§32 Radio trail sources

As it was mentioned in section §5, radio trail sources show considerable ``real'' bending of their jets with deflection angles of about $\unit{90}{\degree}$ in many cases. Fig.(I.3) and fig.(VI.1) show two typical examples of these types of radio sources. As can be seen from both of them, the jets follow semicircular paths and bend up to about $\unit{90}{\degree}$. The jets seem to loose collimation after the complete bending has occurred.

Figure VI.1: Radio trail galaxy 3C 129. The image is an NVSS (Condon et al., 1998) radio map with peak flux of $\unit{7.5414 \times \power{10}{-2} }{Jy \usk \reciprocal {beam} }$. The levels are at $\unit{7.5414 \times \power{10}{-4} \times ( -5, 10, 15, \ldots, 95 ) }{}$ (Leahy & Yin, 2000).

Since it is the proper motion of the host galaxy with respect to the intergalactic medium that produces the curvature of the jets, the deflection angle cannot be greater than $\unit{90}{\degree}$. The results presented in eq.(25.7) show that jets which have a relativistic equation of state and a bulk relativistic motion of the gas within its jet, cannot be deflected more than $\sim \! \unit{50}{\degree}$. Since the deflections of radio trail sources are greater than this value, this result implies that most radio trail sources generate shocks at the end of their curvature. However, observations (see for example de Young, 1991; Eilek et al., 1984; O'Dea, 1985, and references within) show that the velocity of the material of the jets $\lesssim 0.2$-$0.3 \ensuremath{\mathsf{c}}$. Therefore, the bulk motion of the flow is non-relativistic, despite the fact that the gas inside the jet has a relativistic equation of state. The calculations which resulted in eq.(25.7) can be repeated for this circumstances. To do this, a polytropic index of $4/3$ in the calculations made in section §25 has to be used for the classical case. The result is that (Icke, 1991):

$$\theta_{\text{max}} = \unit{134.16}{\degree}.$$ (32.1)

In other words, these type of jets will develop a terminal shock if their jets bend more than $\sim \! \unit{135}{\degree}$. This seems to be the reason why all radio trail sources are able to bend so much without disrupting their internal structure.

As was discussed in section §25, the value of the bending angle in eq.(32.1) is an upper limit. The formation of an internal shock depends on the particular trajectory that a jet follows. In order to see if a specific jet will develop a shock at the onset of its curvature, the upper plot made in fig.(IV.4) is repeated in fig.(VI.2) for the case in which the polytropic index of the flow inside the jet has a value of $4/3$.

Figure VI.2: Plot of the maximum ratio $D / R$ ( $D$ is the with of the jet and $R$ is the radius of curvature of the curved trajectory it follows) as a function of the difference $\theta - \theta_\star$, where $\theta$ is the deflection angle and $\theta_\star$ is the maximum allowed bending angle a jet can have in order not to produce a terminal shock. This plot is similar to the ones produced in Chapter IV (see fig.(IV.4)). The trajectory of the jet was assumed to be a circumference and the plot refers to parameters such that a shock at the onset of the curvature is produced. Jets with parameters such that their values lie down below the curve are stable against the generation of internal shock waves at the beginning of the trajectory. This plot differs from the ones presented in fig.(IV.4) in that the polytropic index in the flow of the gas inside the jet is assumed to be $4/3$ and the bulk motion of the flow inside the jet is considered to be non-relativistic (Icke, 1991). The numbers in the plot label the different values of the Mach number that the flow has.

In order to find out whether radio trail galaxies generate internal shocks at the onset of their curvature due to the deflection of their jets, we can make use of eq.(5.1), which can be rewritten as:

$$\frac{D}{R} = \frac{ \rho_\mathnormal{e} \ensuremath{v}_g^2 }{ \rho_j \ensuremath{v}_j^2 }.$$ (32.2)

In this relation $D$ and $R$ represent the width of the jet and the radius of curvature of its trajectory respectively. The right hand side of this equation can be calculated from observations. However, it is important to mention that it is difficult to know the precise value of the width of the jet $D$ since it is not clear in most observations that the jet is resolved. It is also difficult to fit the radius of curvature of the source because of projection effects. Nevertheless, the symmetric U shape from typical radio trail galaxies imply that the sources do not lie too far away from the plane in the sky and so, the obtained value for the radius of curvature should not differ too much from its real one. The fraction on the left hand side of this relation is the abscissa of the plot in fig.(VI.2). Typical values (de Young, 1991; Forman & Jones, 1982; Eilek et al., 1984) for radio trail galaxies, obtained using a mixture of observations and various different theoretical models for radio trail galaxies, are: a density of the intergalactic medium $n_\mathnormal{e} \! \sim \! \unit{ \power{10}{-3} }{ \centi \meter \rpcubed }$, velocity of the host galaxy $\ensuremath{v}_g \! \sim \unit{ \power{10}{2} }{ \kilo \meter \usk \reciprocal \second }$, particle number density of the jet $n_j \! \sim \! \unit{ \power{10}{-1} }{ \centi \meter \rpcubed }$ and a jet velocity $\ensuremath{v}_j \! \sim \! \unit{ \power{10}{4} }{ \kilo \meter \usk \reciprocal \second }$. With these values, it follows that the ratio $D / R \! \sim \! \unit{ \power{10}{-2} }{ }$. In other words, radio trail sources are typically far below the curve in fig.(VI.2) and they will not generate internal shocks at all. This is the reason why the jets in this type of radio sources can bend so much, maintaining their collimation and avoiding the generation of internal shocks.

Let us analyse in more detail the archetypical elliptical radio galaxy NGC 1265 (3C 83.1B) which lies in the outer parts of the Perseus cluster of galaxies (see fig.(I.4)). The radial velocity of this radio galaxy with respect to the mean velocity of the Perseus cluster is $\sim \! \unit{2000}{\kilo\meter \usk \reciprocal\second}$ and the Mach number of the flow is $M \! \sim \! 2$   -$4$ (O'Dea & Owen, 1987a; Begelman et al., 1979). As can be seen from fig.(I.3), the total bending angle is $\theta \! \sim \! \unit{80}{\degree}$. This is much less than the maximum allowed value of $\theta_{\text{max}} \sim \! \unit{135}{\degree}$ calculated in eq.(32.1). In other words, no terminal shock is present in the radio source. In order to see whether a shock at the onset of the curvature is present, we make use of eq.(32.2) and the observed values for NGC 1265 (O'Dea & Owen, 1987b; Sijbring & de Bruyn, 1998; Begelman et al., 1979; de Young, 1991) of $\ensuremath{v}_j \sim \! \unit{ \power{10}{5} }{ \kilo \meter \usk \per \secon... ...mathnormal{e} \! \sim \! \unit{ \power{ 10 }{ -1 } }{ \centi \meter \rpcubed }$. With these values, the ratio $D / R \! \sim \! \power{10}{-3}$. Comparison of this with the plot in fig.(VI.2) indicates that the jets do not generate an internal shock at the onset of the curvature.

The ratio $D / R$ can also be calculated geometrically from the figures themselves. For example, let us take the case of the radio trail source NGC 1265 shown in fig.(I.4). As can be seen from the figure, two major deflections in each jet are observed. The first deflection is not very strong, and bends the jet only about $\unit{20}{\degree}$. The second one takes the jet from this bent angle up to $\sim \! \unit{80}{\degree}$. As was mentioned above, the value of $\unit{80}{\degree}$ is less than the maximum possible bending angle calculated in eq.(32.1) and so, no terminal shock is produced. From fig.(I.1) the ratio $D / R$ at the onset of the curvature was calculated by fitting a circle to the trajectory using the nucleus of the radio source and two adjacent points around it. The width of the jet is calculated from the resolved image. It then follows that $D / R \! \sim \! 0.05$. This value is of the same order of magnitude as the value found above using eq.(32.2).

We can apply this geometrical technique to various different sources. Fig.(VI.1) shows an NVSS image of the radio source 3C 129. Just as in the radio galaxy NGC 1265, the jets seem to have two major deflections. The first curves the jets up to $\sim \! \unit{ 40 }{ \degree }$ and the second deflects them to an angle of $\sim \unit{ 90 }{ \degree }$. By taking three points on the first curved trajectory, the ratio $D / R \! \sim \! 0.07$. According to the diagram in fig.(VI.2) it follows that the jet will not develop a shock at the onset of the curvature. In other words, the jets in 3C 129 do not generate any internal shocks due to their bending.