised oxygen emitting from a level that, in any laboratory gas, would be emptied by collisions long before it could radiate — a forbidden line, possible only in gas so thin that an excited ion waits undisturbed for the many seconds the transition takes. Those lines became the standard way to count the heavy elements in ionised gas, in nebulae round dying stars, in the regions round young massive stars, and in galaxies across most of cosmic time, where the metals a galaxy keeps are read almost entirely from them.

There is a second way to count the same oxygen. When a three-times-ionised oxygen ion captures an electron, the electron cascades down through the levels of the doubly ionised ion and emits a series of faint lines on the way, the recombination lines. They are a few thousand times fainter than the forbidden lines, and measuring them took the large telescopes and sensitive spectrographs of the last few decades. They should give the same answer. In almost every nebula where both have been measured, they do not: the recombination lines give more oxygen, typically twice as much in star-forming regions and between twice and several times as much in planetary nebulae, and in a few planetary nebulae more than a hundred times as much.

The ratio is called the abundance discrepancy factor, and it is one of the oldest unresolved problems in nebular astrophysics. This essay draws the two leading explanations side by side. Each produces a discrepancy, but they do it by different routes, they need different amounts of something unexpected, and — the part that matters most for the galaxies — they disagree about which of the two counts is the right one.

Why galaxies are counted with the bright lines #

It helps to see how different this is from counting elements in a star. A stellar spectrum reads composition from what is missing: absorption lines formed in an atmosphere whose temperature and pressure are tied together by hydrostatic balance and radiative transfer, so that one model fixes both and the line strengths follow. The strength of an absorption line depends on temperature too, and on gravity, which is the second thing a spectrum says, but the atmosphere’s structure is constrained tightly enough that abundances come out to a few hundredths of a dex for the best-studied stars.

A nebula has no such structure. It is a cloud of gas held at about ten thousand kelvin by the balance between the energy the star’s ionising photons deposit and the energy the gas radiates away, and the balance can differ from place to place. The heavy elements are both the tracer and part of the thermostat: oxygen’s forbidden lines are among the main coolants, so a nebula richer in oxygen runs cooler. The temperature cannot be predicted; it has to be measured from the gas itself, from the very lines that are then used to count the oxygen.

Yet the forbidden lines are what every distant measurement uses, because they are bright. A star-forming galaxy at a redshift of two shows its [O III] and [O II] lines, its hydrogen lines and its [N II] lines in a spectrum taken in an hour; its recombination lines of oxygen, a thousandth as bright, are beyond reach. The oxygen abundance of a galaxy, and so the record of how much of its gas has been processed through stars and how much has been lost or replaced by inflow, is read from forbidden lines everywhere, calibrated against the nearby regions in which both kinds can be measured. If the two disagree in those nearby regions, the calibration of the whole scale is in question.

Two lines from one ion, with opposite tempers #

The two kinds of line respond to temperature in opposite ways. A forbidden line is excited by collisions with free electrons, and only electrons with more energy than the excitation can do it. The green [O III] line at 500.7 nanometres comes from a level 2.5 electronvolts up, so its emissivity per ion goes as the Boltzmann factor $e^{-28\,760\,\mathrm{K}/T}$ and nearly doubles between ten and fourteen thousand kelvin. The auroral line at 436.3 nanometres comes from a level more than twice as high and changes much faster, which is exactly why the ratio of the two is the standard thermometer of an ionised nebula: it measures how many electrons can reach the higher level compared with the lower.

A recombination line works the other way. Capture is most likely for slow electrons, so the emissivity falls gently with temperature, roughly as its inverse. Between ten and fourteen thousand kelvin it drops by a quarter.

That difference makes the two counts sensitive to different things. An oxygen abundance from a forbidden line depends exponentially on the temperature used to interpret it, and a ten per cent error in temperature is a forty per cent error in oxygen. An abundance from a recombination line barely cares about temperature, but it is so faint that it is measured in only the brightest nebulae, and it responds to any gas, however cold, that contains the right ions. A nebula with a single uniform temperature gives the same oxygen from both. A nebula without one need not.

Two thermometers that part company #

Manuel Peimbert proposed in 1967 that the temperature of a real nebula is not uniform. The gas near the star is hotter than the gas at the edge, shocks and dense knots add structure, and the temperature at each point is set by a balance of heating and cooling that need not be the same everywhere. He described the variation by a single number, $t^2$, the mean square of the fractional departure from the mean, weighted by where the hydrogen emits.

If the temperature fluctuates, a thermometer reads a weighted mean, and the weights depend on the thermometer. The auroral-to-nebular ratio of [O III] is dominated by the hot gas, because the auroral line rises so steeply with temperature; it reads high. The Balmer jump — the step in the continuum emitted when free electrons recombine with protons, whose size measures how many slow electrons there are — is weighted towards the cool gas; it reads low. For a nebula with a mean temperature of ten thousand kelvin and $t^2$ of 0.04, the two thermometers differ by nearly two thousand kelvin; at 0.08, by nearly four thousand.

That divergence is observed. In planetary nebulae the Balmer jump temperature is systematically lower than the [O III] temperature, often by one to three thousand kelvin, and the nebulae with the largest gaps tend to have the largest abundance discrepancies. On its face it is evidence for the fluctuation picture. It is also exactly what the other picture predicts, for a different reason: a cold component emits recombination continuum and no forbidden lines, so it drags one thermometer down and leaves the other alone.

How much fluctuation a factor of two needs #

The fluctuation picture explains the discrepancy as an error in the forbidden-line count. The observer measures the [O III] temperature, which reads high; interprets the green line at that temperature, and so assumes each ion emits more strongly than the average ion really does; and concludes there are fewer ions than there are. The recombination count, insensitive to temperature, is nearly right. The discrepancy factor that results grows with $t^2$, and faster in cooler nebulae, where the Boltzmann factor is steeper.

The question is how much fluctuation is needed. In this calculation, with a Gaussian spread of temperatures and every abundance computed at the [O III] temperature, a factor of two needs $t^2$ of about 0.05 at six thousand kelvin, 0.08 at eight thousand and 0.12 at ten thousand — root-mean-square variations of twenty to thirty-five per cent. Analyses done in Peimbert’s first-order formalism, which treats the abundances slightly differently, find values for star-forming regions of 0.03 to 0.06. Either way it is far more than the gas is expected to have. Photoionisation models of a nebula round a hot star, which compute the temperature at each point from the balance of heating and cooling, give $t^2$ of a few thousandths up to about 0.02, which produces a discrepancy of at most twenty per cent.

So the fluctuation picture needs an extra source of temperature structure several times larger than the known physics provides. Candidates have been proposed — shocks, dissipation of turbulence, cosmic-ray heating, dense knots with different cooling, and electrons whose energies follow a distribution with an excess of fast particles rather than a thermal one — and each has been argued to supply some of it. None has been shown to supply all of it everywhere.

A hundredth of the gas, and a factor of ten #

The second picture, developed by Xiaowei Liu and collaborators from the late 1990s, puts the discrepancy in the gas rather than in the thermometer. Suppose a nebula contains, mixed through it, a small amount of gas that is very rich in heavy elements and poor in hydrogen — knots of material processed inside the star and thrown out without being diluted. Gas with that many heavy elements cools very efficiently through their infrared lines and sits at about a thousand kelvin. At that temperature no electron has enough energy to excite the green line: the cold knots are invisible in forbidden lines. But slow electrons are just what recombination needs, and a recombination line emits several times more strongly per ion at a thousand kelvin than at ten thousand.

The arithmetic is lopsided. If the cold, oxygen-rich gas carries one per cent of the hydrogen emission and thirty times as much oxygen per hydrogen atom as the rest, the forbidden-line count is untouched — the cold gas does not emit forbidden lines — and the temperature from the [O III] ratio is untouched too. But the recombination count roughly triples. With a hundred times as much oxygen, a factor of about seven. The extreme discrepancies of planetary nebulae, ten to a hundred and more, need nothing more exotic than this: a percent or two of very metal-rich, very cold gas.

Which gas each line is reporting on #

In that picture the two counts are not two measurements of one quantity. At a cold fraction of one per cent, two-thirds of the recombination line comes from the cold knots, while the forbidden line comes entirely from the warm gas. Each count is a fair measurement of a different gas. Dividing one by the other gives not an error but a ratio of two abundances in two components, and the discrepancy factor is a measure of how much metal-rich material is hidden in the nebula and how rich it is.

This is also the picture that the most extreme cases favour, because of who has them. In the last decade, the planetary nebulae with the largest discrepancies — factors above ten — have turned out overwhelmingly to have close binary stars at their centres, pairs with periods of hours to days that must have passed through a common envelope, in which one star swallowed the other and the shared envelope was ejected. The nebula in such a system is the ejected envelope, and it is plausible that the ejection threw out some highly processed material from deep inside along with the outer layers. Spatially resolved spectra of some of these nebulae show the recombination lines concentrated towards the centre while the forbidden lines are spread through the shell, as two separate gases would be and a single fluctuating gas would not.

That is the surprising connection the discrepancy has turned out to make: in the most extreme cases, a puzzle in atomic spectroscopy is the fossil of an orbital event — the in-spiral of a binary star through its companion’s envelope — and the size of the factor is a record of how that ejection mixed the star’s processed core into its outer layers.

Which count is right depends on which picture is true #

The two pictures make opposite recommendations. With temperature fluctuations, the recombination count is close to right and the forbidden-line count is too low — by a factor of nearly two at $t^2 = 0.08$. With cold metal-rich knots, the forbidden-line count is close to the abundance of the bulk of the gas, and the recombination count is too high — by a factor of two at a one-per-cent cold fraction — because it is dominated by a small amount of unusual material.

For the planetary nebulae the clump picture is persuasive in the extreme cases and the question is open in the moderate ones. For star-forming regions, which have smaller and remarkably uniform discrepancies of about two, the stakes are larger, because those regions are where the oxygen abundance of a galaxy is measured, and the galaxies’ abundances are calibrated against them. If the fluctuation picture is right there, every forbidden-line abundance in extragalactic astronomy is low by a factor of about two — a correction that would move the oxygen content of galaxies, the heavy-element budget of the universe, and the yields of the stars inferred from them. If the clump picture is right, nothing needs correcting except the recombination lines. A metallicity scale for the whole of galaxy evolution rests on which.

There is an outside check. The oxygen abundance of the young stars that formed from the same gas can be measured from their spectra, which form in stellar atmospheres where the temperature is well understood. In the Orion Nebula and a few other regions, the abundances of young B stars agree better with the recombination-line count than with the forbidden-line count, which favours the fluctuation picture for those regions. The comparison is possible in only a handful of places, and the stellar abundances carry their own model uncertainties.

A third suspect: electrons that are not thermal #

Both pictures assume that the free electrons at each point have a Maxwellian distribution of energies at the local temperature. In 2012 it was proposed that they might not. In the solar wind and in planetary magnetospheres, electron energies are measured directly and follow distributions with long high-energy tails — “kappa” distributions, with an excess of fast electrons over a thermal one — maintained by processes that inject energetic particles faster than collisions can thermalise them. If the electrons in a nebula had such a tail, the forbidden lines, which need fast electrons to excite them, would be enhanced relative to a thermal gas at the same mean energy, while the recombination lines, which need slow ones, would hardly change.

The effect mimics temperature fluctuations closely: the auroral line, from the higher level, is enhanced most, so the [O III] thermometer reads high, and the forbidden-line abundance comes out low. It does so with no temperature structure at all. The objection is the collision time. In a nebula of a few hundred electrons per cubic centimetre, electrons share energy with each other in seconds to minutes, far faster than any heating process could rebuild a tail, and the energetic photoelectrons released by ionisation are thermalised long before they would distort the forbidden-line excitation by much. Calculations of how far from thermal the distribution can be under realistic heating find departures too small to matter. The suggestion is not fully closed, because it would need only a small excess at energies of a few electronvolts, but it has not found the source of the energy that would sustain it.

Which leaves the situation where it started: two explanations that each fit part of the evidence, both demanding something the standard picture of a nebula does not provide — a temperature structure several times larger than any smooth model gives, or a hidden component of very metal-rich cold gas.

What the models leave out #

The emissivities are simple temperature laws — a Boltzmann factor for the forbidden lines, power laws for the recombination lines and for Hβ — rather than computed atomic rates, and the auroral line is related to the nebular one by the standard low-density formula, ignoring collisional de-excitation, which matters in the dense inner regions of planetary nebulae. The temperature fluctuations are a symmetric Gaussian, whereas real fluctuations are skewed and spatially correlated. The cold component is a single temperature with a fixed oxygen enhancement; real knots would have a range, and their hydrogen-poor composition changes their own recombination physics. Neither picture includes the third proposal, non-thermal electron energies, which would bias the forbidden lines in a way resembling temperature fluctuations without any actual variation of temperature.

Still open: whether the moderate discrepancies are one thing or two #

The extreme cases look settled in outline: a metal-rich component, plausibly from a binary’s ejection, emitting recombination lines that the bulk of the nebula does not. The moderate discrepancies, a factor of two in nearly every H II region and in most planetary nebulae, are not. They are too uniform, from one region to another and one galaxy to another, to look like the accident of a few knots, and they need more temperature structure than any model of a smooth nebula provides.

Integral-field spectrographs that map both kinds of line across a nebula at high resolution are now able to ask where each comes from, pixel by pixel. If the recombination lines and the forbidden lines come from the same places, with the same kinematics, the discrepancy is a property of the gas’s temperature or its electrons. If they separate — the recombination lines from small, cold, slow-moving knots — it is a property of its composition. The answer decides whether the oxygen content of every galaxy measured from its glowing gas has been underestimated by a factor of two for half a century, or measured correctly all along.