5. Application to Cygnus A, 3C 219 and 3C 215
To test the model predictions for the source environment against direct X-ray observations over a range of different viewing angles, the radio data of three different FRII-type objects are used: the narrow-line radio galaxy Cygnus A, the broad-line radio galaxy 3C 219 and the radio-loud quasar 3C 215. According to orientation-based unification schemes of the various sub-classes of radio-loud AGN (e.g. Barthel 1989), the viewing angle, , of Cygnus A should be greater than those of 3C 219 and 3C 215. Furthermore, 3C 219 and 3C 215 were selected because of their rather irregular radio lobe structure. The model is based on a very regular geometrical shape of the cocoon, Eq. (12), and using 3C 219 and 3C 215 it is possible to estimate to what extent this restriction limits the applicability of the model.
Note, that results of the model fitting for Cygnus A are presented in Fig. 6 and Fig. 7 only at one frequency. However, the model fits are always obtained for all sources using two maps at two different frequencies.
5.1. Cygnus A
For this source at I used raw data from the VLA archive at 1.8 GHz and 5 GHz. The lower frequency observations were made in August/September 1987 in A-array and the 5 GHz data were obtained in January 1984 in B-array. A detailed analysis of these and other observations of Cygnus A can be found in Carilli et al. (1991). Standard calibration and self-calibration was performed using the software package AIPS. This resulted in two radio maps with comparable angular resolution of 1.3" at the two observing frequencies. The CLEAN components were restored in maps with an individual pixel size of 0.3"0.3". The map at 1.8 GHz with the hot spot emission removed (see below) is shown in the upper panel of Fig. 6. The rms noise in the maps is 0.03 Jy beam-1 at 1.8 GHz and 0.006 Jy beam-1 at 5 GHz. All pixels in the maps with a surface brightness below 5 were discarded for the 2-dimensional comparison method. In the case of the 1-dimensional method only pixels below 3- were neglected. The surface brightness distribution along the jet axis, , for the 1-dimensional method was obtained for both radio lobes along a cut from the core of the source to the eastern and western hot spot respectively.
Since the hot spot emission is not modeled, it has to be removed from the maps and the 1-dimensional cuts. In the maps an aperture centered on the surface brightness peak in each lobe and with a radius of 2.6" corresponding to twice the beam width is removed (see Fig. 6). In the western lobe the bright secondary hot spot (Carilli et al. 1991) and the bright ridge connecting this hot spot with the main one are also removed. For the 1-dimensional method, the distance of the hot spot to the edge of the lobe, , was estimated as the distance of the maximum of to the last point at which this function has a value above 3 in the direction away from the source core. To remove the contribution of the hot spots to , all pixels within of the edge of the lobe were neglected in the following 1-dimensional comparison process.
The spatial resolution of the radio maps of Cygnus A is comparatively high. The 1.3" angular resolution corresponds to a spatial resolution of about 1.9 kpc. For many sources, particularly at high redshift, maps of such high quality are not available. In order to estimate the effects of a lower angular and therefore lower spatial resolution, I also convolved the two maps of Cygnus A with a Gaussian beam of 5" FWHM. In the case of the 2-dimensional method the radius of the aperture used to remove the hot spot emission was fixed to 5". For the 1-dimensional method, the surface brightness distribution along the jet axis was extracted from these lower resolution maps in the same way as in the higher resolution case. Note that the emission of the hot spots is smeared out over a larger area in the low resolution maps. To avoid any bias from the enhanced emission of the hot spot region I used the higher value of obtained from the low resolution maps in both, the high and low resolution, 1-dimensional comparison.
5.1.1. The eastern lobe
The eastern lobe of Cygnus A is covered by 46.6 independent telescope beams along the jet while in the widest part there are 22.2 beams across. In the lower resolution maps these numbers decrease to 12.9 and 6.1, respectively. Note, that only a fraction of the lobes has an observed surface brightness above the rms limits, i.e. they do not extend all the way from the hot spots to the core in the observations. This implies that the model fits are based on regions covering less area than the theoretical extend of the cocoons. For both resolutions I find an axial ratio, , of 2.1 at the point where the lobe is widest from the 2-dimensional maps. The length of the lobe, , is 64.4" for the low resolution map and 60.6" for the high resolution map. The prediction of the best-fitting model in comparison with the observations is shown in the lower panels of Fig. 6 and Fig. 7 for the 2-dimensional method and in Fig. 8 and Fig. 9 for the 1-dimensional method at 1.8 GHz. The 5 GHz data is not shown for the 1-dimensional method but is similar to the result at 1.8 GHz. The parameters of the best-fitting models are given in Table 3. The errors on these and for all the following model fits are estimated using the boot-strap method (e.g. Press et al. 1992). It is not possible to calculate error estimates using the -values derived in the minimisation procedure directly as the values of the surface brightness in neighboring pixels are not independent. Roughly 2000 data sets were created by drawing data with replacement from the original set. The same minimisation procedure as in the original model fitting was then applied to them and the error given in the table is the 1 limit on the respective model parameters.
Table 3. The best-fitting model parameters from the 1 and 2-dimensional comparison methods.
Even after subtracting the contribution of the hot spot the deviations of the model from the observations are large. The fact that the model fit is poorer at 5 GHz is mainly caused by the smaller rms noise of the observed 5 GHz map. As expected, the model cannot fit structures which appear as discrete surface brightness enhancements in the maps. This is particularly clear in the case of the bright arc seen just behind the hot spot in radio maps of the eastern lobe (see Fig. 6 and Fig. 7) which also causes the secondary peak in at about (see Fig. 9). Although convolving the maps with a larger beam `draws' some flux from the hot spot into the arc mentioned above, the results for the two different resolutions are very similar for both comparison methods. At both frequencies the observed maps show a concentration of the radio emission towards the centres of both lobes. This region is also extended a long way along the jet axis, particularly in the western lobe (see following section). In Fig. 6 and Fig. 7 it is clear that the model cannot fit this concentration properly. The very smooth lobes of the model are `fatter' than the observed lobes further away from the hot spots and do not extend as far back as the observations indicate. This is particularly striking in the western lobe at 5 GHz (see Fig. 7). This clearly illustrates the limitations of the model assumption of a smooth shape of the cocoon and a regular backflow within the cocoon. The large uncertainty of the viewing angle, , is caused by the model mainly depending on which changes only by a factor 1.2 within the estimated errors. This is comparable to the uncertainties of the other model parameters.
Using Eqs. (2) and (3) The power of the jet, , and the central value of the density distribution of the gas surrounding Cygnus A, , are calculated. The results are given in Table 4. Here I assume that the core radius of the environmental density distribution in Eq. (1) is given by kpc. The viewing angle is inferred from the flux ratio of the jet to the counter-jet in the two lobes of Cygnus A (Hardcastle et al. 1999). This assumes that the two jets are identical and that the observed flux ratio is entirely due to relativistic beaming effects. Furthermore, a constant bulk velocity within the jets, , is assumed and set to 0.62 c. Variations of across the source and asymmetries between the two jet sides will significantly influence the estimate for . However, the model is consistent with the estimate given by Hardcastle et al. (1999).
Table 4. Properties of the source environment derived from the best-fitting model parameters in comparison with observations. For the determination of the core radii of Hardcastle & Worrall (2000) inferred from X-ray observations were used. These are kpc for Cygnus A, kpc for 3C 219 and kpc for 3C 215.
The observed central density of source environment given in Table 4 is derived from X-ray observations ROSAT of the hot gas surrounding Cygnus A (Hardcastle & Worrall 2000). For this, the prescription of Birkinshaw & Worrall (1993) for the conversion of central surface brightness to central proton density was used. The core radius, was estimated by Hardcastle & Worrall (2000) from the X-ray observations and I use their value, kpc, in converting from the density parameter given by the model to . The value thus found from the best-fitting model agrees within the error with the X-ray observations.
5.1.2. The western lobe
The western lobe of Cygnus A is covered by 53.5 independent beams along the jet and 23.2 beams at the widest point perpendicular to the jet. For the lower resolution maps I find 14.6 and 6.3 beams, respectively. Similar to the eastern lobe the observed emission does not extend all the way from the hot spot to the core and so the model fit is based on a smaller area. From the 2-dimensional maps I find for the widest part of the lobe and the length of the lobe is 73.0" and 69.5" for the low and high resolution case respectively. The best-fitting models for the two different resolutions again agree well. The models yield an age for the western lobe somewhat higher than that of its eastern counterpart (see Table 3) which is mainly caused by its greater length. However, the pressure within the cocoon is remarkably similar in both lobes. This implies also good agreement between the estimates for the jet power and the density of the source environment between the two sides of Cygnus A (Table 4).
5.2. 3C 219
For 3C 219 at VLA maps at 1.5 GHz in B-array and 4.9 GHz in C-array were used. The observations were taken in October and December 1998 by Dennett-Thorpe et al. (in preparation) who also performed standard reduction on the data set using AIPS. The resolution of the resulting maps is roughly 4.3" and the rms noise is Jy beam-1 at 1.5 GHz and Jansky beam-1 at 4.9 GHz. The northern lobe of 3C 219 has a rather irregular shape and no clear hot spot (Clarke et al. 1992). This leads to large ambiguities in the determination of its length or the geometrical parameters needed for the model presented here. I therefore only used the southern lobe which has a length of 40.4" and an aspect ratio at its widest point of . The lobe is covered by 9.4 independent beams along the jet axis and by 5.9 beams perpendicular to it. The jet and counter-jet in 3C 219 are unusually bright and so the jet emission was removed from the maps of the southern lobe. For the 1-dimensional comparison method, I extracted along a line off-set by 2" to the south of the line connecting the core of the source with the hot spot of the southern lobe. This avoids contamination of the surface brightness distribution by the jet emission. The hot spot in the southern lobe of 3C 219 is somewhat set back from the edge of the lobe. An aperture with a radius of 4.3", i.e. the width of the telescope beam, centered on the surface brightness peak was removed from the maps. For the 1-dimensional comparison, only values of core-wards of the hot spot were used.
The parameters of the best-fitting model are given in Table 3. The uncertainties of the model parameters is comparable to those found for the two lobes of Cygnus A. The angle to the LOS of 3C 219 is found to be smaller than that of Cygnus A. Since 3C 219 is a broad line radio galaxy this is in the expected sense, but the value of predicted by the model is about double that inferred from the flux ratio of the jet and counter-jet. As mentioned above, the jets of 3C 219 are unusually bright and this may reflect some enhanced disruption of the jet flow by turbulence or even a complete restart of the jets in this source (Clarke & Burns 1991). The latter possibility has let Schoenmakers et al. (2000) to include this source among their examples of Double-Double Radio Galaxies (DDRG). The morphology of these sources strongly suggests restarting jets (Kaiser et al. 2000). The large jet to counter-jet flux ratio of 3C 219 may therefore be caused by effects other than relativistic beaming. The best-fitting value of is consistent with orientation-based unification schemes.
The central density of the gas surrounding 3C 219, , predicted by the model is considerably smaller than that inferred from X-ray observations. To derive I used kpc (Hardcastle & Worrall 1999). This discrepancy will be discussed in Sect. 6.
5.3. 3C 215
This source is a radio-loud quasar at with very irregular morphology (Bridle et al. 1994). I obtained raw observational data from the VLA archive at 1.5 GHz in A-array and 4.9 GHz in B-array. The 1.5 GHz observations were carried out by Miley in May 1986 while the 4.9 GHz observations were taken by Hough in December 1987. Again standard reduction with AIPS was performed on the data and resulted in two maps with an angular resolution of 1.9". The maps were restored using a pixel size of 0.3"0.3" and the rms noise is Jansky beam-1 at 1.5 GHz and Jansky beam-1 at 4.9 GHz. The southern half of 3C 215 is very distorted with the jet bending in various places with enhanced surface brightness (Bridle et al. 1994). This part of the source is not consistent with a regular FRII-type lobe morphology and resembles in its outer regions an FRI-type structure. Therefore no attempt was made to apply the model to the southern part of the source. The northern lobe is more regular, however, the hot spot here is weak and the lobe widens considerably close to the core in a north-eastern direction. The lobe has a length of 26.6" and its aspect ratio at the point where the width of the lobe is greatest is 1.2. The hot spot in the northern lobe is not located at the very edge of the lobe similar to the southern lobe of 3C 219. An aperture centered on the surface brightness peak with a radius 3.8" corresponding to the size of two telescope beams was removed from the map. For the 1-dimensional comparison the surface brightness distribution was extracted along a line off-set by about 2" to the east from the core-hot spot direction to avoid emission from the jet. Again only pixels core-wards from the hot spot were used in the 1-dimensional case. The northern lobe of 3C 215 is covered by 14.0 independent beams along the jet and 11.7 beams at the widest point perpendicular to the jet.
Parameters of the best-fitting model and the derived properties of the environment of 3C 215 are given in Table 3 and Table 4. The uncertainties for the model parameters are considerably larger for this source than for the two previous ones. The viewing angle to the jet axis, , is smaller than for Cygnus A or 3C 219. This is again consistent with the predictions of unification schemes as 3C 215 is a quasar. The smaller observed value is again inferred from the flux ratio of the jet and counter-jet of 3C 215 (Bridle et al. 1994). Similarly to 3C 219 this ratio may be increased in 3C 215 because of the distorted morphology of its large scale radio structure. The southern jet does not seem to be embedded in a cocoon and it is therefore unlikely that the two jets are intrinsically identical. The flux ratio probably reflects physical processes other than purely relativistic beaming.
For the determination of I used kpc from Hardcastle & Worrall (1999). The density of the gaseous environment of 3C 215 is predicted to be much lower than inferred from X-ray observations. Discussion of this point is deferred to Sect. 6.
© European Southern Observatory (ESO) 2000
Online publication: October 24, 2000