The chemical evolution of globular clusters – II. Metals and fluorine

Abstract

In the first paper of this series, we proposed a new framework in which to model the chemical evolution of globular clusters. This model is predicated upon the assumption that clusters form within an interstellar medium enriched locally by the ejecta of a single Type Ia supernova and varying numbers of asymptotic giant branch stars, superimposed on an ambient medium pre-enriched by low-metallicity Type II supernovae. Paper I was concerned with the application of this model to the observed abundances of several reactive elements and so-called non-metals for three classical intermediate-metallicity clusters, with the hallmark of the work being the successful recovery of many of their well-known elemental and isotopic abundance anomalies. Here, we expand upon our initial analysis by (i) applying the model to a much broader range of metallicities (from the factor of 3 explored in Paper I, to now a factor of ∼50; i.e. essentially, the full range of Galactic globular cluster abundances; and (ii) incorporating a broader suite of chemical species, including a number of iron-peak isotopes, heavier -elements and fluorine. While allowing for an appropriate fine-tuning of the model input parameters, most empirical globular cluster abundance trends are reproduced; our model would suggest the need for a higher production of calcium, silicon and copper in low-metallicity (or so-called ‘prompt’) Type Ia supernovae than predicted in current stellar models in order to reproduce the observed trends in NGC 6752, and a factor of 2 reduction in carbon production from asymptotic giant branch stars to explain the observed trends between carbon and nitrogen. Observations of heavy-element isotopes produced primarily by Type Ia supernovae, including those of titanium, iron and nickel, could support/refute unequivocally our proposed framework, although currently the feasibility of the proposed observations is well beyond current instrumental capabilities. Hydrodynamical simulations would be necessary to study its viability from a dynamical point of view.

Full text

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arXiv:1109.1938v1 [astro-ph.GA] 9 Sep 2011
Mon. Not. R. Astron. Soc. 000, 1–15 (2011)Printed 12 September 2011(MN LATEX style file v2.2)
The Chemical Evolution of Globular Clusters - II. Metals
and Fluorine
P. S´ anchez-Bl´ azquez1, A. Marcolini2, B K. Gibson2,3,4, A. I. Karakas5,
K. Pilkington2,3,4and F. Calura2
1Departamento de F´ ısica Te´ orica, Universidad Aut´ onoma de Madrid, E28049, Cantoblanco, Madrid, Spain
2Jeremiah Horrocks Institute, University of Central Lancashire, Preston, PR1 2HE, UK
3Department of Astronomy & Physics, Saint Mary’s University, Halifax, Nova Scotia, B3H 3C3, Canada
4Monash Centre for Astrophysics, School of Mathematical Sciences, Monash University, Clayton, VIC, 3800, Australia
5Research School of Astronomy & Astrophysics, Mt Stromlo Observatory, Weston Creek, ACT 2611, Australia
12 September 2011
ABSTRACT
In the first paper in this series, we proposed a new framework in which to model
the chemical evolution of globular clusters. This model, is predicated upon the as-
sumption that clusters form within an interstellar medium enriched locally by the
ejecta of a single Type Ia supernova and varying numbers of asymptotic giant branch
stars, superimposed on an ambient medium pre-enriched by low-metallicity Type II
supernovae. Paper I was concerned with the application of this model to the observed
abundances of several reactive elements and so-called non-metals for three classical
intermediate-metallicity clusters, with the hallmark of the work being the success-
ful recovery of many of their well-known elemental and isotopic abundance anomalies.
Here, we expand upon our initial analysis by (a) applying the model to a much broader
range of metallicities (from the factor of three explored in Paper I, to now, a factor
of ∼50; i.e., essentially, the full range of Galactic globular cluster abundances, and
(b) incorporating a broader suite of chemical species, including a number of iron-
peak isotopes, heavier α-elements, and fluorine. While allowing for an appropriate
fine tuning of the model input parameters, most empirical globular cluster abundance
trends are reproduced, our model would suggest the need for a higher production of
calcium, silicon, and copper in low-metallicity (or so-called “prompt”) Type Ia su-
pernovae than predicted in current stellar models in order to reproduce the observed
trends in NGC 6752, and a factor of two reduction in carbon production from asymp-
totic giant branch stars to explain the observed trends between carbon and nitrogen.
Observations of heavy-element isotopes produced primarily by Type Ia supernovae,
including those of titanium, iron, and nickel, could support/refute unequivocally our
proposed framework, although currently the feasibility of the proposed observations is
well beyond current instrumental capabilities. Hydrodynamical simulations would be
necessary to study its viability from a dynamical point of view.
Key words:
stars: AGB and post-AGB - stars: chemically peculiar - globular clusters: individual:
NGC 104 (47 Tuc), NGC 6121 (M 4), NGC 6397, NGC 6752, NGC 7078 (M 15).
nuclear reactions, nucleosynthesis, abundances - stars: abundances -
1 INTRODUCTION
Globular clusters (GCs) are gravitationally-bound systems
containing hundreds of thousands of stars, each cluster
orbiting about a parent galaxy. The Milky Way (MW)
itself has more than 150 associated GCs (Harris 1996).
While conceptually simple, internally coeval systems, from a
chemical perspective, GCs are extremely interesting in the
sense that they show a large variation in the abundances
of light elements (i.e, Li, C, N, O, F, Na, Mg, and Al)
(e.g. Smith 1987; Kraft et al. 1993; Grundahl et al. 2002;
Cohen et al. 2002; Yong et al. 2003; Sneden et al. 2004;
Yong et al. 2005; Pasquini et al. 2005; Cohen & Mel´ endez
2005; Smith et al. 2005; Carretta et al. 2006; Gratton et al.
2007; Bonifacio et al. 2007; Marino et al. 2008, and refer-
ences therein) both internally to a given cluster, and between

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2
S´ anchez-Bl´ azquez et al.
clusters. Conversely, the abundances of heavier α (e.g., Si,
Ca, Ti), iron-peak (e.g., Fe, Ni, Cu), and neutron-capture el-
ements (e.g., Ba, La, Eu) do not, in general, show the same
star-to-star variation.
Correlated (and anti-correlated) elemental and iso-
topic (anomalous) trends between these various nuclei are
observed from the main sequence turn-off through to the
tip of the first ascent giant branch, and are not shared
by the corresponding stars in the field (de Silva et al.
2009). For this reason, the primary driver responsible
for these anomalous abundance patterns is thought to
be “external” and related with local environment, while
“internal” mixing mechanisms are thought to be important
only in causing the variations of C and N (and possibly
O) in evolved giants. Such arguments led to the postula-
tion of the “self-pollution” scenario (Cottrell & Da Costa
1981; Dantona et al. 1983) as being perhaps the one most
robust in explaining the chemical abundances anomalies
in clusters (see the review of Gratton et al. 2004, and
references therein). According to the self-pollution scenario,
a previous generation of stars polluted the atmospheres
of stars observed today or provided much of the material
from which those stars formed (e.g. Jehin et al. 1998;
Parmentier et al. 1999; Tsujimoto et al. 2007). Several
models have been proposed with intermediate-mass asymp-
totic giant branch (AGB) stars (e.g., Cottrell & Da Costa
1981; Denissenkov & Herwig
D’Antona et al. 2005;Bekki et al.
from massive stars (e.g., Prantzos & Charbonnel 2006;
Decressin et al. 2007c,a) as the most popular candidate
polluters, as they provide the most simple explanation
for the lack of internal spread seen in Fe and Ca in most
clusters.
Hydrodynamical simulations of GC formation and evo-
lution under the “classical” scenario have been performed
by D’Ercole et al. (2008) and Bekki (2010), although some
fine-tuning to the stellar IMF of the second generation –
as well as the duration of the star formation – were neces-
sary to reproduce the chemical properties and the masses of
the first and second generation of stars and to prevent SN
explosions in the second generation (D’Ercole et al. 2008).
These classical scenarios still possess several problems
that require solution. One of the main problems lies in un-
derstanding how a second generation can form with a current
total mass comparable to the first generation, at least for the
old MW GCs (see D’Ercole et al. 2008 for a discussion of the
problem) without invoking an anomalous IMF (D’Antona &
Caloi 2004; Decressin et al. 2007) or extreme mass loss from
the first generation stars during the GC evolution (by at
least a factor of 10, e.g., D’Antona & Caloi 2004; Bekki &
Norris 2006; Prantzos & Charbonnel 2006; D’Antona et al.
2007; Decressin et al. 2007). External pollution form stars in
the field (Bekki et al. 2007) have been proposed to alleviate
this problem.
Other, more speculative, models have considered, for
example, variations in the shape of the initial mass func-
tion (IMF) (Smith & Norris 1982; D’Antona & Caloi 2004b;
Prantzos & Charbonnel 2006) or significant variations to the
underlying stellar structure and associated nucleosynthesis
(e.g., no hot bottom burning, no third dredge up, extra
mixing, and/or overshooting). Such models have certainly
enjoyed success in explaining certain aspects of anoma-
2003;Fenner et al.
2007)
2004;
winds and
lous abundance patterns in GCs (D’Antona & Ventura 2007;
Bekki et al. 2007), but not in their entirety.
In summary, although the ”classical” scenario succeeds
in reproducing many of the observed chemical abundance
anomalies of GCs, it does still remain problematic, as does
the lack of consensus concerning the dominant formation
processes of GCs. As such, we considered it valuable, and
indeed still do, to consider alternate, potentially viable, so-
lutions.
In Paper I (Marcolini et al. 2009), we proposed a new,
and somewhat unique, framework in which to model the
chemical evolution of globular clusters. Contrary to previ-
ous “self-pollution” models, in our framework the first stars
form with “peculiar” abundance patterns seeded by Popu-
lation III pre-enrichment, while the so-called “normal” stars
form during a second phase of self-pollution from Popula-
tion II Type II supernovae (SNeII). Despite being restricted
to an analytical chemical evolution framework, and admit-
tedly requiring confirmation via fully hydrodynamical sim-
ulations, we showed that our model successfully reproduced
most of the well-known anti-correlations between various
light elements and isotopes, while maintaining both constant
iron and C+N+O abundances, and simultaneously recover-
ing the empirical magnesium isotope patterns. The only ma-
jor problem that the model encountered was in its underpro-
duction of aluminum; in order to reproduce observations, a
factor of 50 more Al production in intermediate-mass AGB
stars was needed. We noted that such an underproduction
could be accommodated within the stellar nucleosynthetic
uncertainties, without compromising the predicted abun-
dance pattern of Al in the Milky Way, via the use of Galactic
chemical evolution modeling.
Our picture for globular cluster formation is predicated
upon the assumption of localised pollution from a single
Type Ia and ∼100 AGB stars. Such conditions give a qual-
itative explanation for the complete absence of Galactic
GCs with [Fe/H]? −2.4 (e.g. Harris 1996), while extremely
metal-poor field stars ([Fe/H]? −3.0) exist in copious num-
bers (e.g. Frebel et al. 2007, and references therein).
In Paper I, emphasis was placed on the evolution of
the light elements within several intermediate-metallicity
([Fe/H]∼−1.4) GCs (NGC 6752, M 13, and NGC 2808).
In Paper II, we now extend the application of the model
to both the more metal-poor (M 15 and NGC 6397, with
[Fe/H]=−2.26 and −1.95, respectively - Harris 1996) and
metal-rich (47 Tuc, with [Fe/H]=−0.76) regimes. In light of
the availability of recent observational work, we also explore
the behaviour of fluorine in M 4, and the iron-peak elements
(and isotopes) for NGC 6752 and 47 Tuc. In § 2, we discuss
our adopted stellar yields, while in § 3, the framework itself
is summarised. Our results, both globally and on a cluster-
by-cluster basis, are presented in § 4−7.
2STELLAR YIELDS
As discussed in Paper I, we employ an IMF-weighted yield
distribution for our chemical evolution predictions, based
upon four sets of SNeII models, spanning the mass range 10-
60 M⊙ (Woosley & Weaver 1995; Chieffi & Limongi 2004;
Kobayashi et al. 2006), as summarised in Table 1. Therein,
we only show the new elements analysed in this paper; we

Page 3

The Chemical Evolution of Globular Clusters
3
Figure 1. Schematic of the model outlined in Paper I. Initially,
a localised volume (inner blue region; solid line) pre-enriched
with Population III ejecta is polluted by a single SNIa and ∼100
intermediate-mass AGB stars. After a new generation of stars is
born, associated SNe II begin to pollute and expand the inner
volume, while mixing concurrently with the surrounding lower-
[Fe/H] ISM.
refer the reader to Tables 1 and 2 of Paper I for the elements
discussed in our earlier work. While in a global sense, the
various yields are in reasonable agreement (particularly for
the α-elements), there are obviously those for which factors
of ∼3 variations exist (e.g., several of the iron-peak elements,
for which the sensitivity to the mass cut is most extreme).
As in Paper I, an average yield is derived by averaging be-
tween the compilations when agreement exists to within a
factor of two, otherwise an admittedly arbitrary decision was
made to adopt the compilation that best reproduces the
abundances of low-metallicity (−2.0 ?[Fe/H]? −1.0) halo
stars (e.g, Gratton et al. 2000). The element with the largest
variation is that of fluorine; this is due to the inclusion of
the neutrino process by Woosley & Weaver (1995) that in-
creases by more than an order of magnitude the production
of light elements (including7Li and19F) (see Timmes et al.
1995, for further discussions). In this case we use the yields
of Woosley & Weaver (1995). Unless stated differently, in
the following we refer to the yields given in Table 1 as the
“Model” yields.
In Table 2, we summarise the mean yields for
intermediate-mass AGB stars (see the complementary Ta-
ble 3 of Paper I, for the elements studied in our earlier work)
calculated by averaging over a Salpeter (1955) IMF, in the
mass range 4−7 M⊙, the Karakas & Lattanzio (2007) yields
with Z=0.0001 (we refer to this model as the “reference
model”). There are several Fe-peak elements which were not
considered by these authors (i.e., Ca, Ti, V, Cu) and, for
this work, we have therefore assumed that these elements
are not synthesised in significant quantities in intermediate-
mass AGB stars.1
1We note here in passing that our preliminary work now suggests
For SNe Ia we use the yields of Iwamoto et al. (1999)
and explore the impact of various stellar physics treatments,
including slow deflagration (W7) and delayed detonation
models (WDD1 and CDD1). While SNe Ia produce small
amounts of light elements (relative to SNe II), they can pro-
duce significant amounts of heavier α-elements (e.g., Si, Ca,
and Ti) and Fe-peak elements. The specific yield of the latter
amount depends upon the deflagration speed and ignition
densities, with the different models producing yields that
span a factor of four (see Table 2 and, more importantly,
the original paper of Iwamoto et al. 1999 for details).
3MODEL
The details of the models are described in Paper I; here, we
briefly review the main characteristics. The model has two
phases: first, the localised effect of intermediate-mass AGB
field stars and a single SNIa explosion produce an inhomoge-
neous pollution that is added to an interstellar medium that
was previously enriched by very low metallicity SNe II. Since
the peak of the SNIa rate has a timescale (∼70−80 Myr; e.g.
Matteucci & Recchi 2001; Mannucci et al. 2006) compara-
ble with the lifetime of a 5 M⊙ star (Schaller et al. 1992;
Karakas & Lattanzio 2007) such pre-enrichment is not a pri-
ori unreasonable (Paper I provides a detailed discussion of
the initial conditions). The formation of a proto-GC takes
place inside this chemically-peculiar region; the stars formed
during this phase will have “peculiar” chemical properties
compared with “normal” field stars of the same metallicity.
As star formation proceeds, new SNe II explode, pol-
luting the gas with the product of their nucleosynthesis.
This represents the second phase of the model. The stars
formed in this phase will have chemical properties that are
more similar to the field stars of the same metallicity (hence-
forth referred to as “normal”). Eventually, the occurrence of
SNe II explosions quenches further star formation and the
GC evolves passively.
During the evolution there are two mechanisms that
govern the chemical properties of the forming stars. We
showed in Paper 1 that the two mechanisms act together
to regulate the [Fe/H] abundance, which is kept essentially
constant throughout a cluster’s evolution. Furthermore, the
general chemical properties of the stars forming in this re-
gion evolve from “peculiar” to that more typical of essen-
tially “pure” SNe II enrichment, which is the opposite to
what is normally assumed in self-enrichment models.
The final chemical properties of the GC are controlled
by three parameters: the size of the inner region where
the SNIa is confined (Rin), the initial number of AGB
stars (NAGB), and the abundance of the interstellar medium
(ISM) of the halo ([Fe/H]ISM). Table 3 lists these parame-
ters for the globular clusters examined in this paper. We
can see that for these models, there is an inverse correlation
that low-metallicity, massive, AGB stars can produce a substan-
tial amount of copper, ie., one 5 M⊙, Z=0.004 model we have
generated produced [Cu/Fe]∼+0.8. A larger grid of models will
be required before we can assess its global importance within the
context of our GC modeling efforts.

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4
S´ anchez-Bl´ azquez et al.
Table 1. Mean SNeII stellar yields averaged over the progenitor mass range 10-60 M⊙for a Salpeter (1955) IMF for several compilations:
W&W=Woosley & Weaver (1995); C&L=Chieffi & Limongi (2004); KOB=Kobayashi et al. (2006). The yields of Fe are given in solar
masses while for different elements we show the [Xi/Fe] ratio.
SNe type Fe[F/Fe][Si/Fe][Ca/Fe][Ti/Fe][V/Fe][Co/Fe] [Ni/Fe][Cu/Fe]
W&W (Z=0.0002)
W&W (Z=0.002)
C&L (Z=0.0001)
C&L (Z=0.0001)
KOB (Z=0.001)
KOB (Z=0.001+HN)
6.1e-2a
6.9e-2a
1.0e-1
1.0e-1
7.5e-2
1.1e-1
−0.29
+0.11
−2.69
−1.57
−1.08
−1.20
+0.45
+0.42
+0.62
+0.60
+0.60
+0.57
+0.31
+0.28
+0.54
+0.51
+0.35
+0.28
−0.06
−0.13
−0.22
−0.15
+0.03
−0.13
−0.29
−0.21
−0.31
−0.47
−0.31
−0.25
−0.01
+0.12
−0.39
−0.26
−0.35
−0.17
+0.26
+0.30
+0.34
+0.23
−0.35
−0.21
−0.55
−0.37
−0.55
−0.68
−0.39
−0.41
Model (SNe II)9.0e-2+0.00 +0.40+0.30
−0.05
−0.30 +0.00+0.00
−0.55
Table 2. Mean SNe II, SNe Ia, and intermediate-mass AGB yields in solar masses. The SNe II entry corresponds to the adopted
“average” Model listed at the end of Table 1. The SNe Ia yields corresponds to the models described by Iwamoto et al. (1999).
The mean AGB yields (from Karakas & Lattanzio 2007) are averaged over the progenitor mass range 4−7 M⊙ assuming a
Salpeter (1955) IMF.
SNe typeFeFSiCaTiV CoNiCu
SNe II (Model) 9.00e-23.00e-5 1.28e-19.40e-32.23e-4 1.33e-52.45e-45.33e-31.33e-5
SNIa (W7)
SNIa (WDD1)
SNIa (CDD1)
7.49e-1
6.72e-1
6.48e-1
5.67e-10
1.70e-9
5.83e-10
1.56e-1
2.74e-1
2.79e-1
1.19e-2
3.10e-2
3.18e-2
3.43e-4
1.13e-3
8.18e-4
7.49e-5
1.33e-4
1.11e-4
1.04e-3
3.95e-4
2.91e-4
1.26e-1
3.40e-2
3.50e-2
3.00e-6
6.92e-6
7.28e-7
AGB (model)2.27e-5 4.98e-9 4.87e-50. 0.0.7.93e-8 3.07e-6 0.
between the value of the surrounding [Fe/H]ISMand the con-
finement radius of the SN Ia, Rin.2Such an inverse correla-
tion is consistent physically with the expected inverse corre-
altion between metallicity and cooling efficiency. The higher
the metallicity of the confining ISM, the more efficient the
cooling and energy dissipation, and the more confined the
SN remnant’s expansion will be. A factor of 100 increase
in metallicity would make the radius at which the remnant
merges with the ISM a factor of three smaller and, there-
fore, the enrichment will be spread over a smaller volume
(Gibson 1994).
To illustrate the effect of the model free parame-
ters on the observed O-Na anti-correlation we show, in
Fig. 2 models changing the number of AGB polluters,
the initial radius (Rin and the initial iron content of the
interestellar medium [Fe/H]ISM, together with a collection
of observational data for different GCs (Kraft et al. 1993;
Sneden et al. 1997; Ivans et al. 2001; Ram´ ırez & Cohen
2003; Sneden et al.2004;
Yong et al. 2005;Carretta et al.
2004; Koch & McWilliam 2008). We also show three differ-
ent models with (with NAGB=100, 160, 250, and Rin=40,
30, and 20 pc. It can be seen that appropriate choices
for the initial conditions are able to reproduce both the
trend and the associated “scatter” in the observed O-Na
anti-correlation.
Cohen & Mel´ endez
2006;
2005;
Sneden et al.
2Admittedly, the correlation is not a particularly strong one, par-
ticularly when the two models from Paper I (bottom two entries
of Table 3) are included.
4 VERY METAL-POOR AND METAL-RICH
GCS
4.1The case of very metal-poor GCs: M 15 and
NGC 6397
In Figure 4 we show the [Na/Fe]–[O/Fe] and the [N/Fe]–
[C/Fe] evolution predicted by our models for M15 and
NGC 6397 together with the corresponding observational
datasets of Sneden et al. (1997), Cohen & Mel´ endez (2005),
and Carretta et al. (2005). Table 3 lists the model param-
eters inferred which best-fit the overall [Fe/H] content of
NGC 6397 and M15. To fit the abundances for these clus-
ters, a larger value of Rin and lower value of [Fe/H]ISM were
needed, relative to those employed for the more metal-rich
GCs. The models recover the O-Na anti-correlation for both
clusters, but are less successful in doing the same for the C-
N anti-correlation. This was discussed in Paper I, where we
showed that while the the main contribution to the produc-
tion of carbon and nitrogen comes from intermediate-mass
AGB stars, the amount of nitrogen produced by these stars
is much larger than that of carbon. The result is that while
the initial AGB production of carbon is offset by the Fe
deposited by the SN Ia, the [N/Fe] ratio can reach values
as high as 1.5 dex. For this reason, while the carbon con-
tent remains practically constant during the evolution (solid
lines), the nitrogen content varies by more than an order of
magnitude (see right panels of Figure 4). As a consequence,
our reference models (solid lines) are able to reproduce the
observed spread in N while keeping C constant. However,
in order to obtain an anti-correlation between the relative
abundance of these elements, the production of C by AGB

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The Chemical Evolution of Globular Clusters
5
Figure 2. Representative model predictions for the O-Na anti-correlation. The first generation of stars are born with peculiar chemical
properties (i.e., low [O/Fe] and high [Na/Fe]). Once the first SNeII start to explode, polluting and expanding the inner region, the chemical
properties of the forming stars evolve toward “normal” values typical of pure SNeII pollution (i.e., high [O/Fe] and low [Na/Fe]). Upper
left panel: the three lines represent models with different initial conditions - i.e., models in which the number of intermediate-mass AGB
stars and the initial radius of the inner pre-polluted region has been changed to match the observational scatter observed in different
GCs (small circles, see text for more details about the data sets). The solid lines represent models with the combinations of values of
NAGB=100, 160, 250 and Rin=40, 30 and 20 pc with the former (latter) values representing the lower (upper) line. Upper right panel:
The three lines correspond to the models with the same inner radius (Rin=30 pc), but with different number of AGB stars: NAGB=100,
160, and 250, from lower to upper line, respectively. Bottom left panel: The three lines reflect the effect of changing the inner radius,
keeping the number of AGB stars fixed to NAGB=160. Bottom right panel: Effect of changing the [Fe/H]ISMkeeping the rest of the
parameters fixed. From top to bottom [Fe/H]ISM=−3.30, −2.55 and −1.40.
stars needs to be reduced by a factor of four (dashed lines
of Figure 4).
It can also be seen that the absolute values of [C/Fe] for
NGC 6397 are somewhat difficult to reproduce in our model,
particularly at low [N/Fe] values. However, the comparison
with carbon and nitrogen measurements need to be done
with caution. The abundances in NGC 6397 have been de-
rived from spectra of slightly evolved stars (subgiants). The
surface abundances of these elements are affected by internal
mixing and may not reflect the compositon of the ISM when
those stars formed (see Smith & Tout 1992; Charbonnel
1994, 1995; Denissenkov & Tout 2000; Weiss et al. 2000;
Gratton et al. 2004, and reference therein). The carbon
abundance measurements of M15 and NGC 6397 differ by
∼0.3 dex (see right panel of Figure 4). This is very difficult
to understand given the similarities in their overall metal-
licity and nitrogen content. We have assumed here that this
offset is artificial and perhaps reflects a systematic error in
the carbon measurements of NGC 6397, but admit that this
remains, for now, just an assumed interpretation. If we apply
an offset to the carbon abundances of the NGC 6397 stars
to match the abundances of M15 stars, our model agrees
better with the observations (Fig. 4).

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6
S´ anchez-Bl´ azquez et al.
Figure 3. Evolution of various element ratios for the model for globular cluster M4; comparison observational data from Ivans et al.
(1999, squares) and Smith et al. (2005, triangles). Upper-left panel: evolution of [O/Fe] versus [Fe/H]. Upper-right panel: evolution of
[Na/Fe] versus [O/Fe]. Lower-left panel: evolution of [O/Fe] versus [F/Fe]. Lower-right panel: evolution of [Na/Fe] versus [F/Fe].
Table 3. Initial conditions of the models for the different glob-
ular clusters
[Fe/H]ISM
[Fe/H]in
Rin(pc)
NAGB
M 15
NGC 6397
NGC 6752
M 4
47 Tuc
−3.30
−3.20
−2.55
−1.65
−1.40
−2.35
−2.05
−1.60
−1.15
−0.75
65
49
36
29
20
150
150
250
130
140
M 13
NGC 2808
−3.50
−2.85
−1.50
−1.10
31
24
170
180
4.2 The case of metal-rich GCs: 47 Tuc
As a final test for the evolution of the abundances of light ele-
ments we consider the case of the metal-rich globular cluster
47 Tuc ([Fe/H]= −0.67) (e.g., Carretta et al. 2004). In order
to match its chemical properties we assume that the initial
SN Ia was quite localised (Rin=20 pc) and that the Fe con-
tent of the ISM of the halo at the epoch of formation was
higher than for the metal-poor GC ([Fe/H]ISM = −1.40).
This value is in the range of iron metallicities encountered
in the Galactic halo (−4.0 ?[Fe/H]?+0.0 peaking near
[Fe/H]≃ −1.7 (e.g., Ryan & Norris 1991)). For the specific
case of 47 Tuc we have used the Z=0.004 yields, consistent
with its higher initial metal content.
The evolution of the O-Na and C-N anti-correlations
are plotted in Fig. 5 together with the observational
data of Carretta et al. (2005), Alves-Brito et al. (2005), and
Briley et al. (2004). Again the anti-correlations are better
reproduced once we assume that the carbon production in
AGB stars is reduced by a factor of four compared with the
theoretical values of Karakas & Lattanzio (2007). There is
also an evident offset of 0.15 and 0.4 dex between the mea-
surements of carbon and nitrogen derived by Carretta et al.
(2005) and Briley et al. (2004). While it is beyond the scope
of this paper to understand the origin of these offsets, our

Page 7

The Chemical Evolution of Globular Clusters
7
Figure 4. Upper-left panel: evolution of [Na/Fe] versus [O/Fe] for the model of M15 (solid line) plotted against the data by Sneden et al.
(1997). Upper-right panel: evolution of [N/Fe] versus [C/Fe] for the same model (reference model, solid line) plotted against the data
by Cohen & Mel´ endez (2005). Dashed line indicate the effect of reducing the production of C by AGB stars by a factor of four. Lower
panel: Same, but showing the evolution of the NGC 6397 model, plotted together with the observational data by Carretta et al. (2005).
Note that in this case most of the lowest [O/Fe] abundances are upper limits. The dotted line in the lower-right panel corresponds to
the same evolution of the dashed line (reference model with a reduced production of C by AGB stars) but offset by 0.3 dex (see text for
details).
assumed yields are in agreement with the values obtained
by Carretta et al. (2005) but, again, the model has no prob-
lem in being able to reproduce the trend from Briley et al.
(2004, dotted line).
4.3[Fe/H] evolution
Figure 7 shows that in agreement with the results of Pa-
per I for intermediate metallicity GCs, the [Fe/H] remains
constant during the evolution of metal-poor and metal-rich
globular clusters. (note that we choose to plot [Na/Fe] in-
stead of the usual [O/Fe] because most of the oxygen values
for this cluster are upper limits).
5M4 AND ITS FLUORINE CONTENT
Fluorine abundances have been derived for the stars of M4
by Smith et al. (2005), finding they vary by a factor of six,
correlate with oxygen, and anti-correlate with sodium and
aluminum. Fluorine yields are a strong function of stellar
mass and, therefore, can be used to constrain the nature
of the polluters in globular clusters. Fluorine, like oxygen,
is scarcely produced in intermediate-mass AGB stars (F is
destroyed in 5−6 M⊙ and produced in the 4 M⊙ models of
Karakas & Lattanzio 2007) and here, is mainly synthesised
by the SNe II models.
As is evident in the lower panels of Fig. 3, where we
plot the observational data alongside the predicted evolu-
tion of our model, fluorine correlates with oxygen and is
anti-correlated with sodium. In our model, [F/Fe] evolves

Page 8

8
S´ anchez-Bl´ azquez et al.
Figure 5. Left panel: evolution of [Na/Fe] versus [O/Fe] for the reference model for 47 Tuc plotted against the observational data
collected by Carretta et al. (2005, triangles) and Alves-Brito et al. (2005, diamonds). Right panel: evolution of [N/Fe] versus [C/Fe] for
the same model plotted against the values presented in Carretta et al. (2005, triangles), Briley et al. (2004, circles). Dashed line indicate
a model where the production of C by AGB stars has been reduced in a factor of 4 with respect to the reference model. In the case of
Carretta et al. (2005) open triangles refer to sub-giant stars while filled triangles refer to dwarf stars. The dotted line in the right panel
corresponds to the dashed-line model offset by 0.15 dex in carbon and 0.40 dex in nitrogen.
from an initially sub-solar value to a slightly super-solar
value, typical of SNe II pollution, and in agreement with
the observational constraints. Smith et al. (2005) argue that
the correlations can be explained within a self-polluting sce-
nario in which AGB stars act as the main polluters. However,
the destruction of fluorine by intermediate-mass AGB stars
does not appear sufficient to explain the “final” extreme low
values of the [F/Fe] ratio. In our model, these values are ex-
pected in the first stars formed, due to the inhomogeneous
SN Ia effect.
6OTHER ELEMENTS
6.1 Heavy α-elements
While there are multiple observations of light element abun-
dances in GCs, only recently have heavier elements be-
come accessible for statistically significant samples of stars.
Yong et al. (2005) measured 20 elements in 38 bright gi-
ants of NGC 6752, while Carretta et al. (2004) reported the
abundances of nine sub-giants and three dwarfs of 47 Tuc,
both for α-elements and Fe-group elements. While light el-
ements show the already discussed variations spanning in
some cases more than an order of magnitude, silicon and
heavier elements (Xheavy) show much smaller star-to-star
scatter in their [Xheavy/Fe] ratio. This smaller scatter is
consistent, in most cases, with measurement uncertainties.
Furthermore, the mean [Xheavy/Fe] ratios are consistent
with field stars of the same metallicity. There is some ev-
idence for heavy α- elements correlating with lighter ele-
ments; Yong et al. (2008) found a statistically significant,
although loose, correlation between silicon and aluminum
in NGC 6752, noting that the Si abundances are roughly
constant from star to star.
In Figure 6 we plot the evolution of the [Xi] ratio
Table 4. Isotope ratios for the averaged yields of SNe II and
SNe Ia. The labels are the same as used in Tables 1 and 2.
48Ti/50Ti
56Fe/58Fe
58Ni/60Ni
SNe II (W&W)
SNe II (KOB)
SNe II (C&L)
1.1e4
2.3e2
0.9e4
3.7e4
1.7e3
6.6e4
0.5
0.36
2.0
Model (SNe II)1.e4 5.e40.5
SNe Ia (W7)
SNe Ia (WDD1)
SNe Ia (WDD3)
1.9
2.0
35.2
2.1e2
1.8e2
6.5e2
8.8
6.2
11.2
of different α-elements versus the [O/Fe] for the model of
NGC 6752, together with the observational values obtained
by Yong et al. (2005). As can be seen, our model is able to
reasonably reproduce the spread in the O-Mg correlation.
The different lines in Figure 6 represent models with dif-
ferent prescriptions for the SN Ia yields of Iwamoto et al.
(1999): W7 (model with slow central deflagration), WDD1
(mode fast deflagration), and CDD1 (model delayed detona-
tion in the outer layer). As can be seen, the abundance of the
Fe-group elements (in particular, the neutron-rich species
such as50Ti,58Fe and58Ni) depend considerably upon the
chosen physics of the model. From Table 2, it is evident that
variations in the SNe Ia yields by up to a factor of four are
possible for these elements.
Intermediate-mass AGB stars produce some Mg -
mostly in the form of the neutron-rich isotopes (Fenner et al.
2003) - and thus even if a variation of ∼1 dex is achieved
in [O/Fe], only a mild depletion of 0.3 dex in [Mg/Fe] is
present, due to the AGB contribution. The Mg production
in different SN Ia models is roughly the same, although it

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The Chemical Evolution of Globular Clusters
9
Figure 6. Evolution of different α-elements ([Mg/Fe], [Si/Fe], [Ca/Fe] and [Ti/Fe]) versus [O/Fe] for the case of NGC 6752. Open circles
correspond to the observational dataset from Yong et al. (2005). The colour-coded lines correspond to models with different SNe Ia yields.
Orange lines: W7; blue lines: WDD1; black lines: CDD1. The green lines represent a model in which the Si and Ca yields have been
increased of a factor of three and two, respectively, compared with the yields of model WDD1. The dashed blue line in the lower-right
panel is the corresponding blue solid line, offset by 0.2 dex.
Figure 7. Left panel: evolution of [O/Fe] versus [Fe/H] for the model of globular cluster M 15 compared with the observational dataset
from Sneden et al. (1997). Right panel: evolution of [Na/Fe] versus [Fe/H] for the model of globular cluster 47 Tuc compared with
the observational datasets of Carretta et al. (2004, triangles) and Koch & McWilliam (2008, squares). For these very metal-poor and
metal-rich GCs, the [Fe/H] remains roughly constant throughout their respective evolution.

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10
S´ anchez-Bl´ azquez et al.
Figure 8. Evolution of different α-elements ([Mg/Fe], [Si/Fe], [Ca/Fe] and [Ti/Fe]) versus [Na/Fe] for the case of 47 Tuc. Open circles
correspond to the observational dataset from Carretta et al. (2005); the filled symbols are the three dwarfs, while the open symbols are
the nine sub-giants. The colour-coded lines correspond to models with different SNe Ia yields. Orange lines: W7; blue lines: WDD1; black
lines: CDD1. The dashed line in the bottom right panel corresponds to the [Ti/Fe] prediction using the WDD1 SNe Ia model, but offset
by ∼0.3 dex (see text for details).
is small compared to the production of Mg in SNe II and
AGB stars.
An apparent failure of our model is that the “shape”
of the O-Mg correlation is “concave”, while it looks to be
more “convex” in the observational data (upper left panel
of Figure 6). This may be due to the fact that we are using
averaged yields for SNe II. Indeed more massive SNe II pro-
genitors produce much larger amounts of Mg and explode
earlier, explaining why the empirical [Mg/Fe] increases quite
quickly, at apparent odds with our model.
While the contribution from the SN Ia is not relevant
for Mg, this is not the case for the other heavier α-elements.
Indeed, Table 2 shows that a SN Ia produces more Si, Ca,
and Ti than a single SN II (note however that their produc-
tion ratio against iron, [Xi/Fe], is lower than for the case
of SNe II and is usually sub-solar). Moreover, variations in
the initial value of the ratio [Xi/Fe] (up to 0.2 dex for Si
and 0.4 dex for Ca and Ti) are possible when testing differ-
ent SNe Ia models, even if all these values remain, more or
less, sub-solar (see Figure 6). In addition, our models pre-
dict a correlation between Si-O and Ca-O, with variations
of up to ∼0.5 dex in the [Si/Fe] abundance and ∼0.3 dex
for [Ca/Fe] (for models CDD1 and WDD1). These predicted
correlations are not in agreement with the observational con-
straints. Adopting the W7 model increases the scatter, only
making the situation worse.
For Ti, the models CDD1 and WDD1 maintain the
[Ti/Fe] abundance at a roughly constant value, consistent
with the observations (even if there is an intrinsic offset of
∼0.2 dex; dashed line in the bottom right panel of Figure 6).
In contrast, model W7 produces [Ti/Fe] abundances that do
not agree with the observations. Since Ca is not synthesized
in intermediate-mass AGB stars, we investigate how much
Ca production we would require in (very) metal-poor SNe Ia
to fit the observational constraints. Indeed, a factor of two
more Ca production in (very) low metallicity SN Ia3com-
pared with the WDD1 model (green line in Figure 6) is
able to reconcile our model with observations. Note that the
fact the [Ca/Fe] ratio in halo field stars is observed to de-
cline with [Fe/H] implies that the Ca yields of Iwamoto et al.
(1999) are consistent with SN Ia at larger metallicity.
The same experiment can be made for Si, but in this
case we should also try to take into account a possible higher
AGB contribution. As already discussed, Yong et al. (2005)
found a statistically significant correlation (at odds with our
3Or just the prompt SNe Ia which explode on timescales shorter
than ∼100 Myr.

Page 11

The Chemical Evolution of Globular Clusters
11
model) between the [Si/Fe] and [Al/Fe] which, in their inter-
pretation, can be explained if the reaction
favoured over27Al(p,α)24Mg. Hot-bottom burning (HBB) in
intermediate-mass AGB stars is expected to produce small
amounts of28Si from proton capture on27Al, though the Si
yields are expected to be small (Karakas & Lattanzio 2003,
2007). The Si production depends on the temperature of
the HBB region and also on the assumed reaction rates.
Again, similar to Ca, the problem of Si can be solved with
a higher production of Si in (very) low metallicity SN Ia4
and/or a higher Si production in AGB stars (a mean value
of 3.5 × 10−3M⊙ (compared to 4.8 × 10−5M⊙ predicted
by Karakas et al. 2008)). Note that this is exactly the value
of Al we needed in Paper I to reproduce the Al-Mg anti-
correlation in this GC. If our framework is correct, it means
that current intermediate-mass AGB models are underesti-
mating the yields of Al and Si by 1-to-2 orders of magnitude.
From Figure 8, we see that are no serious issues in re-
producing the same α-elements for the more metal-rich clus-
ter 47 Tuc.5In this case, the collection of sub-giants and
dwarfs from Carretta et al. (2004) are consistent with our
model. We should note that due to the higher SNe II pre-
enrichment ([Fe/H]ISM), this model does not reach the low
values of [O/Fe] seen in the NGC 6752 model. In this case,
the Iwamoto et al. (1999) yields have no problem in fitting
the observational dataset. In our framework, 47 Tuc should
form later than NGC 6752 (i.e., from a more metal-rich ISM)
and the suggestion that very low metallicity SN Ia (or AGB)
may have slightly different (up to a factor of four) Si and Ca
yields is plausible. In this case, the temporal evolution goes
from high values of [Na/Fe] to low values.
27Al(p,γ)28Si is
6.2Fe-peak elements
SNe Ia and SNe II synthesise a significant fraction of the
Fe-group elements, while intermediate-mass AGB stars con-
tribute very little of the same. The analysis of [Xi/Fe], where
Xi represents an Fe-peak element, can prove extremely use-
ful in understanding the relative contributios of the two
types of supernovae. The ratio of iron produced in SNe Ia-to-
SNe II is FeSNeIa/FeSNeII ≃ 7−8 (depending on the adopted
yields); if [Xi(SNeIa)/Xi(SNII)] <7−8, then [Xi/Fe] will
increase during the GC evolution and viceversa.
Figure 9 compares the results of our models for Fe-group
elements (specifically V, Co, Ni, and Cu) with the corre-
sponding observational data of Yong et al. (2005), for the
case of the globular cluster NGC 6752. [V/Fe] and [Ni/Fe]
remains roughly constant, in good agreement with observa-
tions, when using the WDD1 and CDD1 models. Using the
W7 model for SN Ia, though, produces less V and more Ni
than needed to keep [Ni/Fe] and [V/Fe] constant, leading
to variations of 0.2 and 0.4 dex for [V/Fe] and [Ni/Fe], re-
spectively (see Table 2). The trend in the evolution of Co
4Or, again, prompt SN Ia; as before, the value of [Si/Fe]∼+0.0
in field stars at solar metallicity implies that at higher metallicity,
the Iwamoto et al. (1999) yields are correct (but note that these
yields are calculated for solar metallicity).
5In Figure 8, we have shown the [α/Fe] trends against
[Na/Fe], rather than [O/Fe], as the majority of the stars in the
Carretta et al. (2005) sample either have non-detections or only
upper limits to their oxygen abundances.
is well-reproduced (apart from a small ∼0.1 dex offset) by
using SNe Ia model W7, while the others underestimate its
production by roughly a factor of two.
In the lower two panels of Figure 9, we compare the
[V/Fe] and [Ni/Fe] evolution of our model for the metal-
rich GC 47 Tuc, with the data presented in Carretta et al.
(2005). The models underproduce vanadium by a factor of
two, and overproduce nickel by ∼0.2 dex. We note in pass-
ing that observationally, the two clusters themselves differ
in their mean [V/Fe] by ∼0.3 dex. The origin of this dif-
ference remains unclear and in the lower left panel of Fig-
ure 9, we show our model predictions for the 47 Tuc [V/Fe]-
[Na/Fe] evolutionary trend we show the same curves offset
by ∼0.3 dex, to coincide with the data. We cautiously sug-
gest that our model can (roughly) match the evolution of
most of the Fe-group elements analysed, within the theoret-
ical uncertainties of SNe yields, and those associated with
the observational data.
Having just made that conclusion though, one notable
exception exists, in the form of copper. Indeed, while the
observed Cu content for NGC 6752 seems to be roughly
constant, our model predicts a very strong correlation with
oxygen, independent of the chosen SNe Ia model. From the
middle-right panel of Figure 9, we see that the initial [Cu/Fe]
ratio is greatly underestimated in our models, for any choice
of the SNe Ia yields: all the SNe Ia models analysed here
produce very little copper (see Table 2). The origin of the
copper observed in field halo stars remains a matter of de-
bate and, at this stage, the magnitude of this apparent fail-
ure of the model needs to be confirmed/refuted with ad-
ditional empirical data. Assuming a value for SNe Ia cop-
per production of 1.2 × 10−4M⊙ would bring the mod-
els into agreement with the extant data. It is very inter-
esting to note that such a suggestion is not entirely with-
out precedent; indeed, this value is very similar to the one
(0.5 − 2.0 × 10−4M⊙) proposed by Matteucci et al. (1993)
to reproduce the [Cu/Fe] versus [Fe/H] trend for Milky Way
stars (but see also Romano & Matteucci 2007). In addi-
tion, Mishenina et al. (2002), analysing the copper abun-
dance trend in 90 metal-poor stars in the metallicity range
−3.0 ?[Fe/H]? −0.5, found evidence that SNe Ia must play
a significant role (>65%) in producing copper.
Yong et al. (2008) pointed out that even if the abun-
dance of Fe-peak elements shows a small scatter in
NGC 6752, their values statistically correlate with nitro-
gen. This is exactly what we expect in our model, even if
the exact slope and scatter of this correlation depends upon
the interplay between a SN Ia origin and a SN II elemental
origin.
7ISOTOPE RATIOS AS AN OBSERVATIONAL
TEST TO VALIDATE THE FRAMEWORK
In Paper I, we tested our model predictions for the behaviour
of Mg isotopes in GCs against measured isotope abundances
in NGC 6752 and M 13. Our framework succeeded in repro-
ducing their observed trends, but we require a better test to
verify the SN Ia framework of our model (recall, Mg is not
a strong constraint for the SN Ia framework, as its origin is
linked to SNe II and AGB stars). The purpose of this sec-
tion is to make predictions regarding isotope ratio of heavy

Page 12

12
S´ anchez-Bl´ azquez et al.
Figure 9. Upper Four Panels: Evolution of Fe-group elements ([V/Fe], [Co/Fe], [Ni/Fe], and [Cu/Fe]) versus [O/Fe] for the case
of NGC 6752; open symbols correspond to the observational data from Yong et al. (2005). Bottom Two Panels: Evolution of Fe-group
elements ([V/Fe] and [Ni/Fe]) versus [Na/Fe] for the case of 47 Tuc; open symbols correspond to the observational data from Carretta et al.
(2005). In all six panels, the colour-coded lines refer to models with different SNe Ia yields. Orange lines: W7; blue lines: WDD1; black
lines: CDD1. Dashed lines in the upper right and bottom left panels correspond to the arbitrary ∼0.1−0.3 dex model offsets required to
match the extant data (see text for details). The green line in the middle-right panel corresponds to a model in which the Cu production
in SNe Ia is assumed to be ∼ 1.2 × 10−4M⊙, in order to match the observational constraints.
elements where the imprint of a SN Ia should be more im-
portant; this can constitute a definitive test for the model.
As already noted by Iwamoto et al. (1999), even if dif-
ferent SN Ia models can produce slightly different yields,
they all agree that a SN Ia should produce heavy neutron-
rich isotopes, such as
50Ti,
58Fe, and
58Ni. In Table 4 we
summarise the isotope ratios of these elements for different
SNe II and SNe Ia models in the literature. For the follow-
ing, the SNe II yields we use are labeled “SNe II model”
in Table 4, while we will test the different ratios associated
with different yields for SNe Ia.
In Figure 10, we show the predictions of our NGC 6752

Page 13

The Chemical Evolution of Globular Clusters
13
and 47 Tuc models for the evolution of the isotope ratios
48Ti/50Ti,56Fe/58Fe, and58Ni/60Ni versus [O/Fe]. As can
be seen, variations of up to two orders of magnitude are
predicted and should be observed if inhomogeneous pollu-
tion by the SN Ia is the condition for a GCs formation. In-
deed as50Ti and58Fe are primarily produced in SN Ia, and
marginally produced in SNe II, the56Fe/58Fe and48Ti/50Ti
isotope ratios increase considerably during the formation of
the GC. The case of
both isotopes are produced by both types of SNe. In this
case, the
even if there is only a change of 0.8-1.0 dex in the58Ni/60Ni
ratio, which decreases during the evolution.
While the measurements of such isotope ratios would
help to test our framework, from an observational perspec-
tive such measurements are likely beyond the limit of cur-
rent instrumentation. It might be possible to measure Fe
isotope ratios from the FeH molecular lines; such lines are
most likely to be present in extremely cool dwarfs. This is be-
yond the observational capabilities of current instrumenta-
tion but it might be possible using projected high-resolution
near-infrared spectrographs on 40m-class telescopes. Iso-
tope ratios for Ti have been measured in only a handful
of near-solar metallicity stars (e.g., Lambert & Luck 1977;
Chavez & Lambert 2009). These isotopes can only be mea-
sured from TiO molecular lines, and such lines are only
present in metal-rich giants ([Fe/H] ? −0.7) or very cool
dwarfs. Since NGC 6752 is reasonably metal-poor, the only
possibility of finding TiO lines would be at the very cool
end of the main sequence, beyond current instrumentation.
Owing to its higher metallicity, 47 Tuc ([Fe/H]∼ 0.7) may
be an excellent candidate to test our predictions with cur-
rent capabilities while measurements in other, more metal
poor GC will be possible with the advent of the 40-m class
telescopes.
AsshownbyHughes et al.
Chavez & Lambert(2009)
sub-solarmetallicitygiants
log(48Ti/50Ti)≈1.1,withvery
[Fe/H]≈−0.7 to [Fe/H]≈+0.0. If confirmed in a metal-
rich GC like 47 Tuc, this would suggest that the W7 or
WDD1 SNe Ia models are to be preferred over the CDD1
model (which is ∼0.7 dex higher in log(48Ti/50Ti) in this
metallicity range).
58Ni and
60Ni is a bit different since
58Ni production by SN Ia should be observable
(2008),
that
disc+halo
variation
the
datasuggest
the
little
mildly
show
from
in
8 CONCLUSIONS
In this paper we continue the study of the chemical evolution
of GCs in the new framework of “peculiar pre-enrichment”
presented by Marcolini et al. (2008). We extend the previous
study to the very metal-poor globular clusters M 15 and
NGC 6397 and the metal-rich cluster 47 Tuc. We also study
the chemical evolution of fluorine in M 4. Our major findings
can be summarised as follow:
(i) In our framework we can reproduce the sodium-
fluorine anti-correlation, as well as the oxygen-fluorine cor-
relation observed in M 4.
(ii) The model can reproduce the very low [Fe/H] con-
tents of M 15 and NGC 6397 assuming that they formed
early in the history of the halo, and that the SN Ia+AGB
stars’ inhomogeneous pollution was localised in a larger ra-
dius (compared to the more metal-rich clusters). The Na-O
and C-N anti-correlations are also well-reproduced for these
cases.
(iii) The light-element anti-correlations are also repro-
duced in the case of the metal-rich globular cluster 47 Tuc,
which in our framework should form at a later stage of the
halo’s formation, with the inhomogeneous pollution confined
to a much smaller radius (e.g., ∼20 pc).
(iv) We also tested our model against the observed α-
element abundances (O, Mg, Si, Ca, and Ti) for different
SNe Ia models. In general, the slow deflagration SN Ia mod-
els (WDD1 and CDD1) of Iwamoto et al. (1999) better re-
produce the observational constraints than the fast deflagra-
tion (W7) models. This is because the WDD1 and CDD1
models produce more Si, Ca, and Ti. The model reproduces
the α-element content in the case of the metal-rich globular
47 Tuc, where a slight Na-Si and Na-Ca anti-correlation is
observed. In the case of the intermediate-metallicity globular
NGC 6752 we predict a moderate Si-O and Ca-O correlation
which is not compatible with observations. However, this
problem can be solved by assuming that (very) low metal-
licity (or prompt) SNe Ia produce a factor of four (two)
more Si (Ca) than the yields prescribed by Iwamoto et al.
(1999), that were computed for solar metallicity. Also, more
Si production by intermediate-mass AGB stars would also
help to solve this problem. This last point, though, is not
compatible with current AGB models.
(v) The model predicts that the evolution of most of the
Fe-peak elements should remain approximately constant (V
and Ni), or slightly anti-correlated with oxygen (Co). This
is because SNe Ia yields prescribe a large production of the
Fe-group elements (with the slow deflagration model better
fitting the observations). However, while our model in gen-
eral predicts that Fe-group elements should slightly corre-
late or anti-correlate with lighter elements, the details (and
the scatter) depends strongly on the adopted SNe II yields,
with yields at different metallicities required (especially for
SNe Ia) for a more detailed analysis.
(vi) The only notable exception is copper, which we fail to
reproduce by more than an order of magnitude when using
yields from the literature. This is due to the low Cu pro-
duction in theoretical SNe Ia models. We propose a value
for the copper production of ∼ 1.2 ×10−4M⊙ in SNe Ia, to
match the observational constraints. While copper produc-
tion is still a matter of debate in the literature, it is worth
noting that our proposed value is essentially the same sug-
gested by Matteucci et al. (1993), to reproduce the copper
abundances of the disk+halo of the Milky Way.
(vii) Finally, we propose that the discovery of high iso-
topic ratios (at high [α/Fe]) involving the neutron-rich50Ti,
58Fe, and58Ni species could be used as an observational test
of our model. This is because these are mainly produced by
SN Ia, unlike the Mg isotopes employed in Paper I which
are produced by both SNe II and AGB stars. This test will
likely only be possible for the most metal-rich clusters, such
as 47 Tuc.
(viii) The dynamical feasibility of the proposed scenario,
however, remains to be probed wiht hydrodynamical simu-
lations.

Page 14

14
S´ anchez-Bl´ azquez et al.
Figure 10. Evolution of the isotope ratios for48Ti/50Ti,56Fe/58Fe, and58Ni/60Ni versus [O/Fe], for the NGC 6752 model (upper
panels) and for 47 Tuc (lower panels). The isotopes50Ti,58Fe, and58Ni are mainly produced by SNe Ia (Iwamoto et al. 1999). The
colour-coded lines correspond to models with different SNe Ia yields. Orange lines: W7; blue lines: WDD1; black lines: CDD1.
ACKNOWLEDGMENTS
We kindly thank David Yong for his insights into isotopic
abundance determinations. PSB is supported by the Minis-
terio de Ciencia e Innovaci´ on (MICINN) of Spain through
the Ramon y Cajal programme. PSB also acknowledges
a Marie Curie Intra-European Reintegration grant within
the 6th European framework program and financial support
from the Spanish Plan Nacional del Espacio del Ministerio
de Educaci´ on y Ciencia (AYA2007-67752-C03-01). BKG ac-
knowledges the support of the UK’s Science & Technology
Facilities Council (STFC: ST/F002432/1, ST/G003025/1),
the UK’s National Cosmology Supercomputer (COSMOS),
the University of Central Lancashire’s High Performance
Computing Facility, and the generous financial support of
Saint Mary’s and Monash Universities visitor programs. AIK
acknowledges support from the Australian Research Coun-
cil’s Discovery Projects funding scheme (DP0664105).
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