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Astron. Astrophys. 334, 87-95 (1998)

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3. Calculations and discussion

3.1. The numerical scheme and boundary conditions

For computational convenience to solve the partial differential Eq. (3), we introduce here dimensionless variables, [FORMULA], [FORMULA], [FORMULA] with the scale [FORMULA], [FORMULA], the general boundary conditions as [FORMULA], and zero torque at the origin, [FORMULA], where [FORMULA] and [FORMULA] are the disk's outer radius and inner radius, and the viscous torque G as given by (Pringle 1981).


This choice of boundary conditions guarantees zero viscous coupling between the disk and the central object, and so allows all mass reaching that point to flow freely inward. Evidently, the accretion to the central BH is still limited by the Eddington condition.

3.2. Initial conditions

The procedure of our calculation is at first to develop a disk galaxy, and after the merger, to continue the turbulent viscous accretion disk evolution in the newly formed molecular disk. For this purpose, we set up two reasonable initial density distributions for the gas disks before and after a merger. We will discuss these two initial conditions in paragraph 1) and 2) below.

1) The initial gas disk surface density for disk galaxy formation is dependent on the characteristics of the protogalactic halo and on the details of dissipative collapse of the gas to the rotationally supported disk. As for our disk evolution scenario, our disk is basically a Keplerian selfgravitating disk (Duschl et al., 1997), we will just use its stationary solution [FORMULA] as our initial gas disk surface density distribution for the disk evolution, where [FORMULA] is the outer radius of our disk [FORMULA].

2) In our calculation, we will assume the initial gas distribution of the molecular disk assembled after merger to be a power law [FORMULA], as a good approximation to the equilibrium distribution of mass with a highly flattened axisymmetric, self-gravitating system (Toomre 1963). This law fits the observation result of the distribution of molecular gas in merger remnants, such as, ARP220 (Scoville et al. 1986), where more than [FORMULA] of the CO emission, corresponding to more than [FORMULA], is confined to a galactocentric radius smaller than 1 kpc.

3.3. Numerical calculation and discussion

In our model, we basically consider the case when a merger happens some time after the disk has already been well developed. This does not mean that it is a necessary condition for merger in real life, but it does really exclude the possibility that ellipticals or bulges form through a single collapse of a protogalaxy in high redshift.

In order to understand the merger effects, we first show in Fig. 1 and Fig. 2 (line) a normal disk evolution. We can see in Fig. 1 that after BH evolution starts from a seed BH ([FORMULA]), it will grow a central BH [FORMULA] (for a protogalaxy of [FORMULA]) after a full Eddington evolution of nearly [FORMULA] years. Because the gas was almost used up by the BH evolution and the star formation finally, both activities will quiet down, thus the ratio of the BH mass and the stellar mass will slowly approach a constant. We emphasize that no spheroid exists at all during this disk evolution stage.

According to our model, at a certain time after the disk evolution quiets down, if a violent merger happens between two well developed disk galaxies, it will smash the original stellar disks completely and form the spheroid of a new early type galaxy. Meanwhile, the merger can cause a severe perturbation in the central region and drive the gas inwards to concentrate in the central kiloparsec (Barnes 1992, Casoli et al. 1988). We simulate this effect by redistributing a certain amount of cool gas to that region, setting up a molecular disk, triggering a starburst and nuclear activity there. In the real universe, the violence of the merger to form an elliptical/bulge can have a variety of properties. So, the amount of gas redistributed in the center by the merger effect can be different also. In our simulation, we therefore redistribute the same amount of cool gas as the host galaxy's in its central region, to simulate the effect of a major merger between two disk galaxies with comparable mass, and half the amount of cool gas for the case of a minor merger. We find from our calculation that the amount of gas mass redistributed to the central region has no influence on the mass ratio limitation, but it does affect the final black hole mass.

We show in Fig. 2 (dashed line) and Fig. 3 the evolution of the mass ratio ([FORMULA]), BH mass and the stellar component mass for the case when a major merger happens near [FORMULA]. We can see that black hole grows in full Eddington evolution at the beginning, until some time later, the gas in the disk is drained off by star formation and BH evolution, both activities quiet down, thus constrain the mass ratio firstly around a limited value of [FORMULA]. If a merger happens afterwards, it can reawaken these two activities, with the appearance of a starburst and central AGN. The extreme starburst and BH growth last only for a short time, then the gas in the central region will be used up by these two activities, or some of it is blown away by the wind; till this stage, we see the normal ellipticals/bulges with the mass ratio limited. From Fig. 2 and Fig. 3, we see that a merger dramatically shortens the convergence time scale and grows a massive BH in the center quickly.

[FIGURE] Fig. 3. The time evolution of the black hole mass, spheroidal component mass and the Eddington growth mass limitation with one major merger. The triangles indicate the merger event.

In our model, the prescriptions of the star formation time scale and the accretion time scale are [FORMULA], and [FORMULA]. Considering a reasonable accretion time scale ([FORMULA]) and the star formation rate for a normal galaxy (Dalcanton et al. 1997), we choose [FORMULA] for the normal disk evolution phase. We simulate the starburst and central engine activity by taking [FORMULA], thus shorten both time scales by a factor of 100. So it will dramatically increase the star formation rate to nearly two orders of magnitude in ultraluminous starburst galaxies; it also fits the starburst time scale about [FORMULA] (Rieke et al. 1980, Kronberg et al. 1985, Mihos & Hernquist 1994, Smith et al. 1996). We will show in Fig. 9 and Fig. 10 that the mass ratio is not very sensitive to the exact number of [FORMULA] and [FORMULA], but clearly sensitive to their correlation. We will discuss this in detail below.

We show the mass ratio evolution in Fig. 4, for the case that protogalaxy mass ranges from [FORMULA] to [FORMULA] ; and in Fig. 5, the case when merger happens at a different time after the disk quiets down. We see from our results that the final mass ratio limitation does not depend on these two parameters.

[FIGURE] Fig. 4. The time evolution of mass ratio of the central black hole and its host spheroid for the protogalaxy mass ranges from [FORMULA] to [FORMULA]. The curves coincide after the merger, and converge to the almost same mass ratio [FORMULA].
[FIGURE] Fig. 5. The time evolution of the mass ratio ([FORMULA]) when the merger happens in different times.

In our calculation, we basically choose the star formation mass return rate [FORMULA] (Tinsley 1974), and the inner radius for the molecular disk [FORMULA] (Moran et al. 1995, Barvainis 1995). In Fig. 6 and Fig. 7, we see that the mass ratio limitation will vary moderately if we vary [FORMULA] by a factor of two and [FORMULA] by a factor of 12.

[FIGURE] Fig. 6. The time evolution of the mass ratio of the central black hole and its host spheroid for the mass return rate [FORMULA]. A standard value for this parameter is 0.3 (Tinsley 1974), but various common IMFs can give other values. Varying this parameter by a substantial factor does influence the resulting mass ratio [FORMULA]. It appears as if this final ratio depends on [FORMULA] approximately as [FORMULA]. This means that our results depend on the assumption that the universal spread of this parameter is less than about two.
[FIGURE] Fig. 7. The time evolution of mass ratio of central black hole and its host spheroid when the inner boundary radius of the molecular disk shifts with a factor of 12, i.e. [FORMULA]. We see in the graph that the mass ratio converges to a constant quickly after the merger. The final ratio [FORMULA] varies by a factor of 5 for the variation of [FORMULA] by a factor of 12. This shows a soft dependence.

From our calculation, we see that the mass ratio of the central black hole and its host spheroid is very sensitive to the ratio of the starburst time scale and the turbulent accretion time scale after a merger (see Fig. 10). The mass ratio converges to the value [FORMULA] only when the turbulent accretion time scale approximately equals the starburst time scale (i.e. [FORMULA] 1). This probably hints at a physical reality, which is, a merger will drive a large amount of cold gas inwards to the central region, the increased molecular cloud collisions can trigger a starburst in the central region, the kinetic energy output from these young massive stars and the shocks from supernovae will heat and disturb the ISM locally, thus probably induce turbulent viscous accretion to feed both the AGN and starburst in the central region. These two activities, starburst and central AGN, will interact with each other, feedback, drain the gas in the molecular disk in a short time, thus grow a massive BH quickly there. To some degree, it probably can start a strong wind from the center, blow off the leftover gas in the disk, stop the starburst and the accretion process in the central region. It seems that this evolution scenario can help to explain the observed limited mass ratio region [FORMULA] within a factor of three. In order to see the sensitivity of this mass ratio limitation to the equal time scale assumption, we show the results in Fig. 10 for different ratios of the starburst time scale and the turbulent accretion time scale; we can see the dramatic divergence of the mass ratio when [FORMULA].

All we discussed above is the case where only one major merger happens between two disk galaxies to form an elliptical or the bulge of a spiral. In a real hierarchical universe, multiple minor mergers would also probably occur after the major merger. These could bring in more cool gas to the central region and start another evolution cycle. In order to imitate our universe closely, we also calculated the case that one major merger is followed by another minor merger. We show our result in Fig. 8, and it seems that the multimerger will not change this universal ratio.

[FIGURE] Fig. 8. The time evolution of the mass ratio of the central black hole and its host spheroid with one major merger and another minor merger.
[FIGURE] Fig. 9. The time evolution of the mass ratio of the central black hole and its host spheroid when the parameters [FORMULA] vary with a factor of 10. We see in the graph that the final mass ratio limitation does not strongly depend on the exact value of these parameters, as long as [FORMULA].
[FIGURE] Fig. 10. The time evolution of the mass ratio of the central black hole and its host spheroid with different correlation between starburst time scale and the turbulent accretion time scale. We calculate the cases: [FORMULA]. We see that the variation of the final value [FORMULA] is almost a factor of [FORMULA] for a variation of time scale ratio by a factor of [FORMULA]. This shows that the correlation between these two time scales deeply dominates the final mass ratio limitation. It seems also that the mass ratio can finally converge to a limited region [FORMULA] only when the starburst time scale and the turbulent accretion time scale are almost equal.
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© European Southern Observatory (ESO) 1998

Online publication: May 12, 1998