Contents List of Figures

290 6 MACRO– AND MICRO–JETS DRIVEN BY BLACK HOLES
S i = ρhΓ 2 bv i
(704)
τ = ρhΓ 2 b − D − P . (705)
v i = u i /u t is the three velocity and Γb the Lorentz factor. The system of equations
is closed by means of an equation of state (EOS), which we shall assume to be given
in the form P = P (ρ, ɛ).
In the non-relativistic limit D, S i and τ approach their Newtonian counterparts
ρ, ρv i and ρE = ρɛ + ρv 2 /2, and the equations of system reduce to the classical
ones. In the relativistic case the equations of are strongly coupled via the Lorentz
factor and the specific enthalpy, which gives rise to numerical complications.
An important property of system (5) is that it is hyperbolic for causal EOS.
For hyperbolic systems of conservation laws, the Jacobians have real eigenvalues
and a complete set of eigenvectors. Information about the solution propagates at
finite velocities given by the eigenvalues of the Jacobians. Hence, if the solution is
known (in some spatial domain) at some given time, this fact can be used to advance
the solution to some later time (initial value problem). However, in general, it is
not possible to derive the exact solution for this problem. Instead one has to rely
on numerical methods which provide an approximation to the solution. Moreover,
these numerical methods must be able to handle discontinuous solutions, which are
inherent to non-linear hyperbolic systems.
The simplest initial value problem with discontinuous data is called a Riemann
problem, where the one dimensional initial state consists of two constant states separated
by a discontinuity. The majority of modern numerical methods, the so-called
Godunov–type methods, are based on exact or approximate solutions of Riemann
problems [23, 20].
Although MHD and general relativistic effects seem to be crucial for a successful
launch of the jet, purely hydrodynamic, special relativistic simulations are adequate
to study the morphology and dynamics of relativistic jets at distances sufficiently far
from the central compact object (i.e., at parsec scales and beyond). The development
of relativistic hydrodynamic codes based on HRSC techniques has triggered the
numerical simulation of relativistic jets at parsec and kilo–parsec scales [18]. In Fig.
138 we show the time evolution of a light, relativistic (beam flow velocity equal to
0.99) jet with large internal energy. The logarithm of the proper rest-mass density is
plotted in grey scale, the maximum value corresponding to white and the minimum
to black.
Highly supersonic models, in which kinematic relativistic effects due to high
beam Lorentz factors dominate, have extended over-pressured cocoons. These overpressured
cocoons can help to confine the jets during the early stages of their evolution
and even cause their deflection when propagating through non-homogeneous
environments. The cocoon overpressure causes the formation of a series of oblique
shocks within the beam in which the synchrotron emission is enhanced. In long term