## 2. Black hole to bulge mass correlation in AGNs## 2.1. Illustrative modelAGNs are usually thought to be powered by gas accretion to the central massive black holes. There is now firm evidence that QSOs could be formed during the violent galactic interactions together with central starbursts (Sanders et al. 1988a, 1988b, Taniguchi & Shioya 1998, Heckman et al. 1986, Bahcall et al. 1997). Although only of Seyfert galaxies have companions, there are still many aspects of observations which support a simple unified formation scheme for QSOs and Seyferts (Rafanelli et al. 1995, Taniguchi 1999). In this sense, QSOs and Seyferts are the consequences of either galaxy mergers or tidal interactions, meanwhile a massive black hole may grow during the process of mergers or tidal interactions since the enhanced cloud-cloud collision or star formation could increase the mass inflow to the center. We adopt such a unified formation scheme for Seyferts where we think the tidal perturbation of some satellites not only kinematically heat and thicken the host disks, form the central bulge, but may increase the effect of self-gravity in the molecular disk, enhance the cloud-cloud collision and star formation, thus result in a considerable mass inflow to the central region on a relatively short time (Lin et al. 1988, Quinn et al. 1993, Walker et al. 1996, Veláquez & White 1999). Although we consider in our model the tidal interactions as a formation scheme for most Seyferts, we do not reject a possibility that some of AGNs with low black hole to bulge mass ratio could still be major mergers, but at the earlier evolutionary phase in which the black holes have not had time to reach the asymptotic value. We adopt as the initial mass distribution for protogalaxies in our model, with the typical mass of and the disk scale of , similar as the size of our Galaxy. The evolution of the surface density of a differentially rotating disk with angular velocity and viscosity is governed by (Lüst 1952, Pringle 1981) where for a Keplerian selfgravitating disk (Duschl et al. 2000); the accretion time scale ; and the star formation time scale (Pringle 1981). is the star formation rate in the disk with mass return rate (Tinsley 1974). In this case, the mass influx at the inner boundary is: We adopt for a flat rotation law for the protogalaxies in our calculation with as a normalization velocity. The justification of such simplification for a self-gravitating disk at parsec to kiloparsec region is discussed in detail by Wang & Biermann (1998). The spheroidal Bondi accretion rate is given by , where is the Bondi accretion radius defined as ( is the mass of the central black hole; the effective relative velocity between seed black hole and the ambient gas) (Bondi 1952). In our model, we start from a tiny black hole , with the inner boundary of the accretion disk which is usually thought to be the inner edge of a torus. Changing the scale of inner boundary or the seed black hole mass only influences the early evolution, but no significant effect on the final result. At early time, , we assume the mass shears inwards at the inner radius to form a uniform Bondi flow with the mass distribution ( is the time step of the calculation); afterwards, black hole would grow and reach a stage of , we assume in this case black hole accretes the mass shearing inwards within its influence at via a uniform Bondi flow with the mass distribution . The Bondi parameter is assumed to be in our calculation as a possible reduction factor due to the angular momentum. Thus the accretion rate and the star formation rate in a time step is given by: We could roughly estimate the black hole to bulge mass ratio by , where we take the value of in case of . The estimated ratio shows a strong dependence on the velocity dispersion , the value and the black hole mass . From Eq. (3), we know the accretion rate when , which indicates that the nuclear activity would dim slowly as the central black hole becomes massive enough. In this case, a critical black hole mass would be reached when the nuclear accretion rate decreases to a level below Eddington rate in units of . Assuming , we get the critical black hole mass , with and corresponding to the viscous diffuse time scale of (we will discuss these particular parameters later). Afterwards, the accretion rate would decrease quickly to be much less than Eddington rate, and the black hole will not grow significantly. We say this is an upper limit, simply because we use the initial surface gas density throughout the estimation. Actually, the gas would be depleted during the evolution by star formation and accretion etc., it would thus take a time scale much longer than Hubble time in order to approch such a mass or maybe never could reach such a level if there is no external trigger for the nuclear accretion, such as galaxy mergers or tidal effects. In any sense, the black hole growth would be locked up in a level of in such an evolutionary scheme. Meanwhile, we assume in our model , since the result of numerical simulation by Wang & Biermann (1998) shows a strong correlation between the starburst and accretion in merging galaxies, which could regulate the black hole to bulge mass ratio to a level within a factor of three in QSOs and early type galaxies. In this case, if we consider Seyferts and QSOs are formed in a similar scheme, we could adopt this result as a reasonable assumption for our present model. We now use as input the critical black hole mass , the assumption and to Eq. (3), a mass ratio about 0.001 would be approched. Choosing is based on the observations by Zylka et al. (1990) of molecular line emission in the central region of our Galaxy with the assumption that the velocity field of molecular clouds would be the low limit of the sound speed of hot gas (von Linden et al. 1993, Garcia-Munoz et al. 1977, Reynolds 1989, 1990). We know from Eq. (3) and the numerical calculation, the variation of the velocity dispersion of accreting gas, , would cause a broad distribution of black hole to bulge mass ratio, which is actually shown by the observations. The ratio could reach when , close to the virial velocity of the system. In this case, the black hole evolution would follow Eddington accretion for a longer time, resulting in a QSO phase finally. We should mention here the range () is a crude upper limit, since we could not include the surface density evolution by star formation and accretion in this estimation. The practical values should be given by numerical simulation shown in Fig. 1.
## 2.2. Numerical results and discussionTo numerically solve the partial differential Eq. (1), we
introduce the dimensionless variables,
, ,
with the scale
, .
The general boundary conditions , and
zero torque at the origin, . The
viscous torque This choice of boundary conditions guarantees zero viscous coupling between the disk and the central object, allows all mass reaching the inner boundary to flow freely inward. The numerical results are shown in Fig. 1, where we model the effects of a tidal perturbation in a few kpc region by increasing the viscous friction. In fact, the numerical simulation by Lin et al. (1988) demonstrated that a tidal disturbance could propagate to the nucleus from a distance of a few kpc on a time scale shorter than the tidal interaction by "swing amplication", which induces a rapid increase in the effective viscosity and results in an increase in the rate of mass transfer and energy dissipation. We adopted in the calculation a reasonable set of parameters which could match the observations quite well. The viscous diffuse time scale is in case of a tidal perburbation according to the numerical result of Lin et al. (1988). We consider the tidal perturbation would enhance cloud-cloud collision, thus increase both star formation and accretion in a self-gravitation disk. According to the numerical results of Wang & Biermann (1998) for the QSO evolution, we adopt the assumption (i.e. the star formation approximately scales with the accretion). In this case, , corresponding to a star formation rate in the central disk, which is at a reasonable level for the observed star formation rate in Seyferts. The initial seed black hole in our model is tiny, . We know from the calculation that the black hole grows by a rate at the beginning till ; afterwards, the accretion rate would decrease by and only a mediate black hole of can be formed if there is no external trigger. In this case, a tidal perturbation around could help to grow a massive black hole, which corresponds to in an Einstein-de Sitter Universe with (where we think the tidal interactions happen frequently, also could be a peak of Seyfert activities). Although the parameters, such as initial black hole mass, the inner boundary of the accretion disk etc. could have certain influence on the mass ratio, we notice from the numerical results and the illustrative estimation that velocity dispersion of the accreting gas is another key parameter besides the value of , which could cause a large dispersion of the black hole to bulge mass ratio in AGNs. Fig. 1 shows a correlation between the nuclear black hole to bulge mass ratio and the velocity dispersion of accreting gas to the center from our simulation, which gives a rough proportionality of . In our model, we are aiming to explain a broad distribution of black hole to bulge mass ratio of a similar bulge system in QSOs and Seyferts. If we adopt the masses of the spheroidal systems scale with their luminosities as and the scale length , combining with the virial theorem gives the bulge velocity dispersion (Faber & Jackson 1976, Faber et al. 1987, Peterson 1997). So, we obtain . The assumption that the velocity dispersion of accreting gas is systematically proportional to the bulge velocity dispersion gives a correlation of , which is shallower than the slope of given by Ferrarese & Merritt (2000), but steeper than the best-fit correlation () by Gebhardt et al. (2000). We see from Fig. 1 that the variation of the velocity dispersion of accreting gas would cause a broad distribution of black hole to bulge mass correlation, where the accreting gas with higher velocity dispersion close to the bulge velocity dispersion could lead to an accretion near Eddington limit, and form more massive black holes in QSOs than in Seyferts of similar bulges. Therefore, the black hole to bulge mass correlation in massive systems is close to an upper limit and much tighter than that in medium-bulges. The physical interpretation of such discrepancy could be due to the different formation or evolution environment of the two kinds of systems (Seyferts and QSOs). The violent collision between two galaxies could trigger intense starburst in the center, heat or shock the interstellar medium efficiently than tidal interaction, thus drive a Bondi flow fuelling the central black hole with probably a higher sound speed and result in a higher accretion rate. In this case, it may enhance a QSO evolution with the accretion rate close to Eddington limit, form a massive black hole, and lead to a black hole to bulge mass ratio higher than in case of Seyferts. © European Southern Observatory (ESO) 2000 Online publication: October 2, 2000 |