## 5. The age of the sourcesIn this section we determine the dynamical age of the sources as a
function of its size, that is, the distance 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 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 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
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 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
© European Southern Observatory (ESO) 1998 Online publication: December 16, 1997 |