A&A 377, 23-43 (2001)
DOI: 10.1051/0004-6361:20010862
C. Zier - P. L. Biermann
Max-Planck-Institut für Radiostronomie (MPIfR) Auf dem Hügel 69, 53121 Bonn, Germany
Received 2 March 2001 / Accepted 14 June 2001
Abstract
Observations of HST and groundbased data strongly suggest that most
galaxies harbour central supermassive black holes and that most
galaxies merge
with others. Consequently a black hole binary emerges as the two black
holes are spiraling into the center towards each other. In our work we
are investigating two basic questions of our understanding of the
central activity of galaxies and find that both can be answered with
``yes'': (1) Do the black holes actually merge?
(2) And does the effect of the torque of the black hole binary on the
surrounding stellar distribution help to explain the presence of the
ubiquitous torus of molecular material surrounding apparently all
active galactic nuclei?
The first question is the topic of the present
article, while the second question will be subject of a subsequent
paper.
Simulating the evolution of a stellar cluster in the potential of such
a binary by solving the equations of the restricted three body problem
we obtained the following results: provided that the cluster is
about as massive as the black hole binary the two black holes coalesce
after
due to ejection of stars and finally via
emission of gravitational radiation. Whether a star is ejected or
not crucially depends on its angular momentum. Almost all stars
whose angular momentum is twice as large as that of a star circulating
around the binary in
a distance corresponding to that between the black holes, stay
bound to the binary.
In a sequence of models where the mass of the secondary black hole
increases while M_{1} is
fixed, a bigger fraction of stars is ejected. For a more massive
binary also the cluster has to be more massive in order to allow the
two black holes to coalesce. The merger then proceeds on smaller time
scales.
The cluster is depleted in the central region and the final
distribution of stars assumes a torus-like structure, peaking at
three times the initial distance of the two black holes. The
relationship of the bound stars to the obscuring torus in active
galactic nuclei will be investigated in a subsequent paper.
Key words: galaxies: active - galaxies: nuclei - galaxies: interactions - galaxies: kinematics and dynamics
The bright radiation in the centers of galaxies is generally thought to be produced by a powerful engine, which is located at the dynamical center of its host. Because the region from where this high luminosity is emitted is concentrated to a small volume, non-stellar activity is suspected to cause such a great energy output. It is argued that these active galactic nuclei (AGN) are powered by accretion onto a massive black hole (BH) in the center. While matter is sinking down in the potential of such a black hole, energy is released and this accretion process is thought to be dominated by an accretion disk. The maximum mass of these central objects, of order 10^{-2.5} times the stellar spheroidal mass of their host galaxies (Magorrian et al. 1998; Richstone et al. 1998; Wang & Biermann 1998; Macchetto 1999), is consistent with the mass in black holes needed to produce the observed energy density in quasar light if reasonable assumptions are made about the efficiency of the energy production of the nucleus (Novikov & Thorne 1972; Shakura & Sunyaev 1973; Chokshi & Turner 1992; Blandford 1999). One of the best candidates of galaxies showing strong evidence for harbouring a massive BH in its center is NGC 4258. From VLBI observations of megamasers in its nucleus, Miyoshi et al. (1995) deduce the existence of a mass of about which must be located inside the inner radius of . Peterson & Wandel (1999) use emission-line variability data on NGC 5548, a Seyfert 1 galaxy, to infer the existence of a mass of order within the inner few light days of the nucleus. There are numerous other examples suggesting that most galaxies keep a super-massive black hole in their center. This is the first of our two basic assumptions.
According to hierarchical models a typical bright galaxy has merged a few times since the quasar era (Frenk et al. 1988; Carlberg & Couchman 1989; Baugh et al. 1996; Richstone et al. 1998). For binary black holes with the timescales for a decay in the different regimes indicate that the binary will merge on a scale short compared to the time when the next merger event occurs (Xu & Ostrike 1994). The rate of mergers for bright galaxies in the observable universe may exceed (Richstone et al. 1998). Taking into account that local quasars might have been formed by major mergers between galaxies, Taniguchi (1999) proposes that all active galactic nuclei of the local universe are products either from minor or major mergers. This appears to be consistent with almost all important observational properties of Seyfert galaxies. Recent studies (Mulchaey & Regan 1997; Ho et al. 1997; de Robertis et al. 1998) show that Seyfert galaxies do not have always either bar-structure or companion galaxies, what often had been considered to be the cause of effective gas fuelling to the central engine (see, e.g., Adams 1977; Noguchi 1988; Shlosman et al. 1990; Barnes & Hernquist 1992). An alternative way to supply the central engine with matter is the merger driven fuelling, see for example Toomre & Toomre (1972), Roos (1981), Mihos & Hernquist (1994), de Robertis et al. (1998). In order to complete a minor merger about are needed. This is enough time to smear out the relics so that most of the advanced minor mergers may be observed as ordinary-looking isolated galaxies (Walker et al. 1996) and Seyfert galaxies are observed to possess a statistically significant excess of faint ( ) companion galaxies (Fuentes-Williams & Stocke 1988). The frequent merging of galaxies is our second basic assumption and consequently leads to massive black hole binaries in combination with the first assumption.
Observations show that all ellipticals have central density cusps which can be divided into "weak'' cusps ( , bright ellipticals) and "strong'' ( , fainter ellipticals) cusps (Kormendy et al. 1994; Kormendy et al. 1996; Faber 1997). It has already been suggested by Begelman et al. (1980) that the mass ejection due the merging of two massive black holes should reduce the central density and that the core expands. In several numerical simulations it has been found that the merging of two galaxies and their central black holes can account for the weak cusps, where the sinking of the BH is the dominant mechanism (Makino & Ebisuzaki 1996; Nakano & Makino 1999; Milosavljevic & Merritt 2001).
Because the critical central regions of AGN are strongly nonspherical but spatially unresolved, orientation effects are thought to be important, see e.g. the discussion of quasar spectra by Sanders et al. (1989). Thus the same object seen from other angles would be classified differently. In type 1 objects the radiation emerging from the center is not blocked by matter in the line of sight of the observer. Hence the X-rays in the -range, the big blue bump (BBB) and the broad emission line region (BLR) as well as the narrow emission line region (NLR) can be seen directly. On the other hand, objects classified as type 2, show strongly absorbed X-rays and no clear evidence for the big blue bump and the BLR (Lawrence & Elvis 1982). But the NLR is indistinguishable from that detected in type 1 objects. This region is located at bigger radii from the center and thus not blocked by matter at a smaller distance in the line of sight. Sometimes a type 1 spectrum is seen in the polarized light of type 2 classified objects and thus gives strong evidence for the existence of an obscuring torus, see e.g. Miller et al. (1991): this is interpreted within the unification model as radiation emerging from the center and then scattered by ionized matter in the opening of the torus into the line of sight (Antonucci & Miller 1985; Miller & Goodrich 1990; Tran & Kay 1992). This model ascribes the different appearance of AGN rather to different orientations than to intrinsic differences. Its basic ingredients which are responsible for the non-spherical symmetry of the nucleus are relativistic beaming in a jet, the obscuration of the central engine by a molecular, dusty torus, and possibly the power of the central engine (see e.g the reviews by Antonucci 1993; Urry & Padovani 1995; Madejski 1998; 1999 and articles by Falcke & Biermann 1995; Falcke et al. 1995; Taniguchi 1999; 2000). Recent XMM-Newton observations of the radio-quiet quasar Mrk 205 might provide the first evidence for a dust-torus in a quasar (Reeves et al. 2001). XMM-data of IRAS 13349+2438 suggest column-densities which are consistent with a dusty torus (Sako et al. 2001). Observations impose important constraints on the properties of such tori. From X-ray absorption a column density of is deduced (Mushotzky 1982; Mulchaey et al. 1992; Madejski 1998), i.e. the tori have to be optically thick. Number ratios of types 1 and 2 objects tell us that such tori are also geometrically thick (see e.g. Taniguchi & Anabuki 1999 and references therein). The AGN spectra also show a prominent infrared bump between and (Sanders et al. 1989), which is thought to be generated by dust reprocessing the incident radiation from the center. In about distance the dust in the torus is heated to its evaporation temperature of about and then reemitting the central ratiation in the infrared (Sanders et al. 1989; Haas et al. 2000). Pier & Krolik (1992) and Pier & Krolik (1993) find that a very compact and thick dust torus, supported by radiation pressure, is not able to provide the required broad temperature range of the dust in order to explain the infrared (IR) emission. But a clumpy torus, as suggested by Krolik & Begelman (1988), has the advantage that the dust, when organized in clouds, can survive more easily the strong radiation field of the AGN. Also the dust in the outer parts can be heated to higher temperatures and a broader temperature range can be achieved (Efstathiou & Rowan-Robinson 1995). Such a clumpy torus model seems to be the most promising scenario also to Haas et al. (2000). A torus is also used in high energy physics according to Protheroe & Biermann (1997) who show for Blazars that the TeV -emission site is far from the central engine due to -interactions with the far infrared photon field of the torus.
Now, based on our two underlying assumptions, necessarily leading to a black hole binary, we here investigate the influence of a stellar cluster and a black hole binary (BHB) in its center on each other in order to answer two basic questions in our understanding of the central activity of galaxies:
Since a BHB has a non-spherical symmetry it might cause an initially spherical symmetric stellar distribution to possibly assume a torus-like structure. Hence, to find an answer to both the questions we performed numerical simulations of the stars in the potential of the BHB by solving the three-body problem for each star of the initial distribution. Integrating over all stars of the cluster we obtain the properties of the stellar population and can trail its evolution under the influence of the binary. Following the suggestion by Kazanas (1989), Alexander & Netzer (1994) and Alexander & Netzer (1997) that the BLR is made up by stars and their winds, we propose that the gas and dust of the torus is bound to the stars within their winds. The dynamics of these stars in the potential of the binary then supports the torus in the vertical direction, solving the problem of how to keep the torus geometrically thick.
In this paper we investigate the conditions for which the black holes merge and how long such a merger lasts. We will first use a mass-ratio of q=10 for the black holes to discuss the results in more detail. Afterwards we compare them with those obtained for the ratios q=1 and 100.
In a following paper (Paper II) we will concentrate on the configuration of the stars remaining bound to the black holes, their ability to constitute a dusty torus and the observational consequences in order to answer the second question.
The evolution of the orbit of a comparatively low mass secondary black hole moving around a massive BH M_{1}in the center of a massive galaxy is investigated with analytical and numerical means by Polnarev & Rees (1994). Dynamical friction between both galaxies results in a loss of energy and angular momentum so that the cores of the galaxies quickly approach each other. The probability of merging with a given initial value of pericenter is roughly proportional to the square of this value. Thus small initial pericenters are unlikely and in the most cases Polnarev & Rees (1994) find the eccentricity never to exceed a value of order 0.1. After the stars surrounding the secondary black hole have been stripped off by tidal effects, it moves on bound orbits in the mean gravitational field of the core of the primary galaxy. Dynamical friction extracts kinetic energy from the secondary BH (Chandrasekhar 1943; Binney & Tremaine 1987) which binds to M_{1}at the radius . Below this separation M_{2} is still surrounded by the sea of field stars of the core. The density enhancement of surrounding stars in its wake after gravitational interaction further decreases the velocity of M_{2} so that it keeps on spiralling inwards due to dynamical friction. The kinetic energy is lost to these stars, resulting in a heating of the core, but the stars do not gain enough energy as to leave the system. Eventually, after some , the distance between both the BHs shrinks to the cusp-radius of M_{1}, , where the velocity dispersion of the stars of the core equals the keplerian velocity in the potential of M_{1}(Begelman et al. 1980; Mikkola & Valtonen 1992; Vecchio et al. 1994; Quinlan 1996). Inside the kinematics of the stars and M_{2} is dominated by the potential of M_{1}, and the stars are now interacting with both BHs rather than with M_{2} only. This is the point where we start our calculations. Neglecting the influence of the other stars of the core we are now dealing with the three-body problem for a single star in the potential of both BHs. In the interaction with the black hole binary (BHB) some stars gain enough energy in order to be ejected from the central region. On the other hand the more distant stars just perturb the binary's center of mass and leave its semi-major axis unaffected and the ejection of individual stars becomes the dominant process for further hardening. Thus stars moving on orbits with radii of about the semi-major axis of the binary contribute most to the shrinking of the binary in this stage. This is also found by Hemsendorf et al. (2001) in their numerical calculations. If there are sufficient such stars the hardening continues till eventually the black holes coalesce due to the emission of gravitational radiation. Otherwise the hardening stalls if the inner region is not sufficiently refilled with matter by some other process.
Therefore, once the black holes approached each other as close as the cusp radius, the eccentricity of their orbits is likely to be very small (Milosavljevic & Merritt 2001) so that we assume them to move on circular orbits around each other.
The net force acting on a star in a galaxy is mainly determined by the large structure of the galaxy. Consequently the gravitational force of the mass distribution in the galaxy varies smoothly with space as it acts on a single star.
The number
,
how often an individual star of mass mhas to cross
a system of N identical stars so that the stars velocity v is changed
by order of itself is
The time a star needs to cross the volume of the stellar distribution is of order so that the relaxation time may be defined as . Now, for a total amount of N=10^{8} solar mass stars we obtain for the number of crossing times . If we assume the star to be of solar mass and the cluster to extend to radii of we obtain the typical velocity of a star to be . This allows to compute the relaxation time, which is .
Thus, compared to the trajectory a star would follow if the other matter would be perfectly smoothly distributed, individual stellar encounters perturb this trajectory only over of order crossing times. This means that it takes several crossing times to deflect a star from its mean trajectory and therefore it is possible to obtain some understanding of the dynamics by investigating the orbits of the stars in a suitable mean potential without taking into account individual stellar encounters. Since the time range of our simulation is smaller by a factor of than the relaxation time we can neglect individual stellar encounters in our calculations. Thus the evolution of a stellar distribution in the potential of a BHB can be modelled by solving the equations of motion of the individual stars of a cluster separately. Afterwards the results of the single stars can be combined to model the evolution of the complete cluster.
If we assume that the potential in the central region is dominated by the two black holes we can neglect the mean potential of the stellar distribution when we are solving the equations of motion for the stars. Since the cluster can be treated as a collisionless system, we can solve the equations of motion for each star separately, i.e. we are dealing with the restricted three body problem. In the following all coordinates are given relative to the center of mass, which is identified with the origin of the coordinate system. The axis of rotation is the z-axis and we always assume and correspondingly for the mass-ratio . The two black holes are moving on circular orbits around each other in a fixed distance of .
To write the equations of motion in a dimensionless form we
used the following normalization constants
The numerical integration of the equations is processed faster
in the comoving frame where the BH masses are stationary on the
x-axis and the rotation axis is pointing along the z-axis with the
z=0-plane being the equatorial plane of the BHs.
In this system the equations of motion read (see Appendix A):
= | (3) |
= | |
= |
These have been solved numerically after being supplied with a set of initial conditions for the stars which we will discuss in the next section.
In order to check the accuracy of the numerical integration, we keep track of the deviations of the Jacobian Integral (Appendix B), which is a conserved quantity in this problem.
The surface-brightness profiles of many elliptical galaxies are well fitted by an isothermal sphere out to a few core radii. On the other hand rotation curves of spiral galaxies are often remarkably flat out to great radii, and this suggests that we should construct models that deviate from the isothermal sphere only far from their cores (Mihalas & Binney 1981; Binney & Tremaine 1987.
The structure of an isothermal self-gravitating sphere of gas is identical with that of a collisionless system of stars whose density in phase-space is given by Eq. (C.4), see Appendix C. This correspondence between the gaseous and stellar-dynamical isothermal sphere originates in the velocity distribution of both systems. Integrating Eq. (C.4) over the volume yields the Maxwell-Boltzmann distribution for the velocities. As we know from kinetic gas theory this is the equilibrium distribution for a gas whose particles are allowed to collide elastically with each other. Therefore, if a system of particles is described by a distribution function according to Eq. (C.4), it makes no difference whether the particles collide with each other or not. This makes the isothermal sphere a very well suited initial distribution for our purposes. The mean and the mean-square velocity of the stars in the isothermal sphere, and , are independent of the position. Since the velocity dispersion is isotropic everywhere its components are the same: .
Interestingly Milosavljevic & Merritt (2001) find in their numerical calculations that the radial density profiles of the pre-merger galaxies and of the merged galaxy just after the formation of a hard BHB are essentially the same for a short time. For the initial density distribution they used a powerlaw with index -2.
In our model we distribute the stars according to the singular isothermal sphere (Eq. (C.9)) in the range . In order to obtain the Maxwell-Boltzmann distribution in velocity each component has been distributed according to a Gaussian. Since we are interested in stars which initially are bound by the potential of the BHB (E<0) and do not leave the system immediately after the calculations started we choose the free parameter , the velocity dispersion, to be a third of the escape velocity, .
The initial conditions for 8000 stars have been generated and supplied to Eqs. (3). These are solved with a Runge-Kutta code of fourth order based on the routine rkck by Press et al. (1995). The stepsize is selfadapting to ensure that the desired accuracy is always maintained and that the stepsize does not become too small in order to save computing time. Due to our choice of the initial conditions all stars are bound to the binary in the beginning, having negative energies. Some of them are ejected later during the run time. We define a star as being ejected if the following three conditions are fulfilled simultaneously:
In the next section we present the results we obtained for the simulation using a mass-ratio q=10 ( ). Afterwards we will compare them with the results we obtained for q=1 and q=100.
Figure 1: Initially the central region is dominated by the ejected population (EP), the outer regions by the bound population (BP). Finally, at , the bound star distribution follows a powerlaw with index -4 in the heated region ( ), while an index is maintained in the range . The inner parts are scoured out and a maximum emerges at , showing the torus-like configuration the stars assume. For a cusp of stars bound to M_{1} only is left. is normalized to and such that the area underneath the solid line is 1. | |
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In the following all quantities are given in and related to the observers frame if not indicated otherwise. The origin of the coordinate system is the center of mass of the BHB, with its rotation axis pointing along the z-axis. At both black holes lie on the x-axis, with M_{1} situated on the positive and M_{2} on the negative seminaxis. The y-axis is perpendicular to both the others so that all axes form a right handed tripod. In spherical coordinates denotes the angle to the positive z-axis, and is the azimuth in the x-y plane.
A key feature is the torque exerted by the rotating black holes on the stars, which changes the star's orbit and can eject it from the system.
During the simulation we kept track of the single stars what enables
us at all times to assign them either to the ejected population (EP),
which is constituted
by stars being ejected before the end of the simulation, or to the
bound population (BP) when they remain tied to the binary.
Both these populations combine to
give the "total population'' (TP). Quantities referring to EP, BP and TP are
denoted with the indices "ej'', "bn'' and "total'' respectively.
Figure 2: Cuts through the stellar density in the comoving frame are shown with contours logarithmically scaled. The right panel displays the equatorial plane (BHs marked by black spots). Perpendicular to it the x=0-plane is shown (left panel), with the y-axis drawn as dashed line, so that the BHs are in front and behind the paper-plane. The initial distribution is a Gaussian and the mass-ratio is 10. Time increases from top to bottom (indicated by ). After initially stars close to M_{2}'s orbit (1) are ejected (2nd row) also the polar regions are depleted and a torus in the equatorial plane finally emerges at (3rd row). | |
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The density distribution of both populations EP and BP shows that the bound stars dominate the outer regions ( , dashed-dotted line in Fig. 1). With decreasing radius the fraction of ejected stars (long-dashed line) is increasing with both populations being comparable in number density in a distance , corresponding to . At the density of the BP has a maximum and after a gap in the distribution between 1 and , where M_{2} is circling around M_{1}, the density of bound stars is increasing again towards smaller radii. Niemeyer & Biermann (1993) could reproduce the range of observed mm to infrared spectra of radio-weak quasars. In their model dust is confined to molecular clouds in a disk configuration and is heated by the central engine. According to the model by Barvainis (1987) the dust is heated to its evaporation temperature by the central UV luminosity in distance. This evaporation radius is taken as the inner radius of the torus by Lawrence (1991) and is in very good agreement with the inner edge we find for the bound stellar distribution at about (Fig. 1). Krolik & Begelman (1988) determine the inner radius of the torus from the balance between cloud evaporation by central radiation and inflow by dissipative processes in the torus and also obtain . At distances smaller than one parsec ( ) only 62 stars of the BP (a fraction of 1/80) are found at . These are bound to the primary black hole only, and till the end of the simulation the number of stars in this cusp-region is increased by just one which has been captured by M_{1}.
The evolution of the initial density distribution (a Gaussian) to its final state is shown in Fig. 2. It displays cuts through the 3-dimensional density distribution in the comoving frame of the binary. The right panel shows the equatorial plane (z=0) with the BHs marked by the black points with white frames, M_{1} to the right. The cuts in the left panel are perpendicular to the equatorial plane and contain the x = 0-plane with the y-axis indicated by the dashed line. The time proceeds from top to bottom and corresponds to 10^{4.86}, 10^{6.57} and respectively for q=10. The first row displays the distribution very close to the initial state. After (second row) we see that mainly stars close to the secondary's orbit ( ) have been ejected and a gap appears at this distance in the equatorial plane leaving a shell in the range behind. At about the density has a maximum and we can detect a shallow torus emerging at this radius. But still the polar regions are populated as dense as the equatorial plane, see Fig. 2, second row.
However, as is illustrated in the 3rd row of this figure, after
about
the central regions have been scoured out
and the hole in the distribution has a cylindrical geometry elongated
along the binaries rotation axis. In the equatorial plane finally a
torus with a radius of about
emerges, showing that the
density distribution develops from the initial sphere over a
shell-like distribution (2nd row) to the final torus. Thus
one important condition of the unification scheme is fulfilled, namely
that the stars do assume a torus-like distribution at a distance which
is in agreement with the origin of thermal infrared emission by dust
(Barvainis 1987; Krolik & Begelman 1988; Haas et al. 2000). Whether this torus also satisfies
the other requirements, like optical thickness, will be investigated in
our Paper II.
Figure 3: The component sin of the normalized torque of the binary, which acts on a star moving on a fixed circular orbit with radius is displayed as function of time. The angle enclosed by the rotation-axis of the binary and the orbital spin-axis of the star is denoted by and is varied in the range from to . Perturbations are stronger for orbits through the polar regions; these stronger perturbations lead to secular changes in the orbit, finally leading to ejection for about half of the stars in the polar cap (see Fig. 2). | |
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The basic topology of the final density distribution can be understood in physical terms, where we consider the torque exerted by the two black holes on an orbiting star. Figure 3 shows of the normalized torque the component sin relative to the z-axis of the BHB as a function of time for different angles between the rotation axis of the BHB and the symmetry axis of the star's orbit. The bigger the angle is (i.e. for orbits through the polar regions), the more is the star's trajectory disturbed by the influence of the two BHs. The curves are symmetric since for simplicity it has been assumed that the star is moving on a keplerian circle of radius R around a point mass which corresponds to that of the binary and is located at the origin. Of course real trajectories are disturbed by the torque and the curves will not be symmetric any more. The cumulative effect of these large excursions in the polarcap regions deplete the stellar population leaving a torus behind.
Because of the conservation of the Jacobian Integral
(Eq. (B.4)) the stars which gain energy also gain angular
momentum in the z-component and thus help to enhance the density in
the equatorial plane.
In Paper II we will discuss the distribution of the BP and
investigate in the properties of the torus.
Figure 4: The angular momentum clearly separates the BP (black dots) from the EP (grey dots), which initially have less angular momentum. The line of transition from the EP to the BP is drawn by eye. Its almost constant value is decreasing with increasing mass-ratio q. Along this line both populations diffuse into each other, showing that the transition is smooth. | |
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At the finally ejected stars can be clearly distinguished from the bound population by their total angular momentum or pericenter . The mean value of the angular momentum of the EP is smaller by a factor of 3.22 compared to the BP (see Table 1). This separation is clearly illustrated in Fig. 4, where we have plotted the initial values of angular momentum of the stars versus their radius. The ejected stars, marked by grey dots, have low angular momenta, while stars staying bound to the binary till the end of the simulation (black dots) exceed a certain value. Both populations can be separated from each other by a transition line, as seen in Fig. 4. The angular momentum has been normalized to the value of a star which moves on circular orbits in a distance corresponding to the separation of the BHs ( ) around the combined mass of the BHs M_{1} + M_{1} which is situated at the center ( ). Along this transition-line, which is drawn by eye and approaches a value of about 2 in these dimensionless units, the different populations diffuse into each other so that the transition from one population to the other is smooth.
This value is double the angular momentum of a star moving on a circular
orbit with the radius corresponding to the separation of the
binary. Or, the other way round, this transition-value of corresponds to an orbit with a radius four times as big as the
semi-major axis of the binary.
Figure 5: Ejected stars (grey dots) cover a region with and , marked by the horizontal and vertical lines. This allows them to closely approach the orbit of M_{2}. Bound stars (black dots) avoid this unstable region. One unit corresponds to . | |
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However, due to their much lower angular momenta the finally ejected stars come much closer to the black holes. For the mean pericenters we get and showing that they differ by more than a factor of 10. In Fig. 5 we have plotted the initial apocenters on the x-axis versus the initial pericenters on the y-axis. Without loss of information and to keep the figure distinct, we plotted only 1/4 of each population, chosen by random. While the pericenters of the ejected stars stay below (i.e. , horizontal line), their apocenters exceed ( , vertical line). Hence they cover a region which always allows them to come very close to the orbit of the secondary BH, which rotates at a distance of around the center of mass. Here the stars undergo violent interactions with the binary and eventually gain sufficient energy ( ) to be kicked out. On the other hand the bound stars avoid this region and stay at distances large enough to avoid such violent interactions with the binary. Only very few stars, less than from the BP, have apocenters less than . These are bound to the primary black hole M_{1} only and are not ejected because the influence of the secondary BH is too small. They form the small density peak around M_{1} which is seen in Fig. 1.
For the same apocenter the ejected stars have on average smaller
pericenters than the bound stars and so are moving on orbits with
higher eccentricities. As a consequence the radial velocity component
should exceed the tangential component, leading to a radially
anisotropic velocity. But due to our choice of initial conditions the
velocity anisotropy
Figure 6: The BP is shown to be strongly tangential anisotropic while the EP is radially anisotropic. As the eccentricity the velocity anisotropy increases with for all populations but for the TP which is initially set to zero at all distances. At the BP is radially anisotropic in the heated region . In the cusp region no clear tendency is detectable. | |
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The eccentricity of both populations is increasing with distance from the binary. Close to the center the bound stars have to move on almost circular orbits in order not to come closer at the pericenter and to avoid strong interactions with the black holes. With increasing distance also the eccentricity can increase as long as a big enough pericenter ( ) is maintained i.e. not smaller than twice the separation of the BHs. The EP must have a pericenter below this value so that its stars can interact violently enough with the binary at the point of the closest approch to the black holes in order to become ejected. Thus with increasing apocenters also their eccentricity increases, where as . Consequently these stars have low angular momenta and their orbits become the more radially anisotropic the bigger their distance to the center is. Hence their kinetic energy is stored in radial rather than in tangential motion.
As time proceeds there are almost no changes in the mean value of the total angular momentum for both populations, BP and EP. With the angular momentum being normalized to L_{0} = m a^{2} /t_{0}, where is the mass of a star, the mean values of are -0.045 and 0.063 for the BP and EP respectively. While on average the bound stars are slightly counterrotating, the ejected stars are corotating so that the total population shows no net rotation in the beginning. This matches the results obtained by Innanen (1979) and Innanen (1980). The mean angular momentum does not change for the bound stars as time proceeds, but it increases by a factor of about 4.54 for the ejected population (see Table 1). Such a growth is expected, because of the conservation of the Jacobian Integral (Eq. (B.4)). All ejected stars gain energy and consequently their angular momentum in the z-component has to be increased, at the expense of the BHB. Hence the ejection of stars leads to a net loss of angular momentum and thus to hardening of the binary.
In order to get some quantitative information about the hardening we can use the statistical values tabulated in Table 1. They allow to compute the mean angular momentum that is extracted from the binary by a single star and thus also the number of stars which is required to carry away all the binaries angular momentum so that the black holes coalesce. But, since we made certain assumptions and neglected some effects, these numbers will give just an order of magnitude estimate. On the one hand the feedback of the stars on the binary has not been taken into account because we fixed the distance of the black holes throughout all calculations. As a consequence the "cross-section'' for the stars to be ejected is not shrinking with increasing time, when ejected stars are extracting angular momentum from the binary. Thus the fraction of ejected stars, obtained from the simulation, is bigger and the initial total number of stars needed to allow for merging will be a lower limit. For the same reasons also the time-scale of the merger of the BHs, which we will estimate in this section, will be shorter than in reality. On the other hand we neglected star-star interactions, which would support the loss-cone feeding and therefore increase the fraction of ejected stars. Also we assumed the black holes to move on circular orbits. If the orbits were eccentric, gravitational radiation would become important earlier at a bigger semi-major axis resulting in a faster merger and less stars needed in the cluster. Both the latter effects counteract neglection of the shrinking of the black holes' distance, but as shown previously they are of minor importance.
However, the angular momentum of the BHB is
= | |||
= | (5) |
Figure 7: The loss-rate of the binary's angular momentum is largest in the beginning. For the loss-rate approximates a powerlaw with index , shown by the dotted line (fit 2). The dashed curve (fit 1) is used to calculate the number of ejected stars. One time unit corresponds to . | |
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To get an estimate of the timescales on which the black holes merge and which serve as a lower limit as stated previously, we have to know about the loss-rate of the angular momentum of the binary. This is displayed in Fig. 7 as function of time. The data are normalized to the number of simulated stars so that the curve represents the mean rate at which a total of one solar mass extracts angular momentum from the binary when ejected gradually till the end of the simulation. This mass will be referred to in the following as a "representative star''. A simple fit to the data enables us to estimate the time needed for the black holes to merge and also to calculate the number of stars required so that the black holes can coalesce. This is then compared with the number obtained above by statistical means and should be the same if the fit is sufficiently accurate.
After a delay of
the loss-rate is increasing
steeply, passing through a maximum at
(corresponding to
)
and then
quickly approaches a powerlaw with index
.
This initial
delay is caused by two reasons: first the stars have to interact with
the binary before they gain enough energy to be kicked out since all
stars are bound at the beginning. Afterwards the star, having now a
positive energy, has to move to a distance of
in order to
be registered as ejected by the code.
The exponential decrease of the loss-rate for
(
)
shows that the system evolves faster
in the early stages and the
angular momentum varies very slowly at later times. If the
black holes get sufficiently close due to the ejection of stars, gravitational
radiation will become strong enough to complete the merger.
For
the curve is not very smooth because of the
small number statistics. As we will see later this corresponds to the
time required by the BHs in order to merge,
.
The lossrate can be approximated by a powerlaw with index
,
multiplied with a function which cuts it off
at
:
Comparing both the loss-rates of angular momentum due to gravitational
radiation and ejection of stars allows us to compute the time and
distance where gravitational radiation eventually becomes the dominant
physical process for the
merging of the black holes. As before we assume the number of stars to be
sufficient so that all angular momentum of the binary is lost as
time tends to infinity. Hence the angular momentum lost by the binary
as a function of time is
= | |||
= | (16) |
This rate has now to be compared with the shrinking rate caused by the
emission of gravitational radiation.
Using
and
,
the total
and reduced mass respectively, the energy of the binary reads
(22) |
Figure 8: The shrinking rates due to ejection of stars (solid line) and emission of gravitational radiation (dashed line) are shown to have very different slopes. Therefore the region of the transition is quite narrow. The separation is normalized to . | |
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If
in Eq. (23) is equal to 1,
i.e. the separation where we started the simulation and where the BHs are
apart from each other, the time needed to reduce
by some large fraction because of emission of gravitational
radiation only is of order of
.
Thus in the
beginning, at distances as large as
,
the
hardening of the binary caused by emission of gravitational radiation is
completely negligible and proceeds on time-scales orders of magnitudes
higher
than shrinking due to ejection of stars. To compare these two
rates of hardening we have to rewrite Eq. (18) in its
dependence on distance
instead of
.
Unfortunately, because of
its complicated form, we can not build the inverse function of
.
However, to proceed we first have to further
simplify our fit function. This is done by resetting the parameter
to zero, so that the cut-off funtion in
Eq. (9) vanishes and we are left with
a pure power law only. In order not to have too large differences
between this fit and the data for small times and to take account of
the initial increase we shift the parameter
from
20 to 50. Hence the system function (10) becomes
Only the fraction
of the initial angular momentum of the
binary is left at this distance. With
for the powerlaw
index the relation between the distance and time of the transition is
,
and
Eq. (28) simplifies to
The energy of the binary scales with its separation as . If the distances of the stars in the cluster are increased or decreased by the same factor as the binary's separation, the energy of a star also scales with this factor to the power of -1, i.e. and . Thus the basic results will be unchanged as long as the binary remains "hard'', but the time-scales of the merging due to ejection of stars scale as .
The final merger of two supermassive BHs should lead to one of the most energetic events in the universe. As a result of such a merger, the spin and its direction are expected to change (Wilson & Colbert 1995). Consequently also the direction of the jet, which is aligned with the BH's spin, will jump into the new direction of the spin of the merged BH. This is a possible explanation for the observed X-shaped radio galaxies. This class of objects exhibits four jets, one pair of which shows "young'' synchrotron emission and the other one "old'' emission, such as seen in B2 0828+23 (Parma et al. 1985; Rottmann et al. 1998). The spectral aging of the secondary lobes in this object suggests that a merger has happened about ago (priv. communication with H. Rottmann). This timescale is very close to the one we have found here for the merger of two supermassive BHs and thus supports the idea of mergers inducing a jump of the jet.
As we have shown in Sect. 3.1,
both populations, the ejected and bound stars, can be clearly
distinguished by their angular momentum or their pericenter.
To learn more about their angle distribution is the task of
this section. For this purpose we introduce some angles, as is
illustrated in Fig. 9.
We define
as the angle between
the positive z-axis (the rotation axis
)
and
the angular momentum
of a
star. The angle between
and the negative
z-axis is denoted by
,
so that
.
To the angle between the plane in which the star
rotates and the equatorial plane we refer as
.
Figure 9: This sketch illustrates the meaning of the different angles. The equatorial plane is defined by the black holes rotating around each other, with the angular frequency vector pointing along the z-axis. M_{1} is displayed a little off-center since the coordinate system is centered in the center of mass frame. Tilted to this plane with an angle is the orbital plane of a star, whose angular momentum encloses the angles with (the positive z-axis) and with the negative z-axis. The line from the center to the current position of the star has the angle with . | |
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The angle which is enclosed by the line connecting the origin with the current position of the star and the z-axis is denoted by . The mean and the dispersion of this angle are very similar at for all three populations (EP, BP, TP), namely and . These are just the values we expect for a spherically symmetric distribution, as the total population at . Thus both populations, EP and BP, are also spherically symmetric distributed in the beginning and none of them shows any preferences as to certain polar angles. However, from Table 1 we see that the mean angular momentum of the ejected stars increased by just a factor of 1.1, while is by more than a factor 4.5 bigger at the end of the simulation. Thus we expect the geometry of the distribution of the ejected population to be flattened and not spherically symmetric at the end of the simulation.
In Fig. 10 we plotted the density of the stars in
dependence of
the cosine of
.
As stated above we see within the
fluctuation of the data a uniform
distribution for
(dashed line), according to a spherical
symmetry. But after the stars have been ejected the density has a
maximum at
(solid line). Thus they
are concentrated towards the equatorial plane of the binary.
The same conclusion is drawn from Fig. 11.
It displays the density as a function
of the orientation of the orbital angular momentum of the stars,
.
At
(dashed line) the density is almost constant in
,
thus there
is no special orientation prefered for the orbital plane of the
stars. The density is slightly increasing with
,
showing that the ejected stars initially are weakly corotating on
average.
Figure 10: The density of the ejected stars at (spherically symmetric distributed) and after their ejection ( ) as function of their polar angle is displayed. After being ejected the cluster is more concentrated to the equatorial plane . | |
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Figure 11: The orientation of the angular momenta of the EP at and after the ejection at is displayed. They show a strong concentration towards large positive values, so that the orbital plane is close to the equatorial plane. | |
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The stars which gain the most energy also gain the most angular momentum (see Eq. (B.4)). In Fig. 12 we plot the energy versus after the ejection of stars. The figure shows a strong correlation between the energy and , confirming the prediction of Eq. (B.4): the orbital planes of the stars with the highest (kinetic) energies are more concentrated towards the equatorial plane of the binary. A similar correlation is found when the eccentricity is plotted versus the orientation of the orbits, showing that the most eccentric trajectories ( ) are most aligned with the plane in which the black holes rotate.
The strong changes in and of the ejected stars compared to the almost constant values for the BP show that the BHB is mainly influenced by the ejected stars. Provided that initially there are enough stars, the EP can extract sufficient angular momentum to allow the black holes to merge, as shown in Sect. 3.2.
Finally we want to give in this section a crude estimate of the
stability of the remaining torus-like distribution of the bound stars,
once the BHs have merged. In a first approximation we can use the
formulae from Sect. 2.1 for collisionless systems.
With the following values,
,
,
and the velocity set to the Keplerian in
a distance of
,
,
we obtain for the
ralaxation time
Figure 12: The energy at of the EP is plotted versus the orientation of the angular momentum. The more energetic the stars are, the more they are concentrated to the equatorial plane. | |
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This time scale can now be compared to the time
which a
star needs to spiral from the outer edge to the inner edge of the
torus due to dynamical friction. In fact, the diffusion coefficient is
identical to the deceleration of a test star owing to dynamical friction
by the surrounding field stars (Binney & Tremaine 1987),
We chose for the
parameters
,
and
so
that the volume integration from
to infinity gives the right
number of bound stars. Substituted in the expression for
and using for ,
and
the same
values as above we obtain for the time a star needs to spiral from
distance to the inner edge of the torus at
Figure 13: At the end of the simulation the more stars are ejected the bigger M_{2} is. While for q=1 there is almost no cusp left in the center, the region close to the orbit of M_{2} at is not much depleted for q=100. In the range all density distributions follow the initial powerlaw with index -2 while at bigger distances independent of q. The initial distribution is for all mass-ratios the same and represented by the solid line. | |
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Finally we will discuss the influence of the mass-ratio of the black holes on the ejected and bound star populations. In altering the mass-ratio q = M_{1}/M_{2}we always keep the mass of the primary black hole fixed to . For the mass of the secondary we chose M_{2} = 10^{8} and 10^{6} solar masses corresponding to the ratios q=1, 100respectively. Hence also the total mass of the binary is changed.
As M_{2} increases and q decreases the sphere of influence of
the secondary BH expands and consequently the inner regions become
more unstable. Thus a bigger fraction of stars is ejected and the
density cusp bound to the primary BH only becomes less dense, being made
up by
and
for q=1 and 100 respectively.
Between
and
the
density is not much affected
by variations in q and always follows a powerlaw with
index -2 (see Fig. 13). In the heated region
(
)
too the
density does not depend on the mass ratio and the relation
holds for all q.
Due to the enhanced instability of the central region for a more
massive secondary black hole the bound stars have to be more
tangentially anisotropic in order to diminish the eccentricity of their
orbits and not to come too close to the binary. This causes the inner
edge of the torus at
to be much sharper defined and thus
the maximum of the density at
to be
much more pronounced for
q=1, while it is almost smeared out for q=100(Fig. 13).
This is clearly visible in Fig. 14, where the density
distribution is displayed at
t=10^{7.58} and
and corresponds within a factor of a few to the lower limits of the
merging times
t=10^{6.89} and
of the
BHs for q=1 and 100 respectively. In fact, the bottom row for
q=100 shows no torus,
but the density is diminished around the orbit of M_{2} and weakly
indicates the formation of a shell-like distribution. This is
confirmed if we let the simulation continue till
,
about 10 times longer. Then the density distribution is
comparable to that of q=10 after
(2nd row in
Fig. 2).
Figure 14: Same as Fig. 2 for the final density distributions of the mass-ratios q=1 (top row) and 100 (bottom row). For q=1there is no central cusp left and a pronounced torus with sharp defined edges emerges after . There is no torus seen in case of q=100 after , only close to the orbit of the secondary BH the sensity is diminished. | |
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Thus the central stellar cluster in the potential of two merging BHs does not evolve into a torus-like structure for a sufficiently large mass-ratio q and rather stalls in the phase when it assumes a shell-like structure.
On the other hand we can see that in the case of q=1 the final torus ( ) is much more pronounced than for q=10, compare Figs. 2 and 14. This means that for sufficiently small mass-ratios q a stellar torus forms in the late stages of a merger, while for young mergers and mergers with a big mass-ratio a shell-like density distribution is maintained. For q=1 in Fig. 14 the inner edge of the torus is more sharply defined, and there is actually no cusp left in the center. Consequently the bound stars must have bigger angular momenta as M_{2} increases. If we compare the initial angular momentum of the ejected stars for the mass-ratios q=1 and 100 with the ratio q=10, it turns out that it increases with the mass of the secondary BH (decreasing with q). The relation also holds at the end of the simulation as well as for the EP. Plotting the initial angular momentum versus the radius for each star for q=1 and 100, just in the same way as for q=10 in Fig. 4, gives a very similar separation between the EP and BP. But with increasing q the transition value of , dividing both populations, decreases from a little more than 2 for q=1 to a value below 2 for q=100 with always the same initial conditions. As the value of transition of in Fig. 4 is decreasing with M_{2}, also the mean values of the angular momentum of both populations are decreasing. Stars which have been ejected for small mass-ratios q``defect'' to the BP as q increases and the inner region becomes more stable. Therefore the deviation of the BP increases while it is decreasing for the ejected population (see Table 1).
The smaller the mass-ratio is, the more angular momentum is extracted
from the binary by a single ejected star on average
(
,
see
Table 1). But this is not
enough to compensate for the bigger amount of angular momentum of the
binary due to the more massive secondary BH,
.
Thus the cluster has to be more massive
so that the binary can transfer its angular momentum to a larger
number of ejected stars in order to merge.
To compute the required number of stars and the time
needed to enable the merging of the black holes we proceed in the same way
as before in Sect. 3.2 for q=10.
With decreasing mass-ratio the maximum of the angular momentum
lossrate is more peaked and has a value
times higher for q=1compared to q=10 (see Fig. 15). The maximum is
reached the sooner, the smaller q is (
corresponding to
and
for
respectively). Also the powerlaw with index
-3/2 is approximated earlier by the loss-rate for small mass-ratios
and therefore the system is relaxing faster.
ejected stars | bound stars | ||||||||||
q | |||||||||||
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) |
0 | 0.086 | 1.165 | 0.532 | -0.064 | 3.773 | 1.432 | |||||
1 | 0.422 | 1.334 | 0.481 | -0.063 | 3.780 | 1.429 | 0.667 | 10^{8.58} | 10^{6.79} | 1/117 | |
0 | 0.063 | 1.149 | 0.507 | -0.045 | 3.701 | 1.471 | |||||
10 | 0.284 | 1.264 | 0.497 | -0.043 | 3.711 | 1.467 | 0.614 | 10^{7.97} | 10^{7.15} | 1/197 | |
0 | 0.086 | 1.005 | 0.411 | -0.039 | 3.411 | 1.562 | |||||
100 | 0.239 | 1.072 | 0.435 | -0.037 | 3.418 | 1.561 | 0.395 | 10^{7.35} | 10^{7.67} | 1/265 |
The number of stars and the times required for the merging are
listed in Table 1. The values we computed for the case
q=100 are not as accurate as for the other mass-ratios, since the
data of the angular momentum loss-rate do not allow for a fit as good
as for q=1 and 10.
However, the values show that for a binary with a fixed
mass for the primary BH more stars in the cluster are required
for smaller mass-ratios in order to allow the black holes
to coalesce. On the other hand the binary with the smaller mass-ratio
is merging faster, provided that the cluster is sufficiently
massive. For instance if q=1, the merger is 2.4 times
faster than the one where q=10, while a factor of 3.5 times more
stars in total are required.
For a binary with smaller q the distance
,
where
gravitational radiation starts to dominate the merging process,
is bigger and thus reached after a shorter time due
to the larger mass of M_{2}(
and
for
respectively, see
Table 1).
Figure 15: The smaller the mass-ratio q is, the more peaked is the maximum and the earlier the maximum is reached. Thus the system is relaxing faster. For all q the loss-rates finally approximate a powerlaw with index -3/2. For q=100 the curve is not as smooth due to the smaller number statistics. | |
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Assuming the average stellar mass to be that of the sun, these numbers show that the ejected mass is of order of the total mass of the binary . While is decreasing more strongly with increasing q than the binary mass, the ejected mass always exceeds that of the secondary BH. The ratio of both masses, is increasing with q. Therefore a primary BH with mass M_{1}ejects more mass if it merges with N black holes of mass M_{1}/N than merging with another BH of mass M_{1}. This is the same result which has been obtained by Quinlan (1996) in scattering experiments.
One basic problem in our understanding of the central activity of galaxies is addressed in this paper: do the black holes in the center of two merging galaxies also merge? Based on the two assumptions that galaxies contain supermassive black holes in their centers and that galaxies merge we simulated a stellar cluster in the potential of a BHB. When the distance between the black holes has shrunk to and ejection of stars dominates the merging, we start our calculations. Since the simulated time is much smaller than the relaxation time of the cluster and the potential is dominated by the mass of the two black holes we were dealing with the restricted three body problem. The results we obtained by solving these equations for the stars of the cluster, initially distributed according to the singular isothermal sphere, just give an order of magnitude estimate since in the calculations we neglected further shrinking of the separation of the BHs due to ejection of stars.
We find that the angular momentum divides the stellar cluster into a bound and ejected population. Stars staying bound in the potential of the BHs are distributed in a torus-like shape. Its relaxation time exceeds , so that this structure can be regarded as stable. It can explain lots of the features which are postulated by the unification model for a dusty torus in AGN and will be the topic of Paper II.
Stars which belong to the ejected population have low angular momentum and are moving on highly eccentric orbits, being radially anisotropic in velocity. The distribution of their peri- and apocenter always allows them to come very close to the binary. Here they gain sufficient energy in violent interactions with the black holes to become ejected. The ejected stars gain angular momentum in their z-component at the expense of the binary, which can be understood in terms of the conservation of the Jacobian Integral ( ). Thus the binary continues to shrink. Integrating its loss-rate allows to calculate the number of ejected stars and the time required for the black holes to coalesce.
Starting with distance between the BHs, we find that a stellar cluster with a mass of order of the binary is required to allow the BHs to coalesce. The merging proceeds on timescales of order of and will change the spin of the central BH (Wilson & Colbert 1995). Consequently also the jet, emanating from the center, will jump and flow along the new spin direction after the merger is completed. This might be an explanation for the observed X-shaped radio galaxies, where a spectral aging time of the the secondary lobes suggests that a merger has happened ago (priv. comm. H. Rottmann). The agreement with our merger timescale is in favour of this idea.
Most of the time (factor 4) this merging process is dominated by ejection of stars before emission of gravitational radiation takes over. Thus a stellar cluster with a total mass comparable to that of the binary is needed to allow the black holes to merge due to the ejection of a fraction of of all stars. After the ejection the stars are strongly concentrated to the equatorial plane of the binary, the more the higher their energy is, and the initial spherical geometry of the EP has been transformed into a cylindrical geometry.
As the mass of the secondary black hole is increased, the central region becomes more unstable and a bigger fraction of stars is ejected. In order to extract sufficient angular momentum from the binary, more stars are required, so that the clusters mass still is of the order of magnitude of the binary's mass. The system is relaxing faster and the distance where gravitational radiation dominates the merging process is reached earlier, so that the binary merges on smaller time scales ( and for and 100 respectively). The ratio is increasing with q and therefore a primary black hole with mass M_{1} ejects more mass in N minor mergers with secondary black holes of mass M_{2}=M_{1}/N than in one major merger with mass M_{2}=M_{1}.
In major mergers the mass distribution is likely to be completely rearranged and the resulting galaxy will probably have a rather elliptical than spiral shape. In their overview of the structure of nearby AGN (Malkan et al. 1998) find that Seyfert 1 nuclei reside in more early-type host galaxies than Seyfert 2. This is in agreement with our finding, that the more massive the secondary BH is, the more stars from the cluster are ejected. Thus the remaining torus-like distribution becomes shallower and there is a higher probability to see the type 1 nucleus directly. Combinig this with the conclusions by (Wilson & Colbert 1995), that radio loud objects are possibly the product of major mergers, it is expected that radio loud galaxies are more likely to host type 1 nuclei.
The key result of this paper is: provided that a black hole binary is surrounded by a cluster of solar mass stars with , i.e. the total mass of the cluster is comparable to that of the binary, the black holes merge on time scales of about . In Paper II we will demonstrate that the stellar torus surrounding the merged binary also meets the other requirements of the unification scheme. We will demonstrate that the stellar winds are pushed by radiation from the central source into long tails, and that the ensemble of many stellar wind-tails can constitute the observed torus.
Acknowledgements
C. Z. wishes to thank Matt Malkan, Ski Antonucci, Graeme Smith, David Merritt and Stefan Westerhoff for their helpful discussions and their generous and kind hospitality. We are grateful to our referee Edward J. M. Colbert for his helpful advice and comments.
In the restricted three body system two massive bodies (BHs)
are circulating around each other. A third particle
(a star), whose mass is negligible
(
), is moving in the time dependent gravitational
potential of the massive bodies, which reads in the dimensionless form
(Eq. (3)):
.
The star is located at
while
and
denote the positions of M_{1} and M_{2} respectively.
The origin is identified with the center of mass of the two massive
bodies and their axis of rotation is aligned with the z-axis and at
t=0 with both black holes situated on the x-axis.
The equations of motion for the star are
(A.3) | |||
For the restricted three body problem the energy is not conserved
because the potential is time-dependent and
consequently no integral of motion. The same holds for the angular momentum,
which is not conserved since the potential is not invariant to
rotations. But an integral of motion can still be found
if we multiply the equation of motion for a star (A.3) with
,
= | |||
= | (B.1) | ||
(B.2) |
= | |||
= |
In a spherically symmetric potential there exist four isolating integrals,
the energy E and the three components of the angular momentum
.
According to the Jeans Theorem applied to spherical
systems any non-negative function of these integrals can serve as the
distribution function of a spherical stellar system. The extension of
the Strong Jeans Theorem to spherical systems by
Lynden-Bell (1962) allows to conclude that the distribution function of
any steady state spherical system can be
expressed as function a
.
If the system is spherically
symmetric in all its properties, f is independent on the direction
of
so that
f = f(E, L).
If now the potential
is provided by the stellar system itself
and we understand f as the mass distributionfunction (i.e.
)
the Poisson equation
(C.1) |
Following the notation of Binney & Tremaine (1987) we
define the relative potential and the relative energy by
and
.
The constant
is choosen in a way that f > 0 for
and f = 0 for
.
As
,
the
relative potential
also satisfies the Poisson equation
with the boundary condition
for
.
A distribution function depending on the
relative energy
only further simplifies the spherical model. In such
systems with
the velocity dispersion
tensor is isotropic everywhere because the velocity dispersions are
the same in all spherical components
and .
Using
this distribution function and if we choose
in
a way that
for
,
Eq. (C.2) becomes
In order to find now a solution of Eq. (C.6) we
simply assume a powerlaw for the density (
)
and
substitute it in Eq. (C.6) and find
.
Thus the two parameters must have the values b=2 and
to fulfil this equation and we obtain for the
density