View metadata, citation and similar papers at core.ac.uk brought to you by CORE

provided by CERN Document Server Mon. Not. R. Astron. Soc. 000, 000–000 (2001)

Evolution of globular cluster systems in three of the cluster

R. Capuzzo-Dolcetta, I. Donnarumma Dipartimento di Fisica, Universit`a La Sapienza, P.le Aldo Moro 2, I-00185 Roma, Italy [email protected], [email protected]

Accepted . Received ; in original form

ABSTRACT We studied and compared the radial profiles of globular clusters and of the stellar bulge component in three galaxies of the observed with the WFPC2 of the HST. A careful comparative analysis of these distributions confirms that are more concentrated toward the galactic centres than globular clusters, in agreement with what was observed in many other galaxies. If the observed difference is the result of evolution of the globular cluster system sstarting from initial profiles similar to those of the halo–bulge stellar components, a relevant fraction of their mass (74%, 47%, 52% for NGC 1379, NGC 1399 and NGC 1404, respectively) is disappeared in the inner regions, likely contributing to the nuclear field population, local dynamics and high energy phenomena in the primeval life of the . An indication in favour of the evolutionary interpretation of the difference between the globular cluster system and stellar bulge radial profiles is given by the positive correlation we found between the value of the mass lost from the GCS and the central galactic mass in the set of seven galaxies for which these data are available. Key words: galaxies: globular clusters; galaxies: distribution; galaxies: globular cluster evolution; galaxies: nuclei

1INTRODUCTION by Harris & Racine (1979) and Racine (1991); in their opinion globular clusters originated earlier, when the A lot of elliptical galaxies have globular cluster systems density distribution was less peaked. However, this hy- (hereafter GCSs) that are less concentrated to the galactic pothesis cannot explain why the two distributions are centre than the bulge-halo stars, thus showing radial distri- very similar in the outer galactic regions. butions with larger core radii than those of the bulges. Two (ii) another explanation is based on the simple and reason- classical examples of this phenomenon are the two Virgo able assumption of coeval birth of globular clusters and giants M87 and M49 (Grillmair et al., 1986, McLaughlin, halo stars (McLaughlin, 1995), with a further evolution 1995), to which our Galaxy and M31 have been added later of the GCS radial distribution, while the collisionless (Capuzzo-Dolcetta & Vignola, 1997). The sample of galaxies halo stands almost unchanged. that shows this particular feature has been recently enlarged thanks to 11 elliptical galaxies selected by Capuzzo-Dolcetta The GCS evolution is due to & Tesseri (1999) in a sample of 14 galaxies with kinemat- ically distinct cores (Forbes et al.,1996) observed with the (i) dynamical friction, that brings massive clusters very WFPC2 of the . Though we could close to galactic centre, not yet conclude that this phenomenon is common to all (ii) tidal interaction with a compact nucleus (see for exam- galaxies, we may say that there is no case where the halo ple Capuzzo-Dolcetta, 1993). stars are less concentrated than the GCS. Moreover there is a general agreement on that the difference between the two The combined effect of these dinamical mechanisms acts to radial distributions is real and not due to a selective bias. deplete the GCSs in the central, denser regions, leaving al- Consequently, different hypotheses have been advanced with most unchanged the outer profile, that so remains similar the purpose to explain this feature. Among these, two seem to the halo stars’ one. The efficiency of the mentioned phe- the most probable: nomena is higher in galactic triaxial potentials (see for exam- ple Ostriker et al., 1989, Pesce, Capuzzo-Dolcetta & Vietri, (i) the difference between the two distributions reflects dif- 1992), in which we find a family of orbits, the ‘box-like’, that ferent formation ages of the two systems, as suggested do not conserve any component of their angular momentum 2 R. Capuzzo–Dolcetta and I. Donnarumma

2 and then well sample the central galactic regions. Pesce et al. (fi f0i) pi σ = − (2) (1992) showed as globular clusters moving on box orbits lose 2 f0ipi their orbital energy at a rate an order of magnitude larger P where f e f are, respectively, the fitted and the observed than on loop orbits of comparable size and energy. These i P 0i values of the surface density. We have defined the weights p results showed as the previous evaluations of the dynamical i as the inverse of the error bars of the surface density data. friction efficiency, based on very simplified hypotheses on The modified core model approximates very well the GCS the globular cluster orbit distribution, led to an overstimate density data in the inner regions of the two galaxies NGC of the dynamical braking time scales. On the contrary, it 4 1399 and NGC 1404 (σ 10− ), while the σ value results has been ascertained that massive globular clusters in tri- 3 ∼ a little higher ( 10− ) for NGC 1379. As we will see in axial potentials could be braked during their motion so to the following, this∼ is justified by the somewhat peculiar be- reach the inner regions in a relatively short time(Capuzzo- haviour of the light profile of the latter galaxy. By means Dolcetta, 1993). There they could have started a process of of fitting formula (1) we estimated the number of “missing” merging, giving origin to a central massive nucleus (eventu- clusters in a galaxy and the corresponding mass removed ally a black hole) or fed a preexistent one (Ostriker, Binney from its GCS. In fact, an approximated value of the mass & Saha, 1989; Capuzzo-Dolcetta, 1993). Under the hypoth- fallen to the galactic centre is obtained through the two val- esis that the initial GCS and halo-bulge radial distributions ues, N and m , of the number and of the mean mass of were the same, an accurate analysis of the observations al- l l the missingh (‘losti ’) globular clusters. A priori, the deter- lows an estimate of the number of “missing clusters” and mination of m needs a detailed evaluation of the effects of therefore of the mass removed from the GCSs. l evolutionaryh processesi on the original mass spectrum of the Actually, McLaughlin (1995), Capuzzo-Dolcetta & Vig- GCS. However, the most relevant phenomena (tidal shocking nola (1997) and Capuzzo-Dolcetta & Tesseri (1999), scaling and dynamical friction) act on opposite sides of the initial the radial surface profiles of the halo stars of a galaxy to that mass function then – if the initial mass function is not too its globular cluster system, estimated the number of missing asymmetric – we expect that the mean value of the glob- globular clusters as the integral of the difference between ular cluster mass has not changed very much in time (see the two radial profiles. A study like this led, for example, Capuzzo-Dolcetta & Tesseri, 1997) . Hence we can assume Capuzzo-Dolcetta & Vignola (1997) and Capuzzo-Dolcetta the present mean value of the mass of globular clusters, m , & Tesseri (1999) to suggest that the compact nuclei in our as a good reference value for m . To estimate m it ish thusi galaxy, M31 and M87 as well as in many other galaxies could l necessary to assume a mass functionh i that satisfactorilyh i rep- have reasonably sucked a lot of decayed globular clusters in resents the present distributions of globular clusters in the the first few Gyrs of life. The aim of this paper is an ex- three galaxies of our sample. On the basis of the observed tension of previous (McLaughlin 1995; Capuzzo-Dolcetta & similarity among the mass spectra of elliptical galaxies (Har- Vignola 1997; Capuzzo-Dolcetta & Tesseri, 1999) works to ris & Pudritz 1994; McLaughlin 1994), we have obtained a the globular cluster data of a sample of three elliptical galax- single mass spectrum for the Fornax galaxies normalizing ies in the Fornax cluster (NGC 1379, NGC 1399 and NGC the mass spectrum adopted by McLaughlin (1995) for the 1404) whose HST data were taken by Forbes et al. 1998a,b, Virgo giant M87. We have made this normalization using a hereafter referred to F98a and F98b. mass to light ratio β (M/L)V, equal to 1.5 for the GCS. The resulting mass spectrum≡ is

s N(m) m− (3) ∝ 2 THE DATA AND THE RESULTS with s defined as 5 In this Section we present and discuss our analysis of GCS s0 0.5 0.2 mmin m 1.2 10 M ≡ ± 5 ≤ ≤ × 6 radial distribution in the overmentioned galaxies in the For- s = s1 1.65 0.10 1.2 10 M m 1.5 10 M (4) ≡ ± × ×6 ≤ ≤ × nax cluster. The data come from recent HST WFPC2 obser- ( s2 3.0 0.31.5 10 M m mmax ≡ ± × ≤ ≤ vations (F98a, F98b) and, so, they take the advantage of the where m and m depend explicitly on β.Of course an HST excellent resolution, which allows an accurate analysis min max error in the value of s leads to an error in the evaluation of of the crowded inner regions of galaxies at Virgo and Fornax the mean mass m . Then we think important to check the cluster distance. Moreover it allows the subtraction of most degree of reliabilityh i of our method by estimating (in the way of background galaxies. The available density data are, as described in the Appendix A) the mean error made using the we will see later, at a high level of completeness. mass spectrum (3) to evaluate the mass. We have approximated the observed density profiles with a modified “core model”

Σ0 2.1 NGC 1379 Σ(r)= γ (1) 2 1+ r NGC 1379 is an EO galaxy in the Fornax cluster studied by, rc e.g., Harris & Hanes (1987), Kissler-Patig et al. (1997) and h i where Σ0, γ and rc are free parameters (rc is called core later F98b. Harris & Hanes (1987) compared the radial pro- radius, but, to be precise, it corresponds to the usual defini- file of the GCS in NGC 1379 with the surface brightness of ton of being the distance from the centre where the surface the galaxy itself as given by Schombert (1986). Their study distribution halves its central value just when γ =1.We was limited to the radial range 5-35 kpc, and they did not have chosen the free parameters in (1) minimizing the r.m.s detect any difference between the two profiles. This was con- error, i.e. firmed by Kissler-Patig et al. (1997) that used ground-based Evolution of globular cluster systems in three galaxies of the Fornax cluster 3 data over the 3-10 kpc range. The recent analysis of F98b, 5 based on data of the HST WFPC2, confirms the similarity between the two surface profiles in the outer regions but, at the same time, show as the two differ in the inner regions. 4 Thanks to these latter data, that cover a radial range larger than previously available, we managed to make a compar- ative study between the halo stars’ surface profile and the GCS one. 3

2.1.1 Observed data and the modified core model 2 In F98b detailed plots of the radial profile of the GCS of log surface density NGC 1379 are presented, thanks to the high resolution of HST. The profiles shown in the paper are two: one (contain- 1 ing GCs with B < 25.5) is complete and uncontaminated and the other (composed by objects with B> 26.5andwithout 0.25 0.5 0.75 1 1.25 1.5 1.75 correction for incompleteness) is expected to be composed log radius mostly by background galaxies. The radial profile of the brighter sample begins to de- Figure 1. Surface density profile for the the sample of globular crease from 10 to 80 arcsec ( 1 7 kpc); at 80 arcsec cluster system in NGC 1379 (black dots) with B<25.5from from the centre∼ the profile is lost∼ in− the background (see F98b. The solid line is the modified core model fit for GCS made in our study. The dotted curve is the surface brightness profile of F98b). the underlying galaxy, arbitrarly normalized to match the radial Fig.1 shows the radial profile of clusters with B < 25.5 profile of cluster system in the outer regions. Radial distance is 2 with a background of 2.7 objects per arcmin subtracted. 2 in arcsec (1 arcsec = 89 pc); surface density in arcmin− . The superposed dotted line represents the luminosity profile of the underlying galaxy as provided by Kissler-Patig et al. (1997), scaled vertically to match the GCS profile in the the two profiles differ, i.e. 1.77-37.58 arcsec (see Fig.1). The outer regions. In this way the two profiles agree well at 35- ∼ bulge–halo radial profile has an almost constant slope, -2.4, 70 arcsec (3-6 kpc). The logarithmic slope of the GCS profile in the range 10 to 50 arcsec from the centre, but for our is 2.4atr>35 arcsec. Inwards of 30 arcsec it flattens ∼− purpose it has been necessary to know the star profile also out. ∼ in the region within 10 arcsec.This has been made possible F98b estimated a core radius of 23 6 arcsec (2.0 0.5 thanks to Grillmair, who gave us (1999, private communi- ± ± kpc) assuming it as the distance from the centre where cation) the model used to subtract the stellar light from the surface density halves its central value. In its turn, the globular clusters profile (99 % complete for B<25.5). We galaxy light seems to decline with a constant slope from 10 obtained a logarithmic slope of the stellar profile of about -1 ∼ to 50 arcsec, where it seems to change slope slightly. To in the radial range 1.77-10 arcsec (see Fig.1). In the annulus ∼ be sure of completeness, we have initially analysed only the 1.77-37.58 arcsec, the number of stars is 512, on the other brighter sample (B< 25.5): in a second analysis, we added side, the modified core model fitting the globular cluster the clusters of the fainter sample. Fitting the GCS density distribution gives 132 as number of clusters observed in the data with the law (1), we have the best fit with Σ0 = 220.62 same annulus. Then, in the assumption that the normalized 2 arcmin− ,γ=1andrc = 23 arcsec (solid line in Fig.1). The stellar profile gives the initial distribution of globular clus- modified core model approximates very well the density data ters, we find that the number of “missing clusters” is 380. in the outer regions, while it seems to be an overestimate in Hence NGC 1379 could have lost about 74% of its initial the inner regions. According to F98b the central dip of the population of globular clusters. observed GCS radial distribution is not an artefact of their analysis, but it represents a real drop in the volume density of clusters near the nucleus of galaxy. In fact they estimated 2.1.3 Data completeness and the mass fallen to the a 97% completeness of the density data in the galactic region galactic centre inwards of 10 arcsec. The relevant number of clusters possibly missing in the inner The observed∼ decreasing behaviour toward the galactic region of NGC 1379 corresponds to a great amount of mass centre of the GCS distribution of NGC 1379 seems to indi- supposedly fallen close to the galactic centre. Remembering cate that the dynamical depleting mechanisms are efficient, that the value obtained of N is restricted to clusters with in this galaxy, up to significantly large radii. l B<25.5, we decided to provide two different evaluations of the mass removed from the GCS: one is the missing mass in the magnitude range of the observations, M , and another is 2.1.2 The “missing clusters” in NGC 1379 l the missing mass in the “whole” magnitude range, M˜ l. For As we said in the Introduction, the estimate of the (poten- this purpose it is useful to make some considerations on the tially) missing clusters can be obtained as the integral of GCS luminosity function. the difference between the GCS and halo stars’ surface den- F98b reported B and I photometry for 300 globu- sity distribution, after this latter has been normalized. For lar clusters candidates in NGC 1379. This sample∼ was de- this reason we have considered only the radial range where tected with 4 WFPC2 chips covering a total area of 4.8 4 R. Capuzzo–Dolcetta and I. Donnarumma arcmin 2. ThesamplewithB 26 is almost complete; at 3 B>26 the completeness begins≤ to decrease very sharply. . The principal source of contamination by foreground stars and background galaxies is due to compact, spherical galax- 2.5 ies and F98b provided a background-corrected luminosity function as a gaussian function with a peak magnitude of B =24.95 0.30 and a width σB =1.55 0.21. This lumi- hnosityi function± is almost complete and uncontaminated± in 2 the range 21

B V =0.7andamasstolightratio(M/L)V, =1.5, h − i thus obtaining (5.2 104 M

2.2.2 Number of missing globular clusters in NGC 1399 1.44 109M , which differs of about 10% from that previ-

ously× obtained. The presumed flattening of the GCs distribution in NGC 1399 has been deeply discussed. While Bridges et al. (1991) stated that the GCS has the same slope of the halo stars, Wagner et al. (1991) and Kissler-Patig (1997), on the con- 2.3 NGC 1404 trary, noticed a flattening of the GCS radial profile toward NGC 1404 is an E1 galaxy, about a magnitude fainter than the galactic centre. A weighted average of the slopes pro- NGC 1399, sited at about 10 arcmin south east of NGC vided by studies earlier than that of F98a brings to a value 1399. of 1.47 0.09 for the GCS and 1.72 0.06 for the halo As for NGC 1399, much work has been done on the GCS stars.F98a− ± found a difference of 0.−4 0.22± in the slopes be- ± of this galaxy. The estimates of the number of globular clus- tween their fitted GCS distribution profile and that of the ters of NGC 1404 are made difficult by its vicinity to NGC stellar halo, concluding that the flattening of the GCS den- 1399. Grillmair et al. (1997) speculated that a great number sity profile is a real feature. To provide an estimate of ‘miss- of the NGC 1399 GCs (those of intermediate metallicity) ing’ GCs in NGC 1399, we adopted for the stellar light the have been tidally stripped by NGC 1404. The difficulties in following profile distinguishing the two GCSs did not allow to obtain precise log I(r)= 1.6logr +2.502, results from earlier studies, based on photographic plates − (e.g., Hanes & Harris 1986 and Richtler et al. 1992). This obtained shifting the stellar halo density data (Forbes 1999, problem seems to have been solved by the recent studies by private communication) in the vertical direction, such to F98a, based on data obtained by the HST WFPC2 and the match the GCS profile in the outer regions. Extrapolating 1.5 m CTIO telescope. inward this profile to the galactic radius of 0.1 arcmin and integratig this surface density distribution in the galactic region where the GCS and the halo stars profiles differ, i.e. 2.3.1 The data and their fitting formulas 0.1 8.3 arcmin, we found a number of stars of about ∼9680. On− the other side, in this radial range we find a num- As for NGC 1399, the magnitude and colour intervals of the ber of GCs equal to 5165, which yields to an estimate of 4515 GC sample are 21

To estimate the mass lost in the whole GC× magnitude The modified core model gives a number of globular range, we adopted as total range of globular cluster masses clusters of 744 120 in this galactic region (the error in the 4 7 ± that between mmin =10 M and mmax =10 M . Normal- GC number has been calculated adopting an error in the izing the mass distribution (3) to the number of globular limiting radius equal to 1 arcmin, as suggested by F98a). clusters observed in the range 21

2.5 Table 1. The presently observed number of clusters (N), its ini- tial value (Ni), its fractional variation (∆), the mass lost in the magnitude range of the observations (Ml) and in the whole mag- 2 nitude range (Ml,tot) .

1.5 Galaxy NNi ∆ Ml(M ) Ml,tot(M )

NGC 1379 132 512 0.74 1.30 108 1.50 108 NGC 1399 5165 9680 0.47 1.30 × 109 1.44 × 109 1 NGC 1404 508 1061 0.52 1.37 × 108 1.75 × 108 × ×

log surface0.5 density HST data show that NGC 1404 is dominated by red globu- lar clusters, with a significant tail to the blue. Their colour distribution yields a mean value B I =1.96, but F98a -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 h − i log radius considered necessary to account for the background contam- ination. Using the CTIO data they found a bluer colour dis- Figure 3. Surface density profile for GCSs in NGC 1404 (black tribution, with a mean value B I =1.71. Now, thanks h − i dots) from HST data in F98a. The modified core model fit to the relation between B V and B I given by Grill- − − adopted in our study is represented by the solid line, while the mair et al. (1999) we transformed B I into B V , power law fit to the same data by F98a is shown as the dashed obtaining a B V value of abouth 0.75.− i Thish value− fori line. The underlying stellar light (dotted line) arbitrarly normal- B V togetherh − withi a distance modulus m M =31.2 ized in the vertical direction to match the globular cluster system hmaps− thei magnitude range 21

the number of missing clustersh i equal to× 553, we deduce an amount of the mass lost by GCS of 1.37 108 M . Hanes (1986), 190 80, and that by Richtler et al. (1992), × ± With regard to the total mass lost we have again 880 120. adopted the mass range 104M

9.5

4 THE CORRELATION BETWEEN THE MASS

LOST BY GCS AND THE CENTRAL 9 GALACTIC BLACK HOLE MASS Even if theoretical arguments and observational data are clearly compatible with the explanation of the difference 8.5 between the GCS and stellar bulge radial distributions as a result of dynamical evolution, we still cannot firmly state it. By the way it is interesting, and surely constitutes an 8 argument in favour of this interpretation, to see how our results for the (supposed) mass loss in form of glbobular cluster correlates with the galactic central black hole mass, at least for the set of seven galaxies for which these data are 7.5 available (our galaxy, M 31, M 87, NGC 1399, NGC 4365, NGC 4589, IC 1459).

Sources of data in Fig. 4 are: for Ml this paper and Capuzzo– 7 Dolcetta & Vignola, 1997, while the black hole masses are 678910 from Magorrian et al. (1998) except for our Galaxy and IC 1459 taken from Gebhardt et al. (2000). A recent estimate by Saglia et al. (2000) gives as compatible with kinematic Figure 4. The correlation between the (logarithmic) GCS mass lost and the central galactic black hole mass for 7 galaxies for data of the inner 5 arcsec of NGC 1399 a black hole mass which these data are available. Masses are in solar masses. The 5 108M , i.e. about a factor of 10 less than the value ∼ × straight solid line is the least square fit to all data, the dashed estimated by Magorrian et al. (1998) and reported in Fig. 4. line is the least square fit restricted to data of the most massive Fig. 4 shows, even in the paucity of data available, a black holes (LogMbh > 8) while the curve is the exponential fit clear increasing trend of Ml as function of Mbh. Actually, described in the text the least square straight line fitting the data relative to galaxies with black hole masses heavier than 108 M has . slope 0.99, i.e. the Ml - Mbh relation is remarkably linear, while the straight line connecting the 2 low mass points has In this way we have found that the GCSs of NGC 1379, slope 0.08. The best simple fit to the whole set of data is NGC 1399 and NGC 1404 could have lost the 74%, 47% given by LogMl = a + bexp(LogMbh)witha =7.158 and and 53% of their initial populations, respectively. These re- 4 2 b =1.194 10− , giving r =0.76 and standard error = 0.44 sults are, obviously, limited to the magnitude range of the (the least× square straight line fitting the whole data set observations within which the data reach a sufficiently high yields r2 =0.61 and standard error = 0.56). Incidentally, level of completeness. we note that the steeper slope of the Ml - Mbh relation for The large number of ‘missing’ globular clusters obtained re- high values of the nucleus mass is what expected when the veals that a significant amount of mass could have reached black hole is very massive and it acts as powerful engine of the galactic centre, too. An estimate of the mass fallen to GCS evolution. the galactic center needs the assumption of a value for the initial mean mass of globular clusters, m . It is reliable (see Capuzzo-Dolcetta & Tesseri 1997) toh assumei for m the mean mass value of the presently observed globularh i clus- 5CONCLUSIONS ters. This quantity can be estimated assuming a mass spec- We used data taken with the WFPC2 of the Hubble Space trum, that we took as a piece-wise power law for it seems Telescope to compare the globular cluster system distribu- to represent one of the best approximations for elliptical tion to that of bulge–halo stars in a sample of three elliptical galaxies globular cluster systems (McLaughlin 1994; Har- galaxies in the Fornax cluster. ris & Pudritz 1994). In particular, we adopted as unique The first relevant outcome of this study is that the ra- mass spectrum for the three galaxies of the Fornax cluster dial distributions of the globular cluster systems in NGC that obtained by normalization of the luminosity function 1379, NGC 1399 and NGC 1404 confirm that the GCSs adopted by McLaughlin (1995) for the Virgo elliptical giant in elliptical galaxies are less concentrated to the inner re- M87, and assuming a mass to light ratio (M/L)V, =1.5. gions than the halo stars. Following the scheme of previous We provided two different estimates of the mass lost in works (McLaughlin 1995, Capuzzo-Dolcetta & Vignola 1997, form of globular clusters: the first is that due to clusters in Capuzzo-Dolcetta & Tesseri 1999), i.e. assuming that the the magnitude range of the observations and the other refers globular clusters and halo star distributions were initially the “presumed” whole magnitude range. The first estimate, the same and the present difference is due to evolutionary Ml, is simply obtained as the product of the mean mass and effects acting on the GCS, we computed the number of glob- the number, Nl, of the missing globular clusters. With re- ular clusters lost as the integral of the difference between the gard to the second estimate we made the, quite reasonable, two surface distributions (suitably normalized). assumption that the total number of missing globular clus- 8 R. Capuzzo–Dolcetta and I. Donnarumma ters scales with the total number of the clusters present in Madore B. et al., 1996, BAAS, 189, Abstract No. 108.04 the galaxy. Hence we computed this total number of globu- Magorrian J. et al., 1998, ApJ, 115, 2285 lar clusters assuming 104M

Amm m REFERENCES min 1 s1 s0 ≤ ≤ k(m)= Am1 − m1 m m2 . (A4) Bridges, T.J., Hanes, D.A., & Harris, W.E. 1991,AJ, 101, 469.  s1 s0 s2 s1 ≤ ≤ Am1 − m2 − m2 m mmax Capuzzo-Dolcetta, R. 1993, ApJ, 415, 616.  ≤ ≤ Capuzzo-Dolcetta, R. & Tesseri, A. 1997, MNRAS, 292, 808. where, of course, mmin, mmax, m1 and m2 depend linearly Capuzzo-Dolcetta, R. & Tesseri, A. 1999, MNRAS, 308, 961. on β. To give (A1) a more compact form, we have put Capuzzo-Dolcetta, R., & Vignola, L. 1997, A&A,327, 130. Forbes D.A., Elson, R.A., Grillmair C.J., Willinger G.M., m0 mmin and m3 mmax, ≡ s1 ≡s0 s1 s0 s2 s1 Brodie J. 1998a, MNRAS,, 293, 325. k0 A, k1 Am1 − ,k2 Am1 − m2 − , ≡ mmax≡ s ≡ Forbes D.A., Elson, R.A., Grillmair C.J.,Rabban M., Willinger N = k(m)m− dm. tot m G.M., Brodie J. 1998b, MNRAS,, 295, 240. min Forbes D.A., Franx, M., Illingworth, G.D., Carollo,C.M. 1996, RememberingR that s =(s0,s1,s2) (A1) becomes ApJ, 467, 126. 2 Gebhardt, K. et al., 2000, ApJ, 539 L13 ∆ m ∂ m si ∆si ∂ m β ∆β Grillmair C.J., Forbes D.A., Brodie J., Elson, R.A. 1999, AJ, h i = h i + h i (A5) m ∂si m si ∂β m β 117, 167. h i i=0 h i h i X Grillmair, C., Pritchet, C., & van de Bergh, S. 1986, AJ, 91, ∂ m ∂ m where h i , h i are so defined: 1328. ∂si ∂β Hanes D.A., Harris W.E. 1986, ApJ, 309,564. 2 j j Harris W.E., Hanes D.A. 1987, AJ, 93, 1368. ∂ m 1 ϕi (j +1, 2) ϕi (j, 2) + Harris, W.E., & Pudritz, R.E. 1994, ApJ,429, 177. h i= j − j (A6) ∂si Ntot m ϕi (j +1, 1) ϕi (j, 1) j=0 −h i − Harris, W.E. & Racine, R. 1979, ARA&A, 17, 241. X    Jacoby G.H. et al. 1992 PASP, 104, 599. with  s +s s +s Kissler-Patig M., Kohle S., Hilker M., Richtler T.,Infante L., − j − j j ∂kj mr j mr 1 Quintana H., 1997, A&A, 319, 470. ϕi (r, s)= ∂s s +s + δi s +s kj s +s ln mr ,and i − j − j − j −   Evolution of globular cluster systems in three galaxies of the Fornax cluster 9

2 sj +2 sj +2 m− m− + ∂ m 1 kj j+1 − j h i = j j .(A7) ∂β Ntot β   s +1 s+1  j=0 m mj−+1 mj− X −h i −    