A cooling flow that never arrives

About a seventh of a rich cluster’s ordinary matter is in its galaxies. The rest is hot gas that fills the space between them, at tens of millions of degrees, held up against gravity by its own pressure and shining in X-rays. Most of that gas has been sitting there quietly for billions of years. It is so thin that collisions are rare, and each emission of a photon carries away a tiny part of the energy of a very hot particle, so in the outskirts of a cluster the gas has not noticeably cooled since it fell in.

At the centre the arithmetic is different. Radiation by free electrons goes as the square of the density, while the heat content goes as the density. So the cooling time goes as one over the density and the densest gas cools fastest. In many clusters, the X-ray surface brightness rises to a sharp central peak, and the implied density reaches a few hundredths of a particle per cubic centimetre. That gas has a cooling time much shorter than the age of the universe.

The implication was worked out in the 1970s by Andrew Fabian, Paul Nulsen, Len Cowie, James Binney and others, and it looked inevitable. Gas that cools loses pressure. The pressure is what holds it up, so it sinks inward, and the gas above it follows. A slow, steady inflow should form: a cooling flow. By the 1990s it was a large field, with depositions of hundreds of solar masses a year inferred in the richest clusters, and the question was where all that matter went. Nobody had yet shown that it went anywhere.

How long the centre has #

The cooling time at constant pressure is the enthalpy per unit volume over the rate of radiation:

$$t_{\rm cool} = \frac{\tfrac52(n_e+n_i)kT}{n_e n_i \Lambda(T)},$$

where $\Lambda$ is the cooling function, dominated above a few times $10^7$ K by bremsstrahlung, and growing as $\sqrt{T}$, with a contribution from metal lines that matters more at lower temperatures. For an electron density of 0.05 per cubic centimetre and a temperature of a few times $10^7$ K, that is less than a billion years.

The observation behind this figure is the X-ray surface brightness and spectrum of the gas, obtained in annuli and deprojected. The density comes from the brightness, since the emission scales as the square of the density. The temperature comes from the shape of the bremsstrahlung continuum. So the curve is not a theory. It is what the measured gas would do if nothing were heating it. And the time that is compared with it, here the time since redshift one, is an estimate of how long the core has been left in peace, which is uncertain by no more than a factor of two.

That leaves a boundary, the cooling radius, inside which the gas has had time to radiate away its heat. About a third of nearby clusters have a steep central peak of this kind, and they are now called cool-core clusters; the rest have been recently stirred by mergers, which have mixed the core and raised its entropy.

What the flow would have to deposit #

If each shell of gas inside the cooling radius radiates its enthalpy and sinks, the rate at which mass is cooled comes straight from the luminosity:

$$\dot M = \frac{2}{5}\frac{L\,\mu m_p}{kT}.$$

This is energy conservation. It contains no model of the flow’s structure, only the amount of heat per unit mass that has to go. Summing the shells of the cluster above gives the number.

That has to be compared with what is at the centre. The dominant central galaxy in such a cluster has a star formation rate typically between a fraction of a solar mass per year and some tens. A rate of hundreds of solar masses a year, kept up for billions of years, is the mass of a giant galaxy in cold matter. It would fill the middle of the cluster with young stars, with cold molecular gas, or with something else equally visible.

During the 1980s and early 1990s, a whole search was made for this missing mass. Some cold gas was found, as were optical emission filaments around central galaxies and blue light from some recent star formation, though always a small fraction of what the formula demanded. The usual reply was that the matter must be in forms that could not be seen, such as very low-mass stars or cold clouds, with a dim initial mass function. For a time, explanations of this kind had many adherents, and the rate $\dot M$ was reported from the imaging in catalogues as a measured quantity.

A flow has to pass through every temperature #

The test that finally decided the question is a small argument with a clear structure. Gas cooling from $10^8$ K to a star must pass through every temperature on the way. In a steady flow each parcel radiates its enthalpy at every stage, so the luminosity per interval of temperature is fixed and proportional to the temperature:

$$\frac{dL}{d\ln T} = \frac52\dot M\frac{kT}{\mu m_p}.$$

Between about $3\times10^6$ K and $10^7$ K, that luminosity has to come out mostly in lines of highly ionised iron and other elements. The Fe XVII lines near 15 ångströms are especially strong at a few million degrees. They should be conspicuous in any spectrum of gas going through that range at hundreds of solar masses a year.

The Reflection Grating Spectrometers on XMM-Newton first saw this region of a cluster’s spectrum with adequate resolution in 2001. John Peterson and his colleagues, looking at Abell 1835 and then at a sample, found the gas at the ambient temperature and down to about a third of it, and then nothing like the required amount below: the Fe XVII lines were many times too weak. There may be some gas cooler than that, at a rate perhaps a tenth of the flow’s, or less. But the flow as a flow does not exist. Gas cools from $T$ to $T/3$, and at that stage something is stopping it.

The reasoning is a good model for this kind of argument. The formula for $\dot M$ gave a rate that could be hidden by inventing an invisible destination. A test of what must happen on the way to that destination has no such escape, because the emission of each temperature range is determined by energy conservation, whatever the gas ends up as. The destination never needed to be found.

A bubble’s enthalpy #

Something therefore pays the bill. The total to be supplied is the luminosity inside the cooling radius, $10^{44}$ to $10^{45}$ erg/s for such a system, for several billion years. Supernovae in the central galaxy are short by about an order of magnitude and conduction from the outer gas can help in some systems but does not stop the coolest cores. A central black hole, weighed by the motion of the stars around it, can do it, because accreting a tiny part of a solar mass per year gives this much power; and every cool-core cluster has one, usually with a radio source.

The clinching images came from the Chandra X-ray Observatory, launched in 1999, with its half-arcsecond resolution. The X-ray gas at the centre of the Perseus cluster shows dark holes, one on each side of the centre, each filled with radio emission from the active nucleus. These are bubbles of relativistic plasma that have pushed the hot gas aside. Further out are older detached ones, rising through the gas buoyantly.

The bubble gives a calorimeter that needs no model of the jet. To inflate a volume $V$ against an ambient pressure $p$ takes energy $pV$, and the bubble’s own internal energy is another $3pV$ for relativistic plasma, so the enthalpy is $4pV$. Divide that by the time for the bubble to rise, and the power of the source appears.

Surveys of cavities in tens of clusters and groups, done by Birzan, Rafferty and collaborators, then showed that the cavity power tracks the cooling luminosity over a few orders of magnitude of power, from galaxy groups to the richest clusters, with the same scatter of about a factor of ten. A black hole of a billion solar masses, whose gravity dominates a region of perhaps tens of parsecs, is thus setting the thermal state of a region a thousand times larger. There is a remarkable step across scale in that fact, of the same size as the step between a person and a large country, and it works through the hydrodynamics of a bubble.

A thermostat with the right sign #

To keep a cool core steady for billions of years the heating must match the cooling over a long time and in a stable way. That needs a feedback: if a core overcools, more fuel has to reach the black hole, the bubbles have to grow, and the core must then be heated again. A mechanism of that sort is sometimes thought of as a thermostat, and it needs two things to work: a heating rate that depends on the gas’s state and a way of bringing news of the state down to the black hole.

The news is supplied by the cooling gas itself. Simulations, particularly by McCourt, Sharma, Gaspari and Voit, have shown that a hot atmosphere in hydrostatic balance becomes thermally unstable, forming cold clumps that rain down, when its cooling time falls below about ten times its free-fall time. The cold clumps fall through to the centre and feed the black hole.

This explains a number of things together. The filaments of cold and warm gas seen near central galaxies, including the long ones in Perseus, appear only when the central cooling time drops under about a billion years; there is a sharp threshold observed in the entropy, at about $30\ \mathrm{keV\,cm^{2}}$. The observed star formation in those galaxies comes on at the same threshold. Molecular gas, in amounts of $10^{9}$ to $10^{11}$ solar masses, is found in just those cores. All this suggests that a small fraction of the gas does condense out, by this route, at a rate of about the same as the observed star formation, and the rest is held up.

Note that the correlation does not itself prove feedback. Voit and others take the observed minimum of ten to twenty as evidence of a self-regulated state, but in a sample selected by peaked X-ray cores there may be selection in the same direction, and systems with values below ten may be rare because they quickly go through a burst of accretion. Clusters that are not cool cores also have a much higher ratio and are outside the argument.

The energy has to reach the gas #

There remains the question of how bubble energy turns into heat in the gas, and it is the least settled part of the argument. A bubble rising buoyantly does work. It lifts low-entropy gas with it, a part of which falls back, and it stirs the surrounding atmosphere. Sound waves and weak shocks are visible as ripples in the deepest X-ray images of Perseus, and the ripples dissipate on some scale; turbulence has been proposed as an alternative, and at first it seemed likely that it would be the dominant route.

A direct test came when the Hitomi satellite observed the Perseus cluster in 2016 at a resolution enough to see the widths of the iron lines. The line-of-sight velocity dispersion was found to be about 160 kilometres a second, which is unexpectedly small. That means the kinetic pressure is only a few per cent of the thermal pressure in spite of the violent appearance of the images. Whether that is sufficient to transport the energy across the core in time depends on the scale at which the turbulence is driven, and there has been a controversy between those who find it adequate and those who conclude that the energy must be carried by waves instead. The XRISM mission has continued these measurements since 2023, with results for several more cores that again indicate modest velocities.

The bubble’s enthalpy shows the budget is adequate. It does not show which channel delivers it. Those are different statements, and only the first has so far been firmly established.

What the figures leave out #

The cluster is one representative model. It has a smooth density profile with a core, a temperature falling to a third of its outer value in the centre, and no substructure. Real cool cores have cold fronts from sloshing, bubbles and filaments which are not small perturbations: the assumption of spherical symmetry is crude at the radii where the action takes place.

The cooling function is simplified. Bremsstrahlung with a modest line correction does well above a few times $10^7$ K, and it underestimates the cooling at lower temperatures where metal lines dominate. The drawn temperature range is therefore approximate for the coolest gas.

The match between bubble power and luminosity is a factor of ten at best. The buoyancy time is itself only an estimate, the bubbles may be older or younger than assumed, and the fill of relativistic plasma is inferred rather than observed. A single bubble is a snapshot of a source that varies, so the proper average runs over several generations.

The chart cannot show the time-history. All these figures show a static state, and the whole point of feedback is that it is not static. A given system may go through cycles of overcooling and overheating on scales of about a hundred million years, with no observation able to follow it through one of them.

Where the same bill is due elsewhere #

The argument scales down. A massive elliptical galaxy has hot gas with similar cooling arithmetic and little star formation, and the question of why such galaxies stop forming stars is at the centre of current theory of galaxy formation. Without something heating the gas, models of the galaxy population overproduce massive galaxies by a large factor, and they do so at the same mass scale, about $10^{12}$ solar masses in halo mass and above, where the bright end of the census of galaxies falls off. Black-hole feedback is now a required ingredient in those models, with parameters chosen partly to make that fall-off come out right. So the cluster core is the place where the mechanism can be seen working, and the rest of the galaxy population is the place where it does most of its work and cannot be seen.

This also bears on the correlation between a black hole’s mass and the velocity dispersion of its host, which has been a central puzzle for two decades. The energy required to unbind gas from a galaxy is comparable to a small fraction of the energy released when a black hole grows, and it is at least arguable that the two are linked by something of the same kind.

Clusters also serve as a warning about what a measurement is. For twenty years $\dot M$ appeared in tables, was compiled and compared, and led to theories of low-mass star formation in cooling flows. All of it was derived correctly from observed luminosities and temperatures, and all of it assumed that nothing was heating the gas. It was never a measurement of a mass flow, and it was the name of the quantity that made it seem so.

A quiet centre among noisy ones #

There is one extreme case that fits the mechanism in an unexpected way. The Phoenix cluster, found in 2010 by the South Pole Telescope through its effect on the microwave background, has a star formation rate of several hundred solar masses a year, a value as large as the classical flow would have predicted, and a strong cool core. It is the system that seemed to behave as the 1980s picture said. Its central black hole, as measured, seems to be accreting rapidly, and its bubbles do not yet supply the energy needed. So it may be an instance of what happens when the regulator is late, a brief failure that goes on for some tens of millions of years. Such systems must be rare if the thermostat works most of the time. And they are.

A temperature that weighs the cluster too #

The same gas has a second job, and the comparison limits what can be concluded about either. Assuming hydrostatic balance, the density and temperature profiles that give the cooling time also give the cluster’s total mass, and that mass is a main input to cosmological counts of clusters. So what has been described here matters beyond the core. If bubbles and stirring supply a pressure that the X-ray temperature does not register, the hydrostatic mass comes out low, and a shortfall of that kind is what haunts the counts.

Hitomi’s low velocity in Perseus was welcome news in that respect, since it implies that non-thermal pressure in the core is small, a few per cent. Most cosmological work avoids the core anyway. The bias at larger radii, where the mass is really measured, cannot be read from a core’s dispersions, and it is expected to be larger there.

The cooling gas also connects to the question of where the rest of the ordinary matter is. If feedback heats and expels gas from groups, then the hot gas fraction should fall steadily toward lower masses, and it does. The dispersion measures of fast radio bursts now constrain how far out that gas has been pushed, and early results suggest a stronger feedback than many simulations had assumed. The bubbles visible in a cluster’s centre are one local view of a process with effects at megaparsec scales.

Still open: how a bubble warms what it does not touch #

Bubbles are localised and collimated along the jet axis, and they rise along narrow paths, while the cooling gas fills the whole volume. To stay in balance, heat must reach gas lying at right angles to the jet and at all radii inside the cooling radius. Mechanisms such as turbulence and sound waves spread energy, but the measured velocities are modest, and sound waves dissipate only under conditions in plasma that are not well known. The viscosity of hot, collisionless cluster gas is itself an unresolved issue. In the outermost parts of the core the energy may arrive largely by gas motion and the mixing produced by mergers.

The next X-ray spectrometers will measure velocities in many cores, spatially resolved, on scales down to the size of a bubble. If energy is carried by turbulence, the velocity distribution should show the spectrum characteristic of a cascade. If it is carried by waves, the velocity should appear coherent and rippled. The black hole’s power and the gas’s losses are known to match, but no one has yet traced a joule of energy all the way between them.