Annu. Rev. Astron. Astrophys. 1994. 32: 277-318
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4. THEORETICAL ISSUES ON COOLING FLOWS

4.1 The Global Structure

The original models for cooling flows (Cowie & Binney 1977, Fabian & Nulsen 1977, Mathews & Bregman 1978) assumed that the gas was homogeneous, i.e the gas has a single temperature and density at each radius. The gas density rises inward approximately propto r-1. The temperature drops inward in accord with pressure equilibrium, which means that its variation is weaker than propto r given the gravitational potential of the cluster and central galaxy within the flow. The thermal instability was invoked as a means of creating filaments near the center (Fabian & Nulsen 1977, Mathews & Bregman 1978). One problem with this instability, discussed later, is whether an overdense and therefore cooler (and more rapidly cooling) blob will fall and merge with regions of similar density or entropy (Cowie, Fabian & Nulsen 1980). This would suppress the instability. Instead, it was assumed that if the gas was unstable, then all phases moved together.

The Einstein Observatory X-ray images of the Perseus cluster (Fabian et al 1984b) and of several other clusters (Stewart et al 1984a, b) demonstrated that the mass deposition by the flow was distributed over 100 kpc or more. This requires that the gas is inhomogeneous. A clump must be overdense by about a factor of 2 at the cooling radius if it is to cool completely while flowing only halfway to the center. In a pioneering paper, Nulsen (1986) developed the idea of a multiphase, comoving cooling flow in which the major input parameter was the initial spectrum of density perturbations in the gas. Magnetic fields were invoked to bind the individual density perturbations or clouds both internally and to the mean flow. The clouds would otherwise break apart as they fell through the more tenuous phases.

A numerical multiphase model applied to the X-ray surface brightness profiles of several clusters (Thomas et al 1987) showed that the fractional volume distribution of the initial density perturbations had to haven characteristic shape, the origin of which was not clear but could itself be related to cooling. White & Sarazin (1987a, b, c, 1988) constructed models of cooling flows with distributed mass deposition which they compared to the data. They assumed various dependences for the rate at which matter cools out of the flow as functions of the local density and other properties and highlight the dependence of Mdot on r as a function of the local gravitational profile. (There is general agreement between all approaches to the X-ray data that assume a cooling flow). These studies show that if one assumes the gas to be homogeneous when analyzing the data, the derived results are a good approximation to the (emission-weighted) mean values of temperature and density.

Whether or not the linear thermal instability grows in a gravitational field has been considered by Malagoli et al (1987), White & Sarazin (1987a), Balbus (1988), Tribble (1989a), Loewenstein (1989), and Balbus & Soker (1989) using both particle and wave approaches in Eulerian and Lagrangian systems. The work by Balbus & Soker (1989), which includes the effects of the background flow, indicates that any growth of the linear thermal instability is weak; the medium is only thermally unstable if it is convectively unstable (which a homogeneous cooling flow is not). Numerical work by Hattori & Habe (1990) and Yoshida et al (1991) demonstrate how some blob configurations might fall and mix.

Loewenstein (1990) and Balbus (1991) have shown that weak magnetic fields can destabilize cooling flows and allow the overdense gas to cool. The magnetic field helps to suppress the buoyancy which triggers convective motions in the flow. The magnetic field needs to be strong enough for the Alfven velocity to exceed the phase velocity of buoyancy oscillations.

As already mentioned, the observations require that the gas initially has large density inhomogeneities, with fractional density deltarho / rho > r/R for a blob of size r at radius R in the flow (Nulsen 1986), which makes the amplification of infinitesimal perturbations a much less important issue for observed cooling flows. If denser blobs exist and survive dynamically in the flow, then they will cool well before reaching the center. The main issues (to be discussed later) are the origin and survival of such denser blobs.

Conduction is another effect that has been widely debated (see e.g. Takahara & Takahara 1979, Tucker & Rosner 1983, Friaca 1986, Gaetz 1989, Böhringer & Fabian 1989). The observations of cooling flows demand that cooler regions are immersed in hotter ones. The energy equation for a unit volume of gas in a constant-pressure cooling flow with conduction is

Equation 6 (6)

where n, T, µ, Lambda, and kappa propto T5/2 are the gas density, temperature, mean molecular weight, cooling function, and Spitzer (1962) conductivity coefficient, respectively. It is implausible to make the two terms on the RHS balance, such that the LHS is zero (i.e no mass drop-out) over a wide temperature range; instabilities grow on a timescale shorter than the local cooling time. Either the gas then becomes isothermal, or conduction has a negligible effect relative to radiative cooling (Nulsen et al 1982, Stewart et al 1984b, Bregman & David 1988). For the conditions in most cooling flows, conduction must be suppressed by a factor of < 10-2 (Binney & Cowie 1981, Fabian et al 1991) below the Spitzer value, and for an inhomogeneous flow to occur with small clouds the factor must be much larger.

The conduction models of Bertschinger & Meiksin (1986) fail to account for the cooler X-ray emitting gas that is observed, although they do predict a density rise in the core of the cluster. The conduction model of Sparks (1992) is not relevant to observed flows since the implied large-scale planar geometry would be readily observable in X-ray images (Fabian et al 1994a). His model is one variant of so-called warming flows that postulate a mass of cold matter heated, and possibly evaporated, by the conduction of heat from the surrounding hot gas in the core of a cluster (Bregman 1992, Sparks et al 1989, de Jong et al 1990). Such models do imply a wide range of temperature components but cannot explain the actual observed distribution of temperatures and, in particular, cannot account for the observed strength of the FeX VII line, without requiring impossibly high mass evaporation rates (Canizares et al 1993).

Magnetic fields are observed to occur in the ICM with a pressure of about 1% of the thermal pressure (Kim et al 1991). In a cooling flow the field should be amplified by compression. This is verified by observations of Faraday rotation in those flows surrounding extended luminous radio sources (Dreher et al 1987, Ge & Owen 1993). At small radii where the magnetic field is largest, it may reconnect and therefore inject energy into the hot gas (Soker & Sarazin 1990). Alternatively it may help stifle a homogeneous flow.

Tangled magnetic fields are usually invoked to suppress thermal conduction (as have plasma instabilities; Jafelice 1992). How this takes place is unclear but it seems plausible that mirroring can trap electrons in such a field and so reduce conduction to very low values (Borkowski et al 1990). Tribble (1989b) has studied models in which conduction takes place freely along highly tangled field lines. This leads to a different functional form for the conductivity than the usual one, involving a length-scale dependence. The plasma becomes inhomogeneous as cooling dominates some field lines but not others. It is possible that some such complex conduction model may account for some important aspects of the X-ray data, but it has not been developed so far.

The mass deposition rate can be reduced if a deep gravitational potential is assumed. The gravitational work done on the gas as it flows inward offsets some of the cooling luminosity. The largest factor that can be obtained in this way is about 2 (Arnaud 1988). Changing the potential on a slow timescale as the cluster evolves can have some effect, particularly in staving off the time at which a large flow develops (Meiksin 1990). The idea that the flows have not yet become steady and are only now about to happen is similar to the suggestion by Hu (1988) that strong cooling has only just begun (see also Murray & Balbus 1992). However, such models do not account for the good agreement of Mdotspectral with Mdotimage or for the common occurrence of cooling flows.

Several authors have suggested that there is a heat source that counterbalances the effect of radiative cooling, so that the temperature of the gas does not actually decrease (in contradiction to the X-ray spectra showing a range of gas temperatures, cooling times - covering several orders of magnitude < 1010 yr, and emission measures, all of which are consistent with simple cooling of the gas). No such process has yet been identified. Note that it requires 1061-1062 erg to balance a strong flow over 1010 yr, much higher than the energy residing in the lobes of even the most powerful radio sources. Note too that most heating processes cause the gas to become thermally unstable.

It has been suggested that cosmic rays from a central engine or radio source, for example, could stop or heat the core of a flow (Tucker & Rosner 1983; see also Böhringer & Morfill 1988, Rephaeli 1987). This cannot be a general phenomenon that balances the radiative losses since then the cosmic-ray pressure would have to exceed the thermal pressure. No flow would then be inferred in the first place (Loewenstein et al 1991).

The motion of galaxies through the ICM may also act as a heat source (Miller 1986). Much of the drag energy may cause the core of a cluster to be a noisy place with many large-amplitude sound waves (Binney 1988) and/or internal gravity waves (Balbus & Soker 1990); the inward increasing density in a flow can focus the sound into its core (Pringle 1989). How that energy dissipates is not known, although phenomenological (Heckman et al 1989) and physical (Crawford & Fabian 1992) models for the origin of the optical line emission common in the centers of cooling flows require a source of chaotic or turbulent energy. The observed optical line widths require that the gas has motions of a few 100 km s-1. A turbulent model for cooling flows, including star formation, has been proposed by Westbury & Henriksen (1992). Balbus & Soker (1990) find that the energy transported by gravity waves is unlikely to grossly affect a cooling flow. No stable heating process yet devised is able to counteract the effects of radiative cooling and account for the observed X-ray images and spectra.

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