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Astron. Astrophys. 358, 57-64 (2000)
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 can be
sampled at ( ) times intervals, i.e.
at times with n=1, N and the
variability can be derived as
![[EQUATION]](img116.gif)
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,
depends on the star density
distribution parameters. In each of our computations we have chosen
specific values of k, and
and changed that of
in order to have a `good'
value. For
in the `good' range, the influence
of the other parameters on variability and luminosity values are the
following:
Changes in
and k are crucial for the
resulting luminosity and variability as described in the
following.
Different values of
have been tested in the interval
( ); these changes do not influence
the results if or
. For
the external regions of the cluster
give a big contribution so as has
been settled in order to have a reasonable
( ) number of stars.
The effects of changing the
value of are more complex. In the
framework in which is freely
changed, the presence of a different central mass apparently does not
have any effect on and hence on
. This fact is due to our working
hypothesis of selecting a specific interval of
values. In fact, from the
approximate expression (9) it appears that
depends only on
and
. If
has a value in the `good' range,
independently of ,
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
as
:
![[EQUATION]](img128.gif)
As a matter of fact the dependence from the central black hole mass
is completely hidden in , and through
it, in the choice of star density parameter
(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
(i.e., that of
) with that of the ratio
. In fact, in this case
has a different value and hence
and
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.
Changes in the the mass
distribution parameters, x, ,
do not have a sizable influence on
if the value of
is suitably rearranged.
In conclusion the two parameters which can change the variability
value and that of are those which
define the shape of star density, i.e. k and
(in which is included the dependence
on through the change of
, 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
values, a different star distribution implies different average
velocities of the colliding stars and therefore a different amount of
the available energy, . 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 and
depend on the parameters k
and . 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
(Fig. 2) or the steepest profile
(Fig. 3).
![[FIGURE]](img150.gif) |
Fig. 2. Variability as a function of for , , and different values of . Different symbols correspond to different values of , namely, (squares), (triangles), (circles). The dashed line has slope -1/2.
|
![[FIGURE]](img168.gif) |
Fig. 3. Variability as a function of for , , and different values of k. Different symbols correspond to different values of k, namely, (circles), (asterisks), (squares), (triangles). The dashed line has slope -1/2.
|
In order to obtain Figs. 2 and 3
values have been chosen, for each pair
( ), so as the values of
result in the range
( ). This fact implies that the
values of used for Figs. 2 and 3 are
in the range . 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 ( )
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
for a specific pair
( ), 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. in
our case), the expected relationship for a discrete event model is
found.
© European Southern Observatory (ESO) 2000
Online publication: June 26, 2000
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