6.3. Lobe Spectra and the Age of the Radio Source
Spectral ageing across the radio lobes has been considered in detail by Alexander et al. (1984, 1996) and Carilli et al. (1991a). These studies find a roughly linear increase in age with distance from the hotspots. The implied `separation velocity' is 0.045c h-1, derived using minimum energy fields of 50µG. The oldest electron populations are found in the radio `plumes' extending to the north and south of the center of the radio bridge, with an implied source age of 6 Myr.
As pointed out by Winter et al. (1980) the separation velocity measured from spectral ageing is the sum of two velocities: the source advance speed and the back-flow velocity of material in the lobes. An important question is whether the waste hotspot material is simply left behind by the advancing lobes (back-flow velocity = 0), or whether there is bulk fluid flow back towards the galaxy? An independent estimate of the advance speed comes from the ram pressure calculation. The ram pressure advance speed for the hotspots in Cygnus A is 0.02c, derived using minimum energy fields. In the context of the dentist's drill model Williams (1985) has shown that for sources in which the hotspots cover only a small fraction of the heads of the source such as Cygnus A the overall ram pressure advance speed must be lowered by the ratio of the hotspot area to the area of the heads of the lobes. In Cygnus A this would decrease the advance speed by another order of magnitude. This leads to the problem that the source advance speed derived from ram pressure arguments is well below the separation velocity derived from spectral ageing analysis. A back-flow velocity >> source advance speed contradicts very basic continuity arguments in Cygnus A for which the radio plumes (presumably the catch-basin for the back-flow) constitute only a small fraction of the total volume of the source (Carilli et al. 1991a). This problem is alleviated by considering that the source may still be expanding transversely, and aggravated by the possibility that the source advance may have been slower in the past due to the overall density gradient in the cluster (Arnaud et al. 1984). A possible solution is to allow departures from minimum energy conditions. Carilli et al. find a self-consistent, although admittedly ad hoc, model is possible in which the the fields are systematically below minimum energy by a factor of three resulting in an average source advance speed separation velocity 0.01c, and a source age of 30 Myr.
Alexander et al. (1984, 1996) and Carilli et al. (1991a) argue that expansion losses are large going from the hotspots to the radio lobes but that in the lobes themselves the dominant energy loss mechanism is synchrotron radiation. In terms of the shape of the spectra above the break Carilli et al. find that the lobe spectra steepen more than allowed by continuous injection models, but less steeply than exponential. The best fitting model involves `one-shot' injection with subsequent radiative losses, but without continuous isotropization of the pitch-angle distribution which would lead to an exponential cut-off (Pacholcyzk 1970, Jaffe and Perola 1973, Kardashev 1962).
Overall, the general evolution of spectral steepening with distance from the hotspots, and the fact that the lobe spectra steepen more than allowed in the case of continuous injection, are in good agreement with synchrotron spectral ageing theory and the jet model for powerful radio galaxies. However, there are a few difficulties with this simple analysis. First are the difficulties encountered when considering the relative contribution of back-flow and source advance to the separation velocity as derived under minimum energy conditions, as discussed above. Second, Carilli et al. determine an injection index of -0.7 in the radio lobes, while the best fit value for the hotspots is -0.5. This `injection index discrepancy' is difficult to reconcile with the idea that the principal location of particle acceleration is the hotspot (3) And third, the lack of an exponential cut-off implies no pitch angle scattering of the relativistic particles. This is hard to justify physically since pitch angle scattering of streaming electrons by self-induced Alfven waves is thought to be an efficient process (Eilek and Hughes 1991, Wentzel 1974, Wentzel 1969).
Another important question in spectral ageing analyses is the possibility of a temporally and/or spatially variable magnetic field and their effect on derived synchrotron ages. This question has been considered by a number of authors (Wiita and Gopal-Krishna 1990, Tribble 1993, Siah and Wiita 1990). Tribble shows that allowing for a distribution of magnetic field strength requires the use of a weighted-mean field strength in the age calculation. He also shows that a distribution in field strength solves the exponential cut-off problem (or lack thereof), by `smearing-out' the individual exponentially steepening spectra resulting in observed spectra similar to those found in the lobes of Cygnus A.
3 An interesting aside is the fact that a
`injection index discrepancy' exists for cosmic rays in the disks of
spiral galaxies. The observed index for the integrated emission from
spiral galaxies is typically -0.7, while the average supernova remnant
(presumably the sites of cosmic ray acceleration) has an index of -0.5
The solution proposed in the case of spiral galaxies is
energy dependent cosmic ray diffusion. However, this solution cannot be
invoked in the case of radio galaxies, since diffusion is probably
insignificant compared with convection for relativistic electron
transport in the lobes
(Carilli et al. 1991a).
3 An interesting aside is the fact that a similar `injection index discrepancy' exists for cosmic rays in the disks of spiral galaxies. The observed index for the integrated emission from spiral galaxies is typically -0.7, while the average supernova remnant (presumably the sites of cosmic ray acceleration) has an index of -0.5 (Condon 1992). The solution proposed in the case of spiral galaxies is energy dependent cosmic ray diffusion. However, this solution cannot be invoked in the case of radio galaxies, since diffusion is probably insignificant compared with convection for relativistic electron transport in the lobes (Carilli et al. 1991a). Back.