In this paper an analytical model for the surface brightness distribution of the cocoons of FRII-type radio sources is developed. The model is based on the self-similar model of Kaiser & Alexander (1997, KA) and its extension by Kaiser et al. (1997, KDA). The cocoon is split into small volume elements, the temporal evolution of which is traced individually. These elements are then identified with cylindrical slices which are rotationally symmetric about the jet axis. The bulk backflow of these slices is determined self-consistently from the constraints on the cocoon shape. Thus a 3-dimensional model of the synchrotron emissivity is constructed. Projecting this along the LOS, the model yields surface brightness profiles in one or two dimensions in dependence of several model parameters. It is shown that a number of degeneracies prevent constraints to be placed on all parameters. Comparatively robust estimates can be found for the cocoon pressure, , the source age, t, the angle of the jet to the LOS, , and, if the 2-dimensional comparison method is used, the initial slope of the energy distribution of the relativistic particles, p.
The model may be viewed as an extension of the classical spectral aging methods. However, it accurately takes into account the loss history of the relativistic particles and the evolution of the magnetic field in the cocoon. The age estimates derived from the model are therefore accurate and not only lower limits. It is also shown that diffusion of relativistic particles cannot significantly distort the energy spectrum of the relativistic particles in the cocoon.
In Sect. 5 the model predictions were compared with observational data for the three sources Cygnus A, 3C 219 and 3C 215. The four free model parameters are constrained to within at least a factor 2. The uncertainties of the cocoon pressure are remarkably small, less than 50%, while the source ages are constrained by the model to within 80% or better. The errors on the viewing angle are large as the model mainly depends on which changes in the range of orientation angles found here (31o to 90o) by only 52%. The uncertainties in the model parameters translate to errors in the determination of the jet power, , and the density parameter, . However, the resulting uncertainties still allow at least order of magnitude estimates for these quantities.
The model results do not depend strongly on the resolution of the radio observations used in the fitting procedure. The results for both lobes of Cygnus A derived at 1.3" and 5" resolution are consistent with each other within the errors. However, the uncertainties in the parameter determination increase significantly with decreasing resolution. For radio maps which barely resolve the lobes perpendicular to the jet axis, the 2-dimensional method adds little in terms of constraints on model parameters to the 1-dimensional method. From the examples studied here it is not possible to give a firm lower limit for the resolution of maps used to fit with the model. However, the parameter uncertainties calculated here suggest that at least 4 independent beams along the jet axis are required. As was pointed out in Sect. 3.4, distorted lobe structures may indicate large turbulent flows within the cocoon. As the model is based on the assumption of largely non-turbulent backflow it cannot be used for such irregular sources.
The model can be used to infer the viewing angle of FRII-type sources to the LOS. Despite the large uncertainties in this determination, the three sources modeled here show the dependence of on the spectral type of the host galaxies expected from orientation-based unification schemes.
The central density of the source environments found using this model is consistent with X-ray observations in the case of Cygnus A. In the case of 3C 219 and 3C 215 the estimates are too low. These discrepancies may stem from overestimates from X-ray observations of the core radius, , and slope, , of the density distributions. They are caused by the contribution of the large scale structure, i.e. cocoons and hot spots, of the radio sources to the X-ray emission. The, at present, insufficient spatial resolution of X-ray maps prevents the removal of these contributions before the determination of and .
The model allows the simultaneous determination of a number of key
parameters of extragalactic radio sources of type FRII. Together with
the 1 and 2-dimensional comparison methods, it is straightforward to
analyse large numbers of objects drawn from complete samples of
extragalactic radio sources. The quality of the radio data needed is
only moderate and will already be available for many of the objects in
question. However, the large scale structure of the objects should be
reasonably regular in the sense that large scale turbulent motion
within the cocoon is unlikely. Within these limits the method can
provide order of magnitude estimates for the properties of the sources
and their environments. This may help to answer some of the questions
related to the cosmological evolution of the FRII population and
unification schemes of radio-loud AGN.
© European Southern Observatory (ESO) 2000
Online publication: October 24, 2000