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The Evolution Of Carbon, Sulphur, and Titanium Isotopes from High-Redshift to the Local Universe

Recent observations of carbon, sulphur, and titanium isotopes at redshifts z~1 and in the local stellar disc and halo have opened a new window into the study of isotopic abundance patterns and the origin of the chemical elements. Using our Galactic chemical evolution code GEtool, we have examined the evolution of these isotopes within the framework of a Milky Way-like system. We have three aims in this work: first, to test the claim that novae are required, in order to explain the carbon isotope patterns in the Milky Way; second, to test the claim that sulphur isotope patterns at high-redshift require an initial mass function biased towards massive stars; and third, to test extant chemical evolution models against new observations of titanium isotopes that suggest an anti-correlation between trace-to-dominant isotopes with metallicity. Based upon our dual-infall galactic chemical evolution modelling of a Milky Way-like system, and the subsequent comparison with these new and unique datasets, we conclude the following: novae are not required to understand the evolution of 12C/13C in the solar neighbourhood; a massive star-biased initial mass function is consistent with the low ratios of 12C/13C and 32S/34S seen in one high-redshift late-type spiral, but the consequent super-solar metallicity prediction for the interstellar medium in this system seems highly unlikely; and deficient isotopes of titanium are predicted to correlate positively with metallicity, in apparent disagreement with the new datasets; if confirmed, classical chemical evolution models of the Milky Way (and the associated supernovae nucleosynthetic yields) may need a substantial overhaul to be made consistent.


INTRODUCTION
Galactic chemical evolution models are employed to study the spatial and temporal evolution of elements and isotopes throughout the Universe. When coupled to phenomenological representations of galaxy assembly, such models can be compared directly with exquisite elemental and isotopic abundance patterns observed locally in the Milky Way. From these comparisons, conclusions can be drawn regarding the veracity of the underlying micro-physics governing stellar evolution and nucleosynthesis, in addition to the macro-physics governing the assembly of galaxies, the redistribution of the interstellar medium (ISM) over galactic timescales, and the relative birthrate of stars of various masses (the so-called initial mass function, or IMF). While most galactic chemical evolution models to date have concentrated on predictions related to the total elemental abundance patterns, in unique circumstances, the availability of detailed isotopic patterns can enhance the predictive power of these models; such isotopic patterns afford additional leverage for discriminating between the various origin sites for the chemical elements. A full literature review of the field would be unwieldy, but we refer the reader to any number of comprehensive reviews and the many important references therein -e.g. Timmes, Woosley & Weaver (1995, hereafter TWW95), Prantzos, Aubert & Audouze (1996), Fenner & Gibson (2003), Romano & Matteucci (2003, hereafter, RM03), and Chiappini et al. (2008).
Recent observational work has opened a new window into the study of isotopic abundance patterns -specifically, the identification (for the first time) of carbon and sulphur isotopes at redshifts z∼1 (Muller et al. 2006, hereafter M06; Levshakov et al. 2006), coupled with the recent determination of titanium isotopic abundances in the local stellar disc and halo (Chavez 2008). The abundances of each of these isotopes, and their evolution with redshift, hold clues as to the relative importance of supernovae versus asymptotic giant branch stars versus novae in seeding the Universe with these important elements.
In this work, we present predictions for isotopic ratios of carbon, sulphur, and titanium within the framework of a classical Milky-Way-like disc galaxy model. The paper is organised as follows: the fundamental nucleosynthetic origins of the relevant carbon, sulphur, and titanium isotopes are first (briefly) reviewed in § 2; in § 3, we introduce the chemical evolution code (GEtool) and the four stellar yield compilations employed in our modelling; finally, the results are presented and summarised in § 4 and § 5, respectively.

ORIGIN OF CARBON, SULPHUR, AND TITANIUM ISOTOPES
Because we in the astronomical community are generally more accustomed to discussing elemental abundance patterns, rather than isotopic patterns, we felt it would be useful to provide an overview of the nucleosynthetic origins of these relevant isotopes. Much of this section has been derived and summarised from Clayton's (2007) exceptional handbook, to which the reader is referred for definitive and comprehensive descriptions. Complementary discussions of the relevant nucleosynthesis processes and their products can be found in , Arnett (1996), Pagel (1997), and Matteucci (2001).

Carbon-12
As the initial product of helium burning (the classical tripleα process), 12 C is the second most abundant nucleus formed by nucleosynthesis in stars. While the exact accounting remains uncertain, significant carbon production can likely be traced to massive stars that eventually become Type II supernovae (e.g. Chiappini et al. 1997), with a substantial contribution also derived from intermediate mass asymptotic giant branch (AGB) stars (e.g. Carigi et al. 2005). The newly created 12 C nuclei in these latter stars is convected to the surface, often leading to the formation of a carbon star; ultimately, this carbon-enriched envelope is lost to the ISM through the process of stellar winds associated with planetary nebulae. The amount of 12 C produced depends critically upon the 12 C + 4 He → 16 O reaction.

Carbon-13
The 13 C isotope is considered a secondary nucleus, produced not from the nuclear fusion of hydrogen and helium, but as a secondary process involving "seed" nuclei of 12 C. It is thought to originate within stars not massive enough to become supernovae, in particular, AGB stars. At sufficient temperatures, 13 C is produced via the capture of a proton by the 12 C nucleus, to form 13 N, which itself undergoes βdecay to form 13 C. The relative abundance depends upon both the relevant reaction and destruction rates. Having said that, there is also evidence to suggest that 13 C nucleosynthesis also has a primary component (i.e., a production pathway exists which does not depend upon seed nuclei of 12 C). For example, within AGB stars, periodic dredge-up episodes bring newly-formed 12 C to the surface, converting AGB stars to carbon stars, as noted above. If the temperature at the base of the envelope is sufficient, this primary 12 C can be partly converted to (primary) 13 C and 14 N by the first two reactions of the CN-cycle. However, if the star is massive enough (M > 4M⊙) then hot bottom burning can occur, delaying or preventing the AGB star from turning into a carbon star, and therefore this process.

Sulphur-32
32 S is formed mostly through oxygen burning, two 16 O nuclei colliding to form 28 Si and 4 He, with these products subsequently fusing to yield 32 S. Almost all 32 S is produced in Type II supernovae, which eject approximately ten times the quantity synthesised within Type Ia supernovae, and occur roughly fives times as often.

Sulphur-34
34 S originates as a byproduct of oxygen-burning. 34 S is partly a secondary isotope as it is formed from newly-made 32 S and 33 S by neutron captures, which itself is aided if the star that creates them also has carbon and oxygen in its composition. 18 O and 22 Ne created from this initial carbon and oxygen produce extra neutrons that are needed by heavier sulphur isotopes, but excess neutrons are also produced during oxygen burning by positron emissions, and as such, 34 S is also partly a primary isotope. As for 32 S, 34 S is produced primarily from supernovae.

Titanium-46
46 Ti originates from oxygen-and silicon-burning in massive stars. Two 16 O nuclei collide and subsequent α-captures produce 44 Ti, while the capture of two free neutrons results in 46 Ti. This can also be viewed as the addition of an α-particle to 42 Ca, so 46 Ti becomes abundant in the same oxygenburning zone that synthesises 42 Ca. If the burning continues into that of silicon, then the 46 Ti abundance erodes quickly. 46 Ti cannot be labelled as a secondary nucleus as there are positive β-decays during the oxygen burning.

Titanium-47
47 Ti also originates from oxygen-and silicon-burning in massive stars, but in this case, three free neutrons are captured. This can be viewed as the addition of a neutron to 46 Ti, so 47 Ti becomes abundant in the same oxygen-burning zone that synthesises 46 Ti. Some models of Type Ia supernovae also contribute to interstellar 47 Ti. There does appear to be an apparent problem though, in the sense that models of both supernovae types have been claimed to be insufficient producers of 47 Ti with respect to observations (Timmes et al. 1995).

Titanium-48
The production of 48 Ti is traced to the nucleosynthesis of 48 Cr in stellar explosions; two subsequent β-decays after its ejection leads to 48 Ti. This occurs mostly in explosive silicon-burning and during helium fusion. The latter could occur either by the α-rich freezeout of shock-decomposed nuclei near the core of a Type II supernova, or as part of explosive helium burning associated with Type Ia supernovae.

Titanium-49
49 Ti is produced mainly by the nucleosynthesis of radioactive 49 Cr in stellar explosions of both types of supernovae. The isotope 49 Cr is the result of the explosive fusion of helium as outlined above; 49 Cr is also synthesised during silicon-burning, as for the lighter titanium isotopes.

Titanium-50
It has been suggested that 50 Ti is produced primarily in sub-Chandrasekhar mass Type Ia supernovae (Timmes et al. 1995), during which electron capture turns the composition neutron-rich. Some fraction of the 50 Ti is likely also made by slow neutron capture within the the burning shells of pre-supernovae massive stars and AGB stars. It would appear that explosive burning in Type II supernovae does not produce significant quantities of 50 Ti.

THE CHEMICAL EVOLUTION MODEL
Throughout our work, we have used the GEtool (Fenner & Gibson 2003;Fenner, Murphy & Gibson 2005) galactic chemical evolution package, employing its default "dual infall" (halo + disc) mode (similar in spirit to the seminal models of Chiappini, Matteucci & Gratton 1997). Within this framework, the halo phase occurs on a rapid timescale and enriches the initially primordial gas to a metallicity of ∼10% solar. The second (disc) phase is delayed by ∼1 Gyr with respect to the first, and acts over a more prolonged timescale. We assume the infall of fresh material during this second phase to be somewhat metalenriched (10% solar) and α-enhanced (0.4 dex), consistent with patterns seen in metal-poor halo/thick disc stars (e.g. Ryan, Norris & Beers 1996) and present-day high-velocity infalling halo gas (e.g. Gibson et al. 2001).
The rate at which material is accreted is assumed to decline exponentially. The evolution of total surface mass density σtot(r, t) is given by: where the infall rate coefficients A(r) and B(r) are chosen in order to reproduce the present-day surface mass density of the halo and disk components, which we take to be 10 and 45 M⊙ pc −2 , respectively. The adopted timescales for the infall phases in the solar neighbourhood (r⊙=8.5kpc) are τH = 0.05 Gyr and τD = 10.5 Gyr, with the functional form for the latter being τD(r)=1.38r − 1.27, reflecting the "inside-out" formation framework in which the dual-infall model operates. These timescales and coefficients provide model predictions consistent with various local observational constraints such as the metallicity distribution function, age metallicity relation, and present-day gas surface density distribution, and are consistent with those employed by Fenner et al. (2005). We adopt a fairly conservative star formation prescription based upon a "Schmidt Law" of the form: where the value of the star formation efficiency ν is constrained by the present-day gas fraction (for this work, ν=0.06 Gyr −1 ).

Initial Mass Function
The shape of the initial mass function (IMF) controls the fraction of material locked-up in stellar generation, which in turn determines the rate at which different elements are released into the ISM. Our default assumption is that of the three-component IMF of Kroupa, Tout & Gilmore (1993;hereafter, KTG), with lower-and upper-mass limits of 0.08 and 60 M⊙, respectively; the KTG IMF lies between those of Salpeter (1955) and Scalo (1986), in terms of mass fraction tied up in Type II supernovae progenitors (e.g. Table 7 of Gibson 1997). Unless otherwise stated, we assume that the mass fraction of the IMF which is tied up in SNe Ia progenitor binary systems (total binary masses in the range 3−16 M⊙) is 4%; such an assumption yields, within our adopted model formalism, a disc-averaged present-day ratio between SNe II and SNe Ia rates of 4.1 (consistent with the disc-averaged empirical SNe II to SNe Ia ratio of 3.7van den Bergh 1988). We supplement this by exploring a range of single power-law IMF slopes to isolate the relative contributions of low-and high-mass stars.

Stellar Yields
In order to sample the range of uncertainties inherent to stellar evolution modelling, we explore the use of several sets of metallicity-dependent nucleosynthetic yields, in this work -those of , Chieffi & Limongi (2004), Kobayashi et al (2006), 1 for Type II supernovae, 2 and those of Karakas & Lattanzio (2007), for low-and intermediate-mass single stars (hereafter, WW95, CL04, K06, and KL07, respectively). For Type Ia supernovae, the yields of Nomoto et al. (1997) have been assumed.
As these yields only have data for M 40 M⊙, and our GEtool models assumed an upper-mass limit of 60 M⊙, a linear extrapolation was used, to extend the yields to the highest masses. We stress though that the results do not depend upon the assumption of linear, as opposed to logarithmic or, indeed, "flat" extrapolation. A detailed isotope-by-isotope comparison between the compilations is beyond the scope of our work; indeed, as "end-users" only of the products of the aforementioned sophisticated stellar evolution and nucleosynthesis codes, one could argue it is not even a feasible undertaking. What is important though, from our perspective, is that by employing a range of yield compilations, we are in some sense sampling the "convolution" of a fair range of stellar "micro-physics" uncertainties intrinsic to these codes. This is, realistically, the best compromise for end-users to take.
As noted earlier, we will be concentrating upon the chemical evolution of the primary isotopes of carbon, sulphur, and titanium, in response to recent advances pertaining to their observation locally and in the distant Universe. To foreshadow the chemical evolution results described in § 4, in Figs. 1 and 2 we show the 12 C/ 13 C, 32 S/ 34 S, and 46,47,49,50 Ti/ 48 Ti isotope ratios, as a function of stellar mass (at solar metallicity) for each of the yield compilations adopted here.
We can see from Fig. 1 that, for carbon, there is little difference between supernovae and hypernovae (K06-SNe and K06-HNe, respectively), whereas for sulphur, the explosion energetics lead to a factor of ∼3 variation in the predicted ejecta ratios. We will return to the issue of the impact of apparently "discrepant" individual stellar models in § 4, but we should note now, for example, the obvious "outlier" seen in the K06 M=25 M⊙ solar metallicity model (upper panel of Fig. 1); this two-orders-of-magnitude outlier is also seen in the K06 M=18 M⊙ Z=0.004 (not plotted) model. In both cases, this can be traced to the respective models producing ∼100× the 13 C of the "flanking" models. In Fig. 2, it is also seen that there is a factor of ∼3 variation in the predicted ejecta ratios of K06-SNe and K06-HNe. Figures 3 and 4 show the variation of 12 C/ 13 C and 32 S/ 34 S, with mass, for different metallicities. For clarity, only the WW95 and KL07 yields are shown, as the CL04 and K06 yields demonstrate a similar metallicity-dependence as that of WW95; the general trend of decreasing isotope ratio with increasing metallicity can be seen. In massive stars of decreasing metallicity, it is expected that the amounts of 12 C and 32 S would be similar, but the amounts of 13 C and 34 S would decrease (WW95).

New Boundary Conditions
The observational data used in the models in this paper have been taken from RM03, M06, Levashakov et (2006), Anders & Grevesse (1989), Mauersberger et al (2004), Spite et al (2006) and Chavez (2008). The data for the carbon and sulphur isotopes are summarised in Table 1. The ages used have been derived under the assumption of the concordant ΛCDM cosmology.

Local Constraints
RM03 have used the solar system value of carbon from Cameron (1982), and derive a local ISM value based upon the average of a range of observational data; the local ISM value employed in our work is taken directly from RM03. Assuming a present-day age of 13 Gyr, the solar system is taken to have formed at t = 8.5 Gyr (i.e., 4.5 Gyr ago). Wherever possible and relevant, we have used the solar values of Anders & Grevesse (1989).
For 32 S/ 34 S, we use the data compilation from Mauersberger et al. (2004), itself based upon Ding et al. (2001) for the solar system value (derived using the Canyon Diablo triolite) and Chin et al. (1999) for the local ISM.
Finally, Chavez (2008) has recently measured, for the first time, titanium isotope ratios in nearby low-mass stars by studying isotopic shifts seen in TiO spectra of M-dwarfs of the halo, and thin and thick discs. This was done using the 2d-coudé spectrograph at the 2.7m telescope at McDonald Observatory, with a nominal resolving power of ∼120k. In total, the isotopic ratios for 11 stars in the metallicity range −1<[Fe/H]<+0 have been derived. The data is shown in Fig. 8 and tabulated (and fully described) in Chavez (2008).

Higher Redshift Constraints
The values obtained recently by M06 for a spiral galaxy at redshift z=0.89 are listed in Table 1; the average value of the two spiral arm features seen in absorption towards a background radio-loud QSO is used. From M06 (figure 1), the mean galactocentric distance for these absorption features within the foreground galaxy's spiral arms is ∼4.5 kpc.
In Levshakov et al (2006), C I features associated with a damped Lyα system at z=1.15 seen in the spectrum of HE 0515−4414 were analysed to derive the 12 C/ 13 C ratio. The inferred lower limit of 12 C/ 13 C > 80 suggested, for the first time, that the abundance of 13 C in extragalactic clouds was very low. In order to infer the isotopic abundance patterns of the ISM at the time of formation of the Milky Way, we used a sample of extremely metal-poor giants ([Fe/H] −3) from Spite et al. (2006). We employed the "unmixed" stars from their sample (i.e., stars which have not had their surface abundances affected by deep mixing), and taken the mean 12 C/ 13 C value (19.38) as representative of time t∼1 Gyr (roughly the timescale of formation for the Galactic halo). Figure 5 shows the time evolution of the 12 C/ 13 C and 32 S/ 34 S ratios in our default dual-infall model for the solar neighbourhood ( § 3) using three different SNeII yield compilations. The observed values for the solar system (RM03), local halo stars (Spite et al. 2006), and the local interstellar medium (RM03;Mauersberger et al. 2004) are shown for comparison. The values from M06 are also plotted; as mentioned earlier, the mean galactocentric distance probed by the background QSO towards this high-redshift galaxy is ∼4.5 kpc; as such, besides the "solar neighbourhood" prediction, we have also plotted the results of our default model at a galactocentric radius of 4.5 kpc, using the WW95 yields.

RESULTS
The evolution of 12 C/ 13 C, employing the WW95 and CL04 yields, is essentially indistinguishable after ∼5 Gyrs, while the discrepant nature of the models employing the K06 yields is self-obvious. We have not shown the models using the K06 hypernovae yields, as their impact for the evolution of 12 C/ 13 C is negligible. In the case of 32 S/ 34 S, each of the yield compilations results in fairly self-consistent evolutionary trends, although the hypernovae do reduce the predicted ratio by a factor of two relative to models neglecting them.
From Fig. 5, we also note the existence of predicted positive radial gradients in both 12 C/ 13 C and 32 S/ 34 S, reflecting both the inside-out galaxy formation framework and the consequent increased importance of the secondary production of 13 C in the inner regions of the galactic model (e.g. RM03). From the lower panel of Fig. 5, we can see that our template dual-infall model is consistent with the extant galactic 32 S/ 34 S data (solar system and local ISM); the lower value observed at z=0.89 (M06) can be partially reconciled with it being nearer its respective galaxy's centre than the solar neighbourhood, 3 a point to which we return in § 4.2.

The Need For Novae
Previous galactic models for 12 C/ 13 C (e.g. RM03), while consistent with the local ISM, predicted an increase in 12 C/ 13 C over the past 5-10 Gyrs, driven in part by the use of the older van den Hoek & Groenewegen (1997) yields. To ameliorate that apparent discrepancy, RM03 introduced an important additional source of 13 C, in the form of novae. While successful in recovering the decline in 12 C/ 13 C with time, the overproduction of 13 C resulted in a significant mismatch between the model and the observations, as shown in Fig. 6, which required an a posteriori re-scaling of the model to the solar system value.
Conversely, the predicted decline in 12 C/ 13 C over the past 5-10 Gyrs in our model is naturally consistent with the observed solar value and that of the local ISM. This behaviour is driven by the new KL07 yields (which obviously RM03 did not have access to) without the need of recourse to any additional 13 C novae contribution. The putative need for novae might even be exacerbated if one were to include, for example, "born again" (i.e., re-ignited) stars such as Sakurai's Object; it has been suggested recently that such objects might be the dominant source of 13 C in the Universe (Hajduk et al. 2005). In the future, a phenomenological treatment of both novae and such born-again objects will be implemented within GEtool.

Isotopes At High-Redshift
As alluded to earlier, it can be seen in Fig. 5 that it is difficult for our template model to reproduce the isotopic ratios observed in the spiral galaxy at z=0.89. Admittedly, the K06 yields appear to provide a better fit to 12 C/ 13 C at t=5.7 Gyr (upper panel) although, at the same time, they are less successful for 32 S/ 34 S. We have examined two possible alternatives to our template solar neighbourhood model which could explain the lower values observed at highredshift: (i) varying the galactocentric distance of the model; (ii) varying the IMF.
First, as noted earlier, the data of M06 probes a galactocentric distance closer to 4.5 kpc, rather than the solar galactocentric distance of 8.5 kpc; our template Milky Way model for this radial "bin" would predict a 12 C/ 13 C evolution offset by ∼20% from the solar neighbourhood value, as shown in Fig. 5. This is, however, insufficient to reproduce the observed values.
Second, we explored the dependence of the predicted isotope ratios upon the relative proportion of massive stars in the IMF by flattening (significantly) the high-mass end of the IMF (representing the IMF by a single power-law of slope 0.9). Figure 7 shows the time evolution of the 12 C/ 13 C and 32 S/ 34 S isotope ratios using both this massive-star biased IMF and the default KTG IMF. The immediate conclusion to be drawn is that both ratios decrease dramatically (by a factor of approximately four) when adopting the massive star-biased IMF. Having said that, it is important to be aware that in large part, this dramatic decrease is a consequence of the accelerated global chemical enrichment produced by the flatter IMF, as these isotopic ratios decrease rapidly with increasing metallicity (recall the earlier discussion surrounding Figures 3 and 4), as opposed to any mass-dependency in the yields. While an IMF slope of 0.9 appears to be a viable solution to the low ratios seen at z=0.89, the predicted global metallicity would be ∼4× so- Figure 1. Top panel: Carbon isotope ratios as a function of stellar mass at solar metallicity for the sets of yields indicated in the inset. Bottom panel: As above, but sulphur isotopes. In both panels, the Type Ia supernova (SNe Ia) values were derived from Nomoto et al. (1997). The grey horizontal lines correspond to the various observational constraints described in § 3.3. Figure 2. Titanium isotope ratios as a function of stellar mass at solar metallicity for the sets of yields indicated in the inset. In all panels, the Type Ia supernova (SNe Ia) values were derived from Nomoto et al. (1997).   . Evolution of carbon and sulphur isotope ratios in the solar neighbourhood (r = 8.5 kpc) using the KTG IMF for the Type II supernovae (SNeII) models of WW95, CL04, and K06. For the carbon isotopes: The asterisk, arrow, and plus sign represent the solar value from RM03, the high-redshift Levshakov et al (2006) lower limit, and the Spite et al (2004) halo star data, respectively; the error bar at 5.7 Gyr represents the M06 data, and the error bar at 13 Gyr correspond to the local ISM values, as reported by RM03. For the sulphur isotopes: The asterisk represents the solar system value of Mauersberger et al. (2004), while the error bars at 5.7 and 13 Gyr correspond to the high-redshift data of M06 and the local ISM (Mauersberger et al. 2004), respectively. lar; such an extreme value seems unlikely for a fairly average looking late-type spiral at high-redshift. 4 We note that neither scenario appears consistent with the low 12 C/ 13 C seen in the local halo stars within the Milky Way (i.e., the t=1 Gyr datum in Fig. 7). To try and reach such low values, we examined a range of halo infall timescales and disk infall "delays", for example, having a 5 Gyr delay between the first and second infall phase. All of these attempts to reach the low values seen in halo stars were to no avail, however, and the models could not be made to reproduce the observational data. This is not completely surprising, as none of the yields used in this work have ratios as low as 20, as shown in Fig. 1, other than the 5M⊙ value of KL07. Therefore, there is no combination of parameters in this work that would lead to such low values, and the answer must lie in additional physics. One such solution, and perhaps the most likely, is the idea of rotationally-induced mixing at low-metallicity, as demonstrated by Chiappini et al (2008), to which we refer the reader.

Titanium Isotopes
Until recently, the chemical evolution of titanium isotopes has been more of an academic exercise than an experimentally-driven one, in the sense that observational constraints outside the pre-solar nebula did not exist (e.g. see the lack of data in TWW95; figure 28). The classical TWW95 models suggested that all of the titanium isotopes Figure 6. Comparison between the evolution of carbon isotope ratios using the WW95 yields (without novae: solid curve), for the template model described here, alongside the RM03 models both with (dashed curve) and without (dotted curve) the inclusion of a novae contribution. An important difference between the RM03 curves shown here, and those in the corresponding figure 1a of RM03, is that the models have not been re-scaled a posteriori to match the solar neighbourhood data. Contemporary low-and intermediatemass stellar yields, such as those of KL07, obviate the need for any such "re-calibration". Symbols are as in Fig. 5.   Figure 7. The evolution of carbon and sulphur isotope ratios in the solar neighbourhood using a single power-law IMF slope of 0.9 (cf. slope of 1.35 for Salpeter 1955) for the SNeII models of WW95 and CL04. The metallicities compared to the solar value for each of the WW95 models used are also shown. Symbols are as in Fig. 5. were underproduced with respect to the solar values, with 47 Ti and 50 Ti being particularly problematic. Two developments over the past decade make it timely to revisit this issue: (1) dual-infall models such as RM03 and our own have supplanted the simpler monolithic-like collapse models of TWW95, and (2) recent observational data from Chavez (2008) have, for the first time, provided stellar values (outside the pre-solar nebula) against which to confront models.
In Fig. 8 Figure 8 (bottom panel) again shows the dual-infall model predictions as well as the ratios observed in the 11 disc and halo stars from Chavez (2008). We can identify three interesting conclusions from our preliminary analysis of this dataset: (i) the underproduction of 50 Ti claimed by TWW95 appears to be, in large part, an artifact of the single-infall model; (ii) 47 Ti remains problematic; whether this reflects an important missing nucleosynthetic source from our chemical evolution models, such as helium detonation in sub-Chandrasekhar mass Type Ia supernovae, remains unclear; (iii) most importantly, our models predict a positive correlation between metallicity and [ 46,47,49,50 Ti/ 48 Ti] over the range of metallicity sampled by the observations, while the data are more suggestive of a lack of correlation (or even a slight anti-correlation). If confirmed, this would be very difficult to understand within the context of existing galactic chemical evolution models.
We have explored the evolution of carbon, sulphur, and titanium isotopes in both the local-(Milky Way) and highredshift Universe, using a suite of chemical evolution models generated with GEtool. We examined the need for a novae contribution to explain the carbon isotope patterns in the Milky Way, the evidence for a massive star-biased IMF at high-redshift based upon sulphur isotope ratios, and the impact of new observations of titanium isotope patterns in nearby stars upon our picture of galactic chemical evolution. We have found: • In contrast to earlier studies, the necessity for a significant contribution of 13 C from novae is ameliorated when employing contemporary models of asymptotic giant branch stars. 6 • A massive star-biased IMF at high-redshift results in a significant decrease in the predicted 12 C/ 13 C and 32 S/ 34 S, consistent with those observed, but at the expense of predicting highly super-solar metallicity in otherwise normallooking spiral galaxies.
• Earlier studies which suggested that 50 Ti was significantly underproduced were, in part, led to this conclusion by the adoption of older monolithic-style collapse models of galactic evolution. Our dual-infall model eliminates these apparent problems, although it remains true that 47 Ti is problematic (i.e., underproduced).
• Our chemical evolution models predict a positive correlation between trace-to-dominant titanium isotope ratios and metallicity, while the data is more suggestive of a lack of correlation.