3. Parameter choice
3.1. The star density
The star density distribution shown in expression (2) holds for while for larger distances is assumed to have a very steep gradient so that no contribution comes from the regions outside . The choice of the density parameters , and k is not a trivial one. From the observational point of view no indication exists about the star density profile near galaxy nuclei.
Theoretical models of cluster evolution around a black hole (Bahcall & Wolf 1976, Lightman & Shapiro 1978, Murphy et al. 1991) give more information about the density profile but the number of assumptions used in this procedure recommends to take these values only as a suggestion about the possible parameter ranges. For example, Murphy et al. 1991 starting from a density distribution with a slope for find, after 15 Gyr of evolution, an internal () slope of which may become harder () in the outer shell () if the central black hole mass is smaller than .
Since a sure choice of the density parameters is not possible we decided to test our model for different values of k and , namely and . has been set equal and is a free parameter.
3.2. The black hole mass
Black hole masses in active galactic nuclei are usually thought to be in the range . The central mass amount has different consequences on the structure of our model. First of all the mass value is strictly linked to the typical dimension of the physical system through the scaling parameter . This fact implies that, for example, a star cluster core extended around a black hole of mass is only cm wide while that around a black hole of mass is cm wide. Such a difference implies a factor between the two cluster volumes and, therefore, it will have a non-negligible influence on star collision rate values and on the total number of stars in the cluster. The importance of this point, related with the choice of star density values in the cluster, will be evident in the next section.
The second consequence of a different central mass on the model structure is the following. Our model requires the presence of stars near the central black hole, hence the above mass range must take into account the condition that stars must survive to the black hole tidal effect. Assuming that stars outside must not be disrupted, this new constraint reads
Making use of expression (5) and performing a mean over the stellar mass distribution, the above condition becomes
Since this limit has been obtained by weighting over the mass distribution it does not assure that massive stars orbiting near the black hole are not destroyed. However a more detailed calculation shows that already for the fraction of disrupted massive stars inside is less than . Our model is therefore consistent for black hole masses in the range . For lower values of the model should take into account tidal disruption of stars.
3.3. Star mass distribution
The parameters entering the definition of the star mass distribution (Eq. (1)), have been chosen following the classical values, i.e. , and . Probably near a black hole these values are not the good ones since, for example, an enhanced massive star formation can be supposed (Williams & Perry 1994). However, more suitable values are not known and so we chose to assume these ones and to vary them in a second time to test their influence on the model.
© European Southern Observatory (ESO) 2000
Online publication: June 26, 2000