§22 Discussion

Once an observed deflection is given, it is possible to work backwards and find useful properties concerning the initial interaction of a jet with a stratified density region. For example, by taking the ``standard'' values mentioned in sections §18 and §19 for the pressure and density in the stratified gas it is possible to calculate the initial azimuthal angle $\varphi_0$ for a given initial velocity of the jet. In order to illustrate this, consider eq.(17.16) and eq.(17.17). Because the derivative $\left( {\mathrm d}\eta / {\mathrm d} \varphi \right)_e$ has a negative value at the point at which the jet leaves the cloud, it is possible to find the value of the deflection angle $\cos\psi$. This angle is a function of the velocity of the jet $\ensuremath{v}_0$ and the initial azimuthal angle $\varphi_0$. To visualise this, an example is shown in fig.(III.7) for the case in which a relativistic jet interacts with an isothermal cloud. The contour levels for which $\cos\psi \! = \textrm{const}$ give the required relation between the initial velocity and azimuthal angle. Fig.(III.8) shows two examples of these contours.

Figure III.7: Three dimensional plot showing the variations of the deflection angle $\psi$ which is defined as the azimuthal angle the velocity of the flow makes with the $x$ axis at the moment it leaves the cloud. The deflection angle is a function of the initial velocity of the jet $\ensuremath{v}_0$ and the initial azimuthal angle $\varphi_0$. The plot was produced for the case in which a relativistic jet interacts with an isothermal cloud.

Different combinations of the various parameters involved (or the known observables) in the problem can be assumed so that, for a given deflection, the other quantities can be calculated. For instance one can ask for the values of the central density of the gas in the cloud, the density in the jet, etc.

Figure III.8: Variations of initial azimuthal angle $\varphi_0$ as a function of velocity $\ensuremath{v}_0$ in units of the speed of light $c$ for constant values of the deflection angle $\psi$. The angle $\psi$ is the azimuthal angle the velocity vector of the flow in the jet makes with the $x$ axis at the moment it leaves the cloud. Every plot was calculated for $\cos\psi \! = \! const$ with values of $\psi$ given by $\unit{175}{\degree}, \unit{170}{\degree}, \ldots, \unit{155}{\degree}$. The gradient of $\psi$ decreases towards the lower left part of each diagram. In other words, deflections become stronger as the curves approach this region on the diagram. An isothermal sphere and an isothermal gas in hydrostatic equilibrium with a dark matter halo were assumed for the top and bottom diagrams respectively in the case of a relativistic jet.

The most important consequence of the calculations presented in this chapter is the sensitivity of the deflection angles to variations in velocity -see for example fig.(III.8). This sensitivity is due to the fact that the force applied to a given fluid element in the jet (due to pressure and gravitational potential gradients) is the same independent of the velocity of the flow in the jet. 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 soon enough, giving rise to very straight jets.