Evolution of globular cluster systems in elliptical galaxies — I. Log-normal initial mass function

Abstract

We study the evolution of globular cluster systems (GCS) in elliptical galaxies and explore the dependence of their main properties on the mass and the size of the host galaxy.

We focus our attention on the evolution of the GCS mass function (GCMF), on the fraction of surviving clusters and on the ratio of the final to initial total mass in clusters; the dependence of these GCS properties on the structure of the host galaxy as well as their variation with the galactocentric distance inside individual host galaxies is thoroughly investigated. We adopt a log-normal initial GCMF with mean mass and dispersion (log M_i = 5.25 and σ_i = 0.6) similar to those observed in the external regions of elliptical galaxies where memory of initial conditions is likely to be well preserved.

After a survey over a large number of different host galaxies we restrict our attention to a sample of galaxies with effective masses, M_e, and radii, R_e, equal to those observed for dwarf, normal and giant ellipticals.

We show that, in spite of large differences in the fraction of surviving clusters, the final mean masses, log M_f, of the GCMF in massive galaxies (log M_e ≳ 10.5) are very similar to each other (log M_f≃5.16, M_V=−7.3, assuming ML_V = 2) with a small galaxy-to-galaxy dispersion; low-mass compact galaxies tend to have smaller values of log M_f and a larger galaxy-to-galaxy dispersion. These findings are in agreement with those of recent observational analyses.

The fraction of surviving clusters, N_fN_i, increases with the mass of the host galaxy, ranging from N_fN_i∼0.9 for the most massive galaxies to N_fN_i∼0.1 for dwarf galaxies. We show that a small difference between the initial and the final mean mass and dispersion of the GCMF and the lack of a significant radial dependence of log M_f inside individual galaxies do not necessarily imply that evolutionary processes have been unimportant in the evolution of the initial population of clusters. For giant galaxies most disruption occurs within the effective radius, while for low-mass galaxies a significant disruption of clusters takes place also at larger galactocentric distances.

The dependence of the results obtained on the initial mean mass of the GCMF is also investigated, and it is shown that outside the interval 4.7 ≲ log M_i ≲ 5.5 both the spread and the numerical values of log M_f are not consistent with those observed.

celestial mechanics, stellar dynamics, globular clusters: general, galaxies: elliptical and lenticular, cD, galaxies: star clusters

1 Introduction

The study of the properties of globular cluster systems (hereafter GCS) in our Galaxy and in external galaxies can provide important clues on the formation and the evolution of individual globular clusters and of their host galaxies. A large number of observational investigations have been carried out (see, e.g., Ashman & Zepf 1998 and Harris 2000 for recent reviews), and the wealth of data collected have provided several indications on the relations between the properties of GCS and those of the host galaxies, as well as on the dependence of the properties of individual GCS on the position inside their host galaxies.

On the theoretical side, a firmly established theory for the formation of globular clusters is still lacking, and the interpretation of a number of observational results is a matter of debate. Several theoretical studies addressing the evolution of the properties of GCS have concluded that evolutionary processes lead to the disruption of a significant number of globular clusters, and can change the shape and the parameters of the initial mass function of the system (see, e.g., Fall & Rees 1977, Fall & Malkan 1978, Caputo & Castellani 1984, Chernoff, Kochanek & Shapiro 1986, Chernoff & Shapiro 1987, Aguilar, Hut & Ostriker 1988, Vesperini 1994, 1997, 1998, Okazaki & Tosa 1995, Capuzzo Dolcetta & Tesseri 1997, Gnedin & Ostriker 1997, Murali & Weinberg 1997a, 1997b, Ostriker & Gnedin 1997 and Baumgardt 1998; see also Meylan & Heggie 1997 for a recent review on the dynamical evolution of globular clusters).

On the other hand, many observational investigations have shown that the mass functions of globular cluster systems (hereafter we will indicate the mass function of a globular cluster system by GCMF, and the luminosity function of a globular cluster system by GCLF) of galaxies with structures markedly different from each other, in which the efficiency of evolutionary processes should be different, are very similar to each other, and they are all well fitted by a log-normal function with approximately the same mean value and dispersion (see, e.g., Harris 1991); the turnover of the GCLF has been often used as a standard candle calibrated on the value of the turnover of GCS of galaxies in the Local Group for the determination of extragalactic distances (see, e.g., Jacoby et al. 1992).

As to the radial variations of the GCMF properties inside individual galaxies, while in some observational analyses it has been claimed that inner clusters tend to be more luminous than outer clusters (see Crampton et al. 1985 for the M31 GCS, van den Bergh 1995a, 1996 for the Galactic GCS, Gnedin 1997 for the GCS of the Milky Way, of M31 and of M87; see also Ostriker & Gnedin 1997 for a theoretical analysis of the data presented in Gnedin 1997, and Chernoff & Djorgovski 1989, Djorgovski & Meylan 1994 and Bellazzini et al. 1996 for evidence of differences in the structure of inner and outer Galactic clusters), most observational studies fail to report any significant radial trend in the properties of the GCMF (see, e.g., Forbes et al. 1996a, Forbes, Brodie & Hucra 1996b, 1997, Kavelaars & Hanes 1997, Harris, Harris & McLaughlin 1998 and Kundu et al. 1999; see also Gnedin 1997 for a discussion of the possible role of the statistical methodology used in the determination of the difference between the GCLF parameters of inner and outer clusters).

The apparent universality of the GCMF in galaxies with different structures and the lack of a strong radial gradient of the properties of the GCMF inside individual galaxies have been often interpreted, at odds with the conclusions of the theoretical studies mentioned above, as an indication that evolutionary processes do not play a relevant role in determining the current properties of GCS.

In some recent investigations the issue of the universality of the GCMF has been addressed more in detail (Whitmore 1997; Ferrarese et al. 2000; Harris 2000), and the absolute magnitude of the turnover of the GCLF has been determined using primary standard candles; these studies have shown that, in fact, there is an intrinsic scatter in the properties of GCMF of galaxies of the same type, and that there are some non-negligible differences between the mean properties of the GCMF of galaxies of different type. In particular, the results of the most recent of these studies (Harris 2000) show that (1) giant ellipticals have a mean turnover magnitude, equal to −7.33, with a galaxy-to-galaxy rms scatter equal to 0.15 (corresponding, for ML_V = 2, to log M = 5.16 and a scatter of 0.06), (2) dwarf ellipticals have a mean turnover about 0.4 mag fainter than giants, with a galaxy-to galaxy rms scatter of 0.6 mag (corresponding to log M = 4.99 and a scatter of 0.24), and (3) for disc galaxies, the mean turnover of the GCLF, appears to be about 0.15 mag brighter than that of giant ellipticals, with a dispersion of about 0.2 mag (log M = 5.21 and a scatter of 0.08).

In this paper we will focus our attention on elliptical galaxies, and we will explore the dependence of the evolution of the GCMF, of the fraction of clusters surviving after one Hubble time, and of the ratio of the final to the initial total mass of clusters on the structure of the host galaxy; the efficiency of evolutionary processes in producing a significant radial gradient of the properties of the GCMF inside individual galaxies will be studied. The results of our simulations will be compared with the observational data available for giant, normal and dwarf elliptical galaxies.

We will adopt a log-normal initial GCMF with parameters equal to those observed in the external regions of some galaxies where evolutionary processes are unlikely to have altered the initial conditions of the GCS; we will not consider here younger globular cluster populations which could have formed during mergers or interactions. In a companion paper (Vesperini 2000) we will consider the evolution of GCS with a power-law initial GCMF similar to that of young cluster systems observed in merging galaxies.

The layout of the paper is as follows. In Section 2 we describe the method adopted for this investigation. In Section 3 we explore the evolution of GCS located in a large set of host galaxies with different effective masses and radii, and derive a number of general results on the dependence of the evolution of GCS on the host galaxy; then we show the implications of our results for a sample of host galaxies with effective masses and radii equal to those estimated observationally for a number of ellipticals; the results obtained are compared with the observed trends in the properties of GCS in elliptical galaxies. In Section 4 we study the dependence of our results on the initial mean mass of the GCMF. In Section 5 we discuss and summarize all the results.

2 Method

In order to calculate the evolution of the masses of individual globular clusters in a GCS, we will adopt the expressions derived by means of large set of N-body simulations by Vesperini & Heggie (1997) and already used to study the evolution of the Galactic GCS in Vesperini (1998). The evolutionary processes we will consider are the mass-loss associated to the evolution of individual stars in a cluster, two-body relaxation, the presence of the tidal field of the host galaxy, and dynamical friction. The effects due to the time variation of the tidal field for clusters on elliptical orbits (see, e.g., Weinberg 1994a,b,c and Gnedin, Hernquist & Ostriker 1999) were not considered in the simulations of Vesperini & Heggie and are not included here. For details on the N-body simulations we refer to Vesperini & Heggie.

Fitting the time evolution of the mass of individual clusters as obtained by N-body simulations, Vesperini & Heggie (1997) derived the following analytical expression for a cluster with initial mass M_i and located at a Galactocentric distance R_g:

 

(1)

where t is time measured in Myr, ΔM_st.ev.M_i is the mass-loss due to stellar evolution (see equation 10 in Vesperini & Heggie 1997), and F_CW is a parameter introduced by Chernoff & Weinberg (1990) which is proportional to the initial relaxation time of the cluster and is defined as

 

(2)

where N is the total initial number of stars in the cluster, and v_c is the circular velocity around the host galaxy.

For the host galaxy we will assume a simple isothermal model with constant circular velocity.

The effects of dynamical friction at any time t are included by removing, at that time, all clusters with time-scales of orbital decay (see, e.g., equation 7.26 in Binney & Tremaine 1987) smaller than t.

Each host galaxy we have considered is characterized by a value of the effective radius, R_e, and of the effective mass M_e. For each GCS we have explored, we have drawn 20 000 random values of M_i according to the initial GCMF chosen, and the distances of clusters from the centre of the host galaxy are such that the number of cluster per cubic kpc is proportional to with R_g ranging from 0.16R_e to 5R_e. The adopted profile for the radial distribution is similar to that observed for Galactic halo clusters; a similar slope for the initial radial distribution has been obtained by Murali & Weinberg (1997a) from detailed models of the evolution of the GCS of M87.

We have studied the evolution of GCS in a large set of different host galaxies: Fig. 1 shows all the pairs (log M_e, log R_e) considered.

Figure 1.

Set of values of the effective mass, M_e (M_⊙), and of the effective radius, R_e (kpc), of the host galaxies considered in the paper.

The evolution of each GCS has been followed for one Hubble time, here taken equal to 15 Gyr.

3 Results

3.1 General results

In this section we discuss the results obtained adopting a log-normal initial GCMF with mean mass log M_i = 5.25 and dispersion σ_i = 0.6, which is similar, for example, to the GCMF of outer (and thus unlikely to be significantly affected by evolutionary processes) clusters in M87 (see, e.g., McLaughlin, Harris & Hanes 1994 and Gnedin 1997). Figs 2(a) and (b) show the contour plots of the final (at t = 15 Gyr) mean mass and dispersion of the GCMF, log M_f and σ_f, in the log M_e−log R_e plane. In most host galaxies with low values of M_e, for which dynamical friction is more important than evaporation, log M_f < log M_i, while log M_f > log M_i for high-mass host galaxies where mass-loss and disruption due to two-body relaxation are more important. It is interesting to note that for a large number of host galaxies, log M_f and σ_f do not differ significantly from their initial values.¹Figs 2(c) and (d) show the contour plots of the ratio of the final to the initial total number of clusters, N_fN_i, and of the ratio of the final to the initial total mass of clusters, M_GCS,f /M_GCS,i. The four panels of Fig. 2 clearly show that a small difference between the initial and the final parameters of the GCMF does not necessarily imply that the number of clusters disrupted by evolutionary processes is negligible. As already shown in Vesperini (1998) for the Galactic globular cluster system, the Gaussian shape is always well preserved during evolution, and the disruption of a large number of clusters does not always lead to a strong variation in the values of the initial mean mass and dispersion.

Figure 2.

Contour plots of (a) log M_f, (b) σ_f, (c) N_fN_i and (d) M_GCS,fM_GCS,i in the log M_e−log R_e plane.

The minima of the curves of constant N_fN_i and M_GCS,fM_GCS,i correspond approximately to the transition from a regime dominated by evaporation to one dominated by dynamical friction: if only the effects of evaporation were taken into account, the curves of constant N_fN_i and M_GCS,fM_GCS,i would go down from the upper right to the lower left region of the log M_e−log R_e plane; as dynamical friction takes over as the dominant evolutionary process, the curves of constant N_fN_i and M_GCS,fM_GCS,i turn up, leading to values of these quantities smaller than those one would have if evaporation were the only process included.

Fig. 3 shows the contour plot of the difference, Δlog M_inn-out, between the mean mass of inner clusters (defined as those with galactocentric distances smaller than R_e) and outer clusters (those located at galactocentric distances larger than R_e).

Figure 3.

Contour plot of the difference between the final mean mass of inner clusters (R_g < R_e) and of outer clusters (R_g > R_e), Δlog M_inn-out, in the log M_e−log R_e plane.

Since the efficiency of evolutionary processes depends on the galactocentric distance, the formation of a radial gradient in the parameters of the GCMF has to be expected; nevertheless, as shown in Fig. 3, in several cases this gradient is very weak. Moreover, since the observed positions of globular clusters in external galaxies are not real galactocentric distances but projected distances, any radial gradient will appear even weaker, and its detection is likely to be difficult (see, e.g., Harris 2000).

Inner clusters are those more affected by evolutionary processes: depending on whether the dominant evolutionary process is dynamical friction or evaporation and on the balance between disruption of clusters by these evolutionary processes and the evolution of the masses of the clusters which survive (see Section 3.1 in Vesperini 1998 for a detailed discussion of the effects of disruption and evolution of the masses of surviving clusters) Δlog M_inn-out can be positive or negative and, as shown in Fig. 3, in a number of cases it is very close to zero. Another point to note is that, in principle, the radial gradient of log M, if investigated in finer radial bins, does not need to be monotonic: it is conceivable that as R_g increases, the dominant evolutionary process and the balance between disruption and evolution of the masses of the surviving clusters which are responsible for the evolution of log M, can change in a way leading to a non-monotonic behaviour of log M.

Fig. 4 shows Δlog M_inn-out versus the fraction of surviving clusters, N_fN_i: it is important to remark that, while all GCS characterized by a large value of Δlog M_inn-out have had their initial population of clusters significantly depleted by evolutionary processes, a weak or negligible radial gradient of the mean mass of the GCMF does not imply that evolutionary processes have not played an important role in modelling the properties of individual clusters and in disrupting a significant fraction of the initial number of clusters; as Fig. 4 shows, there are a large number of systems in which much less than 50 per cent of the initial number of clusters have survived, but still Δlog M_inn-out is equal to or close to zero.

Figure 4.

Difference between the final mean mass of inner clusters (R_g < R_e) and of outer clusters (R_g > R_e), Δlog M_inn-out, versus the fraction of surviving clusters after 15 Gyr, N_fN_i.

3.2 Implications

We now turn our attention to the implications of the results discussed above for a sample of host galaxies with structural properties known from observations. We will refer to the sample of elliptical galaxies for which the values of M_e and R_e are provided in Burstein et al. (1997). Here we will adopt values of M_e and R_e corresponding to H₀ = 75 km s⁻¹ Mpc⁻¹. Figs 5(a)–(d) show the contour plots of log M_f, σ_f, N_fN_i and M_GCS,fM_GCS,i in the log M_e−log R_e plane with the observational values of log R_e and log M_e superimposed; these figures illustrate the expected final properties of the GCS of the galaxies considered.

Figure 5.

Contour plots of (a) log M_f, (b) σ_f, (c) N_fN_i and (d) M_GCS,fM_GCS,i in the log M_e−log R_e plane (already shown in Fig. 2), with the observational values of log M_e and log R_e for elliptical galaxies (data from Burstein et al. 1997) superimposed as filled dots.

The fraction of the initial population of clusters which survive after one Hubble time and the ratio of the final to the initial total mass in clusters for the sample of galaxies considered span the entire range of possible values; in contrast with such a broad range of values of N_fN_i and M_GCS,fM_GCS,i, Fig. 5(a) shows that most galaxies (in particular, those with log M_e ≳ 10) occupy a region of the log M_e−log R_e plane corresponding to a very narrow range of log M_f: a small range of values oflog M_fin galaxies with different structures implies neither that the fraction of the initial number of clusters which have been disrupted in these galaxies is similar, nor that evolutionary processes did not alter the initial conditions of individual clusters. This point is further illustrated in Figs 6(a)–(d), where we have plotted log M_f, σ_f, N_fN_i and M_GCS,fM_GCS,i versus the mass of the host galaxy. Fig. 6a shows that log M_f is approximately constant for log M_e ≳ 10–10.5, while it decreases for smaller galaxies; this result is perfectly consistent with the findings of the observational analysis by Harris (2000) discussed above in the Introduction, which shows that the mean luminosity of globular clusters is approximately constant in giant ellipticals, and that the mean luminosity of clusters in dwarf galaxies is fainter than that of giant ellipticals.

Figure 6.

(a) log M_f, (b) σ_f, (c) N_fN_i, and (d) M_GCS,fM_GCS,i from the simulations discussed in Section 3.2 versus the logarithm of the effective mass of the host galaxy for a set of globular cluster systems located in host galaxies with values of R_e and M_e equal to the observational values plotted in Fig. 5.

Figs 6(c) and 6(d) show that N_fN_i and M_GCS,fM_GCS,i increase with the mass of the host galaxy. The trend of N_fN_i and M_GCS,fM_GCS,i to increase with the mass of the host galaxy results from the relation between the observational values of M_e and R_e (for log M_e ≳ 10, see Burstein et al. 1997 for further details on the Fundamental Plane relations for the sample of elliptical galaxies considered here), which is such that both the efficiency of relaxation and that of dynamical friction decrease as M_e increases (for example, implies ). This is clearly illustrated in Fig 5(c): the slope of the relation between log M_e and log R_e is steeper than the slope (referring, for example, to the high-M_e side of the plot) of the curves of constant N_fN_i and, as M_e increases, observational points in the log M_e−log R_e plane cross curves corresponding to larger and larger values of N_fN_i.

Fig. 6(c) is also important for its implications for the dependence of the number of clusters on the mass of the host galaxy. Several investigations have been carried out to study the abundance of globular clusters as a function of the properties of the host galaxies: the specific frequency, S_N, defined as the number of clusters per unit luminosity (Harris & van den Bergh 1981), has been determined for a number of galaxies and, more recently, Zepf & Ashman (1993) have introduced a mass-normalized specific frequency defined as the ratio of the total number of clusters to the total mass of the host galaxy. In general, observational analyses (see, e.g., Ashman & Zepf 1998 and Elmegreen 2000, and references therein) have shown that ellipticals tend to have specific frequencies larger than spirals and, within the sample of ellipticals, the specific frequency tends to increase with the luminosity of galaxies (see, e.g., Djorgovski & Santiago 1992, Santiago & Djorgovski 1993, Zepf, Geisler & Ashman 1994 and Kissler-Patig 1997). A similar trend is observed also for the mass normalized frequency (see Ashman & Zepf 1998). For nucleated dwarf galaxies this trend is reversed, and the specific frequency tends to increase with decreasing luminosity of the host galaxy (van den Bergh 1995b; Durrell et al. 1996; Miller et al. 1998). The origin of the observed trend between specific frequency and luminosity (or mass) of the host galaxy is still not clear (see, e.g., Elmegreen 2000 and Harris 2000 for recent reviews). In a recent work, McLaughlin (1999; see also Blakeslee 1999) has shown that the observed trend is consistent with a constant efficiency of cluster formation and a varying efficiency of star formation per unit gas mass available; the trend for nucleated dwarf galaxies could be ascribed, according to McLaughlin's analysis, to a smaller fraction of gas retained (and thus of stars formed) in these galaxies after the formation of globular clusters.

Our analysis can provide information only on the ratio of the final to the initial number of clusters; only with the additional knowledge of the dependence of N_i on the mass of the host galaxy would it be possible to determine the dependence of the current number of clusters on the host galaxy mass. Fig. 6(c) shows that the fraction of surviving clusters depends on the mass of the host galaxy; though with a large scatter, a power-law scaling fits our theoretical results well. This implies that for N_i∼M^α and ML∼L^β, N_f∼M^α+0.35 or N_f∼L^{(1+β)(α+0.35)}; for N_i∼L^λ, or N_f∼L^{0.35(1+β)+λ}.

We do not aim at a detailed fit of our results with the observational correlations, but it is interesting to note that, for example, either assuming N_i∝M or N_i∝L, our results imply that evolutionary processes will lead to a dependence of the current specific frequency on the mass (and on the luminosity as well) of the host galaxy: in particular, for N_i∝L we obtain N_f∼M^1.16 or, adopting ML∼L^0.24 (see, e.g., Faber et al. 1987), N_f∼L^1.43; for N_i∝M we obtain N_f∼M^1.35 or N_f∼L^1.67. We thus conclude that, starting with a mass- (or luminosity-) independent specific frequency, evolutionary processes can lead to a trend mass(or luminosity)-specific frequency consistent with that observed. The possible role of evolutionary processes in producing a luminosity-specific frequency correlation has been pointed out also by Murali & Weinberg (1997a), and this result is in agreement with their conclusion.

It is important to remark that our analysis shows that evolutionary processes can lead to a trend mass(or luminosity)-specific frequency without producing, at the same time, any correlation (which would be in contrast with observations) between the mean mass of the GCMF and the mass of the host galaxy for galaxies with log M_e ≳ 10.5.

Figs 7(a)–(c) show the histograms of log M_f, N_fN_i and M_GCS,fM_GCS,i for galaxies with log M_e > 10.5: the average value of log M_f for these galaxies is equal to 5.16, with a galaxy-to-galaxy rms dispersion of about 0.03; these values are in good agreement with those found by Harris (2000) for giant galaxies. It is interesting to emphasize again the contrast between the relatively narrow distribution of log M_f and the much broader distributions of N_fN_i and M_GCS,fM_GCS,i.

Figure 7.

Distribution of (a) log M_f, (b) N_fN_i, and (c) M_GCS,fM_GCS,i from the simulations for globular clusters systems located in host galaxies with values of R_e and M_e equal to the observational values plotted in Fig. 5 and having log M_e > 10.5.

In order to illustrate the evolution of the distribution of log M for galaxies with log M_e > 10.5, we have plotted in Fig. 8 the time evolution of the values of log M corresponding to the 10th, 50th and 90th percentiles of the log M distribution. The initial rapid decrease of log M is caused by the mass-loss due to stellar evolution, which does not depend on the host galaxy; the subsequent evolution, which is essentially determined by two-body relaxation and dynamical friction, depends on the structure of the host galaxy and broadens the distribution of log M with a slight increase of the median log M.

Figure 8.

Time evolution of the mean mass corresponding to the 10th (bottom curve), 50th (central curve) and 90th (top curve) percentiles of the distribution of mean masses of globular clusters systems located in the subsample of host galaxies with values of R_e and M_e equal to the observational values plotted in Fig. 5 and log M_e > 10.5 (see Fig. 7a for the complete distribution of log M at t = 15 Gyr for the same sample of host galaxies).

For galaxies with log M_e < 10.5, plots like those shown in Figs 7 and 8 are less significant because of the smaller number of objects for which data are available and of the non-uniform distribution of these objects in the log M_e−log R_e plane. The important point to note is that, in general, as is evident from Fig. 6a, there is a clear trend for low-mass galaxies to have a log M_f smaller than that of high-mass galaxies and a larger galaxy-to-galaxy dispersion; these differences are perfectly consistent with the findings of the observational analysis by Harris (2000).

In order to further explore the dependence of the properties of the GCMF on the host galaxy and to better illustrate the differences between giant, normal and dwarf galaxies, we have considered the subsample of all the initial conditions investigated in Section 3.1 corresponding to the region where most observational data are located (see Fig. 9). For four representative cases, shown in Fig. 9 by filled dots, a further detailed analysis of the time evolution of the properties of the GCMF and of their dependence on the galactocentric distance has been carried out, and the results are discussed below in Section 3.3. We have divided the sample of host galaxies into three subsamples according to their masses: high-mass galaxies with log M_e > 10.5, intermediate-mass galaxies with 9.5 < log M_e < 10.5, and low-mass galaxies with log M_e < 9.5.

Figure 9.

Subsample of values of R_e and M_e considered for the investigation described in the final part of Section 3.2. Filled dots indicate the values of log M_e and log R_e for the host galaxies considered in Section 3.3 for a more detailed study of the time evolution and of the dependence on the galactocentric distance of the main GCS properties.

The panels in the left column of Fig. 10 show the distribution of log M at t = 2, 5, 10, 15 Gyr for the three subsamples considered. These plots further illustrate the expected differences in the mean mass of the GCMF of different classes of galaxies and the intrinsic galaxy-to-galaxy dispersion in the distribution of log M in galaxies of the same type; the result discussed above is confirmed: low-mass galaxies tend to have lower values of log M and a larger galaxy-to-galaxy dispersion.

Figure 10.

Evolution of the distribution of log M (left-hand panels), of the ratio of the total number of clusters survived at time t to the total initial number of clusters, N(t)/N_i (central panels), and of the ratio of the total mass of clusters survived at time t to the total initial mass of all clusters, M_GCS(t)/M_GCS,i, (right-hand panels) for the set of host galaxies with values of R_e and M_e plotted in Fig. 9. Each row corresponds to the time indicated on the first panel on the left. In each panel the solid line shows the distribution for globular cluster systems in host galaxies with log M_e > 10.5, the dashed line that for host galaxies with 9.5 < log M_e < 10.5, and the dotted line that for host galaxies with log M_e < 9.5.

The panels in the central and in the right columns of Fig. 10 show the distributions of the fraction of surviving clusters and of the ratio of the total mass in clusters at t = 2, 5, 10, 15 Gyr to the total initial mass in clusters for the same samples of host galaxies discussed above.

3.3 Detailed analysis of the evolution of GCS in four fiducial host galaxies

In this subsection we focus our attention on four fiducial galaxies with values of log M_e and log R_e shown by filled dots in Fig. 9, and we study in more detail the evolution and the final properties of their GCS.

In Fig. 11 we show the time evolution of log M and σ, of the fraction of surviving clusters and of the ratio of the total mass of surviving clusters to the total initial mass of clusters. While log M evolves significantly only in the first 2–3 Gyr, individual clusters are continuously disrupted, and those which survive keep losing mass (see Figs 11c and 11d): after the initial evolution, the gross properties of the GCMF adopted do not evolve significantly, despite the continuous disruption of clusters and the evolution of the masses of the surviving clusters.

Figure 11.

Time evolution of the mean mass of the GCMF log M (a) of the dispersion of the GCMF σ, (b) of N(t)/N_i, (c) of M_GCS(t)/M_GCS,i, and (d) for globular cluster systems in host galaxies with effective masses and radii plotted in Fig. 9 as filled dots [open dots are for (log M_e,log R_e)=(12,1.4), crosses for (log M_e,log R_e)=(11,0.7), triangles for (log M_e,log R_e)=(10,0.1), and filled dots for (log M_e,log R_e)=(9, −0.2)].

In Figs 12(a)–(c) we plot log M_f, N_fN_i and M_GCS,fM_GCS,i versus the galactocentric distance. For the two most massive galaxies considered, evolutionary processes are efficient only within a galactocentric distance approximately equal to R_e, while most clusters survive at larger radii where most of the mass-loss is that associated to the effects of stellar evolution. For less massive and more compact galaxies, a strong disruption and mass-loss occur well beyond R_e, and even at R≃2–3R_e about 50 per cent of the clusters are disrupted. In spite of the significant disruption, only in the least massive and most compact host galaxy considered, a strong radial gradient of log M_f extending beyond R_e is produced by the effects of evolutionary processes.

Figure 12.

(a) log M_f, (b) N_fN_i and (c) M_GCS,fM_GCS,i versus the galactocentric distance (normalized by the effective radius of the host galaxy) for globular cluster systems in host galaxies with effective masses and radii plotted in Fig. 9 as filled dots (symbols for different host galaxies as in Fig. 11).

The plot of N_fN_i versus R_g is of interest for the interpretation of the data on the radial dependence of the specific frequency. Observational data (see, e.g., McLaughlin 1999, and references therein) show that GCS have, in general, a spatial distribution less centrally concentrated than the stellar haloes of their host galaxies with cores larger than those of the stellar halo, while in the outer regions the spatial distribution of clusters usually matches that of the stellar halo; this implies that the local specific frequency increases with the galactocentric distance in the central regions, and tends to a constant value in the outer parts of the host galaxy.

Our results show that the increase of the specific frequency with R_g for R_g ≲ 1–2R_e can be, at least in part, the result of the effects of evolutionary processes (see also Capuzzo Dolcetta & Tesseri 1997 and Murali & Weinberg 1997a), while for R_g ≳ 1–2R_e evolutionary processes have been unimportant in giant galaxies, and any observed radial trend has to be ascribed to the processes of cluster formation. Of course, it is not possible to exclude the possibility that the decrease of the specific frequency in the central regions was not in part imprinted by the formation processes and subsequently further strengthened by evolutionary processes.

4 Dependence of the results on the mean mass of the initial GCMF

We conclude our investigation by exploring the dependence of the main final properties of GCS on the initial value of log M (as to the initial dispersion, we will keep the value adopted until now, σ = 0.6).

We will restrict our attention to galaxies with log M_e > 10.5. We have adopted the following values of log M_e and log R_e: (log M_e, log R_e)=(12,1.4), (11.75,1.18), (11.5,1), (11.25,0.8), (11.0,0.64), (10.75,0.52), (10.5,0.4); these values span the entire strip of the log M_e−log R_e plane covered by real galaxies with log M_e > 10.5.

The main evolutionary process responsible for the disruption of clusters is evaporation due to internal relaxation for GCS with low values of log M_i, or, as log M_i increases, disruption of high-mass clusters by dynamical friction. As discussed in Vesperini (1998), log M_f is larger or smaller than log M_i, depending on the balance between disruption of low-mass clusters by evaporation, disruption of high-mass clusters by dynamical friction and evolution of the masses of the clusters which survive: for low values of log M_i most of the disrupted clusters are low-mass clusters and, in general, log M_f > log M_i, while, as log M_i increases, disruption by dynamical friction becomes the most important process and most of the disrupted clusters are those in the high-mass tail of the initial GCMF which leads to log M_f < log M_i.

Fig. 13 shows log M_f, N_fN_i and M_GCS,fM_GCS,i as functions of log M_i for the seven fiducial host galaxies considered. The range of values spanned by log M_f is larger for low and high values of log M_i while for intermediate values of log M_i (4.7 ≲ log M_i ≲ 5.5), in spite of the significant spread of values of N_fN_i and of M_GCS,fM_GCS,i, the range of values of log M_f is very small.

Figure 13.

(a) log M_f, (b) N_fN_i and (c) M_GCS,fM_GCS,i versus the initial value of log M (assuming a log-normal initial GCMF with σ = 0.6) for the seven host galaxies with log M_e > 10.5 discussed in Section 4. In each panel the seven curves shown correspond, from the top to the bottom one, to host galaxies with decreasing values of M_e [in panel (a) the curves cross each other, and we refer to the order of the seven curves on the right-hand side of the plot].

As shown in Figs 13(b) and (c), low-mass galaxies are generally those where disruption processes are more efficient and where the difference between log M_i and log M_f is larger. The dependence of log M_f on M_e varies with log M_i: for low values of log M_i, log M_f decreases as M_e increases, while log M_f increases with M_e for high values of log M_i. Since here we have restricted our attention to galaxies with log M_e > 10.5, for which observational analyses show that log M spans a narrow range of values, our results seem to exclude the possibility that the initial GCMF was a log-normal function with low (log M_i ≲ 4.7) or high (log M_i ≳ 5.5) values of the mean mass. Moreover, the values of log M_f obtained for log M_i ≲ 4.7 and for log M_i ≳ 5.5 do not fall in the range of observed values.

5 Summary and conclusions

In this paper we have studied the evolution of globular cluster systems in elliptical galaxies. We have followed the evolution of globular cluster systems with a log-normal initial GCMF with initial mean mass and dispersion similar to those currently observed in the external regions of some galaxies where evolutionary effects are unlikely to have significantly altered the initial conditions (log M_i = 5.25 and σ_i = 0.6 corresponding, for ML_V = 2, to M_V=−7.5 and σ_V = 1.5); a large set of N-body simulations carried out by Vesperini & Heggie (1997) have been used to determine the evolution of the masses of individual globular clusters.

In the first part of the paper we have carried out a survey over a large sample of values of the effective mass, M_e, and radius, R_e, of the host galaxy, and we have investigated in detail the dependence of the final mean mass and dispersion of the GCMF, of the fraction of surviving clusters, and of the ratio of the final to the initial total mass of clusters on M_e and R_e. The difference between the properties of the GCMF of inner and outer clusters has been investigated too. Contour plots of log M_f, σ_f, N_fN_i, M_GCS,fM_GCS,i, Δlog M_inn-out in the log M_e−logR_e plane (see Figs 2, 3 and 4) have allowed us to determine the relation between the evolution of the GCMF and the fraction of clusters disrupted, and to show the dependence of the evolution of a GCS on the structure of the host galaxy.

In the second part of the paper we have focused our attention on a subset of values of M_e and R_e equal to the observational values available for giant, normal and dwarf ellipticals, and we have compared our theoretical results with the results of some recent observational analyses (see Figs 5–10).

• (i)

  For galaxies with log M_e ≳ 10.5, our results are in general qualitative and quantitative agreement with the findings of several observational studies: log M_f is approximately constant (log M_f≃5.16 or M_V=−7.3 for ML_V = 2) with a small galaxy-to-galaxy dispersion. We have shown that the narrow range spanned by log M_f for galaxies with different structures and the lack of a strong radial gradient in the properties of GCMF inside many individual galaxies do not imply that evolutionary processes have been unimportant: in most of the host galaxies we have considered, a significant fraction of clusters have been disrupted, and the masses of many of those which survived have changed; in contrast with the narrow range of values of log M_f, the fraction of surviving clusters for different host galaxies, N_fN_i, spans almost the entire range of possible values.

• (ii)

  As to dwarf galaxies, our results show that evolutionary processes lead to values of log M_f smaller than those of giant ellipticals, and to a larger galaxy-to-galaxy dispersion. Both these findings are in agreement with those of observational analyses (Harris 2000).

• (iii)

  The fraction of surviving clusters increases with the mass of the host galaxy, and it ranges from N_fN_i∼0.9 for the most massive galaxies to N_fN_i∼0.1 for dwarf galaxies. A mass-(or luminosity-)specific frequency correlation can result from the effects of evolutionary processes (see also Murali & Weinberg 1997a). The ratio of the final to the initial total mass of clusters ranges from M_GCS,fM_GCS,i∼0.8 for the most massive galaxies to M_GCS,fM_GCS,i∼0.01 for dwarf galaxies.

• (iv)

  The time evolution of the distribution of log M, N(t)/N_i, M_GCS(t)/M_GCS,i for all the galaxies considered, as well as the detailed time evolution of the GCMF parameters, of N(t)/N_i, and of M_GCS(t)/M_GCS,i for four fiducial galaxies, has been investigated (see Figs 8, 10 and 11); this allows us to determine the expected properties of GCS in elliptical galaxies younger than the adopted age of 15 Gyr. Our analysis shows that while the number of clusters and their total mass, N(t)/N_i and M_GCS(t)/M_GCS,i, continue to decrease for 15 Gyr, the mean mass of clusters, log M, evolves significantly only in the first few Gyr.

• (v)

  For some fiducial galaxies we have studied in greater detail the dependence of the properties of the GCMF and of the fraction of surviving clusters on the galactocentric distance (see Fig. 12). We have shown that in massive galaxies disruption occurs mainly within R_e, while in low-mass compact galaxies a significant fraction of clusters are disrupted also at R_g > R_e. The difference in concentration between the spatial distribution of clusters and that of stars is, at least in part, due to the disruption of inner clusters.

  As discussed above, we have shown that there are several systems in which the radial gradient of log M_f is negligible, in spite of a significant disruption of clusters.

• (vi)

  For a sample of fiducial galaxies with log M_e > 10.5, we have studied the dependence of the final properties of the GCMF on the initial value of log M, log M_i. We have shown how the dependence of log M_f on the mass of the host galaxy, the spread of values of log M_f and the fraction of surviving clusters vary with log M_i (see Fig. 13). log M_f increases as log M_i increases; for a given value of log M_i, the spread of values of log M_f is larger for log M_i ≲ 4.7 and log M_i ≳ 5.5, while for intermediate values of log M_i (4.7 ≲ log M_i ≲ 5.5) the range of log M_f is small.

A small spread of values of log M_f in different galaxies is never the result of negligible disruption or of a similar fraction of surviving clusters in the host galaxies considered. The properties of the final GCMF are always determined by the interplay between disruption of clusters and evolution of the masses of the clusters which survive.

In this paper we have focused our attention on GCS with a log-normal initial GCMF. In a companion paper (Vesperini 2000) we will study the evolution of GCS, starting with a power-law initial GCMF similar to that observed in young cluster systems in merging galaxies, and we will investigate whether the final properties of GCS starting with such a functional form for the initial GCMF are consistent with those currently observed in old globular cluster systems.

Acknowledgments

I thank Giuseppe Bertin and the referee, Oleg Gnedin, for many useful comments on the paper.

Support from a Five College Astronomy Department fellowship is acknowledged.

¹

In comparing initial and final values of log M, one has to consider that, no matter how efficient dynamical friction and evaporation are, all GCS have their initial total mass and mean mass decreased because of the mass-loss associated with stellar evolution; for the initial stellar mass function used in the simulations by Vesperini & Heggie (1997), on which this investigation is based, after 15 Gyr the mass lost by each cluster because of stellar evolution amounts to about 18 per cent of the initial total mass; this means that also in those GCS in which no cluster is disrupted, the final total mass and the final mean mass are always smaller than or equal to approximately 0.82 times the corresponding initial values.