E. VILKOVISKIJ

Copyright © 2012 SciRes. IJAA

122

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