The influence of galaxy mergers on the mass dispersion of brightest cluster galaxies
aa r X i v : . [ a s t r o - ph . C O ] J un The influence of galaxy mergers on the mass dispersion ofbrightest cluster galaxies
Th. Jagemann Astronomisches Rechen-Institut Heidelberg,M¨onchhofstraße 12-14, D-69120 Heidelberg, Germany [email protected]
ABSTRACT
The absolute magnitude of the brightest galaxy of clusters varies remarkablylittle and is nearly independent of all other physical properties of the clusteras, e.g., its spatial extension or its richness. The question arises whether theobserved small scatter is compatible with the assumption of dynamical evolutionof the cluster. This is investigated with the help of statistical analysis of theresults of cluster simulation. The underlying interaction process is merging (andalso destruction) of smaller galaxies forming the giant galaxy. The cluster itselfis supposed to be in virial equilibrium.We find that the evolutionary importance of merger processes grows withdecreasing scale. Rich clusters as well as their brightest members evolve merelyslowly whereas compact groups as well as their brightest members evolve morerapidly and more violently. We also find that the number of merger processesleading to the growth of the brightest cluster galaxy (BCG) is small enoughto keep the BCG mass dispersion below the measured value. Our simulationsubstantiate that just the combination of the initial distribution function and thefollowing merging to form the BCG can explain the remarkably small varianceof mean BCG masses between clusters of different size and different number ofgalaxies.
Subject headings:
Galaxies: clusters: general – Galaxies: evolution – Galaxies:interactions – Galaxies: mass function Current address: ESA/ESTEC, Postbus 299, 2200 AG Noordwijk, The Netherlands
1. Introduction
The mean absolute magnitude of brightest cluster galaxies is h M VI i = − . σ ( M VI ) = 0 . σ ( L ) L = 0 . . (1)Batcheldor et al. (2007) show that the luminosity dispersion in the near infrared is evenless. Due to this small intrinsic dispersion BCGs are ideal tools for cosmological distancedetermination (Sandage (1972); Postman & Lauer (1995)). Furthermore their theoreticalinvestigation offers the possibility to understand the properties and evolution of the clusteritself and its members.Sandage (1972) points out that there is a variety of possible accounts for the observedscatter in BCG apparent magnitudes, but all contributions but intrinsic dispersion (i.e.scatter in absolute magnitude) seem to remain small, in particular is there no systematictrend with redshift. At least the small total dispersion shows that any single dispersioncontribution remains small.There are many kinds of suggestions trying to explain the observed small scatter inmagnitudes of brightest cluster galaxies. One possibility is the existence of a common lu-minosity function all galaxies of a cluster (including the brightest one) are drawn from(Schechter & Peebles (1976)), either with or without further development. The alternativeconsists of a unique creation process, a “standard mold”, or a not yet discovered phys-ical effect that delimits the luminosities of galaxies and cuts off the bright end of theluminosity function. This hypothesis is underpinned by the astonishing fact that bright-est galaxies of rich clusters are just as luminous as the brightest observed field galaxies(Humason, Mayall, & Sandage (1956)).How special are BCGs? It has been argued that surface brightness profiles of BCGs arewell fit by the same S´ersic law that describes less-luminous spheroids and obey the same rela-tions between fitting parameters that characterise E/S0 galaxies generally (Batcheldor et al.(2007)). On the other hand, v.d.Linden et al. (2007) show that BCGs contain a larger frac-tion of dark matter as compared to non-BCGs of the same stellar mass, influenced by thecluster environment.X-ray observations of the intracluster medium (e.g., Peres et al. (1998)), in particularof the cluster core, has given rise to the assumption of cooling flows depositing up to 100 M ⊙ yr − , building up the BCG over many Gyrs. However, Motl et al. (2004) support the alter- 3 –native that cores of cool gas are being built from the accretion of discrete stable subclusters.At the same time they realise that the common presence of substructure in galaxy clustersargues for accretion or merger events that occured sufficiently recently not to be erased byrelaxation of the cluster ( Bird (1994)).Galactic cannibalism, i.e. the accretion of the existing galaxy population through dy-namical friction and tidal stripping predominantly in the evolved cluster, has been pro-posed by several authors (e.g., Ostriker & Hausman (1977)). Further investigations byMerritt(Merritt (1983), Merritt (1984), Merritt (1985)) and others (e.g., Lauer (1985),Dubinski (1998)) showed, however, that galaxies are moving too fast and are significantlytidally truncated thereby suppressing dynamical friction. They concluded that the dynami-cal friction timescales are generally too long such that galaxies could therefore only accretea small fraction of their total luminosity within a Hubble time.Galactic mergers remain as the most likely dynamical effect that dominates BCG evo-lution. But given a vivid merger history, it is problematical how a uniform magnitude of thebrightest cluster galaxy can get over frequent encounters while being exposed to mergers withother galaxies, or deletion. Tightly related to this puzzle, the question comes up whetherthe common luminosity function of all up to date analysed clusters is a direct consequenceof the initial mass fluctuations of the early universe – or otherwise, if the mass spectrum atthat time was an entirely other one, and evolution processes provide for the universality ofthe luminosity function.However, the scenario seems less agitated. Hoessel (1980) estimates, correlating spatialextension and luminosity of BCGs, that mergers of galaxies of comparable size lead to theformation of the first-ranked member. As he divides the collapse time of a cluster by the timethat pass between two close encounters of two relatively big galaxies, he obtains about fourmerger events forming the BCG. This result coincides with the number of observed cores incD-galaxies. In the same way, the luminosity function changed only little, because mergerprocesses operate effectively only on large massive galaxies (Tr`evese, Cirimele, & Appodia(1996)).Numerical simulations of large groups of 50-100 spherical galaxies in an equilibrium clus-ter have been done by Funato, Makino, & Ebisuzaki (1993), Bode et al. (1994), Garijo, Athanassoula, & Garc´ıa-G´omez(1997). These simulations find a runaway merging process occuring near the cluster centerforming a BCG. Recent analyses of the Millenium Simulation (Springel et al. (2005)) byLucia et al. (2006) indicate that, on the one hand, the most massive elliptical galaxies havethe oldest and most metal rich stellar populations. On the other hand, massive ellipticals arepredicted to be assembled later than their lower mass counterparts, and they have a largereffective number of progenitor systems (reaching up to N eff ≈
2. The model
Numerical simulations require the masses of the galaxies. On the presumption of a givenmass-to-light ratio Υ, Eq. (1) is directly translatable into the BCG mass dispersion. Thoughmasses resulting from dynamical simulations are still dependent on Υ once compared withthe observed luminosities, the relative variation does not depend on Υ.The following subsections briefly describe the model assumptions that enter the simu- 5 –lation computations. For further details see Jagemann (1997).
The cluster simulation starts with N galaxies homogeneously distributed in a sphere ofradius R and total mass M . R and M are kept constant in time in the course of development;in other words, we assume that the cluster is isolated and decoupled from the Hubble flow.At the beginning the number of galaxies in the cluster equals N = N . N diminishes inthe course of time due to mergers and destruction of galaxies. The mass of the destroyedgalaxies is counted to the intergalactic medium, so that the total mass of the cluster remainsconstant.The galaxy masses are initially drawn from the exponential distribution function dis-cussed in Sect. 3. The model starts from the assumption that the total mass of the galaxycluster is proportional to the total mass of all galaxies: M tot = N M · k (2)Here N is the initial galaxy number and M is the mean galaxy mass (Sect. 3). Then theconstant k is the ratio of cluster mass to galaxy mass, i.e. the ratio of mass-to-light ratio ofthe cluster (in the following denoted by Υ cluster ) and mass-to-light ratio of the galaxies: k = Υ c luster Υ g alaxies (3)The total mass of a (rich) cluster is of the order of 10 M ⊙ . Therefore the mass-to-light ratio of a cluster is within the range [200; 500] Υ ⊙ , and so the discrepancy between themass-to-light ratios of present clusters as a whole and their individual members is at least k = 5 and probably more. An interesting and crucial question concerning the interactionbetween galaxies of a cluster is how much of the cluster dark matter is really gravitationallybound within galaxies (Merritt (1985)), i.e. the value of k . Two possible extreme cases areimaginable: either the dark matter hides within individual halos of single galaxies, or thesehalos are smeared over the whole intergalactic background due to tidal processes or similareffects. As virial velocities of the galaxies are determined by the total cluster mass, whereasindividual galaxy masses enter decisively those equations that determine the interactionbetween two encountering galaxies, the trend resulting from the different extreme cases canbe anticipated: 6 –If dark matter exists in individual halos, the merger cross section is much larger thanin a cluster containing significantly intergalactic matter, so the model develops more rapidlyand more violently. This work considers these two cases. In this paper individual galaxies are described by two parameters: their radius r i andtheir mass M i ( i = 1 . . . N ). Mass and radius are changing in the course of cluster evolution;at the beginning, the masses of cluster galaxies are characterized by a distribution functionwith characteristic mass M and an associated distribution function in galaxy radii withcharacteristic radius r .The evaluation of galaxy encounters is done under the assumption that galaxies arerepresented by Plummer spheres: ρ ( r ) = 34 π M a ( r + a ) / (4)Here a is the Plummer radius, M is the total mass of the galaxy and G is the gravitationalconstant. Later, for the energy transfer in the tidal approximation, the root mean square(RMS) radius of the extended perturbed mass distribution is needed. So we further assumea cut-off radius r RMS ≈ . a . In the following, we call this proportional factor β := r RMS /a .The galaxy radii r i = r RMS are related to the drawn masses via the Faber-Jackson-law.Calculating interaction of cluster galaxies, the significant parameter is not the radius butrather the binding energy which computes for a Plummer model to: E = − π GM a . (5) Encounters of galaxies let the model evolve. In this work, encounters are described bythe two parameters v and B . v is the relative velocity, and the maximum impact parameter B defines the maximum separation between two passing galaxies that is evaluated to be anencounter. Fixing the total cluster mass M tot and characteristic radius R to be independentmodel parameters, the mean square of the spatial velocity v is set by the virial theoremthat is assumed to be applicable to the cluster: 7 – v = f G M k NR (6)This procedure is in contradistinction to the procedure of Krivitsky & Kontorovich(1997). They find a kind of “explosive” merging without the additional constraint of virialequilibrium.The radius R of the sphere is assumed to be proportional to the gravitational radiuswith proportionality factor f . In addition, we assume the spatial virial velocity v (thatalso determines the rate of encounters) equals p h v − i that has to be put into the formulaof energy transfer.The impact parameter b of the actual encounter is chosen within the limits of [0; B ] suchthat encounters are uniformly distributed upon the maximum encounter cross-section πB .Using the impulse approximation, in this model B is defined the way that the error dueto neglecting encounters with impact parameters greater than B is less than one per centfor a galaxy of standard size r ( r corresponds to M via the Faber-Jackson-law).The mean time between two encounters is given by∆ t = 8 R vB N ( N −
1) (7)and diminishes with decreasing galaxy number N during the cluster development. The totaldevelopment time is set to T = 10 yrs. Gravitational interaction is the origin of cluster development. We compare the kineticenergy from orbital motion of two encountering galaxies with their internal energies. Thebinding energy of an individual galaxy increases at the expense of the orbital energy, i.e.at the expense of the total kinetic energy of their relative motion. As long as the totalenergy transfer ∆ E i + ∆ E j is less than the kinetic energy of the relative motion, the pairof galaxies is assumed to stay unbound. The energy input leads to an increase in bindingenergy of a galaxy which therefore expands. If energy input exceeds the binding energy ofthe concerning galaxy, this leads to its destruction.But if the total energy transfer is higher than their kinetic energy,∆ E i + ∆ E j > E kin , (8) 8 –the encounter ends up in a bound system, and it is assumed that the galaxies merge. If theencounter leads to a merger event, both participating galaxies are replaced by a single one(the remnant), and its mass and radius has to be recalculated. The remnant mass is simplythe sum of both galaxy masses, and the radius is computed from energy conservation: E R = E i + E j + E kin . (9)It is possible that the remnant binding energy becomes E R >
0, if E kin > − ( E i + E j ) . (10)In this case no remnant is created and both galaxies break up.To calculate energy exchange, the energy transfer is expressed using the impulse andtidal approximation (Spitzer (1958)) combined with a smooth interpolation of the two lim-iting cases of b = 0 and b ≥ ˆ b (ˆ b = 5 max( r h ,i , r h ,j ), r h is the median radius of the respectiveperturbed system).The energy transfer of a particle of mass M j passing an extended particle system atdistance b with velocity v according to impulse and tidal approximation is∆ E i = 4 G M j M i r i v b , (11)where r i is the RMS-radius of the perturbed system i ( Spitzer (1958)).In the case of central penetration ( b = 0) the energy transfer between two identicalPlummer models is given by ∆ E max = G β m j m i v r i (12)where β was introduced in Subsect. 2.2. The interpolation formula is then assumed to be∆ E ∆ E max = 11 + β r i b (13)(Fig. 1).Proceeding this way we use two terms of energy transfer in high speed encounters, i.e. inencounters where relative velocities are higher than the galaxy internal velocity dispersion. 9 – Fig. 1.— Energy transfer with interpolation formula Eq. 13. Abscissa: b/ ˆ b , ordinate:∆ E/ ∆ E max ; dashed line: impulse approximation, solid line: interpolation. 10 –This approximation is not too crude (Aguilar & White (1985)) although galaxy relativevelocities are about as high as their star velocities.Other dynamical processes, especially dynamical friction between galaxies or galaxiesand the intergalactic background, tidal interaction between galaxies and the cluster potentialas well as mass-loss due to ram-pressure are neglected.
3. Mass dispersion
Trying to understand the BCG mass dispersion, it is necessary to know more aboutthe stochastic variance, inherent in the distribution function. When drawing N galaxiesfrom a given frequency distribution, there is already a dispersion in the mass of a distinct,e.g. the brightest, galaxy depending on the kind of distribution and the number of drawngalaxies. An analytical approximation of the global luminosity distribution of galaxies intoday’s clusters is Schechter’s law (Schechter (1976)): N ( L ) d L = N ∗ (cid:18) LL ∗ (cid:19) α exp (cid:18) − LL ∗ (cid:19) d LL ∗ (14)with α = − / L ∗ = 1 . · h − L ⊙ in the visual band N ∗ ∈ [20; 115] .α is given within an error of 20%, but there are also votes for an exponential distributionin compact clusters ( α = 0) differing from that of field galaxies (Geller & Peebles (1976)).In any case, solely the number of the brightest (cD-) galaxies in the centres of galaxy clus-ters are not exactly accounted for (Dressler (1978), and references therein). This fact byitself is an indication of the answer to the central question: Is Schechter’s law the result ofdevelopment in kind of merger and destruction processes or does it reflect the mass spec-trum in the early universe (Lauer (1985))? Since the BCG is in an exceptional positionwith respect to Schechter’s law and does not fit the global distribution, it is not probablethat Schechter’s distribution accrues from a different distribution (except that other thandevelopment processes would affect the BCG luminosity). Otherwise the distribution wouldinclude the BCG.With the assumption of a specific mass-to-light ratio of individual galaxies, galaxymasses can be drawn from a universal mass distribution. Because of better analytical han- 11 –dling and to avoid divergences galaxy masses of the model cluster are drawn from an expo-nential distribution. In this case, the initial probability distribution is p BCG ( M ) d M ≡ N (cid:16) − e − MM (cid:17) N − M e − MM d M, (15)shown in Fig. 2.In this case the expected value of the most massive initial galaxy can be written as anharmonic sum that depends on the mass M characterizing the distribution function and onthe number of drawn galaxies N : h M BCG i = M N X k =1 k . (16)The BCG mass dispersion is then computed to σ BCG = M vuut N X k =1 k . (17)With N = 1000, e.g., the relative deviation computes to σ BCG / h M BCG i = 0 .
17. Thoughthe expected value of the BCG masses increases with galaxy number, the relative deviationdoes not. Generally the fact is remarkable, that initially BCG masses depend on the numberof galaxies, whereas in reality such a dependence is not found, and the BCG relative massdispersion is distinctly smaller for all numbers of galaxies than the observed one. Thesedifferences may be due to development processes, although additional correlations among R and N may cause selection effects in ( N, R )-space.Statistical statements about clusters of galaxies are derived from model assumptionsdepending on several parameters. Thus the resulting quantities to be analysed are onlyestimates of the mean sampled over infinite runs. The accuracy, given as relative 1 σ variance,is proportional to the square root of the number of simulation runs. Differently set up modelclusters with the same set of initial parameters are here called an ensemble. So since all oursimulation results are based on ensemble sizes of 100 runs, they are accurate to within 10%.Additionally, any dispersion in absolute magnitudes (or masses, respectively) in a modelthat depends on variable parameters will generally depend on these parameters, too. Forexample, BCG masses – as well as their dispersions – that are drawn from a distributionfunction depend on a characteristic mass M scaling this function and depend also on thetotal number of galaxies N drawn from this distribution. 12 – Fig. 2.— Frequency distribution of the brightest cluster galaxies with exponential massdistribution and N = 500 and M = 1. 13 –Fig. 3.— BCG masses and their variance at the beginning of the simulation. Initial param-eters are: M = 4 · M ⊙ , r = 15 kpc , v = 800 km / s , R = 3 Mpc , T = 10 Gyr .
14 –Fig. 4.— BCG masses and their variance at the end of the simulation. The initial parametersare the same as in Fig. 3. 15 –Figures 3 and 4 illustrate the dependence of the mean of BCG masses on the initial num-ber of galaxies. Likewise, standard deviations belonging to any value of N are shown, too.Figure 3 describes the initial state, Fig. 4 the evolved state. The curve in Fig. 3 is smooth,because the expected value of BCG mass and dispersion are theoretically computable. InFig. 3 the stochastic 10% noise of the mean values is obvious.In order to calculate the total dispersion of BCG masses, let these masses be a functionof a vector ~x in parameter space. Furthermore, assume that a distribution function p ~x ( M )is given. The expected value of M then averages the ensemble and the parameter space to hh M ii := Z ~x h M ( ~x ) i p ( ~x ) d ~x , (18)where h M ( ~x ) i means an ensemble average. The mass dispersion averages the ensemble andthe parameter space to (∆ M ) = h (∆ M ) i + hh M i i − hh M ii , (19)where the inner brackets always denote an ensemble average. Thus, the total dispersion isthe expected value of the individual dispersion in ~x plus the scatter of the expected valuesof M in ~x : (∆ M ) = h (∆ M ) i + (∆ h M i ) . (20)This expression is used later on to calculate the mass dispersion of the simulations. At thebeginning of the dynamical development, any scatter in BCG absolute magnitudes or massesis the sum of different kinds of model dispersion:1. The statistical variation at constant model parameters within an ensemble, wherebyone gets a convolution of two distributions: the stochastic variance at a constant vectorof model parameters and2. the variance caused by the distribution of clusters upon the model parameters.Hence the observed relative mass dispersion is the upper limit of the statistical error.In this context it is astonishing that there is definitely no correlation between M BCG andrichness (meaning the final galaxy number) in today’s observed clusters (Sandage (1972);Postman & Lauer (1995)). But drawing galaxies from a distinct distribution (except from a δ -function), there is always – within the bounds of resolution – such a dependence, so that 16 –there inevitably has to be another process to explain the independence of M BCG on N . Thesimulations try to answer the question whether mergers can account for such a behaviour.The observed scatter in absolute magnitudes is in any case an upper limit of the intrinsicBCG mass dispersion within any ensemble.The dependence of the BCG absolute magnitudes on the scaling mass M is morecomplicated. M is positively correlated with M BCG with high statistical significance. Inorder to eliminate any photometric calibration error as well as a global shift of the luminosityfunction, Tr`evese, Cirimele, & Appodia (1996) compare M BCG − M ( M is the magnitudeof the 10th brightest galaxy) with M − M . They find, that both differences are negativelycorrelated.
4. Results
Table 1 lists the input parameters of the simulation.Here the two parameters R and N are allowed to vary independently (2) in the giveninterval while the pair ( M ; r ) remains constant. Then, the mean relative velocity computedusing the virial theorem adjusts automatically. Afterwards a new pair ( M ; r ) is fixed fromthe variation interval, and N and R are varied again. M and r are not correlated. k is setto 1 or 10 (the probable value of present clusters).The underlying idea is that there is a fixed value of M for all clusters in nature withinnarrow limits (corresponding to the fixed L ∗ in the Schechter distribution), that the mass-to-light ratio of all clusters is initially the same and that all galaxies of equal initial massare of the same size, independent of cluster membership.On the other hand, there are clusters of different size and variable richness in naturethat enter the measurements of mass dispersion of the brightest cluster galaxies and areequally put together.At the end of the simulation the following ensemble mean data are derived:1. The mean relative velocity calculated from the virial theorem,2. the number of galaxies at the end of the simulation,3. the total number of encounters within the maximum impact parameter,4. the number of constructive merger processes, 17 –5. the number of destructive merger processes,6. the number of destroyed galaxies without forming a remnant,7. the number of merging events that lead to the formation of the BCG,8. the BCG mass,9. the BCG mass ensemble dispersion.Except for the last two items giving the relative mass dispersion, the data serve asinvestigation of the model and are not connected to observations. M = 5 · M ⊙ , r = 20kpc and k = 1In the following, results of this parameter set are discussed in detail. Concerning othervalues of M , r and k only occuring changes in contrast to this set are mentioned.The choice of M = 5 · M ⊙ means a moderate mass-to-light ratio of about Υ g alaxies =10 of today’s observed clusters, i.e. the absence of dark coronae, and k = 1 means the initialabsence of intergalactic matter.The number of encounters of all galaxies is shown in Fig. 5. This total number ofencounters increases within the bounds of parameter variation to 10 and corresponds es-sentially to Eq. 7, substituting the total development time T for ∆ t . Therefore the numberof encounters is proportional to N / and to R − / and changes in the course of evolutiononly at small radii R , where the galaxy number decreases due to merger and destructionprocesses (cf. Fig. 10).Figure 6 shows the mean BCG masses at the end of the simulation. They are locatedbetween about 30 and 70 · M ⊙ and show a clear peak at small radii and small initialgalaxy numbers.In order to exclude the influence of the initial mass distribution in this histogram, thepure mass increase is shown in Fig. 7. Generally the BCG is growing in the course ofdevelopment. The increase amounts to 40 · M ⊙ at the peak which stands out even moreclearly than in Fig. 6.The main result of the simulation, i.e. the mean values of the BCG relative massdispersion, is shown in Fig. 8. The dispersion is nearly constant at about thirty per cent andincreases at the small clusters to a maximum of 57 %. 18 –Table 1: Simulation parametersParameter Value(s) RemarkEnsemble size n = 100 constantTotal time T = 10 Gyr constantChar. mass M ∈ {
5; 50 } · M ⊙ variable (1)Char. radius r ∈ {
20; 100 } kpc variable (2)Υ ratio k ∈ {
1; 10 } variable (1)Cluster radius R ∈ [0 .
25; 3 .
00] `a 0.25 Mpc variable (2)Richness N ∈ [100; 1000] `a 100 variable (2)Fig. 5.— Logarithm of the total number p of encounters. 19 –Fig. 6.— Mean values of BCG masses after the development. M = h M BCG i [10 M ⊙ ]. 20 –Fig. 7.— Mean values of BCG mass increase. M − M h M BCG , afterwards i −h M BCG , previously i [10 M ⊙ ]. 21 –Fig. 8.— Mean values of BCG mass dispersion after the simulation. d = h σ BCG i / h M BCG i . 22 –Fig. 9.— Mean values of the increase of BCG mass dispersion. d − d h σ BCG , afterwards i / h M BCG , afterwards i − h σ BCG , previously i / h M BCG , previously i . 23 –Fig. 10.— Mean number of mergers leading to the BCG formation. 24 –In order to investigate the increase in mass dispersion as in Fig. 7 and to exclude theinfluence of the initial distribution, the relative mass distribution after the simulation issubtracted by the relative dispersion before the simulation in Fig. 9. As in Fig. 7, the peakstands out more clearly against the stochastic noise.To answer the question how many mergers are necessary to form the BCG this parameter f is shown in Fig 10. The maximum value lies at about six mergers and decreases steeplyfrom there to a value of about one.Not only merger events but also destructions of galaxies are part of cluster development.Figure 11 shows how many galaxies are left at the end of the simulation. Here a clear trendis recognizable: the percentage of destroyed or merged galaxies increases considerably withdecreasing cluster radius. In the case of rapid evolution the number of galaxies may decreasedown to twenty per cent of the original value.The difference between Fig. 11 and Fig. 13 provides roughly for the number of galaxiesdestroyed in encounters (Fig. 12). It turns out that the biggest fraction of galaxies is de-stroyed when the energy transfer is less than the sum of binding energies of the 2 galaxies.The percentage of disrupted galaxies is always less than 65 per cent and decreases with in-creasing radius. At small radii the fraction of destroyed galaxies is roughly independent ofthe initial galaxy number, but increases with galaxy number at increasing radii.Finally, to address the question about the fate of the remaining galaxies, the number ofall the merger events of all galaxies is divided by the final galaxy number in Fig. 13. Theparameter t in this diagram denotes the percentage of those galaxies that are involved in amerger event. In the same way t is a measure of the frequency describing how often a galaxyhas been involved in a merger event averaged over all cluster galaxies. In the peak vicinityevery galaxy has experienced one merger on the average. t sharply decreases with increasing N and R . M = 5 · M ⊙ , r = 20kpc and k = 10The choice of M and r corresponds to the choice in Sect. 4.1, k = 10 corresponds (at M = 5 · M ⊙ ) roughly to the maximum value of the present fraction of intergalactic darkmatter in galaxy clusters.The mass distribution of single galaxies corresponds exactly to the one of Sect. 4.1, butbecause of the addition of intergalactic matter the virial velocity is increased by the factor √
10 according to Eq. 6. This is why the number of encounters is about this factor higher 25 –Fig. 11.— Mean number of destroyed/merged galaxies per initial galaxy number after de-velopment. 26 –Fig. 12.— Percentage of destroyed galaxies. 27 –Fig. 13.— Mean number of merger events per final galaxy number. 28 –in comparison with Fig. 5. The high spatial velocity-dispersions a little above the diagonalof more than 2400 km/s are not observed in reality; therefore this part of the simulation isexcluded in the following discussion.The final galaxy numbers drop to values of significantly less than hundred per centat small radii. The qualitative behaviour follows the one of Fig. 11, but the maximumis, however, clearlier marked at small radii and galaxy numbers, and about half as manygalaxies less have been destroyed at a fixed pair ( R , N ) compared with Sect. 4.1. Thedominant development process here is not merging but destruction of encountering galaxies.The mean mass distribution of BCGs after the simulation has not changed much: Theinitial mass distribution can easily be recognized. Deviations in comparison to the initialdistribution arise in about half of the cases, these deviations are about ± · M ⊙ for eachhalf.Only little signs of evolution are present in the BCG mass dispersion after the simulation:The initial mass scatter is still present. The relative dispersion does not exceed a value of 25per cent. The trend of the initial distribution having smaller spreads with increasing galaxynumbers still exists. Just as for the mean mass values there is an irregular change in thescatter of 1–2% compared with the initial scatter. M = 50 · M ⊙ , r = 100kpc and k = 1The total mass of the galaxy cluster is the same here as in Sect. 4.2, the galaxies howeverare ten times more massive. k = 1 means the initial absence of intergalactic matter as inSect. 4.1. The dark matter is now hidden in individual halos instead of spread throughoutthe intergalactic medium. Therefore more extended galaxies with a characteristic radius of r = 100 kpc are chosen at this parameter set.Figure 14 shows the BCG mass mean values after the development. In comparison withFig. 6 the shift of the maximum toward smaller galaxy numbers but significantly greatercluster radii is striking.Figure 15 makes it clear that the relative mass dispersion remains always below fortyper cent and even decreases significantly going to greater galaxy numbers, particularly atthis set of parameters.Fig. 16 shows that the number of mergers leading to the formation of the BCG is non-zero for a wider range of parameters N and R . The maximum mean number of mergers isfour as in Sect. 4.1. 29 –Fig. 14.— Same as Fig. 6, but now for model 4.3. 30 –Fig. 15.— Same as Fig. 8, now for model 4.3. 31 –At smaller galaxy numbers the percentage of those galaxies that have been involved inmerger processes increases up to sixty per cent (Fig. 18), cf. also Fig. 13. From a galaxynumber of about four hundred onwards there are practically no merger events to be noted.
5. Discussion
There are two processes with opposite effects which rule the evolution of the model:On the one hand the number of encounters decreases with increasing cluster radius at fixedgalaxy number (cf. Fig. 5). On the other hand the virial velocity increases with increasingtotal cluster mass by increasing richness at fixed cluster radius. Although the number ofencounters rises with velocity, the equations of energy transfer (Eqs. 11 and 12) dependon v − , and the kinetic energy term of the relative motion depends on v . So high energytransfers are only effective for mergers at low velocities. Taking both effects together, onlycompact groups evolve noticeably in contrast to clusters. The number of encounters g of asingle galaxy for the total development time can be estimated to g = N − πR / πB vT (21)if richness does not change dramatically. The merger probability can be estimated in thesame way: let b m be the impact parameter so that the merger criterion is fulfilled within itsenclosed area: b m = r √ Gm r v . (22)Then the fraction of merger events (caused by the stochastic choice of impact parameters)is b m /B . This fraction multiplied by the number of encounters then reveals the number ofmerger events V of one galaxy (e.g. the BCG): V = 8 f − r T p Gm · r NR (23)So the number t of mergers depends strongly on the cluster radius and decreases withincreasing R (cf. Fig. 11). On the other hand t increases proportional to the square root of theinitial number of galaxies (assumed that it remains nearly constant, e.g. Fig. 11 at R = 0 . πβ r f · RN > . (24)The remnant has a chance to survive only at high cluster radii and small galaxy numbers.Together with the merger criterion this leads to the occurring maximum development at smallcluster radii and small numbers of galaxies (Figs. 7, 10, 13): Small radii are necessary toform remnants, and their binding energies are negative only at small galaxy numbers.These estimates expressed in simple formulae (Eqs. 21-23) are possible because oursimulation show that there is no agravating change of the initial mass distribution.The simulation computations of Sect. 4.3 show a merger behaviour of all galaxies andespecially of those forming the BCG comparable to Sect. 4.1. A higher galaxy mass has apositive effect on the number of mergers because the mean energy transfer rises with thethird power of the characteristic mass, whereas the kinetic energy of the relative motion stillrises linearly. On the other side a higher cluster mass in Sect. 4.3 in contrast to Sect. 4.1causes the virial velocity to increase, suppressing energy transfer such as in Sect. 4.2.Interpreting simulation 4.2 to represent cluster states further evolved in time comparedwith simulation 4.1 as the mass of the intergalactic medium increases due to tidal stripping(Merritt (1984)), a picture of cluster evolution is formed: In the early phases of clusterevolution merger processes dominate the development of mass spectrum and BCG, thenpaling into insignificance as time goes by (cf. Albinger (1995)). Comparing both simulationsit should be noted that the cluster mass is the same in simulation 4.2 at N = 100 andsimulation 4.1 at N = 1000. An increase of the mass fraction of intergalactic matter leads toa drastic decrease in the merger rate at equal galaxy mass distributions, because the meankinetic energy controlled by the virial theorem (Eq. 6) outgrows the expected energy transfersin Eq. 13, so that mergers become improbable. The amount of galaxy binding energy is notinfluenced by a variation of k , and so a (less steep) decrease in galaxy destructions is expectedon the basis of Eq. 13 as shown by the simulations.A closer consideration of Fig. 7 together with Fig. 10, dividing the mean increase ofBCG mass by the mean number of mergers leading to the BCG formation, shows that thebrightest galaxy must have been built by mergers of galaxies of uniform mass M independentof general conditions of the cluster. This is in accordance with the observation results ofHoessel (1980).Figure 13 gives information about the number of constructive mergers per final galaxynumber, i.e. the fraction of galaxies that merged with other galaxies during the simulation.This percentage is equivalent to the averaged number of mergers a galaxy has undergone. 36 –This fraction rises going from high to small cluster radii and amounts to between someand about hundred per cent. Toomre (Shu (1982)) speculated that perhaps all ellipticalcluster galaxies have been built by mergers. From the fraction of galaxies that are presentlyinvolved in gravitational interactions and the probable life time of this passing phenomenahe calculated that merging of spiral galaxies happens for a fraction of about ten per cent ofall cluster galaxies, in good agreement with the results of this work. The fact that especiallyrich compact clusters which are assumed to be dynamical older systems than spiral rich oneshave a significantly larger fraction of elliptical galaxies (up to 35 per cent in the Coma cluster(cf. Oemler (1974))) may have multiple reasons. First a comparison of the simulations inSects. 4.1 and 4.2 shows that mergers grind to a halt at rising fraction of intergalactic matter,as the fraction of prospective interacting galaxies decreases at constant total cluster massdue to mergers and destructions. Hence the merger rate is higher in a cluster of earlierstate of development (corresponding to irregular spiral rich clusters) than in the final stateof compact clusters. So the fraction of elliptical galaxies in Toomre’s estimation has to becorrected upwards. Secondly elliptical galaxies may be a result of disturbances caused bynearby passages without merging that would lead to its rise to the presently observed value.The percentage of galaxies destroyed in encounters (Fig. 12) yields an estimate of thedark matter fraction in clusters provided by destroyed galaxies. This fraction amounts upto 60 per cent corresponding to a ratio of cluster mass and total galaxy mass of k = 2 .
6. Conclusions
There is a common result from all simulations of Sect. 4: the cluster evolution is tightlyrelated to the evolution of the brightest galaxy of this cluster. Only at small galaxy num-bers and simultaneously at small cluster radii the dynamical development due to the modeldescribed in Sect.2 plays a noticeable role (cf. e.g. Fig. 10). At the same time increasingmerger action is a concomitant of increasing destruction of galaxies. So, in the parameterrange of rapid evolution, the number of galaxies depleted such that the evolved small clusterlooks like a compact group rather than an Abell cluster. Nevertheless merging is always thedominant process (at least for the most massive galaxies). For clusters of presently observedrichness and their most luminous member (being the comparative value to our simulation) 37 –the dynamical development of the mass spectrum and of correlated quantities as the BCGmass causes only little changes with respect to the initial conditions.Considering Fig. 6 and Fig. 8 the steadiness of BCG masses and their ensemble disper-sion attract attention (irrespective of compact groups). This approximate independence ofmasses on cluster radius is observed in reality, and there are two causes that balance eachother: At the beginning of the simulation the BCG masses are small and rise with increasingrichness. In the course of evolution mergers take place preferentially at low galaxy numbersresulting in a relatively constant BCG mass level after the development in dependence of theparameters R and N . Taking into account an additional correlation between cluster radiusand richness (e.g. Oemler (1974)) the BCG mass steadiness would turn out almost clearer,just as their dispersion would be narrower than stated in this work because of the restrictionin parameter space.Why does the BCG mass scatter remain significantly narrow even at the end of clusterdevelopment? First of all the number of mergers is relatively small for the mentioned reasons(between one and six merger events depending on the parameter set), and secondly theidentity of a galaxy to be BCG is fixed only when the cluster development is complete.So, not the mass dispersion of a distinct galaxy fixed before the starting development isinvestigated, but the mass dispersion of a galaxy that is a posteriori identified to be themost massive one. Therefore the BCG mass dispersion turns out to be smaller than in thecase of any individual galaxy chosen at the beginning of the simulation and following itsevolution.I thank Prof. Dr. R. Wielen and Prof. Dr. B. Fuchs for stimulating discussions andDr. S. Frink for help in the preperation of this work for publication. REFERENCES
Aguilar, L.A., White, S.D.M., 1985, ApJ, 295, 374Albinger, C., 1995, degree dissertation at Ruprecht-Karls-Universit¨at Heidelberg, GermanyBatcheldor, D., Marconi, A., Merritt, D., Axon, D.J., 2007, ApJ, 663, L85Bird, C.M., 1994, ApJ, 107, 1637Bode, P.W., Berrington, R.C., Cohn, H.N., Lugger, P.M., 1994, ApJ, 433, 479Dressler, A., 1978, ApJ, 222, 23 38 –Dubinski, J., 1998, ApJ, 502, 141Funato, Y., Makino, J., Ebisuzaki, T., 1993, PASJ, 45, 289Garijo, A., Athanassoula, E., Garc´ıa-G´omez, C., 1997, AA, 327, 930Geller, M.J., Peebles, P.J.E., 1976, ApJ, 206, 939Hoessel, J.G., 1980, ApJ, 241, 493Hoessel, J.G., Gunn, J.E., Thuan, T.X., 1980, ApJ, 241, 486Humason, M.L., Mayall, N.U., Sandage, A.R., 1956, AJ, 61, 97Jagemann, Th., 1997, degree dissertation at Ruprecht-Karls-Universit¨at Heidelberg, Ger-manyKrivitsky, D.S., Kontorovich, V.M., 1997, A&A, 327, 921Lauer, T.R., 1985, ApJ, 292, 104von der Linden, A., Best, P.N., Kauffmann, G., White, S.D.M., 2007, MNRAS, accepted forpublicationDe Lucia, G., Springel, V., White, S.D.M., Croton, D., Kauffmann, G., 2006, MNRAS, 366,499Merritt, D., 1983, ApJ, 264, 24Merritt, D., 1984, ApJ, 276, 26Merritt, D., 1985, ApJ, 289, 18Motl, M.M., Burns, J.O., Loken, C., Normann, M.L., Bryan, G., 2004, ApJ, 606, 635Oemler, Aug.Jr., 1974, ApJ 194, 1Ostriker, J.P., Hausman, M.A., 1977, ApJL, 217, 125Peres, C.B., Fabian, A.C., Edge, A.C., Allen, S.W., Johnstone, R.M., White, D.A., 1998,MNRAS, 298, 416Postman, M., Lauer, T.R., 1995, ApJ, 440, 28Sandage, A., 1972, ApJ, 178, 1 39 –Schechter, P., 1976, ApJ, 203, 297Schechter, P.L., Peebles, P.J.E., 1976, ApJ, 209, 670Shu, F.H., 1982, The physical universe, University science booksSpitzer, L., 1958, ApJ, 127, 17Springel, V., White, S.D.M., Tormen, G., Kauffmann, G., 2001, MNRAS, 328, 726Springel, V., et al., 2005, Nature, 435, 629Tr`evese, D., Cirimele, G., Appodia, B., 1996, A&A, 315, 365