2. Jet Propagation in a Clumpy Medium
The interaction of an extragalactic jet with dense clouds has been studied by De Young (1991, 1993) through a series of hydrodynamical simulations. As we said previously, we shall try to build an analytical model that, in spite of its simplicity, is able to describe the phenomenon reasonably well. We shall investigate two possibilities, the first one being that during the collision both the cloud and jet behave as if they were solid bodies and are scattered one by the other (Model A). In that case one can treat the problem as an elastic shock.
The second possibility (Model B) is that the jet crosses the cloudy medium by drilling its way through the clouds. One may expect this seeing that the jet cross-section is in many cases much smaller than the cloud cross-section and thus, the low density and high velocity stream of particles will just penetrate the much heavy cloud without affecting too much its structure or moving it from its initial position. Of course the real situation should be a compromise between the two extremes just described although, as we argue below, it resembles the first one more closely.
2.1. Model A: Jet-cloud scattering
That the clouds are partially deflected by the jet can be seen from the numerical simulations by De Young (1991), although they refer to clouds at a distance of a few kpc from the galaxy, the scale being bigger than the one envisaged here. For a cloud large compared with the size of the jet, during the first moments of the collision we see that the behavior of the cloud is approximately that of an almost rigid body being struck by a much smaller mass. After some time, the part of the cloud that has interacted more directly with the jet suffers considerable erosion. By then, the cloud has already moved a distance greater than its radius perpendicularly to the jet trajectory and the path of the jet is almost free. If the size of cloud is of the same order as the jet radius, although considerable erosion takes place at a much early time, the cloud still moves aside. It is thus reasonable to assume as a rough approximation that the interaction between the jet and cloud can be described as an elastic shock.
Let us suppose that in the uniform intercloud medium the jet propagates with velocity determined by the balance between the ram pressure and internal pressure. We shall assume a spherical shape for the cloud and that it is initially at rest. If its mass is M and m is the mass of the jet that takes part in the collision, the velocity of the cloud after the collision will be
Here and is the angle between the velocity V and the direction of propagation of the jet. In order to estimate the mean velocity of the jet going through many encounters like these with a cloud, we must first calculate the characteristic time of the collision. Since this is essentially the time it takes for the cloud to move a distance equal to , where b is the impact parameter and its radius, we have
From Eq. (1) we obtain
where we assumed that the cloud is much heavier than the jet, that is, .
The masses entering in are the mass of the cloud and that of the jet, which can be estimate from the expression . Here and are respectively the radius and density of the jet, the cloud density and is the mass of the hydrogen atom. Substituting M and m in (2) we obtain the collision time
The factor comes from averaging over the values of the impact parameter b from to . It is weakly dependent upon the ratio and, for instance, for we have .
Supposing now that the jet encounters on average clouds per unit length, and under the simplifying assumption that it stays at rest during the collision, which is a good approximation provided , then the average propagation velocity of the jet is given by
2.2. Model B: Jet perforates clouds
The second possibility referring to the jet-cloud interaction corresponds to the other extreme situation in which the jet perforates the cloud. In this case the jet is decelerated when going through the cloud just by the fact that the density is increased as compared with that of the ambient medium. Let be the velocity of the jet in a medium filled with clouds whose density is . If the jet encounters clouds per unit length, then the average velocity is given by
where is the density of the external medium and the factor comes from averaging over all values of the impact parameter. It is interesting to note that the factor does not depends explicitly on the radius of the cloud (see Eq. (11) below) and therefore, is independent of as well.
As we shall see later in Sect. 5, this situation is not equivalent to the case where the propagating medium, although homogeneous, has a high density equal to the mean density obtained by spreading the clouds uniformly (Model C).
© European Southern Observatory (ESO) 1998
Online publication: December 16, 1997