§5 Bending of jets in radio galaxies

As jets in radio galaxies expand, they interact with their surrounding environment. There are many examples in which the jets, rather than having a straight trajectory, show changes in their morphology due to interaction with their surroundings induced by different physical mechanisms.

Dramatic curvature is observed in certain FR II radio sources, particularly those known as radio trails or head-tail objects. A good illustration of of such a source is NGC 1265 shown in fig.(I.3) (see O'Dea & Owen (1986) for detailed multifrequency VLA maps of the source). The jets in this radio galaxy are strongly curved giving the source a semicircular shape, with the host galaxy at the pole. The curvature of the jet is attributed to the motion of the host elliptical galaxy through the high density intracluster medium which results in a significant ram pressure on the radio emitting material of the jet (Begelman et al., 1984). Indeed, the path of the jet can be determined from Euler's equation of hydrodynamics in its classical form (Begelman et al., 1979; de Young, 1991; Christiansen et al., 1981; Jaffe & Perola, 1973) or its relativistic generalisation (O'Dea, 1985). In the classical case, the bending equation takes the form:

$$\frac{ \rho_j \ensuremath{v}_j^2 }{ R_{\text{bend}} } = \frac{ \rho_\mathnormal{e} \ensuremath{v}_g^2 }{ R_p },$$ (5.1)

for a jet with density $\rho_j$ and internal gas constant velocity $\ensuremath{v}_j$. The density of the intergalactic (or external) medium is $\rho_\mathnormal{e}$ and $\ensuremath{v}_g$ is the velocity of the host galaxy with respect to the intracluster medium. The radius of curvature of the jet is $R_{\text{bend}}$ and the pressure scale length, or the scale over which the ram pressure acting on the beam changes, is $R_p$. The pressure scale height is assumed to be equal to the width of the jet (de Young, 1991). The meaning of eq.(5.1) is that the centrifugal acceleration exerted by the jet as it curves has to balance the gradient in the ram pressure acting on the jet over a pressure scale height. Integration of eq.(5.1) gives the required shape of the jets. Numerical simulations by de Young (1991) have shown that the simplified bending equation is in very good agreement with these more detailed computations.

Figure I.3: The prototype radio tail galaxy NGC1265 (Begelman & Rees, 1996). An intergalactic wind directed upwards in the image hits the radio galaxy and produces enough ram pressure on the jets to cause them to bend.

Another way of inducing deflections in radio jets is caused by a combination of kinetic and geometrical effects. For example, consider the case in which the proper motion of the host galaxy through the intergalactic medium is attributed to the gravitational field of a companion galaxy. In this case each fluid element travelling along the jet follows a straight trajectory, but since the galaxy producing the jet is moving in a Keplerian orbit about its companion, the jet appears to be curved. Sources for which this kind of physical mechanisms occur are called mirror symmetric. The prototype radio galaxy 3C 449 shown in fig.(I.4) illustrates this type of symmetry.

Figure I.4: The radio source 3C 449 shows mirror symmetry which is attributed to the motion of the host galaxy in orbit about a companion (Begelman & Rees, 1996). As the galaxy ``rotates'' about its close companion the jets bend sharply giving rise to mirror symmetry.

The models developed for mirror symmetric radio sources are based in the idea that the material of the jet moves ballistically. This is a first order approximation, but reproduces quite well the shape of the radio sources (Blandford & Icke, 1978).

Radio sources with jets can possess another peculiar shape due to a combination of kinetic and geometrical effects. This occurs if the jets precess about a defined axis. The precession can result in the jet being curved as observed in the plane of the sky, although any fluid particle of the jet always follows a straight trajectory. This behaviour is manifested in the plane of the sky as inversion (or 180$^\circ$ rotation) symmetry. For example, a bend to the right in one jet becomes a bent to the left in the opposite jet. A typical example is the radio galaxy NGC 326 which is shown in fig.(I.5). It is very likely that this precession originates at the very base of the jet (see Begelman et al., 1984, and references therein), close to the central engine.

Figure I.5: Inversion symmetry in the radio galaxies source associated with NGC 326. The radio image of the galaxy shows a bend in the top left jet implies a bend to the bottom right jet. This peculiar shape arise because the jets precess about a certain axis, resulting in a cone-like radio structure. The projection on the plane of the sky of this motion produces inversion symmetry.

A radio source which presents inversion symmetry can be modelled as follows. In its simplest form, the advancing jet moves under the influence of a dynamic pressure force $\boldsymbol{F}$ per unit mass which is given by $\boldsymbol{F} \! = \! \left( \boldsymbol{w} - \boldsymbol{\ensuremath{v}} \right) / T$ where the velocity of the jet beam is $\boldsymbol{\ensuremath{v}}$, the velocity of the central source is $\mathbf{w}$ with respect to the intracluster medium and $T$ is a stopping time. In other words, the equation of motion of the beam is:

$$\frac{ \mathrm{d} \boldsymbol{\ensuremath{v}} }{ \mathrm{d} t } = \frac{ 1 }{ T } \left( \boldsymbol{\ensuremath{v}} - \mathbf{w} \right).$$ (5.2)

Integration of eq.(5.2) gives:

$$\boldsymbol{\ensuremath{v}} = \mathbf{w} - \left( \mathbf{w} - \boldsymbol{\ensuremath{v}}_0 \right) e^{-t/T},$$ (5.3)

where $\boldsymbol{\ensuremath{v}}_0$ is the initial velocity of the jet. Again, integration of eq.(5.3) gives the position $\mathbf{r}$ of the beam as a function of time:

$$\mathbf{r} = \mathbf{w} t + T \left( \boldsymbol{\ensuremath{v}}_0 - \mathbf{w} \right) \left( 1 - e^{-t/T} \right),$$ (5.4)

in which $\mathbf{r}( t \! = \! 0 ) \! = \! 0$ has been chosen as the base of the jet. The initial velocity $\boldsymbol{\ensuremath{v}}_0$ is usually assumed to be constant in magnitude, but it can precess on a cone with opening angle $\theta$ and period $P$ (Icke, 1981):

$$\boldsymbol{\ensuremath{v}}_0 = \ensuremath{v}_0 \begin{pmatrix}\... ...ft[ 2\pi \left( t_0 - t \right) / P \right] \ \cos\theta \quad \end{pmatrix},$$ (5.5)

in spherical polar coordinates $( r, \theta, \varphi \! = 2 \pi \left( t_0 - t \right) / P)$, with a precession period $P$. With the aid of eqs.(5.4)-(5.5) the inversion symmetry can be modelled to great accuracy for radio galaxies with this particular shape (Icke, 1981).

Evidence exists for deflections of galactic and extragalactic jets when they interact with high density clouds in the interstellar and extragalactic environment surrounding them (see for example Burns, 1986; Bachiller et al., 1995; Lehnert et al., 1999; McNamara et al., 1996; Best et al., 1998).

The first suggestion that a deflection of a jet could be due to its interaction with a cloud was made by Burns (1986), to explain the contradictions encountered when trying to apply the arguments of radio trail sources to wide angle tail (WAT) sources. WAT radio galaxies have a similar C-shaped structure, like radio trails but they appear distorted. Indeed, in most sources the tails bend at the point where the jet disrupts, and some others bend after this disruption (O'Donoghue et al., 1990). Burns (1986) came to the conclusion that large bends in WAT's approaching $\unit{1}{\mega pc}$ in scale can not be achieved unless unphysically high speeds through an extremely dense intracluster gas occurs.

More recently, Lehnert et al. (1999) have presented direct evidence for a jet-cloud interaction in the quasar PKS 1318 + 113 (fig.(I.6)). The radio image of PKS 1318 shows a strong deflection of the southern radio jet. The deflection begins to occur exactly at the point at which two separated Ly$\alpha$ clouds seem to be surrounding the radio jet. As explained by Lehnert et al. (1999), it seems that the jet has drilled a hole through the cloud. This is the reason why they appear as two independent clouds around the jet at the point where it begins to curve.

Figure I.6: Quasar PKS 1318 + 113. The figure shows the narrowband Ly$\alpha$ image with VLA $\unit{ 2 }{ \centi \meter }$ radio map contours superimposed (Lehnert et al., 1999). The southern jet of the quasar is deflected as it bores a hole through what seems to be two independent Ly$\alpha$ clouds seen at the southwest of the centre of the source.

Pressure stratification can cause a jet to curve. This was first proposed by Icke (1991) and extended by (1996) and (1996). These authors carried out their analysis using non-relativistic jet velocities and assuming that the cloud was a Gaussian sphere, an isothermal plane parallel atmosphere and an isothermal sphere respectively in their publications. In Chapters III-IV a generalisation of their models is analysed together with a description of the stability of the bending jets.