6 The energy densities and pressures of the radio lobes

The pressure in the bridges of radio lobes of FRII-type sources are higher than that of their environment (e.g. Subrahmanyan & Saripalli 1993; Subrahmanyan et al. 1996). Radio source evolution models predict that the pressure in the lobes decreases with increasing source size (e.g. Kaiser & Alexander 1997), so that the lobes of large radio sources should be closer to pressure equilibrium with their environment. Since GRGs are the largest radio sources known, they are best suited to constrain the pressure in the ambient medium. In this section we investigate the energy densities in the lobe of the FRII-type GRGs in our sample and we relate this to the properties of their environment, the IGM.

  

Table 7: The equipartition magnetic field strengths and energy densities of the two lobes of the FRII-type sources in the 1-Jy sample. We have omitted the source 0309+411 since it is strongly core dominated which did not allow an accurate measurement. Column 1 gives the name of the source. Column 2 gives the sidedness indicator of the lobes "A'' and "B'', where "N'' stands for north, "E'' for east, etc. Columns 3 to 6 give the lengths, l, and widths, w, of the lobes (note, that lonly gives the part of the lobe from which radio emission has been detected). Column 7 gives the reference for the observations we used to determine l and w. Columns 8 and 9 give the equipartition magnetic field strength and Cols. 10 and 11 give the equipartition energy densities

$$\begin{tabular}{l c r@{$\,\pm\,$}r r@{$\,\pm\,$}r r@{$\,\pm\,... ....09 & 8.1 & 0.9 & 10.1 & 1.2 \\ \hline \hline\\ \end{tabular}$$

References: (1) 1.4-GHz WSRT data (Paper I); (2) Jägers 1986; (3) Vigotti et al. 1989; (4) Lara et al. (in preparation); (5) Mack et al. 1997; (6) FIRST survey; (7) 1.4-GHz WSRT data (unpublished); (8) Parma et al. 1996; (9) NVSS survey; (10) Röttgering et al. 1996; (11) Riley & Warner 1990.

   
6.1 Energy densities of the radio lobes

We have calculated the equipartition energy densities and magnetic field strengths in the lobes of the FRII-type GRGs in our sample. We have used a method similar to that outlined by Miley (1980). To determine the luminosity of each lobe, we have used the integrated 325-MHz flux densities (see Table 3). The spectral index of each lobe has been calculated between 325 MHz and 10.5 GHz. In case of significant steepening of the radio spectrum at frequencies below 10.5 GHz this value will be too low, but the influence of this on the result is only marginal (see Miley 1980).

Figure 9: Profiles of the equipartition energy density distribution along the major axes of several of the radio sources. The zeropoint is the position of the radio core. The numbers along the upper axis denote the distance from the core in units of kpc. Negative values are on that side of the source which is mentioned first in column two of Table 7

The volumes of the radio lobes have been estimated assuming a cylindrical morphology. The width has been taken as the deconvolved average full width of the lobe, measured between the 3$\sigma$contours on a radio contour map. The length of the lobe, whenever a clear gap exists between the core and the tail of the radio lobe (see, e.g., B0648+733, B2147+816), has been taken as the distance between the innermost and outermost edge of the lobe at 325 MHz. The radio axes have been assumed to be in the plane of the sky; although this is certainly not the case for the majority of sources, the correction factors are not large compared to the uncertainties introduced by assuming cylindrical morphologies. These used lobe dimensions are quoted in Table 7. Further assumptions are a filling factor of the radiating particles of unity and an equal distribution of energy between electrons/positrons and heavy particles such as protons. The low and high-frequency cut-offs in the radio spectra have been set at 10 MHz and 100 GHz, respectively.

In Table 7 we present the equipartition magnetic field strengths and energy densities of the lobes of the FRII-type sources in our sample. We find that the magnetic field strengths are low, typically a few $\mu$G. Consequently, also the energy densities, which are directly derived from the magnetic field strengths ( $u \propto B^2$), are low. This is as expected since the GRGs are extremely large and not exceptionally powerful.

   
6.2 Pressure gradients in the lobes

The equipartition energy densities in Table 7 are the lobe volume averaged energy densities. Within the lobes, however, large variations may be present. To investigate this, we have made profiles of the equipartition energy density for the FRII-type sources above $7\hbox{$^\prime$ }$ in size. We have used the same method as we have used to find the spectral index profiles in Sect. 5.1, i.e. by superposing on the source an array of boxes of width half the FWHM beam-size. In each box we have determined the average deconvolved width of the lobe by measuring the distance between the $3\sigma$ outer contours, and we have integrated the 325/354-MHz flux density. The spectral index of each box was calculated between 325/354 MHz and 10.5 GHz. In the case of B0813+758 and B1209+745 we used the 1.4-GHz NVSS map as the high-frequency map in order to employ the higher resolution available. In all cases the radio maps were first convolved to the lowest resolution map used. Only in boxes where the integrated flux density at both frequencies exceeded $3\sigma$, the spectral index was calculated and the equipartition energy density determined. In case of the sources B0050+402 and B0648+733, the deconvolved lobe widths were below zero, and we have therefore omitted these sources from further analysis. The reason for this is not entirely clear to us.

The profiles of the equipartition energy density for the remaining sources are presented in Fig. 9. In some cases, such as B0813+758, B0945+734 and B1312+698, we find the largest values of $u_{\rm eq}$ at the heads of the lobes. In some other cases, such as B1209+745 and B2043+749, we clearly detect the influence of the radio core. When we compare the lobe averaged energy densities as presented in Table 7 with the average value of the profiles in the plots, we find that the differences, as expected, are relatively small. In only a few cases we find discrepancies, exceeding a factor of two, and these are mostly due to the difference in the assumed geometry of the lobes.

In a relativistic plasma, such as which constitutes the radio lobes, the pressure, p, is directly related to the energy density by $p = \frac{1}{3}u_{\rm eq}$. Therefore, the plots in Fig. 9 also show the behaviour of the lobe pressure as a function of position along the radio axis. Since we have used a relatively low frequency for the flux density measurements and we have measured the spectral index for each plotted point separately, the results should not be highly sensitive to the increased effect of spectral ageing of the electrons towards the radio core.

If the host galaxy is at the center of a cluster, or if it has an extensive gaseous halo, hydrostatic pressure equilibrium requires a radial decrease of the pressure. The gradient should be strongest at small radii, around the core radius of the gas distribution. If the pressures in the bridges of the lobes are close to pressure equilibrium with their surrounding, it is expected that this gradient should be reflected in the lobe pressures at small radii.

We do not find such a behaviour in the majority of our sources. In the few cases where the energy density appears to increase near the position of the host galaxy this can be attributed to the presence of a radio core and/or radio jets (e.g. B1209+745 and B2043+749). Only in B1543+845 and in the eastern lobe of B0813+758 we observe a small increase in the pressure at small radii which cannot be related to a strong radio core or a jet (L. Lara, priv. comm.), and which might thus indicate the presence of a pressure gradient in the environment. In most sources the energy density actually increases with increasing distance from the host galaxy. This indicates that the lobes must be overpressured with respect to their environment, even at small radii, and that there are pressure gradients in these lobes: The heads of the lobes have much higher pressures than the bridges.

6.3 The density of the environment of the lobes

From the ages, advance velocities and energy densities of the lobes we can estimate the density of the environment. We assume that the propagation of the head of the lobe is governed by a balance between the thrust of the jet and the ram-presssure exerted by the environment. In this case, the external density, $\rho _{\rm a}$, is given by $\rho_{\rm a} = \Pi_{\rm j} / (A_{\rm h} v_{\rm h}^2)$, where $\Pi_{\rm j}$ is the thrust of the jet, $A_{\rm h}$ is the area of the bowshock and $v_{\rm h}$ is the advance velocity of the head of the lobe. The thrust $\Pi_{\rm j}$ is given by $Q_{\rm jet} / v_{\rm j}$, with $v_{\rm j}$ the velocity of the material in the jet, which we assume to be c, the velocity of light, and $Q_{\rm jet}$, which is the amount of energy delivered by the jet per unit time, or the jet power. We can estimate the jet power by dividing the total energy contents of the lobes of the radio source by the age of the source. An important factor in this is the efficiency with which jet energy is converted to radiation. A conservative choice is given by Rawlings & Saunders (1991): Half of the energy goes into the expansion of the lobe and the other half into radiation. Blundell et al. (1999), on the basis of radio source modelling, estimate values between 0.3 and 0.6 which agrees with a value of 0.5. We therefore assume that the jet power is twice as high as the energy content of a radio lobe, divided by its age. The energy content can be estimated by multiplying the equipartition energy density with the volume of the lobe.

Since $\rho_{\rm a} \!\propto\! A_{\rm h}^{-1} \!\propto\! D_{\rm h}^{-2}$ (where $D_{\rm h}$is the diameter of the impact area), $D_{\rm h}$ plays an important role in determining the ambient density. We cannot constrain it directly from our observations, but a good estimate is given by the size of the observable hotspot (e.g. Hardcastle et al. 1998). High-resolution observations of the hotspots in B1312+698 can be found in Saunders et al. (1987). These show that the size of the hotspots is less than 15 kpc. Arcsecond resolution observations of the source B2043+748 give similar constraints (Riley et al. 1988). Further, Hardcastle et al. (1998) find a positive correlation between the linear size of a source and the diameter of the hotspots in a sample of powerful 3CR FRII-type radio sources at z<0.3. This is confirmed for a larger sample of sources by Jeyakumar & Saikia (2000). Based on these results, we have used a diameter of 5 kpc to calculate the ambient densities of the lobes. For the age of a source, we have used the average age of both its lobes, and of both, CI and JP, ageing models. We have used the ages calculated with an internal magnetic field strength of $B_{\rm MWB}/\sqrt {3}$, since these are upper limits to the age and thus provide lower limits to $Q_{\rm jet}$ and $v_{\rm h}$. Using this average age we have calculated the average jet power of the two lobes, which we assume to be equal on both sides of the source. The resulting densities are presented in Table 8.

We find particle densities between $1\ 10^{-5}$ and $1\ 10^{-4}$ cm^-3. They have been calculated using a mean atomic mass of 1.4 amu in the environment of the radio sources. The densities are in good agreement with the results of Mack et al. (1998). If we assume a temperature in the IGM of a few 10⁶ K (e.g. Cen & Ostriker 1999), the thermal pressure in the environment would be $\sim\!2\ 10^{-14}$ dyn cm^-2 for a particle density of $4\ 10^{-5}$ cm^-3. Such low pressures support the indications in Sect. 6.2 that the radio lobes must be overpressured with respect to the IGM.

  

Table 8: Properties of the environment of the seven GRGs with useful lobe velocity and age determination. Column 1 gives the name of the source. Column 2 gives the volume, V, of the radio source. Column 3 gives the total energy, $u_{\rm tot}$, of the radio lobes, defined as the equipartition energy density times the volume. Column 4 gives the average age, t, of the lobe. Column 5 gives the jet power, $Q_{\rm jet}$. Column 6 gives the ambient density, $\rho _{\rm a}$, of the lobe and Col. 7 gives the particle density, $n_{\rm a}$, assuming a mean particle mass of 1.4 amu. The densities have been calculated using the mean of the jet powers of the lobes printed in Col. 5 and a diameter of the jet impact area of 5 kpc

$$\begin{tabular}{l c r@{$\,\pm\,$}l r@{$\,\pm\,$}l r@{$\,\pm\,... ....40 & \phantom{1}1.93 & 0.16\\ \hline \hline \\ \end{tabular}$$