7 Discussion

Figure 10: The radio power against the lobe advance velocity for several samples of sources. Crosses: Liu et al. (1992); Circles: Leahy et al. (1989); Squares: Alexander & Leahy (1987); Triangles: Parma et al. (1999); Diamonds: GRGs from this paper; Star: The GRG B0313+683 from Schoenmakers et al. (1998)

   
7.1 The relation between lobe velocity and linear size

In Sect. 5.2 we have derived the lobe advance velocities of seven of the GRGs in our sample. We have compared these velocities to those found in three samples of powerful FRII-type 3CR radio sources by Alexander & Leahy (1987), Leahy et al. (1989) and Liu et al. (1992), and in a sample of lower luminosity sources presented by Parma et al. (1999). The sizes and radio powers of the sources in Leahy et al. and Parma et al. have been converted to a Hubble constant of 50 kms^-1Mpc^-1. Of the Parma et al. sample we have only used sources which are not of FRI-type and which show a steepening of the radio spectrum from the hotspot to the core ("type 2'' sources, in their notation). Also, the radio powers have been converted from 1.4 GHz to 178 MHz using a spectral index of -0.8. The sample of Alexander & Leahy consists of radio sources with an angular size above $45\hbox{$^{\prime\prime}$ }$, that of Leahy et al. of sources larger than $20\hbox{$^{\prime\prime}$ }$ and that of Liu et al. of sources larger than $4\hbox{$^{\prime\prime}$ }$. The sources 3C154 and 3C405 (Cyg A) are studied by both Alexander & Leahy and Leahy et al.; we have used the better constrained values of the latter group. For all sources we have used the average age and velocities of the two lobes. For the GRGs we have taken the velocities derived using the JP ageing model, since this model has been used by Alexander & Leahy, Leahy et al. and Liu et al. Lastly, we have added the advance velocity of the GRG B0313+683 (Schoenmakers et al. 1998), which has been derived in a similar way as the GRGs presented here.

Figure 11: The linear size of the radio sources against the lobe advance velocities. Symbols as in Fig. 10. The dashed line indicates the size a source can attain at an age of 108 yr. a) (left) All sources. b) (middle) Sources with 178-MHz radio powers below 1026.5WHz-1ster-1, only. c) (right) Sources with 178-MHz radio powers above 1026.5WHz-1ster-1

If we plot the radio power at 178 MHz against the lobe advance velocity for all sources (Fig. 10) we find a strong correlation between these parameters. This correlation was noted before by Alexander & Leahy (1987) and confirmed by Liu et al. (1992). We find that the GRGs agree with the correlation, but that they have the highest advance velocities for their radio powers. To find out whether the advance velocity of the radio lobes is related to the linear size of a source, we have plotted these parameters for all sources in Fig. 11a. No correlation is found, although the spread in velocities appears to decrease with increasing linear size. However, when we separate the low luminosity from the high luminosity sources, we find a strong correlation for the low-luminosity sources (see Fig. 11b). We have set the separation at a radio power of 10^26.5WHz^-1ster^-1, which is in between the highest radio power in the low-luminosity sample of Parma et al. (1999) and the lowest in the high-luminosity samples of Liu et al. (1992) and Leahy et al. (1989). For sources with higher radio powers the relation between velocity and source size seems to be inverted, although the correlation is much weaker (Fig. 11c).

At first sight, the correlation between linear size and lobe advance velocity for the low-luminosity sources suggests that radio sources increase their lobe advance velocity as they increase in size. However, the correlation can also be the result of two selection effect, provided that radio sources decrease in radio power as they increase in size and age (cf. Kaiser et al. 1997; Blundell et al. 1999). First, assuming that the radio power is proportional to the jet power and decreases with increasing linear size, large radio sources must have higher jet powers than small radio sources of similar observed radio power. Therefore, that fraction of large radio sources which are in a flux density limited sample must have the highest jet powers of all. If all radio sources are in similar environments, a higher jet power will most likely lead to a faster lobe advance; this yields the observed correlation. Second, since radio sources have a limited life-time, sources with low velocities will never become as large as sources with high lobe velocities. This is indicated by the dashed lines in Fig. 11, which represents an age of 10⁸ yr; not many sources are found to the right of this line. Of course, sources may also live longer than this, but they then have to drop below the sensitivity limit in order to disappear from the plot.

Figure 12: The linear size of the radio sources against their spectral age. The symbols are as in Fig. 10. a) (left) All sources. b) (middle) Sources with 178-MHz radio powers below 1026.5WHz-1ster-1, only. c) (right) Sources with 178-MHz radio powers above 1026.5WHz-1ster-1. The diagonal lines indicate the relation for an expansion velocity of 0.01c (upper line), 0.1c (middle line) and 1c (lower line)

The above cannot explain the behaviour observed for the high power radio sources, though. There are several possible reasons for this, among which the following (see also Parma et al. 1999). First, the velocity estimates in these sources depend strongly on the assumed magnetic field strength in the lobes. This was taken as the equipartition field strength, but as shown by Liu et al. (1992), lowering this will drastically decrease the estimated velocities. Second, there may be strong backflows present in the lobes of the powerful sources (see also Scheuer 1995). This will also lead to an overestimation of the advance velocities since one actually measures the separation velocity between the head of the lobe and the material flowing back. Further, the mixing of old material with much younger, backflowing material near the radio core will lead to a further underestimation of the age of the source, and thus to an overestimation of the growth speed. Scheuer (1995) suggests that backflows are important in powerful sources such as present in the sample of Liu et al. (1992). It is unknown if strong backflows occur in less powerful sources as well. Lastly, in a flux density limited sample such as the 3CR, high-power sources are at higher redshifts than low power sources. Blundell et al. (1999) point out that this leads to the selection of younger and smaller radio sources with increasing redshift. Assuming that the radio power is directly related to the jet power, and that this is directly related to the advance velocities of the lobes, the lobes of powerful radio sources must have high velocities. The low-power sources discussed above are all at low redshift, so that this selection effect is not in operation.

7.2 Spectral ages of radio sources

Parma et al. (1999) find that the spectral ages of low power radio sources are well correlated with their linear sizes. In Fig. 12a we have plotted the linear sizes and spectral ages of the sources from the samples introduced in Sect. 7.1. We confirm the correlation between linear size and age, but we also find that the high-power sources have a stronger dependence on the linear size (see Figs. 12b and c). This is due to the higher lobe velocities, the cause of which we have discussed in Sect. 7.1. We find that the GRGs are among the oldest sources in this plot, although their ages are not extreme as compared to the sources from Parma et al. This is related to the correlation between linear size and lobe velocity (see Fig. 11b). Only large sources with high jet powers will end up in our GRG sample, which probably implies that their lobes advance faster (see previous section). This will introduce a bias in our sample towards sources which have higher advance velocities and thus lower ages. A multi-frequency study of GRGs at lower flux densities (such as those presented in Paper I) would be valuable to test the strength of this bias.

   
7.3 Homogeneity of the Mpc-scale enviroment of GRGs

In Sect. 4.1 we have presented the armlength and bending angle asymmetries of the GRGs and that of a sample of smaller-sized 3CR sources from Best et al. (1995). We find that the GRGs are slightly more asymmetric in their armlengths, although the bending angle distributions are similar. The radio lobes of GRGs have expanded well out of any extended emission-line gas regions, which may contribute significantly to the observed distribution of armlength-asymmetries in smaller sized radio sources (e.g. McCarthy et al. 1991). Therefore, it can be expected that the asymmetries are dominated by orientation effects. If, however, the asymmetries in GRGs are environmental, they provide information on the homogeneity, on Mpc-scales, of the intergalactic medium (IGM) surrounding their host galaxies.

We have attempted to fit the distribution of armlength asymmetries for the 3CR sources and the GRGs (see Fig. 3a) using orientation only (cf. Best et al. 1995). We find that the best fit for our subsample of 3CR sources is obtained for a maximum angle of $43\hbox{$^\circ$ }$ between the plane of the sky and the line of sight. This assumes that the lobe velocities in a single source are equal and distributed as determined by Best et al. Since all sources in this 3CR subsample are radio galaxies, this agrees with the orientation dependent unification scheme (e.g. Barthel 1989). For the GRGs no satisfactory fit can be found. The main reason for this is the smaller number of sources in the first bin (see Fig. 3a), as compared to the second bin. To explain the observed distribution of the armlength asymmetries with equal advance velocities of the two lobes would require the introduction of a minimal allowed angle of $\sim 14\hbox{$^\circ$ }$ between the radio axis and the plane of the sky. We therefore conclude that orientation effects alone cannot explain the observed distribution of armlengths in GRGs.

Best et al. (1995) show how the presence of non-linearity in radio sources alters the distribution of armlength asymmetries. For the GRGS, however, the incorporation of the bending angles presented in Table 4 does not improve the fit significantly. As shown by Best et al., bending angles have the strongest effect on the observed armlength asymmetry distribution for sources oriented well away from the plane of the sky. For sources close to the plane of the sky, an increase of the maximum bending angle shifts the distribution of armlength asymmetries more towards x=0, which is opposed to what we observe.

These results suggest that the observed asymmetries must, at least partly, be due to asymmetries in the lobe advance velocities of the GRGs. This does not necessarily imply the existence of large-scale inhomogeneities in the IGM. Other possibilities are a difference in the efficiency of the energy transport down the jet, or of the jet opening angle. Both can influence the thrust produced by the jet material at the head of the lobe which influences the advance velocity.

In Sect. 4.1 we have investigated if the armlength asymmetry is related to the lobe flux density and spectral index asymmetries. Our most significant result is that the shorter arm tends to have the steepest radio spectrum. Can this be explained by environmental asymmetries? If there is a higher density on one side of the source, the radio lobe on that side may be slowed down and better confined than its counterpart on the other side. Provided that the magnetic field is frozen into the plasma, adiabatic expansion of a radio lobe results in a decrease of the magnetic and particle energy density (since $B\propto (r/r_0)^{-2}$ and $E \!\propto \! (r/r_0)^{-1}$, where r₀ is the initial radius and r the radius after expansion, assuming spherical expansion; Scheuer & Williams 1968). This implies that the particles responsible for the emission at a fixed frequency must have higher energies themselves (emitted frequency $\propto E^2 B$), of which there are less available. These two mechanisms lead to a shift of the emitted spectrum to both lower intensities and lower frequencies. Since spectral ageing itself produces a convex spectral shape, the shift of the spectrum as a whole to lower frequencies leads to a steeper observed spectrum in case of an adiabatically expanded lobe as compared to a lobe which has not expanded, or by a lesser amount (e.g. Scheuer & Williams 1968). This is contrary to what we find for the GRGs, arguing against a purely environmental origin for the shortest arm to have the steepest spectrum.

A second possibility is that maybe not the expansion of the lobe as a whole is important, but only the expansion of the radiating plasma as it leaves the overpressured hotspot area. In a higher density environment, and thus presumably higher pressure environment, the pressure in the hotspot will be larger as well (strong shock condition). If the pressure in the lobes is roughly equal in both lobes, then the drop in pressure of the plasma leaving the shock (or hotspot) will be larger on the side with the higher ambient density. Hence, the plasma will undergo a larger amount of expansion. This expansion will be relatively fast and will thus closely resemble the case of adiabatic expansion described above, i.e. the somewhat aged particle energy spectrum will be shifted towards lower energies. This will result in a steeper lobe spectrum on the side of the source with the shortest arm, as observed.

Since we have not yet enough information on the intrinsic properties of the jets and hotspots of the GRGs, we can not conclude that there are large scale inhomogeneities in their Mpc-scale environment. More detailed radio, and perhaps X-ray observations are necessary.

7.4 Evolution of the environment of GRGs

In an adiabatically expanding Universe filled with a hot, diffuse and uniform intergalactic medium, the pressure, $p_{\rm IGM}$, should increase as a function of redshift, z, as $p_{\rm IGM} \!\propto\! (1+z)^5$(e.g. Subrahmanyan & Saripalli 1993). In order to measure this effect using radio lobes, it is necessary to investigate which sources have the lowest radio lobe pressures. Radio source evolution models predict a decrease in the lobe pressure with increasing linear size (e.g. Kaiser & Alexander 1997), so that it can be expected that the largest radio sources have the lowest lobe pressures.

Cotter (1998) uses a large sample of radio sources, among which are GRGs out to redshifts of $\sim 1$, and shows that indeed the energy densities in the radio lobes of GRGs are lower than those in smaller sized radio sources. Also, he shows that the lower limit for the lobe pressures evolves so strongly with redshift that it does not contradict a (1+z)⁵ relation. This prompts the question if the equipartition pressures in the radio lobes of GRGs indeed indicate a strong evolution in their environments.

In Table 7 we have presented the energy densities in the lobes of the FRII-type GRGs in the 1-Jy sample. Although the range of redshift spanned by our sample is small ( ), a truly strong evolution of the energy density with redshift should show up. In Fig. 13a we have plotted for each source the average energy density of the two lobes against its redshift. We have used different symbols for sources with a linear size below and above 2 Mpc. The dotted line in the figure is the expected energy density in the lobes if the IGM has a present-day pressure, $p_{(\rm IGM,0)}$, of 10^-14 dyncm^-2 as suggested by Subrahmanyan & Saripalli (1993), a redshift evolution as described above and if the lobes are in pressure equilibrium with the IGM. We find that the energy densities in the lobes of the sources with a size above 2 Mpc are systematically lower than those in the lobes of the smaller sized sources, confirming the trend, already noted by Cotter (1998), for the largest radio sources. Also, the energy densities of the lobes of the GRGs indeed increase strongly with increasing redshift. We find that no source has a lobe energy density which would place it below the line predicted by a (1+z)⁵ IGM pressure evolution with a present-day pressure of 10^-14 dyncm^-2.

Figure 13: Lobe-averaged equipartition energy densities of the FRII-type GRGs against redshift. a) (left) GRGs in the 1-Jy sample. Closed symbols are sources with a (projected) linear size between 1 and 2 Mpc, open symbols are sources larger than 2 Mpc. The dashed line indicates the expected behaviour for a source with a flux density, volume and spectral index as given by the median values of the sample. The dotted line indicates the lower limit if the pressure in the lobes is dominated by relativistic particles, the lobes are in pressure equilibrium with the IGM and the pressure, p, of the IGM follows the relation $p = 1.0\ 10^{-14}\,(1+z)^5$ dyn cm-2. b) (right) The lowest measured energy densities in the lobes (see Fig. 9) against redshift. Symbols are as in (a)

An important issue is how the selection effects affect the results. The equipartition energy density, $u_{\rm eq}$, is related to the total power, P, and the volume, V, of a radio source by $u_{\rm eq} \!\propto\! (P/V)^{4/7}$. The sources plotted are from a flux density limited sample ($S \!>\! 1$ Jy at 325 MHz), which implies that sources at larger redshift have, on average, higher radio powers and thus higher energy densities at equal physical source dimensions.

In Fig. 13a we have plotted the expected equipartition energy density as a function of redshift (dashed line) for a source with a flux density equal to the median flux density of the sample, a volume equal to the median volume and a spectral index equal to the median spectral index of the sources. Clearly, the slope and position of this line follows the redshift relation of the measured energy densities well. This suggests that the observed relation of the energy density with redshift is most likely a result from our use of a flux density and linear size limited sample, rather than of a cosmological evolution.

The energy densities are the intensity weighted averages over the two lobes. Since these values are sensitive to the contributions from the hotspots, the values in the fainter bridges of the lobes may be lower. From the profiles of the equipartition energy density presented in Fig. 9 we have taken the lowest found value for each source and plotted this in Fig. 13b. We find that most sources still lie well above the line describing the assumed (1+z)⁵ pressure evolution with $p_{\rm (IGM,0)} = 10^{-14}$dyncm^-2. Also, the dichotomy between sources smaller and larger than 2 Mpc still exists, which indicates that even the lowest energy densities that we measure in the lobes are systematically lower in larger radio sources.

We can only conclude that there is no evidence in our sample for a cosmological evolution of the energy density in the lobes of GRGs, and that there is therefore also no evidence for a cosmological evolution of the pressure in the IGM. On the other hand, we are also aware that we cannot reject the hypothesis that the pressure of the IGM evolves as $p_{\rm IGM} \propto (1+z)^5$, provided that the present-day value dyn cm^-2.

The existence of a population of GRGs with lobe energy densities of erg cm^-3 at redshifts of at least 0.6 would challenge this hypothesis. Assuming a length-to-width ratio of 3, a size of 1.5 Mpc and a spectral index of -0.8, this requires that GRGs have to be found with a flux density of mJy at 325 MHz at these redshifts, which is well above the sensitivity limit of WENSS and thus feasible. The flux density limit scales with the aspect ratio, R, as R^-3 and will thus rapidly drop below the survey limit of $\sim 25$ mJy (Rengelink et al. 1997) for more elongated sources. The only published sample of higher redshift GRGs is that of Cotter et al. (1996), which has been selected at 151 MHz with flux densities between 0.4 and 1 Jy. These flux densities are higher than the above limit. The lobe pressures for the sources in this sample are presented by Cotter (1998). Indeed, he finds an apparent evolution of the lobe pressures in this sample, but this is therefore most likely also the result of the use of a flux density and linear size limited sample and not due to a true cosmological evolution of the IGM pressure.

Lastly, we have to mention the possibility that the environments of GRGs may be particularly underdense with respect to the environments of other radio sources. This has sometimes been suggested as the cause of the large size of the GRGs. If also the pressures in the ambient medium of GRGs indeed are typically lower, then the selection of radio sources as large as GRGs would be strongly biased towards regions with low pressures. Since this must be valid at all redshifts, we may be sampling the lowest pressure regions of the Universe only, which may be untypical of the general IGM.