Inflating the cavities

As discussed in Heinz et al. (2002), the current radio source is likely too weak to produce notable cavities, and it is likely that the observed ICM cavities are ``ghosts'' of a previous and more powerful period of activity. In this picture, the cavities are in a passive phase of evolution (see Reynolds, Heinz & Begelman 2002). The X-ray cavities, which were created by a past phase of supersonic lobe expansion, have decelerated to sub-sonic velocities. Any shocks once bounding the lobes have weakened into mere compression waves. The fact that this activity produces an expanding shell of ICM implies that gas from the core regions will be lifted to higher points in the cluster, thereby adiabatically cooling as it de-pressurizes. This cooling effect can largely offset the heating from the ICM compression and (certainly to within the accuracy of our data) mask any remaining signs of compressional heating. This explains the lack of hot gas in or around the cavities.

In this evolutionary phase, the cavities will buoyantly rise within the cluster potential on a timescale a factor of a few longer than the sound crossing time of $\sim 2 \times 10^7\,{\rm yrs}$. As they rise buoyantly and expand, the relativistic electron population will undergo synchrotron, inverse Compton, and adiabatic energy losses. The synchrotron and inverse Compton losses result in a high-frequency cut-off that gradually marches to lower and lower radio frequencies. Using the standard formulae for synchrotron losses (e.g., Rybicki & Lightman 1979), it is readily shown that, assuming an isotropic relativistic electron distribution evolving in a constant or decreasing strength magnetic field, the high-frequency cut-off of the synchrotron spectrum will obey

$$\nu_{\rm cut} \lesssim 26\,\left(\frac{B}{60\,\mu{\rm G}}\right)^{-3}\left(\frac{t}{20\,{\rm Myr}}\right)^{-2}\,{\rm MHz},$$ 1

where approximate equality corresponds to the case where the magnetic field and the particle pressure are constant in time. This expression assumes no fresh injection or acceleration of relativistic electrons (which would turn the cut-off into a spectral break), and hence only applies once the radio-lobes are no longer supplied by active jets (i.e., after the radio-source ``dies''). The ICM pressure at the location of the ghost-cavities is measured to be approximately $p \approx 10^{-10}\,{\rm ergs\,cm^{-3}}$. If we assume that the synchrotron emitting plasma is in pressure equilibrium with the surrounding ICM (which is very likely to be true for the ghost cavities) and furthermore, that the magnetic field in the plasma has approximately equipartition strength and is tangled on scales small compared to the cavity size, this pressure gives us a field strength of $B \approx 60\,\mu{\rm G}$. Thus, assuming ICM/cavity pressure balance and equipartition magnetic fields, we can see from eqn.(1) that the cavities will fade out of the 1.4 GHz band only 4 Myr or so after the outburst of the radio-galaxy activity has ceased. Since we believe the ghost cavities to be approximately 20 Myr old (Heinz et al. 2002), we see that there has been ample time for the plasma filling the cavities to fade out of the higher frequency radio bands if the magnetic field posesses roughly equipartition strength.

Studies with ROSAT, Chandra, and XMM-Newton have allowed the magnetic field strengths of several radio lobes to be estimated through the direct detection of the X-rays thought to be produced by inverse Compoton scattering of the Cosmic Microwave Background (CMB) by the relativistic electrons (Leahy & Gizani 2001; Hardcastle et al. 2002; Grandi et al. 2003; see also Wilson, Young, & Shopbell 2001 for related arguments in the hot spots of Cygnus A). In these studies, it is typically found that the magnetic field is at least a factor of two lower than the equipartition value. Even if the magnetic field has half of the equipartition field strength, there is sufficient time for the 1.4GHz emission from the ghost cavities to fade.

Having put forward a fairly traditional hypothesis for the formation and evolution of the X-ray cavities, we now proceed to consider the complexities special to Abell 4059.