Annu. Rev. Astron. Astrophys. 2003. 41: 191-239
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8.1. Chandra Observations and Entropy Floors

Instead of devising esoteric constructions to hide the cooling gas, why not simply heat the gas and avoid cooling altogether? This is a good idea, but it is far from proven that it can work. There is certainly enough energy in garden variety active galactic nuclei (AGN), ~ tau 1044 ergs s-1, even for duty cycles tau of a few percent, to balance the total energy radiated from galactic/group cooling flows, Lx ~ 1041.5±1 ergs s-1. The heating idea has received enormous additional support from high resolution Chandra X-ray images that reveal an amazing variety of structural disturbances near the flow centers where, according to current orthodoxy, supermassive black holes can be activated by inflowing gas to become AGNs. So far most Chandra observations are of cluster-centered, E galaxies having strong radio sources, but similar hot gas irregularities are also apparent in the centers of more normal, group-centered E galaxies and seem to be an almost universal phenomenon [M84 (Finoguenov & Jones 2001), NGC 507 (Forman et al. 2001), NGC 4636 (Jones et al. 2002, Loewenstein & Mushotzky 2002), NGC 5044 (Buote et al. 2003a).

Of particular interest are the X-ray cavities, regions of much lower thermal emission that are often coincident with radio lobes. In Hydra A/3C295 (McNamara et al. 2000; David et al. 2001; Allen et al 2001) the non-thermal energy density in the radio lobes has apparently displaced the thermal gas. In other cases, the radio-X-ray connection is less clear: Perseus/NGC 1275 (Böhringer et al. 1993; Fabian et al 2000; Churazov et al. 2000), A2052 (Blanton et al. 2001b) or A4059 (Heinz et al. 2002). For example, only one of the two high-contrast central X-ray cavities in Perseus/NGC 1275 is coincident with a maximum in the double-lobed radio emission. The other radio lobe has apparently not displaced the local thermal gas, although relativistic electrons appear to be flowing from this radio lobe into an adjacent X-ray hole. Perseus/NGC 1275 is a particularly interesting case because it contains multitude of X-ray holes located at random azimuthal orientations and distances from the center. Most of the "ghost cavities" at larger distances from the galactic center no longer produce (~ GHz) radio emission; either the relativistic electrons have lost their energy or the holes have a different origin. Whether the cavities are dominated by thermal or non-thermal plasma, they must be buoyant on timescales tbuoy ~ 107 - 108 yrs (e.g. Churazov et al. 2001). In any case, X-ray cavitation and other irregularities in the central hot gas suggest energy sources capable of heating the cooling flows.

Another manifestation of central heating may be the high entropy typically observed in group scale flows. In a perfect, starless, metal-free, LambdaCDM, hierarchical universe filled with adiabatic gas and self-similar dark halos, the bolometric X-ray bremsstrahlung luminosities of galaxy groups and clusters would scale in a self-similar fashion with gas temperature, Lx propto T2 (Kaiser 1986; 1991; Evrard & Henry 1991). However, in our particular universe this relation is somewhat steeper, Lx propto T3 (e.g. Arnaud & Evrard 1999), and may become very steep (at least Lx propto T4) for groups having kT ltapprox 1 keV (Helsdon & Ponman 2000a, b). Evidently the gas in E-dominant groups has somehow acquired an additional "entropy" S ident T / ne2/3 ~ 100 keV cm2 in excess of that received in the cosmic accretion shock near the virial radius. This excess has been referred to as the "entropy floor" (Ponman et al. 1999; Lloyd-Davies et al. 2000). More recent studies (Mushotzky et al. 2003; Ponman et al. 2003) claim that the entropy at 0.1 of the virial radius varies as S0.1 propto T0.65 with no pause at the 100 kev cm2 floor for groups and E galaxies; S0.1 propto T would be expected in a perfect adiabatic universe.

The universality of entropy-enhancement in groups has led to the hypothesis that the gas experienced some additional early heating before (or as) it flowed into the group dark halos. Early "pre-heating" by ~ 1 keV/particle could explain the aberrant behavior of groups in both the Lx - T and S - T plots. Although this level of heating would only be apparent in the shallow central potentials of galaxy groups, many authors have adopted a stronger assumption: that all baryonic gas, including gas currently in both groups and clusters, experienced some "non-gravitational" heating (star formation, Pop III stars, AGN, etc.) prior to its entry into the dark halos (e.g. Tozzi & Norman 2001; Borgani et al 2001, 2002). Heating at early cosmic times when the density is low is attractive because the entropy can be increased with a smaller energy expenditure per particle. The energy required exceeds that expected from Type II supernovae associated with the formation of the old stellar population in elliptical galaxies and spiral bulges, ~ 0.2 keV per particle, assuming a Salpeter IMF. Perhaps more attention should be paid to the possibility that much of the low entropy gas was removed by star formation and the remaining gas was heated by star formation (Loewenstein 2000; Brighenti & Mathews 2001) and by a central AGN after the baryonic gas entered the dark halos.

8.2. AGN Heating

The absence of XMM spectral evidence for cooling at the expected level and obvious disturbances in Chandra images are the primary motivations for introducing some type of vigorous heating in the flow models. At the present time, however, both of these motivations are better documented for E galaxies in cluster environments than for similar galaxies in small groups that are more highly constrained. The gasdynamical consequences of heating have been investigated on two spatial scales: (1) attempts to arrest or retard radiative cooling throughout the hot gas and (2) studies of the evolution of small heated regions. We shall discuss these two approaches in turn.

In some models of heated "cooling flows" the physical heating mechanisms - shocks, cosmic rays, Compton heating, thermal conductivity etc. - are explicitly invoked, but in others an ad hoc heating is simply imposed on the flow. Thermal conduction has been a popular heating option for many years (e.g. Bertschinger & Meiksin 1986; Meiksin 1988; Bregman & David 1988; Rosner & Tucker 1989). The uncertain influence of magnetic field geometry on the conductivity provides another adjustable parameter to fit the data. If the thermal conduction term in Equation (5) is comparable to the radiative emission term, ne2 Lambda ~ f 2kappa T / r2, the magnetic suppression factor must be f ~ 56(ne rkpc)2 T7-7/2 ~ 0.2rkpc-0.36 T7-7/2 where ne approx 0.055rkpc-1.18 cm-3 applies to NGC 4472 for rkpc gtapprox 0.3 (Figure 2a). This level of magnetic suppression is in excellent agreement with f ~ 0.2 determined by Narayan & Medvedev (2001) for thermal conduction in a hot plasma with chaotic magnetic field fluctuations. Conduction could be very important in cluster-centered E galaxies such as M87 that are surrounded by much hotter gas; these galaxies are of interest because they represent galactic scale flows with different outer boundary conditions. Regardless of the heating mechanism, it is essential that the density and temperature profiles in heated flows agree with observations. In particular, the heating must preserve the positive temperature gradients universally observed within several Re (Figure 2b).

8.3. Cooling Flow Models with Heating

Nonthermal radiofrequency emission from cosmic ray electrons is observed in the centers of ~ 50 percent of giant E galaxies (e.g. Krajnovic & Jaffe 2002). It is likely that proton cosmic rays are also present or even dominant as in the Milky Way. Cosmic rays could heat the thermal gas either by Coulomb collisions (Rephaeli 1987; Rephaeli & Silk 1995) or by dissipation of Alfven waves excited as cosmic rays stream along field lines (e.g. Tucker & Rosner 1983; Rose et al. 1984; Rosner & Tucker 1989; Begelman & Zweibel 1994). Loewenstein, Zweibel & Begelman (1991) studied the propagation of cosmic rays in a static hot gas galactic atmosphere having an idealized globally radial magnetic field. They suppose that outwardly streaming cosmic rays heat by wave dissipation at small galactic radii (also see Böhringer & Morfill 1988) whereas gas at larger radii is heated by thermal conduction. Although their model is successful in producing central regions with dT / dr > 0, the central gas density gradient dn / dr is flatter than observed because the gas is both supported and heated by cosmic rays. They conclude that this type of cosmic-ray heating cannot balance radiative cooling.

Tabor & Binney (1993) develop steady state flows in which a convective core heated from the center is attached to an outer cooling inflow. However, in the convective core the temperature gradient is determined by the gravitational potential, (k / µ mp)(dT / dr) = - (2/5)(dPhi / dr), and must always be negative, contrary to observation. The gas velocity in the convective core is problematical: if it is negative, all the gas shed from evolving stars, ~ 1010 Modot, would cool at the very center; if it is positive or zero, cooling would occur as a galactic drip (Mathews 1997) in compressed gas at the outer boundary of the convective core. Binney & Tabor (1995) discuss time dependent hot gas flows in E galaxies without dark halos. In addition to central AGN heating, a large number of Type Ia supernovae at early times drive a galactic outflow throughout much of the evolution and help to reduce the mass of cooled gas (as in Ciotti et al. 1991). These supernovae are also likely to enrich the hot gas with iron far in excess of observed abundances. In the supernova heated flows dT / dr is negative throughout the flow evolution, unlike observed profiles, and the most successful models agree with observed gas density profiles only during a brief ~ 108 year period preceding cooling catastrophes when the central gas density increases rapidly due to radiative losses. Binney & Tabor also consider intermittent AGN heating within ~ 1 kpc triggered by gas inflow into the center. As SNIa heating subsides and the flow evolves toward a central cooling catastrophe, the AGN heating is activated but the central gas density still continues to increase. As Binney & Tabor suggest, this may result in distributed cooling and star formation in the central 1-2 kpc. Such radiative cooling might not be possible in a strict interpretation of the XMM spectra.

Tucker & David (1997) study time dependent heated models of cluster scale flows without cooling dropout or mass supply from stars. Gas near a central AGN is intermittently heated by relativistic electrons when gas flows into the origin. The temperature and entropy gradients become negative and the density gradient is positive. Such flows are both convectively and Rayleigh-Taylor unstable. When Tucker & David include thermal conduction, the outward heat flux results in stable solutions having cores of nearly flat density that are nearly isothermal. However, recent Chandra and XMM observations of the cluster-centered galaxies M87 in Virgo and NGC 4874 in Coma show that the gas temperature drops precipitously from 3 - 9 keV to ~ 1 keV between 50 and 10 kpc of the galactic centers (Böhringer et al. 2001; Molendi & Gastaldello 2001; Arnaud et al. 2001; Vikhlinin et al. 2001). These steep thermal gradients can be understood with old-fashioned time-dependent cooling flow models with distributed cooling dropout (Brighenti & Mathews 2002a), but only if thermal conductivity is suppressed (f ltapprox 0.1).

Ciotti & Ostriker (1997; 2001) assume that gas in the cores of isolated elliptical galaxies is Compton heated by powerful isotropic gamma rays (Lbol ~ 1046 - 1048 erg s-1; TC approx <hnu> / 4 ~ 109 K) whenever mass is accreted into an AGN at the center. The explosive expansion of this ultrahot gas drives strong shocks out to ~ 5 kpc and beyond. The hard continuum is only activated about ~ 10-3 of the time, so only one in ~ 1000 of known elliptical galaxies should be a gamma-dominant quasar at any time. For these models Ciotti & Ostriker also assume a high Type Ia supernova rate and no additional circumgalactic gas. Such hot gas atmospheres - less deeply bound and with small inertial masses - can be significantly modified by AGN heating, reducing the mass of gas that cools near the galactic core. However, dT / dr is generally negative and the gas density has a broad central core, although the density and temperature profiles vary with time. The central core in the X-ray surface brightness distribution varies in size from 1 to 40 kpc which is much larger than the ~ 0.6 kpc core in NGC 4649, an elliptical having an LB similar to that used in the Compton heated models. It is unlikely, moreover, that gamma ray QSOs exist in all elliptical galaxies or that this radiation is isotropic. AGN sources dominated by gamma rays, discovered with EGRET, are blazars or OVV (optically violent variables) type QSOs having flat radio spectra with superluminal VLBI activity, suggesting that the hard radiation is highly collimated along the line of sight (e.g. von Montigny et al. 1995; Impey 1996; Hartman et al. 1999). Finally, the Compton temperature TC of more typical, more isotropic AGNs and QSOs is less than 107 K, so cooling flow gas is more likely to be Compton cooled than heated (Nulsen & Fabian 2000).

Kritsuk, et al. (1998; 2001) describe 1D and 2D heated galactic flows having convective cores. The initial state for these flows is a static hot gas atmosphere in which the source terms in equations (3) and (5) dominate, i.e. gas ejected from stars is exactly consumed by distributed radiative dropout (- q rho / tcool) and radiative losses are balanced by stellar and supernova heating (Kritsuk, et al. 1998). When additional central heating is supplied, the core of the flow becomes convective. After 3 × 108 years the temperature gradient becomes negative in the convecting core a few kpc in size, unlike the profiles in Figure 2b. At this time an incipient cooling catastrophe seems to develop where the slowly outflowing convective core confronts the stationary gas beyond, similar to the thermal instabilities in galactic drips (Mathews 1997). Brighenti & Mathews (2002b) consider a wide variety of 1D and 2D cooling flows that are heated within some radius by an unspecified process and triggered by gas flow into the origin. Of the wide variety of heating scenarios considered, none are in reasonable agreement with observed hot gas temperature and density profiles. Even for poor fits to the observations, these models require finely tuned heating scenarios. Idealized flows in which radiative cooling is perfectly balanced by global heating are grossly incompatible with observations. AGN feedback heating often results in spontaneous and spatially distributed cooling produced by non-linear compressions in turbulent regions. Turbulence-induced cooling is associated with quasi-cyclic variations in the hot gas density profile. When cooling flows are partially supported by nonthermal pressure, similar cooling instabilities develop. The global mass cooling rate is not altered by any form of heating considered - in apparent violation of XMM spectra.

Ruszkowski & Begelman (2002) describe a cluster flow model that is heated by conduction from the outside and the entire flow is instantaneously heated when gas flows into the central AGN. The temperature gradient in these solutions is positive, and this much desired outcome is relatively insensitive to the scale and shape of the region heated by the central AGN. Successful Ruszkowski-Begelman flows require that the thermal conductivity suppression factor be in the range 0.1 ltapprox f ltapprox 0.5. Fortunately this range includes the value f approx 0.2 recently advocated by Narayan & Medvedev (2001). Flows on galactic scales also benefit from the combined effects of AGN heating and conduction, but the agreement is generally less satisfactory than for cluster flows (Brighenti & Mathews 2003). Because of the lower temperature in galactic flows, kT ~ 1 keV, marginally acceptable, but not ideal, flow solutions are possible only if the conductivity is close to its full Spitzer value, f approx 1. However, in these solutions the hot gas iron abundance is several times solar throughout most of the flow. This is much higher than observed, even though the current Type Ia supernova rate used is low, SNu(tn) = 0.06 SNu. Therefore, to fully accept the beneficial combination of heating plus conduction in galactic flows, it is also necessary to hypothesize some means of reducing the computed iron abundance. Some (but not all) of the iron produced by Type Ia may cool before entering the hot gas, some selective cooling removes the excess iron, etc.

In summary, all current attempts to reduce the mass cooling rate dot{M} in cooling flows by factors or 5 - 10 with various heating mechanisms are inadequate for one reason or another, particularly for flows in E-dominant group gas. When computed for many Gyrs, the heating often has little effect on dot{M} and the global density and temperature profiles disagree with the observations. Thermal conduction has a helpful role, but is less effective in galactic flows. Essentially static flows cannot be correct either because the hot gas iron enrichment by Type Ia supernovae inside the central E galaxy would be enormous after several Gyrs. Abundances are a major constraint on galactic flows.

8.4. Models of X-ray Cavities

Several recent theoretical studies of X-ray cavities have examined the consequences of introducing heated gas in some localized region away from the center of the cooling flow. If the energy density in the cavities is sufficiently large, they will expand supersonically, producing shocks that heat the adjacent cooling flow gas (Clarke, Harris, & Carilli 1997; Heinz, Reynolds, & Begelman 1998; Begelman & Cioffi 1989; Rizza et al. 2000; Reynolds, Heinz, & Begelman 2001; Soker, White, David & McNamara 2001). While such violent heating may occur in some situations, recent Chandra and XMM observations indicate that the gas temperature in the rims around the cavities and radio holes is typically cooler, not hotter, than average (Fabian 2001; McNamara 2001). These low-entropy rims cannot be understood as local gas that was shocked and subsequently lost entropy by radiation (Nulsen et al 2002; Soker, Blanton & Sarazin 2002).

The 2D hot buoyant bubbles computed by Churazov et al. (2001) slowly float upward in the cooling flow atmosphere (also: Saxton et al. 2001; Brüggen & Kaiser 2001; Brüggen et al. 2002). These rising cavities are accompanied by a column of colder (low entropy) gas that moves radially upward near the center of the bubble. Unless this colder gas is subsequently heated, however, it should eventually fall back. Quilis et al. (2001) studied the evolution of a nearly axisymmetric 3D bubble produced by heated gas at some finite radius, simulating jet heating. They noticed that the buoyant bubble is surrounded by a shell of slightly colder gas which they attributed to cooling expansion as low-entropy gas is pushed by the bubble toward regions of decreasing ambient pressure (Nulsen et al 2002; Soker, Blanton & Sarazin 2002). Heating by jets can also create buoyant regions with slightly cooler rims that float upwards approximately along the jet axis (Reynolds, Heinz, & Begelman 2001; 2002). Brighenti & Mathews (2002c) show that central density irregularities and large, randomly oriented X-ray holes may be a natural result of the evolution of a single spherically heated region at the center of a massive E galaxy. The gaseous rims around the holes are cool, as observed, provided the heating occurs in the region of low entropy near the center of the flow and this is illustrated with an analogous similarity solution.

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