§21 Isothermal cloud and dark matter halo

As in section §18, consider the case of an isothermal cloud being penetrated by the jet. In this case, it is possible to find an exact solution to the problem, since eq.(18.2) and eq.(20.2) can be used to show that:

$$\frac{{\mathrm d}\eta}{{\mathrm d}\varphi} = \pm \frac{1}{ \sin \... ...ta\gamma_0^{-2} \right) - \eta^2 \sin^2\negthinspace \varphi_0 \right\}_.^{1/2}$$ (21.1)

In other words, the solution is the same as that already found in eq.(17.14) and eq.(17.15) but with:

$$a=\left(\frac{\ensuremath{\mathsf{c}}}{\ensuremath{v}_0}\right)^2... ...thsf{c}}}{ \ensuremath{v}_0 } \right)^2 \gamma^{-2}, \qquad e=-\sin^2\varphi_0.$$ (21.2)

Fig.(III.5) shows plots of the trajectory of the jet for different values of its initial velocity.

Figure III.5: Deflection of a relativistic jet produced by its collision with an isothermal cloud (semicircle). The jet is assumed to travel parallel to the $x$ axis at the moment it enters the cloud from the right. In each plot different trajectories are shown for different values of the initial height of the jet $y/r_0\!=\!0.05$, $0.15$,...,$0.95$ as measured from the $x/r_0$ axis. The top, middle and bottom panel plots were calculated for values of the initial velocity of the jet $\ensuremath{v}_0$ in units of the speed of light $\ensuremath{\mathsf{c}}$ of $0.99$, $0.97$, $0.95$ respectively.

Let us now consider the case in which the gas in a galaxy is in hydrostatic equilibrium with a dark matter halo. As it was shown in section §19, for $r_0 \! \gg \! \mathsf{a}$, the variation of the pressure in the galaxy is given by eq.(19.5) and the trajectory of the path of the jet is given by eq.(17.14) and eq.(17.15) together with eq.(19.7) and the substitution $\Gamma \! \rightarrow \! 1/2$. Fig.(III.6) shows plots of this for $\mathit{k} \! = \! -3$ and different values of the initial velocity of the jet $\ensuremath{v}_0$.

Figure III.6: Trajectory of a relativistic jet as it crosses a galaxy. The gas in the galaxy is assumed to be in hydrostatic equilibrium with a gravitational potential given by a dark matter halo in the galaxy. It is assumed that the jet enters the galaxy parallel to the $x$ axis at a height of $y/r_0\!=\!0.05$, $0.15$,...,$0.95$ in different cases. The plots were calculated for the case in which the parameter $\mathit{k} \! = \! -3$ and the initial velocity of the jet in units of the speed of light is $0.999$, $0.995$ and $0.99$ from top to bottom. Continuous lines are analytic approximations to the problem and dashed ones are direct numerical solutions.