ASTRONOMICAL OXYGEN ISOTOPIC EVIDENCE FOR SUPERNOVA ENRICHMENT OF THE SOLAR SYSTEM BIRTH ENVIRONMENT BY PROPAGATING STAR FORMATION

We find that the most likely explanation for the anomalous [¹⁸O]/[¹⁷O] of the solar system is that the Sun and planets formed from molecular cloud material enriched in exogenous ¹⁸O-rich oxygen ejected from stars in a nearby cluster that ended their lives as SNe II. Measurements of the relative abundances of all three stable oxygen isotopes place critical constraints on mixing between normal Galactic oxygen and this ¹⁸O-rich ejecta. We explore this proposition, and its consequences for the origin of the solar system, below.

With three isotopes of oxygen we can compare the isotopic composition of the ISM at the time the Sun was formed with different supernova products in order to identify plausible sources of exogenous oxygen. The oxygen isotopic composition of the local ISM 4.6 Gyr before present can be estimated from ages of between 13.6 Gyr and 10 Gyr for the Galaxy and the 4.6 Gyr age of the solar system. As described above, both [¹⁸O]/[¹⁶O] and [¹⁷O]/[¹⁶O] are ratios of secondary to primary nuclides that have risen linearly, to first order, with time in the Galaxy at nearly constant [¹⁸O]/[¹⁷O]. The rise in the abundances of ¹⁸O and ¹⁷O relative to that of ¹⁶O linearly with time leads to the relationship Δage/age ∼ (Δ[¹⁸O]/[¹⁶O])/([¹⁸O]/[¹⁶O]). We therefore expect both [¹⁸O]/[¹⁶O] and [¹⁷O]/[¹⁶O] ratios in the ISM to have risen by between 35% and 46% in the past 4.6 billion years. More complicated models for oxygen GCE are consistent with this estimate (Prantzos et al. 1996). The ∼350‰–460‰ increase in both ratios can be subtracted from the composition of the present-day ISM to obtain an estimate of the oxygen isotopic composition of the ISM at the time of the formation of the Sun. The precision of this estimate is limited by an uncertainty of at least ∼20% (±200 ‰) in [¹⁸O]/[¹⁶O] and [¹⁷O]/[¹⁶O] for the present-day ISM. With this uncertainty, the present-day nominal [¹⁸O]/[¹⁶O] and [¹⁷O]/[¹⁶O] values of the local ISM (Wilson 1999) are reasonable upper limits for the values for the ISM 4.6 Gyr ago.

Previous work has emphasized that [¹⁸O]/[¹⁷O] of oxygen liberated by the explosion of an SN II varies systematically with the mass of the progenitor star (Gounelle & Meibom 2007). One can illustrate the inputs of individual stars to the composition of stellar ejecta from star clusters using mass fractions over small intervals of the initial mass function (IMF). Starting with the IMF, yielding the number of stars N of mass m

$$\begin{equation} \xi (m) = \frac{{d\!N}}{{dm}} = \beta m^{ - \alpha } \end{equation} \tag{ 1 }$$

integration over some mass interval m_l to m_u (lower to upper) yields for number of stars

$$\begin{equation} N_{m_l \rightarrow m_u } = \beta \int _{m_l }^{m_u } {m^{ - \alpha } } dm \end{equation} \tag{ 2 }$$

and for mass

$$\begin{equation} m_{m_l \rightarrow m_u } = \int _{m_l }^{m_u } {m\;\xi (m)\;dm}. \end{equation} \tag{ 3 }$$

The fraction of mass contained in stars of masses m_l to m_u is then

$$\begin{equation} X_{m_l \rightarrow m_u } = \frac{{\int _{m_l }^{m_u } {m\;\xi (m)\;dm} }}{{\int _{0.08\,M \odot }^{100\,M \odot } {m\;\xi (m)\;dm} }}. \end{equation} \tag{ 4 }$$

From these mass fractions of the total stellar system we obtain mass fractions of oxygen ejected from a generation of star formation by fitting existing models for mass loss as functions of progenitor mass. For stars >8 M_☉ we used the SNe II yields (by yields we mean production and not net yield that results from considering the nuclides remaining in the stellar core) from Rauscher et al. (2002, hereafter RHHW02, including supplemental tables available online) taking into account also the calculations of Woosley & Weaver (1995, hereafter WW95) and Woosley & Heger (2007, hereafter WH07; yield and mass cut tables kindly provided by A. Heger). These studies exhibit large differences in ¹⁷O production. A reduction in the ¹⁷O yield from WW95 to RHHW02 reflects the revision to the destruction rate for that nuclide (Blackmon et al. 1995). A further reduction in the¹⁷O yield from RHHW02 to WH07 resulted from lower initial CNO abundances of the progenitor stars. The mass of oxygen ejected from the larger progenitors (⩾30 M_☉) depends on the choice of the intensity of the SN "piston" employed by RHHW02. For the calculations presented here, we made use of the higher energy piston models. The resulting yields are similar to those of WW95. The oxygen yields as a function of progenitor mass are shown in Figure 4 along with the fits to the calculations used in the present calculations. The fit to the RHHW02 calculations for mass of oxygen ejected relative to mass of progenitor, M_O/M_*, is

$$\begin{eqnarray} \frac{{M_{{\rm O}} }}{{M_* }} &=& 1.0250\times 10^{ - 5} M_*^3 + 5.4290\times 10^{ - 4} M_*^2 \nonumber \\ &-& 1.6617\times 10^{ - 4} M_* - 1.1800\times 10^{ - 3}, \end{eqnarray} \tag{ 5 }$$

where masses are in solar units. For comparison with the SNe II calculations, and for calculating mass fractions of oxygen relative to all of the oxygen injected into the ISM by a stellar cluster, we include the oxygen produced by mass loss from AGB stars for M_* ⩽ 8 M_☉ as calculated by Karakas & Lattanzio (2007, hereafter KL07; Figure 4). The equation for the fit to the KL07 calculations for mass of oxygen released relative to mass of progenitor is

$$\begin{eqnarray} \frac{{M_{{\rm O}} }}{{M_* }} &=& 5.5687\times 10^{ - 5} M_*^3 + 8.4985\times 10^{ - 4} M_*^2 \nonumber \\ &+& 3.9956\times 10^{ - 3} M_* + 1.3560\times 10^{ - 3}. \end{eqnarray} \tag{ 6 }$$

From Equations (4) through (6), the fraction of oxygen attributable to a progenitor mass M_* in the total ejecta from a generation of stars, X_O,Δm, represented by a specified IMF is

$$\begin{equation} X_{{{\rm O,}}\Delta {{\rm m}}} = \frac{{\frac{{M_{{\rm O}} }}{{M_* }}X_{\Delta m } }}{{\sum \limits _{\Delta m} {\frac{{M_O }}{{M_* }}X_{\Delta m} } }}, \end{equation} \tag{ 7 }$$

where Δm represents the mass interval m_l to m_u, the summation is over all mass intervals, and M_* is the mean mass for that interval. We are concerned here with both the total oxygen ejected and the isotopic composition of that oxygen. Oxygen isotope abundances for the ejecta, expressed as the mass fraction of oxygen for nuclide i, $X^{i}{{\rm O}} = M_{^i{{\rm O}}}/(M_{^{16}{\rm O}}+M_{^{17}{\rm O}}+M_{^{18}{\rm O}})$, are shown in Figure 5. The fit to the RHHW02 calculations used here for ¹⁸O is

$$\begin{equation} X{}^{18}{{\rm O = }}\frac{{0.0048}}{{(1 + \exp (- (M_* - 18.3973)/(- 2.1545)))}}. \end{equation} \tag{ 8 }$$

For ¹⁷O we fit the RHHW02 calculations and then scaled the results by a factor of 0.5 to bring them into line with the more recent results from WH07. The fit prior to scaling (Figure 5) is

$$\begin{eqnarray} X{}^{17}{{\rm O}} &= &3.1995\times 10^{ - 4} - 1.9676\times 10^{ - 5} M_* \nonumber \\ &+&3.043\times 10^{ - 7} M_*^2. \end{eqnarray} \tag{ 9 }$$

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Using the calculations described above we can plot the isotopic compositions and mass fractions of oxygen ejected from stars comprising a stellar cluster in three-isotope space (Figure 6). The result shows that SNe II produce a wide range of compositions from the high [¹⁸O]/[¹⁶O]–low [¹⁷O]/[¹⁶O] ejected by smaller progenitors with masses less than 30 M_☉ to the low [¹⁸O]/[¹⁶O]–high [¹⁷O]/[¹⁶O] ejected by the more massive progenitors. It is clear that exogenous oxygen from low-mass SNe II (<20M_☉) could explain the anomalously high [¹⁸O]/[¹⁷O] of the Sun compared with the more normal compositions of the Galaxy. The implication is that one or more stars that exploded and enriched the protosolar cloud with [¹⁸O]/[¹⁶O]-rich oxygen were B stars. The more massive O stars, on the other hand, eject oxygen with [¹⁸O]/[¹⁶O] too low to allow for mixing with ancient ISM to produce the solar oxygen isotope ratios while AGB stars, the sources of presolar grains in meteorites, produce oxygen too low in [¹⁸O]/[¹⁷O] to explain the solar values.

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Supernova yield calculations are inherently uncertain and this uncertainty is difficult to quantify. In order to illustrate the level of uncertainty, we show in Figure 7 examples of ejecta oxygen isotope ratios as functions of SNe II progenitor masses from a variety of recent studies. In all cases, one sees that the lower mass progenitors in all of the calculations do indeed tend to produce higher [¹⁸O]/[¹⁶O] and [¹⁷O]/[¹⁶O] ejecta than the higher mass progenitors. However the exact locations of the yields relative to the local ISM in three-isotope space vary considerably from model to model. Of the calculations considered here, those of Nomoto et al. (2006, hereafter NTUKM06) exhibit the greatest disparity relative to the others, with substantially greater abundances of both secondary oxygen nuclides compared with the results of WW95, RHHW02, Limongi & Chieffi (2003, hereafter LC03), and WH07. Our smoothed yield versus mass function is consistent with the most recent of these models (WH07) by design (Figure 7). The veracity of what follows depends on the extent to which current calculations faithfully represent supernova oxygen isotope yields, but in all cases the inverse relationship between [¹⁸O]/[¹⁶O] and SN II progenitor mass is robust.

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We note that there is a discrepancy in the production of ¹⁸O between WH07 and RHHW02. The yield given by WH07 is substantially greater than that given by RHHW02 for stellar masses greater than 25 M_☉ (Figure 7). However, we have performed the calculations described below using fits to both sets of oxygen isotope yields and found no appreciable difference in the results. This is in part because we are concerned with supernova progenitor masses considerably less than 25 M_☉.

For the purposes of these calculations, the precise form of the IMF is unimportant. This is because the main discrepancies between different IMFs are for the very low and very high masses, while, as we show below, we are primarily concerned with the mass range from 8 to 30 M_☉.

Given the apparent requirement that exogenous oxygen that affected the isotopic composition of the solar system came from B stars but not O stars, the question then arises as to how oxygen from exploding B stars would be "selected" by the solar precursor in greater abundance than oxygen from other sources. The answer lies in the stochastic nature of star formation in general, as described below.

The relative age of the cluster proximal to the protosolar molecular cloud is limited by the constraint on the average supernova progenitor star mass. B stars having masses <20 M_☉ and >8 M_☉ (the minimum mass to produce an SN II) require 10–30 Myr to evolve prior to explosion as SNe II (Prantzos 2008; Schaller et al. 1992). The stars that were the source of the high [¹⁸O]/[¹⁷O] oxygen therefore belonged to a generation of star formation that predated the solar system by at least 10 Myr. We emphasize this conclusion. The oxygen isotope data indicate an episode of enrichment that involves a generation of star formation that predated that which produced the Sun. This conclusion is in contrast to, and inconsistent with, earlier models for supernova enrichment of the solar system (e.g., Schramm & Olive 1982) in which it is assumed that the exploding supernova and the Sun formed together as siblings in the same star cluster.

Enrichment of a region of star formation by explosions of B stars from an earlier generation of star formation is consistent with protracted star formation in molecular cloud complexes in the Galaxy spanning 10–20 Myr (e.g., Gounelle et al. 2009). With this scenario of propagating star formation in mind, we used a statistical analysis to examine the likelihood for [¹⁸O]/[¹⁷O] enrichment of the protosolar molecular cloud by oxygen ejected from B stars that evolve to become SNe II. We adopted 20 Myr as a conservative upper limit for the time interval over which B stars reside in a cloud complex after formation, corresponding to a minimum stellar mass of 11 M_☉. Current evidence indicates that young clusters of moderate size that produce one or more B stars disrupt their parental molecular clouds on timescales of ∼3–10 Myr (Leisawitz et al. 1989; Elmegreen 2007), a time span less than the 10 Myr required for the most massive, and therefore the most short-lived, B stars (18 M_☉, ∼ B0) to explode as SNe II. However, molecular clouds are usually present in cloud complexes extending over hundreds of parsecs, and these complexes are sites of protracted episodes of star formation lasting tens of millions of years (Elmegreen 2007; Hartmann et al. 2001). With velocity dispersions of ∼10 km s⁻¹ between stellar subgroups within a complex (de Bruijne 1999), the more massive B stars (B0–B1) can drift only 100–200 pc during their lifetimes. Such a star cannot leave the vicinity of the giant molecular cloud (GMC) complex before exploding as an SN II; B0–B1 stars formed in one cloud are likely to be in close proximity to other clouds within the same extended complex when they become supernovae. Indeed, such supernovae are often invoked as triggers for successive generations of star formation (Briceño et al. 2007). Evidence that star clusters can encounter multiple clouds is provided by the presence of multiple generations of stars within some well-studied clusters (Mackey 2009). The net result is that complexes hundreds of pc in size persist for up to 50 Myr, GMCs within the complexes survive for 10–20 Myr, and GMC cores that produce individual clusters last ∼3 Myr (Elmegreen 2007).

For the statistical analysis we used the mass generation function of Kroupa et al. (1993) modified by Brasser et al. (2006) to obtain 300 random realizations of star clusters of various sizes ranging from tens to tens of thousands of members. With this function the mass of the jth star of the cluster is obtained from the expression

$$\begin{equation} M_{j}/M_{\odot } =0.01+(0.19x^{1.55} +0.05x^{0.6})/(1-x)^{0.58}, \end{equation} \tag{ 10 }$$

where x is a uniformly distributed random number between 0 and 1. Sampling of the IMF, as simulated with Equation (10), results in systematic relationships between occurrences of SNe II, the maximum size of SNe progenitors, and cluster size. Figure 8 shows that the average fractional number of stars that produce SNe II within 20 Myr of the birth of the cluster is 1.4 × 10⁻³ for all cluster sizes. The fact that all of the simulated clusters produce the same average fractional number of supernovae illustrates that clusters of all sizes do indeed represent random samplings of the IMF. However, the discrete, stochastic sampling produces important differences in populations as a function of cluster size. Clusters composed of relatively few numbers of stars can sometimes produce no SNe II at all simply because the progenitor stars (those with M_* > 8 M_☉) are relatively rare. This is illustrated in Figure 9 where it can be seen that, with the 20 Myr time constraint where we consider only stars with masses greater than 11 M_☉, only about 15% of the clusters composed of 100 stars, a practical minimum size for clusters (Lada & Lada 2003), produce SNe II whereas 80% of the clusters of 1000 stars produce SNe II in these simulations. The maximum stellar mass, and therefore most massive supernova progenitor, also varies with cluster size as a fundamental consequence of discrete sampling of the IMF in which the more massive the star, the more rare its occurrence. As cluster size increases, the maximum in the frequency of most massive members of the cluster shifts to higher mass (Figure 10). In other words, smaller clusters tend to produce smaller supernovae and larger clusters tend to produce larger supernovae.

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We emphasize two important points in this regard. First, the histograms in Figure 10 do not depict the frequency distributions of all masses, only that of the most massive member of each cluster. Second, that the propensity for larger clusters to more reliably produce the more massive stars is a natural consequence of stochastic sampling of any IMF in which frequency varies inversely with stellar mass. Although debate surrounds whether sampling of the IMF by star clusters is truly random (Weidner et al. 2010), the positive correlation between the mode in maximal star mass and cluster size is evident nonetheless (e.g., Weidner et al. 2010, Figure 4).

We combined supernova oxygen isotope yields with the statistical analysis described above in order to examine the relationship between star cluster size and the oxygen isotopic composition of the oxygen ejected by SNe II from the cluster. The results are depicted in oxygen three-isotope space by contouring the relative probability of occurrence of supernova oxygen ejecta of a given isotopic composition. The isotope ratios of the ejecta are integrated from time 0 to 20 Myr after the (instantaneous) formation of the cluster. Assignment of probabilities is an expediency that amounts to a Gaussian smearing of each model datum. In this way, clusters of adjacent points receive greater weight (greater probability) than individual points. For this purpose, each of the 300 time-integrated oxygen isotopic compositions of SNe II ejecta is smeared by a Gaussian distribution:

$$\begin{equation} P_{{}^{17}{{\rm O}}} = \frac{1}{{\sigma \sqrt{2\pi } }}\exp \left({ - \frac{1}{2}\left({\frac{{\delta {}^{17}{{\rm O}^{\prime }} - \delta {}^{17}{{\rm O}^{\prime }}_{{{\rm Model}}} }}{\sigma }} \right)^2 } \right) \end{equation} \tag{ 11 }$$

and

$$\begin{equation} P_{{}^{18}{{\rm O}}} = \frac{1}{{\sigma \sqrt{2\pi } }}\exp \left({ - \frac{1}{2}\left({\frac{{\delta {}^{18}{{\rm O}^{\prime }} - \delta {}^{18}{{\rm O}^{\prime }}_{{{\rm Model}}} }}{\sigma }} \right)^2 } \right), \end{equation} \tag{ 12 }$$

where δ¹⁷O'_Model and δ¹⁸O'_Model are the model supernova ejecta isotope ratios in δ' notation and σ is taken to be slightly larger than the grid spacing. The normalized joint probability for a given grid square in three-isotope space is then

$$\begin{equation} d\!P_{{{\rm grid}}} = \frac{1}{n}{P_{{}^{{{\rm 17}}}{{\rm O}}} P_{{}^{{{\rm 18}}}{{\rm O}}} \;d\delta {}^{17}{{\rm O}^{\prime }}\;d\delta {}^{18}{{\rm O}^{\prime }}}, \end{equation} \tag{ 13 }$$

where dδ¹⁷O' = dδ¹⁸O' = σ and n is the number of points (300 in this case). For the results shown here, we set σ equal to a convenient but arbitrary level of smearing of 40‰  compared with a grid spacing of 33‰. Altering σ and the grid spacing has no substantive effect on the results.

Probability density contours for the isotopic compositions of oxygen ejected over a 20 Myr period from clusters of 500 and 5000 stars are shown in Figure 11. The contours quantify the inverse relationship between cluster size and [¹⁸O]/[¹⁶O] of oxygen ejected by SNe II. Oxygen ejected en masse from a cluster of 5000 stars (gray contours) is more likely to extend to lower [¹⁸O]/[¹⁶O] than is oxygen ejected from a cluster composed of 500 stars (black contours).

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We also show in Figure 11 calculated mixing curves between estimates for the ISM 4.6 Gyr before present and the most likely compositions produced by a cluster of ca. 500 stars that would produce the composition of the Sun. One curve is based on mixing with the most probable composition for a 500 star cluster. This peak in the probability density for the 500 star cluster occurs at the highest [¹⁸O]/[¹⁶O] permitted by the 20 Myr constraint (i.e., the lowest possible mass for SNe II progenitors). The other mixing curve is based on the lowest [¹⁸O]/[¹⁶O] ratio consistent with the oxygen isotopic composition of the ISM 4.6 Gyr before present (i.e., the present-day [¹⁸O]/[¹⁶O] ISM, see above). The mixing curves demonstrate that smaller clusters, represented here by our statistical representation of clusters of 500 stars, were more likely sources of oxygen isotope enrichment for the solar system than larger clusters (e.g., the 5000 star clusters represented by gray contours).

Results of this analysis show that oxygen ejecta with the composition required to explain the oxygen isotope ratios of the solar system relative to the ISM 4.6 Gyr before present is more than twice as likely to have come from a star cluster of several hundred stars than from a cluster of several thousand stars. Larger clusters tend to produce oxygen too low in [¹⁸O]/[¹⁶O] while considerably smaller clusters produce too few SNe II. In our analysis, 19% of the clusters of 500 stars produce oxygen isotope ratios suitable to explain the solar composition by supernova enrichment. For comparison, the corresponding probability for a cluster of 5000 stars is 7%. Relationships between the mixing curves constrained by the contours in Figure 11, the oxygen isotopic composition of the ISM, and the oxygen isotopic composition of the solar system are shown in Figure 12.

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The mass fraction of exogenous oxygen in the solar parental cloud can be estimated from the mixing curves in Figures 11 and 12. The masses of ejected oxygen represented by the two mixing curves in Figures 11 and 12 are 0.7 and 7.5  M_☉ for the high- and low-[¹⁸O]/[¹⁶O] cases, respectively. The minimum mass of the enriched protosolar cloud material, M_MC, can be calculated from the assumption of a 100% efficiency of injection and simultaneous solution of the mass balance equations

$$\begin{equation} C_{{}^{17}{{\rm O}}, \odot } = \frac{{({M_{{}^{17}{{\rm O,MC}}} + M_{{}^{17}{{\rm O,EJ}}} })}}{{M_{{{\rm MC}}} }} \end{equation} \tag{ 14 }$$

$$\begin{equation} C_{{}^{18}{{\rm O}}, \odot } = \frac{{({M_{{}^{18}{{\rm O,MC}}} + M_{{}^{18}{{\rm O,EJ}}} })}}{{M_{{{\rm MC}}} }}, \end{equation} \tag{ 15 }$$

where $C_{^{17}{\rm O},\odot }$ and $C_{^{18}{\rm O},\odot }$ are the solar concentrations by mass of the specified isotopes, $M_{{}^{17}{{\rm O,MC}}}$ ($M_{{}^{18}{{\rm O,MC}}}$) is the unknown initial mass of ¹⁷O (¹⁸O) in the cloud material, and $M_{^{17}{{\rm O,EJ}}}$ ($M_{^{18}{{\rm O,EJ}} }$) is the calculated mass of ¹⁷O (¹⁸O) added to the cloud by the SNe II ejecta constrained by the mixing curves. With the additional constraint that the initial [¹⁸O]/[¹⁷O] of the molecular cloud was on the Galactic line, such that $M_{{}^{17}{{\rm O,MC}}} = M_{{}^{18}{{\rm O,MC}}} (17/18)/4.1$, we obtain M_MC ∼ 700 to 5000 M_☉. The mass fraction of SNe II oxygen of 0.1% (e.g., 100 × 0.7/700), together with SNe II oxygen mass fractions 5–10 times solar (Rauscher et al. 2002), suggests a total contribution of SNe II ejecta to the protosolar molecular cloud of ∼1% by mass.

Enrichment of the protosolar cloud by SNe II ejecta may help to explain other puzzling aspects of the isotopic composition of the solar system. We consider here the implications for the isotopes of silicon, carbon, and the short-lived radionuclides ⁶⁰Fe and ²⁶Al as well as for the interpretation of presolar oxide grains. A caveat to any such calculation that attempts to use a single supernova source for numerous isotopic and elemental systems is the prospect for heterogeneous mixing between the supernova ejecta and the target cloud material. Nonetheless, in order to assess the collateral implications of 1% enrichment of the protosolar molecular cloud by SNe II ejecta for other isotope systems, we derive a general mixing equation that illustrates explicitly the relative importance of elemental abundances and isotope ratios in producing mixtures of molecular cloud and SNe ejecta. Mass balance for the number of atoms of isotope 1 of element E, $n_{E_1}$, between pre-enrichment molecular cloud (MC) material, supernova ejecta (SNe), and the final mixture (solar, ☉) can be written in terms of the total atoms for each reservoir j, N_j, and the atomic fractions of nuclide E₁ in reservoir j, $(x_{E_1})_j$, such that

$$\begin{eqnarray} (x_{E_1 })_ \odot &=& \frac{{n_{E_1 } }}{{N_{{{\rm MC}}} + N_{{{\rm SNe}}} }} \nonumber \\ &=& (x_{E_1 })_{{{\rm MC}}} x_{{{\rm MC}}} + (x_{E_1 })_{{{\rm SNe}}} x_{{{\rm SNe}}}, \end{eqnarray} \tag{ 16 }$$

where x_SNe = N_SNe/(N_SNe + N_MC), the atomic fraction of supernova ejecta in the mixture, and x_MC is the atomic fraction of original molecular cloud material. Recognizing that x_MC = 1 − x_SNe and taking the ratio of Equation (16) for two isotopes of element E we obtain

$$\begin{eqnarray} {}^{2/1}R_ \odot &=& \frac{{(x_{E_2 })_ \odot }}{{(x_{E_1 })_ \odot }} \nonumber \\ &=& \frac{{x_{{{\rm SNe}}} (x_{E_2 })_{{{\rm SNe}}} + (1 - x_{{{\rm SNe}}})(x_{E_2 })_{{{\rm MC}}} }}{{x_{{{\rm SNe}}} (x_{E_1 })_{{{\rm SNe}}} + (1 - x_{{{\rm SNe}}})(x_{E_1 })_{{{\rm MC}}} }}. \end{eqnarray} \tag{ 17 }$$

Equation (17) is rearranged to obtain an expression for the atomic fraction of supernova ejecta in terms of elemental abundances and isotope ratios for supernova ejecta, pre-enrichment molecular cloud, and the final solar mixture:

$$\begin{equation} x_{{{\rm SNe}}} = \frac{{(x_{E_1 })_{{{\rm MC}}} \left({{}^{2/1}R_ \odot - {}^{2/1}R_{{{\rm MC}}} } \right)}}{\Gamma }, \end{equation} \tag{ 18 }$$

where

$$\begin{eqnarray} \Gamma &=& (x_{E_1 })_{{{\rm SNe}}} {}\, ^{2/1}R_{{{\rm SNe}}} - (x_{E_1 })_{{{\rm MC}}} {}\, ^{2/1}R_{{{\rm MC}}} \nonumber \\ &-& {}^{2/1}R_ \odot ({(x_{E_1 })_{{{\rm SNe}}} - (x_{E_1 })_{{{\rm MC}}} }). \end{eqnarray} \tag{ 19 }$$

In practice, we can equate atomic fractions with atoms per hydrogen in applying Equation (18). We will assume that the relative abundances of the elements (as opposed to isotope ratios) in the precursor cloud were indistinguishable from solar values. Numerical experiments in which fictive molecular cloud elemental abundances were used confirm that deviations from this simplifying assumption have negligible effects on the results.

4.3.1. Carbon Isotope Ratios

4.3.2. Silicon Isotope Ratios

A long-standing problem has been an apparent excess in ²⁸Si in the solar system relative to expected values. An excess in ²⁸Si over ²⁹Si and ³⁰Si in the solar system relative to the ISM 4.6 Gyr before present is evident by comparisons with presolar SiC grains (Alexander & Nittler 1999). Mainstream SiC grains come from AGB stars that predate the Sun by hundreds of millions to billions of years. GCE should therefore have resulted in the younger Sun having greater [²⁹Si]/[²⁸Si] and [³⁰Si]/[²⁸Si] than these earlier-formed AGB stars, yet solar values are lower, not higher, by 11%–12% (Alexander & Nittler 1999). SNe II expel an excess of ²⁸Si relative to the heavier Si isotopes (Rauscher et al. 2002) and it has been suggested previously that the solar system might have been enriched by ²⁸Si from supernovae (Alexander & Nittler 1999). Models for SNe II ejecta tend to produce too much ³⁰Si relative to ²⁹Si compared with the silicon isotopic compositions necessary to explain the relationship between solar system and mainstream SiC presolar grains (Figure 13), but this may be a problem of spurious overproduction of ³⁰Si in the calculations (Alexander & Nittler 1999). Using Equation (18), we find that addition of 1% by mass of ejecta from the s19 model of RHHW02 decreases [²⁹Si]/[²⁸Si] and [³⁰Si]/[²⁸Si] in the pre-enrichment solar system molecular cloud by 12% and 11%, respectively, making the ISM 4.6 Gyr before present greater in [²⁹Si]/[²⁸Si] and [³⁰Si]/[²⁸Si] than the majority of presolar mainstream SiC grains. Therefore, the same enrichment process that explains the aberrant [¹⁸O]/[¹⁷O] of the solar system could also explain most of the excess in ²⁸Si in the solar system (Figure 13). A caveat is that the oxygen isotopic composition of the s19 supernova ejecta model is not ideal for explaining the anomalous [¹⁸O]/[¹⁷O] of the solar system as it is slightly lower in [¹⁸O]/[¹⁶O] than progenitors depicted with the mixing curves in Figures 11 and 12, and would therefore require the ISM 4.6 Gyr before present to have been higher in [¹⁸O]/[¹⁶O] than today (violating expectations from GCE). This problem is not severe, however, given the uncertain dispersion in ISM [¹⁸O]/[¹⁶O] along the Galactic slope-1 line in three-isotope space.

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4.3.3. Short-lived Radionuclides

4.3.4. Presolar Grains

The oxygen isotope ratios of presolar oxide grains found in meteorites bear on the question of the GCE of oxygen. Nittler (2009) argues that models for the origins of these grains are consistent with [¹⁸O]/[¹⁷O] equal to the solar value of 5.2 for all of the AGB stars from which they derive. In all cases, however, fundamental problems arise with interpretations of presolar grain oxygen isotope ratios when the solar system [¹⁸O]/[¹⁷O] is not equal to the ISM today (e.g., Alexander & Nittler 1999). Nittler and colleagues have interpreted the presolar oxides as having come from a range of masses of AGB stars from about 1.2 to 2.2 M_☉. This corresponds to a sampling of Galactic oxygen over 5.5 Gyr based on the lifetimes of the progenitors stars between 6.6 and 1.1 Gyr (e.g., Schaller et al. 1992). That is to say, a 1.2 M_☉ star that enters the AGB phase of evolution at the time the solar system was forming sampled oxygen from the ISM 6.6 Gyr prior to the formation of the Sun, or by any measure of stellar lifetimes, very early in the evolution of the Galaxy. A 2.2 M_☉ star entering the AGB phase at the time the Sun was forming represents a sampling of oxygen from 1.1 Gyr prior to the formation of the Sun. Therefore, if the interpretation of the presolar grain data is taken at face value, the ISM was characterized by a solar [¹⁸O]/[¹⁷O] of 5.2 for 7 billion years leading up to the formation of the Sun, then sometime between 4.6 Gyr ago and now, the bulk [¹⁸O]/[¹⁷O] of the ISM changed by nearly 30% after having been constant for the prior 7 Gyr. Such a change might be expected once the Milky Way ages to the point where high-mass stars are no longer being made, but that is not the case yet. Therefore, we conclude that it may be necessary to modify models for the origins of presolar grains. We note that the presolar grain oxygen isotope data cluster about a Galactic [¹⁸O]/[¹⁷O] of about 4.1 (Figure 14), suggestive of a causal relationship between the peak in the presolar grain data and the Galactic [¹⁸O]/[¹⁷O] (though a coincidence cannot be discounted). In addition, there are presolar oxide grains that fall below the solar [¹⁸O]/[¹⁷O] line (with higher than solar [¹⁸O]/[¹⁷O]) that are not consistent with AGB or supernova predictions and so are as yet unexplained by any model, suggesting that the origin of high [¹⁸O]/[¹⁷O] grains may not be understood in general.

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The peculiar nature of the solar [¹⁸O]/[¹⁷O] suggests that the type of chemical and isotopic enrichment during star formation proposed here is not the norm. The likelihood for the enrichment by B stars but not O stars can be estimated from the probability for the occurrence of clusters of stars having the requisite number of stars (ca. 500 as opposed to several thousands) and the fraction of those clusters that produce supernovae. The cluster mass function in terms of the distribution (f) of cluster sizes (specified by the number of stars comprising the cluster, N_*, rather than mass for consistency with our statistical analysis) is characterized with a power law (e.g., Elmegreen & Efremov 1997; Parmentier et al. 2008) such that

$$\begin{equation} \frac{{df(N_*)}}{{d\!N_* }} \propto N_*^{ - \alpha }. \end{equation} \tag{ 20 }$$

The distribution function is therefore f(N_*) ∝ N^1−α_*. The likelihood of a cluster having from N_min to N_max stars is given by the integral of the distribution function

$$\begin{equation} P_{\Delta N_* } = a\int _{N_{\min } }^{N_{\max } } N_*^{1-\alpha }{dN_* }, \end{equation} \tag{ 21 }$$

where a is the normalizing factor. The value for α in Equation (20) that characterizes the cluster mass function is generally found to be near 2.0 (Elmegreen & Efremov 1997; Haas & Anders 2010). In this case a = 0.109 for a total range in N_* from 100 to 10⁶. However, reported values for α vary from about 1.8 to 2.4 (Haas & Anders 2010) with a varying accordingly. For clusters of 100 to 1000 members and α = 2.0, $P_{\Delta N_{*} } = 0.25$. For α ranging from 1.8 to 2.4 the range in $P_{\Delta N_{*} }$ is 0.11–0.62.

The mass generation function used here (Equation (10)) yields a best-fit relationship between the number of stars comprising a cluster and the fraction of those clusters that produce supernovae within the 20 Myr timescale (Figure 9):

$$\begin{equation} X_{\rm SNe} =1-\exp (-bN_{*}), \end{equation} \tag{ 22 }$$

where b = 1.537 × 10⁻³. From Equations (21) and (22), the joint probability for occurrences of SNe II within 20 Myr from a cluster of an appropriate range in N_* is

$$\begin{equation} P_{\Delta N_*, {\rm SNe}}=\int _{N_{\min } }^{N_{\max } } {a N_*^{ 1-\alpha } \left(1-\exp (- bN_*)\right)} \;dN_*. \end{equation} \tag{ 23 }$$

Evaluation of Equation (23) yields $P_{\Delta N_*, {\rm SNe}} = 0.10$ for α = 2.0 with a range due to uncertainty in α from 0.05 (α = 1.8) to 0.23 (α = 2.4). Our 5%–20% probability is consistent with previous estimates for the fraction of stars formed under the influence of massive stars adjacent to molecular clouds (Mizuno et al. 2007; Hennebelle et al. 2009). We must also consider, however, that not every cluster in this size range that produces supernovae has the right oxygen isotopic composition because occasionally some of these clusters do produce the larger O stars that are too low in [¹⁸O]/[¹⁶O] to be consistent with enrichment of the protosolar cloud. The simulation shown in Figure 11 suggests that roughly half of the clusters of ca. 500 stars that produce supernovae also have B stars as their maximal stellar mass and so produce oxygen isotope ejecta suitable to explain the solar system [¹⁸O]/[¹⁷O]. The probability derived above should therefore be halved, yielding an estimate of 2.5%–10% for the occurrence of a suitable star cluster. These conclusions are invalid if SNe II oxygen isotope yields are closer to the high-[¹⁸O]/[¹⁶O] values calculated by NTUKM06 rather than the values reported by RHHW02, LC03, and WH07, for example.

Another source of oxygen to consider in a star-forming region is that produced by the prodigious winds of Wolf–Rayet (W-R) stars (in particular WC stars with He-burning products exposed at the surface). We conclude that this is a less likely alternative to SNe II ejecta. First, W-R stars evolve from more massive O stars generally (Crowther 2007), making them by their very nature less common than less massive B stars that end their lives as SNe II. Indeed, the ratio of the rates of occurrence of SNe II to the rates of occurrence of Type I b/c SNe, the likely endpoint for W-R stars, is about 5 (Vanbeveren 2005). Second, the W-R phase of evolution lasts for 10⁵ years (e.g., Maeder & Meynet 1994, and references therein) and so the chances of catching winds from this phase of evolution alone, without also capturing the ensuing collapse supernova debris, are small. Third, rates of mass loss from W-R stars are of order 10⁻⁴ to 10⁻⁵ M_☉ yr⁻¹, and these rates multiplied by the duration of the W-R phase of evolution yields of order one solar mass of total wind material (e.g., Binns et al. 2006). The mass of oxygen released will be considerably less than the total mass in these winds. Therefore, the mass of oxygen released is ≪1 M_☉ compared with the minimum of approximately 1 M_☉ of oxygen liberated by low-mass SNe II. When considering the brevity of the W-R phase of evolution, the relative rarity of O star W-R progenitors, and the relatively low oxygen yields from the winds, it seems that enrichment of a star-forming region in oxygen isotopes from WC stars is not as likely as enrichment by SNe II ejecta.