§4 Fluid dynamics of jets

The simplest way to understand the flow of these relativistic jets is to use conservation laws (Blandford, 1990; Begelman et al., 1984). In what follows we will use some of the basic equations derived in Chapter II. This section is included here for consistency with the discussion of jets.

Let us first discuss the case of non-relativistic jet flow, with a polytropic index $\gamma \! = \! 5/3$. It is simplest to approximate the jet as a one dimensional flow of variable cross section area $A$. The rate at which mass is injected into the jet in time $t$, the discharge, is $\dot \mathsf{M}$. The flow inside the jet has a velocity $\ensuremath{v}$. Using eqs.(10.2)-(10.3) it follows that the total energy (kinetic and internal) of the flow inside the jet with respect to time -the power, is given by (Blandford, 1990):

$$L = \frac{ 1 }{ 2 } \dot \mathsf{M} \ensuremath{v}^2 \left( 1 + \frac{ 3 }{ M^2 } \right),$$ (4.1)

where $M$ is the Mach number of the flow. It follows from the derivation of this relation that the first term inside the parenthesis on the right hand side of eq.(4.1) is the bulk kinetic energy transported by the fluid. The second term is the transport of the enthalpy, that is, the internal energy plus the work $p V$, where $V$ is the volume. If the jet is steady, the thrust, that is the rate of change of momentum with respect to time, is given by (Blandford, 1990):

$$P = \dot \mathsf{M} \ensuremath{v}\left( 1 - \frac{ 3 }{ 5 M^2 } \right).$$ (4.2)

The first term on the right hand side of eq.(4.2) is the rate of supply of bulk momentum flux and the second term is contributed by the gas pressure $p$. When the flow is highly supersonic, that is $M \gg 1$, it follows from eqs.(4.1)-(4.2) that the bulk motion dominates. The simplest assumption to make is that the jet expands adiabatically and that it is stationary. Under these circumstances, the power and discharge, given by $\dot \mathsf{M} \! = \rho A \ensuremath{v}$ are constants. The mass density is represented by $\rho$. We assume that the jet is in pressure equilibrium with its surroundings.

Since the discharge is constant and because the gas is by assumption adiabatic, it follows that $\rho \propto A^{-1} \ensuremath{v}^{-1} \propto p^{3/5}$. Thus, the Mach number and the velocity within the jet scale as:

$$M \propto p^{-4/5} A^{-1}, \ensuremath{v}\propto p^{-3/5} A^{-1},$$ (4.3)

respectively. It follows from eq.(4.3) that the area, as a function of the pressure for a given luminosity, passes through a minimum. This minimum occurs when the velocity of the jet attains the local velocity of sound (Blandford, 1990). In other words, a jet accelerates to supersonic speeds by passing through a converging-diverging or De Laval nozzle (Landau & Lifshitz, 1995; Begelman et al., 1984). It can be shown that in the subsonic portion of the flow, the pressure and density are approximately constant and so, according to eq.(4.3), the area decreases inversely proportional to the velocity: $A \propto 1 / \ensuremath{v}$. In contrast, for the supersonic regime the velocity is almost constant. Using eq.(4.3), this implies that the area increases as $A \propto p^{ -3 / 5 }$. Very often (Binney & Tremaine, 1997) the pressure profile in a galaxy scales with the inverse of the square of the radial coordinate $r$ measured from the centre of the galaxy, that is $p \propto r^{ -2 }$. This implies that the angle made by the jet width and the axis of the radio source decreases as $A^{ 1/2 } r^{-1} \propto r^{ -2/5 }$. Thus, we have just proved that a supersonic jet can be collimated as the pressure diminishes, regardless of the expansion of its cross sectional area.

The thrust in an adiabatic jet is not constant. It actually diminishes during subsonic propagation and increases in the supersonic regime. This occurs because the surface of the jet is not exactly parallel to the mean flow velocity and an external pressure force acting parallel to the jet changes its momentum (Blandford, 1990; Begelman et al., 1984).

Let us consider now a fully relativistic jet. The assumptions made are that the fluid inside the jet is an ultra-relativistic plasma with an equation of state $p = e / 3 \propto n^{ 4 / 3 }$, where $e$ is the proper internal energy per unit proper volume and $n$ the number of particles per unit proper volume. The relativistic power $L$ can be written down directly using the fact that, if $\mathcal{T}^{km}$ is the energy-momentum tensor and $\ensuremath{\mathsf{c}}$ the speed of light, then $\ensuremath{\mathsf{c}}\mathcal{T}^{0\alpha}$ is the energy flux density vector (see section §8):

$$L = 4 p \gamma^2 \ensuremath{v}A = ,$$ (4.4)

where $\gamma$ is the standard Lorentz factor. Since the particle number has to be conserved, it follows from the definition of the continuity equation, eq.(9.1), that

$$n \gamma \ensuremath{v}A = .$$ (4.5)

Substitution of eq.(4.4) into eq.(4.5) implies that:

$$A \propto \frac{ \gamma^2 }{ \ensuremath{v}}$$ (4.6)

As in the non-relativistic case, the area $A$ is again minimised at the point where the flow is transonic. The thrust is given by the space components of the energy-momentum tensor:

$$P = \left( 4 \gamma^2 \frac{ \ensuremath{v}^2 }{ \ensuremath{\mathsf{c}}^2 } + 1 \right) p A.$$ (4.7)

The thrust changes in exactly the same way as its non-relativistic counterpart does.

Real astrophysical jets do not satisfy necessarily the assumptions we made above. This is because in a real jet, there will be a velocity shear across the jet and most probably turbulence combined with internal shocks inside the jet will develop. There will also be radiative losses and internal dissipation causing particle acceleration. Not even the mass density flux will be constant along the jet, because it is expected that jets entrain gas from the surrounding environment. Finally, perhaps the most important omission from the considerations discussed above is the fact that jets might well be hydromagnetic (Blandford, 1990). Nonetheless, the whole picture provides a framework for studying more detailed effects of the physics of these sources.