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

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4. The emission

The emission process is the same as outlined in Paper I, to which we refer. For the generic "j" collision the emission [FORMULA], in erg s-1, is given by the star available kinetic energy, [FORMULA], times a function of the impact parameter which takes into account the efficiency of the process, times a temporal profile:

[EQUATION]

Here the cooling time, [FORMULA], is defined as the time the plasma sphere takes to become optically thin:

[EQUATION]

with [FORMULA] the Thompson cross section, [FORMULA] cm s-1 the cloud expansion velocity and where it has been assumed that [FORMULA] solar masses of expanding plasma are involved. The possibility of pair creation has been taken into account in deducing [FORMULA]. It has not been reported here since it does not have influence on the values of the emission variability. How pair creation modifies the emission light curve is shown in Paper I.

If all collisions with all possible combinations of parameters take place, the mean emission in erg s-1 will be:

[EQUATION]

whose order of magnitude and parameter dependence can be derived as

[EQUATION]

However, in practice we shall only observe the contribution of the collisions which take place in a certain time interval. Hence we can derive the emission light curve adding at each time t the contribution of all J collisions taking place from [FORMULA] to [FORMULA] and having a non-vanishing light curve at that time:

[EQUATION]

The observed mean luminosity of the source is then

[EQUATION]

For a sufficiently large number of collisions [FORMULA] clearly approaches H and hence [FORMULA] can be approximated, using expression (8), as

[EQUATION]

In conclusion the procedure used to derive the emission from the star cluster orbiting around the black hole is as follows. For each set of random chosen parameters, i.e. for each collision, the light curve [FORMULA] is computed. From the sum of all these contributions the overall light curve [FORMULA] and the mean luminosity [FORMULA] are deduced.

As the approximate expression (9) for the luminosity shows, the mean luminosity value is directly proportional to [FORMULA] and, on the other hand, the light curve shape obviously depends on the collision rate. In Paper I, the authors arrived to the conclusion that, in order to reproduce the observed AGN features, collision rates in the range [FORMULA] collisions/yr are required. An easy but qualitative way to arrive to an analogous conclusion is from variability observations, which report intensity variations on time scales [FORMULA] in the range (some year [FORMULA] days). In the framework in which these intensity variations are due to star collisions occurring at intervals of ([FORMULA]), it follows [FORMULA] collisions/yr. As an example Fig. 1 shows a light curve with a reasonable value of [FORMULA].

[FIGURE] Fig. 1. Light curve for the case of a collision rate [FORMULA] and for [FORMULA].

Hence, in the scenario in which AGN luminosity variations are explained by means of stellar collisions, the model must account for collision rate values of the order of ten. However, obtaining collision rate values of the order of ten independently of the AGN physical system, i.e. of the central mass amount, is not obvious. To better explain this point the parameter dependence of [FORMULA] can be derived from expression (6) assuming that the star density is constant inside [FORMULA] and vanishing outside:

[EQUATION]

[EQUATION]

So, for different [FORMULA] values we can obtain the same [FORMULA] value using almost identical values of [FORMULA] and [FORMULA] (which means different values of the quantity [FORMULA]) or, if [FORMULA] is fixed, using different values, suitably chosen, of [FORMULA]:

[EQUATION]

Hence, [FORMULA] values in the `good' range can be obtained either changing the system scaled dimensions [FORMULA] or the central star density [FORMULA] or both. So the star cluster core extension, if defined in terms of [FORMULA], will be different in systems with different central masses: we can suppose that the lowest central masses ([FORMULA], see Sect. 3.2) are related to larger [FORMULA] ratios in order to avoid too high star density. For example we obtain the same collision rate [FORMULA] collisions/yr for [FORMULA], [FORMULA] stars/pc3, [FORMULA] and for [FORMULA], [FORMULA] stars/pc3, [FORMULA]. We note that, in spite of the high central star densities, the total number of stars in the two clusters (i.e. stars lying inside the shell of radius [FORMULA], corresponding to 1 pc for [FORMULA] and to 10 pc for [FORMULA]) is not a large one, since it is [FORMULA] and [FORMULA] stars, respectively.

The consequences of the above analysis are twofold: i) high star densities are generally required in order to have suitable [FORMULA] values; ii) a change in the density parameters [FORMULA] and [FORMULA] can also be representative of a change of the central black hole mass.

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© European Southern Observatory (ESO) 2000

Online publication: June 26, 2000
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