5. The age of the sources
In this section we determine the dynamical age of the sources as a function of its size, that is, the distance R of the hot spot to the central object. This is given by
The lower limit pc is quite appropriate considering that our attention will be focused on compact sources whose size is in the range 20 - 100 pc.
In the case of a cloudy medium, the age of the source when the jet scatters the clouds () is obtained substituting the velocity in the above expression by Eq. (4). For the case where the jet perforates the clouds the age is obtained by using Eq. (5) for the velocity . In order to evaluate how effective the clouds are in decelerating the jet, we also study the propagation of the jet through a massive but uniform medium whose density is equivalent to that of the clouds (Model C). In this case an age is obtained when the velocity is given by (6) but with the external density being substituted by . Finally, the age of a source propagating through a low density medium is also calculated and should represent the age of an extended source. In this case the velocity is just given by Eq. (6).
In what follows we evaluate the age as a function of the size of the source for different values of the parameters. As far as the geometry of the jet is concerned we make and 0.5, with the parameter determining the density within the clouds and 2. The characteristic luminosity of the central engine and erg s-1 and its mass . The radius of the clouds at the reference radius, which we put pc, is picked in the range 1 - 25 pc.
Of course, the combination of all these parameters will not always give a physically meaningful picture. For example, it may happen that the number of clouds in the line of sight up to a distance R, that is , is either too small and there could be no collision of the jet with them, or that it is so large that the clouds will overlap. To avoid this we require that, on the one hand, the jet has encountered at least one cloud when it grows to a size of 20 pc and, on the other hand, that is always less than 0.5. We also require the ratio between the radius of the jet and that of the cloud to be less than 1/2. Thus, although the parameters are in the range given above, some combinations of them will not appear in the results we give below.
In Fig. 1a and 1b we show the results for and respectively and for , erg s-1, and pc. The first thing to note is the effectiveness of the clouds in slowing down the jet. As it enters the cloudy region that lies between 3 and 300 pc the age of the source increases significantly. We see how Model A, in which the jet scatters the clouds, is much more effective () than Model B where it perforates them. We also note that both Models A and B are better than the uniform, high density model (). For instance, in sources whose distance of the hot spot to the central object is 100 pc, is approximately yr and yr while yr. If there is no density enhancement due to the clouds the age would be yr. As one would expect, whereas is more than one order of magnitude greater than , Model B gives an age not too different of Model C although still a few times greater.
The ages are generally smaller for than for . In Fig. 1b we have for pc, yr, yr, yr and yr. The main reason is that the jet with is more collimated than in the case and advances more rapidly trough the medium.
Figs. 2a and 2b show the age respectively for and but now for . The ages are smaller by approximately a factor 2.5, being yr and yr for and yr and yr for . This illustrates how the radial distribution of the density of the clouds affects the age. From Eq. (12) we see that, if , the density of the clouds is constant throughout the entire region and they will be more efficient in braking the jet even at large distances from the center. On the other hand, for the clouds become rapidly less dense as R increases and the jet will be able to maintain its velocity much longer, resulting in a lower age. As we mentioned before, this should be a more realistic case since is closer to the theoretical value () for which tidal disruption of the cloud can be avoided. Comparing Figs. 1 and 2 we also observe that the difference between Models B and C becomes less pronounced as the value of increases.
Therefore, the group of GPS sources like that have, beside the compact structure on the tens of parsec scale, an extended emission suggesting ages yr, may well be explained by a model lying between the two extreme cases represented by our Model A and B.
On the other hand, we see that the high-density uniform model (Model C) gives for ages around yr which is compatible with that found by Readhead et al. (1996a) for . For this "compact symmetric object" that has an overall size 120 pc ( pc), they estimated an age of yr assuming that the jet has a low velocity of 0.02c and is being confined by a uniform cloud 200 pc in radius whose density is cm-3. This is almost equivalent to our Model C where the jet advances at an average velocity of approximately 0.04c between 3 and 300 pc and the density at 100 pc is cm-3.
Curves C and D of Figs. 1 and 2 should represent the evolution of more extended ( kpc) sources. Indeed, for pc the age is around yr. However, as we have already pointed out in Sect. 3, we did not take into account the variation of luminosity with the age of the source as one must expect after yr, that is kpc (Carvalho, 1985; Fanti et al., 1995; Readhead et al., 1996b). Thus, extrapolating the models to sources larger than 1 kpc must be done with caution.
In Fig. 3 we give an overall view of the age of a 100 pc source in Model A as a function of the radius of the clouds at pc, that is, , and how it depends upon the different parameters. The curves begin or finish at distinct values of to satisfy the constraints mentioned early in this section. As we have already seen, the effect of increasing the parameter is to decrease the age of the source. As for the luminosity L, the less powerful the central object is the more the age increases, since according Eq. (6) the jet advances more slowly into the medium. The figure also shows how the mass of the central object affects the age. A large value of allows the existence of denser clouds as suggested by Eqs. (12) and (13), which in turn will decelerate the jet more efficiently. Seemingly, the age goes up if one increases the radius of the clouds as we must expect. Models B, C and D do not depend on the size of the cloud and in Fig. 4 we compare their ages with that of Model A calculated for , again for a source whose hot spot is at 100 pc from the galactic nucleus. The figure shows the ages in the various models for and and for varying in the range . Here, erg s-1 and .
© European Southern Observatory (ESO) 1998
Online publication: December 16, 1997