Astron. Astrophys. 358, 57-64 (2000)
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
, in erg s-1, is given by
the star available kinetic energy, ,
times a function of the impact parameter which takes into account the
efficiency of the process, times a temporal profile:
![[EQUATION]](img70.gif)
Here the cooling time, , is
defined as the time the plasma sphere takes to become optically
thin:
![[EQUATION]](img72.gif)
with the Thompson cross section,
cm s-1 the cloud
expansion velocity and where it has been assumed that
solar masses of expanding plasma are
involved. The possibility of pair creation has been taken into account
in deducing . 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]](img76.gif)
whose order of magnitude and parameter dependence can be derived as
![[EQUATION]](img77.gif)
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
to
and having a non-vanishing light curve at that time:
![[EQUATION]](img80.gif)
The observed mean luminosity of the source is then
![[EQUATION]](img81.gif)
For a sufficiently large number of collisions
clearly approaches H and
hence can be approximated, using
expression (8), as
![[EQUATION]](img83.gif)
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 is computed. From the sum of
all these contributions the overall light curve
and the mean luminosity
are deduced.
As the approximate expression (9) for the luminosity shows, the
mean luminosity value is directly proportional to
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
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 in the range (some year
days). In the framework in which
these intensity variations are due to star collisions occurring at
intervals of ( ), it follows
collisions/yr. As an example Fig. 1
shows a light curve with a reasonable value of
.
![[FIGURE]](img96.gif) |
Fig. 1. Light curve for the case of a collision rate and for .
|
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
can be derived from expression (6)
assuming that the star density is constant inside
and vanishing outside:
![[EQUATION]](img98.gif)
![[EQUATION]](img99.gif)
So, for different values we can
obtain the same value using almost
identical values of and
(which means different values of the
quantity ) or, if
is fixed, using different values,
suitably chosen, of :
![[EQUATION]](img102.gif)
Hence, values in the `good' range
can be obtained either changing the system scaled dimensions
or the central star density
or both. So the star cluster core
extension, if defined in terms of ,
will be different in systems with different central masses: we can
suppose that the lowest central masses
( , see Sect. 3.2) are related to
larger ratios in order to avoid too
high star density. For example we obtain the same collision rate
collisions/yr for
,
stars/pc3, and for
,
stars/pc3, . 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
, corresponding to 1 pc for
and to 10 pc for
) is not a large one, since it is
and
stars, respectively.
The consequences of the above analysis are twofold: i)
high star densities are generally required in order to have suitable
values; ii) a change in the density
parameters and
can also be representative of a
change of the central black hole mass.
© European Southern Observatory (ESO) 2000
Online publication: June 26, 2000
helpdesk.link@springer.de  |