## 4. Comparison with observationsThe model presented here depends on a large number of parameters. I assume that each volume element only contains the relativistic particle population and therefore . The other model parameters can be roughly grouped into -
geometrical parameters: the constants describing the cocoon shape , and and the orientation of the jet axis with respect to the Line-Of-Sight (LOS), , -
properties of the initial energy distribution of the relativistic electrons and/or positrons: the slope of the distribution, *p*, and the high energy cut-off, , -
properties of the source and its environment: the pressure within the cocoon, , the age of the source, *t*, and the slope of the power law density distribution of the environment, .
In the following I discuss the methods employed in comparing the model predictions with observations. Using these methods, I will then show that because of the nature of the model several degeneracies between model parameters exist. Eliminating these requires further assumptions to be made but also reduces the complexity of the parameter estimation. ## 4.1. 2-dimensional comparisonA radio map of an FRII source is composed of pixels which contain
information on the monochromatic radio surface brightness of the
source, , at a given position
projected onto the plane of the
sky. Here I use the The surface brightness is calculated at a frequency which is given by the observing frequency in the frame of the observer, , and the redshift of the source, i.e. . Finally, to account for cosmological effects the result of the LOS integration must be multiplied by , where is the luminosity distance to the source. The model map can be calculated at arbitrary resolution. However, before comparing the result with the observed map it must be convolved with the beam of the radio telescope used for the observations. This was done assuming a 2-dimensional Gaussian shape for the telescope beam. Once a virtual map is compiled for a set of model parameters this map can be compared pixel by pixel to a map resulting from observations of an FRII radio source using a -technique, Here, is the measured monochromatic surface brightness with rms error and is the model prediction. It is then possible to find the best-fitting model by varying the model parameters and thereby minimising the resulting -difference between virtual and observed map. The minimisation routine uses a n-dimensional downhill simplex method (Press et al. 1992), where n is the number of model parameters to be fitted. The minimisation can be done separately for the two halves of each cocoon since the model describes one jet and the associated half of the cocoon. Although in principle the minimisation can be done with one observed map at a single observing frequency, the constraints on the model parameters are improved by using two maps at two different frequencies. In this case the -differences resulting from the two maps are simply added together. In principle this is equivalent to compiling spectral index maps from two observed maps and comparing these with the model predictions. However, using the two maps directly has the advantage that pixels below the rms limit in one map but not in the other are not entirely lost for the fitting procedure. Furthermore, information on the absolute surface brightness in a given location is not contained in a spectral index map. The model would have to be normalised `by hand' and the various possibilities to do this would lead to ambiguities in the estimation of model parameters. In Sect. 5 I use two individual maps at two frequency for each source. Using the best-fitting model parameters
and Preliminary results using this comparison technique applied to radio observations of Cygnus A was presented in Kaiser (2000). In Sect. 5 this analysis is extended to also include 3C 219 and 3C 215. ## 4.2. 1-dimensional comparisonThe 2-dimensional comparison method described in the previous section requires a ray-tracing algorithm for the projection of the 3-dimensional model. For many maps of radio sources the number of pixels are so large that this method can become computationally very expensive. Furthermore, for maps of lower resolution the cocoon may not be resolved in the direction perpendicular to the jet axis. In such maps the surface brightness gradient along the jet axis can be extracted by taking a cut through the map along the line connecting the source core and the radio hot spot in the cocoon on one side. This yields a 1-dimensional curve of surface brightness as a function of projected distance from the source core, . The off-axis pixels will not add much information. The function can then be compared with the model predictions and a best-fitting model may be found using two maps at two observing frequencies as outlined in the 2-dimensional case. This 1-dimensional comparison involves a much smaller number of pixels for which a model prediction must be calculated than the 2-dimensional method. ## 4.3. Model parameters
To study the influence of the individual model parameters on the
surface brightness distribution, I define a fiducial model with a set
of fixed model parameters given in Table 2. These model
parameters imply a magnetic field just behind the hot spot of 13 nT or
130
For simplicity and ease of comparison I use the 1-dimensional model which only predicts the surface brightness distribution along the jet axis (see previous section). The variation of the model predictions with varying parameters for the 2-dimensional case are essentially similar but the differences between maps are more difficult to visualise. The solid line in Fig. 1 shows at GHz assuming a pixel size of 0.3"0.3", corresponding to 0.8 kpc0.8 kpc for , appropriate for a telescope beam of 1.2" FWHM. For simplicity and in contrast to the 2-dimensional comparison method the averaging effects of the observing beam which extends over four pixels on the curve was not taken into account. In any case, in this section the only interest is in gross trends of the model predictions when the model parameters are varied and the effective smoothing of the beam on the already rather smooth curve is small. A continuous curve is shown, since the pixel size is small compared to the scale of the plot. The length of the cocoon corresponds to more than 100 pixels. As pointed out in Sect. 3.2, the emission of the hot spot is not modeled in the approach presented here. The cocoon surface brightness shown in this and the following figures is caused by the cocoon material only after it has passed through the hot spot region. The emission from the hot spot will in general dominate the total emission from the end of the cocoon at . In a comparison with observed maps the predictions of the model can therefore not be used in this region.
Also shown in Fig. 1 are the results for the same model with a higher and lower cocoon pressure. In the model the strength of the magnetic field and the energy density of the relativistic particles in the cocoon is tied to the cocoon pressure. A higher pressure therefore causes a higher peak of the surface brightness distribution. However, the increased synchrotron energy losses of the relativistic electrons also lead to a stronger gradient of the distribution towards the core of the source, i.e. the older parts of the cocoon. In Fig. 2 the effects of the source age on the distribution of
is shown. The population of
relativistic particles at a given relative distance,
The same source viewed at different angles of the jet axis to the LOS, as shown in Fig. 3, results in a scaling of the surface brightness similar to the effects of changing the cocoon pressure. Note however, that the overall normalisation of depends sensitively on the value of the cocoon pressure. The effects of a higher pressure also include a steepening of starting from the peak of this function towards the source core (Fig. 1). This is not seen for variations of the viewing angle (Fig. 3). In practice therefore, the best-fitting values for the cocoon pressure is set mostly by the overall surface brightness of the cocoon while the viewing angle is mainly determined by the behaviour of close to the core.
Fig. 5 summarises the effects of the remaining model parameters on the surface brightness distribution. Changing the aspect ratio of the cocoon, , results in a scaling of very similar to the effects of a changing viewing angle, . Decreasing leads to the end of the cocoon to become more blunt which in turn implies a lower value of , i.e. a slower backflow within the cocoon (Eqs. 15 and 16). This is analogous to an older particle population at a given distance from the hot spot and the resulting surface brightness distribution is similar to that of an older source. Changing the second shape parameter has similar effects as that of a variation of and is therefore not shown. The main effects of changing the slope,
Finally, a different value of the slope of the gas density distribution in the source environment, , has a negligible effect on . The relevant curves for and are indistinguishable from that for shown in Fig. 1 to Fig. 5. ## 4.4. Degeneracy of parametersBoth, and , control how pointed the shape of the cocoon is. The ends of the cocoon are dominated by the emission from the hot spots which are not part of the model. Furthermore, if the jet direction is not stable over the lifetime of the source, they may at times advance ahead of the rest of the cocoon which distorts the cocoon shape. This has been referred to as the `dentist drill' effect (Scheuer 1982). In the previous section (see also Eq. 15) it was already shown that changing and/or results in a change of the profile and magnitude of the backflow in the cocoon. This in turn causes changes to the surface brightness profiles similar to changes of the source age. For these reasons the model cannot provide strong constraints on either of these parameters, at least not independent of the source age. Therefore I set in the following and . The surface brightness distribution predicted by the model is
almost independent of the slope of the density profile of the external
medium, . I therefore set
without influencing the model
results significantly. Note that because of Eq. (15) this then
also implies a fixed value for .
This may seem a rather significant restriction of the model but as was
shown in the previous section or
, through their influence on
, and the age of the source,
The slope of the initial energy distribution of the relativistic
particles, As was pointed out above, the effects of varying the initial slope
of the energy distribution of the relativistic particles, Finally, the degeneracy between , the width of the cocoon, and the viewing angle, , is resolved by determining the projected cocoon width from the observed map and use in the model calculations. Using these additional assumptions, the number of model parameters
which are fitted by comparison to the observations decreases to four:
The pressure in the cocoon, , the
source age, © European Southern Observatory (ESO) 2000 Online publication: October 24, 2000 |