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

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1. Introduction

The `standard model' of extragalactic classical double radio sources by Scheuer (1974) explains these sources as twin jets emerging from the Active Galactic Nucleus (AGN) and impinging on the surrounding gas. The compact, high surface brightness regions or hot spots at the end of the jets are interpreted as the sites of the interaction between the jets and the environment. After passing through the hot spot region, the jet material inflates the radio lobes or cocoon which is observed as diffuse emission in between the hot spots and the source core (e.g. Muxlow & Garrington 1991 and references therein). This picture forms the basis of virtually all more recent attempts at modeling the dynamics and radio emission properties of powerful radio galaxies and radio-loud quasars (Begelman & Cioffi 1989, Falle 1991, Daly 1994, Nath 1995, Kaiser & Alexander 1997, Chyzy 1997, Kaiser et al. 1997, Blundell et al. 1999).

These models for the evolution of individual classical doubles or FRII-type objects (Fanaroff & Riley 1974) can be used to study the cosmological evolution of the population as a whole. One of the more important trends is the apparent decrease of the mean linear size of the radio structures with increasing redshift (see Neeser et al. 1995 and references therein).

Two recent attempts in fully explaining the linear size - redshift anti-correlation were presented by Blundell et al. (1999) and Kaiser & Alexander (1999a). Blundell et al. argue that a specific form of pre-aging of the relativistic particle population in the hot spots in connection with the lower flux limit of complete observed samples causes this anti-correlation (the so-called youth-redshift degeneracy, Blundell et al. 1999, Blundell & Rawlings 1999). Alternatively, Kaiser & Alexander propose that the apparent smaller sizes of radio sources at high redshift could be caused by a denser environment of these objects at high redshift. Clearly, the evolution or non-evolution of the radio source environments is of great interest to decide this and other important cosmological questions.

Various methods have been employed in determining the properties of the radio source environments. X-ray observations of the hot gas around radio sources in clusters yield direct estimates of the gas density (Crawford et al. 1999, Hardcastle & Worrall 1999). However, such studies are at present confined to objects at low redshifts. Furthermore, the AGN (e.g. Sambruna et al. 1999) and the large-scale radio structure contribute to the X-ray emission (Brunetti et al. 1999, Kaiser & Alexander 1999b, Sect. 6). The properties of the density distribution inferred from X-ray observations may therefore be considerably influenced by the presence of the radio source itself.

Faraday rotation of the polarisation angle of the synchrotron emission and the related depolarisation can be used to determine the gas density of the material surrounding the radio lobes (e.g. Garrington et al. 1988, Laing 1988). Unfortunately, this method does not provide a direct measure of the gas density as the rotation measure also depends on the strength of the magnetic field in the source environment. Usually it is not possible to break this degeneracy because the strength of the magnetic field is only poorly known.

Constraints on the density of the radio source environments also come from optical or infrared galaxy counts around the host galaxies (e.g. Hill & Lilly 1991). It is not straightforward to decide whether galaxies in the field of the radio source host are associated or chance background objects. Resolving the ambiguity would ideally require the spectroscopic measurement of the redshifts of all objects in question. This is very time consuming. In any case, the method provides only indirect constraints on the gas density around the radio source as this has to be inferred from comparison with low redshift clusters or groups of similar richness.

The ages of powerful radio sources can in principle be determined from their radio spectrum (e.g. Alexander & Leahy 1987). The various energy losses of the relativistic particles depend on time and so the shape of the radio spectrum contains an encoded history of the source. In practice the time-dependence of the energy losses complicates the estimation of the spectral age because different parts of the source have different ages. Even when radio maps at various frequencies which fully resolve the radio lobes are used, it is difficult to disentangle the effects of the various loss mechanisms and possible bulk backflow of the cocoon material along the jet (e.g. Rudnick et al. 1994).

The model developed in this paper aims at tracing the individual evolution of parts of the cocoon and thereby providing more accurate estimates for the source age. At the same time the model also constrains the density in the source environment and other parameters like the energy transport rate of the jets. It is solely based on radio observations which are available for a large number of objects, even at high redshift. This model may therefore provide an important step in determining the cosmological evolution of the FRII radio source population.

In Sect. 2 I show that diffusion of relativistic particles in the cocoons of FRII-type objects should not significantly change the energy distribution of particles. In Sect. 3 the dynamical model of Kaiser & Alexander (1997, hereafter KA) and its extension by Kaiser et al. (1997, hereafter KDA) to include synchrotron emission are briefly summarised. A 3-dimensional model of the synchrotron emissivity of the cocoon based on this analysis is constructed in Sects. 3.3 and 3.4. Methods for comparing the model predictions with observations are developed in Sect. 4. The degeneracy of model parameters resulting from the comparison method is also discussed here. The model is then applied to three FRII-type radio sources, Cygnus A, 3C 219 and 3C 215, in Sect. 5. The results are discussed in Sect. 6. The main conclusions are summarised in Sect. 7.

Throughout this paper I use [FORMULA] km s-1 Mpc-1 and [FORMULA].

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Online publication: October 24, 2000