Astron. Astrophys. 324, 449-456 (1997)

5. Conclusions

From the energetic viewpoint, our models could account for typical filament systems with but not for the extreme ones (e.g., Perseus) with ( is the number of recombinations per cooling proton required to yield the observed H luminosity). Our models are better at reproducing the luminosities of the weaker class I filaments than the more luminous class II systems. AH could be a dominant heating mechanism for the class I filaments, and even for the less luminous class II filaments.

It should be noted that there are evidences that matter is dropping out of the cooling flows at radii larger than kpc, so that only a fraction of the overall cooling flow rate (over typical radii of kpc) reaches the region where the filaments are seen. A fraction reaching kpc is suggested from a variation (Thomas et al. 1987). This value is derived from a simple model in which the flow is stationary and the X-ray emission comes from a homogeneous gas. Taking into account nonsteady flow (the flow time from 100 to 10 kpc is comparable to the age of the system - for instance, the values of n and T adopted in the present calculations imply a flow velocity of 5-50 km s^-1 at 10 kpc for yr^-1) and the contribution of nonlinear blobs to the X-ray emission makes the reduction of less drastic than that given by the model (Meiksin 1990; Friaça 1993). These effects, however, do not prevent matter from being removed from the flow, and the resulting is . Therefore, mass removal from the cooling flow limits the energetic input of AH, and the luminosities given in the previous section should be reduced by a factor .

Even if AH cannot be invoked as the sole mechanism powering the optical line emission of class II filaments, it could be, however, an important ingredient to explain the emission coming from low ionization lines in these systems. The models with nonlinear heating and high AH efficiencies exhibit a strong [OI] 6300 emission, and, therefore, AH could be an additional component together with other mechanisms for producing most of the emission in H and higher ionization lines (as [NII] 6583).

The combination of two distinct heating mechanisms, each one being responsible for certain emission lines and/or explaining one class of filaments, has allowed to build hybrid models for optical filaments, which have been particularly succesful at reproducing the observations of optical filaments. Shocks models tend to reproduce the locus in the line-ratio diagram of class II filaments, although the energetic requirements are stringent. A hydrid model could, therefore, combine shocks and an additional heating source which explains the higher ionization class I filaments. In this way, Crawford & Fabian (1992) invoke shocks between cold clouds to explain the class II filaments (and produce their [OI] emission) and mixing layers to account for the class I filaments with lower H luminosities and higher levels of ionization (higher [NII] 6583/H ratios). Other additional heating sources are magnetic reconnection (Jafelice & Friaça 1996), and Alfvén heating (this work). Jafelice and Friaça (1996) suggest a hydrid model involving magnetic reconnection (with an efficient producion of [OI] emission). and self-absorbed mixing layers with high flux and low temperature for the incident radiation (producing most of the H and [NII] emission).

Whether AH is at work together with shocks or with mixing layers, it efficiently supplies [OI] emission. For a representative erg s^-1 of model E, one obtains a [OI] 6300 luminosity erg s^-1 (see Fig. 2). Given the average ratio [OI] 6300/H of the optical filaments, AH could account for the [OI] 6300 emission of systems with up to erg s^-1. The AH keeps the gas relatively warm, giving rise to strong emission of [OI] 6300.

We note that the energetic requirements concerning the [OI] emission of the more luminous systems, like Perseus, with erg s^-1 (considering a H luminosity of erg s^-1 (Heckman et al. 1989) and a cooling flow rate of 183 yr^-1 (Fabian 1994)), are not satisfied by our model even in the most favorable case. For these extreme systems, an additional heating mechanism is required.

Whereas there is undoubtedly a connection between cooling flows and optical filaments, this connection is troubled by some facts, and our model can shed light in this issue. In the first place, while there is a strong correlation between the H luminosity and the mass accretion rate (at a 99.94% confidence level according to Heckman et al. 1989), the relation cannot be direct because of the large (more than two orders of magnitude) scatter in the relation and the fact that some massive cooling flows (e.g., A2029) show no detectable line emission. The existence of a cooling flow, therefore, is a necessary but not sufficient condition for detectable line emission. Secondly, except for a few filament systems (e.g., Perseus and A1795) extending over several tens of kpc, the optical line emission is confined to the inner 10 kpc of cooling flows, whereas the X-ray images of clusters reveal that the cooling gas is distributed throughout the cooling radius ( kpc).

The high spatial concentration of optical filaments towards the center could be explained by a greater magnetic field strength in the central regions (Soker & Sarazin 1990) and by a higher level of turbulence in these regions (Loewenstein & Fabian 1990; Loewenstein 1990; Begelman & Fabian 1990). As a consequence, an intense Alfvén wave generation is boosted only in the inner cooling flow. On the other hand, the level of turbulence could be the "second parameter" besides mass accretion rate, regulating the luminosity of optical filament systems. Turbulence, caused, for instance, by stirring of the central ICM by a galaxy moving through the cluster core, or by interactions between the ICM and the relativistic plasma of an extended radio source, could trigger, via generation and dissipation of Alfvén waves, optical emission lines. In this scenario, the generation of Alfvén constitutes a step in the tapping of the turbulent energy of the ICM into line luminosity.

Additional support for the idea that turbulence can be crucial for producing the line luminosity comes from the fact that the kinetic energy flow, determined from the observed velocity dispersion of the filaments, correlates much tighter (a scatter of one order of magnitude) with the H luminosity (Heckman et al. 1989) than does. This indicates that it is the combination of mass flow and high levels of turbulence that is relevant for the line luminosity, rather than a large mass accretion rate alone.

The present calculations are intended to apply to the inner kpc of the cooling flow s, where the optical filaments are most commonly seen. The turbulent velocities of the cooling flow increase toward the center of the cluster, as inferred from the increase of the line widths near the center (Heckman et al. 1989). Since the turbulent energy of the cooling flow is the energy source of the Alfvén waves, it is only in the inner kpc that the density of turbulent energy is high enough to make AH efficient. In this region, blobs condensating out of the cooling flow are kept warm enough by AH to produce strong line emission. Blobs cooling down in the outer regions are subject to low levels of AH and cool quiescently without noticeable optical emission (similarly to model A). In this way, the AH mechanism is consistent with the fact that line emission does not extend to kpc.

A further issue raised by the results of our models concerns the growth of thermal instabilities in cooling flows. It has been suggested that a number of processes could inhibit the growth of thermal instabilities in the presence of gravitational fields and background flow in cooling flows (Malagoli et al. 1987; Balbus 1988; Balbus & Soker 1989; Tribble 1989; Loewenstein 1989; Brinkmann et al. 1990; Hattori & Habe 1990; Yoshida et al. 1991; Malagoli et al. 1990; Reale et al. 1991). The fact that one of our models (with nonlinear heating, , and ) has reached thermal stability shows that also Alfvén heating can constitute a mechanism for suppressing strong growth of thermal instabilities in cooling flows. On the other hand, Gonçalves et al. (1993a; 1996a) have considered the possibility that the broad line regions of quasars are formed via thermal instability in the presence of AH. They have investigate three damping mechanisms of Alfvén waves - resonance surface, nonlinear, and turbulent -, and found that AH could establish a stable two-phase medium in the broad line regions of quasars. The thermal stability of the model mentioned above suggests that, also in the case of the cooling flow medium, AH could induce a stable two-phase equilibrium.

© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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