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

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2. Model

2.1. Formation of early-type galaxies, star formation and black hole evolution in a hierarchical universe

We consider a hierarchical universe. In this scenario, an elliptical/bulge does not form in a single collapse and burst of star formation at high redshift. Instead, in our model, gas cools and condenses at the center of a virialized halo of dark matter, and forms a centrifugally-supported gas disk there; star formation takes place mainly in the disk at a modest rate and gas is accreted inwards to the central object at the same time; a stellar disk and a central black hole may finally develop. If two disk galaxies merge at some time, the violent interaction between the two galaxies will smash the original stellar disks completely, and form the spheroidal component of the new early type galaxy. Meanwhile, the activity drives the outer cold gas to the central region of the newly formed galaxy and sets up a new molecular disk there; the triggered starburst and central AGN will put their kinetic energy into the ISM and induce turbulent viscous accretion in the central region.

In the following, we describe the background for our model, and then the model itself.

1) In the early universe, tidal torques spin up both the dark matter and the baryonic matter in some region of space; when the region becomes overdense, it will eventually collapse into a galaxy. This process together with the cooling can finally form a primordial gas disk, possibly with a seed black hole in the center, surrounded by a dark halo.

2) When this primordial gas disk is formed, the viscosity can start to transfer angular momentum to the outside and accrete material into the central region of a galaxy, while at the same time, star formation is taking place in the disk. The two activities, accreting material inwards i) to feed the central object and ii) to form a stellar disk, occur together and outline a picture of evolution of the disk galaxy.

3) With the hierarchical galaxy formation model, the violent merger of two disk galaxies will destroy the original stellar disks completely. All the stars in the disks are transferred to the bulge or spheroidal component of the new galaxy (Baugh et al. 1996, Kauffmann et al. 1993, Kauffmann 1995b, 1996), and thus form ellipticals/bulges.

4) During or after the violent mergers, the fate of the gas component is very uncertain. Some gas may be shock-heated by collisions between galaxies and injected back into the intergalactic medium. Alternatively, a large fraction of the gas may lose its orbital angular momentum, decrease its orbital radius, and be driven to the center (Toomre & Toomre, 1972). We propose here that most of the gas contained within the central kiloparsec or so is in the form of dense clouds which are on more or less circular orbits; self-gravity may play a critical role in the subsequent evolution of this molecular gas disk.

5) Both the concentration of molecular clouds towards the merger nucleus and an increased efficiency of star formation due to cloud-cloud collisions (Scoville et al. 1986) will result in the appearance of a nuclear 'starburst'. This merger triggered starburst probably can stir up turbulent accretion in the new molecular disk by heating or shocking the ISM from supernovae (SN), and thus help to grow a quasar black hole in a very short time. The possible fate of cool gas could be to form stars in a starburst, to feed a quasar black hole by turbulent accretion, or to be blown out of the disk by a wind.

There are several key concepts and basic assumptions in our model:

1) By the disk formation scenario in the early universe (Dalcanton et al. 1997), we basically assume that gas and baryons from a protogalactic halo cool, collapse or settle into a rapidly rotating gas disk on a time scale, that is much shorter than the star formation time scale [FORMULA] and the angular momentum transport time scale due to spiral density waves etc. (Zhang 1996, Gnedin et al. 1995, Olivier et al. 1991). Nevertheless, this assumption does not restrict our results here. Because, even if there is still some gas in a hot phase when the merger takes place, this gas could cool and together with the cool molecular gas from the outer disk set up the new molecular disk in the central region of the elliptical. This can change the total mass of the newly formed molecular disk, increase both star formation and accretion to the central black hole, but it will not affect the final mass ratio of the black hole and its stellar component.

2) We basically assume that the first merger event between two disk galaxies which possibly form the elliptical/bulge happens around [FORMULA] or later after the gas and baryons from a protogalactic halo cool, collapse and initially set up a gaseous disk. At this later evolution stage, the drastic disk evolution has already quieted down, or in other words, a stellar disk has already been well developed (Rieke et al. 1980, Kronberg et al. 1985). This merger time scale fits also the cosmological pattern of the merger event and the formation history of ellipticals/bulges near [FORMULA] (Baugh et al. 1996), but we can see in Fig. 5 that the final mass ratio limitation does not depend on the exact merger time.

3) With the basic assumption that [FORMULA] is proportional to [FORMULA] (i.e. [FORMULA]), Lin & Pringle (1987) got a nice fit for the exponential stellar disk evolution in a spiral galaxy. We will adopt the same assumption for our disk evolution and also for the starburst and AGN stage after the merger. We found that the final mass ratio limitation [FORMULA] strongly depends on this assumption of equal time scales (see Fig. 10).

4) As for the uncertainty of the detailed physical interaction between starburst and AGN, the purpose of this paper is just to sketch out an indicative picture for these activities; we will basically imitate the coexistent 'starburst' and the subsequent AGN evolution with the starburst time scale and the turbulent accretion time scale much shorter than their normal time scales before mergers (Mihos & Hernquist 1994).

5) We adopt the notion that accretion onto the central BH is restricted to a rate at which the Eddington luminosity is reached. In our calculation, we assume that the central BH is a seed BH ([FORMULA]) at the outset, the growth of this seed BH is limited by the Eddington limit. The extra gas supply to the center can form gas clouds there shrouding the central BH, or be blown away by a wind.

According to the scenario sketched here, disk evolution including the star formation in the gas disk and a viscous accretion to the central region will start in a protogalaxy before the merger, thus develop a stellar disk and a black hole in the center. We will show the BH evolution, stellar mass evolution and mass ratio evolution for a normal disk galaxy in Fig. 1 and Fig. 2 (line). By our model, we assume, at a certain time, after the disk evolution quiets down, two disk galaxies merge, firstly, destroy the original stellar disks and form the spheroidal component, secondly, drive the cool gas left to the center, redistribute a molecular disk there. A starburst can be triggered in this newly formed molecular disk by the violent interaction, and it probably will induce a turbulent viscous accretion to feed both the central AGN and starburst there. These two activities will grow together rapidly, compete for the gas supply and drain the gas in the molecular disk in a very short time, thus restrict the growth of the black hole to a certain ratio relative to the newly formed stellar component. We will show that the mass ratio of these two components ([FORMULA]) converges to a limited range of value [FORMULA] from our calculation in Fig. 2 (dashed line).

[FIGURE] Fig. 1. The temporal evolution of the central black hole mass and the stellar mass in a normal disk galaxy (without any merger). The short dashed curve is the Eddington limitation for the growth of BH. We note that BH growth at the Eddington limit in a log-log plot of mass versus time is exponential.
[FIGURE] Fig. 2. The time evolution of the mass ratio ([FORMULA]) in a normal disk galaxy (without any merger), and that with one merger. We note that both cases lead to a very similar final mass ratio.

2.2. Fundamental equations and viscosity

We take the gas surface density [FORMULA], to be the sum of HI and H [FORMULA] surface density (plus the small amount of heavy elements). The evolution of [FORMULA], is governed by the standard viscous accretion disk equations of continuity and conservation of angular momentum (Lüst 1952, Pringle 1981), with a sink term due to the conversion of gas to stars ([FORMULA]). The basic equations are

[EQUATION]

[EQUATION]

If we assume [FORMULA] for a stable accretion, and replace the radial velocity [FORMULA] from Eqs. (1) and (2), we get,

[EQUATION]

Where [FORMULA] is the mass return of gas through mass loss from stars. In our calculation, we take [FORMULA] (Tinsley 1974), but we show in Fig. 6 how sensitive the mass ratio limitation is to this fraction. [FORMULA] is the angular velocity, and [FORMULA] is the star formation time scale. As for the viscosity [FORMULA], we will take a new viscosity prescription [FORMULA] (Duschl et al. 1997) for our Keplerian selfgravitating molecular disk. In this case, the accretion time scale [FORMULA] (Pringle 1981).

In our model, we basically assume that [FORMULA] is proportional to [FORMULA], i.e. [FORMULA] for the disk evolution stage before merger and also the starburst & AGN stage after a merger. In this case, with a flat rotation law, we get the star formation rate as: [FORMULA], which is exactly the observed star formation law for disk galaxies (Dopita et al. 1994). We will show in Fig. 10 that the relationship of these two time scales [FORMULA] is a key parameter for the final mass ratio limitation.

The evolution of the stellar surface density [FORMULA] and the central black hole [FORMULA] are given by the prescriptions as below:

[EQUATION]

[EQUATION]

Where [FORMULA] is the inner boundary radius of the molecular disk. We basically choose [FORMULA] in our calculation, a value which we adopt for the inner edge of the molecular disk (Moran et al. 1995, Barvainis 1995). We will show in Fig. 7 that the final mass ratio limitation weakly depends on this radius. [FORMULA] is a standard velocity in our calculation, with the relationship [FORMULA].

The stars are assumed to be on approximately circular orbits and to stay at the radii at which they are formed, so they are effectively frozen out of the viscous evolution. For that gas left still in the disk, it will continue the viscous accretion process.

As for most disk galaxies, their rotation curves are usually quite flat outside the parsec scale, which corresponds well to the region of our interests (Linden et al. 1993, Yoshiaki et al. 1989, de Blok et al. 1996, Genzel et al. 1987, 1994). Our own galaxy shows rather well, that a flat rotation is a fair approximation in the inner region. Also, the recent HST work by Faber et al. (1997) shows that even the general class of "core galaxies" with an inner surface brightness law of a powerlaw in radius with an index 0. to -0.3 still give a rather shallow radius dependence of their circular velocity with [FORMULA], which corresponds to an angular velocity law of [FORMULA], still far from rigid rotation. The other class of "power law galaxies" all have an inner rotation law which is quite flat. So, we adopt the approximation of a flat rotation curve in our calculation, and solve the Eqs. (3), (4) and (5) by a standard first-order explicit finite difference method.

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© European Southern Observatory (ESO) 1998

Online publication: May 12, 1998

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