§20 Relativistic analysis

Let us consider the case in which relativistic effects are included in the collision between a relativistic jet and a stratified high density region. In order to simplify the problem, the self gravity of the cloud acting on the jet is ignored. For this case, the relativistic generalisation of Euler's equation is given by eq.(9.7), in which $\omega$ and $p$ are the enthalpy per unit volume and the pressure of a given fluid particle in its proper frame of reference respectively. The speed of light is $\ensuremath{\mathsf{c}}$ and $\gamma$ is the standard Lorentz factor.

The arguments used to prove the conservation of the angular momentum of the jet in section §17 can be generalised for the relativistic case in the following way. The term in brackets in the left hand side of eq.(9.7) represents the classical force per unit mass acting on an element of fluid as it moves. By considering steady flow and because the pressure depends only on the radial coordinate $r$, vector multiplication of the radius vector $\mathbf{r}$ with eq.(9.7) shows that the quantity $\mathbf{l} \! = \mathbf{r} \times \boldsymbol{\mathit{v}}$ is conserved during the motion of the fluid. This quantity corresponds to the specific angular momentum in classical mechanics, but is not its relativistic counterpart, which is given by $\mathbf{r} \times \gamma\boldsymbol{\mathit{v}}$. The constancy of $\boldsymbol{l}$ implies that motion is two dimensional and so eqs.(17.3)-(17.4) are valid in the relativistic case as well.

Multiplication of eq.(9.7) by the unit tangent vector $\hat{ \boldsymbol{\ensuremath{v}} }$ for steady adiabatic flow, shows that the trajectory of the jet is described by Bernoulli's law (eq.(9.8)):

$$\int{{\mathrm d}\left(\frac{\gamma w}{n}\right)} =0.$$ (20.1)

The line integral is taken from the initial position of a given fluid particle to its final position. The particle number per unit proper volume is $n$ and we assume that the electrons in the jet are ultrarelativistic, so that the equation of state is given by $p \! = e/3$ with $e$ being the internal energy density of the plasma. The requirement that the pressure of the jet equals that of the cloud, together with the fact that $p \! \varpropto \! n^{4/3}$, makes it possible to integrate eq.(20.1) giving:

$$\frac{{\mathrm d}\eta}{{\mathrm d}\varphi} = \pm \frac{\ensuremat... ...^2\varphi_0 - \gamma_0^{-2} \left( \frac{p}{p_0} \right)^{1/2} \right\}_,^{1/2}$$ (20.2)

in which $\eta \! \equiv \! r_0/r$. The sign of ${\mathrm d}\eta/{\mathrm d}\varphi$ in eq.(20.2) varies as the jet crosses the cloud. For example, for the ultrarelativistic case, in which a straight trajectory is expected, it is positive for $\eta_* \! > \! 1 / \sin\varphi_0$, and negative when the inequality is inverted. A general analytic solution of eq.(20.2) can be found because, for high relativistic velocities, the third term on the right hand side of eq.(20.2) is important only for $\eta \! = \! 1/\sin\varphi_0$. In other words, the pressure stratification of the cloud can be written as eq.(17.12) with the substitution $\Gamma \! \rightarrow \! 1/2$. We can therefore expand eq.(20.2) about $\eta \! = \! 1/\sin\varphi_0$ to obtain a relation like eq.(17.14) but with:

$$a = 1 - \frac{ \alpha }{ \gamma_0^{2} },$$

$$b = - \frac{ \beta }{ \gamma_0^{2} } \sin\varphi_0,$$ (20.3)

$$e = -\left( \frac{ \zeta }{ \gamma^2 } + 1 \right) \sin^2 \negthinspace \varphi_0.$$