The environment of formation as a second parameter

The environment of formation as a second parameter for globular cluster classification

D. Fraix-Burnet, E. Davoust11footnotemark: 1 and C. Charbonnel11footnotemark: 1
Université Joseph Fourier, CNRS, Laboratoire d’Astrophysique de Grenoble, BP 53, F-38041 Grenoble cedex 9, France
Université de Toulouse, CNRS, Laboratoire d’Astrophysique de Toulouse-Tarbes, 14 av. E. Belin, F-31400 Toulouse, France
Observatoire Astronomique de l’Université de Genève, 51, chemin des Maillettes, 1290 Versoix, Switzerland
E-mail: fraix@obs.ujf-grenoble.fr; davoust@ast.obs-mip.fr; Corinne.Charbonnel@unige.ch
Accepted 2009 June 11. Received 2009 June 10; in original form 2009 April 29
Abstract

We perform an evolutionary multivariate analysis of a sample of 54 Galactic globular clusters with high-quality colour-magnitude diagrams and well-determined ages. The four parameters adopted for the analysis are: metallicity, age, maximum temperature on the horizontal branch, and absolute V magnitude. Our cladistic analysis breaks the sample into three novel groups. An a posteriori kinematical analysis puts groups 1 and 2 in the halo, and group 3 in the thick disc. The halo and disc clusters separately follow a luminosity-metallicity relation of much weaker slope than galaxies. This property is used to propose a new criterion for distinguishing halo and disc clusters. A comparison of the distinct properties of the two halo groups with those of Galactic halo field stars indicates that the clusters of group 1 originated in the inner halo, while those of group 2 formed in the outer halo of the Galaxy. The inner halo clusters were presumably initially the most massive one, which allowed the formation of more strongly helium-enriched second generation stars, thus explaining the presence of Cepheids and of very hot horizontal-branch stars exclusively in this group. We thus conclude that the “second parameter” is linked to the environment in which globular clusters form, the inner halo favouring the formation of the most massive clusters which subsequently become more strongly self-enriched than their counterparts of the galactic outer halo and disc.

keywords:
Globular clusters: general – Galaxy: formation – Galaxy: evolution – Galaxy: halo – method: statistical
pagerange: The environment of formation as a second parameter for globular cluster classificationApubyear: 2009

1 Introduction

Globular clusters are touchstones of astrophysics.galaxies. The oldest of them witnessed the formation and early evolution of their host galaxies and of their substructures, and their study has historically coloured the different scenarios of galaxy formation. However, as a collective population in a galaxy, they present unsolved problems. In particular their origin is not firmly established despite the large amount of work devoted to the analysis of correlations among their observable properties. It has long been realised that part of the difficulty arises from sheer dynamical evolution undergone by these objects since the time of their formation. Indeed, any star cluster is the subject of a long list of erosive mechanisms that operate at different rates depending on the cluster’s location and orbit within the Galaxy, and on its initial mass (Djorgovski & Meylan (1994), Gnedin & Ostriker (1997)).

Related to these difficulties, the search for a “second parameter”, beyond metallicity, to explain the distribution of stars along the horizontal branch, has met with a limited success. Zinn (1993) suggested that the halo globular clusters break down into two groups according to their horizontal-branch properties: the two groups have different ages, kinematics and radial distributions. Rey et al. (2001) found that an age difference can explain different horizontal-branch morphologies at a given metallicity. But Dotter (2008), following the seminal work by Rood (1973) on the impact of stellar mass loss on the horizontal-branch morphology, has shown that considering -element enhancement and metallicity-dependent mass loss along the red-giant phase produces similar effects. On the other hand Lee, Gim & Casetti-Dinescu (2007) have found that globular clusters with extended horizontal branch are more massive than normal clusters and are dominated by random motions with no correlation between kinematics and metallicity. Multivariate analyses have been performed by Fusi Pecci et al. (1993) who find that more concentrated clusters have bluer and longer horizontal-branch, and by Recio-Blanco et al. (2006) who find that more massive clusters have a horizontal branch that extends to higher temperatures.

We can point to several reasons why previous studies have not been completely successful. For example, empirically separating the clusters into disc and halo populations solely on the basis of one parameter (metallicity) cannot be satisfactory, as the environment at birth must influence other cluster parameters. This was already recognized by Zinn (1985), who combined metallicity and kinematics to establish two major groups (halo versus bulge/disk). The properties of the horizontal branch have been used, but its usual characterisation by the parameter HBR = (B-R)/(B+V+R) is also unsatisfactory. In this respect, the parameter (maximum effective temperature on the horizontal branch) introduced by Recio-Blanco et al. (2006) is a welcome innnovation as shown in the present paper.

The classical paradigm describing globular clusters as fairly simple systems of coeval stars of homogeneous chemical composition has been seriously challenged recently, and this may bring a crucial piece to the puzzle. One fundamental characteristic of these systems is their metallicity [Fe/H] (identified as the so-called “first-parameter” (van den Bergh (1967))) that is generally inferred from their integrated photometric colours, and that varies strongly from cluster to cluster; in our Galaxy, globular clusters have [Fe/H] ranging between -2.2 and 0 (Harris (1996)). Spectroscopy reveals that, within individual clusters, stars present very homogeneous contents in Fe, but also in and s-elements, indicating that protoclusters formed from gas pre-enriched in heavy metals (James et al. (2004); Prantzos & Charbonnel (2006)). This is in agreement with the predictions of quantitative models that rule out stochastic self-enrichment in most globular clusters as a significant contributor to their heavy metals, leaving pre-enrichment as the dominant contributor to metallicity [Fe/H] (Bailin and Harris (2009)).

However globular cluster stars exhibit extremely scattered light-element (Li, F, C, N, O, Na, Mg, and Al) contents that are not seen among their field counterparts (see e.g. Carretta (2006)). This points to early internal chemical evolution (i.e., self-enrichment) in the globular cluster driven by first-generation massive and fast evolving stars which polluted with their hydrogen-burning products (among which helium in very important quantities) the intracluster gas out of which second generation stars formed (Decressin, Charbonnel & Meynet (2007), Prantzos & Charbonnel (2006)). Recent findings of double or even multiple stellar populations in the colour-magnitude diagrams of several globular clusters, as well as the complexity of the horizontal branch morphology (namely the wide colour distribution, i.e., effective temperature, of the stars presently burning helium in their core) constitute further evidence for internal evolution (Piotto (2009)). All these features can indeed be related to the presence of a second generation of He-enriched stars. Importantly, the star formation history depicted by these features seems to vary from cluster to cluster (Milone et al. (2008)) in a way which is still far from being understood. However we have now firm evidence that Galactic globular clusters have undergone internal chemical evolution and complex star formation histories during their infancy that shaped their properties and in particular their present total mass (Decressin, Charbonnel & Meynet (2007)). This new paradigm has opened a novel route for a better understanding of the origin and history of globular clusters.

It thus appeared to us that a multivariate analysis which simultaneously takes into account any cosmic variance due to evolving physical conditions and groups objects according to environment of formation would be very valuable. Cladistics provides such a methodology. It differs from other clustering analyses in that it focuses on evolution within and between groups rather than on similarities between objects (Wiley et al. (1997)). Cladistics is very commonly used in evolutionary biology and has been pioneered in astrophysics by Fraix-Burnet et al. (2006a, b); Fraix-Burnet (2009) and successfully applied to the dwarf galaxies of the local group (Fraix-Burnet et al. (2006c)).

This paper presents a multivariate analysis based on the method of cladistics of a large sample of Galactic globular clusters. After presenting the data and the method of analysis (Sect. 2), we describe the three groups found by the cladistic analysis (Sect. 3.1), and discuss two important results, evidence for self-enrichment (Sect. 4) and a possible luminosity-metallicity relation (Sect. 5). We then compare the properties of the three groups with those of Galactic halo field stars (Sect. 6), before proposing a scenario for the formation of the three groups (Sect. 7).

2 Data and method of analysis

The choice of parameters is a crucial step in any multivariate analysis. Djorgovski & Meylan (1994) have shown that the manifold of Galactic globular cluster properties has a dimension larger than 4, but that a subset of parameters linked to morphology and dynamics forms a three-dimensional family. Including properties of the stellar populations (e.g. a horizontal-branch parameter or metallicity) will increase the manifold by 1 or 2 dimensions (Fusi Pecci et al. (1993)). Finally, using a large number of photometric and structural parameters, Recio-Blanco et al. (2006) found that 4 eigenvectors account for 79% of the total sample variance.

Taking advantage of this indication, we selected the following four parameters for analysis: relative ages, metallicity ([Fe/H]), absolute V magnitude (), and maximum effective temperature () on the horizontal branch. The age parameter is related to the secular evolution of the stellar populations. [Fe/H] reflects the chemical composition of the environment when and where globular clusters formed and is the “first parameter” for the horizontal branch morphology is a structural parameter that measures the present total baryonic mass of the globular clusters111The absolute magnitude of globular clusters in the Milky Way spans a vast range (, Harris (1996)), and reflects a large mass range (, McLaughlin & van der Marel (2005)).. Finally is a measure of both the pristine chemical composition of the protocluster ([Fe/H] being the first parameter) and of the helium enrichment during early internal chemical evolution, since stars with higher helium content are expected to reach higher effective temperatures on the horizontal branch (D’Antona et al. (2002); Recio-Blanco et al. (2006)). The last three quantities describe truly intrinsic properties of globular clusters. As the age parameter evolves in all clusters, it cannot be used to classify them at the same level as the other three parameters. We thus gave it a lower weight in the cladistic analysis (see Appendix A).

We performed our analysis using the large sample (54 objects) of Recio-Blanco et al. (2006) based on homogeneous Hubble Space Telescope photometry. We used the values obtained uniformly from this database by Recio-Blanco et al. (2006), as well as the relative ages and values they adopted (i.e., taken respectively from De Angeli et al. (2005) and the 2003 on-line revision of Harris (1996)). We did not include more parameters in the cladistic analysis as we preferred to avoid the unwanted effect of redundancies, which give more weight to correlated parameters. Note also that we did not use any kinematical information in the cladistic analysis. However we used other parameters a posteriori to characterize the different groups found by the cladistic analysis: the radial velocities and structural parameters were taken from the 2003 on-line revision of Harris (1996), the orbital parameters from Dinescu, Girard & van Altena (1999); Dinescu et al. (2003); Casetti-Dinescu et al. (2007). In this respect, the distinct orbital properties of group 3 found a posteriori (see §3) are independent of the methodology.

More recent age estimation was published by Marin-Franch et al. (2009) after most of this project was completed, but for only 35 out of 54 globular clusters of our sample. Using these produces an inhomogenous data set, from different sources, relying on different values of Fe/H. The ages of Marin-Franch et al. (2009) have been determined from colour-magnitude diagrams and values of Fe/H which are different from those of Recio-Blanco et al. (2006). Marin-Franch et al. (2009) themselves point out the importance of using a homogeneous set of Fe/H to derive the ages. Nevertheless, it is instructive to perform analyses using both sets and compare the results, so as to determine how sensitive they are to the specific choice of parameter values. One then has the problem of combining two sets of ages, and it is not obvious how this should be done. It turns out that 9 out of 11 globular clusters that calibrate relative ages of Marin-Franch et al. (2009) are in common with De Angeli et al. (2005), and using the relative ages of De Angeli et al. (2005) for these 9 globular clusters gives a mean value of 1.0055 instead of 1.00, which is fine. However, comparing the ages of all the globular clusters in common suggests a non-linear systematic effect, which should perhaps be taken into account. Furthermore, we cannot simply convert each set of relative ages to absolute ages with the zero-point of each set, because the two zero-points are rather different (11.2Gyr for De Angeli et al. (2005) and 12.8Gyr for Marin-Franch et al. (2009)). We thus simply used the relative ages without any attempt at homogenizing them, and the zero-point of one author for all ages. This additional analysis is compared to the main one in Sect. 3.2.

The multivariate analysis was performed using the method of cladistics. In short, the method works as follows. One first builds a matrix with values of the four parameters for each clusters. The values must be discretized, and the number of bins (here 10) depends on the resolution one wants for the analysis. One then chooses a cluster which represents the most unevolved state, in the present case the metal-poorest cluster (NGC 6934), and the software classifies all the other clusters in order of increasing diversification of properties (in other words, by increasing distance in the manifold of parameters). Clusters which are diverging from the original cluster in the same direction are put on the same branch. We refer to Appendix A and Fraix-Burnet et al. (2006a, b, c); Fraix-Burnet (2009) for more details on the principles of the method.

3 Three groups of clusters

3.1 The main tree

The main result of our analysis is presented in the form of a tree structure, a usual form of representation in graph theory. The properties of the sample can be read from the structure of the tree shown on Fig. 1.

Figure 1: The tree resulting from the cladistic analysis. The sample break into three distinct branches. Group 1 is composed of inner halo clusters, group 2 of outer halo clusters, and group 3 of disc clusters

The tree has been rooted with group 2 which has the lowest metallicity on average, and as such is supposedly made of more primitive (or ”ancestral”) material. The tree divides into three main branches, which define three groups with quite distinct properties. Age increases roughly monotonically along each branch, as expected. There are some subbranches sharing similar values of the adopted parameters, such as NGC 5904, 2808, 6388 and 6441, and which set them apart in the four-dimensional space of parameters, implying that they might in fact belong to a fourth small group with properties similar to those of group 1. Specialists will immediately remark that the latter three clusters indeed share peculiar properties, in particular a very Helium-rich stellar population (Pumo, D’Antona & Ventura (2008)). We emphasize that the Helium abundance is not one of the parameters included in the cladistic analysis: it thus must influence one way or another the four parameters used in the analysis.

Group 1 Group 2 Group 3
inner halo outer halo thick disc
Number of clusters 25 11 18
(kpc) 9.4 (7.4) 12.9 (8.0) 4.2 (2.9)
(kpc) 4.8 (4.5) 8.6 (7.6) 1.9 (2.0)
Fe/H -1.40 (0.35) -1.92 (0.16) -0.92 (0.35)
(mag) -8.5 (0.7) -7.6 (0.6) -7.1 (0.9)
(km/s) -7. +46. +119.
(km/s) 120 (107) 151 (107) 69 (74)
Age 9.98 (0.96) 11.17 (0.70) 10.18 (0.48)
Age- + x -
Age- [Fe/H] - - +
Age- x x x
- [Fe/H] + x +
- [Fe/H] + + +
- + x x
Number of clusters 12 8 5
(Myr) 353 (212) 391 (254) 142 (35)
() -691 (341) -649 (339) -1027 (111)
0.63 (0.18) 0.54 (0.18) 0.21 (0.10)
(kpc) 17 (12) 19 (13) 6 (1.3)
(kpc) 7.2 (6.2) 9.6 (8.5) 1.5 (1.0)
(deg) 32 (12) 38 (17) 21 (15)
886 (802) 941 (575) 866 (314)
(km/s) 100 (78) 77 (40) 34 (16)
(km/s) +19 (141) -33 (124) -3 (22)
(km/s) +15 (121) +89 (129) +170 (33)

average of 21 values; average of 15 values;
average of 13 values; average of 6 values.

Table 1: Properties of the three groups of globular clusters. No age estimate is available for NGC 6139, 6229, 6304, 6388, 6441, 6569 and 6642. The middle part of the Table gives the correlations: + means a correlation, - means an anticorrelation, x no correlation. The orbital properties presented in the lower part of the Table are only available for a subset of each group. Numbers in brackets are rms dispersions

The properties of the three groups are presented in Table 1. The first 7 rows give the characteristics of the 3 groups: number of clusters in each group, mean distance from the Galactic center , height above the Galactic plane , metallicity [Fe/H], absolute magnitude (), mean velocity of rotation in the Galactic plane (computed with the equations given by Frenk & White (1980) using a velocity of the Sun of 220 km/s), radial velocity dispersion . The next 6 rows describe the correlations between the four parameters. The bottom part of the Table lists orbital parameters taken from Dinescu, Girard & van Altena (1999); Dinescu et al. (2003); Casetti-Dinescu et al. (2007), which were not available for all clusters of each group: successively number of clusters, period of rotation , total energy , eccentricity of the orbit , apocentric distance , maximum distance reached above the Galactic plane , inclination angle with respect to the Galactic plane , angular momentum , and finally the velocity components in cylindrical coordinates: vertical velocity , radial velocity and tangential velocity .

Paired t-tests showed that the differences of the means of the groups taken two by two is not equal to 0 () for the parameters , , [Fe/H], . This is also the case between group 1 and group 2 and between group 1 and group 3 for and , and between group 2 and group 3 for and age. There is also evidence () for different means of the latter two groups for , , , and , as well as between group 1 and group 2 for age and between group 1 and group 3 for . We emphasize that the rotational properties of Galactic globular clusters are very uncertain, since they are derived from projected radial velocities or numerical simulations of orbits in a model Galaxy.

These properties show that the first two groups belong to the halo population of clusters, while the third group belongs to the thick disc population. Hereafter, the thick-disc clusters will simply be called disc clusters. The average velocity of rotation of group 1 and 2 together is = 9 km/s. Group 3 is confined to the Galactic plane, and has a high and low . If we separate group 3 into two subgroups of equal size according to their distance from the Galactic center, we find that is 88 km/s for the inner subgroup ( kpc) and 187 km/s for the outer subgroup. There is also evidence that group 3 has a shorter , lower , and , and no radial motion, as expected from clusters that partake in the overall rotation of the disc.

One cluster has certainly been misclassified. NGC 6981 is in group 3 although it is at = 9.1 kpc and has a low velocity of rotation. Since NGC 6981 is a borderline cluster in all the Figures, the value of one of the four parameters may be erroneous. Indeed, raising from –7.04 to –7.27 brings it into the next bin in our cladistic analysis (which requires that the data be discretized into a limited number of bins), and running the cladistic analysis again moves the cluster to group 2. Moving any of the other three parameters by one bin and redoing the cladistic analysis does not change the status of the cluster. Since we found no reason for an erroneous , we left it in group 3.

Another possible discrepant cluster is NGC 6266, which is in group 1, but which, according to Dinescu et al. (2003), belongs to a rotationally supported system, on the basis of its kinematics (but without precise orbital determination), despite its low metallicity. However, it is on the same subbranch as (and undistinguishable from) NGC 7089, which definitely belongs to the halo, according to our analysis and that of Dinescu, Girard & van Altena (1999). In addition, as pointed out by the referee, an isotropic distribution of orbits will statistically produce one or several ones in or near the Galactic plane. We are thus confident that NGC 6266 belongs to the halo.

We now compare the statistical properties of the two halo groups with those of the Galactic halo field stars. The dichotomy of the Galactic halo stellar population has been suspected for some time. The most quantitative study in that respect, that of Carollo et al. (2007), clearly identifies two broadly overlapping structural components corresponding to an inner and an outer halo. Stars of the inner halo are in highly eccentric orbits, in slightly prograde rotation, and have an average metallicity of [Fe/H] = –1.6. The outer halo stars have a uniform distribution of eccentricities, are in highly retrograde orbits, and of lower metallicity [Fe/H] = –2.2. These properties are also among those that distinguish the two groups of halo clusters, and indicate that the clusters of group 1 may have originated in the inner halo: they have higher eccentricities and metallicities than group 2, while group 2 formed in the outer halo: they have the largest , , and , lowest metallicities; is positive, but not significantly so.

3.2 Additional analysis using an inhomogeneous set of ages

We also applied our analysis to the sample of 54 globular clusters with composite ages as explained in Sect. 2. It leads to roughly the same three groups as before, with several obvious halo clusters moving into G3 (which is in principle composed of thick-disk globular clusters), and two globular clusters of G1 moving into G2, among them NGC 6218 which has a Cepheid (and should thus be in G1). In other words, the number of misclassified globular clusters rises from one (NGC 6981) to only about 6. This indicates that our groups are fairly robust. Of course, their detailed contours depend on the choice of data and we believe that the slight discrepancy is due to the inhomogeneous data of this additional analysis: the ages of Marin-Franch et al. (2009) have been determined from colour-magnitude diagrams and values of Fe/H which are different from those of Recio-Blanco et al. (2006), while logTe comes from Recio-Blanco et al. (2006) and is derived from diagrams using the corresponding set of Fe/H values.

4 Evidence in favour of self-enrichment

Self-enrichment by a first generation of stars is frequently advocated to explain the chemical anomalies in globular clusters (e.g. Prantzos & Charbonnel (2006)) in relation with horizontal-branch morphology (D’Antona & Caloi (2008)). We list below distinctive properties of the three groups of globular clusters pointing to such a process.

The well-known correlation between metallicity and extent of the horizontal branch is clearly present in Fig. 2: the more metallic the cluster, the less extended its horizontal branch at a given metallicity. Also, the more luminous (that is the more massive) clusters tend to have more extended horizontal branch (Recio-Blanco et al. (2006); Lee, Gim & Casetti-Dinescu (2007)). This latter point is easily understood in the self-enrichment framework, in the sense that more massive globular clusters retain the helium-rich ejecta of massive polluter stars in their deeper potential well more efficiently than less massive globular clusters .

The crucial new information brought by our analysis in this context is that the present mass of the clusters seems to depend on their origin, the inner halo clusters being presently more massive than their outer halo counterparts (Fig. 2). In fact, in order to have extreme light-element abundance patterns, and in particular extreme O-Na anticorrelation (Carretta (2006)), which is linked to the extent of the horizontal branch (Carretta et al. (2007)), the inner halo clusters must have been even more massive in the past, before they lost a huge number (96%) of first-generation low-mass stars in their early dynamical evolution (Decressin, Charbonnel & Meynet (2007)). We thus expect the mass difference between the inner and outer halo globular clusters to have been even larger in the past.

The disc and halo clusters are well separated in the age-metallicity diagram, shown in Fig. 3. For the halo clusters, metallicity decreases with age. For the disc clusters, on the contrary, it marginally increases with age, if at all. The spread in age is 4 Gyr for the halo component and only 1.5 Gyr for the disc one. The figure confirms that the metallicity of NGC 2808 should indeed be about 0.5dex lower. The two other He-rich clusters of group 1 do not appear on this plot because no age estimate is available for them, but their metallicity is indeed about 0.5dex higher than the highest metallicity of the rest of the halo population. This supports the claim by Caloi & D’Antona (2008) and Prantzos & Charbonnel (2006) (see also Decressin, Charbonnel & Meynet (2007)) that He-enrichment must be associated with the build-up of abundance anomalies of light elements during the phase of self-enrichment.

The presence of multiple stellar populations in the colour-magnitude diagram is another evidence for self-enrichment. Unfortunately, only three of the clusters of the sample, NGC 1851, 2808 and 6388, all belonging to group 1, are known to have such a feature (Piotto (2009)). We predict that two other known such clusters, NGC 5139, 6656 which are both metal-poor and massive, are also inner halo globular clusters.

Additional clues to the self-enrichment scenario can be gathered from the RR Lyrae and Cepheid contents of the three groups. Globular clusters have historically been divided into two groups (Oosterhof I and II) according to the properties of their RR Lyrae stars. Using the compilation of Clement et al. (2001), we find that the two Oosterhof types are equally present in group 1 and 2. More interestingly, we find that the distribution of periods of RR Lyrae stars in group 2 is more sharply peaked than the corresponding distribution for group 1, which presents a minor secondary peak at a higher period. The narrower period distribution in group 2 implies a small dispersion in mass loss along the red-giant branch (Caloi & D’Antona (2008)), while the wider distribution in group 1 implies several generations of stars, each with a narrow distribution of mass loss along the red-giant branch, and with increasing helium abundance (D’Antona & Caloi (2008)), reinforcing the necessity of self-enrichment.

Furthermore all the population II Cepheids are found in clusters of group 1, and none in clusters of group 2. Population II Cepheids result from the evolution of post-horizontal branch stars which start from the higher temperatures of the zero-age horizontal branch and move toward the asymptotic giant branch or leave that branch on rapid blueward loops (Wallerstein (2002)); this explains their absence in halo clusters with low .

We thus reach the conclusion that the inner halo favours the formation of very massive clusters, which retain more easily the products of first-generation stars and thus become more strongly self-enriched, giving rise to more extended horizontal branches.

Figure 2: Metallicity-logTe diagram with symbol size indicating visual magnitude. The halo clusters are represented by open symbols: squares for group 1 and triangles for group 2. The disc clusters (group 3) are represented by full squares
Figure 3: Age-metallicity diagram. Same symbols as Fig. 2. No age estimate is available for NGC 6139, 6229, 6304, 6388, 6441, 6569 and 6642. Metallicity decreases with age in the halo clusters while it very marginally increases with age in the disc clusters

5 A luminosity-metallicity relation for globular clusters?

A mass-metallicity or luminosity-metallicity relation is found among galaxies, but is not expected and has not been found in the Galactic globular cluster system (Djorgovski & Meylan (1994)), nor in numerical simulations of globular cluster formation (Kravtsov & Gnedin (2005)). However, it has been found, in the form of a ”blue tilt”, in colour-magnitude diagrams of globular cluster systems in bright ellipticals (Brodie & Strader (2006); Harris et al. (2006); Harris (2009); Mieske et al. (2006)). Such a relation, where metallicity increases with Mv, is present in our sample (Fig. 4), if we consider the disc and halo clusters separately.

We have plotted a line separating the disc and halo clusters in Fig. 4, which can be used as a criterion for distinguishing the two types of clusters, in conjunction with other criteria, since the separation is not perfect (NGC 104 is a notable exception). The disc clusters are on average fainter than the halo ones by about 3 magnitudes at a given metallicity. Correlation lines were adjusted to the two subsamples: they have comparable slopes of -2.0 and -2.8 for the halo and disc clusters respectively, with admittedly low correlation coefficients of 0.53 and 0.32. These slopes are much lower than that of -6.75 predicted and found for the dwarf galaxies of the Local Group (Dekel & Silk (1986)). They were however probably much steeper when the globular clusters formed, since we expect some of these objects to have lost a large fraction of their initial mass in the self-enrichment framework.

A remarkable property of our disc clusters is that they extend to metallicities lower than the conventional limit of [Fe/H] = – 0.8 (Zinn (1985)). This is not very surprising per se, since Dinescu, Girard & van Altena (1999) have found three metal-poor Galactic globular clusters with thick-disc kinematics, and in M31 there is also evidence for metal-poor globular clusters with disc kinematics (Morrison et al. (2004), although see Fusi Pecci et al. (2005)).

If we divide group 3 along the conventional limit, we find that the metal-rich clusters have = 184 km/s whereas the other ones have = 71 km/s, rather low, but still significantly larger than that of the halo clusters. We have checked that this low mean velocity is not due to one cluster in particular. The two subgroups do not distinguish themselves otherwise; in particular they have the same spatial distribution (same mean and ). Two of the low-metallicity disc clusters, NGC 6171 and 6362, have orbits determined by Dinescu, Girard & van Altena (1999), which confirm that they do belong to the disc population. In fact Dinescu, Girard & van Altena (1999) state that their most significant result is to have shown the existence of metal-poor clusters with orbits consistent with the thick-disc motion. Our analysis confirms this finding.

The second important result of the present paper is that the disc and halo globular clusters should not be separated on the basis of metallicity, but rather of a multivariate analysis, taking into account other parameters. We propose to use the magnitude at a given metallicity as a rough criterion, with a limit such that , together with other criteria, such as location in the Galaxy and velocity of rotation.

Figure 4: Metallicity- diagram. Same symbols as Fig. 2. The solid line is the limit between disc and halo clusters according to our criterion ( = –2.4Fe/H – 10.45). The disc cluster above this line is NGC 104

6 Comparison with other studies and with halo field stars

Before interpreting the differences between the three groups in terms of formation history, we compare them to the traditional disc, young halo and old halo groups of Zinn (1993), to emphasize that they are rather different. There are 4, 4 and 1 young halo clusters in group 1, 2 and 3 respectively, and the numbers are 17, 10 and 8 for the old halo clusters. Two of Zinn’s disc clusters (NGC 6388 and 6441) are in our group 1. Since they are located near the Galactic centre and have no determined orbits, it is not possible to decide if they kinematically belong to the disc or the halo. Furthermore, as mentioned in Sect. 3.1, these two clusters might in fact belong to a fourth small group, maybe of Galactic bulge clusters. Turning to the metallicity vs HBR diagram, we find that group 1 extends to lower metallicities at a given HBR than the old halo clusters, and that there are clusters of group 1 and 2 among the old halo clusters with the reddest HBR. Comparing our groups to those of Lee, Gim & Casetti-Dinescu (2007), we find that all the clusters of their group with extended horizontal branch are in our group 1, except NGC 4833 (which is in our group 2).

If we now compare (Table 2) our grouping with that of Harris (2001) (see his Table 1.6), we find that G3 dominates in metal-rich clusters class (MRC) and G1 and G2 dominate in metal-poor cluster class (MPC). In MPC alone there is no clear separation in R between G1 and G2, while G3 tends to be in the inner regions. G2 tends to be among the more metal-poor globular clusters and tends to be at larger galactocentric distances. In summary, the MPC/MRC dichotomy corresponds roughly to our halo/disk separation, and our G2 populates the very metal-poor and distant MPC. Table 2 also shows that a classification based on arbitrary criteria does not quite retrieve the groups obtained with a multivariate analysis.

We now compare the content in -elements of the two groups of halo globular clusters with those of halo field stars. Several studies have shown that the field stars in the outer halo, identified through their kinematical or orbital parameters, tend to have lower and/or more dispersed relative abundances in -elements (Stephens & Boesgaard (2002); Gratton et al. (2003)), which points to a difference in star formation rates of their birth environment: the -elements are indeed almost exclusively provided by core-collapse supernovae, which arise from ephemeral massive stars, while Fe is essentially produced in type Ia supernovae, which arise from stars with longer lifetimes. We find marginal evidence for a similar difference between groups 1 and 2 (see Fig. 5). [/Fe] decreases with increasing [Fe/H] in group 1, while it is more dispersed and shows no clear trend with [Fe/H] in group 2. This confirms a similar origin for the -elements in globular clusters and field stars of the halo, which thus has to occur prior to protocluster formation.

Figure 5: Metallicity versus light-element abundance. Same symbols as previous figures. The line is a least-squares fit to the globular clusters of group 1. Small triangles are disc (green) and halo (blue) field stars. Small red open squares are dwarf galaxies of the Local Group (from Venn et al. 2004)
Table 2: Comparison between Table 1.6 of Harris (2001) and our grouping by number of clusters for each class.
Total G1 G2 G3
MRC All [Fe/H] -1 14 2 0 12
MRC R = 0 - 4 kpc 9 2 0 7
MRC R = 4 - 9 kpc 5 0 0 5
MPC All [Fe/H] -1 40 23 11 6
MPC R = 0 - 4 kpc 10 6 1 3
MPC R = 4 - 8 kpc 11 5 4 2
MPC R = 8 - 12 kpc 7 6 1 0
MPC R = 12 - 20 kpc 7 4 2 1
MPC R 20 kpc 5 2 3 0
MPC -2.30 [Fe/H] -1.85 9 1 8 0
MPC -1.85 [Fe/H] -1.65 6 4 2 0
MPC -1.65 [Fe/H] -1.50 8 6 1 1
MPC -1.50 [Fe/H] -1.32 8 5 0 3
MPC -1.32 [Fe/H] -1.00 9 7 0 2
MPC All [Fe/H] -1.70 27 20 1 6
MPC HBR0, R 8kpc 13 7 5 1
MPC HBR0, R 8kpc 5 5 0 0
MPC HBR0 18 6 0 12

7 Origin of the three groups of globular clusters

It is generally assumed that the Galaxy assembled in a hierarchical fashion from collapsing haloes of dark matter (Bertschinger (1998). Small proto-galactic clumps formed first, from initial small-scale density fluctuations, and collapsed in a dissipationless way (because their gas was quickly consumed or blown away), or else they merged and grew in size to form larger clumps which spiralled toward the inner regions of the Galaxy by dynamical friction and experienced a dissipational collapse. This scenario, or variants of it, first proposed by Searle & Zinn (1978) and verified by cosmological simulations(Bekki & Chiba (2001)), has repeatedly been invoked to explain the properties of the Galactic globular clusters and of the stellar halo. Furthermore, cosmological simulations have shown that globular clusters can form in the densest regions of collapsing subhaloes of dark matter and giant molecular clouds, when the clouds reach a critical density and are under a high external pressure. The mass distribution function of the clusters is similar to that of the clouds (Kravtsov & Gnedin (2005)).

It has often been proposed that some clusters, in particular those identified as young halo clusters by Zinn (1993), which tend to be in the outer halo and counter-rotating, were formed by accretion and disruption of satellite galaxies. But the chemical homogeneity of the halo, as well as substantial differences in chemical composition between field stars in the halo and dwarf spheroidal galaxies, argue against the accretion scenario (Stephens & Boesgaard (2002); Pritzl, Venn & Irwin (2005); Geisler et al. (2007)).

The properties of our three groups of clusters can be interpreted in the following way, without resorting to an external origin for any of the groups:

– The clusters of the outer halo (group 2) formed during the initial dissipationless collapse of the proto-galaxy, from material already polluted by earlier generations of stars, but not well homogenized and thus inhomogeneous in -elements. Contrary to the outer halo stars, they lost their initial average retrograde rotation by dynamical friction and gravitational encounters. As suggested by the referee, this group could also have originated in some ”pregalactic dwarfs” (i.e. metal-poor, gas-rich satellites that soon afterward began hierarchical merging).

– The clusters of the inner halo (group 1) formed later, during the dissipational phase of Galactic collapse, which continued in the halo after the formation of the thick disc and its globular clusters. Since the formation of group 1 occurred later, the molecular clouds from which they formed had time to grow by accretion of smaller clumps. These clouds were already enriched at the same level in -elements. Thanks to the strong potential well in the clusters (as evidenced by their high central velocity dispersions), the He-rich ejecta of first generation massive stars were not blown away and found their way into a more strongly helium-enriched second generation of stars, favouring the production of hot horizontal-branch and Cepheid stars.

– As indicated by their short range in age (1.5 Gyr, see Fig. 3), the disc clusters (group 3) formed in a more rapid fashion than the two other groups, before many clusters of group 1. This could presumably be due to the higher densities and external pressure in the thick disc. This group shows significant average prograde rotation, because the dissipational collapse of the disc conserved angular momentum. The metal-poor disc clusters seem to rotate more slowly and have larger eccentricities and inclinations than the metal-rich disc clusters. Since there is no significant age difference between the two subgroups, we assume that the metal-poorer clusters formed further away from the Galactic plane, and thus retained a larger vertical velocity component.

Acknowledgements

Paired t-tests were performed with the VOStat software (http://vo.iucaa.ernet.in/ voi/VOStat.html). We thank Asis Chattopadhyay for advice. We would like to thank the referee, W. Harris, for his constructive comments and suggestions.

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Appendix A Cladistics applied to globular clusters

Multivariate clustering methods compare objects with a given measure and then gather them according to a proximity criterion. Distance analyses are based on the overall similarity derived from the values of the parameters describing the objects. The choice of the most adequate distance measure for the data under study is not unique and remains difficult to justify a priori. The way objects are subsequently grouped together (this is called the linkage) is also not uniquely defined. Cladistics uses a specific measure that is based on characters (a trait, a descriptor, an observable, or a property, that can be given at least two states characterizing the evolutionary stages of the object for that character) and compares objects in their evolutionary relationships. Here, the ”distance” is an evolutionary cost. Groupings are then made on the basis of shared or inherited characteristics, and are most conveniently represented on an evolutionary tree.

Character-based methods like cladistics are better suited to the study of complex objects in evolution, even though the relative evolutionary costs of the different characters is not easy to assess. Distance-based methods are generally faster and often produce comparable results, but the overall similarity is not always adequate to compare evolving objects. In any case, one has to choose a multivariate method, and the results are generally somewhat different depending on this choice (e.g. Buchanan & Collard (2008)). However, the main goal is to reveal a hidden structure in the data sample, and the relevance of the method is mainly provided by the interpretation and usefulness of the result. In the present paper, the use of cladistics is justified a priori by the evolutionary nature of globular clusters, and a posteriori by the strong astrophysical significance of the grouping found.

In astrophysics, cladistics has already been applied to galaxies because they can be shown to follow a transmission with modification process when there are transformed through assembling, internal evolution, interaction, merger or stripping (Fraix-Burnet et al. (2006a, b, c); Fraix-Burnet (2009)). For each transformation event, stars, gas and dust are transmitted to the new object with some modification of their properties. For globular clusters, interactions and mergers are probably rare. It was previously thought that once they assembled, only the stellar ageing would affect their properties. Nowadays, we have firm evidence that internal evolution can create another generation of stars, and clusters can lose mass. Basically, the properties of a globular cluster strongly depend on the environment in which it formed (chemical composition and dynamics), and also on the internal evolution which includes at least the ageing of its stellar populations. To compare globular clusters, it is thus necessary to take into account the different stages of evolution of both the objects and their environments of formation. Since the clusters form in a very evolving environment (evolution of the Universe and the dynamical environment of the parent galaxy), the basic properties of different clusters are related to each other by some evolutionary pattern. In particular, the dust and gas from which the stars of the globular clusters form have been ”polluted” (enriched in heavy elements) by more ancient stars, being field stars or belonging to other globular clusters. This results in a kind of transmission with modification process, which justifies a priori the use of cladistics. It must be clear that this is not a ”descent with modification” in the sense that there is no replication. But evolution does nevertheless create diversity. We are dealing with phylogeny (relationships between species), not with genealogy (relationships between individuals). Since a multivariate classification of globular clusters is not yet available, we assume in the present work that each cluster represents a “species” that will have to be defined later on.

As our work on galaxies has shown us, it is important to remove parameters that are redundant. Since previous studies of the manifold of Galactic globular clusters have shown that 4 parameters are sufficient to describe their diversity, we selected 4 parameters, 3 of which are intrinsic characteristics of the environment of formation. The fourth one, age, is particular in the sense that it does not inform on the conditions when the clusters formed, and is not discriminant for clustering because it evolves similarly for any cluster (parallel evolution). However, age is useful to rank the clusters within each group. Consequently, we applied to age a weight half that of the other three parameters. In addition, a stepmatrix was employed to impose the irreversibility constraint on the age parameter (age can only increase). In contrast to multivariate distance methods, undocumented values are not a problem in cladistics analyses. This is why the seven galaxies that have no age determination (see Fig.3 and Table 1) have not been excluded in our work.

In this paper, we use parsimony as the optimisation criterion. This works as follows. One first builds a matrix with values of the four parameters for all clusters. The values for each parameter are discretized into 10 bins representing supposedly evolutionary states. Discretization of continuous variables is quite a complex problem, especially in the evolutionary context (see e.g. Goloboff et al. (2006); Thuillard & Fraix-Burnet (2009)). The choice of the number of bins cannot be made in a simple objective way. Here, we took equal-width bins, and considered a compromise between an adequate sampling of continuous variables and the uncertainties on the measurements. The first constraint is given by the software (32 in this case). The second one would a priori give a lower limit of something like total range / uncertainty, but Shannon’s theorem would multiply this by 2. Hence, 10 bins would account for about 20% measurement errors, which is rather large, but Recio-Blanco et al. (2006) do not provide precise error estimates, especially for logTe. Even so, border effects always imply that some objects could belong to a bin or its neighbour, a process that add some more artificial noise. The best way to avoid this effect is to make several analyses with different number of bins and check that the result does not depend on this number. We have done this for 3, 5, 8, 10, 12, 15 and 20, and the result is identical, to at most one misplaced cluster, for numbers higher than 8. For 3, no structure is found, and for 5 bins the groups are not well defined.

Then, all possible arrangements of clusters on a tree are constructed, and using the discretized matrix, the total number of state changes is computed for each tree. The most parsimonious tree is finally selected. If several such trees are found, then a consensus (strict or majority rule) tree is built. The whole procedure is computerised since the number of arrangements is here very large. The result is a diversification scenario that should be confronted to other knowledge and parameters. Maximum parsimony heuristic searches were performed using the PAUP*4.0b10 (Swofforf (2003)) package. The results were interpreted with the help of the Mesquite software (Maddison & Maddison (2004)).

The tree presented in Fig. 1 is a majority rule consensus tree of 20000 trees, the strict consensus tree showing exactly the same three groups but with group 1 and 2 slightly less resolved. To further assess the robustness of the tree, it was not possible to make bootstrapping due to the irreversibility constraint on the age parameter, and it would not have been very significant with only 4 parameters. We performed other analyses using only 3 parameters, excluding the age. They all gave essentially the same three groups, but they were individually slightly less resolved, as expected. All these convergences yield strong confidence on the tree shown in Fig. 1. In the end, the most important point is the astrophysical interpretation we are able to give of the results.

On Fig. 1, the tree is rooted with group 2 as an outgroup. This is not strictly necessary in cladistics, and we find here the same three groups whatever the root chosen or even on the unrooted tree. But we know that a low metallicity is an ancestral state for stars in general, this is why we have chosen group 2 a posteriori because it has a homogeneously low metallicity.

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