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Astron. Astrophys. 362, 447-464 (2000)

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2. Comparison to spectral aging methods

The age of extragalactic radio sources can be estimated by the use of spectral aging arguments. This method relies on the determination of the break in the radio spectrum caused by the time-dependent energy losses of the relativistic electrons within the cocoon (e.g. Alexander & Leahy 1987). At frequencies higher than the break frequency, [FORMULA], the spectrum significantly steepens due to the radiative energy losses of the electrons. The spectral age of a electron population, [FORMULA], with observed break frequency [FORMULA] within a magnetic field of strength B which does not vary in time is proportional to [FORMULA]. Despite the implication of source dynamics that the magnetic field does vary in time, the constant field relation is usually used in determining the spectral age of a given source. In general the strength of the magnetic field will decrease while the source expands and so [FORMULA] will be an overestimate of the true source age, t. However, because of variations of the magnetic field in the cocoon and the backflow of gas in this region, [FORMULA] is usually found to be lower than the dynamical ages inferred from the advance speed of the cocoon by various methods (Alexander & Leahy 1987).

The model presented in the following can be viewed as an extension of the spectral aging formalism. Within the cocoon it traces the evolution of the local magnetic field in time. This allows the accurate determination of the energy distribution function of the relativistic particle population and thus the emitted spectrum at a given location along the cocoon. It is therefore not surprising that the model predicts an older age for Cygnus A (see Sect. 4) than the classical spectral index analysis of Carilli et al. (1991). The combination of a dynamical model with the accurate treatment of the local evolution within the cocoon of the radio emission properties implies that the spectral and dynamical ages are identical.

2.1. Diffusion of relativistic particles

In the context of spectral aging methods other processes changing the energy distribution of the relativistic electrons and thus invalidating the age estimates have been put forward. These are summarised and discussed by Blundell & Rawlings (2000). They show that most of these processes are rather inefficient and will not strongly influence spectral aging methods or the model discussed here. However, Blundell & Rawlings (2000) claim that the anomalous diffusion mechanism of Rechester & Rosenbluth (1978) can lead to very fast diffusion of relativistic particles through the tangled magnetic field in the cocoon. So much so that the radio spectra observed at any point along the cocoon essentially arise from the electrons accelerated by the jet shock at the hot spot within the last [FORMULA] years of the observation. This would imply that no information on the source age can be derived from the spatial properties of the observed emission. Support for a universal particle energy spectrum may come from the interpretation of observations of Cygnus A by Katz-Stone et al. (1993), Rudnick et al. (1994) and Katz-Stone & Rudnick (1994). They find that a single, non-power law spectrum shifted in frequency by the local strength of the magnetic field is emitted by all parts of the lobes of Cygnus A. Note however, that the spectral shape they find may be caused by the free-free absorption in our own galaxy of the emission of Cygnus A at low frequencies (Carilli et al. 1991). In this case, the universal spectrum is simply explained by shifting an aging energy spectrum of relativistic particles which is not a single power law. Alternatively, changes in the strength of the magnetic field along a given line of sight may also introduce additional curvature in the observed spectrum (Rudnick et al. 1994). The mentioned interpretation of the radio observations of Cygnus A in itself is therefore no proof of efficient diffusion acting in the lobes of radio galaxies.

In the appendix I show that anomalous diffusion is probably much less effective in the cocoon plasma as previously thought. In general we do not observe any signature for diffusion losses of the cocoons of radio sources and so diffusion will not alter the distribution of relativistic particles within the cocoon. This allows us to use the spatial distribution of the synchrotron radio emission of FRII sources to infer their age. The model developed in the following can be viewed as an extension to the classical spectral aging methods in that it takes into account the evolution of the magnetic field in the lobe.

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© European Southern Observatory (ESO) 2000

Online publication: October 24, 2000