Oxygen has three stable isotopes, 16O, 17O, and 18O, which are present on the Earth in fractional abundances of about 99.76, 0.038, and 0.2%, respectively. In geochemistry and cosmochemistry, O-isotope ratios are commonly expressed as δ-values, fractional deviations from standard (normalizing) ratios, and data are often presented in a “three-isotope” plot, e.g., with δ18O on the abscissa and δ17O on the ordinate. Delta-values can be defined either as linear (e.g., δ18O = 103 × [(18O/16O)/(18O/16O)0− 1], where “0” denotes a normalization ratio) or logarithmic (e.g., δ18O′ = 103 × ln[(18O/16O)/(18O/16O)0]) functions of the normalized isotope ratio. Logarithmic delta-values have the advantage that data with the same 18O/17O ratios lie along a line of slope-one in a three-isotope plot and were used by Young et al. (2011) in their discussion of O-isotope GCE and the solar composition. However, they are not commonly used in the astronomical literature, and for large deviations from zero their quantitative meaning is not always intuitively obvious. For example, a linear δ18O value of −1000 corresponds to 18O/16O = 0, but a logarithmic δ18O’ value of −1000 means 18O/16O = 0.37 times the normalizing ratio. Thus, for clarity in this article, where we present data as logarithmic delta values, we also show the normalized isotopic ratios without the natural logarithm applied.
Models of oxygen isotopic abundances in our galaxy have roots in nearly a half-century of theoretical studies of GCE (Audouze and Tinsley 1976; Prantzos et al. 1996); one recent review is by Meyer et al. (2008). As stars evolve and contribute newly synthesized nuclei to the ISM, the galactic inventories of heavy elements (metallicity) increase. However, because different isotopes are made by different nuclear processes in different types of stars, which may evolve on very different timescales, isotopic and elemental ratios also vary with metallicity. GCE theory distinguishes between primary16O, whose synthesis in stars is independent of metallicity, and secondary17O and 18O, which require pre-existing C, N, and/or O. Simple analytical GCE models (Clayton 1988) predict that the ratio of a secondary to a primary isotope, e.g., 17O/16O, should increase linearly with metallicity, and the ratio of two secondary isotopes, e.g., 18O/17O, should remain constant. In a three-isotope plot, the O-isotopic composition of the galaxy should evolve along a slope-one line toward the upper right. This simplest analysis is based on the assumption that all three O isotopes are produced in short-lived massive stars. If this accurately describes GCE, then the average 18O/17O ratio of the ISM has not changed since 4.6 Gyr ago, and the unusual 18O/17O ratio of the Sun compared with the present-day ISM (Fig. 1), requires another explanation, e.g., local pollution of the Sun’s parental cloud by the ejecta from one or more supernovae (Prantzos et al. 1996; Young et al. 2011).
Figure 1. Ratio of 18O to 17O measured in molecular clouds (Penzias 1981; Wouterloot et al. 2008) and young stellar objects (Smith et al. 2009) as a function of distance from the galactic center. The Wouterloot et al. (2008) study included multiple measurements of some MCs (e.g., Orion KL), and for these we combined the data into a single, averaged data point here so as not to bias the overall distribution. Because estimates of the locations of the studied MCs has changed greatly since 1981, Penzias’s data are only shown for clouds in common with the Wouterloot et al. (2008) study and the galactocentric distances are shifted to the modern values. Gray dashed line indicates schematically a possible gradient observed in the data set of Wouterloot et al. (2008). YSO data are plotted with 2-σ error bars. The reported uncertainties on the individual MC measurements are typically smaller than 10% relative. The 1-σ uncertainty on the solar system value (5.2 ± 0.2; Young et al. 2011) is smaller than plot symbol.
Download figure to PowerPoint
Radio observations of molecular clouds in the galaxy over the last few decades have supported the view outlined above of O-isotopic GCE (Wannier 1980; Penzias 1981; Wilson and Rood 1994). Namely, there is evidence for gradients of increasing 17,18O/16O ratios with decreasing distance from the galactic center, but relatively constant 18O/17O ratios throughout the galaxy. Since the overall metallicity of the galaxy is observed to increase toward the galactic center, this supports the idea that 16O is primary and 17O and 18O are secondary. However, the radio measurements entail relatively large uncertainties, especially for most measurements of 17O/16O and 18O/16O, which require measuring double-isotopic ratios (e.g., 12C18O/13C16O) and making assumptions regarding 12C/13C ratios.
Figure 1 shows the 18O/17O ratios of molecular clouds throughout the galaxy from the pioneering study of Penzias (1981) and the much more recent work of Wouterloot et al. (2008). Both studies are consistent with this ratio being basically constant across much of the Galaxy and lower on average than the well-defined solar ratio. Moreover, independent infrared measurements (Smith et al. 2009) of O-isotopes in two young stellar objects (also shown on Fig. 1) show that they have similar 18O/17O ratios to those derived from the radio data, demonstrating that the offset between the solar ratio and typical ISM compositions is not due to some systematic uncertainty in the reduction of the radio data. There is a systematic offset toward higher 18O/17O ratios in the more recent radio work, most likely reflecting refinement of observational and analysis techniques over several decades. The Wouterloot et al. (2008) data show that at least one MC at the same galactocentric distance as the Sun has an 18O/17O ratio close to solar, hinting that cloud-to-cloud scatter may be as important as overall average trends. More importantly, the Wouterloot et al. (2008) study finds significantly lower 18O/17O ratios than earlier work in the Sagittarius-B MC close to the galactic center and higher ratios in the outer Galaxy. While clearly needing additional investigation, this suggestion of a radial gradient is qualitatively consistent with the evolution of the Galaxy toward more 17O-rich compositions with time.
To derive a galactic gradient in 18O/16O ratio, Young et al. (2011) combined the 13C16O/12C18O line ratios reported by Wouterloot et al. (2008) with an independent estimate of the galactic gradient in 12C/13C ratios from Milam et al. (2005). Caution is of course necessary when combining disparate data sets, each with its own uncertainties and potential biases, in this way, as it can be very difficult to realistically estimate the uncertainty in the result. For example, Wouterloot et al. (2008) report 13C16O/12C18O line ratios for two different CO rotational transitions (J = 1–0 and J = 2–1), and these data sets indicate significantly different galactic gradients in 18O/16O even if combined with the same galactic 12C/13C gradient (Fig. 2). In fact these authors state that chemical fractionation, not a trend in bulk galactic isotope composition, is a likely explanation for the observed 13C16O/12C18O gradients.
Figure 2. Ratios of 16O to 18O of molecular clouds derived from radio observations of transitions in different molecules plotted against distance from the galactic center. CO data were derived following Young et al. (2011) by combining line ratios of 13C16O/12C18O reported by Wouterloot et al. (2008) with the galactic 12C/13C gradient derived by Milam et al. (2005) from CO observations. One-sigma error bars are estimated from the scatter in CO line ratios and the reported uncertainty in the 12C/13C gradient. Formaldehyde data are from the compilation of Wilson and Rood (1994) and OH data are from Polehampton et al. (2005). The thick black line is the gradient derived by Wilson and Rood (1994); the thin solid curve is an exponential fit to the CO J = 1–0 data and dashed curve is an exponential fit to CO J = 2–1 data.
Download figure to PowerPoint
In Fig. 2, we compare the galactic gradients in 16O/18O derived as described above from the studies of Wouterloot et al. (2008) and Milam et al. (2005) with those compiled by Wilson and Rood (1994), from double-isotope measurements of formaldehyde, and with those measured by Polehampton et al. (2005), from OH measurements. The latter study is important as it is a direct measurement, not requiring a correction for any other isotopic ratios (i.e., 12C/13C for the CO and H2CO studies). The gradient derived from the CO J = 1–0 data is clearly steepest, whereas the OH data show little evidence for any gradient at all. A striking feature of Fig. 2 is the large scatter in the data, both between different studies, and for different clouds within a single study. For example, the CO J = 1–0 data show a factor of approximately three range in 16O/18O ratio for clouds close to the solar galactocentric radius. Such scatter may be real; there is mounting evidence for large scatter in metallicity among stars of the same age in the solar neighborhood (Edvardsson et al. 1993; Nordström et al. 2004) and this could translate directly into O-isotopic variations assuming that the O-isotopic ratios scale with metallicity via GCE processes. We emphasize that a spatial gradient (with galactocentric radius) need not imply the same evolution in time: The ISM that we observe is everywhere comparatively “young” compared with stars and the Galaxy itself.
Moreover, the evident scatter in O-isotopic ratios for relatively nearby MCs has other important implications for our discussion. First, if it is real, it indicates that it is too simplistic to consider that a single O-isotopic composition pertains to even relatively nearby and contemporaneous regions of the Galaxy. Second, the data clearly show that the local ISM 16O/17O and 16O/18O ratios are highly uncertain by at least a factor of two, far greater than the ±20% assumed in the SN enrichment model of Young et al. (2011).
Noncanonical Models of GCE: A Two Phase ISM
Nearly all models of O-isotopic GCE adopt the instantaneous mixing approximation (IMA), which assumes that the ISM consists of a single homogeneous reservoir. The IMA cannot be absolutely correct; winds from AGB stars and SN ejecta are hot and tenuous compared with molecular clouds and will not mix efficiently, and SNe drive bubbles and “galactic fountains” that eject hot gas into the halo. Gilmore (1989) suggested that molecular clouds may be self-enriched by the very massive stars they spawn. Malinie et al. (1993) proposed that star formation was both heterogeneous and bursting (episodic), and different parts of the galactic disk mix only after 0.1–1 Gyr had elapsed. Thomas et al. (1998) relaxed the IMA, dividing the ISM into separate “active” (cool, dense, and star-forming) and “inactive” (hot, less dense) phases. Stellar ejecta first enter the inactive phase and only enter the “active” phase after a characteristic time τI. They found that τI = 0.1 Gyr gave only a slight improved description of the metallicity evolution of the galactic disk compared with an IMA, and only for the first few Gyr. A value of τI = 1 Gyr was ruled out by the existence of very old metal-rich stars. Spitoni et al. (2009) also examined time delays of up to 1 Gyr due to the ejection of gas in galactic “fountain” and the time for it to return by cooling. They found that such delays have only a small effect on elemental chemical evolution and radial abundance gradients. They also found that the use of highly metallicity-dependent yields of WW95 does induce an effect, but this choice also predicts elemental ratios (i.e., [O/Fe]) that are contradicted by observations. Note that the IMA is independent of assumptions concerning radial transport and mixing in the galactic disk (Schönrich and Binney 2009).
More subtle influences of heterogeneity and noninstantaneous mixing might be detected in O-isotopic compositions. Gaidos et al. (2009) used a two-phase model analogous to Thomas et al. (1998) and consisting of a diffuse ISM in which no star formation occurs, plus star-forming molecular clouds. Their model produces two effects: an O-isotopic offset between the two reservoirs, and trajectories in a three-isotope plot that deviate from the canonical slope-one line, with a general trend to lower 18O/17O ratios with time. The latter is explained both as a result of depletion of the ISM of 18O as a result of its preferential incorporation into low-mass stars born in self-enriched star-forming regions, and preferential injection of 17O-rich AGB ejecta into the diffuse ISM. To more fully illustrate and explore the outcomes of these effects on solar and ISM 18O/17O ratios, we consider here some simple two-phase ISM models of GCE.
We model the ISM as consisting of two phases: a diffuse, inactive ISM (phase “I”) and denser, star-forming regions (SFRs, or phase “S”). No mass is lost, except through the formation of long-lived stars and stellar remnants. In steady-state this loss is balanced by the infall of primordial, metal-free gas into the diffuse ISM. The governing equations for nucleosynthetic isotopes in this two-phase model are
where Σ is a mass surface density (constant in steady-state); xi and yi are the mass fractions of the ith isotope in the diffuse ISM and SFRs, respectively; pi and qi are the respective rates of injection of the isotope by both SN and AGB stars; F and D are the rates of formation and dissipation of SFRs; and R is the total rate of star formation. This model differs from that of Thomas et al. (1998) in that SN ejecta is assumed to be first added to the residual gas of SFRs before it is dispersed back into the diffuse ISM.
In steady state, mass balance in the SFR population requires F=D+R. If we assign a characteristic lifetime τs to SFRs before they are recycled back into the diffuse ISM, then D = Σs/τs and F = Σs(1 + s)/τs where s = Rτs/Σs is the star formation efficiency. Then,
to which the solutions are
Thus, in steady state the isotopic composition of SFRs is displaced by an amount qis/[R(1 + s)] with respect to the diffuse ISM. This offset represents contamination by stars that inject isotopes into SFRs after a lifetime shorter than τs. If τs = 10 Myr (Williams et al. 2000), this includes O stars more massive than 15 M⊙. Such massive stars produce comparatively little 17O and 18O and thus the displacement of the SFR trajectory relative to the ISM trajectory in a three-isotope plot is primarily along the slope-one line. The low efficiency of star formation (s ∼ 0.1) (Williams et al. 2000) also means that the amplitude of the displacement is small. This solution essentially assumes that the typical SFR undergoes the local SN enrichment invoked by Young et al. (2011) to explain the anomalous O-isotopic composition of the Sun, but as we shall see, the average effect on 18O/17O is smaller than that predicted by those authors. Is this solution stable? The eigenvalues of the system of differential Equations 1 and 2 are
where m = Σs/ΣI = 1. They are real and negative for all values of m and s, thus the steady-state solution is stable; transients decay on short (τs) and long (τI ≈ τs/(mf)) timescales.
Figure 3 plots solutions using IMF-integrated yields of oxygen isotopes for four different values of metallicity. The assumption of quasi-steady-state holds if the changes in metallicity or star formation rate are small compared with the lifetime of the longest-lived significant contributors to O inventories (perhaps about 2 Gyr) and must break down at early epochs and very low metallicities (see below). For the adopted set of yields (listed in the caption to Fig. 3), the predicted 18O/16O ratio at solar metallicity is within 7% of the solar value; however, 17O is overpredicted by a factor of two (Kobayashi et al. 2011, hereafter K11, arrived at a like result with a similar set of yields). A discrepancy of this magnitude is often assigned to uncertainties in the yields of 17O.
Figure 3. Predicted quasi-steady-state oxygen isotope ratios in a two-phase model of the ISM for metallicities of 10−4 (not seen), 0.01, 0.1, and 1Z⊙. Gas in star-forming regions (solid line plus stars) collects ejecta from stars with lifetimes <10 Myr; all longer-lived stars contribute to the diffuse ISM (dashed line plus circles). The dotted line is the expected slope-one line of galactic chemical evolution. The inset shows details around the solar-metallicity calculations in units of ‰. The calculations used Equations 5 and 6, a star formation efficiency of 10%, and IMF-integrated yields of 16O, 17O, and 18O based on the IMF of Kroupa (2002), the AGB yields of Karakas (2010), the “super-AGB” yields of Siess (2010), the yields of Kobayashi et al. (2006) with the corrections of Kobayashi et al. (2011) for SN progenitors between 12 and 40 M⊙, and the wind plus ejecta yields of Portinari et al. (1998) for less massive or more massive SN progenitors.
Download figure to PowerPoint
The predictions are plotted as logarithmic δ-values normalized to the calculated composition of solar-metallicity SFRs. This procedure essentially normalizes out the uncertainty in the absolute predictions and allows relative trends to be examined (Timmes and Clayton 1996). The main effect is that SFRs are 16O-enriched with respect to the diffuse ISM (inset of Fig. 3), and the diffuse ISM is only slightly (about 1.5%) 17O-rich compared with SFRs. The latter difference is far smaller than the observed difference in 18O/17O between the solar system (5.2) and ISM (about 4) and stands in contrast with explanations based on self-contamination in OB associations (e.g., Prantzos et al. 1996; Young et al. 2011). One major difference between these calculations and those of Gaidos et al. (2009) is the use of revised AGB star yields from Karakas (2010). However, larger O-isotopic variations could occur in individual SFRs due to the stochastic effects of small numbers of SNe in smaller molecular clouds.
Production rates pi and qi of secondary isotopes such as 17O and 18O increase with metallicity, so xi, yi, and the offset between them will also increase with metallicity. If the yields of the two isotopes depend on metallicity in the same way, the trajectories of the ISM and SFRs in a log-δ three-isotope plot will be parallel and have a slope of one (pure secondary/secondary behavior). In Fig. 3 the slopes deviate from unity, especially at low metallicity. This is because the metallicity-dependence of the production of one heavy isotope differs slightly from the other. Although the ideal nucleosynthetic production of a secondary isotope is proportional to the abundance of the primary isotope, stars of different masses produce the isotopes in different relative abundances and differences can arise that depend on metallicity. For example, metallicity partially controls the opacity of stellar atmospheres and hence stellar evolution and mass loss by winds. If 17O and 18O are preferentially produced during different phases of stellar evolution, differences in metallicity will produce differences in the ratio of 17O to 18O yields. For example, the mass ejected in the 17O-enriched winds of AGB stars increases with metallicity while no analogous enhancement occurs for 18O-enriched SN ejecta. Figure 4 shows that the metallicity dependencies of the IMF-integrated yields of 17O and 18O are neither identical nor directly proportional to each other, thus at least small departures from slope one are expected. These by themselves, however, appear insufficient to explain the difference between the solar system and the present-day ISM (Fig. 3). Moreover, trajectories have slopes smaller than one, at least near solar-metallicity, indicating a gradual increase in 18O/17O with time, opposite in sign to that required to explain the solar-present ISM difference.
Figure 4. IMF-integrated yields of 17O and 18O for metallicities of 10−4 (lower left), 0.01, 0.1, and 1Z⊙, showing that the two secondary isotopes do not exactly track each other. Solid lines are the contributions of stars more massive than 15 M⊙, which have main-sequence lifetimes <10 Myr and thus (hypothetically) contribute to star-forming regions, while dashed lines are for less massive, longer-lived stars that contribute to the diffuse ISM. The standard model (triangles) uses the same sets of yields as in Fig. 3. Lines with crosses use the SN yields of Woosley and Heger (2007) at solar metallicity, and circles use the SN yields of Woosley and Weaver (1995) for progenitors up to 40 M⊙.
Download figure to PowerPoint
Equation 6, summed over isotopes, also predicts the absolute steady-state abundance of oxygen in solar-metallicity SFRs, i.e., equal to the total IMF-averaged yield (p + q)/R. Our adopted set of yields over-predicts the solar value (Asplund et al. 2009) and clearly GCE is not near steady state, i.e., the inflow of fresh gas exceeds the rate at which mass is locked up in stellar remnants. Moreover, some gas in SFRs is fragmented and heated by SNe and Wolf-Rayet stars into hot, tenuous superbubbles and fountains and may permanently escape the disk. We can incorporate this effect in the model by multiplying the last term in Equation 1 by a retention factor f < 1. The steady-state solution becomes:
For s = 0.1, small values of f will produce a diffuse ISM that is 17O-enriched with respect to SFRs (Fig. 5). However, the observed offset is between the Sun and current SFRs, the diffuse ISM not being observationally accessible. GCE itself does not depart significantly from a slope-one trajectory in a three-isotope plot and we conclude that a heterogeneous ISM cannot, by itself, explain the observed offset in 17O/18O.
Figure 5. Predicted galactic isotopic evolution as in Fig. 3, except only 10% of mass in star-forming regions is returned to the diffuse ISM. Unlike the closed model, this reproduces the total oxygen abundance and the 18O/16O ratio of the Sun (i.e., a star-forming region at solar metallicity). This produces a 30% elevation in the diffuse ISM 18O/17O ratio relative to the Sun. However, the observed offset is between the Sun and molecular clouds and protostellar cores, i.e., star-forming regions; there is too little CO in the diffuse ISM to measure isotopic abundances.
Download figure to PowerPoint
Time-Dependent Star Formation
If the secondary isotopes are produced in different amounts in stars of different masses, which have different main-sequence lifetimes, variation in the rate of star formation can result in an uneven contribution to the inventory of oxygen isotopes (Gaidos et al. 2009). Early calculations of O-isotope GCE, e.g., Timmes et al. (1995), assumed that all three O-isotopes are produced in massive stars based on the nucleosynthesis yields available at the time and predicted O-isotopic evolution that closely follows the slope-one behavior expected from the simplest GCE expectations. However, subsequent refinement of nuclear reaction rates has shown that 17O is not efficiently made in massive stars (Blackmon et al. 1995; Woosley and Heger 2007) and must largely come from AGB stars and/or classical novae. Both types of sources evolve on longer timescales than do massive stars, indicating that the production of 17O is delayed with respect to the 18O made by massive stars. The 17O/18O ratio would thus be expected to increase with time as longer-lived stars evolve away from the main sequence and eventually add 17O-enriched ejecta to the ISM, causing deviations from a slope-one trajectory in a three-isotope plot.
The AGB yields adopted here (Karakas 2010) indicate that the largest production of 17O occurs in stars of >2 M⊙, which have lifetimes <2 Gyr (Fig. 6); departure from quasi-steady-state will scale with changes in star formation rate or metallicity over such an interval. In the GCE model of Gaidos et al. (2009), a constant gas infall rate combined with the Schmidt–Kennicut formula for star formation rate (Fuchs et al. 2009) gives rise to a decreasing 18O/17O ratio with time, one sufficient to explain the Solar-ISM difference (Fig. 7). The calculations of Kobayashi et al. (2011), which included up-to-date nucleosynthesis yields for both AGB stars and supernovae, also predict an increase (about 5%, Figs. 7 and 8) in 17O/18O over the past 4.5 Gyr, but not enough to explain the Solar-ISM difference (about 25%, Fig. 1). Note that neither of these models makes predictions for O-isotopic gradients across the galactic disk, which could be compared with the observations in Figs. 1 and 2.
Figure 6. Yields of 17O (solid) and 18O (dashed) per unit log interval of time in Gyr versus main-sequence lifetime of the source stars. This plot illustrates that, while most 18O is injected within 20 Myr of star formation, a large fraction of 17O is not introduced into the ISM until after about 1 Gyr. The same yields as Fig. 3 were used, except those of Portinari et al. (1998), which were excluded.
Download figure to PowerPoint
Figure 7. Predicted O-isotope ratios from three GCE models, normalized to the values predicted at solar metallicity. Ellipse shows present-day local ISM based on observations of molecular clouds (estimated from Figs. 1 and 2). Open diamonds are from the model of Gaidos et al. (2009) and circles are from the model of Kobayashi et al. (2011); symbols are plotted at 500 Myr intervals. These models deviate from canonical slope-one evolution (solid line) expected if secondary isotopes 17O and 18O are produced at identical rates in the Galaxy. Thick curve is the predicted single-phase GCE in a hypothetical “star burst” scenario in which 3% of the mass of the disk instantaneously forms a generation of stars from nearly solar metallicity gas around 4.6 Gyr ago. Subsequent isotopic evolution is dictated solely by the postmain sequence evolution of these stars and ejection of O-isotope-enriched gas. The instantaneous mixing assumption (IMA) is adopted in the calculations and the Sun forms 9 Myr after the main burst and before AGB stars begin ejecting 17O-rich material into the ISM. The choice of 3% and 9 Myr is purely to show that that the magnitude of the excursion is sufficient to produce the observed offset between the Sun and the present star-forming regions/ISM, although the scenario itself is unrealistically simplistic.
Download figure to PowerPoint
Figure 8. Predicted evolution of O-isotopic ratios in the solar neighborhood in the last 5.6 Gyr from three GCE models: TWW95: model of Timmes et al. (1995); 17O/16O and 18O/16O trends lie on top of each other, indicating “slope-1” behavior on an O three-isotope plot. K11: Model of Kobayashi et al. (2011); 17O/16O evolves slightly more rapidly since solar birth than 18O/16O due to delayed input of 17O from AGB stars (Fig. 7). Nova-1 and Nova-2: GCE model of Romano and Matteucci (2003) assuming all 17O production is from novae. Nova-1 is case where nova 17O production is assumed to be secondary, Nova-2 corresponds to primary nova production of this isotope.
Download figure to PowerPoint
The effect of time delay can be modeled by decomposing the injection of the ith isotope into the ISM into instantaneous and delayed parts proportional to the present and past rates of star formation:
where τ is the time delay between the formation of the stars and when they eject the isotope. The first factor in the time-delay term of the RHS accounts for changes in metallicity; the second accounts for changes in the star formation rate. If we assume that (1) time delay affects 17O production but not 16O or 18O production, (2) the mean metallicity of the ISM and SFRs has remained constant over the past 5 Gyr (i.e., creating the “G-dwarf problem”), and (3) that fractional changes in isotopic composition are <<1, then Equation 6 can be linearized to give an expression for the fractional change in the 17O/18O ratio of SFRs since the Sun formed:
where f is the fraction of 17O produced in the delayed component, and the derivatives are in Gyr−1. The difference depends on the curvature of the logarithmic star formation rate and will be positive (i.e., present ISM enriched in 17O) only if the star formation rate is decelerating (convex). The reason for this can be understood as follows: a decreasing rate of star formation means that the ISM and SFRs will receive 17O-rich gas from longer-lived, earlier-forming stars that are more numerous than the shorter-lived, 18O-contributed stars that formed more recently. However, to produce a positive difference in the ISM or SFRs between the present and past (i.e., 4.6 Gyr ago), that decrease must be larger now, i.e., star formation is decelerating.
The magnitude of the excursion depends on the rate of deceleration but even if most 17O production is delayed (f∼ 1), Equation 11 shows that the star formation rate must change on a timescale of no more than a few Gyr to explain the approximately 25% ISM-solar difference; a constant or exponentially decreasing star formation rate will not affect the 17O/18O ratio. The nominal model of Gaidos et al. (2009) assumes a constant rate of gas infall and the Schmidt–Kennicut law and produces a convex star formation history. The estimated rate of star formation approaches an asymptotic value in the last few Gyr. Star formation in the Kobayashi et al. (2011) is also convex, but gas infall peaks 8.5 Gyr ago, and star formation does not crest until about 2 Gyr ago (K11 Fig. 11). These star formation histories could partially explain the origin of the 17O/18O offset predicted by the models.
Similar effects may also occur if the metallicity of the ISM has substantially changed in the past 5 Gyr. If the yield of 17O, as a secondary isotope, scales with metallicity then the change in 17O/18O due to metallicity evolution and the time delay of 17O production is approximately:
Again, positive evolution in 17O/18O requires decelerating change (in metallicity), which is plausible if the age–metallicity relationship is flat at recent time (the so-called “G dwarf problem”). Even if f∼ 1, a deceleration of about 0.07 dex/Gyr is required over the past 4.6 Gyr to explain the observed offset. Thus, both decelerations in star formation rate and metallicity evolution may have contributed to the isotopic offset between the Sun and the ISM.
Sufficiently rapid changes in star formation rate and metallicity could occur during episodes of enhanced star formation (“starbursts”), specifically around the time of the Sun’s formation. Clayton (2003) invoked the merger of a metal-poor satellite galaxy with the Milky Way 5–6 Gyr ago and a subsequent burst of star formation to explain the statistics of silicon isotopes in presolar SiC stardust grains. Such an event would also leave its mark on oxygen isotopes (Clayton 2004); elevated levels of star formation immediately prior to the Sun’s formation could enrich the galactic disk with 18O before the subsequent postmain sequence evolution of AGB stars would return the isotopic trajectory to the steady-state situation (Gaidos et al. 2009, Fig. 2).
Figure 7 plots an illustrative but unrealistically simple case where a single generation of stars forms instantaneously from a gas of nearly solar metallicity and 4 Myr later begins adding oxygen back to the gas. No star formation occurs subsequent to the burst. The IMA is used and we do not differentiate between the diffuse ISM and SFR phases. The yields are the same as in the previous calculations. To map GCE onto a three oxygen-isotope plot it is also necessary to specify the fractional mass processed by this generation of stars and the epoch at which the Sun forms relative to the starburst. Values of 3% and 9 Myr, respectively, cause the oxygen isotopic composition of the ISM to evolve to its current 17O-rich position 4.6 Gyr after the formation of the Sun. In the first 10 Myr after the starburst, the isotopic trajectory of the ISM is toward 16O-rich conditions as the most massive stars explode. During the subsequent 15 Myr the ISM is enriched with 17O and 18O by SN of 12–18 M⊙ progenitors. At around 25 Myr, “super” AGB stars begin evolving off the main sequence and ejecting 17O-enriched gas. This phase continues as “normal” AGB stars continue to add 17O-rich material and is largely complete by 2 Gyr after the starburst. Obviously, the IMA should be relaxed and a burst must be superposed on a background level of star formation and metal-poor gas infall. However, we find that these trajectories are sensitive to yields that are very uncertain and from phases of progenitors which have not yet been established. Specifically, nucleosynthetic models predict that the yields of very massive (>40 M⊙) stars are sensitive to uncertain parameters including the degree of fallback onto the remnant black hole or neutron star, the effects of mass loss, and the effects of rotation (Woosley and Weaver 1995; Meynet et al. 2010), and the existence of “super” AGB phases (7–12 M⊙) has yet to be established. Our calculations show that the 17O-richness of the ISM could be explained by the delayed contribution of AGB stars to the ISM, but a quantitative test of any model must await a sounder foundation of yield calculations.
Contribution of Classical Novae to O-Isotope GCE
Classical novae may also be important but poorly quantified contributors to “delayed”17O. Novae are thermonuclear explosions occurring on white dwarfs (WDs) due to accretion of H and He from less-evolved binary companions. Overall, they eject relatively little mass into the ISM and hence are unimportant for the GCE of most isotopes. However, the high-temperature H-burning that occurs is predicted to produce very large amounts of the light isotopes 7Li, 13C, 15N, and, of importance here, 17O, as well as the radioactive nuclei 26Al and 22Ne (José and Hernanz 1998; Starrfield et al. 1998), and thus may play an important role in the galactic production of all of these (Romano and Matteucci 2003). Because they require both the evolution of the low- or intermediate-mass parent star to the WD stage and about 1–2 Gyr of WD cooling to ensure a strong nova outburst (Romano et al. 1999; Romano and Matteucci 2003), novae are expected to first make a significant contribution to GCE several Gyr after AGB stars. This may lead to a delayed elevation of the galactic 17O/18O ratio, one that might help explain the difference between the solar composition and that of the present ISM.
Novae are rarely included in models of GCE due both to their unimportance in the production of most elements and to the poor understanding of many key parameters, including the fraction of WDs in nova systems throughout galactic history, the total mass ejected by novae and, critically, the nucleosynthetic yields. To our knowledge, there is only one study addressing the role of novae in the GCE of O isotopes (Romano and Matteucci 2003). These authors incorporated novae into a standard (i.e., single-phase ISM) numerical GCE model and used the observed present-day nova rate in the Galaxy to constrain a parameterization of the nova rate in the past. They considered a range of nucleosynthetic prescriptions for low-mass stars, supernovae and novae. Unfortunately, 18O was not included in this model, so we cannot directly infer predictions for the GCE of the 17O/18O ratio. However, since novae are not thought to be a significant source of 18O (José et al. 2012), we can gain some insight by comparing the predicted 17O/16O ratio evolution of Romano and Matteucci (2003), including novae, to other models of 18O/16O evolution calculated with GCE models not including novae.
Figure 8 compares the time evolution of the O isotopic ratios in the solar neighborhood predicted by several GCE models. The isotopic trends have all been “renormalized” so that they are forced to have solar composition at the time of solar birth. As discussed at length by Timmes and Clayton (1996), the renormalization procedure compensates for the large uncertainties in predicted absolute isotopic abundances, and allows one to compare relative isotopic trends with each other and with high-precision data. On this plot, canonical slope-one evolution corresponds to identical trends for 17O/16O and 18O/16O, as observed in Fig. 8 for the model of Timmes et al. (1995), which was based on pure secondary synthesis of 17O and 18O in massive stars and for which the curves are indistinguishable on the plot.
Two predictions of Romano and Matteucci (2003) for the evolution of 17O/16O are shown in Fig. 8. Both models assume that 17O is produced solely by novae, so these calculations provide a limiting case since some 17O is certainly made by AGB stars. The lower trend (“Nova-2”) assumes that 17O synthesis in novae is primary (all novae have the same 17O yield, regardless of the metallicity of the progenitor stars) whereas for the upper one (“Nova-1”) the nova yields are scaled according to metallicity. Both models predict a stronger increase in 17O/16O since solar birth than GCE models that do not include novae. The effect of novae on the evolution of the 17O/18O ratio can be estimated by comparing these trends to the plotted 18O/16O trends. Taking the Timmes et al. (1995) calculation as representative of 18O/16O evolution and the “Nova-1” GCE trend for 17O/16O gives the most extreme increase in 17O/18O since solar birth: about 70%. In contrast, considering the “Nova-2”17O/16O trend and the Kobayashi et al. (2011)18O/16O trend would suggest a much smaller increase of about 10%. Thus, the limited modeling work done to date suggests that nova production of 17O over the past 4.6 Gyr of GCE may readily explain the observed difference in 17O/18O between the Sun and local ISM, but this depends on many uncertain details and it will require more investigation. Moreover, there is no quantitative model prediction for the shape of a 17O/18O gradient across the Galaxy expected if novae are primary 17O producers, and therefore the molecular cloud observations in Fig. 1 are not yet a good diagnostic test.