E. VILKOVISKIJ
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simulation (1 × 32 K, the green lines) are in the both cases
smaller for m = 4 (N2 = 8 K, blue lines) and larger for m =
16 (N2 = 2 K, red lines), where m is the number of the
representing systems in the A-K ensemble. So, N2 = 8 K
can be taken as the minimal number of particles in the
small systems to represent the N1 = 32 K large system,
and (admittedly) to represent any N > N1 system with
m=N/N2 subsystems having each N2 particles. We also
find out that both the inaccuracy and the duration of the
A-K simulations are strongly increased with N < Nmin = 2
K (that is m > 16).
The calculated distributions of cosines of the orbital
inclination angle’s to the accretion disc plane, and the
distributions of particle’s eccentricities both for the inner
orbits (in the region of size equal to the accretion disk size)
and for the whole system are shown in Figures 2-5.
To check the A-K method in more details, we varied the
mass accretion rate. For the case presented in Figures 1-5,
we supposed that all growth of the SMBH mass is due to
stellar captures only (the absence of the gas inflow from
the gas disc to the black hole is an artificial admission,
acceptable for the formal testing of our tasks only). Below
we show results of investigations of another case: The
SMBH mass is increased with both the accreted gas and
captured stars (which supposedly are the stars crossing the
sphere with radius R = 0.01). In that case the gas supply
during one relaxation time was supposed to gain mass ΔM =
Mbh0, where Mbh0 is the initial mass of the black hole (Mbh0 =
0.1 NBU in our model). The results of the calculations are
shown in Figures 6-8.
One can see from Figures 7 and 8 that in this case (in-
creased mass inflow to the SMBH) the stellar orbits tend
to be more circular in the inner region of the system, and
eccentricities are diminished due to the increased BH
mass. The A-K results are presented with solid lines, and
direct calculations with dashed lines; both go close to
each other.
3. Application of the A-K Method to N-Body
Simulations of AGN without Accretion
Disk
Another interesting task is calculations of the star capture
rates to the central SMBH without accretion disk in the
stellar systems with zero rotational moment. This drasti-
cally diminishes the accretion rate of the stars. The reason
is that the disruption radii (DR) of stars by the central
SMBH are small and the “loss cone” (the region of the
orbits inside the DR) has to be filled by the stars’ moment
diffusion [3]. We adopt DR = 8.0e–07 from [4] to calcu-
late the stellar capture rate using the A-K method. The
total number of stars in the system was taken N1 = 105 and
the central BH mass MBH = 0.01 (as in the [4]), but in our
case the initial structure of the stellar system is defined
with the Plummer distribution.
As the direct N-body simulation of N = 105 particles
would take too much time with our computer, we per-
formed the simulations using the A-K method only, that is
the simulations of evolution of the ensemble of m = 50
“small” stellar systems with the central BH (each con-
taining N2 = N1/m = 2 K stars-particles), representing the
evolution of the “large” systems with N1 = 100 K equal-
mass stars, surrounding the SMBH with MBH = 0.01 (one
percent of the stellar system’s mass in NBU, as was
supposed in [4]). We choose the “capture radius” of stars
to the central BH rcap = DR = 8e–7 to compare our result
to that of [4]. The results of calculations are shown in
Figures 9 and 10.
As seen from Figure 9, each individual run with N = 2
K particles yields a few accretion events, presented with
vertical “jumps” at the ladder-like thin lines. In every
capture event, the BH mass increases by m = N1/N2 = 50
stars, equal to 5 × 10–2 of the initial BH mass, which gives
the jumps. We had 15 colors only to draw the 50 “ladder-
tracks”, so the resulting picture is a very tangled one. But
averaging over 50 such runs (the A-K method) produces a
rather smooth curve (shown with the red thick line) with
small enough fluctuations, since it is equal to one run with
100 K particles, in accordance with the A-K paradigm.
In the Figure 9, each ladder-like track represents an
evolution of the BH mass in each of the “small” subsys-
tem of the ensemble. The deviations from the average
thick red line characterize the mean-square deviations of
the individual evolution tracks of the small systems. One
can see that in spite of the large deviations in the behavior
of the “small” subsystems of the A-K ensemble, the ave-
raged curve remains smooth enough.
We remind that in the Figures 9 and 10 the time scale is
presented in units of the “evolution time”, Tev = t/trx,
equally defined for the systems with any different num-
bers of particles (see [1]), which permits to plot and com-
pare them at the same pictures, where the essence of A-K
approach is quite visible.
Figure 10 shows how the fluctuations vary with dif-
ferent numbers of runs m, taken for averaging, which
permits comparison of the “full” (complete) A-K simula-
tion (red line) with the “truncated” A-K simulations, ave-
raged correspondently over 30 (green line), 10 (blue) and
5 (agenda) representing subsystems. The average slope of
the lines (well visible in the red and green lines) looks
evidently increased at the moment of evolution time close
to one relaxation time, possibly as a result of a cusp for-
mation in the stellar system. Note, that all the “evolution
tracks” are close enough at the moments of the evolution
time Tev = t/trx ~ 1.5 - 2, which points to possible applica-
tion of the “truncated” A-K method for the snap-shot
estimations of the large stellar system evolution rate.
From this result, we have calculated the star capture
rates CR per N-body time unit (one crossing time, as was