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Astron. Astrophys. 327, 930-946 (1997)

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4. The central galaxy

4.1. Definition of the central galaxy

We first need to define the central galaxy in an unambiguous way. The information about the particles in each galaxy is stored in a sequential way. Thus the first 900 particles always correspond to the particles initially bound to the first galaxy, the second 900 particles to those initially bound to the second galaxy and so on. The process used to decide which particles are bound to the central galaxy at a given time is as follows: For each galaxy, we take the initially bound 900 particles and calculate the binding energy of each particle with respect to this subsystem. We discard all particles with positive energy and we repeat the process until we get a stable number of particles. Usually only two iterations are needed. We thus define the secondary galaxies at a given time step of the simulation. The discarded particles are not immediately incorporated into the background. As a second step we consider the possibility of some mass transfer between galaxies. We check the possibility that some of the particles that have escaped their initial parent galaxy are now bound to another one of the galaxies. For each galaxy we calculate the energies of all the particles inside a sphere of radius [FORMULA] centered on the galaxy. All the particles which are not part of another galaxy and which have negative energies are ascribed to this galaxy. If after these two steps we find a galaxy with less than [FORMULA] of the number of points it had initially, we do not consider it as a single entity and its particles are assigned to the background. After these two steps we are left with two sets of particles. The first set consists of the particles bound to some of the galaxies and the second is the set of particles not bound to any galaxy. We consider this second set of particles separately, and, in order to distinguish which, amongst these particles, constitute the central object and which the diffuse background, we calculate the energy of these particles relative to this subsystem. Only particles with negative binding energy are considered as the particles forming the central object.

Finally we also take into account the possibility of merging between the satellite galaxies and the central galaxy, or between two galaxies. We check if a galaxy of core radius [FORMULA] and central velocity dispersion [FORMULA] has merged with a second galaxy with parameters [FORMULA] and [FORMULA] placed at a distance [FORMULA] and moving at a relative velocity [FORMULA] to the first galaxy. If both conditions

[EQUATION]

are satisfied simultaneously, the two galaxies are merged and all the particles of the smaller galaxy are ascribed to the bigger one, conserving at the same time the momentum of the binary system. After some tests the parameters A and B were given the values 1.4 and 0.6 respectively.

The core radius of a given galaxy is defined as follows: We first sort the particles in a galaxy as a function of their binding energy and consider the [FORMULA] which are most bound. The core radius is then defined as the smallest radius containing these particles. The value of [FORMULA] has been obtained by imposing that the core radius thus obtained for the initial galaxies coincides with their initial core radius, i.e. 0.2. With these particles we calculate the mean position and velocity of the galaxy. Its central velocity dispersion is defined as the dispersion of the particles within the core radius. For the case of the central object, we used only the [FORMULA] most bound particles, instead of [FORMULA] as in the satellite galaxies, in order to avoid an excessive number of mergings between the satellite galaxies and the central object.

4.2. Three dimensional properties

The central object is divided into three concentric shells as follows: We rank particles in order of decreasing binding energy. We neglect particles in the top 5th percentile, i.e. the most bound particles, because they could be affected by the softening length of our simulations. In the first shell we include particles with binding energy between the 5th and 30th percentiles; in the second shell particles with binding energy between the 30th and 60th percentiles and in the last shell particles with binding energy between the 60th and 90th percentiles. We exclude the last 10% of particles because these could be recently accreted material, and thus not in equilibrium with the rest of the galaxy. Using only three shells we have enough particles in each one. In this we follow the method used by Barnes (1992) in his study of the merger remnants of the collision between two spiral galaxies, with the difference that he used only the [FORMULA] most bound particles, while we include in our analysis the [FORMULA] most bound particles, since we are interested mainly in the external parts.

4.2.1. Shape and orientation

Porter, Schneider & Hoessel (1991) showed from isophotometry of 175 brightest cluster ellipticals that in most cases their ellipticities increase with increasing radius. Ryden, Lauer & Postman (1993) derived mean isophotal axis ratios for 119 brightest cluster ellipticals in Abell clusters and found from best fitting models that their most probable shape parameters are [FORMULA] and [FORMULA]. Furthermore an increasing wealth of evidence shows that the orientation of the brightest cluster ellipticals is not random, but correlates well with that of the cluster in which they are found (Sastry 1968, Rood & Sastry 1972, Austin and Peach 1974, Carter & Metcalfe 1980, Bingelli 1982, Struble & Peebles 1985, Rhee & Katgert 1987, Lambas et al. 1988)

In order to compare the orientation and the shape of our simulations with that of brightest cluster members we calculate for each of the three shells discussed in the beginning of this section the normalised inertia tensor defined as:

[EQUATION]

This normalised tensor gives similar results as the non-normalised one, while avoiding some problems with the more distant particles (Barnes 1992). On the other hand, this tensor could have problems with the particles that are near the center, but these particles are among the [FORMULA] most bound particles that are discarded from our analysis.

This tensor can be diagonalised to obtain the eigenvalues [FORMULA] and their associated eigenvectors. The axial ratios of each shell are defined as follows:

[EQUATION]

We do this for each shell and for each snapshot. In this way we are able to study the shape of the different shells, their relative alignment and their time evolution.

In Fig. 8 we show the results of the time evolution of the axial ratios for the three concentric shells of the central objects obtained in the simulation of collapsing systems. In the left panels we show the [FORMULA] ratios and in the right panels the [FORMULA] ratios. All objects are oblate or mildly triaxial, the triaxiality being most evident in the external shell, in good agreement with the observational results of Mackie, Visvanathan & Carter (1990) and Porter, Schneider & Hoessel (1991). The most triaxial system is obtained in the case of the oblate initial conditions of run Co. In this case there is also a clear difference between the ellipticity of the three shells, the inner shell being the roundest. There are some differences in the evolution of the shape parameters for the central objects formed in runs C1 and C2. The inner parts of the object of run C1 get rounder with time, while this is not the case for the object formed in run C2. This might be explained by the different merging histories of the two simulations.

[FIGURE] Fig. 8. Time evolution of the axial ratios of the central galaxy in our collapsing simulations. The solid line indicates the evolution of these ratios for the shell containing the particles with binding energy between the 5th and 30th percentiles. The dot-dashed line corresponds to the intermediate shell containing particles with binding energies between the top 30th and 60th percentiles, and the dotted line corresponds to the outermost shell, containing particles with binding energies between the 60th and 90th percentile.

In Fig. 9 we can see the time evolution of the axial ratios of the central objects formed in the initially virialised systems. Note that these spherically virialised systems form rounder objects than those formed by spherically collapsing simulations. In the latter cases, the galaxies in the collapse follow mainly radial orbits and enter the central galaxy in some particular direction thus affecting the shape of the central object. On the other hand, in the virialised systems an important fraction of the mass of the central galaxy comes from material stripped from the secondary galaxies. This material accumulates onto the central object at a slower rate and, coming from any direction, gives rise to these rounder objects. The differences between runs Vc1 and Vc2, which are different realisations of the same initial conditions, are rather small.

[FIGURE] Fig. 9. Same as Fig. 8 but for the virialised simulations. In this case all the resulting objects are nearly spherical. Lines as in Fig. 8.

Observations indicate that the brightest cluster galaxies seem to be more triaxial than what our simulations show, and that their projected axial ratios have mean values lower than the values obtained in our simulations (Ryden et al. 1993). This may indicate that the initial conditions for the formation of clusters of galaxies were far from spherical and/or very anisotropic.

Does the orientation of the central object reflect the orientation of the cluster from which it initially formed? Two of our simulations, Cp and Co, have non-spherical initial conditions. As we already saw they form non-spherical central objects. Fig. 10 shows the angle between the minor axis of the initial configuration and that of each shell of the central object of run Co as a function of time. We see that for all three bins there is a very good alignment of the central object with the initial cluster. The central object formed in run Cp is much more spherical. For that reason, in Fig. 11 we have plotted the angle between the major axis of the central object and the major axis of the initial configuration, as well as the angle between the median axis of the central object and the major axis of the initial configuration. Again the three shells of the central object are treated separately. The two major axes coincide well at all times for the outer shell. They correspond well most of the time for the median shell, and for some times for the innermost one. For these two shells and for the times when the two major axes do not coincide, it is the intermediate axis that corresponds to the direction of the initial major axis, especially in the inner, more spherical regions. Similar results have been found by Rhee & Roos (1990) in their simulations of collapsing small clumps of galaxies, although they find that the orientation is better preserved in initially prolate, rather than initially oblate systems.

[FIGURE] Fig. 10. Time evolution of the angle between the minor axis of the initial configuration and that of the central galaxy for Run Co. In the upper panel the angle refers to the shell containing the particles with binding energy between the 5th and 30th percentiles. The middle panel corresponds to the intermediate shell containing particles with binding energies between the top 30th and 60th percentiles, and the lower panel corresponds to the outermost shell, containing particles with binding energies between the 60th and 90th percentile.
[FIGURE] Fig. 11. Time evolution of the angle between the major and median axes of central galaxy with the major axis of the initial configuration for Run Cp. The solid line corresponds to the major axis and the dotted line corresponds to the median axis.

4.2.2. Volume density

In order to study the properties of the central galaxy as a whole we define the principal directions of the whole galaxy as the principal axes of the inertia tensor of particles with binding energy between the 5th and 60th percentiles. The last shell is discarded because the outermost particles often have a substantial asymmetry. The eigenvalues of this system were used to define the ellipticity of the galaxy as a whole and we take its principal directions as the directions of the eigenvectors. Using these values we fitted the three dimensional density profile to a Hernquist profile (Hernquist 1990) using the expression given by Dubinski & Carlberg (1991):

[EQUATION]

where [FORMULA] is the total mass, [FORMULA] is a scale length, related to the half mass ellipsoidal surface

[EQUATION]

q is the ellipsoidal coordinate

[EQUATION]

and [FORMULA] and [FORMULA] are the axial ratios of the whole galaxy.

The particles in the central object were sorted according to their ellipsoidal coordinate and binned in shells, each containing 200 particles. A good fit to the Hernquist profile indicates that the mass distribution is stratified on similar ellipsoids at all radii. This profile was fitted for all the central objects in each snapshot of the simulations. The results are shown in Fig. 12 for the collapsing groups. All the galaxies formed in the collapsing simulations are well fitted by the Hernquist profile. In Fig. 13 we show the same plot for the initially virialised systems. We can see that the objects formed in the strongly bound systems (runs Vc1 and Vc2) are also well fitted by the Hernquist profile. Run V is the most interesting case. The central object formed in this simulation is not well fitted by the Hernquist profile, thus indicating that the object formed under these initial conditions has a different structure.

[FIGURE] Fig. 12. Fits of the three dimensional density distribution of the central objects formed in the collapsing simulations. In the left panel we show fits of Hernquist profiles to the data corresponding to the last integration step. In the right plane we show the deviations of the real density from the fitting function for all the radii and all the snapshots. The small values of these deviations indicate good fits by this law for the entire system and thus that these central objects are homologous.
[FIGURE] Fig. 13. Same as Fig. 12, but for the virialised simulations. Note that the central object formed in run V is not well fitted by the Hernquist profile, indicating a non-homologous nature for this object.

4.2.3. Velocity dispersion and anisotropy

To measure the degree of isotropy in the central galaxies we use the mean velocity dispersions in each of the principal directions. For a spherically virialised central object we expect similar values along each of the principal axes. In Fig. 14 and 15 we plot the results for the collapsing groups and virialised groups respectively. For the collapsing groups we see that the values in any direction vary in a very irregular way, while for the initially virialised simulations they remain nearly constant during the time span of the simulation. These differences are due to the different evolutionary histories of the two classes of systems, as discussed in Sect. 3 and shown in Fig. 4. In the collapsing groups the mergings occur at a roughly constant rate all through the evolution, and at all times there is material that has not settled yet to some equilibrium. On the other hand, for the virialised systems the central objects are to a large extent the result of mergings during the initial stages of the evolution, between the galaxies forming the central seed. Thus the material has had more time to settle to equilibrium. The addition of new material, both by merging and in the form of stripped material, comes at a slower rate, presumably slow enough so as not to alter the existing equilibrium in any crucial way. The presence or absence of irregularities is not the only difference between collapsing and virialised cases. For collapsing groups the velocity dispersion along the X axis is systematically higher than the velocity dispersion along any other direction. This is in agreement with the fact that these are non-spherical systems supported by anisotropic velocity dispersion tensors. Note also that the radial motions dominate over the tangential ones in all cases. This is due to the particular process of formation of these objects, whereby merging galaxies enter the central object following mainly radial orbits. On the other hand for initially virialised cases the velocity dispersion along the X axis does not dominate over the velocity dispersion along the rest of the principal directions, in agreement with the fact that these objects are less ellipsoidal. The three components of the velocity dispersion in spherical coordinates are also nearly equal during all the simulations, indicating that they are isotropic systems in equilibrium and that there is no ordered motion of the particles which constitute these central objects.

[FIGURE] Fig. 14. Time evolution of the mean value of the velocity dispersion of the central object formed in the collapsing simulations. In the left panel we show the time evolution along the three principal directions. These values indicate that the central galaxies have anisotropic velocity dispersion tensors. In the right panel we show the median value of the dispersion using spherical coordinates. Radial motions dominate in the objects formed by collapse.
[FIGURE] Fig. 15. Same as Fig. 14 but for the central objects formed in the virialised simulations. In this case there are no differences between the velocity dispersion along any of the principal directions, in agreement with the fact that these objects are nearly spherical. Contrary to the case of collapsing simulations, radial motions do not dominate, indicating a higher degree of isotropy.

4.3. Two dimensional properties

In order to compare our simulations with the observations of cD galaxies we use the following procedure. We first choose a random projection of the central object. We fix the Z axis and make a rotation about it with a random angle between 0 and [FORMULA]. Then, we fix the Y axis and repeat the same procedure and finally we do the same with the X axis. Next, we select a random number between 0 and 1 and if this number is less than [FORMULA] we project the galaxy on the [FORMULA] plane, if the number is greater than [FORMULA] and less than [FORMULA] we project the galaxy onto the [FORMULA] plane and, if the random number is greater than [FORMULA] we project onto the [FORMULA] plane. Then, for each particle, we keep its projected position and the vertical velocity. The projected object is placed at its center of mass according to these two dimensional positions and the two dimensional inertia tensor is calculated. Using the two eigenvalues [FORMULA] of this tensor we define for each particle the quantity

[EQUATION]

and the particles are ordered according to this value in increasing order. Then they are grouped in bins of 200 particles and we compute the surface density of each bin, except for the particles in the innermost 1.5 kpc, which are the ones mainly affected by the softening of our simulations. We do this for 9 random projections of the central object in each simulation and for each timestep. This procedure allows us to study the time evolution of the projected density profiles of the central galaxies formed in our simulations, while checking at the same time for possible dependencies on the viewing angle.

4.3.1. Time evolution of the surface density profiles

The main bodies of D and cD galaxies have surface brightness profiles which are well fitted by a de Vaucouleurs law (Lugger 1984, Schombert 1986). cD galaxies show an additional luminous halo and the external parts of these galaxies no longer follow the same de Vaucouleurs law as their main bodies (Oemler 1976, Schombert 1986). The colour profiles of these halos seem to be essentially flat and there is no evidence for breaks or discontinuities at the start of the cD envelope, nor for excessive blue colours in the envelope itself (Mackie 1992).

In order to be able to compare our results with the observations we study the time evolution of the surface density profiles using as a reference the [FORMULA] law. We start our study at the time corresponding to half the total time span of the simulation. At this time the central object has already formed and contains more than 10000 particles. Using the procedure described above we compute the surface density profiles of the central galaxies formed in our simulations. These two-dimensional density profiles can be grouped into three categories. In the first category we find the profiles that can be well described by the de Vaucouleurs law. These are the typical profiles of elliptical galaxies, but they are also typical of the brightest cluster members found in poor AWM and MKW clusters and in some Abell clusters, for example NGC 2329 in A569, or the central galaxy in A2029 (see the Schombert 1986 profiles). In the second category we find the profiles that fall systematically below the [FORMULA] law. This is also the case for some brightest cluster members, like the ones in A665, A1228 and A2052 (Schombert 1986). Finally we come to the category of profiles typical of cD galaxies. In a galaxy's external parts, the profiles in this category are systematically above the [FORMULA] law. This is the case for the central galaxies in A779, A1413 and A2199 (Schombert 1986).

We find that the density profile of the central galaxy is determined by the initial conditions of the simulation and does not depend on the viewing angle. The tightly bound and virialised groups (runs Vc1 and Vc2) give central objects that can be well described by the [FORMULA] law. The time evolution of the density profile of run Vc2 is shown in Fig. 16. The surface density profile of this galaxy is well fitted at all times by a de Vaucouleurs law and the same holds for the central galaxy formed in run Vc1. As shown in the previous section, the three dimensional profile of these objects is well described by the Hernquist profile. As the projection of the Hernquist profile gives good fits to the [FORMULA] law for a large range radii (Hernquist 1990) these good fits are not surprising. Note also the good agreement between the profiles obtained in the different projections at each timestep, indicating that there is no dependency of the surface density profile on the viewing angles.

[FIGURE] Fig. 16. Time evolution of the surface density profiles of the central object formed in run Vc2. For each time we show the profiles of nine random projections and the mean best fitting de Vaucouleurs law, which is a good description during all the times.

The second category of surface density profiles arises in the collapsing simulations, especially the ones with anisotropic initial conditions. Fig. 17 gives the time evolution of the profile of the central galaxy of run Cp. We can see that the [FORMULA] law gives good fits only in the main parts of the galaxy, while the external parts fall systematically below this law. This is true for all projection angles, but is more pronounced for the profile along the minor axis and less so for the profile along the major axis. The profiles of the central objects formed in the spherically collapsing simulations also have this feature, but it is not as pronounced, and the profiles can be well fitted by a [FORMULA] at some timesteps. As the objects formed in the collapsing simulations are the ones which show more signs of triaxiality, especially in the anisotropic collapses and in the external parts, this behaviour may be an effect of the triaxiality of the central galaxies. Thus, the presence of profiles falling below the [FORMULA] law may be indicative of objects with strong departures from spherical symmetry.

[FIGURE] Fig. 17. Same as Fig. 16 but for the object formed in run Cp. In this case the [FORMULA] law is a good fit only for the main body of the object, while the external parts fall systematically below this law. This can be a signature of the triaxiality of these objects.

The most interesting cases belong to the third type of surface density profiles, the ones typical of cD galaxies, shown in Fig. 18. These profiles are obtained only in the simulation of the more extended virialised group (run V). The central object formed in this simulation displays strong differences between the outer shell of material and the inner parts. Moreover, as we saw in the previous section, its three dimensional density profile is not well described by the Hernquist law. This is a result of the particular formation process of this object, where the mass coming from stripped material is more important, and leads to surface density profiles typical of cD galaxies. It is important to note that such profiles are not transient, as was the case in the simulations of merging galaxies by Navarro (1990), and that they are independent of the viewing angles. In our simulations, the deviation from a single [FORMULA] law appears as the central object is formed. The inner parts, which correspond to the most bound particles, are well fitted by an [FORMULA] law, while the external parts, which correspond mainly to accreted material, form a halo that can be associated with the halos of cD galaxies found in the central parts of clusters of galaxies. The mass of the system is distributed evenly between the central parts, well fitted by an [FORMULA] law, and the external parts forming the halo.

[FIGURE] Fig. 18. Same as Fig. 16 but for the object formed in run V. In this case the [FORMULA] law fits the profile only in the main body of the object, while the external parts have systematically higher values. This is the profile typical of a cD galaxy. This effect is not a transient phenomenon and is linked to the structure of the central galaxy.

4.3.2. Position in the [FORMULA] plane

One of the best studied correlations between the global properties of elliptical galaxies is the relation between the parameters defining the best fitting [FORMULA] law, i.e. the effective radius [FORMULA] and the effective surface brightness [FORMULA]. These two parameters are found to be mutually dependent, with a relation of the form [FORMULA] constant (Kormendy 1977). Brightest cluster members seem to be an extension of the elliptical sequence towards greater effective radii and lower effective surface brightness. These galaxies, however, have a tendency to be located above the mean relation defined by normal ellipticals (Schombert 1987) and even to have a shallower slope in this relation (Hoessel et al. 1987).

How are the central objects formed in our simulations distributed in the [FORMULA] plane? In the preceding subsection we discussed the fits of the [FORMULA] law to the projected density of our central objects at different timesteps. From these we obtain the values of the corresponding [FORMULA] and [FORMULA], assuming an [FORMULA]. We used nine different random projections, but, since their results are very similar, we randomly select one of the set of values and we plot it on the [FORMULA] plane. We note that different values of [FORMULA] will of course shift the points along the Y axes, while maintaining their relative positions. This can also be achieved by another rescaling of the computer units. We will thus be mainly interested in the slope of the correlation. The objects formed in our simulations follow a relation in this plane similar to the relation for elliptical galaxies. In the top panel of Fig. 19 we show the correlation for the central galaxies formed in the collapsing groups and in the bottom panel the correlation for the central galaxies formed in the virialised systems. As the simulation evolves the central objects get denser and more extended and the corresponding points in this [FORMULA] plane are displaced towards greater [FORMULA] and to smaller [FORMULA]. Both groups of galaxies follow a relation of the form [FORMULA] constant. This slope is somewhat higher than the one found for elliptical galaxies. Using all the data together we obtain a correlation with a slope of 3.9. The objects formed under virialised initial conditions are less dense objects and so they fall systematically towards higher surface brightnesses than the objects formed in collapsing systems and this bias gives the higher slope when all the data are used together.

[FIGURE] Fig. 19. Correlation between the surface brightness [FORMULA] and effective radii [FORMULA] for the central galaxies of our simulations at different timesteps. In the top panel we show the relation for the galaxies formed in collapsing groups and in the bottom panel the relation for galaxies formed under virialised initial conditions. The solid line is in both cases a line with the same slope as the Kormendy (1977) relation. The dotted line is the correlation for our data. The dot-dashed line in the bottom panel shows the correlation for the halos of the cD-like objects formed in run V. Symbols for collapsing groups: run C1 filled triangles, run C2 crosses, run Cp circles, run Co diamonds. Virialised groups: run V halo swiss crosses, run Vc1 lozenges, run Vc2 triangles, run V stars.

Another interesting point in this respect concerns the halos of cD galaxies. The properties of these halos can be characterised by fitting an [FORMULA] law to the outer parts of the surface brightness profile, i.e. the part outside the region which is well fitted by the [FORMULA] corresponding to the main body of the galaxy. Schombert (1988) finds that, on the [FORMULA] plane, these halos form an extension of the relation found for ellipticals and brightest cluster galaxies towards still lower surface brightnesses and larger effective radii, perhaps with a steeper slope. We repeated this for the halos of the central galaxies formed in our run V and give the results in the bottom panel of Fig. 19. They have the same properties with respect to their parent objects as the halos of cD galaxies with respect to their parent galaxies. Schombert (1988) has argued that these halos, which also follow the luminosity profiles of other material in the cluster, like the diffuse background, can not form by mergers but have to form by a process separate from that of first-ranked ellipticals. This is not borne out by our simulations which show that, although the halo has in many respects different properties from the main body of the galaxy, there is no distinct discontinuity in the formation process. Schombert models these halos as a separate entity using a two-component model combining an elliptical galaxy and a separate halo component with different [FORMULA] ratios and velocity dispersions, but the models are fitted with a wide range of values for these parameters. If the halo is a separate entity, we would expect it to have the same velocity dispersion as the system of secondary galaxies in the cluster. At first, data from the central galaxy in A2029 (Dressler 1979) seemed to be in agreement with this idea. Recent data, however, suggest that, while the central galaxy in A2029 does have a rising velocity dispersion profile, this is not a feature common to first-ranked galaxies (Fisher et al. 1995). The projected velocity dispersion profiles of the galaxies obtained in our simulations are in agreement with the profiles of real brightest cluster members. This can be seen in Fig. 20, where we show the profile of the central galaxy formed in run V at the end of the simulation. This is a mean profile obtained by adding the profiles of the nine random projections of this object. The error bars indicate the dispersions over the mean values. The profiles for the rest of the galaxies obtained in our simulations are of the same nature and are independent of the viewing angles. The gradient in velocity dispersions is also in agreement with the gradients in the profiles of real galaxies. Thus, our simulations suggest that, the material that forms the halo of cD galaxies does not need to be material with high velocity dispersion. Deeper observations are needed to confirm this result.

[FIGURE] Fig. 20. Projected velocity dispersion profile of the central galaxy formed in run V at the end of the simulation. The gradient in velocity dispersion is comparable to the gradient found for real galaxies. The profiles for the rest of the galaxies in our simulations are of similar nature.

4.3.3. The Faber-Jackson relation

The Faber-Jackson relation (Faber & Jackson 1976) is a relation of the form [FORMULA] between the total luminosity of elliptical galaxies L, and their central velocity dispersions [FORMULA]. The value of p is still controversial but the most commonly accepted one is [FORMULA] (Terlevich et al. 1981). The brightest cluster members do not follow this correlation very well, and tend to be brighter than predicted from their central velocity dispersions using the relation [FORMULA] (Malumuth & Kirshner 1981, 1985).

The relations for the galaxies formed in our simulations are shown in Fig. 21. Instead of luminosity we use the total mass. This seems to be a good approximation, as the [FORMULA] ratio for ellipticals seems to be independent of luminosity (Tonry & Davis 1981), or a weakly dependent function of the luminosity of the form [FORMULA] (Oegerle & Hoessel 1991). Objects formed in collapsing simulations are located on the [FORMULA] plane very differently from the objects formed under virialised initial conditions. The galaxies formed in collapsing groups do not follow a Faber-Jackson relation and give a scatter diagram in the [FORMULA] plane, while the data corresponding to the galaxies formed from virialised initial conditions show much less scatter. This can be explained if the Fundamental Plane is a consequence of the virial theorem (Pahre et al. 1995). As we have seen in Fig. 14, the velocity dispersion profiles of these galaxies indicate that these systems are not in virial equilibrium. On the other hand, the galaxies formed under virialised conditions are fully isotropic systems and give better correlations. The solid line shown in both diagrams corresponds to a line with the same slope as the Faber-Jackson relation. The dashed line shown in the panel of collapsing groups is a least squares fit, while the dashed line in the panel of virialised groups corresponds to the least squares fit of the galaxies formed in runs Vc1 and Vc2. These objects, which are fully virialised systems, give a slope of 3.6, i.e. in the range of the Faber-Jackson relation. This value, however, is very uncertain, as can be seen from the location of the corresponding points in the lower panel of Fig. 21. It is interesting to note that the objects formed in run V, which can be associated with the cD galaxies in clusters, fall systematically above the line corresponding to the correlation for runs Vc1 and Vc2 which resemble elliptical galaxies, as is the case for real cD galaxies (Schombert 1987). As stated in the beginning of this section, Malumuth & Kirschner (1985) find that brightest cluster members are systematically brighter than what could be expected by the Faber-Jackson relationship. They furthermore find that this effect is stronger for the subset of their galaxies classified by Morgan and his coworkers as cD. It is tempting to draw an analogy between this result and our simulations. Unfortunately the remainder of the Malumuth & Kirschner sample could also contain some cD galaxies. Thus more observational work is needed for a better comparison.

[FIGURE] Fig. 21. Faber-Jackson relation for the central galaxies in our simulations. In the top panel we show the results for the galaxies formed in the collapsing groups and in the bottom panel the results for the galaxies formed in virialised groups. The solid line is a line with a similar slope as the one for elliptical galaxies. The dotted line is the correlation obtained from our data. The different symbols correspond to different simulations and several timesteps are shown for each simulation. Symbols as in Fig. 19.
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Online publication: April 6, 1998
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