Annu. Rev. Astron. Astrophys. 1994. 32: 277-318
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4.2 The Local Structure of Cooling Flows

Spatially-distributed mass deposition implies inhomogeneous flows which in turn require that the gas is multiphase. The gas must consist of a multitude of cooling (and cooled) clouds at all radii. How such clouds are supported against gravitational infall and retain their integrity against breakup into the hotter phase is not clear (Loewenstein & Fabian 1990, Tribble 1991). Any dense gas cloud released into the ICM would fall toward the center of the cluster and fall apart under the ram pressure forces that develop unless something binds it together.

Magnetic fields may be responsible for this. The geometry of the field is uncertain and could range from field lines penetrating both clouds and hot ICM, to fields concentrated in the clouds, or some skin around the clouds (Daines et al 1994). The geometry of the clouds themselves is of course unknown and may be sheetlike rather than spherical. The sizes of clouds and in particular their column densities determine the terminal velocity. The smaller a cloud is, the slower it will fall. As mentioned above, conduction must also be suppressed, this time by very large factors.

The origin of the density inhomogeneities is unclear. They may be fossils of the past stripping of galaxies (Soker et al 1991) or of earlier mergers with denser cooler clusters. The spectrum of densities required to generate the observed Mdot propto r is close to that from cooling gas (Nulsen 1986, Thomas et al 1987, Thomas 1988, Tribble 1991) and may be generated by stirring of an old more homogeneous flow (Daines 1994). Chun & Rosner (1993) have studied the nonlocal behavior of thermal conduction in cluster halos and find that inhomogeneities may be expected.

We do not know how clouds evolve. They may coagulate if they collide at velocities below the internal sound speed of a cloud. They may break apart in falling relative to the hot phase and then mix into it, which has the effect of causing the cooling time of the newly mixed gas to be reduced and so promote formation of a new rapidly cooling cloud. In the center of the flow, cloud-cloud collisions may be common since the density of clouds will be highest there.

Despite these many uncertainties, we can identify some of the many stages in the cooling of a cloud. As the gas cools rapidly below ~ 106 K, it drops out of ionization equilibrium (Edgar & Chevalier 1986, Canizares et al 1988). Large clouds may drop out of pressure equilibrium with the surrounding gas at those temperatures also, since the sound crossing time of a cloud exceeds the cooling time (Cowie et al 1980). Smaller clouds lose pressure equilibrium at lower temperatures. The gas remains essentially optically thin in cooling; where lines are optically thick to resonance scattering (Gil'fanov et al 1987, Wise 1993) then they are also thick in the gas beyond the cooling flow. Numerical computations of collapsing clouds have been presented by David et al (1988) and David & Bregman (1989). Over the range 3 x 104 < T < 3 x 106 K, the cooling time tcool ~ 2 x 106 T65/2 P5-1 yr, where T = 106 T6 K and the thermal pressure P = nT - 105 P5 cm-3 K (P5 ~ 1-100 as the radius drops from ~ 100 kpc to 1 kpc). It therefore becomes very short as the gas cools. Observations of the same value Mdotspectral from different X-ray lines characteristic of different temperatures give good evidence that flows are steady. The sound crossing time of a cloud of size r = r0 kpc is tcross ~ 107 r0 T61/2 yr, so a cooling cloud can easily drop out of pressure equilibrium with the surrounding gas (whether it does so depends on its geometry and internal density distribution). If the cooling remained in ionization balance (which is unlikely) then only clouds smaller than 1 pc would be in pressure equilibrium below 105 K.

The mass of gas expected in a cooling flow as a function of temperature should roughly resemble the inverse of the cooling function, i.e. where the cooling is strong and the cooling times short, there is little gas. How the gas cools below 104 K as the gas recombines and forbidden and fine-structure lines dominate the cooling depends on the ionization state and opacity of the gas. Ferland et al (1994) show that a cloud exposed to the X-ray flux in a flow has a thin outer warm layer and a cold interior which is increasingly molecular with depth. The core quickly drops to the temperature of the microwave background. The longest timescale is the gravitational collapse time of the cloud (assuming it exceeds the Jeans mass) and is tJ ~ 106 T11/2 P5-1/2 yr. The minimum mass of cooled gas remaining in a steady flow is at least Mdot tJ or ~ 108 T1-1/2 P5-1/2 Mdot2 Msun, even if the efficiency of collapse was 100%.

The mass above which a cloud collapses under its own weight is the Jeans mass MJ ~ 3T12 P5-1/2 Msun, only if thermal pressure opposes gravity. This is ~ 0.1 Msun in cloud cores throughout much of a cooling flow, if T ~ 3 K. When a magnetic field dominates the pressure, the equivalent critical mass rises as (n/n0)-2 where n is the density in the cloud and n0 is the density it would have attained at that temperature if the field had been absent. The mass can then be in the range of Mcrit 10-100 Msun, and perhaps more if the magnetic field came into equipartition with the thermal pressure at a much higher temperature. If B propto r-2 and the cloud collapses spherically then the magnetic pressure PB propto T-4/3. If the magnetic field initially contributes about 1% of the total pressure at the mean cluster temperature, as observations indicate (Kim et al 1991), then it can easily dominate the total pressure by the time the gas has cooled to ~ 104 K (see also discussion by David & Bregman 1989). In this case, Mcrit ~ 106 Msun, comparable to the mass range of globular clusters. The magnetic field is therefore important in cooled clouds, determining both their evolution and structural integrity.

The excess X-ray absorption suggests a low efficiency (or longer timescales for cloud collapse) by a factor of about 100 or more. What is unknown so far is whether this can be due to the rate at which clouds lose their magnetic field, or the time taken to accumulate enough mass to exceed the Jeans mass. There are also many other unknown or vague properties of cold clouds which we now briefly outline.

Even when the core of a cloud has cooled, recombined, and becomes mostly neutral, there is still residual ionization due to X-ray irradiation from the surrounding flow. This may provide the link for the magnetic field to bind the cloud; even when the field is unimportant in the core of the cloud it can still give the tension necessary to bind the warm layer at the surface. The low, but significant, ionization in the core of the cloud also enables H2 to form through the presence of H-, which in turn promotes the formation of other molecules. In this case the Jeans mass can be sub-stellar (Ferland et al 1994). In such conditions it may also be possible for grains to form (Fabian et al 1994b) perhaps through the higher acetylenes (B. Draine, private communication). Studies of emission-line nebulae often found at the centers of cooling flows, and continuum color maps do reveal the presence of dust (Hu 1992, Sparks et al 1993, Donahue & Voit 1993). The excess blue light is a further potential indicator of distributed grains, if the light is scattered (polarization measurements can test this). Molecules may then freeze onto cold grains, causing clouds to become very dust-rich and quite different from Galactic clouds.

Such possibilities are highly speculative until further observational data on the cooled gas can be obtained. The column density of gas derived from X-ray absorption, NH ~ 1020-1021 cm-2, corresponds to a thickness of only 1016-1017 T1 P5 cm, in gas at a temperature 10 T1 K and pressure 105 P5 cm-3 K, which is so much smaller than the size of the flow that it must consist of a mist of clouds around the central galaxy. At large radii a long-lived flow should be relatively quiet and undisturbed and the clouds may form grains and/or very low mass stars. The behavior of the core of a flow is different, since the hot gas within the inner 10 kpc or so can only support its own column density of cold clouds without being significantly affected by their weight. If matter drops out of the flow such that Mdot propto r then the mean density of cooled gas builds as r-2, whereas the density in the hot gas is distributed as r-1. This means that the hot gas in the core of any long-lived flow cannot support the full weight of the accumulated cold gas which dominates the dynamics of that region (Daines et al 1994). The clouds there may fall, collide, be shocked, and coagulate. This may be the source of the emission-line nebulae and of any normal star formation, since the bulk of the clouds may be warmer than those suspended in the hot gas at larger radii.

The tight HI and CO limits obtained for many cooling flows severely restrict the nature and form of accumulated cold clouds. Since HI can easily be optically thick (Loewenstein & Fabian 1990) and cold, its emission can therefore be negligible, but it ought to be detectable in absorption. The observational limits thus appear to rule out atomic hydrogen as a major constituent of the X-ray absorbing matter. The gas may therefore be almost completely molecular, which confronts the CO limits unless it either has not formed, or forms into more complex molecules. Alternatively, if the X-ray absorption is from very dust-rich clouds, the current observational limits may be overcome since the cooled matter may be mostly H2 and dust, with most of the other elements adsorbed onto grain surfaces, embedded in a thin partially ionized and magnetic skin. Note that the dust cannot be similar to that in our Galaxy, since it would then have been detected if warmer than 10 K, and if its total mass exceeded ~ 108 Msun (Annis & Jewitt 1993). In the above picture the grains would be larger, perhaps corresponding to a significant fraction of the X-ray absorption column density (a few µm).

No satisfactory theory or picture of cold, cooled clouds in cooling flows yet exists. The chemical and physical conditions are sufficiently different from those in our Galaxy that a simple extrapolation from the situation in our Galaxy may be inadequate.

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