Summary and Conclusions

The interaction of the jets on extragalactic double radio sources has been the central theme of this work. In Chapter I it was discussed that ``real'' bending of jets, ie. bending not produced by projection effects, can be achieved by two mechanisms. Firstly, ram pressure produced by a wind in the intergalactic medium is able to bend jets which are non-relativistic. This causes the jets to have a semicircular, or U, shape with the host galaxy in the pole. Radio galaxies which have this structure are called radio trails. Secondly, deflections of jets can be achieved if jets pass through a stratified high density region. In this case it is the gradients of the pressure that make the jet to follow a non-straight trajectory. Gravitational effects might be important on the deflection of jets depending on the mass of the stratified density region.

In order to understand this second possibility, the hydrodynamical interaction between a jet and a cloud was analysed in detail, once the steady state has been reached. This was carried out in two steps.

In the first case, the bulk motion of the flow in the jet was assumed to be non-relativistic and the self gravity of the gas in the cloud acting on the jet was included in the description of the problem. Because the jet was assumed to expand adiabatically it was possible to calculate the trajectory of the jet by means of an energy equation (Bernoulli's law). In addition, assuming that the jet Mach number was sufficiently high, analytic solutions to the problem were found for the cases in which the cloud was assumed to be an isothermal sphere and when it was considered to be a gas in hydrostatic equilibrium with a galactic dark matter halo. Dimensional analysis showed that when gravity is included in the relevant equations, the whole problem is described by the initial dimensionless Mach number inside the jet $M_0$ and the polytropic index $\kappa$ of the gas in the jet. Gravity, through the gravitational constant $G$, adds another dimensionless parameter $\Lambda \equiv G \rho_0 r_0^2 / M_0^2 c_0^2$ when the stratified density region is an isothermal sphere and $\rho_0$ is the density of the cloud at the radius $r_0$, where the jet penetrates it. The Mach number at this point is $M_0$ and the velocity of sound in the cloud is $c_0$. The parameter $\Lambda$ is an indicator of how large the deflections produced by the gravitational field of the isothermal sphere are from its straight trajectory. The greater $\Lambda$, the more curved the trajectory is. In contrast, when the cloud is assumed to be a sphere of gas in hydrostatic equilibrium with a dark matter halo in a galaxy, the problem also involves the dimensionless number $\mathit{k} \equiv - 4 \pi G \rho_{\mathrm{d}_\star} \mathsf{a}^2 / c^2_\star$. The core radius is represented by $a$, the density of the dark matter halo at its centre is $\rho_{\mathrm{d}_\star}$, and the velocity of sound is $c_\star$. Just as in the previous case, the parameter $\mathit{k}$ is an indicator as to how big deflections induced by the pressure gradients in the gas around the dark matter halo are.

The parameter $\Lambda$ has the following physical meaning. Because $\Lambda$ has the same value of $10^{-2} / M_0^2$ for galactic jets embedded in molecular clouds and extragalactic jets inside cluster of galaxies, this result provides a clue as to why jets in such different environments as giant molecular clouds and the gaseous haloes of clusters of galaxies have similar properties.

In the second case, the bulk motion of the jet was considered to be relativistic, but the gravitational effects induced by the gas in the cloud were not taken into account. Under the same assumptions as in the non-relativistic case, it was possible to show that the trajectory of jets is also determined by the relativistic version of Bernoulli's equation. Analytic solutions for jet velocities with a high relativistic Mach number were found for an isothermal sphere cloud and for gas in pressure equilibrium with a dark matter halo. Since the flow inside the jet was assumed to have an ultrarelativistic equation of state, the only parameter that plays an important role in the solutions is the initial velocity of the flow.

In both, the relativistic and non-relativistic cases, the solutions are extremely sensitive to variations in velocity. This occurs because the pressure and gravitational force fields applied to a certain fluid element in the jet are the same at a given position. However, as the velocity of the flow in the jet increases, there is not enough time for this force to change the curvature of the jet fast enough, giving rise to very straight jets.

When supersonic flow bends, its characteristics intersect at a certain point in space. Since the hydrodynamical quantities have a constant value on characteristic lines, this means that the point of intersection is such that the value of every hydrodynamical quantity is multivalued. This situation cannot happen in nature and a shock wave is created in order to bypass it. In Chapter IV the problem of the formation of shock waves inside bent jets was analysed in detail. If the intersection of the characteristic lines occurs outside the jet, the jet bends without any generation of internal shock waves. However, if this intersection occurs inside the jet, a shock wave forms and it is potentially dangerous to the jet. This is because behind a shock wave the normal component of the velocity is subsonic and collimation can not be achieved. By using a relativistic generalisation of steady simple waves in classical fluid dynamics, it was possible to describe in detail the generation of shock waves inside a jet due to this mechanism.

The proper Mach number in a supersonic flow that bends through a continuous curved trajectory decreases along its path. This implies that at some point, when $M \gtrapprox 1$, the rate of change of the Mach angle with respect to the bending angle increases without limits. This means that the characteristic lines tend to intersect at the end of the bending, when the flow is near the transonic point. Jets with Mach numbers between this lower limit and the maximum allowed value ( $M = \infty$) do not generate shocks at the end of the curvature. The difference between the bending angles evaluated at these last two values of the Mach number give upper limits for the bending angle of jets. For instance, it was shown in Chapter IV that for non-relativistic bulk motion of jets with a polytropic index of $5/3$, appropriate to Herbig-Haro jets, this limiting angle has a value of $\sim \unit{ 75 }{ \degree }$. However, if the polytropic index has a value of $4/3$, for example radio trail sources, the upper limit for the bending angle is $\sim \unit{ 135 }{ \degree }$. Relativistic bulk motion of jets with a polytropic index of $4/3$, appropriate for classical double radio sources, have an upper limit of $\sim \unit{ 50 }{ \degree }$. The reason why relativistic jets can bend less than non-relativistic ones is because the region of transmission of information from a perturbation, bounded by characteristics, is closer to the streamlines in relativistic flow. This increases the chances of an intersection of the characteristics as the jet bends.

In Chapter I the radio-optical alignment effect in powerful double radio sources was discussed. The optical radiation shows an alignment with the radio source axis rather than an elliptical structure. The effect fades away as the sources grow larger ( $\sim \unit{ 10 }{ \mega yr }$) and small sources ( $\sim \unit{ 50 }{ \kilo pc }$) show strongest aligned optical structures. The morphology, kinematics and ionisation properties of the emission line gas of these small radio sources are dominated by the intense effect of shock waves associated with the expansion of the radio jet through the interstellar and intergalactic medium. It seems that cold clouds ( $\sim \unit{ \power{ 10 }{ 4 } }{ \kelvin }$) embedded in the intergalactic and extragalactic medium of the host galaxy are able to interact with the bowshock of the expanding radio source in such a way as to generate sufficient strong shock waves in this interaction to produce the shock ionisation observed.

In Chapter V the one dimensional interaction between a shock and a cold high density region bounded by two tangential discontinuities (a cloud) was analysed in detail. It was shown that as a result of the interaction a discontinuity in the initial conditions was formed and a penetrating shock wave into the cloud and a reflected shock were produced. The shock wave that penetrates the cloud is able to hit, accelerate and compress the cloud until it reaches its opposite boundary giving rise to another (second) initial discontinuity. As a result, a shock wave is transmitted to the external medium and a rarefaction wave bounded by two tangential discontinuities is reflected back to the cloud. The rarefaction wave re-expands and cools the cloud. The important conclusion of this analysis is that most of the energy that the incoming shock was carrying before the collision is injected to the cloud. Very little energy is transmitted to the other side of the cloud, regardless of the strength of the original incoming shock.