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5 The advance velocities and ages of the radio lobes

In this section we derive the ages and velocities of the lobes of the GRGs by making use of the often observed trend that the radio spectrum in the radio lobes of FRII-type sources steepens along the line connecting the hotspots with the core. This can be explained as a result of ageing of the radiating particles after they have been accelerated in the hotspots (e.g. Alexander & Leahy 1987). By fitting spectral ageing models, the spectral index as a function of distance from the hotspot then yields the age as a function of hotspot. From the velocity of the hotspot and the size of the source an estimate of the spectral age of the radio source is obtained. Using ram-pressure equilibrium at the head of the jet, we can then also estimate the density in the ambient medium of the radio lobes.

5.1 Spectral index profiles

We have used the WENSS or 325-MHz WSRT maps of Mack et al. (1997), and the 10.5-GHz Effelsberg data to produce profiles of the spectral index, $\alpha_{325}^{10500}$, of the GRGs along their radio axes. We have selected the 15 FRII-type sources with an angular size above $7\hbox{$^\prime$ }$, only. This ensures that the radio lobes are covered by several independent beams. We have omitted the source B1918+516 since it is confused with two unresolved sources in our low frequency maps (see Paper I). We have convolved the maps at 325/354 MHz to the resolution of the 10.5-GHz Effelsberg maps ( $69\hbox{$^{\prime\prime}$ }\times 69\hbox{$^{\prime\prime}$ }$ FWHM). In case that the declination of a source is below $+51\hbox{$^\circ$ }$, we have convolved both the 325/354-MHz and the 10.5-GHz maps to a common resolution of $69\hbox{$^{\prime\prime}$ }
\times (54\cdot{\rm cosec}\,\delta)\hbox{$^{\prime\prime}$ }$ (FWHM). Additionally, for sources with an angular size $\le 10\hbox{$^\prime$ }$ and for which reliable 1.4-GHz NVSS maps are therefore available, we have also made profiles of the spectral indices between 325/354 MHz and 1.4 GHz (B0050+402, B0813+758, B1312+698, B1543+845), or between 1.4 GHz and 10.5 GHz (B1358+305; B0050+402 was omitted here because only few reliable datapoints could be found). For this, the NVSS maps have been convolved to the resolution of either the WENSS or the Effelsberg observations.

We have superposed rectangular boxes on the radio source, with the long side oriented perpendicular to the radio axis and with a width along the radio axes of 0.5 times the FWHM beam size of the convolved map. The whole array of such boxes has been centered on the radio core position. Possible confusing sources have been blanked from the radio maps. In each box, we have integrated the flux density at each frequency and calculated the spectral index of the box from this. If the flux density inside a box was not significant (i.e. $<\! 3\sigma_{\rm I}$) we used a $3\sigma_{\rm I}$ upper flux density limit to limit the spectral index.

The observations we use have been done using different instruments (i.e. VLA, WSRT, 100-m Effelsberg) which may lead to artefacts in the spectral index profiles. In general, these artefacts are most prominent in areas of low signal-to-noise (background level effects) and in areas with large intensity gradients (beam-size effects), such as along the edge of a radio source. Boxes with a low signal-to-noise have been presented as limits in the spectral index profiles, so background effects should not be serious in these areas. Large intensity gradients occur mostly at the outer edges of the sources, near the hotspots. Therefore, the outermost point on each side of a source is usually not reliable.

The profiles of the spectral index between 325/354 MHz and 10.5 GHz are presented in Fig. 5, those between 325/354 MHz and 1.4 GHz in Fig. 6. We find that only a few of the GRGs show a significant steepening of their radio spectra towards the core (e.g. B2043+749). In other sources, such as B2147+816, such a behaviour is not observed. In some cases a likely explanation is that a flat-spectrum radio jet contributes to the extended lobe emission. The source B1209+745 is known to have a prominent one-sided jet pointing towards the north (e.g. van Breugel & Willis 1981), which explains the flat-spectrum "plateau'' north of the radio core position in Fig. 5. The same may be true for B1312+698, which shows jet-like features in an (unpublished) 1.4-GHz WSRT radio map. However, for other sources such a scenario is less probable since no jet-like features appear in any of our radio maps. Among the possible causes for the apparent absence of ageing in these sources can be mixing of the lobe material due to backflows in the lobes, non-uniform magnetic fields and changes in the energy distribution of the accelerated particles during the sources lifetime.

\begin{tabular}{l l l}
\end{figure} Figure 5: Profiles of the spectral index distribution along the major axes of the radio sources, between 325 (or 354) MHz and 10.5 GHz. The zero point on the x-axis is the position of the radio core. The numbers along the upper axis denote the distance from the radio core in units of kpc. Negative values are on that side of the source which is mentioned first in column two of Table 7

\begin{tabular}{l l l}
\end{figure} Figure 6: Profiles of the spectral index distribution along the major axes of the sources smaller than $10\hbox {$^\prime $ }$, between 325 (or 354) MHz and 1.4 GHz (except for B1358+305, which is between 1.4 and 10.5 GHz). The axes have been annotated similarly as in Fig. 5

5.2 The advance velocities of the radio lobes

We have fitted advance velocities and ages of the radio lobes to the spectral index profiles using the method described in Schoenmakers et al. (1998). In short, this method works as follows: First, we recognize that Inverse Compton scattering of the Microwave Background (MWB) radiation is an important energy loss mechanism of the radiating particles in the lobes of GRGs. Its influence on the energy losses can be described by imposing an additional magnetic field, $B_{\rm MWB}$, whose energy density equals that of the MWB radiation field, i.e. $B_{\rm MWB} = 3.24\,(1+z)^2\ \mu$G, where z is the redshift of the source. In Fig. 7 we have plotted the ratio of the equivalent magnetic field strength of the MWB radiation, $B_{\rm MWB}$, and the equipartition magnetic field strengths in the radio lobes, $B_{\rm eq}$, averaged for both lobes, against both redshift and linear size of the GRGs (see also Ishwara-Chandra & Saikia 1999). See Sect. 6 for the calculation of the equipartition magnetic field strength. We find that in all GRGs $B_{\rm eq} \mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
...rlineskip\halign{\hfil$\scriptscriptstyle ..., which indicates that the IC-scattering process dominates the energy losses of the radio lobes. If we neglect energy losses resulting from the expansion of the lobes, then the radiating particles loose energy only due to synchrotron radiation and IC scattering of MWB photons. In this case, the radiative lifetime of particles is maximized when the internal magnetic field strength of the lobe $B_{\rm int} = B_{\rm MWB}/\sqrt{3}$ (e.g. van der Laan & Perola 1969). In other words, for a given time since the last acceleration of the radiating particles, this magnetic field strength gives the least amount of spectral steepening due to radiation and IC scattering. Using this value for the internal magnetic field strength therefore provides an upper limit to the spectral age of a radio source.

\end{figure} Figure 7: The ratio of the equivalent magnetic field strength of the microwave background radiation, $B_{\rm MWB}$, to the averaged equipartition magnetic field strengths, $B_{\rm eq}$, in the lobes of the GRGs against redshift (left) and projected linear size (right) of the radio source. The dashed line in the right plot indicates the expected behaviour of $B_{\rm MWB}/B_{\rm eq}$ for increasing linear size of a source with constant radio power and redshift ( $B_{\rm eq} \propto V^{-2/7} \propto D^{-6/7}$, where V and D are the volume and linear size of the radio source, respectively). This line only illustrates that the apparent correlation in this diagram is due to selection effects

We have derived the velocities and ages of the radio lobes using both the equipartition magnetic field strength (see Table 7) and the magnetic field strength which gives the maximum age ( $B_{\rm MWB}/\sqrt {3}$). We have calculated the velocities and ages for two ageing models: The Jaffe-Perola (JP) and the Continuous Injection (CI) models (see Schoenmakers et al. 1998, and references therein, for details). These are the extreme cases in the sense that the JP model gives the largest change in spectral index for a given age, whereas the CI model gives the smallest change. We find that in many cases our data are not good enough to decide which model is better applicable. It can be expected, however, that this is the JP model unless reacceleration of the particles in the lobes is important. Last, we assume that the advance velocity of the radio lobes, the injection spectral index of the radiating particles and the magnetic field strength in the lobes have been constant during the life-time of the radio source, and that the magnetic field strength is uniform throughout the lobe.

The velocity we find with our method is the separation velocity between the head of the lobe and the material flowing back in the lobe. This backflow may be important in very powerful radio sources (e.g. Liu et al. 1992; Scheuer 1995), but whether this is also the case in GRGs is not clear. Some of our sources have radio lobes which do not cover the whole area between the hotspot and the radio core. Although this may be caused by the increased effect of spectral ageing at a large distance from the hotspot, it may also indicate that backflows are not important in these radio lobes. Although the issue is far from clear, we will assume for the remainder of this discussion that backflows are unimportant in the lobes of the GRGs.

Many of our sources either do not show the expected ageing signature, i.e. a steepening of the spectrum towards the radio core, or have too few (less than five) or too poorly determined datapoints to provide a meaningful constraint for the fitting process (see Figs. 5 and 6). Only seven sources have spectral index profiles that we could use to fit the velocities and ages. These are B0109+492, B0813+758 (between 354 and 1400 MHz), B1003+351 (western lobe only), B1209+745 (between 354 and 1400 MHz), B1312+698, B1543+845 and B2043+749. The results for these are presented in Table 6, and the model fits in Fig. 8. The plots in Fig. 8 have been made for the case that $B_{\rm int} = B_{\rm MWB}/\sqrt{3}$. In the table we present for each lobe the best fit velocity, v, and injection spectral index, $\alpha _{\rm inj}$, and the reduced $\chi ^2$ of the fit. In case of the source B2043+749 we are able to fit the spectral index profile of the southern lobe using either five or seven spectral index points. The results are significantly different and we therefore present them for both these cases. The values of the reduced $\chi ^2$ presented in Table 6 are almost all smaller than unity. We believe that this is largely due to our conservative error estimates in the spectral indices.

We find that the advance velocities of the heads of the lobes of the GRG are in the range of 0.01c - 0.1c, with an average value of $\sim\!0.04c$. The many sources that could not be fitted using our method clearly question the validity of our method and the results. If some of the mechanisms that prevent the presence of a clear ageing signature in all these sources are also at work in the seven sources that we were able to find the age for, then the velocities that we have obtained are only upper limits to the true velocities. They indicate a general trend, though, that the advance velocities of the lobes of GRGs are below 0.1c.

Table 6: Results of the spectral aging analysis. The top part of the table presents the results for an internal magnetic field strength equal to $B_{\rm MWB}/\sqrt {3}$, which should give the maximum age; the bottom part is for an internal field strength equal to the average equipartition field strength in the two lobes. Column 1 gives the name of the source and the component. Column 2 gives the number of spectral index points used in the fitting. Columns 3 to 5 give the velocity, v, in units of c, the injection spectral index, $\alpha _{\rm inj}$ and the reduced $\chi ^2$ of the fit using the CI model. Columns 6 to 8 give the same for the JP model. Columns 9 and 10 give the age of the lobes resulting from the fitted velocities and the size of the lobes, for each of the two models

\begin{displaymath}\begin{tabular}{l c c r@{$\,\pm\,$}r r@{$\,\pm\,$}r r r@{$\,\...
....64 & 5.4 & 0.3 & 2.7 & 0.2 \\
\hline \hline \\

5.3 The spectral ages of GRGs

The method to find the velocities of the lobes which we employed in Sect. 5.2 also yields the spectral age of the radio source. Using the lobe advance velocities in Table 6 and the known armlengths of the lobes (Table 4), we have calculated the lobe ages. We have calculated the ages for each of the two spectral ageing models and for both the equipartition magnetic field strength and the one that should maximize the age of the lobe, $B_{\rm MWB}/\sqrt {3}$.

In Table 6 we present the spectral ages of the lobes of the seven GRGs in our sample for which we could find the lobe velocities. In general, the differences in the spectral age for the two values of the magnetic field strength are small. In reality the age of the two lobes of a single source should be equal. In all sources where we could fit both lobes, we indeed find that the ages are close to each other. In the southern lobe of the source B2043+749, which we have fitted using both five and seven spectral index points, the best agreement between the ages of the two lobes is found for the fit which uses five spectral points only.

The spectral ages we find all lie in the range between 30 and 150 Myr, depending on the spectral ageing model and, to a lesser degree, on the used value of the internal magnetic fieldstrength. Also, they are comparable to those found for other Giant radio sources, using similar techniques, e.g. 40 Myr for B0136+396 (Hine 1979) and B0821+695 (Lacy et al. 1993), 180 Myr in B0319-454 (Saripalli et al. 1994), 140 Myr in B0313+683 (Schoenmakers et al. 1998). We find an average age of 80 Myr.

\begin{tabular}{l l l l}
\end{figure} Figure 8: The results of the spectral ageing analyses. The plotted points are the spectral index points used for the fit in each lobe of the seven sources and the lines are the best fits for the two ageing models. The dotted line represents the CI model and the dashed line the JP model. The numbers along lower axes denote the distance in kpc from the point where we assume that the age is zero. The numbers along the top axis indicate the distance from the host galaxy in kpc. The results plotted here are those using an internal magnetic field strength of $B_{\rm MWB}/\sqrt {3}$, which should yield the maximum age. For the source B1003+351 only the western lobe could be used for fitting. For the southern lobe of B2043+749 we present two fits, one using seven and one using five spectral index points, since these give quite different results (see Table 6)

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