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Astron. Astrophys. 358, 57-64 (2000)

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5. Emission variability

In order to compare the results of this model to the observed variability values, we derive an explicit expression for this quantity in terms of the parameters of our model.

The computed light curve [FORMULA] can be sampled at ([FORMULA]) times intervals, i.e. at times [FORMULA] with n=1, N and the variability can be derived as

[EQUATION]

This quantity has been numerically computed for different sets of input parameters in order to investigate their influence on the final result. The results of this analysis are described hereafter.

As discussed in the previous section, [FORMULA] depends on the star density distribution parameters. In each of our computations we have chosen specific values of k, [FORMULA] and [FORMULA] and changed that of [FORMULA] in order to have a `good' [FORMULA] value. For [FORMULA] in the `good' range, the influence of the other parameters on variability and luminosity values are the following:

[FORMULA] Changes in [FORMULA] and k are crucial for the resulting luminosity and variability as described in the following.

[FORMULA] Different values of [FORMULA] have been tested in the interval ([FORMULA]); these changes do not influence the results if [FORMULA] or [FORMULA]. For [FORMULA] the external regions of the cluster give a big contribution so as [FORMULA] has been settled in order to have a reasonable ([FORMULA]) number of stars.

[FORMULA] The effects of changing the value of [FORMULA] are more complex. In the framework in which [FORMULA] is freely changed, the presence of a different central mass apparently does not have any effect on [FORMULA] and hence on [FORMULA]. This fact is due to our working hypothesis of selecting a specific interval of [FORMULA] values. In fact, from the approximate expression (9) it appears that [FORMULA] depends only on [FORMULA] and [FORMULA]. If [FORMULA] has a value in the `good' range, independently of [FORMULA], [FORMULA] only could show a dependence on the black hole mass. However, this is not the case as one can check by rewriting R in Eq. (3) in terms of [FORMULA] as [FORMULA]:

[EQUATION]

As a matter of fact the dependence from the central black hole mass is completely hidden in [FORMULA], and through it, in the choice of star density parameter [FORMULA] (see Sect. 4 for details). The same conclusion does not hold for the other possibility presented in Sect. 4, namely that of compensating the change of [FORMULA] (i.e., that of [FORMULA]) with that of the ratio [FORMULA]. In fact, in this case [FORMULA] has a different value and hence [FORMULA] and [FORMULA] change. However, this difference is qualitatively analogous to that due to a change in the density distribution which is discussed hereafter and in Sect. 8.

[FORMULA] Changes in the the mass distribution parameters, x, [FORMULA], [FORMULA] do not have a sizable influence on [FORMULA] if the value of [FORMULA] is suitably rearranged.

In conclusion the two parameters which can change the variability value and that of [FORMULA] are those which define the shape of star density, i.e. k and [FORMULA] (in which is included the dependence on [FORMULA] through the change of [FORMULA], as discussed above). This means that different density profiles can produce similar collision rates but different luminosities and variabilities. Looking again at the approximate expression (9) the above conclusion can be easily understood. In fact, for equal [FORMULA] values, a different star distribution implies different average velocities of the colliding stars and therefore a different amount of the available energy, [FORMULA]. Hence the more the star distribution is concentrated around the center, the higher the luminosity. This fact can be confirmed by Figs. 2 and 3 where it is shown how [FORMULA] and [FORMULA] depend on the parameters k and [FORMULA]. In both figures the points on the right hand side of the picture correspond to more peaked star distributions i.e. to the distributions having the smallest [FORMULA] (Fig. 2) or the steepest profile (Fig. 3).

[FIGURE] Fig. 2. Variability as a function of [FORMULA] for [FORMULA], [FORMULA], [FORMULA] and different values of [FORMULA]. Different symbols correspond to different values of [FORMULA], namely, [FORMULA] (squares),[FORMULA] (triangles), [FORMULA] (circles). The dashed line has slope -1/2.

[FIGURE] Fig. 3. Variability as a function of [FORMULA] for [FORMULA], [FORMULA], [FORMULA] and different values of k. Different symbols correspond to different values of k, namely, [FORMULA] (circles), [FORMULA] (asterisks), [FORMULA] (squares), [FORMULA] (triangles). The dashed line has slope -1/2.

In order to obtain Figs. 2 and 3 [FORMULA] values have been chosen, for each pair ([FORMULA]), so as the values of [FORMULA] result in the range ([FORMULA]). This fact implies that the values of [FORMULA] used for Figs. 2 and 3 are in the range [FORMULA]. These values are very high if compared with the star density observed in the center of our galaxy. However, the context here is different, since in this model we are dealing with active galaxies in which we presume that the activity itself is due to the presence of a great number of stars near the galaxy center. In this context the high star concentration is a necessary ingredient of the model and the star density value can be lower only around very massive black holes, i.e. when the typical dimension of the system ([FORMULA]) increases.

In Figs. 2 and 3 a straight line of slope -1/2 has been drawn. As it has been explained in the introduction, this is the slope expected from a series of independent discrete events. It is evident from the above figures that each group of points obtained varying [FORMULA] for a specific pair ([FORMULA]), is aligned along a line with the same slope -1/2. A small scatter, presumably due to the procedure of random choice of some parameters, is present. This alignment confirms the correctness of all the treatment since it shows that, when increasing the event number (i.e. [FORMULA] in our case), the expected relationship for a discrete event model is found.

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© European Southern Observatory (ESO) 2000

Online publication: June 26, 2000
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