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

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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 [FORMULA] 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 ([FORMULA]) is obtained substituting the velocity [FORMULA] in the above expression by Eq. (4). For the case where the jet perforates the clouds the age [FORMULA] is obtained by using Eq. (5) for the velocity [FORMULA]. 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 [FORMULA] is obtained when the velocity [FORMULA] is given by (6) but with the external density [FORMULA] being substituted by [FORMULA]. Finally, the age [FORMULA] 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 [FORMULA] and 0.5, with the parameter determining the density within the clouds [FORMULA] and 2. The characteristic luminosity of the central engine [FORMULA] and [FORMULA] erg s-1 and its mass [FORMULA]. The radius [FORMULA] of the clouds at the reference radius, which we put [FORMULA] 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 [FORMULA], 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 [FORMULA] and, on the other hand, that [FORMULA] 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 [FORMULA] and [FORMULA] respectively and for [FORMULA], [FORMULA] erg s-1, [FORMULA] and [FORMULA] 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 ([FORMULA]) than Model B where it perforates them. We also note that both Models A and B are better than the uniform, high density model ([FORMULA]). For instance, in sources whose distance of the hot spot to the central object is 100 pc, [FORMULA] is approximately [FORMULA] yr and [FORMULA] yr while [FORMULA] yr. If there is no density enhancement due to the clouds the age would be [FORMULA] yr. As one would expect, whereas [FORMULA] is more than one order of magnitude greater than [FORMULA], Model B gives an age not too different of Model C although still a few times greater.

The ages are generally smaller for [FORMULA] than for [FORMULA]. In Fig. 1b we have for [FORMULA] pc, [FORMULA] yr, [FORMULA] yr, [FORMULA] yr and [FORMULA] yr. The main reason is that the jet with [FORMULA] is more collimated [FORMULA] than in the case [FORMULA] [FORMULA] and advances more rapidly trough the medium.

[FIGURE] Fig. 1. Age of GPS sources as a function of the jet size. The labels A, B, C and D correspond to the models described in the text. The values of the parameters are [FORMULA], [FORMULA] erg s-1, [FORMULA] pc and a   [FORMULA] and b   [FORMULA].

Figs. 2a and 2b show the age respectively for [FORMULA] and [FORMULA] but now for [FORMULA]. The ages are smaller by approximately a factor 2.5, being [FORMULA] yr and [FORMULA] yr for [FORMULA] and [FORMULA] yr and [FORMULA] yr for [FORMULA]. This illustrates how the radial distribution of the density of the clouds affects the age. From Eq. (12) we see that, if [FORMULA], 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 [FORMULA] 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 [FORMULA] is closer to the theoretical value ([FORMULA]) 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 [FORMULA] increases.

[FIGURE] Fig. 2. As in Fig. 1 but for [FORMULA].

Therefore, the group of GPS sources like [FORMULA] that have, beside the compact structure on the tens of parsec scale, an extended emission suggesting ages [FORMULA] 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 [FORMULA] ages around [FORMULA] yr which is compatible with that found by Readhead et al. (1996a) for [FORMULA]. For this "compact symmetric object" that has an overall size 120 pc ([FORMULA] pc), they estimated an age of [FORMULA] 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 [FORMULA] 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 [FORMULA] cm-3.

Curves C and D of Figs. 1 and 2 should represent the evolution of more extended ([FORMULA] kpc) sources. Indeed, for [FORMULA] pc the age is around [FORMULA] 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 [FORMULA] yr, that is [FORMULA] 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 [FORMULA] pc, that is, [FORMULA], and how it depends upon the different parameters. The curves begin or finish at distinct values of [FORMULA] to satisfy the constraints mentioned early in this section. As we have already seen, the effect of increasing the parameter [FORMULA] 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 [FORMULA] 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 [FORMULA], 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 [FORMULA] and [FORMULA] and for [FORMULA] varying in the range [FORMULA]. Here, [FORMULA] erg s-1 and [FORMULA].

[FIGURE] Fig. 3. Age of GPS sources in Model A for [FORMULA] pc as a function of the radius of the clouds at [FORMULA] pc. The labels 0, 1 and 2 are the values of the parameter [FORMULA] and solid and dashed lines correspond to [FORMULA] and [FORMULA] erg s-1 respectively.

[FIGURE] Fig. 4. Age of GPS sources for [FORMULA] pc in models A, B, C and D described in the text. The parameters are [FORMULA], [FORMULA] erg s-1 and [FORMULA] pc.

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© European Southern Observatory (ESO) 1998

Online publication: December 16, 1997