Our knowledge of single stellar evolution and nucleosynthesis lays the groundwork for understanding chemical evolution, i.e., how the interstellar medium changes in chemical composition as it is enriched by dying stars. Observationally, the historical composition of the interstellar medium can be traced by low-mass stars. This is because of their extremely long lifetimes (tens of billions of years) and the fact that their atmospheres preserve their chemical compositions at birth, making them ideal ’stellar fossils’.
1.5.1. Chemical evolution modelling
By accounting for the major processes that affect star formation and the composition of the interstellar medium, we can construct theoretical models whose abundance evolution can be compared with the abundances from real stars measured by spectroscopy. The basic components of a chemical evolution model are:
• initial conditions: the starting gas mass and chemical composition,
• φ(m): the initial mass function,
• ψ: the star formation rate (in M/year) as a function of time or gas density,
• τ(m): the stellar lifetime (in years) as a function of initial mass,
• mre m(m): the remnant mass (in M) as a function of initial mass, and
• qi(m): the stellar yield of speciesifrom a star with initial massm(in M).
The initial mass function is a distribution function for the initial masses of stars produced by star formation. For chemical evolution purposes, it is typically normalised such that
φ(m)dm is the number of stars with masses between m and m+dm per M of star
formation. Common choices for the initial mass function include the classical Salpeter (1955) power law, or more recent forms such at the broken power law of Kroupa et al. (1993).
The star formation rate is often estimated by a power law function of the gas density (Schmidt 1959; Kennicutt & Evans 2012) or even more simply, an exponentially decreasing rate that models a single initial burst.
The results from a grid of stellar evolution and nucleosynthesis models spanning a range of initial masses at the relevant metallicity are interpolated to fix several more of the required functions: the stellar lifetime function, the remnant mass function, and the stellar yields. We ignore the dependence on metallicity here for simplicity (and because star formation ends before the metallicity increases in our models), but more complex models can use interpolated yields from a grid that spans a range of metallicities.
With these ingredients, we can write a set of differential equations to be solved in order to model the chemical composition as a function of time. Here we describe a single-zone chemical evolution model for simplicity, although the model can be straightforwardly extended to multiple zones or two or three dimensional models. This would be achieved by solving the chemical evolution equations separately for each zone, with extra terms that account for the transfer of gas and stars across zone boundaries. The following summary of the equations of chemical evolution is based on Pagel (2009, Chapter 7).
The total mass of the system is the divided up into the gas mass (g) and the stellar mass (s). These variables evolve according to
ds
dt ψ(t)−e(t),
and
dg
dt F(t)−E(t)+e(t)−ψ(t),
whereFis the galactic accretion rate,Eis the galactic ejection rate,e is the rate of mass ejection from the stellar population, and ψis the star formation rate. The stellar mass ejection rate,e(t)is
e(t)
Z mU
mτt
(m−mre m(m))ψ(t−τ(m))φ(m)dm,
wheremτt is the initial mass of a star with a lifetime oft,mU is the upper limit mass,
mre m(m)is the mass of a remnant left by a star with an initial massm, andτ(m)is the stellar lifetime.
For each chemical speciesithe ejection rate is
ei(t)
Z mU
mτt
qi(m)ψ(t−τ(m))φ(m)dm,
whereqi(m)is the stellar yield ofifrom a star with an initial mass ofm. The gas mass fractionsXi(t)evolve according to:
d
dt[gXi]ei−Xiψ+Xi,FF−XiE
1.5.2. Globular clusters
Some of the oldest objects in the local universe are globular clusters, which are very dense, roughly spherical groups of about 105–106stars located mostly in the halo of galaxies. Our Milky Way galaxy hosts over 150 globular clusters (Harris 1996, 2010 edition) and many of these clusters are over ten billion years old (Dotter et al. 2009).
Historically, the ages of globular clusters have provided a useful constraint on cosmology by setting a lower limit on the age of the universe (Chaboyer et al. 1996; Dotter et al. 2010). Their simplicity as single stellar populations has also made them extremely valuable for testing theories of stellar evolution (Johnson & Sandage 1955), where a simple stellar population is defined as a group of stars that formed at the same time in a single burst of star formation within a gas cloud of a uniform chemical composition. The assumption that this described globular clusters made the age dating of clusters relatively straightforward; the age of the cluster is simply the lifetime of the stars that are currently leaving the main sequence, and these stars can be identified from a colour-magnitude diagram.
Detailed abundances studies have revealed that globular clusters show significant spreads in the abundances of light elements such as He, C, N, O, F, Na, Mg, and Al (Cohen 1978; Peterson 1980; Gratton et al. 2004). The variations of these elements are correlated in a way that is almost exclusive to globular clusters (i.e., it is rarely seen in field stars and open clusters, as shown by Kraft et al. 1982; Shetrone 1996; Kraft et al. 1997; Gratton et al. 2000; De Silva et al. 2009) with anti-correlations between the abundances of C and N, Na and O, and sometimes Mg and Al (Shetrone 1996), typically with a C+N+O abundance that is constant within observational errors. These abundance patterns point to a H-burning process at high temperature (&80 MK) and dilution with varying amounts of unprocessed material, although the stellar sites where this burning takes place and the mechanism of dilution are presently not understood (Prantzos et al. 2007). Some current candidates for the hot hydrogen-burning environment are massive AGB (5–8 M) stars (Cottrell &
Da Costa 1981; Ventura et al. 2001), rotating massive stars (Prantzos & Charbonnel 2006; Decressin et al. 2007), massive binaries (de Mink et al. 2009; Bastian et al. 2013), and supermassive (>104M) stars (Denissenkov & Hartwick 2014; Denissenkov et al. 2015b).
The Milky Way globular clusters have metallicities ranging from [Fe/H]≈ −2.4 to−0.3, with remarkably little variation within each cluster in almost all cases. A survey of 19 globular clusters by Carretta et al. (2009a) found no more than 0.05 dex scatter in [Fe/H] measurements, which is consistent with all of the clusters in their sample being mono-metallic. There are a few (∼10%) exceptional clusters that do feature metallicity variation, and these are also some of the most massive clusters. Examples include M22, which has an intrinsic scatter in [Fe/H] of 0.10–0.15 dex (Marino et al. 2009; Da Costa et al. 2009) andωCentauri, although the latter could be the tidally-stripped nucleus of a dwarf galaxy (Freeman 1993; Bekki & Freeman 2003) and might therefore formed differently to
other globular clusters (see the discussion in Gratton et al. 2004).
The abundances of neutron-capture elements (Z>30) are constant within most clusters (Gratton et al. 2004; Yong et al. 2006, 2008a; D’Orazi et al. 2010). However, there are exceptions to this, including M15, which shows variations inr-process elements such as Eu (Sneden et al. 2000). M22 shows two distinct populations with distinct abundances of s-process elements (Y, Zr, Ba) (Marino et al. 2009; Villanova et al. 2010b). Among globular clusters with constant abundances of neutron-capture elements, there exist cluster-to-cluster differences. For example, M4 is a fairly typical mono-metallic metal-poor GC ([Fe/H]−1.18; Carretta et al. 2009a), except that it has super-solar abundances of
s-process peak elements (e.g., Rb, Y, Zr, La, Ba, Pb; Brown & Wallerstein 1992; Ivans et al. 1999). This makes M4 more enriched withs-process elements than M5, another GC with a similar metallicity.
In Chapter 3, we will construct new chemical evolution models that incorporate yields from AGB stars and rotating massive stars to investigate the internals-process enrichment within the cluster M22, as well as the globals-process enrichment of M4.
1.5.3. Chemical evolution code
The candidate has developed a basic one-zone chemical evolution code in modern Fortran named ‘Evel ChemEvol’.
The Evel ChemEvol code solves the equations of chemical evolution (described in Section 1.5.1) with a Simpson rule integration and an adaptive time step. To estimate the integration error at each step, the Simpson rule calculation is compared with a lower-order integration using the mid-point rule. If any of the integration stages exceed their configured error thresholds, the current step is recalculated with a shorter time step.
To ensure that the Evel ChemEvol code produces valid output, the software must be able to reproduce the results of an existing chemical evolution model. We chose to reproduce the model of the globular cluster NGC 6752 by Fenner et al. (2004, hereafter, F04). Following F04, the initial gas abundances are set by polluting primordial gas (roughly 75% H and 25% He) with the ejecta of massive stars until [Fe/H] increases to−1.4, the observed metallicity of NGC 6752. Many of initial these abundances were then manually adjusted to match F04, since the massive star yields of Kobayashi et al. (2006) that we used were different to those of Chieffi & Limongi (2002) that were used in F04. The initial gas mass is set to 1.4×105M, and star formation is triggered with an exponential falloff on a
timescale of 107years. Figure 1.7 illustrates the output of the chemical evolution model. As shown in Figure 1.7, the stellar ejection rate is separated into retained ejecta from stars
M
ʘor M
ʘ
/y
ear
Log
10(time [Gyr])
Figure 1.7 Time evolution of Evel ChemEvol model variables. The plot shows the gas mass (Mgas,
in M), the stellar mass (Mstars, in M), the star formation rate (SFR, in M/yr), the stellar ejection
rate of retained and lost ejecta (SERret and SERlost, in M/yr). In this model, ejecta from stars <
6.5 Mis assumed to retained, while ejecta from more massive stars is lost.
less massive than 6.5 M, and ejecta from more massive stars, whose high-speed winds
are assumed to escape the cluster system. The cutoff at 6.5 Mis chosen for consistency
with F04.
We use the same Kroupa et al. (1993) initial mass function and AGB stellar yields specified in F04. The stellar lifetime function is not described in F04 and is approximated here by a power law with the index−2.9. The results are in good agreement when assuming the same AGB yields for C, N, O (Figure 1.8), Na and O (Figure 1.9), and Al and Mg (Figure 1.10).
20 Introduction
bimodal C and N abundances with one group having roughly solar
[C/Fe] and [N/Fe], and the other group being N-rich and C-poor
(e.g. Da Costa & Cottrell 1980; Norris et al. 1981; Smith & Norris
1993). The empirical data point to an anticorrelation between C and
N that has previously been explained in terms of the operation of
a deep-mixing mechanism in evolved stars. The dependence of the
molecular CN-band strength on stellar luminosity (e.g. Suntzeff &
Smith 1991) is readily understood as a result of increased mixing
of N-rich, C-poor material from within the stellar interior to the
surface, as a function of evolutionary stage. More recently, however,
Grundahl et al. (2002) found an anticorrelation between [C/Fe] and
[N/Fe] in NGC 6752 stars that is independent of luminosity. Cannon
et al. (1998) found a similar CN bimodality and anticorrelation in
47 Tuc from a sample of stars that included not just giants, but also
unevolved main-sequence stars whose shallow convective layers
preclude dredge-up of CNO-cycled material. The presence of the
same CN trend in both dwarfs and giants led Cannon et al. (1998)
to conclude that deep mixing was not singularly responsible for CN
abundance anomalies in 47 Tuc. It seems likely that the CN patterns
in NGC 6752 stars arise from a combination of deep mixing and
some form of external pollution.
Fig. 3 casts doubt on AGB stars being a major source of external
pollution. All the AGB models from 1.25–6.5 M!
expel material
that is enhanced in both Nand
C. From Fig. 3 it is clear that main-
sequence stars severely polluted by AGB material are also expected
to exhibit heightened C and N abundances. The positive correlation
between C and N in the intracluster gas predicted by our GCCE
model fails to match the empirical trend; however, we note that our
calculations only reflect the chemical evolution of the intracluster
medium. The abundance pattern of the gas at a particular time corre-
sponds to the initial composition of stars born at that time, whereas
the observed abundances of elements like CNO in evolved red gi-
ants are likely to differ from their starting abundances due to internal
synthesis and mixing.
Fig. 4 reveals an almost order of magnitude rise in [C+N+O/Fe]
in the intracluster gas within 1 Gyr of formation. A slight drop in the
O abundance is more than compensated for by a dramatic increase
Figure 4.
Temporal evolution of [C/Fe] (dotted line), [N/Fe] (short dashed
line), [O/Fe] (long dashed line) and [C+N+O/Fe] (solid line). In contrast to
observations in NGC 6752 and other globular clusters, C+N+O abundance
is predicted to vary by an order of magnitude if AGB stars are responsible
for the Na and Al enhancement.
in C and N. This robust prediction is based on intermediate-mass
stellar nucleosynthesis and poses further difficulties for the AGB
pollution scenario as an explanation for globular cluster abundance
anomalies, since C+N+O is found to be approximately constant
in many GCs (Ivans et al. 1999). AGB stars have been proposed in
the literature as promising candidates for producing the observed
GC abundance anomalies because they exhibit the required hot H
burning via hot-bottom burning. However, because these stars also
dredge up the products of He-burning, C+N+O is not conserved
in the models, in conflict with the data.
We note that in order to achieve an order of magnitude increase in
the intracluster abundance of Al via the AGB pollution scenario, our
models simultaneously predict an intracluster medium helium mass
fraction,
Y, approaching 0.3. This represents a
∼0.05 increase inY
over the primordial value. D’Antona et al. (2002) have found that the
effects on stellar evolution due to this level of He enrichment would
be difficult to measure observationally but could lead to extended
blue tails in the horizontal branch morphology.
4 C O N C L U S I O N S
The presence of variations in C, N, O, Mg, Na and Al in globular
cluster stars yet to ascend the red giant branch provides compelling
evidence for these chemical patterns already being in place in the
gas from which cluster stars formed, or in gas that later polluted
their atmospheres. Otherwise, one would expect to see chemical
homogeneity in stars below a certain luminosity, unless our under-
standing of deep mixing is seriously flawed. We have constructed
a self-consistent model of the chemical evolution of the intraclus-
ter gas, that included custom-made detailed stellar models, to test
whether the observed inhomogeneities may be caused by contam-
ination from material processed by intermediate-mass stars during
their AGB phase. This model is compatible with either a scenario in
which there is coeval and stochastic sweeping of the intermediate-
mass stellar ejecta by existing lower-mass stars, or one in which
new stars form from AGB polluted material. In the latter case, there
would be a small age spread, with Na-rich stars being a few hundred
million years younger than Na-poor stars.
We find that, regardless of either the mechanism for polluting
cluster stars with AGB material or the level of dilution of Population
III material by AGB ejecta, intermediate-mass stars are unlikely to
be responsible for most of the abundance anomalies. While metal-
poor AGB models generate large quantities of Na and Al that may
account for the observed spread in these elements in NGC 6752, the
AGB pollution scenario encounters a number of serious problems:
(i) O is not depleted within AGB stars to the extent required by
observations;
(ii) Mg is
producedwhen it should bedestroyed;
(iii) C+
N
+
O does not remain constant in AGB processed
material; and
(iv)
25Mg is correlated with
26Mg in the modelled AGB ejecta,
conflicting with the observations of Yong et al. (2003).
Note that all of these problems stem from the addition of helium-
burning products into the AGB star ejecta. Perhaps a generation of
AGB stars which experience HBB but almost no dredge-up would
fit the data better!
The model presented in this paper could be generalized for ap-
plication to other globular clusters by varying three main param-
eters: the initial metallicity; the upper mass limit beyond which
stellar winds are too energetic for the cluster to retain the ejecta (this
C #
2004 RAS, MNRAS353,789–795
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
log10(time [Gyr])1.0
0.5
0.0
0.5
1.0
1.5
2.0
2.5
[C/Fe]
[N/Fe]
[O/Fe]
[(C+N+O)/Fe]
Figure 1.8 Evolution of C, N, and O abundances from Fenner et al. (2004) (top) and with Evel ChemEvol (bottom).
792
Y. Fenner et al.
Figure 1. Predicted trend of [Na/Fe] versus [O/Fe] (a) and [Al/Fe] versus
[Mg/Fe] (b) (thick curve) shown against observational data from Grundahl et al. (2002) (squares and pluses) and Yong et al. (2003) (circles). Data from Yong et al. were shifted on to the Grundahl et al. scale. Diamonds correspond to the 1.25, 2.5, 3.5, 5.0 and 6.5 M!stellar models of Campbell (in prepara- tion), where the size of the symbol indicates the stellar mass. Arrows indicate the effects of changing the mass-loss law for 2.5- and 5.0-M!, stars (see text for details). In the lower panel, the evolution of [24Mg/Fe], [25Mg/Fe] and
[26Mg/Fe] are shown by dotted, dashed, and dot-dashed lines, respectively.