6.1. 1-D versus 2-D method
In general it is expected that an increase in the amount of information available to a given model fitting procedure should decrease the uncertainty in the determination of the model parameters. Comparing the results of the 1 and 2-dimensional methods for the lobes of Cygnus A one would therefore expect that the parameter uncertainties decrease for the higher resolution maps. This is indeed the case for the 2-dimensional method, particularly for the western lobe. However, the 1-dimensional method shows the opposite. Here the uncertainties are larger for the higher resolution maps. This is caused by the attempt to fit a very smooth model for the surface brightness to observational data which shows considerably greater local variations than the model. The fact that the model is unable to fit the local surface brightness structure observed was already noted in the previous Sect. In the 2-dimensional case the off-axis parts of the cocoons provide additional information and the influence of local structure is therefore to some extent averaged out in the fitting procedure. In other words, the 2-dimensional method is able to make use of the larger amount of information in higher resolution maps. The 1-dimensional method is restricted to a cut through the lobe. Here, the averaging effect of a larger telescope beam provides for a smoother surface density profile and the model fits the data better.
In the case of radio maps of low resolution which do not or only barely resolve the lobes perpendicular to the jets along most of their lengths, the 2-dimensional maps add little information to 1-dimensional cuts along the lobes. In these cases the additional model parameter p in the 2-dimensional method presented here allows the model to fit the data with a large range of parameter combinations. The uncertainties of the model parameters are then larger than for the 1-dimensional method. This effect can be seen for 3C 215 and, to a lesser extent, for the low resolution maps of the western lobe of Cygnus A. It is of course possible to fix the value of p for the 2-dimensional method as well but this does not significantly improve the constraints on the model parameters compared to the 1-dimensional method.
Which method is the best to use for a particular set of radio maps depends on the quality of the maps. If the radio lobes are well resolved along the jet axis as well as perpendicular to it then the 2-dimensional methods will provide better constraints. However, it is computationally expensive. In the case of poorer resolution the 2-dimensional method will not add anything to the results obtained from a 1-dimensional comparison. For heavily distorted lobe structures both methods will fail but the 1-dimensional method may still provide order of magnitude estimates if the lobes are not entirely dominated by bright localised structure.
6.2. Determination of viewing angles
Orientation-based unification schemes attempt to explain radio galaxies and radio-loud quasars as essentially the same type of objects albeit viewed at different angles to the jet axis (Barthel 1989). The broad line radio galaxies may then be identified as the low redshift analoga to radio-loud quasars. These unification schemes imply that the viewing angle, , to the jet axis is greater than about for radio galaxies and smaller for quasars. Tests of this hypothesis are often inconclusive as the determination of is difficult in practice.
As described above, the viewing angle may be determined from radio observations alone. However, since the model depends mostly on , the uncertainties are large particularly for . Fortunately, the other model parameters and the source and environment properties inferred from these do not depend strongly on and its error in this range. This agrees with the results of Wan & Daly (1998) who study the effects of the viewing angle on the determination of a variety of source properties in great detail. The source orientation can only be determined accurately if the 2-dimensional comparison method is used on radio maps well resolved perpendicular to the jet axis. This agrees with the findings of Wan & Daly (1998) who use a different source model. Despite the large uncertainties in the model results presented here it is interesting to note that for the three sources studied the best-fitting values of are smaller for the quasar (3C 215) and the broad line radio galaxy (3C 219) than for the radio galaxy (Cygnus A).
6.3. Environments of FRII sources
As I showed above, the model parameters of the best-fitting models can be used to infer the density parameter, , describing the density distribution in the environment of FRII objects. It is impossible using just this model to separate this into the central density, , and the core radius, . Furthermore, the model is insensitive to the slope, , of the external density profile (see Sect. 4.3). To infer central densities, values for the core radius and must be taken from other observations. In Table 4 for the three example sources is shown for the core radii given by Hardcastle & Worrall (1999) and . As noted in Sect. 5, the central densities found for 3C 219 and 3C 215 are inconsistent with those derived from the X-ray observations.
In several studies it was found that the pressure of the FRII source environment derived from X-ray observations apparently exceeds the pressure inside the radio lobes (e.g. Hardcastle & Worrall 2000 and references therein). The discrepancy in density found here is essentially the same phenomenon in the framework of isothermal density distributions for the source environments as described by the -model, Eq. (1). All recent models for the evolution FRII sources, including the one discussed here, are based on the assumption that the cocoon is overpressured with respect to its surroundings. The X-ray observations seem to contradict this assumption.
Can the discrepancies be resolved? In the model described here I approximate the density distribution in the source environment, assumed to follow the -model, by power laws. For cocoons extending well beyond the core radius the exponent of the power law is given by . In the case of Cygnus A this is justified as both lobes extend over more than 100 kpc while kpc. In the analysis was used which is identical with the result of Hardcastle, . For 3C 219 and 3C 215 from the X-ray observations and in both sources the radio lobes only extend to about . Formally the model cannot be applied to these two sources because the underlying dynamical model is valid only for or . However, as the cocoons in both sources do not extend much beyond the core radius, the density distribution in the environments of 3C 219 and 3C 215 are well represented by a power law with exponent . In these cases found from the model is not identical to the central density of the -model. From Eq. (1) it can be seen that the latter is given by , where x is the length of the radio lobe in units of . This correction is insufficient to make the model results consistent with the X-ray observations of 3C 219 and 3C 215.
X-ray observations of the hot gaseous environment of AGN are influenced by the presence of the active galaxy. Before the environment properties can be extracted, the bright X-ray emission of the AGN itself appearing as a point source must be carefully removed. In the case of radio-loud objects the large scale structure caused by the jets can also alter the X-ray emission of its surroundings. The magnitude of these effects is difficult to estimate if the X-ray observations do not fully resolve the scale of the cocoon. The hot spots at the end of the jets are strong sources of inverse Compton scattered X-ray photons. When resolved, these inverse Compton scattered photons are found to distort the X-ray contours of the extended emission (Cygnus A, Carilli et al. 1994; 3C 295, Harris et al. 2000). The more extended cocoon material itself may also act as a scatterer of CMB photons or of AGN emission (Brunetti et al. 1999). Finally, the bow shock surrounding the cocoon compresses and heats the gas in the source environment. Kaiser & Alexander (1999b) give an estimate for the expected X-ray luminosity from this shocked layer of gas,
where two typographical errors are corrected. Expressions for , and may be found in Kaiser & Alexander (1999b). The square brackets mean the difference of the exponential function at the limits of the observing band and in the rest frame of the source. The resulting X-ray luminosity can be converted to a ROSAT count rate using the internet version of PIMMS 1. For the best-fitting model parameters for Cygnus A this implies that roughly 2% of all counts of the ROSAT observations presented by Hardcastle & Worrall (2000) attributed to the entire extended emission come from the layer of shocked gas in between bow shock and cocoon of this source. For 3C 219 this region around the southern lobe alone contributes about 10% of the relevant counts. The contribution of the northern lobe of 3C 215 is negligible (%). However, in this case at least three very compact radio emission regions along the distorted jets are detected which may be powerful inverse Compton sources (Bridle et al. 1994). Only X-ray observations resolving the scale of the radio structure in this and other sources will allow us to determine the contribution to the total X-ray emission from such compact regions and hot spots.
Even without detailed observations it is clear that powerful radio sources can contribute significantly to the extended X-ray emission in the central part of their environments. Helsdon & Ponman (2000) show for the case of loose groups of galaxies that such an overestimate at the centre of an X-ray surface brightness profile may lead to overestimates of the core radius and also of when fitted with a -model. Using these overestimated values of and then yields values for which are too low compared to the `real' central density. An exact analysis of the magnitude of this effect is beyond the scope of this paper. In any case, this effect may explain the discrepancies found between the two methods of estimating the central densities and pressures of the FRII environments.
© European Southern Observatory (ESO) 2000
Online publication: October 24, 2000