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

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4. Comparison with observations

The model presented here depends on a large number of parameters. I assume that each volume element only contains the relativistic particle population and therefore [FORMULA]. The other model parameters can be roughly grouped into

  • geometrical parameters: the constants describing the cocoon shape [FORMULA], [FORMULA] and [FORMULA] and the orientation of the jet axis with respect to the Line-Of-Sight (LOS), [FORMULA],

  • 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, [FORMULA],

  • properties of the source and its environment: the pressure within the cocoon, [FORMULA], the age of the source, t, and the slope of the power law density distribution of the environment, [FORMULA].

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 comparison

A radio map of an FRII source is composed of pixels which contain information on the monochromatic radio surface brightness of the source, [FORMULA], at a given position [FORMULA] projected onto the plane of the sky. Here I use the projected distance along the jet axis, [FORMULA], and perpendicular to the jet, [FORMULA]. Both are measured in units of the projected length of the cocoon, [FORMULA]. In optically thin conditions generally present in radio source lobes this projection corresponds to a LOS integral of the synchrotron emissivity, [FORMULA], through the 3-dimensional source at each pixel location. Following the analysis in Sects. 3.1 and 3.2, [FORMULA] is only a function of the unprojected dimensionless distance from the core measured along the jet axis, l, for a given set of the eight source and environment parameters. Using the model described above, it is therefore possible to construct `virtual radio maps' for a given set of model parameters by projecting the 3-dimensional model onto the plane of the sky. Since a radio source is in general viewed with its jet axis at an angle [FORMULA] to the LOS, the value of l, and therefore that of [FORMULA], changes along the path of the LOS integral. Also, the distance from the core to the tip of the cocoon measured on an observed radio map, [FORMULA], is not equal to the physical size of the cocoon, [FORMULA]. This, of course, also implies that the aspect ratio of the cocoon, [FORMULA], measured in the observed map is smaller than the `real' value [FORMULA], i.e. radio sources appear `fatter' than they really are. All these projection effects have to be taken into account when comparing virtual radio maps resulting from these models to observations.

The surface brightness is calculated at a frequency [FORMULA] which is given by the observing frequency in the frame of the observer, [FORMULA], and the redshift of the source, i.e. [FORMULA]. Finally, to account for cosmological effects the result of the LOS integration must be multiplied by [FORMULA], where [FORMULA] 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 [FORMULA]-technique,


Here, [FORMULA] is the measured monochromatic surface brightness with rms error [FORMULA] and [FORMULA] is the model prediction.

It is then possible to find the best-fitting model by varying the model parameters and thereby minimising the resulting [FORMULA]-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 [FORMULA]-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 [FORMULA] and t, the energy transport rate of the jet, [FORMULA], and the parameter combination [FORMULA] describing the density distribution in the source environment can be calculated from Eqs. (2) and (3).

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 comparison

The 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, [FORMULA]. The off-axis pixels will not add much information. The function [FORMULA] 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 µG. The highest frequency of the emitted synchrotron spectrum is then [FORMULA] GHz which is close to the usually assumed high frequency cut-off in the spectrum of radio galaxies in minimum energy arguments (e.g. Miley 1980). The density parameter [FORMULA] kg [FORMULA] which corresponds to a central density of [FORMULA] kg m-3 or 0.3 particles per cm-3 if [FORMULA] kpc. I also assume that the jet of the sources is 100 kpc long and is viewed at an angle of [FORMULA] to the line of sight. This implies that it would be observed to have a length of [FORMULA] kpc corresponding to 31.2" at a redshift [FORMULA]. For a measured [FORMULA] the aspect ratio of the cocoon is [FORMULA] for the assumed viewing angle.


Table 2. Model parameters of the fiducial model (see text).

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 [FORMULA] at [FORMULA] GHz assuming a pixel size of 0.3"[FORMULA]0.3", corresponding to 0.8 kpc[FORMULA]0.8 kpc for [FORMULA], 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 [FORMULA]. In a comparison with observed maps the predictions of the model can therefore not be used in this region.

[FIGURE] Fig. 1. The influence of the cocoon pressure on the radio surface brightness along the jet axis. All curves are plotted for the parameters of the fiducial model except for a variation in the cocoon pressure. The relative distance from the core of the source is given in units of [FORMULA] kpc (see text). The surface brightness is plotted as it would be extracted by an observer from a map of a source at redshift [FORMULA].

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 [FORMULA] is shown. The population of relativistic particles at a given relative distance, l, from the core of the source has spent more time in the cocoon in an old source compared to a young source. This implies stronger energy losses and the peak of the surface brightness distribution is therefore located closer to the hot spot in an old source. Furthermore, in an observation with a given detectable threshold of [FORMULA] a larger fraction of the cocoon of a young source will be visible than of an older source.

[FIGURE] Fig. 2. The influence of the source age on the radio surface brightness. All model parameters as in Fig. 1 except for a variation in the source age.

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 [FORMULA] depends sensitively on the value of the cocoon pressure. The effects of a higher pressure also include a steepening of [FORMULA] 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 [FORMULA] close to the core.

[FIGURE] Fig. 3. The influence of the viewing angle on the radio surface brightness. All model parameters as in Fig. 1 except for a variation in [FORMULA]. Note that the measured length of the cocoon, [FORMULA], changes from 86.6 kpc for the fiducial model to 98.5 kpc for [FORMULA] and 64.3 kpc for [FORMULA].

Fig. 5 summarises the effects of the remaining model parameters on the surface brightness distribution. Changing the aspect ratio of the cocoon, [FORMULA], results in a scaling of [FORMULA] very similar to the effects of a changing viewing angle, [FORMULA]. Decreasing [FORMULA] leads to the end of the cocoon to become more blunt which in turn implies a lower value of [FORMULA], 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 [FORMULA] has similar effects as that of a variation of [FORMULA] and is therefore not shown.

The main effects of changing the slope, p, and the high energy cut-off of the initial energy distribution of the relativistic electrons, [FORMULA], is a change in the scaling of [FORMULA] (see Fig. 5). This is caused by the dependence of the normalisation of the energy spectrum on p and [FORMULA], Eq. (11). Both scaling effects are similar to the effects of a variation of the viewing angle, [FORMULA]. The effect of a steeper energy spectrum also causes an off-set in the distribution of the spectral index as a function of l as shown in Fig. 4. However, in the cocoon region closest to the source centre and least influenced by the hot spot emission, this effect is small compared to that of a variation of the viewing angle, [FORMULA].

[FIGURE] Fig. 4. Spectral index between 1.5 GHz and 5 GHz for the fiducial model and models with varying viewing angle, [FORMULA], and slope of the initial energy spectrum of the relativistic particles, p. All other model parameters as in Fig. 1.

[FIGURE] Fig. 5. Effects of other model parameters on the radio surface brightness. Model parameters are as in Fig. 1 unless otherwise indicated in the legend.

Finally, a different value of the slope of the gas density distribution in the source environment, [FORMULA], has a negligible effect on [FORMULA]. The relevant curves for [FORMULA] and [FORMULA] are indistinguishable from that for [FORMULA] shown in Fig. 1 to Fig. 5.

4.4. Degeneracy of parameters

Both, [FORMULA] and [FORMULA], 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 [FORMULA] and/or [FORMULA] 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 [FORMULA] and [FORMULA].

The surface brightness distribution predicted by the model is almost independent of the slope of the density profile of the external medium, [FORMULA]. I therefore set [FORMULA] without influencing the model results significantly. Note that because of Eq. (15) this then also implies a fixed value for [FORMULA]. This may seem a rather significant restriction of the model but as was shown in the previous section [FORMULA] or [FORMULA], through their influence on [FORMULA], and the age of the source, t, are degenerate model parameters. In the absence of geometrical constraints on [FORMULA] or [FORMULA], setting them to reasonable values allows the determination of the source age from the fitting method. The ages derived from the model will therefore always depend somewhat on the choice for the source geometry.

The slope of the initial energy distribution of the relativistic particles, p, mainly influences the spectral index distribution in the cocoon. Observations suggest that [FORMULA] and the model is therefore restricted to values in this range. It follows that the model is rather insensitive to the exact value of the high energy cut-off of the distribution, [FORMULA]. A value of [FORMULA] much smaller than the [FORMULA] used in the fiducial model will, in addition to a change of the overall scaling, cause the emission region to shorten along the jet axis at a given observing frequency. However, because the highest frequency of the synchrotron radiation of the cocoon, [FORMULA], depends on [FORMULA] a small value of the high energy cut-off also implies a significantly smaller [FORMULA]. Values substantially below [FORMULA] GHz, which is assumed, here are unlikely in view of observations. I therefore set [FORMULA].

As was pointed out above, the effects of varying the initial slope of the energy distribution of the relativistic particles, p, are small in general. In Sect. 5 the 1 and 2-dimensional comparison methods are applied to observations of three FRII-type radio sources. It is shown there that in almost all cases the best-fitting model parameters for the 2-dimensional method require p to be close to 2. To prevent the degeneracy between p and [FORMULA] to influence the model fits in the 1-dimensional method which involves fewer degrees of freedom, I set [FORMULA] in this case.

Finally, the degeneracy between [FORMULA], the width of the cocoon, and the viewing angle, [FORMULA], is resolved by determining the projected cocoon width [FORMULA] from the observed map and use [FORMULA] 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, [FORMULA], the source age, t, the viewing angle, [FORMULA], and the initial slope of the energy distribution of the relativistic particles, p. In the case of the 1-dimensional comparison method p is fixed to a value of 2.

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© European Southern Observatory (ESO) 2000

Online publication: October 24, 2000