Astron. Astrophys. 329, 845-852 (1998)

3. Motion of the jet in a uniform medium

Suppose that the jet propagates under ram pressure equilibrium with the external medium. Its velocity in a uniform medium can be calculated from

where is the medium density and u the jet internal energy density. If the central engine supplies energy to the jet at a rate L (erg s-1) we have (Scheuer, 1974; Carvalho, 1985)

Here R is the distance from the center of the host galaxy, c is the velocity of light and is the jet solid angle at a reference distance . In the above expression we also take into account a non-linear dependence of the radius of the jet on the distance R according to the law

where is a free parameter.

It has now been well established that in order to explain the evolutionary sequence in which compact symmetric sources evolve into medium-sized and later into large symmetric objects, some sort of luminosity evolution has to be taken into account (Carvalho, 1985; Fanti et al., 1995; Readhead et al., 1996b). More recently O'Dea & Baum (1997) have studied a combined complete sample of GPS and CSS sources. They concluded that they evolve in a different way than the extended classical doubles as they undergo strong luminosity evolution. However, this is expected to happen for sources larger than kpc or, equivalently, for times greater than yr when the luminosity should decrease as , where is a constant. Therefore, this will only affect the compact GPS sources to a small extent and, to avoid the introduction of a new parameter into the model we assume that the power supply L is constant.

Of course, the model is not realistic enough to cope fully with the physics on the parsec scale encountered in compact sources and (6) may give too large a value for the velocity. This is due to the fact that the jet geometry yields an extremely high energy density near the central object because its volume tends to zero. In order to overcome this problem we limit our calculations to a region and choose pc. At this distance the velocity attains its maximum value , which for powerful classical doubles is or the order of . Now, if we substitute and into (6) we obtain the minimum value of the jet opening angle at which ensures that is always less than .

Another problem arises at large distances R. Depending upon the value of and on how fast the density decreases, there could be a region where the jet speed increases with distance from the central object. This happens because, besides encountering less resistance from a medium whose density decreases rapidly, the high collimation of the jet is maintained since a is kept constant. Would there be any observational evidence for such an increa.se of the jet velocity? Indeed, according to Fanti et al. (1995), the linear size distribution for steep spectrum sources can be fitted by a power law of the type , where . This includes compact sources with size pc. They conclude that the expansion velocity of the sources should increase with their size, being proportional to .

Provision has to be made in the model to avoid that the velocity exceeds at larger R. However, due to the form of the gas density distribution that we shall use (Eq. (9) below), this increasing of the jet speed only occurs in a small interval of distances around 100 pc and for kpc, which is already out of the size range of the compact sources. Therefore, in order to make the model as simple as possible, we just choose such that the velocity is less than for most of the range of values of R and limit it to otherwise. For instance, if for , is calculated by substituting and into (6), for we have to take two times greater than the value calculated in this way.

Finally, the density of the jet can be estimated by equating its internal energy density u to and supposing the internal jet speed . We then obtain

Apart from a numerical factor of order unity, an identical expression is reached using Eq. (2.1) of Begelman et al. (1984), for the case in which the internal speed is much larger than the speed at which the jet advances through the medium.

© European Southern Observatory (ESO) 1998

Online publication: December 16, 1997