arXiv:0904.0526v1 [astro-ph.CO] 3 Apr 2009 aie(99 n aie(91;i hi pno globular opinion their distributio density in the (1991); when earlier, Racine originated Har clusters and by suggested (1979) as Racine systems, two & the of ages formation ferent h ups fepann hsfaue mn hs,tosee two these, Among probable: feature. most this the explaining a of by purpose caused the not and real is di distributions the bias. radial selective is that two present, agreement GCS the at general the tween case, than a concentrated no is less say there are Moreover, may stars halo we the galaxies, where all known to common is Harri to up (2004) al. et (2009). Peng al. (2003), Spitler, et (200 Zepf (2006), & al. al. Rhode et et (2008), L.R. Rhode al. (2006 Spitler, (2008), al. (2006), al. al. et et et Bassino Lee Sikkema of (2007), papers al. recent et ra- example, Goudfrooij GCS of re sake the just the to we review, s for Regard mentioned the the (1979). from Racine starting since profiles, & properties, dial Harris their i by of identification review GCS inal study to and dedicated s galaxies literature bulge-halo external of the amount than huge centre le A galactic usually, glob the are, towards that popolous GCSs), centrated less (hereafter or systems more cluster lar contain galaxies elliptical Many Introduction 1. oebr1 2018 1, November ⋆ Astrophysics & Astronomy i h di the (i) di Consequently, characteristic this that concluded yet be cannot it Though eevd.. cetd... accepted ...; Received 5 Moro A. P.le Roma, of e-mail: University Sapienza, Physics, of Dep. GS)rda itiuin n hs fteglci stell galactic the of those and distributions radial (GCSs) eta einmyhv culymre rudtegalactic the around merged actually di have the may region If central evolution. and compa quantitative formation A cluster relevance. astrophysical an has which c galactic the toward flatter significantly is profile density as fGs‘iapae’b ipenraiainproced normalization simple a Results. by ini ‘disappeared’ was GCs which gala of distribution the mass) radial to a peaked from less starting is evolution profile system cluster globular the Methods. Aims. Context. ih aaiseaie.Tecrepnigms ott the to lost mass corresponding The examined. galaxies eight ciiy(G)adlclmsiebakhl omto n gr words. and Key formation hole black massive importan local an and had (AGN) likely activity have regions galactic central toward n htbtentecnrlmsiebakhl asadtet the and mass hole black massive central betw the correlation between clear that the and is this to support observational atrpol fterda itiuino lblrclust globular of distribution radial the of profile flatter A [email protected] ff lblrCutrSse rso nelpia galaxies elliptical in erosion System Cluster Globular h ubro isn lblrcutr ssgicn,rang significant, is clusters globular missing of number The nti ae eaayedt f8elpia aaisi orde in galaxies elliptical 8 of data analyze we paper this In rnebtentetodsrbtosrflcsdif- reflects distributions two the between erence efitdteselradgoua lse ytmrda profi radial system cluster globular and stellar the fitted We aais litcl–cutr:globular clusters: – elliptical galaxies: ff rn yohsshv enavne with advanced been have hypotheses erent aucitn.capuzzodolcetta no. manuscript .Capuzzo-Dolcetta R. ff rnei u oeoino h lblrcutrsse,tem the system, cluster globular the of erosion to due is erence ff rnebe- erence a less was n e-mail: e h aayitgae antd n h ubro globul of number the and magnitude integrated galaxy the een scon- ss ..et L.R. nr hnta fstars. of that than entre ilyidsigihbefo hto tr,w a evaluat may we stars, of that from indistinguishable tially rcmoet nalteglxe tde eetegoua c globular the here studied galaxies the all In component. ar mind, edako h nems aatcrgo,icuigisv its including region, galactic innermost the on feedback t tccnr hnteselroei.Asmn hsdi this Assuming is. one stellar the than centre ctic aieaayi ftepolsmygv ih nbt aaya galaxy both on light give may profiles the of analysis rative ABSTRACT 1 tars. em- eteadfre,o tlaticesdi as h galacti the mass, in increased least at or formed, and centre owth. 7), ure. eta aatcrgo s7 is region galactic central [email protected] ris n .Mastrobuono-Battisti A. and rsse epc ota fteglci tla opnn i component stellar galactic the of that to respect system er u- tlms fgoua lseslost. clusters globular of mass otal m -08,Rm,Italy Roma, I-00185, , ), n s . eann iia otepol ftehl tr.Tee The stars. halo the of profile the unaltere dense to almost central, similar profile the remaining outer in the GCSs leaving the regions, deplete galactic exampl to for centre acts (see mechanisms galactic cal nucleus e combined to compact The a 1993). close Capuzzo-Dolcetta with very interaction tidal clusters and massive brings which collisi the unchanged. fu while almost a distribution, stands with radial halo stars, GCS halo the and of clusters evolution globular of birth coeval of region two galactic outer the the why in similar explain very cannot are distributions hypothesis this However, peaked. vlain ftednmclfito e friction of previo dynamical orbits the a the loop showed rate of results in a evaluations These moving at energy. those and energy than size orbital comparable larger their magnitude lose c central of globular orbits order the that box showed in populate (1992) moving well al. ters then et componen Pesce and any regions. conserve momentum galactic not angular o do the family which a of orbits, is ‘box’ there Pesc the which po 1989; in orbits, Saha triaxial 1992), & Vietri galactic Binney & in Ostriker, Capuzzo-Dolcetta example higher for is (see phenomena tentials mentioned the of oinadtu ec h ne ein narltvl short relatively a in regions inner during the braked reach globul strongly thus massive are and that potentials motion ascertained triaxial been in has clusters it time-sc contrary, braking dynamical the distrib the On of orbit overestimate an cluster to globular led that the on hypotheses simplified n rm2%t 1 fteriiilpplto bnac nt in abundance population initial their of 71% to 21% from ing e nasto aaisosre thg eouin eswth saw We resolution. high at observed galaxies of set a in les osuytedi the study to r h C vlto scue ybt yaia friction, dynamical both by caused is evolution GCS The assumption simple the on based is explanation another (ii) ff rnebtentergoua lse systems cluster globular their between erence × 10 7 -1 1 . 85 × 10 9 sigGsi h galactic the in GCs issing M ⊙ ff l hsms carried mass this All . rnea u oGCS to due as erence ffi iny ae nvery on based ciency, h ubr(and number the e ff c fteedynami- these of ect ⋆ oettransient iolent rcutr lost clusters ar ulu.An nucleus. c utrsystem luster di a s dglobular nd

c ff erence S 2018 ESO ffi he s; at ,then d, ciency onless ution rther their ales. lus- us ar e, r, n e f - t , 2 R. Capuzzo-Dolcetta and A. Mastrobuono-Battisti: Globular Cluster System erosion in elliptical galaxies time (Capuzzo-Dolcetta 1993). There, they could have started data from, that may change case by case, checking how good are a process of merging, giving origin to a central massive nucleus the approximations to observed data. (eventually a black hole) or fed a pre-existent massive object The “initial” distribution of GCs, ΣGC,0(r), (assumed to be equal (Ostriker et al. 1989; Capuzzo-Dolcetta 1993). Under the hy- in shape to the present stellar profile), can be practically ob- pothesis that the initial GCS and halo-bulge radial distributions tained by a vertical translation of the stellar profile, Σs(r), to the were the same, an accurate analysis of the observations would present GCS distribution, ΣGC (r) as given by Eq. 1. We calculate allow an estimate of the number of ‘missing clusters’ and there- the number of missing (lost) clusters as the surface integral of fore of the mass removed from the GCSs. the difference between ΣGC,0(r) and ΣGC(r) over the radial range Actually, McLaughlin (1995), Capuzzo-Dolcetta & Vignola [0, rmax]ofdifference of these two profiles: (1997), Capuzzo-Dolcetta & Tesseri (1999) and Capuzzo- r Dolcetta & Donnarumma (2001), scaling the radial surface max Nl = 2π ΣGC,0(r) − ΣGC (r) rdr. (2) profiles of the halo stars of a galaxy to that of its GCS, estimated Z0 the number of missing globular clusters as the integral of the
Σ difference between the two radial profiles. Capuzzo-Dolcetta The present number of GCs, N, is obtained integrating GC (r) & Vignola (1997) and Capuzzo-Dolcetta & Tesseri (1999) over the radial range, [rmin, R], covered by the observations (for suggested that the compact nuclei in our galaxy, M31 and M87, the value of R we rely on the papers where we got the GCs as well as those in many other galaxies, could have reasonably distribution data from). The values of N obtained with this ff sucked in a lot of decayed globular clusters in the first few Gyrs method are usually di erent from those given by the authors of of life. the papers, but for the purposes of this paper what is important is the difference bewteen N and Nl and, so, it is crucial a In this paper we enlarge the discussion of the comparison of homogeneous way to determine them. GCS and star distribution in galaxies, dealing with eight galaxies The initial number of GCs in a galaxy is, indeed, estimated as = + for which good photometric data are available in the literature Ni Nl N. such to draw reliable radial profiles. The numerical values of Nl, Ni and N are functions of the fitting In Sect. 2 we resume the way, discussed deeply in previously parameters and of the integration limits. These dependences and mentioned papers, to get from the compared GCS-stars radial their contribution to the errors on the final results are discussed profilesin a galaxythe numberand mass of GCs disappeareddue in Appendix. to time evolution of the GCS; in Sect. 3 we present and discuss An estimate of the mass removed from the GCS (Ml) can be the observational data, as well as the analytical fit expressions obtained through the number of GCs lost, Nl, and the estimate to the density profiles; in Sect. 4 we present the extension to the of the mean mass of the missing globular clusters, hmli. A data of this work the correlation found by Capuzzo–Doletta & priori, the determination of hmli needs the knowledge of the ff Donnarumma(2001) between the mass lost by the GCS, the host initial mass spectrum of the CGS, which has su ered of an galaxy luminosity and the mass of the galactic central massive evolutionary erosion. However, the most relevant evolutionary black hole. Finally, in Sect. 5 we summarize results and draw phenomena (tidal shocking and dynamical friction) act on general conclusions. An error analysis of the methods used is opposite sides of the initial mass function, and so we expect that presented in Appendix. the mean value of the globular cluster mass has not changed very much in time (see Capuzzo-Dolcetta & Tesseri, 1997 and Capuzzo-Dolcetta & Donnarumma 2001). Hence, we can 2. The estimate of the number and mass of globular assume the present mean value of the mass of globular clusters, clusters lost hmi, as a good reference value for hmli. For NGC 4374, NGC 4406 and NGC 4636 we calculated hmi Under the hypothesis that the flattening of the GCS distribution using their GC luminosity functions (GCLFs) and assuming in the central region compared to the distribution of the stars the same typical mass-to-light ratio of GCs in our Galaxy, i.e. in the galactic bulge, is due to an evolution of the GCS, the (M/L)V,⊙ = 1.5forNGC 4406andNGC 4636or (M/L)B,⊙ = 1.9 number of GC lost to the centre of the galaxy is obtained by (25) in the case of NGC 4374. For NGC 4636 we used also the the simple difference of the (normalized) density profiles inte- mass function that represents its present distribution of GCs grated over the whole radial range (see McLaughlin 1995). A (see Sect. 3.8). For the remaining galaxies (NGC 1400, NGC general discussion of the problems in evaluating the number and 1407, NGC 4472, NGC 3258 and NGC 3268) there is no better mass of missing clusters in this way can be found in Capuzzo- way to estimate the total mass of ‘lost’ GCs than adopting as a Dolcetta & Vignola (1997), Capuzzo-Dolcetta & Tesseri (1999) ‘fiducial’ reference value for their mean mass, hmli, the value, 5 and Capuzzo-Dolcetta & Donnarumma (2001) and so it is not hmMW i = 3.3 × 10 M⊙, of the present mean GC mass in our worth repeated here. Galaxy. In this paper, we assume as reliable fitting function for the GCS projected radial distribution a “modified core model”, i.e. a law Σ Σ (r) = 0 (1) 3. Data and results GC 2 γ 1 + (r/rc) The data for the study of this paper have been collected from the   where Σ0, rc and γ are free parameters. The choice of this func- literature. We will analyze a set of 8 galaxies for which the GC tion to fit the GCS profiles is motivated by the good agreement content and the radial profile has been reliably determined and found with almost all the data used for the purposes of this pa- apt to our purposes. per, as confirmed by, both, the local maximum deviation of the The galaxies are: NGC 1400, NGC 1407, NGC 4472 (M 49), fitting formula from the observed data and the computed χ2. For NGC 3268, NGC 3258, NGC 4374, NGC 4406 and NGC 4636. the galaxy stellar profile, Σs(r), we rely on the fitting formulas These galaxies go to enlarge the set of seventeen galaxies (Milky provided by the authors of the various papers where we got the Way, M 31, M 87, NGC 1379, NGC 1399, NGC 1404, NGC R. Capuzzo-Dolcetta and A. Mastrobuono-Battisti: Globular Cluster System erosion in elliptical galaxies 3

3

4 2

2 1

0 0

-2 -1 0 -1.5 -1 -0.5 0 0.5 1

4 4

3 3

2 2

1 1

0 0

-2 -1 0 1 -1 0 1

Fig. 1. Surface number density for NGC 1400, NGC 1407, NGC 4472 (M 49) and NGC 3268. Black squares represent the observed GC distribution; the solid line is its modified core model fit. The dashed curve is the surface brightness profile of the underlying galaxy (a power law and a central flat core for NGC 1400 and NGC 1407, a Sersic core model for NGC 4472 and a Nuker law for NGC 3268), vertically normalized to match the radial profile of the GCS in the outer regions.

1427, NGC 1439, NGC 1700, NGC 4365, NGC 4494, NGC luminosity profile of the galaxy for r ≤ rb is almost flat, and 4589, NGC 5322, NGC 5813, NGC 5982, NGC 7626, IC 1459) linked to the external power law. We fitted the GCS distribution −2 whose GCS radial profiles have been compared to the stellar dis- by a modified core model with Σ0 = 14.1 arcmin , rc = 0.7 ar- tribution in previous papers ((30), Capuzzo-Dolcetta & Vignola cmin and γ = 0.88. Integrating ΣGC (r) in the radial range where 1997, Capuzzo-Dolcetta & Tesseri 1999, Capuzzo-Dolcetta & GCs are observed, i.e. from rmin = 0arcmin to R = 2.8 arcmin, Donnarumma 2001). we obtain N = 73 as the present number of GC in NGC 1400. The initial GCS distribution results to be approximated by:

5 −2 3.1. NGC 1400 Σ = 1.37 × 10 arcmin r ≤ rb GC,0(r) −1.88 (3) ( 7.76r r > rb The surface density profile of this galaxy is given by Forbes et al. (2006) (hereafter F06) who fitted it by mean of a power law (see Fig. 1). −1.88 Σs(r) ∝ r . This fitting law is reliable outside the galactic Using the general method described in Sec. 2 and the estimated core, i.e. for r > rb = 0.0055arcmin (Spolaor et al. 2008). The value rmax = 2.3arcmin we have that the number of missing 4 R. Capuzzo-Dolcetta and A. Mastrobuono-Battisti: Globular Cluster System erosion in elliptical galaxies

4 4 3

3 2

2 1

1 0

-2 -1 0 1 -1 0 1

4 4

3

2 2

0 1

-2 -1 0 1 -1.5 -1 -0.5 0 0.5 1

Fig. 2. Surface number density for NGC 3258, NGC 4374, NGC 4406 and NGC 4636. Black squares represent the observed GC distributions; the solid lines are their modified core model fit. The dashed curves are the surface brightness profile of the underlying galaxy (a Nuker law for NGC 3268, a power law and a central flat core for NGC 4374 and NGC 4636 and a Sersic core model for NGC 4406), vertically normalized to match the radial profile of the cluster system in the outer regions.

clusters in this galaxy is Nl = 183, i.e. about 71% of the initial r ≤ rb, a flat distribution matched to the external power law. The −2 population of globular clusters, Ni = Nl + N = 256. GCS modified core model has, in this case, Σ0 = 17.8arcmin , An estimate of the mass lost by the GCS is Ml = Nl hmMW i = rc = 1.02arcmin and γ = 0.85 as better fitting parameters. The 7 6.04 × 10 M⊙. normalizing vertical translation of the stellar profile leads to

3 −2 Σ = 1.04 × 10 arcmin r ≤ rb GC,0(r) −1.42 (4) 3.2. NGC 1407 ( 12.6r r > rb As for NGC 1400, data for this galaxy and its GCS are taken (see Fig. 1). from F06. The luminosity profile of the galaxy stars is fitted bya Integrating the difference of the GCS “initial” and present radial −1.42 power law, Σs(r) ∝ r . This power law fit fails in the inner re- profiles in the galactic region where these differ, i.e. up to rmax = gion where luminosity shows a core of radius rb ≃ 0.045arcmin 2.34arcmin (see Eq. 2), we obtain Nl = 84. The present number (Spolaor et al. 2008). As for NGC 1400 we thus assume, for of GC obtained integrating ΣGC(r) in the radial range covered by R. Capuzzo-Dolcetta and A. Mastrobuono-Battisti: Globular Cluster System erosion in elliptical galaxies 5 the observations (i.e. from 0 to R = 7.3arcmin) is N = 314. The The departure,visible in Fig. 1, of the profile given by Eq.8 from GCS has therefore lost 21% of its initial population, Ni = 398. the modified core model profile in the external galactic region Also in this case, we can evaluate the mass lost by the system as is mainly due to the incompleteness of the GC detection in the 7 Ml = Nl hmMW i = 2.77 × 10 M⊙. outermost regions. Integrating ΣGC (r) from rmin = 0arcmin to R = 7.94arcmin we have N = 505 as present GC number. The number of globular clusters lost is found to be N = 398 3.3. NGC 4472 (M 49) l (from 0 to rmax = 2.3arcmin), that means about 44% of the ini- = Data for the GC distribution in this giant elliptical galaxy in tial abundance, Ni 913. Virgo are taken from Cˆot´eet al. (2003). The galaxy star lumi- Also in this case, to evaluate the mean mass of lost GC in = = nosity distribution, according to Ferrarese et al (2006), is well NCG 3268 we have to assume hmli hmMW i, obtaining Ml = 8 reproduced by a Sersic core model (Trujillo et al. 2004), i.e. by Nl hmMW i 1.31 × 10 M⊙.

1 1 γ rb n r n rb bn −bn Σs(r) = Σb θ(rb − r) + e re θ(r − rb)e re , (5) 3.5. NGC 3258 ( r      ) As forNGC 3268theGCS densityprofiledata forNGC 3258are where Σs(rb) = Σb, θ(x) is the usual Heaviside function, rb (break taken from Dirsch, Richtler & Bassino (2003). The best modi- −2 radius)divides the profileinto an inner (r ≤ rb) power-law region fied core model fit is given by the values Σ0 = 16.9arcmin , and an outer (r ≥ rb) exponential region; re is the ‘effective’ rc = 3.1arcmin and γ = 2.4. The analytical fit to the luminos- radius and bn = 1.992n − 0.3271, with n free fitting parameter. ity profile of the galaxy is, again, obtained with the “Nuker” law For M 49 the parameter values are (Cˆot´eet al. 2003): γ = 0.086, (Eq. (7)) with α = 2.10, β = 1.51, γ = 0 and rb = 0.0192arcmin. n = 5.503, bn = 10.635, re = 208 arcsec (= 16.89 kpc), rb = By mean of the usual procedure, the initial GCS profile is ob- 1.94 arcsec (= 0.158 kpc). tained: In this case the core model fit to the observed GCS distribution 2.10 −0.719 −2 r gives Σ0 = 6kpc , rc = 3.57kpc and γ = 0.652 as optimal ΣGC,0(r) = 8489 1 + . (9) parameters. The usual vertical translation leads to " 0.0192 # The present number of GCs is N = 343 (with R = 7.94arcmin). 0.158 0.086 Performing the surface integral of the difference of the initial and Σ (r) = 722 θ(0.158 − r)+ GC,0  present distribution in the radial range up to rmax = 2.16arcmin  r !  (see Eq.2) we have Nl = 212, correspondingto 38% of the initial 0.18  −10.635 r = +94.7θ(r − 0.158)e ( 16.89 ) . (6) GCS population, Ni 555. = = 7  For this galaxy, we obtained Ml Nl hmMW i 7.00 × 10 M⊙. Fig. 1 shows how the modified core modelprofile does not repre- sent very well the observed GCS external profile, leading, there, 3.6. NGC 4374 (M 84) to overestimate. The surface integral (Eq. 2), performed with r = 42.34kpc, Gomez & Richtler (2004) studied the GCS of this giant ellip- max tical galaxy, using photometry in the B and R bands, to draw gives Nl = 5598. Integrating ΣGC (r) up to R = 100kpc we have that N = 6334. Hence, M 49 has lost 47% of the initial popula- its radial surface distribution. Also in this case, the profile of tion of its GCs, N = 11, 932. the GC number density is flatter than the galaxy light (see Fig. i 2). The best modified core model fit to the GC data is given by As for NGC 1400 and NGC 1407, no better estimate of hmli is −2 9 Σ = 58.4 arcmin , r = 0.31arcmin and γ = 0.278. available, so we evaluate M as M = N hm i = 1.85 × 10 M⊙. 0 c l l l MW The galaxy light is characterized by a central core of radius rb ≃ 0.0398arcmin (Lauer et al., 2007); for r > rb, it is well −1.67 3.4. NGC 3268 fitted by the power law Σs(r) ∝ r (Gomez & Richtler 2004). The usual normalization leads to The GC distribution of this galaxy is discussed by Dirsch, 2.94 × 104 arcmin−2 r ≤ r Richtler & Bassino (2003). The resulting core model fit parame- Σ (r) = b (10) −2 GC,0 −1.67 ters have the values: Σ0 = 24.4 arcmin , rc = 2.6arcmin and ( 135r r > rb, = γ 1.9. The stellar luminosity profile, following Capetti & as GCS initial radial profile. Integrating our core model up to Balmaverde (2006), is well represented by a “Nuker” law (in- R = 11.8arcmin we get N = 4655 as present number of troduced by Lauer et al. 1995): GCs. The usual integration of the difference of the initial and = − present GC distribution (Eq. 2 with rmax 3.84arcmin) leads to r γ r α (γ β)/α = Σ = (β−γ)/αΣ b + Nl 2361. Hence NGC 4374 has lost 34% of its initial popula- s(r) 2 b 1 (7) =  r  " rb ! # tion of globular clusters, Ni 7016. In the case ofNGC 4374 thevalueofthe mean mass ofa GC has where β is the slope of the external region of luminosity profile, been evaluatedusing the GCLF in the R band givenby Gomez & Σ rb is the “break radius” (corresponding to a brightness b) where Richtler (2004).The mean color h(B − R)0i = 1.18 of GCs in this the profile flattens to a smaller slope, measured by the parameter galaxy (Gomez & Richtler 2004) allows us to estimate the mean γ; α sets the sharpness of the transition between the inner and B absolute magnitude and the mean luminosity of GCs in the B = = = outer profile. In the case of NGC 3268: α 2.49, β 1.64, γ band, h(L/LB)⊙i assuming m − M = 31.61 (Gomez & Richtler = 5 0.13 and rb 0.0252arcmin. By the usual vertical translation of 2004). It results h(L/LB)⊙i = 1.75 × 10 (with MB,⊙ = 5.47, Cox this profile, the initial GCS profile is obtained 2000). Adopting the mass to light ratio M/L)B,⊙ = 1.9 obtained 0.13 2.49 −0.606 Σ = 0.0252 + r by Illingworth (1976) for 10 galactic globular
clusters, we get GC,0(r) 13692 1 . (8) hm i = 3.33 × 105 M and M = N hm i = 7.86 × 108M . r ! " 0.0252 # l ⊙ l l l ⊙ 6 R. Capuzzo-Dolcetta and A. Mastrobuono-Battisti: Globular Cluster System erosion in elliptical galaxies

Galaxy Σ0 rc γ rmax R NGC 1400 14.1 0.7 0.88 2.3 2.8 NGC 1407 17.8 1.02 0.85 2.34 7.3 NGC 4472 142 0.73 0.652 8.69 20.52 NGC 3268 24.4 2.6 1.9 2.3 7.94 NGC 3258 16.9 3.1 2.4 2.16 7.94 NGC 4374 58.4 0.31 0.278 3.84 11.8 NGC 4406 26.76 3.52 1.19 5.36 24 NGC 4636 77.66 0.823 0.691 4.75 6.6 Table 1. Col. 1: galaxy name; col. 2, 3 and 4: parameters of the modified core model fit for all the galaxies studied; col. 5: upper limit in the integral giving the number of the lost GCs (rmax); col. 6: upper limit of the integral performed to estimate the present −2 number of GCs (R). Σ0 is in arcmin ; rc, rmax and R are in arcmin.

Galaxy Model η rb re bn n α β γ NGC 1400 lc 7.76 5.5 × 10−3 - - -1.88 - - NGC 1407 lc 12.6 0.045 - - - 1.42 - - NGC 4472 cS 17139 0.0323 3.47 10.635 5.503 - - 0.086 NGC 3268 N 13692 0.0252 - - - 2.49 1.64 0.13 NGC 3258 N 8489 0.0192 - - - 2.10 1.51 0 NGC 4374 lc 135 0.0398 - - - 1.67 - - NGC 4406 cS 13490 0.012 6.86 13.649 7.02 - - 0.021 NGC 4636 lc 70.8 0.0573 - - - 1.5 - - Table 2. Galactic luminosity fitting parameters. Col. 1: galaxy name; col. (2) key identifyingthe galaxy light profile model (lc=linear with a flat core in the inner region, cS=core-S´ersic, N=Nuker); col. 3-10: parameters of the various profile models (see Sect. 2 and −2 3 for details). η is in arcmin ; rb and re are in arcmin.

3.7. NGC 4406 (VCC 881) ter values. The galactic light profile shows an inner flat distribution (a core NGC 4406 is another giant elliptical in Virgo; its GCS has been with radius r ≃ 0.0573arcmin (Lauer et al. 2007)), while for studied by mean of the Mosaic Imager on the 4m Mayall tele- b r > rb the light distribution is well fitted by the power law fit scope at the KPNO (Rhode & Zepf 2004) in the B, V and Σ (r) ∝ r−1.5 (Kissler Patig et al. 1994). R bands. The resulting best fit core model is characterized by s Σ = −2 = = The vertical translation of the stellar profile gives the initial GCS 0 26.76 arcmin , rc 3.52 arcmin and γ 1.19. The galaxy profile: light is well fitted by a Sersic core model (Eq. 5), whose parame- ters have been determined by Ferrarese et al. (2006). Its vertical 3 −2 translation gives the initial GCS radial profile Σ = 5.16 × 10 arcmin r ≤ rb GC,0(r) −1.5 (12) ( 70.8r r > rb. 0.012 0.021 ΣGC,0(r) = 13490 θ(0.012 − r) +  r ! Integrating the present surface density profile of the GCS up to   0.142 R = 6.6arcmin, we obtain N = 1411. Performing the surface in-  −13.649 r + θ − . ( 6.86 ) . 250 (r 0 012)e (11) tegral given in Eq. 2 (with rmax = 4.75arcmin), we estimate that  the number of GCs disappeared is N = 746, i.e. 35% of the ini- = l Integrating the present distribution of GCs, from rmin 0 arcmin tial population, Ni = 2157. In the case of this galaxy we obtained to R = 24arcmin, we have N = 2850. The surface integral given two differentestimates of the mass lost by the GCS, starting from in Eq. 2, with rmax = 5.36arcmin, gives the number of globular data taken from Kissler Patig et al. (1994). The first estimate has clusters lost, Nl = 1359, i.e. about 32% of the initial GC popula- been obtained using the GCLF (Kissler Patig et al. 1994). As tion. for NGC 4406 we calculated the mean absolute V magnitude of = Using the GCLF of this galaxy and its distance modulus m−M GCs, hMV i = −8.07, (given m − M = 31.2 by Kissler Patig et al. 31.12 (Rhode & Zepf, 2004),we evaluated the mean value of the 1994). Assuming for GCs in NGC 4636 the same M/L ratio of = V absolute GC V magnitude, hMV i −8.42 which corresponds to galactic GCs, (M/LV)⊙ = 1.5, the deduced mean luminosity of h i = × 5 = = 5 = 5 the mean luminosity L/L⊙ V 1.98 10 (MV,⊙ 4.82 from GCs, hL/L⊙iV 1.43 × 10 , gives ml,1 2.15 × 10 M⊙, and Cox 2000). = = 8 so Ml,1 Nl ml,1 1.29 × 10 M⊙. Assuming (M/L) = 1.5, we obtain hm i = 2.97 × 105M . V,⊙ l ⊙ Another estimate is found using the mass distribution of GCs This estimate leads to the value of the mass lost by the GCS, obtained in Kissler Patig et al. (1994) transforming the magni- 8 Ml = Nl hmli = 4.04 × 10 M⊙. tude bins of the GCLF candidates into masses using the relation given by Mandushev et al. (1991): log(M/M⊙) = −0.46MV + 1.6 ≃ 3.8. NGC 4636 (corresponding to a mean mass to light ratio (M/L)V,⊙ 2.0). Knowing the mass distribution we can directly calculate the 5 The GC content of this galaxy has been studied by Kissler Patig mean mass of GCs, ml,2 = 3.79 × 10 M⊙, and thus Ml,2 = Σ = = 8 et al. (1994). The modified core model fit has 0 77.66 Nl ml,2 2.97 × 10 M⊙. −2 = = = 8 arcmin , rc 0.823 arcmin and γ 0.691 as optimal parame- The averages of our two estimates gives Ml 2.22 × 10 M⊙. R. Capuzzo-Dolcetta and A. Mastrobuono-Battisti: Globular Cluster System erosion in elliptical galaxies 7

Tables 1, 2 resume the parameters of the radial profile fitting eight elliptical galaxies observed by various authors. We find functions for the studied galaxies, while 3 resume the results in that GCS distributions flatten toward the centre, showing a broad terms of estimated number and mass of GC lost. core in the profile, contrarily to the surrounding star field. This result agrees with many previous findings, indicating, indeed, that GCs are usually less centrally concentrated than stars of the 4. The correlation between M , M and M l V bh bulge-halo. A debate is still open on the interpretation of this The evolutionary explanation of the difference between the ini- observational issue. The “evolutionary” interpretation is partic- tial and present GC distribution implies a correlation between ularly appealing; it claims that, initially, the GCS and stellar the (supposed) mass lost by GCS with the mass of the galactic profiles were similar and, later, GCS evolved to the presently central supermassive black hole (Mbh) and. likely, with the host flatter distribution due to dynamical friction and tidal interac- galaxy luminosity (MV ). Tab. 4 reports the whole set of galaxies tions (Capuzzo-Dolcetta 1993). In this picture, the flatter cen- for which we have the estimate of MV , Mbh and Ml. tral profile is due to the erosion of the inner GCS radial profile. Fig. 3 shows a plot of Tab. 4 data which clearly indicate an in- Many GCs are, consequently, packed in the inner galactic re- creasing trend of Ml as function of Mbh (left panel) and of MV gion, where they influence the physics of the host galaxy. Many (right panel). of the galaxies studied so far have massive black holes at their In particular, the linear fit of data in the left panel is given centres, whose mass positively correlates with our estimates of by log Ml = a log Mbh + b with a ± σ(a) = 0.47 ± 0.20 and number and mass of GC lost. This is a strong hint to the validity b ± σ(b) = 3.9 ± 1.8, giving r = 0.45 and χ2 = 9.5. The, alter- of the mentioned evolutionary scenario, together with the other native, exponential fit gives log Ml = α exp(log Mbh) + β where evident correlation between number and mass of GC lost and α±σ(α) = (7.7±3.3)×10−5 and β±σ(β) = 7.45±0.28, r2 = 0.20 their parent galaxy luminosity. The evolutionary hypothesis is and χ2 = 9.6. also supported by the positive (although statistically uncertain) We performed also fits for for the correlation between Ml and correlation between the (rough) estimate of the galactic central Mbh excluding the data which have a great residual from the phase-space density and integrated magnitude. At the light of mean square (those of NGC 1439 and NGC 1700) obtaining the these encouraging findings, we think that much effort should be following parameters of the linear fit: a ± σ(a) = 0.61 ± 0.15 and spent into deepening the observational tests of this astrophysical b±σ(b) = 2.9±1.3, with r = 0.69 and χ2 = 4.0. The exponential scenario. fit parameters are in this case: α ± σ(α) = (1.17 ± 0.22) × 10−4 and β ± σ(β) = 7.31 ± 0.18, giving r2 = 0.60 and χ2 = 3.1. The least square, straight-line fit to the whole set of data Appendix A: The error on the estimates of number shown in the right panel of Fig. 3 is given by log Ml = aMV + b of lost GCs with a ± σ(a) = −0.62 ± 0.15 and b ± σ(b) = −5.3 ± 3.2, giving Here we describe how we evaluated the errors, ǫ , given in Table r = 0.67 and χ2 = 6.6. The exponentialfit on the same data gives l 3. As explainedin Sect. 2, the number of GCs lost in the galaxies log M = α exp(−M )+β where α±σ(α) = (2.20±0.49)×10−10 l V of the sample has been evaluated as the integral of the difference and β ± σ(β) = 7.33 ± 0.19, r2 = 0.48 and χ2 = 6.2. between the (estimated) initial and present GCS radial distribu- The correlation seen in the right panel of Fig. 3 between M l tions over the radial range [r , r ] where the two profiles dif- and M reflects, both, an expected physical dependence on the min max V fer. The absolute errors on N (∆N ) are given by the sum of the total galactic mass of evolutionary processes acting on GCSs l l error on Ni (∆Ni) and the error on N (∆N), yielding the relative and, simply, the positive correlation between Mbh and MV for ∆Nl error ǫl = of Table 3. We may estimate ∆N and ∆Ni as fol- the same set of galaxies. Actually, the Mbh-MV correlation for Nl the set of galaxies in Tab. 4 has a clearly positive slope, as lows. ∆ shown also by the least square fit in Fig. 4. The least square i) estimate of N For all the galaxies the number N(r , r ) is given by fit is log Mbh = aMV + b with a ± σ(a) = −0.56 ± 0.15 and min max 2 b ± σ(b) = −3.41 ± 3.22, giving χ = 6.54. On the other ‘phys- r max r ical’side, the energy and angular momentum dissipation caused N(r , r ) = 2πΣ dr = min max 0 2 γ by dynamical friction should depend on the inner galaxy phase Zrmin + r 1 r space density (∝ ρ/σ3, where ρ and σ are the galactic mass den-  c    r sity and velocity dispersion, respectively). A stronger dynami- max Σ π(r2 + r2) cal friction causes a faster GC decay toward inner galactic re- = 0 c (A.1) 2 γ gions where the tidal action of a massive black hole depletes (1 − γ) 1 + r rc the GC population. Were brighter galaxies also denser in the   rmin 3   phase-space, the Mbh − MV correlation would have a ρ/σ vs. which is a function of the parameters Σ0, rc, γ, rmin and rmax MV counterpart. Using data available in the literature for a set of 428 galaxies (the largest part coming from a combination of whose indetermination is data available in the catalogue by Prugniel & Simien (1996)) we ∆ = ∂N ∆Σ + ∂N ∆ + ∂N ∆ + find the distribution shown in the right panel of Fig. 4 which, N 0 rc γ ∂Σ0 ∂rc ∂γ far from being conclusive, shows indeed a trend of higher cen- ∂N ∂N tral phase space density in brighter galaxies. The least square fit + ∆rmin + ∆ rmax (A.2) 3 = + = ∂r ∂r is log(ρ/σ ) aMV b with a ± σ(a) −0.07 ± 0.019 and min max b ± σ(b) = −1.65 ± 0.39, giving χ2 = 194.44. where:

rmax 5. Conclusions rmax π 2 + 2 ∂N = r = (r rc ) 2π γ dr γ , (A.3) We presented the comparative discussion of radial distribution ∂Σ0 Z r 2 r 2 rmin 1 + (1 − γ) 1 + of the globular cluster systems and of the stars in a sample of rc rc     rmin    

8 R. Capuzzo-Dolcetta and A. Mastrobuono-Battisti: Globular Cluster System erosion in elliptical galaxies

Fig. 3. The correlation between the GCS (logarithmic) mass lost and the central galactic black hole mass (left panel) and integrated V magnitude of the host galaxy (right panel) for the set of galaxies in Table 4. Masses are in solar masses. Black circles represent the eight galaxies whose GCS data are discussed in this paper, black triangles refer to the others. The straight lines and curves are the approximation fits discussed in Sect.4.

For NGC 1400, NGC 1407, NGC 4374, NGC 4636 we have (see Sect. 3.1, 3.2, 3.6, 3.8 and Tab. 2 for the meaning and the values rmax Σ 3 ∂N = 4π 0γr = of the parameters) + dr 2 (1 γ) ∂rc Zrmin 3 r rc 1 + rc rb rmax     = −α + 1−α = rmax Ni(rmin, rmax) 2πηrb rdr 2πη r dr Σ 2 + 2 Zrmin Zrb 2π 0(rc γr ) = − , (A.4) 2−α rmax γ − rb r r 2 = πηr αr2 + 2πη . (A.8) rc(γ − 1) 1 + b r rc min 2 − α r rb     min

In Eq. A.8, η represents the parameter obtained by the vertical −γ shifting of the luminosity profile. rmax 2 2 ∂N r r The error ∆N is thus given by: = −2πΣ0 r 1 + ln 1 + dr = i ∂γ Zr  rc !   rc !  min        2  rmax ∂N ∂N ∂N ∂N Σ 2 + 2 +   + r  ∆ = i ∆ + i ∆ + i ∆ + i ∆ π 0(r rc ) 1 (γ − 1)ln 1 Ni η α rmin rmax (A.9) rc ∂η ∂α ∂r ∂r =      , (A.5) min max 2 γ (γ − 1)2 1 + r where: rc   rmin   rb rmax ∂Ni = −α + 1−α = 2πrb rdr 2π r dr ∂N rmin ∂η Zrmin Zrb = −2πΣ0 , (A.6) 2 γ 2−α rmax ∂rmin rmin r r 1 + = −α 2 b + rc πrb r 2π , (A.10)   rmin 2 − α   rb

(we set r = 0.1arcmin). rb min ∂N r i = − −1−α = − −1−α 2 b 2πηαrb rdr πηrb r r , (A.11) ∂N rmax ∂rb Z min = 2πΣ . (A.7) rmin 0 2 γ ∂rmax rmax 1 + r r rc ∂N b max     i = −2πηr−α r ln(r )dr − 2πη r1−α ln(r)dr = ∂α b b The fitting parameters used to calculate ∆N are summarized in Zrmin Zrb Tab. 1. = −α 2 rb + −2πrb r ln(rb) ii) estimate of ∆N rmin i rmax 2πηr2−α {1 + [α − 2] ln(r)} The fitting formulas to the initial distribution of GCs change + , (A.12) for the various galaxies studied. (γ − 2)2 rb

R. Capuzzo-Dolcetta and A. Mastrobuono-Battisti: Globular Cluster System erosion in elliptical galaxies 9

Fig. 4. Left panel: the correlation between the logarithm of the central galactic black hole mass (in solar masses) and the integrated V galactic magnitude (see Table 4). Right panel: the correlation between the value of the parameter proportional to the central galactic phase-space density (in arbitrary units) and the galactic integrated V magnitude. The straight lines are the least-square fits to the data (see Sect. 4.)

1 1 1 1 r = . rmax b b n − r n n n Also in this case we assumed rmin 0 1arcmin.For all the galax- ∂Ni n re re rb r = 2πη re "    # − dr, (A.18) ies analyzed, rmin > rb and rmax > rb; so we have ∂b  r ! r !  n Zrb  e e    ∂Ni 1−α   = −2πηr , (A.13) rb γ−1   min ∂Ni rb ∂rmin = 2πηγ dr + ∂ rb Zrmin  r  ∂Ni − 1 1 1 = 1 α r −1 2πηr . (A.14) rmax b b n − r n n max ηbn n re re rb ∂rmax + 2π re "    # dr = nre Zr re ! For M 49 and NGC 4406 (Sect. 3.3 and Sect. 3.7) we have b γ−1 rb 2πηγ rb = = r + Ni(rmin, rmax) 2 − γ r 1 1   rmin r rmax γ b n − r n 1 1 1 rb bn r r r n n −1 = + " e e # = rmax b b − r n 2πη r θ(rb − r) e     θ(r − rb) dr ηbn n re re rb  r  + 2π re "    # dr, (A.19) Zrmin      nre Zrb re ! 1 1 γ rb rmax b rb n − r n  2πη 2 rb n re re  = r + 2πη re "    #dr (A.15) (A.20) 2 − γ r Zr rb γ   rmin b ∂N r r i = 2πη r b ln b dr = where b = 1.992 n − 0.3271. The second row of the previous ∂γ n Zrmin  r   r  expression is justified by the fact that, both for M 49 and NGC rb 2πηr2 r γ r 4406, r > r . Thus the error on N = b 1 + (2 − γ) ln b (A.21) b min l 2 (γ − 2)  r    r  r min ∆ = ∂Ni ∆ + ∂Ni ∆ + ∂Ni ∆ + ∂Ni ∆ + Ni η bn rb re 1 1 rmax rb n r n ∂η ∂bn ∂rb ∂re ∂N ηb bn − i = −2π n re " re re # × ∂N ∂ N ∂N 2     i i i ∂re nr + ∆n + ∆rmin + ∆rmax (A.16) e Zrb ∂n ∂rmin ∂rmax 1 1 n −1 n −1 rb r ∆ = ∂ bn ∆ × rb − r dr, (A.22) where bn ∂n n, is evaluated by the following expressions  re ! re !  of the individual error contribution:      1 1 1 1 r r n r n rb γ rmax rb n r n max b b − ∂N r bn − ∂Ni ηbn n re re i = b + " re re # = = − " # × 2π r dr 2π re     dr 2π 2 re     ∂η r ∂n n Zrb Zrmin   Zrb 1 1 1 1 γ rb rmax rb n r n n n 2π r bn − rb rb r r 2 b " re re # × − = r + 2π re     dr, (A.17)  ln ln  dr, (A.23) 2 − γ r Zr re ! re ! re ! re !   rmin b       10 R. Capuzzo-Dolcetta and A. Mastrobuono-Battisti: Globular Cluster System erosion in elliptical galaxies

Rememberingthat, for both NGC 4472 and NGC 4406, rmin > rb All the integrals from Eq. A.15 to Eq. A.32 must be calcu- and rmax > rb we can estimate the following contributions lated numerically using the values of the parameters given in

1 1 Table 2. The results listed in Table 1 are obtained assuming an ∂N r n r n error of 1% on each independent parameter used. Only in the i = − πη b − min , 2 rmin exp bn   (A.24) case of NGC 3258 we assumed ∆γ = 0.001 because Capetti & ∂rmin  re ! re !     Balmaverde (2006) obtained γ = 0 from their Nuker fit, and so      1 1  it is impossible to give an estimate of the error as a percentage ∂N r n r n i = b − max of γ. 2πηrmax exp bn   . (A.25) ∂rmax  re ! re !        Acknowledgements. See Tab. 2 for the values of the parameters used in the Eq. A.15- Eq.A.25. Last for NGC 3268 and NGC 3258 (see Sect. 3.4, Sect. 3.5 References and Tab. 2 for the values of the parameters) we have Balmaverde, B., Capetti, A., 2006, A&A, 447, 97, BC06 γ−β Bassino, L. P., Faifer, F. R., Forte, J. C., Dirsch, B., Richtler, T., Geisler, D., & rmax γ α α = rb + r Schuberth, Y., 2006, A&A, 451, 789 Ni(rmin, rmax) 2πη r 1 dr, (A.26) Bertin, G., Ciotti, L., & Del Principe, M., 2002, A&A, 386, 149 Z r " rb ! # rmin   Capetti, A., Balmaverde, B., 2005, A&A, 440, 73 thus the error Capuzzo Dolcetta, R., 1993 ApJ, 415, 616 Capuzzo Dolcetta, R., ASP Conference Series, 285 ∆ = ∂Ni ∆ + ∂Ni ∆ + ∂Ni ∆ + ∂Ni ∆ Capuzzo Dolcetta, R., Donnarumma, I., 2001, MNRAS, 328, 645 Ni η rb γ α Capuzzo Dolcetta, R., Tesseri, A., 1997, MNRAS, 292, 808, CDT99 ∂η ∂rb ∂γ ∂α Capuzzo Dolcetta, R., Tesseri, A., 1999, MNRAS, 308, 961 + ∂N i ∆ + ∂ Ni ∆ + ∂Ni ∆ Capuzzo Dolcetta, R., Vicari, A., 2005, MNRAS, 356, 899 β rmin rmax (A.27) Capuzzo Dolcetta, R., Vignola, L., 1997, A&A, 327, 130, CD97 ∂β ∂rmin ∂rmax Carter, D., Jenkins, C. R., 1993, MNRAS, 263, 1049, CJ93 with Cˆot´e, P., McLaughlin, D. E., Cohen, J. G., Blakeslee, J. P., ApJ, 2003, 591, 850 Cox A. N. editor, 2000, Allen’s Astrophysical Quantities, 4th ed., Springer- α γ−β Verlag ∂N rmax r γ r α i = 2π r b 1 + dr, (A.28) Dirsh, B., Richtler T., Bassino L. P., 2003, A&A, 408, 929, D03 ∂η Eisenhauer, F. et al., 2005, ApJ, 268, 246, E05 Zrmin  r  " rb ! # Ferrarese, L., Ford H., 2005, SSRv, 116, 523, Fe05 Ferrarese, L., Cˆot´e, P., Jord???n, A., Peng, E. W., Blakeslee, J. P., Piatek, S., α γ−β Mei, S., Merritt, D., Milosavljevic, M., Tonry, J. L., West, M. J., 2006, ApJ, ∂N rmax r α r γ−1 i = 2πη 1 + γ b + (A.29) 164, 334 ∂ Forbes, D. A., Franx, M., Illingworth, G. D., Carollo, C. M., 1996, ApJ, 467, rb Zrmin " rb ! # (  r  126 + α −1 (γ − β) r γ−α 1 r Forbes, D. A., Sanchez- Blazquez, P., Phan, A. T. T., Brodie, J. P., Strader, J., − r2 b 1 + dr, 2  Spitler, L., 2006, MNRAS, 366, 1230, F06 r  r  " rb ! # b  G´omez, M., Richtler, T., 2004, A&A, 415, 499, GR04 Goudfrooij, P., Schweizer, F., Gilmore, D., Whitmore, B. C., 2007, AJ, 133,  γ−β 2737 rmax γ α α Harris, W. E., Kavelaars, J. J., Hanes, D. A., Pritchet, C. J., Baum, W. A., 2009, ∂Ni rb r = 2πη r 1 + × AJ, 137, 3314 ∂γ Zrmin  r  " rb ! # Harris, W. E., Racine, R., 1979, ARA&A, 17, 241 r 1 r α Illingworth, G., 1976, ApJ, 204, 73 × ln b + ln 1 + dr, (A.30) Kissler, M., Richtler, T., Held, E. V., Grebel, E. K., Wagner, S. J., Capaccioli, (  r  α " rb ! #) M., 1994, A&A, 287, 463, KR94 Lauer T. R., Gebhardt K., Faber S. M., Richstone D., Tremaine, S., Kormendy

γ−β J., Aller M. C., Bender R., Dressler A., Filippenko A. V., Green R., Ho L. C., rmax γ α α 2007, ApJ, 664, 226 ∂Ni − rb r = 2πηα 1(γ − β) r 1 + × Lee, M. G., Park, H. S., Kim, E., Hwang, H. S., Kim, S. C., Geisler, D., 2008, ∂α r r Zrmin   " b ! # ApJ, 682, 135L α r r α Magorrian, J., Tremaine, S., Richstone, D., Bender, R., Bower, G., Dressler, A., ln r Faber, S. M., Gebhardt, K., Green, R., Grillmair, C., Kormendy, J., Lauer, T., × rb rb − α−1 ln 1 + dr, (A.31)   r  α  r  1998, AJ, 115, 2285, M98  1 + " b ! # McLaughlin, D. E., 1995, AJ, 109, 2034  rb  h   i Peng, E. W., Ford, H. C., Freeman, K. C., 2004, ApJ, 602, 705    γ−β  Prugniel P., Simien F., 1996, A&A, 309, 749 rmax γ α α Rhode, K. L., Zepf, S. E., 2003, AJ, 126, 2307 ∂Ni rb r = 2πηα−1 r 1 + × Rhode, K. L., Zepf, S. E., 2004, AJ, 127, 302, RZ04 ∂β Zrmin  r  " rb ! # Rhode, K. L., Katherine, L., Zepf, S. E., Kundu, A. , Larner, A. N., 2007, AJ, r α 134, 1403 × ln 1 + dr. (A.32) Richstone, D., Ajhar, E. A., Bender, R., Bower, G., Dressler, A., Faber, S. M., " rb ! # Filippenko, A. V., Gebhardt, K., Green, R., Ho, L. C., Kormendy, J., Lauer, T. R., Magorrian, J., Tremaine, S., 1998, Nature, 395, A14, R98 γ−β Sikkema, G., Peletier, R. F., D.Carter, Valentijn, E. A., and Balcells, M., 2006, γ α α ∂Ni rb rmin A&A, 458, 53 = −2πηrmin 1 + . (A.33) ∂rmin rmin ! " rb ! # Spitler, L. R., Forbes, D. A., Strader, J., Brodie, J. P., Gallagher, J. S., 2008, MNRAS, 385, 361 Spitler, L. R., Larsen, S. S., Strader, J., Brodie, J. P.; Forbes, D. A., Beasley, with rmin = 0.1arcmin, and M. A., 2006, AJ, 132, 1593 γ α γ−β Spolaor Ma., Forbes D. A., Hau G. K. T., Proctor R. N., Brough S., 2008, ∂N r r α i = 2πηr b 1 + max . (A.34) MNRAS, 385, 667 max Trujillo, I., Erwin, P., Asensio Ramos, A., Graham, A. W. 2004, AJ, 127, 1917 ∂rmax rmax ! " rb ! # R. Capuzzo-Dolcetta and A. Mastrobuono-Battisti: Globular Cluster System erosion in elliptical galaxies 11

Van der Marel, R. P., 1999, ApJ, 117, 744, VM99 Verdoes, G., Van der Marel, R. P., Carollo, C. M., de Zeeuw, P. T., 2000, AJ, 120, 1221, VV00 Zhang, Z., Xu, H., Wang, Y., An, T., Xu, Y., Wu, X. P., 2007, ApJ, 656, 805, Z07 12 R. Capuzzo-Dolcetta and A. Mastrobuono-Battisti: Globular Cluster System erosion in elliptical galaxies

Galaxy N Ni Nl δN ǫl Mi Ml NGC 1400 73 256 183 0.71 0.40 8.45 × 107 6.04 × 107 NGC 1407 314 398 84 0.21 0.12 1.31 × 108 2.77 × 107 NGC 4472 6334 11932 5598 0.47 0.20 3.94 × 109 1.85 × 109 NGC 3268 505 903 398 0.44 0.15 2.98 × 108 1.31 × 108 NGC 3258 343 555 212 0.38 0.16 1.83 × 108 7.00 × 107 NGC 4374 4655 7016 2361 0.34 0.050 2.34 × 109 7.86 × 108 NGC 4406 2850 4209 1359 0.32 0.23 1.25 × 109 4.04 × 108 NGC 4636 1411 2157 746 0.35 0.11 6.41 × 108 2.22 × 108

Table 3. col. (1): galaxy name; col. 2-8: the present number of GCs (N), its initial value (Ni), the number of GCs lost (Nl), the percentage of GCs lost and the estimated relative error on Nl (ǫl, see Appendix), the estimate of the initial mass of the whole GCS (Mi) and of the mass lost by each GCS (Ml).Mi and Ml are in solar masses.

Galaxy MV Mbh Ml Sources MW −20.60 3.61 × 106 1.80 × 107 E05, VV00, CDV97 M 31 −19.82 6.19 × 107 2.30 × 107 M98 , CDV97 M 87 −22.38 3.61 × 109 2.33 × 109 M98, CDV97 NGC 1427 −20.43 1.17 × 108 8.86 × 107 VM99, CDT99 NGC 4365 −22.06 7.08 × 108 7.48 × 107 VM99, CDT99 NGC 4494 −20.94 4.79 × 108 2.98 × 107 VM99, CDT99 NGC 4589 −21.14 3.09 × 108 7.58 × 107 VM99, CDT99 NGC 5322 −21.90 9.77 × 108 -6.51 × 107 VM99, CDT99 NGC 5813 −21.81 2.82 × 108 1.03 × 108 VM99, CDT99 NGC 5982 −21.83 7.94 × 108 8.86 × 107 VM99, CDT99 NGC 7626 −22.34 1.95 × 109 3.59 × 107 VM99, CDT99 IC 1459 −21.68 2.60 × 109 1.57 × 108 Fe05, VM99, CDT99 NGC 1439 −20.51 1.95 × 108 4.79 × 106 VM99, CDT99 NGC 1700 −21.65 4.37 × 109 3.66 × 106 VM99, CDT99 NGC 1399 −21.71 5.22 × 109 1.44 × 108 M98, CDD01 NGC 1400 −20.63 3.71 × 108 8.45 × 107 VM99, F06, CDM NGC 1407 −21.86 5.55 × 108 1.31 × 108 Z07, F06, CDM NGC 4472 −23.10 2.63 × 109 3.94 × 109 M98, RZ04, CDM NGC 3268 −22.07 4.68 × 108 2.98 × 108 BC06, D03, CDM NGC 3258 −21.40 2.14 × 108 1.83 × 108 BC06, D03, CDM NGC 4374 −22.62 1.41 × 109 2.34 × 109 R98, GR04, CDM NGC 4406 −22.30 1.40 × 108 1.25 × 109 CJ93, RZ04, CDM NGC 4636 −21.70 3.63 × 108 6.41 × 108 VM99, KR94, CDM Table 4. col. (1) galaxy name; col. (2), (3) and (4): the V absolute magnitudes, the galactic central black hole masses and the mass lost by GCSs (both in solar masses), respectively; col. (5): bibliographic reference sources for entries in col. (2), (3) and (4); CDM is the present paper, the other ackronyms are defined in the References.