The reconnection time in the intracluster medium is long (Soker & Sarazin 1990). For conditions typical for the bulk of the intracluster gas, the reconnection can proceed with an effective velocity no greater than v_A where v_A is the Alfvén speed (Parker 1979). This gives a timescale t_rec for reconnection over a length scale l of
t_rec 3x10^9 ( / 0.1 )^-1 (l/10 kpc) (B / 1 )^-1 ( n_e / 2x10^-3 cm^-3 )^1/2 yr. (1)
Note that this estimate is probably too low as most reconnection mechanisms only work efficiently when the field is relatively strong, whereas intracluster fields are weak. While clearly uncertain, the important point is that the reconnection time is long.
The magnetic field in clusters is ~1 so that the dominant energy loss mechanism for relativistic electrons is inverse Compton scattering of microwave background photons. The typical aging timescale for radiation observed at a wavelength _obs is
t_age ~ 10^8 (_obs / 30 cm) (B / 1) (1+z)^-9/2 yr. (2)
Note the very strong redshift dependence due to the increased energy density of the microwave background at higher redshift. The shape of the spectrum should be close to that of the Jaffe-Perola spectral aging model (Jaffe & Perola 1973), with an exponential decline at high frequencies. Therefore, if radio halos are created (in some process) and then have no further injection of relativistic electrons or reacceleration, they will fade from view within about 10^8 yr. It is difficult to make radio emission last much longer than this, for if the field is stronger then energy losses are greater and the source will fade more rapidly, and if the field is weaker then the spectral break moves to lower frequencies.
We can come to a simple and robust conclusion by comparing the two timescales. The radio spectral aging time is short, whereas the magnetic field decay time is long. Therefore, if cluster mergers input energy into relativistic particles and magnetic fields, the relativistic particles will lose energy quickly but the magnetic field will decay only slowly. Successive mergers will gradually build up the magnetic field, but the cluster will only have a radio halo in the 10^8 yr immediately following a merger event. We are therefore led to the conclusion that unless some mechanism can continually input relativistic electrons into the intracluster medium, radio halos are transient. Their rarity is then a consequence of their short lifetimes relative to the time between mergers of ~2--4x10^9 yr (Edge, Stewart & Fabian 1992).
In almost all cases radio halos are in fact found in merging or merged clusters. As noted by Edge et al. (1992), clusters without cooling flows tend not to have a single central dominant galaxy, and tend to contain radio halos (Coma, A2255, A2256). The large radio halos similar to that in the Coma cluster are not found in cooling flow clusters. Hanisch (1982) listed five radio halos - Perseus, Coma, A2255, A2256, and A2319. Of these, Perseus has a cooling flow and a minihalo much smaller in extent than the others that is localized to the cooling flow region. As noted by Hanisch (1982), although A2319 is classified as a cD cluster the cD galaxy is not very dominant and the central cooling time is long (Edge et al. 1992), suggesting A2319 is a recent merger. Both Edge et al. (1992) and Watt et al. (1992) also note the observational correlation between cooling flows, radio halos, and mergers.
The primary example of a radio halo is of course that in the Coma cluster, with two bright central galaxies. Coma has no cooling flow and is thought to be dynamically young (Fitchett & Webster 1987; Mellier et al. 1988), the central regions being still unrelaxed and evolving to an equilibrium state. Watt et al. (1992) argue that the difference in the X-ray properties of the Perseus and Coma clusters is caused by the latter having undergone a recent major merger event. This is supported by the presence of X-ray substructure (Briel, Henry & Böhringer 1992), and the finding of significant subclustering in the galaxy distribution by Escalera, Slezak & Mazure (1992).
Another example is given by A2256 which has been shown to be merging by ROSAT observations (Briel et al. 1991) and contains diffuse radio emission possibly associated with the merger (Fabian & Daines 1991). A2256 is a younger merger than the Coma cluster, so that the newly forming halo has not yet had time to spread throughout the cluster. In addition, the main A2256 cluster does not contain a cooling flow and is probably the result of a previous merger - the main cluster contains a diffuse radio halo of its own that is just visible at 610 MHz (Bridle & Fomalont 1976). The low frequency emission from this cluster has spectral index 1.9 (Costain, Bridle & Feldman 1972) which is very steep, just as would be expected for an old dying halo.
This view of single burst creation followed by spectral aging and fading is supported by 11 cm observations of the Coma cluster by Schlickeiser, Sievers & Thiemann (1987), together with the 6 cm upper flux limit of Waldthausen (1980), which show that the radio spectrum breaks sharply, as expected for a Jaffe-Perola type spectrum. In addition, the model makes the extent of the halo simpler to understand, in that electrons are accelerated over large regions of the cluster and can easily diffuse to fill the whole volume, rather than being accelerated in a small number of localized regions and having to diffuse over large distances.
Thus, in contrast to the view proposed by Burns et al. (1992) in which radio halos are not seen in cooling flow clusters because cooling flow clusters have weaker magnetic fields at large radii, radio halos are rare and occur in recently merged clusters because the relativistic electrons responsible for the radio emission are accelerated in the merger event. Radio halos are not commonly seen in cooling flow clusters because cooling flows are not found in rich clusters that appear to have recently merged, possibly because the cooling flow is suppressed or disrupted by a major merger (McGlynn & Fabian 1984). By the time the cooling flow has reestablished itself the radio halo will have faded and will no longer be visible. Even in cooling flows, there will be a reservoir of low energy electrons. This reservoir can be important in two ways, as a source of particles that give observable radio emission in much stronger fields (for example, in cooling flows), and also as a source of high energy particles that can then be reaccelerated in the next merger event.