Distributions of Short-Lived Radioactive Nuclei Produced by Young Embedded Stellar Clusters
aa r X i v : . [ a s t r o - ph . S R ] M a y DISTRIBUTIONS OF SHORT-LIVED RADIOACTIVE NUCLEIPRODUCED BY YOUNG EMBEDDED STAR CLUSTERS
Fred C. Adams , , Marco Fatuzzo , and Lisa Holden Physics Department, University of Michigan, Ann Arbor, MI 48109 Astronomy Department, University of Michigan, Ann Arbor, MI 48109 Physics Department, Xavier University, Cincinatti, OH 45255 Department of Mathematics, Northern Kentucky University, Highland Heights, KY 41099
ABSTRACT
Most star formation in the Galaxy takes place in clusters, where the most mas-sive members can affect the properties of other constituent solar systems. Thispaper considers how clusters influence star formation and forming planetary sys-tems through nuclear enrichment from supernova explosions, where massive starsdeliver short-lived radioactive nuclei (SLRs) to their local environment. The de-cay of these nuclei leads to both heating and ionization, and thereby affects diskevolution, disk chemistry, and the accompanying process of planet formation.Nuclear enrichment can take place on two spatial scales: [1] Within the clusteritself ( ℓ ∼ ℓ ∼ − ∼
10. For the second scenario, we find that distributedenrichment of SLRs in molecular clouds leads to comparable abundances. Forboth the direct and distributed enrichment processes, the masses of Al and Fedelivered to individual circumstellar disks typically fall in the range 10 − pM ⊙ (where 1 pM ⊙ = 10 − M ⊙ ). The corresponding ionization rate due to SLRs typi-cally falls in the range ζ SLR ∼ − × − sec − . This ionization rate is smallerthan that due to cosmic rays, ζ CR ∼ − sec − , but will be important in regionswhere cosmic rays are attenuated (e.g., disk mid-planes). Subject headings: stars: formation — planetary systems: formation — openclusters and associations: general 2 –
1. Introduction
Within our Galaxy, most star formation takes place within embedded stellar clusters,where these systems display a wide range of sizes and other properties (e.g, Lada & Lada2003; Porras et al. 2003; Allen et al. 2007). These background environments can influencethe evolution and properties of planetary systems forming within them through a varietyof processes (Hester & Desch 2005; Looney et al. 2006; Levison et al. 2010; Adams 2010;Pfalzner 2013), including dynamical scattering by other stellar members (Adams & Laughlin2001; Adams et al. 2006; Malmberg et al. 2007), evaporation of circumstellar disks by radi-ation from massive stars (St¨orzer & Hollenbach 1999; Adams et al. 2004), accretion of clus-ter gas onto the disks (Throop & Bally 2008), and the injection of short-lived radioactivenuclei (hereafter SLRs) into circumstellar disks and/or collapsing regions (Cameron 1993;Meyer & Clayton 2000; Ouellette et al. 2007). This paper focuses on this latter issue of nu-clear enrichment. In particular, we construct probability distributions for the abundances ofSLRs that are expected to be delivered to forming solar systems in cluster environments, aswell as to the larger, surrounding regions in molecular clouds.Previous work regarding nuclear enrichment has focused on two related but somewhatdifferent goals. The first approach considers the largest spatial scales, where a great dealof work has been carried out to estimate the steady state production rates and abundancesof SLRs in the Galaxy (Timmes et al. 1995a,b). For example, gamma-ray emission due tothe radioactive decay of Al has been observed for the 1808.65 keV line (Diehl et al. 2006;Smith 2003). Since the Galaxy does not appreciably attenuate such emission, it can beused to determine the galactic inventory of Al; the abundances of other nuclear speciescan be assessed in similar fashion. Such measurements, in conjunction with stellar evolutioncalculations that determine nuclear yields (Woosley & Weaver 1995; Woosley et al. 2002;Rauscher et al. 2002; Limongi & Chieffi 2006), can then be used to estimate (or constrain)the Galactic star formation rate. In an similar vein, gamma-ray observations have alsomeasured the emission due to decay of Fe. By comparing the ratio of line strengths,one can test whether or not the inferred abundances of Fe and Al are consistent withpredictions of stellar nucleosynthesis models (Prantzos 2004). The results are consistent atthe factor-of-two level, which is comparable to the uncertainties in these quantities.On smaller scales, our own Solar System was apparently enriched in SLRs during itsearly formative stages. A great deal of previous work has been carried out to explain thenuclear abundance patterns deduced from meteoritic studies. Evidence for the enrichment of Al is well-established, and the proposal that such enrichment arises from nearby supernovaedates back to Cameron & Truran (1977), or even further. Although Al is readily producedin supernovae, it can also be synthesized through spallation. More recently, meteoritic 3 –evidence for live Fe in the early solar system has been reported (Tachibana et al. 2006)and bolsters the case for supernova enrichment of SLRs (because Fe can only be producedvia stellar nucleosynthesis). In contrast to the case of Al, however, the evidence for Feis controversial and elusive (Moynier et al. 2011). For example, the inferred abundances canbe biased for low count rates, such as those found for Fe (Telus et al. 2012). On the otherhand, a recent study (Mishra & Goswami 2014) concludes that the two SLRs Al and Fewere co-injected into the early solar system from same stellar source. In any case, for thepurposes of this paper, we consider the two nuclear species to be on a nearly equal footing.As discussed below, stellar evolution models predict comparable abundances for Al and Fe, and both species are expected to make substantial contributions to ionization andheating within enriched regions.The apparent need for nuclear enrichment of the early Solar System places constraintson the birth environment of the solar system (Adams & Laughlin 2001; Hester & Desch2005; Williams & Gaidos 2007; Gounelle & Meibom 2008; Gounelle et al. 2009; Adams 2010;Pfalzner 2013; Dauphas & Chaussidon 2011). The requirement that the Sun is born neara high mass progenitor favors the scenario where the Sun is formed within a large cluster,but such environments can also lead to disruption through dynamical scattering interactions(Adams et al. 2006; Malmberg et al. 2007; Dukes & Krumholz 2012) and intense radiationfields (Fatuzzo & Adams 2008; Holden et al. 2011; Thompson 2013). For the nuclear en-richment of our Solar System, these previous studies (along with many others, includingLooney et al. 2006; Williams 2010; Parker et al. 2014) suggest that the probability of en-richment is low, perhaps 1–10%, where the estimated value depends on how the accountingis done. On the other hand, observations of accreted extrasolar asteroids in white dwarfatmospheres indicate that the elevated levels of Al inferred for the formation of own SolarSystem are not abnormal (Jura et al. 2013).Building upon the aforementioned previous work, this paper considers nuclear enrich-ment on an intermediate scale. Instead of focusing on the enrichment of our own SolarSystem, we consider the general problem of determining the distribution of enrichment lev-els that are expected for the whole population of forming stars within a cluster. We alsoconsider nuclear enrichment on the next larger scale of the molecular cloud, but do not focuson the scale of the Galaxy. Nonetheless, this work informs the larger scale picture, as theresults must be consistent with galaxy-wide estimates for nuclear production and the accom-panying star formation rate. On the scale of our own Solar System, this work constrains theprobability of attaining the levels of nuclear enrichment inferred for our own solar nebula.The decay of SLRs is important for star and planet formation for two related reasons:ionization and heating. The nuclear decay products are highly energetic, with E ∼
2. Stellar and Cluster Mass Distributions2.1. Initial Mass Function for Stars
The distribution of SLRs will depend on the stellar initial mass function (IMF). Thisdistribution has been studied intensively (see Kroupa 2001; Chabrier 2003 for recent re-views). In its most basic form, the stellar IMF has the form of a log-normal distributionwith a power-law tail on the high-mass end and perhaps another tail at the low-mass end(Adams & Fatuzzo 1996). In any case, the distribution is heavily weighted toward stars oflow mass (see also Salpeter 1955; Scalo 1998). In this work, however, we are interested inthe high-mass end of the stellar IMF, as only stars with mass above the threshold M ∗ > ∼ M ⊙ contribute to the supply of SLRs through supernova explosions.For the present application, one useful way to parameterize the stellar IMF is to de-fine F SN to be the fraction of the (initial) stellar population with masses greater than theminimum mass ( M ∗ = 8 M ⊙ ) required for a star to ends its life with a supernova explosion. 5 –Current observations indicate that F SN ≈ . dN ∗ dm = F SN γ (cid:16) m (cid:17) − ( γ +1) , (1)where m = M ∗ / (1 M ⊙ ) is the mass in Solar units and the canonical value of the index γ ≈ γ appears to have significant scatter fromregion to region (e.g., Scalo 1998), such that γ is evenly distributed within the range γ =1 . ± .
5. For the sake of definiteness, we use γ = 1.5 as the default value to characterizethe high-mass end of the IMF, but allow the index to vary.As written, the probability distribution of equation (1) is normalized so that Z ∞ dN ∗ dm dm = F SN . (2)As a result, the distribution is normalized to unity for the entire mass distribution (not justthe high mass end), where this statement holds in the absence of a maximum stellar mass. Inpractice, one expects the stellar IMF to have a maximum stellar mass m ∞ , although the valueof m ∞ remains uncertain. In order of magnitude, however, both theory and observationssuggest that m ∞ ≈
120 is a good approximation. If we include this upper mass limit inthe integral of equation (2), we obtain a correction factor [1 − (8 /m ∞ ) γ ] ≈ .
983 in thenormalization. Note that the relative size of this correction factor is much smaller thanthe uncertainties in the other parameters that specify the IMF. As a result, we ignore thiscorrection for the remainder of this work.To complete the specification of the stellar IMF we also need the average stellar mass,i.e., h m i ≡ Z dN ∗ dm mdm . (3)For example, since stellar clusters are often described by their stellar membership size N and/or their mass in stars M , we often need to convert between the two (where M = N h m i ).Similarly, star formation rates can be specified in terms of ‘stars per unit time’ or ‘solarmasses per unit time’, with the conversion factor h m i .To summarize, this paper characterizes the stellar IMF in terms of the three parameters( γ, F SN , h m i ). Since the stellar IMF is steeply declining for all possible choices of γ , mostsupernova progenitors have masses in the range m = 8 −
25. As a result, the most important 6 –parameter that determines radioactive yields is the fraction of stars F SN that are above themass threshold for supernovae. Stars form within embedded clusters, where the number of members in these systemsspans a wide range (e.g., see the reviews of Lada & Lada 2003; Porras et al. 2003; Allen et al.2007). Unfortunately, current observations are not sufficiently complete to specify the dis-tribution of cluster membership sizes N . Studies of embedded clusters in the Solar neigh-borhood indicate that the distribution of cluster sizes N is close to a power-law, so that dN C dN = C N N a , (4)where C N is a normalization constant and where the index a ≈
2. Here, N C is the num-ber of clusters and N is the number of stellar members in the clusters. Studies thatconsider more distant clusters also find power-law distributions, again with a ≈ N = 1 − . For sake of definite-ness, we use a single power-law in the present analysis (but one should keep in mind that thepower-law distribution observed for small clusters nearby and that observed for large clustersat large distances have not been shown to match up). Further, we adopt a benchmark valueof the index a = 2. In this case, the constant C N ≈ P ∗ ( N ) of a star being born within a cluster ofmembership size N . This probability distribution is obtaining by multiplying dN C /dN byanother factor of N and hence is given by P ∗ ( N ) = C P N , with C P = 1log[ N max ] , (5)where we have taken a = 2 and assume that clusters range from N = 1 to N = N max . Weexpect N max ≈ and hence C P ≈ / (6 log 10). The cummulative probability P for findingstars in clusters with membership size N or smaller is thus given by P ( N ) = C P log[ N ] = log[ N ]log[ N max ] . (6)As a result, the probability of finding stars in various sized clusters is evenly distributed ina logarthmic sense. 7 –Next we must specify the radial extent of the cluster. To a reasonable degree of ap-proximation, the cluster radius R can be written as a power-law function of the clustermembership size N , i.e., R = R ( N ) = R (cid:18) NN (cid:19) α . (7)For the clusters found in the solar neighborhood (Lada & Lada 2003; Porras et al. 2003),equation (7) works well with R = 1 pc, N = 300, and α = 1/2. The cluster sample in thesolar neighborhood is limited to the lower end of the cluster membership size distribution– the sample is not large enough to include the largest clusters. If one applies equation(7) to the entire cluster sample (the entire range of N ), then the predicted radii are toolarge for high-mass clusters. We obtain an adequate fit over the entire cluster size range,10 ≤ N ≤ N max , by using index α = 1/3.Finally, we need to specify the spatial distribution of stars within the cluster. Forsimplicity, we assume that the stars in the cluster follow a simple power-law distributionof density. Numerical (N-body) simulations of early cluster dynamics (Adams et al. 2006)show that this assumption is reasonable and indicates that the power-law index of the densitydistribution falls in the range 1 ≤ p ≤
2. As a result, the probability distribution for theradial distance (at a given time, including the time of the supernova explosion) is given by dPdr = 4 πr N n ∗ ( r ) = 3 − pR (cid:16) rR (cid:17) − p , (8)where R is the cluster radius (which varies with cluster membership size N – see equation[7]).
3. Production of Short-lived Radioactive Nuclei3.1. Synthesis of SLRs in Supernovae
Supernovae produce a wide variety of radioactive nuclei. In this treatment, we consideronly the production and distributions of the five most important species of SLRs, namely Al, Cl, Ca, Mn, and Fe. These species all have half-lives shorter than 10 Myr, andrelatively large abundances; these properties, in turn, make them useful for constraining theearly history of our Solar System (Cameron 1993; Meyer & Clayton 2000; Ouellette et al.2007). The abundances of these SLRs are shown as a function of stellar mass in Figures 1 and2, with the yields taken from the calculations of Woosley & Weaver (1995), Rauscher et al.(2002), and Limongi & Chieffi (2006). 8 –For purposes of finding nuclear enrichment levels for typical solar systems, we oftenfurther limit our focus to the two species Al and Fe, because they provide the largestcontribution to the ionization and heating rates. Figure 1 shows the yields for these twoSLRs, where we include results from two different sets of stellar nucleosynthesis calcula-tions. The first group (Woosley & Weaver 1995) considers the range of progenitor masses M ∗ = 11 − M ⊙ , where these results have been updated (Rauscher et al. 2002) for themore limited range in stellar masses M ∗ = 15 − M ⊙ (see also Timmes et al. 1995a,b;Woosley et al. 2002). For the yields shown in Figure 1, we use the updated yields for therange of masses where they are available; hereafter results from this set of papers are de-noted as WW. The second set of results (Limongi & Chieffi 2006), hereafter LC, considersa wider range of masses M ∗ = 11 − M ⊙ . As shown in the Figure, the two different setsof nucleosynthesis calculations are not in perfect agreement. Although the predicted yieldsused here only vary by a factor of ∼
2, and this level of uncertainty is often quoted, wenote that even larger variations are possible. The predicted abundances of both Al and Fe are extremely sensitive to variations in the triple- α reaction (Tur et al. 2010), and thecorresponding reaction rates are not precisely known. As a result, the uncertainties in theyields for SLRs can be larger than a factor of two. Finally, we note that the yields for theWW models are only given up to m = 40. Here we extrapolate to larger values assuming M A = constant for m >
40; although stars more massive than this threshold are rare, thischoice provides another source of uncertainty.For completeness, we show the expected yields of the isotopes Cl, Ca, and Mn inFigure 2 (where these results are taken from the WW group). The abundance of Cl is lowerthan those of Al and Fe by roughly an order of magnitude, so that its contribution to theionization rate is correspondingly smaller. On the other hand, while Ca and Mn haverelatively large abundances, they both decay via electron capture, and do not (immediately)emit ionizing energy.We note that this analysis assumes that all of the SLRs produced by supernova ex-plosions are actually ejected outward and become available for enrichment. In practice,however, calculations of supernovae remain challenging (M¨osta et al. 2014), so that the de-gree of mixing and fallback of synthesized material is not completely known. Indeed, somecosmochemical models suggest some fallback is necessary to explain the particular isotopicabundances observed in the solar system (e.g., Liu 2014). A significant amount of fallbackwould lower the SLR abundances determined in this paper. 9 –
10 20 30 40 50 60 70 80 9020 10010 20 30 40 50 60 70 80 9020 100
Fig. 1.— Yield M A of short-lived radioactive nuclei as a function of progenitor mass (bothquantities are given in units of M ⊙ ). The curves marked by solid squares indicate the yieldsfor Al; the curves marked by open circles indicate yields for Fe. Results are presented fortwo different sets of stellar evolution calculations: Yields from the WW models are shown asblue solid curves (Woosley & Weaver 1995; Rauscher et al. 2002), whereas yields from theLC models are shown as red dashed curves (Limongi & Chieffi 2006). 10 –
10 20 30 40
Fig. 2.— Yield M A of short-lived radioactive nuclei as a function of progenitor star mass (inunits of M ⊙ ). Solid magenta curve marked by solid squares indicates the yields for Cl; thedashed blue curve marked by open circles provides the yield for Ca; the dotted green curvemarked by open triangles provides the yields for Mn. Results are taken from the WWstellar evolution calculations (Woosley & Weaver 1995; Woosley et al. 2002; Rauscher et al.2002). 11 –
For a cluster of given membership size N , and for a given stellar IMF, we want to findthe abundance distribution for each type of radioactive species (here we denote an arbitraryradionuclide as A k ). As outlined below, we can determine these abundance distributionsvia sampling. To leading order, however, the distributions can be characterized by theirexpectation values and widths (or variance). At higher order, however, the distributionsshow significant departures from gaussians.To start, we define the yield weighted by the stellar IMF. More specifically, for a givenIMF and radionuclide A k , we define the expectation value per star h M [ A k ] i ∗ of the yield tobe h M [ A k ] i ∗ ≡ Z m ∞ m min M [ A k ; m ] dN ⋆ dm dm , (9)where M [ A k ; m ] is the yield of species A k for a progenitor star of mass m . The upper limit ofintegration is set by the maximum stellar mass, which is taken here to be m ∞ = 120; since thestellar IMF is a steeply decreasing function of stellar mass, and since the expected yields arenot steeply increasing with mass, most of the support for the integral in equation (9) occursfor the smaller masses, so that results are not overly sensitive to the upper limit. The lowerlimit of integration is set by the minimum mass required for a star to explode and therebyprovide nuclear enrichment; this criterion thus implies that m min ∼
8. However, the main-sequence lifetime of stars with m ∼
8, about 20 Myr, is longer than the expected time forwhich circumstellar disks retain their gas (3 – 10 Myr; Hern´andez et al. 2007), and is longerthan the lifetime of most embedded clusters (3 – 10 Myr; Allen et al. 2007; Gutermuth et al.2009). More recent studies based on
Spitzer observations (Cizea et al. 2007), and SCUBA-2surveys (Williams et al. 2013), suggest that the lifetime of circumstellar disks could be evenshorter, 1 – 3 Myr. As a result, although we use the full range of stellar masses (with m min = 8) to compute expectation values, these results can be subject to a reduction factorbecause of constraints on stellar lifetimes. Finally, we note that the expectation value definedvia equation (9) is normalized so that it provides the expected radioactive yield per star .Because only the massive stars contribute to the yields, this expectation value per star ismuch smaller than the radioactive yield per supernova; these yields are smaller by the factor F SN . For the sake of definiteness, we use F SN ≈ .
005 as a standard benchmark value; thenuclear yields can be scaled upward, or downward, for alternate values of F SN .For the two SLRs of greatest interest, Al and Fe, Figure 3 presents the expectationvalue for the yield per star. These yields are plotted here as a function of the index γ ofthe stellar initial mass function. Note that the yields are given in units of “microsuns” µM ⊙ (where 1 µM ⊙ = 10 − M ⊙ ). For the WW models of stellar evolution, the nuclear yields vary 12 –slowly with index γ and the values for Al and Fe are nearly equal (see the blue solidcurves in Figure 3). In contrast, for the stellar models of LC, the nuclear yields vary by afactor of ∼ γ (see the red dashed curves in the Figure).In addition, the yield for Al is significantly larger than that for Fe. The variations shownin Figure 3 provide a measure of the uncertainty in the yields, due to possible variations inthe index of the IMF and/or uncertainties in the stellar nucleosynthesis calculations.Another source of variation in the nuclear yields arises because the lifetimes of massivestars vary with stellar mass, and these timescales are comparable to the times over whichboth clusters and disks remain intact. If we consider sufficiently short time scales for diskand cluster evolution, we need to calculate the nuclear yields produced by only those starswith the highest masses. To quantify this effect, Figure 4 presents the nuclear yields perstar, calculated using equation (9) with different values for the minimum stellar mass m min .As expected, the yields (per star) decrease steadily with increasing minimum mass. Manyprevious studies for nuclear enrichment of our Solar System use a progenitor with m = 25 asa standard value; a star with this mass spends 6.7 Myr burning hydrogen and a total timeof about 7.5 Myr before experiencing core collapse (Woosley et al. 2002). Figure 4 indicatesthat the yields determined with m min = 25 are lower than those obtained with the fullspectrum of stellar masses by a factor of ∼ Al yields from the WW models and the Fe yields from the LC models vary by less than this factor, whereas the Al yields fromthe LC models and the Fe yields from the WW models vary by a larger factor.We can also plot the nuclear yields as a function of time, as shown in Figure 5, whereonly stars that have evolved far enough to explode in time t are included in the yield. Thisfigure is essentially equivalent to Figure 4, where the minimum mass m min is converted intothe time required for a star of the given mass to evolve and explode as a supernovae. Starsof the highest mass ( m ∼ t ∼ t < Al and Fe), all of the yields reach about half of their asymptotic values by t ∼
10 Myr.A related quantity of interest is the ratio of the mass in Fe produced to that of Al.Figure 6 shows this mass ratio as a function of the IMF index γ . Here, the yields of both 13 – Fig. 3.— Radioactive yields per star for Al and Fe versus index γ of the stellar IMF. Theyields, which are given in units of µM ⊙ = 10 − M ⊙ , are proportional to the fraction of starsabove the supenova mass threshold, taken here to be F SN = 0 . Al are marked by solid square symbols,whereas the corresponding curves for Fe are marked by open circles. 14 –
10 15 20 25 30
Fig. 4.— Radioactive yields per star for Al and Fe versus minimum mass of progenitorstar included in the distribution. The yields are given in units of µM ⊙ = 10 − M ⊙ and theindex of the stellar IMF γ = 1.5. Results are shown for the two different sets of stellarevolution calculations, WW (blue solid curves), and LC (red dashed curves). The curvespresenting yields for Al are marked by solid square symbols, whereas the correspondingcurves for Fe are marked by open circles. 15 –
Fig. 5.— Radioactive yields per star for Al and Fe versus time (in Gyr). For a giventime, only those stars that have evolved enough to explode as supernovae are included in theintegral over the stellar mass distribution. The yields are given in units of µM ⊙ = 10 − M ⊙ and the index of the stellar IMF γ = 1.5. Results are shown for the two different sets of stellarevolution calculations, WW (blue solid curves), and LC (red dashed curves). The curvespresenting yields for Al are marked by solid square symbols, whereas the correspondingcurves for Fe are marked by open circles. 16 –
Fig. 6.— Ratio of radioactive yields per star for Fe and Al, plotted here as a functionof the index γ of the stellar IMF (where the yields are given in units of mass). Results areshown for the two sets of stellar evolution calculations, WW (blue solid curve), and LC (reddashed curve). Horizontal line depicts the observed ratio (Diehl et al. 2006; Diehl 2013),where the shaded region delineates the uncertainty in the measurement. 17 –
10 15 20 25 30
Fig. 7.— Ratio of radioactive yields per star for Fe and Al, plotted here as a function ofthe minimum mass of progenitor star included in the distribution. The yields are given inunits of mass, and the index of the stellar IMF γ = 1.5. Results are shown for the two setsof stellar evolution calculations, WW (blue solid curve), and LC (red dashed curve). 18 –nuclear species are averaged over the stellar IMF, and then the mass ratio is found (reversingthe order of these operations would produce a different result). This mass ratio also dependson the minimum mass included in the determination of the radioactive yields; the resultingmass ratio is shown as a function of the minimum mass in Figure 7. Whereas the overallyields must decline with increasing minimum mass (Figure 4), the mass ratio displays morecomplicated behavior. As the index γ increases, the stellar IMF is weighted more towardstars of lower masses; the mass ratio for the WW models increases, whereas the mass ratiofor the LC models decreases. Similarly, as the minimum mass increases, the predicted massratio decreases for the WW models of stellar evolution, but increases for the LC models. The different species of radioactive nuclei decay at different rates, so their relatativeabundances vary with time. Since one important implication of this work is the ionizationprovided by SLRs, we use the ionization rate to illustate this time dependence. In addition,both molecular clouds and circumstellar disks, the two environments of interest, are com-posed primarily of H , so that we focus on the ionization of molecular Hydrogen. Ionizationrates for other species can be scaled accordingly. For a given nuclear species, labeled bythe index ‘ k ’, the ionization rate ζ k per hydrogen molecule (Umebayashi & Nakano 2009) isgiven by the expression ζ k = E k ω I τ − k X k A − k exp[ − t/τ k ] , (10)where E k is the energy per decay ( ∼ τ k = τ / / ln 2 is the decaytime, X k is the mass fraction, and A k is the atomic weight. The parameter ω I ≈
36 eV is theaverage energy required for an energetic particle to produce an electron-ion pair by passingthrough H gas (Umebayashi & Nakano 2009).The two most important SLRs are Al and Fe, which have abundances that areroughly equal when averaged over the stellar IMF (see Figure 3 for further detail). Becauseof the difference in half-lives, however, Al (with τ / = 0.72 Myr; Rightmire et al. 1958;Norris et al. 1983) will dominate the ionization rate at early times (measured from the timeof the supernova explosion), and Fe (with τ / = 2.6 Myr; Rugel et al. 2009), will dominateat later times. The third species of possible interest, Cl, has a smaller abundance and ashorter half-life ( τ / = 0.3 Myr; Eckstr¨om & Firestone 2004). In addition, the net energyper decay for Cl is only about 0.286 MeV, which is appreciably smaller than that of Al(3.065 MeV) and Fe (2.741 MeV); these values were obtained by averaging the decay energyover the various channels, weighted by the branching ratios, by using the data presented inTable 2 of Umebayashi & Nakano (2009). 19 –The resulting time dependence of the ionization rate is illustrated in Figure 8. Here, wemodel the averaged expected behavior by considering an effective “supernova” that producesthe IMF-averaged yields at t = 0. The resulting ionization rate is normalized so that thesum of the contributions from the three SLRs ( Al, Cl, and Fe) is unity at t = 0. Thetime evolution of the ionization rate then shows the expected behavior: The contribution of Cl is minimal and becomes less important with time. Ionization due to Al dominates atearly times, whereas that due to Fe dominates at later times, with the crossover occurringat about t ≈ . M ⊙ of material, for example, the ionization rateper hydrogen molecule would be about ζ ≈ × − sec − . This value is comparable to,but somewhat larger than, the ionization rates due to cosmic rays in the interstellar medium(where typical estimates imply ζ ≈ − × − sec − ; e.g., van der Tak & van Dishoek2000). Since the star formation efficiency ǫ sfe is low, the amount of material that the ejectamix with is expected to be larger by a factor of 1 /ǫ sfe ∼ ζ ∼ − sec − (see Section 6). Notice also that one expects a wide range of values for the ionization ratesdue to both SLRs and cosmic rays (Fatuzzo et al. 2006; Cleeves et al. 2013a,b).For sufficiently short spans of time, the contribution of SLRs to ionization rates dom-inates over that of long-lived radioactive species. For the abundance patterns deduced forthe Solar Nebula, for example, the contribution from long-lived nuclei is smaller by a factorof ∼ (Umebayashi & Nakano 2009) at the start of the epoch ( t = 0). For the benchmarkcase illustrated by Figure 8, the short-lived nuclear species continue to dominate until time t ≈
16 Myr. As a result, we ignore the contribution of long-lived nuclei for the remainder ofthis paper. Nonetheless, this issue should be examined in the future.
4. Distributions of Nuclear Yields for Clusters4.1. Yields for Clusters in the Large-N Limit
In the limit of large clusters, N → ∞ , the distributions of nuclear yields will approacha gaussian form (see below). For this regime, this section determines the mean values forthe distributions and their expected widths, where these quantities are a function of stellarmembership size N . 20 – Fig. 8.— Contributions of the three most important nuclear species to the SLR-inducedionization rate (using nuclear yields from the WW stellar evolution models). The ionizationrate is normalized to unity at t = 0, the time of the supernova explosion. The relativeabundances are determined by using the yields per star, which have been averaged over thestellar IMF. The curves correspond to Al (solid), Cl (dotted), Fe (dashed), and thetotal (heavy solid). 21 –Here, the radioactive yield for a cluster with size N is given by the sum M [ A k ; N ] = N X j =1 M [ A k ; m j ] , (11)where M [ A k ; m j ] is the radioactive yield of species A k from the jth cluster member (withmass m j ). Only the massive stars (with m j >
8) explode as supernovae at the end oftheir lives and contribute to the radioactive yield of the cluster. As a result, the quantity M [ A k ; m j ] = 0 for most cluster members.In this treatment, we assume that the radioactive yield for a given star is determined bythe stellar mass, which is drawn independently from a specified stellar IMF. The sum fromequation (11) is thus the sum of random variables. Here the variables are the radioactiveyields of the individual stars, so that the variables are drawn from a known distribution,which is in turn determined by the IMF and by stellar nucleosynthesis. In the limit N ≫ h M [ A k ; N ] i of the radioactive yield for the cluster is given by h M [ A k ; N ] i = N h M [ A k ] i ∗ , (12)where h M [ A k ] i ∗ is the expectation value of the yield of radioactive species A k per star, asdefined via equation (9). Keep in mind that the radioactive yield for a cluster will convergeto the value implied by this expectation value in equation (12) only in the limit of large N . The minimum value of cluster membership N required for this convergence is discussedbelow. Small clusters often display large departures from the expectation value.In the limit of large N ≫
1, the central limit theorem implies that the distribution ofyields M [ A k ; N ] must approach a gaussian form (e.g., Feller 1968). In practice, however,the convergence is rather slow. In addition, since only a small fraction of stars contribute tothe nuclear abundances, large N values are necessary for convergence. One of the issues ofinterest here is the value of stellar membership N required for statistical considerations tobe valid. In the large N limit, where the central limit theorem applies, the resulting gaussianform for the composite distribution is independent of the form of the initial distributions,i.e., it is independent of the stellar IMF and the mass-luminosity relation. The width of thedistribution also converges to the value given by h σ i = 1 N N X j =1 σ j ⇒ h σ i = √ N σ ∗ , (13)where σ ∗ is the width of the individual distribution and is defined by σ ∗ ≡ h M [ A k ] i ∗ − h M [ A k ] i ∗ . (14) 22 –The expectation values h M [ A k ] i ∗ and widths σ ∗ of the distributions of radioactive yieldsare listed in Table 1 for the five species of radionuclides considered in this paper (and forour chosen form of the stellar IMF). Results for given for all five isotopes using the WWmodels of stellar evolution, whereas results are only given for Al and Fe using the LCmodels (results are not available for the other nuclear species). Both the expectation valuesand the widths are given in units of µM ⊙ . For each radioactive species, the expectationvalue h M [ A k ] i ∗ and the width σ ∗ of the distributions are given for three values of the index γ of the stellar IMF. The results are not overly sensitive to the slope of the IMF in themass range m >
8, primarily because the radioactive yields are not sensitive functions ofprogenitor mass (see Figures 1 and 2). However, the yields are directly proportional to thefraction F SN of stars above the supenova mass threshold; for the cases shown in Table 1 wehave used F SN = 0 . N X in a cluster required for thewidth of the distribution for the cluster to be smaller than the expectation value of the yieldof the cluster (for the WW yields). Since the expectation value of the yield for a clusteris proportional to N (see equation [12]) and the width of the distribution is proportionalto √ N (see equation [13]), the benchmark cluster size N X = ( σ ∗ / h M [ A k ] i ∗ ) . Notice alsothat this cluster size N X is that necessary to make the distribution of yields narrower thanits expectation value. Even larger membership sizes N are required for the distribution toapproach a pure gaussian form.Table 1 shows an interesting discrepancy between the results obtained from the two nu-clear models. The expectation values for the nuclear yields per star show relatively moderatedifferences between the WW and LC models (compare columns 3 and 6 for Al and Fe),as expected from the results shown in Figures 3 – 7. However, the widths of the distributions(compare columns 4 and 7) are much wider for the LC models than for the WW models,especially for Fe. Part of this difference arises because the LC models provide results forlarger progenitor masses, and the yields increase with mass. The WW models end at m = 40,and we use the yields for the m = 40 model for all higher masses; an alternate extrapolationscheme could resolve part of this difference. However, the LC models also show a steeperdependence of nuclear yields with progenitor mass, especially for Fe (see Figure 1). Asshown in the following subsection, the wider distributions for the LC models require largerclusters (in stellar membership size N ) to approach gaussian forms for the distribution ofradioactive yields per cluster. 23 – Table 1: Parameters for Radio Isotope Distributions
Nuclear Species γ h M [ A k ] i ∗ ( W W ) σ ∗ ( W W ) N X h M [ A k ] i ∗ ( LC ) σ ∗ ( LC ) Al 1.5 0.195 3.88 394 0.297 6.921.7 0.180 3.59 398 0.267 6.131.3 0.0142 0.623 1940 Cl 1.5 0.0137 0.611 19901.7 0.0131 0.594 20601.3 0.0699 2.76 1560 Ca 1.5 0.0697 2.74 15401.7 0.0686 2.70 15501.3 0.329 7.75 556 Mn 1.5 0.326 7.63 5491.7 0.318 7.44 5491.3 0.179 3.52 387 0.213 10.1 Fe 1.5 0.175 3.48 396 0.168 8.531.7 0.169 3.42 409 0.132 7.16Table 1: The first column of the table gives the species of radionuclide and the second columnlists the index of the stellar IMF. In the third column, h M [ A k ] i ∗ is the expectation valueof the radioactive yield per star using the WW models, whereas σ ∗ (fourth column) is thewidth of the corresponding distribution of yields; both quantities are given in units of µM ⊙ = 10 − M ⊙ . In the next column, N X is the number of stars in a cluster required for thewidth of the distribution of yields for the cluster to be smaller than the expectation value.For Al and Fe, the table also lists the expectation value of the yield per star and thecorresponding width of the distribution for the LC models.
Table 2: Radio Isotope Properties for the Early Solar Nebula
Nuclide Daughter Reference Half-life Mass Fraction Mass Uncertainty A k D k R k τ / (Myr) X k M k ( pM ⊙ ) (∆ M k ) /M k Al Mg Al 0.72 3 . × −
190 0.11 Cl Ar Cl 0.30 8 . × −
44 0.46 Ca K Ca 0.10 1 . × − Mn Cr Mn 3.7 4 . × −
20 0.13 Fe Ni Fe 2.6 1 . × −
55 0.35 24 – Y C (cid:144) N H Μ M Ÿ L N H Y L Fig. 9.— Distribution of radioactive yields of Al (using WW results) for clusters and N = 10 (narrow curves), N = 3000 (wider curves), and N = 1000 (widest, irregularcurve). Solid curves show the distributions obtained from sampling the IMF for a largecollection of clusters with fixed membership size N . Dotted curves show the correspondinggaussian profile predicted analytically. The yields for all distributions are scaled by thestellar membership size N . 25 – Y C (cid:144) N H Μ M Ÿ L N H Y L Fig. 10.— Distribution of radioactive yields of Fe (using WW results) for clusters and N = 10 (narrow curves), N = 3000 (wider curves), and N = 1000 (widest, irregularcurve). Solid curves show the distributions obtained from sampling the IMF for a largecollection of clusters with fixed membership size N . Dotted curves show the correspondinggaussian profile predicted analytically. The yields for all distributions are scaled by thestellar membership size N . 26 – Next we determine the distributions of nuclear yields for clusters by direct numericalsampling of the stellar IMF. Figure 9 shows the distribution of radioactive yields of Alfor clusters with fixed stellar membership size N = 1000, 3000, and 10 . The solid curvesshow the distributions obtained by sampling the IMF for a large number of clusters withfixed N , where the nuclear yields are determined by the WW stellar models. Note that theradioactive yields, shown on the horizontal axis, are scaled by the cluster size N (so that thepeak and mean values are nearly independent of N ). For comparison, the dotted curves showthe gaussian profiles calculated using the mean value from equation (12) and the variancefrom equation (13). The relative widths of the distributions decrease with increasing N , asexpected (see Section 4.1). For the larger stellar membership sizes ( N = 3000 and 10 ),the clusters contain enough massive stars so that the distributions are close to the gaussianbenchmarks. Nonetheless, the true (sampled) distributions are slightly asymmetric, withthe peak value somewhat smaller than the expectation value. Although these departuresare small for large N , clusters with smaller membership N display large departures fromgaussian profiles. The figure also shows the result obtained by sampling clusters with only N = 1000 members (shown as the irregular, wide curve). In this case, many clusters onlyhave one or two massive stars large enough to explode, so that the low end of the distributionshows a great deal of structure (which reflects the irregular structure of the radioactive yieldsas a function of progenitor mass, as shown in Figure 1). Although the spikey nature of thedistibution (for N = 1000) is visually prominent, perhaps the most important departure fromfrom a gaussian form is the asymmetry toward lower values. As expected, in the oppositelimit where N → ∞ , the distributions approach true gaussian forms.Figure 10 shows the corresponding distributions of yields for Fe, again using the WWnuclear models and for cluster membership sizes N = 1000, 3000, and 10 . These distribu-tions are analogous to those obtained for Al (compare with Figure 9). As expected, thelarger clusters (with N = 3000 and 10 ) display nearly gaussian profiles, as shown by thedotted curves in the figure. On the other hand, slightly smaller clusters with N = 1000 showcomplicated, irregular structure, for the same reasons discussed above. The distributions arealso asymmetric, with the peak value smaller than the mean value; this trend is small for N ≥ N = 1000.Next we consider the effect of the stellar IMF on the resulting distributions of nuclearyields. Figure 11 shows the distributions of yields for clusters with N = 10 stars and forthe WW nuclear models. Results are shown here for both Al (red curves) and Fe (blackcurves). For each isotope, distributions for shown for three choices of the index γ of thestellar IMF, where γ = 1.25 (dashed curves), 1.5 (solid curves), and 1.75 (dotted curves). 27 –The effect of varying the index γ is modest: The distributions shift their mean values slightlyas γ varies, as expected given the dependence of the expectation values shown in Figure 3.The expected yields per star decrease with increasingly index γ , so that the distributionsmove to the left as the stellar IMF becomes steeper. For clusters with different stellarmembership sizes N (not shown), one obtains analogous distributions; they are relativelywider for smaller N and narrower for larger N , as illustrated in Figures 9 and 10.Figure 12 compares the two models for stellar nucleosynthesis used in this work. Here wepresent distributions of nuclear yields for Al and Fe, as determined using both the WWand LC results. The cluster membership size is taken to be N = 10 for this comparison.For the WW models, the distributions are relatively narrow and approach gaussian forms,as depicted by the black curves in the figure (and as shown previously). For the LC models,however, the distributions are markedly wider, as depicted by the red curves. This behavioris in keeping with the larger widths σ ∗ given in Table 1. For Al, the distribution for theLC yields is shifted to the right compared to that for the WW yields — consistent with thelarger expectation value found using the LC models — and is close to gaussian. For Fe,however, the distribution retains a significantly non-gaussian form, even for this relativelylarge stellar membership size N . For the LC models, we thus find that larger clusters (larger N ) are required for the distributions of nuclear yields to become gaussian, with the effectmore pronounced for Fe. This behavior results from the steep dependence of the Fe yieldswith progenitor mass (see Figure 1) coupled with the steepness of the stellar IMF: A few,rare large stars can contribute an enormous amount of Fe, so that large stellar populations(large N ) are required to fully sample the distribution. In this case, the stellar membershipsize required for a gaussian distribution is larger than N = 10 . In this section we find the expectation values for various quantities of interest. We firstconsider the expectation value for the radioactive yield Y C of an entire cluster. To obtain thisquantity, we integrate over the distribution of cluster sizes and the distribution of possibleradioactive yields (per cluster): h Y C i = Z N max C N N dN Z ∞ dY = Z N max C N N dN N h M [ A k ] i ∗ = C N h M [ A k ] i ∗ log N max . (15)Since C N ≈ N max ≈ , we find that h Y C i ≈ h M [ A k ] i ∗ , (16) 28 – Y C (cid:144) N H Μ M Ÿ L N H Y L Fig. 11.— Distribution of radioactive yields for difference values of the index γ of the stellarIMF. Distributions are shown for both Al (red curves) and Fe (black curves) using theWW nuclear model. The cluster size is taken to N = 10 . For each isotope, distributionsare shown for three values of the index γ = 1.75 (dotted curves), 1.5 (solid curves), and 1.25(dashed curves). 29 – Y C (cid:144) N H Μ M Ÿ L N H Y L Fig. 12.— Comparison of the two nucleosynthesis models for the distribution of radioactiveyields in clusters with N = 10 . The black curves show the distributions for the WW models,whereas the red curves show the distributions for the LC models. The distributions of yieldsfor Al are shown as the solid curves, whereas the distributions for Fe are shown as dashedcurves. 30 –where h M [ A k ] i ∗ is the radioactive yield per star for a given nuclear species (as given in Table1). For example, a “typical” cluster produces only about 2 . × − M ⊙ of Al. This value issmall because the typical cluster is small: The expectation value for the cluster membershipsize is given by h N i = Z N max C N N N dN = C N log N max ≈ . (17)This expectation value is small because we take the distribution of cluster membership sizesto extend all the way down to N = 1. The resulting distribution will thus have many smallclusters, which leads to the small value of h N i . These small clusters contain only a smallfraction of the stellar population, however, so that most stars reside in much larger clusters.A related quantity is the radioactive yield per cluster that a typical star experienceswithin its birth cluster. This quantity, denoted here as h Y C ∗ i , is given by h Y C ∗ i = Z N max C P N dN Z ∞ dY = Z N max C P N dN N h M [ A k ] i ∗ = C P h M [ A k ] i ∗ N max . (18)In this case, C P = 1 / log N max and we obtain the estimate h Y C ∗ i = N max log N max h M [ A k ] i ∗ ≈ . × h M [ A k ] i ∗ ≈ . M ⊙ . (19)As considered in the next section, only a small fraction of the nuclear yield from a clusterwill be delivered to any given solar system. Before considering that issue in detail, however,it is useful to obtain a rough estimate: If we use a typical distance of a solar system to thecluster center of d ∼ f ∼ × − . If a typical star is born in acluster with nuclear yield described by equation (19), the expected mass of Al impingingon a typical star/disk systems is thus about 4 × − M ⊙ . For comparison, the estimatedmass fraction of Al in the early Solar Nebula is X = 4 × − , so that the mass of Al isabout 2 × − M ⊙ (five times larger than the canonical value — see below). Although thisargument uses only typical values, it suggests that clusters can provide nuclear enrichmentto their constituent solar systems at levels comparable to (but often somewhat less than)those estimated for our Solar Nebula.
5. Distributions of SLR Yields Delivered to Solar Systems
A typical star in a typical cluster will intercept a only fraction f of the radioactive yieldproduced by the entire cluster. To start, we ignore timing issues. In this limiting case, the 31 –fraction f is given by the geometrical factor f = f ( r ) = πr d πr cos θ , (20)where r d is the disk radius, and r is the distance from the solar system to the cluster center(where the high mass stars, and hence the supernova ejecta, originate). The factor of cos θ takes into account the fact that the disk is not, in general, facing the supernova blast wave;the distribution of angles is expected to be uniform in µ ≡ cos θ with a mean of 1/2. Theradius r must be larger than the radius for which the disk (with disk radius r d ) is strippeddue to the blast; for a disk radius r d = 30 AU, and for typical supernova energies, thisminimum radial distance r min ≈ . ∼ . µ m (see Ouellette et al. 2010,in particular their Figure 6). Unfortunately, however, the expected size of grains producedduring supernovae remain uncertain. A recent study advocates dust grains with sizes smallerthan ∼ . µ m (Bianchi & Schneider 2007), which would lead to lower injection efficiencies.In this work, we assume 100 percent efficiency, but the results can be scaled (downward) forany choice of this parameter. Another important issue is the transport of the SLRs afterthey are acquired; simulations carried out to date suggest that mixing is indeed efficient(Boss 2011, 2013). The expectation value h Y ∗ i for the yield of SLRs intercepted by a given solar systemtakes the form h Y ∗ i = Z N max C P N dN Z ∞ dY Z Rr min dPdr f ( r ) dr , (21) 32 –which can be written h Y ∗ i = Z N max C P N dN N h M [ A k ] i ∗ Z Rr min − pR (cid:16) rR (cid:17) − p πr d πr dr , (22)where we have used equation (8) to specifiy the probability distribution of the radial positionsand we have taken the mean value of cos θ . This result can be simplified to the form h Y ∗ i = C P h M [ A k ] i ∗ (3 − p ) r d Z N max dNR − p Z Rr min drr p . (23)We can evalute the above result for any value of p and any form for the cluster radius function R ( N ), which is specified by power-law index α (from equation [7]). For the canonical choices p = 3/2 and α = 1/3, we obtain h Y ∗ i ≈ C P h M [ A k ] i ∗ r d r min − / R − / N / N max1 / ≈ . × − h M [ A k ] i ∗ , (24)where we have used typical values to obtain the final approximate equality. Since the yieldsof both Al and Fe are of order 0.2 µM ⊙ (see Figure 3 and Table 1), the typical yield forthese SLRs is about 10 – 20 pM ⊙ (or 1 − × − M ⊙ ).The expectation values for SLR yields discussed here are comparable to — but somewhatsmaller than — the SLR masses that are inferred for the early Solar Nebula. For comparison,Table 2 lists the isotopes of interest for this paper, along with the daughter products, thereference isotopes, the half-lives, the mass fractions, and the total masses. The abundanceslisted in the table are inferred from meteoritic data, which has been compiled by numerousprevious authors (e.g., see Umebayashi & Nakano 2009; Looney et al. 2006; Young et al.2005; Dauphas & Chaussidon 2011; and references therein). For each SLR, the total massis estimated from the mass fraction, where we assume a typical mass for the Solar Nebula of M d = 0 . M ⊙ . We note that the abundance for Fe listed here (taken from Tachibana et al.2006) has been re-measured by other workers (Tang & Dauphas 2012), who found lowerabundances. In any case, the total SLR masses listed in Table 2 are somewhat larger thanthe expectation values discussed above. Taken at face value, this finding implies that SolarSystem abundances result from the high end of the distribution of possible values. To asssessthe probabilties, we need the full distribution, which is determined in the next subsection.
Using the probability distributions for cluster yields and for the radial positions of starswithin clusters, we can find the distribution of radioactive yields delivered to the constituent 33 –solar systems. Let Y C be the nuclear yield for a given cluster, where the value of Y C isdistributed according to a probability distribution dP/dY . Let ξ = r/R be the radialposition of the recipient solar system within the cluster, where R is the cluster radius; theposition is distributed according to dP/dξ , which depends on the density profile of the cluster.The mass M ss of radioactive material delivered to a solar system is given by M ss = Y C f ( r ),which can be written as M ss = Y C πr d πR ξ µ , (25)where µ = cos θ . To find the distribution of the mass M ss of intercepted nuclear material,we need to specify the distributions dP/dY , dP/dξ , and dP/dµ . The projection factor µ hasuniform-random distribution ( dP/dµ = 1) on the interval [0,1]. To start, however, we ignoreprojection effects by setting µ = 1 (projection effects will be reinstated later).As shown above, for clusters with sufficiently large stellar membership size N , thedistribution of yields is nearly gaussian, i.e., dPdY = A Y exp (cid:20) − ( Y C − h Y i ) σ (cid:21) , (26)where A Y is the normalization constant and is given byA Y = 2 √ πσ (cid:20) (cid:18) h Y i√ σ (cid:19)(cid:21) − , (27)where Erf( z ) is the error function (Abramowitz & Stegun 1972). Further, the expectationvalue h Y i and the width σ of the distribution are given by h Y i = N h Y i ∗ and σ = √ N σ ∗ , (28)where the quantities with starred subscripts denote the values per star, calculated fromconvolving the nuclear yields with the stellar IMF.As discussed earlier, the distribution of radial positions depends on the cluster densityprofile, i.e., dPdξ = (3 − p ) ξ − p , (29)where the index p of the cluster density profile lies in the range 1 ≤ p ≤
2. Note that theradial position has a uniform (constant) distribution for the choice p = 2.Next we define a new variable y ≡ Y C h Y i , (30) 34 –so that the mass M ss can be written M ss = h Y i πr d πR yξ = M X , (31)where the second equality defines the composite variable X ≡ yξ , (32)and the benchmark scale M ≡ h Y i πr d πR . (33)For typical values we obtain M = 0 . pM ⊙ (cid:18) N (cid:19) (cid:18) h Y i ∗ . µM ⊙ (cid:19) (cid:16) r d
30 AU (cid:17) (cid:18) R (cid:19) − . (34)The benchmark mass scale is thus of order 1 pM ⊙ = 10 − M ⊙ ; further, cluster radii areobserved to vary as R ∝ N α , where 1 / ≤ α ≤ /
2, so that this mass scale is a slowlyvarying function of stellar membership size N . For comparison, the inferred abundances ofSLRs for the early Solar Nebula correspond to masses in the range 20 – 200 pM ⊙ (Table2). Note that the expected value for the enrichment mass is given by the expectation value M h X i , which will be larger than M . We can evalulate the quantity h X i to obtain h X i = Z ∞ X dPdX dX = (3 − p )( p − " ξ − ( p − min − − ξ − pmin Bλ exp( − / λ ) (cid:3) , (35)where the constants B and λ are defined below. We also have introduced a minimumradius ξ min = r min /R for the solar system location within the cluster, and adjusted thenormalization constant accordingly. Without this cutoff, the solar system could lie arbitrarilyclose to the enrichment sources and the integral would diverge. In practice, solar systemsthat are too close to the cluster center, where the supernova explosions occur, will havetheir disks destroyed; as a result, blast wave physics enforces a minimum radius of 0 . − . ξ min ∼ . − . λ ≪ ξ min ≪
1; in this limit, the expectation value reducesto the simpler form h X i → (3 − p )( p −
1) 1 ξ p − min . (36)This expectation value thus has a typical value h X i ∼
10. The corresponding expectedvalue for radioactive mass enrichment is thus about 10 pM ⊙ , roughly comparable to, butstill somewhat smaller than, the levels inferred for the early Solar Nebula. 35 –The distribution for the scaled variable y takes the form dPdy = A Y h Y i exp (cid:20) − h Y i σ ( y − (cid:21) , (37)where the normalization constant A Y is given by equation (27). If we define the quantity λ ≡ σ h Y i = σ ∗ √ N h M [ A k ] i ∗ , (38)the distribution simplies to the form dPdy = B exp (cid:20) − ( y − λ (cid:21) , (39)where the normalization constant is given by B = 2 √ πλ (cid:20) (cid:18) √ λ (cid:19)(cid:21) − . (40)Now we need to determine the cummulative probability P ( X ) for the variable X . Theprobability is given by the double integral P ( X ) = Z dξ dPdξ Z Xξ dPdy dy , (41)which can be written in the form P ( X ) = Z dξ (3 − p ) ξ − p Z Xξ dy B exp (cid:20) − ( y − λ (cid:21) . (42)The differential probability is then given by dPdX = (3 − p ) B Z dξ ξ − p exp (cid:20) − ( Xξ − λ (cid:21) . (43)For large X ≫
1, this result reduces to the power-law form dPdX ≈ (3 − p )1 + Erf (cid:2) √ / (2 λ ) (cid:3) X − (5 − p ) / λ √ π Z ∞ u a du e − ( u − / λ ≡ CX − (5 − p ) / , (44)where the second equality defines the normalization constant C and where we have defined a = (3 − p ) /
2. In the limit where λ ≪
1, the expression simplifies to the form dPdX = CX − (5 − p ) / where C = 12 (3 − p ) . (45) 36 –One quantity of interest is the fraction of solar systems that will be enriched in SLRsabove a given threshold specified by X ∗ . Note that the mass of SLRs is given by M = M X ,where the benchmark mass is of order 1 pM ⊙ . The fraction of systems that receive X > X ∗ ( M > M ∗ ) can be written in the form P ( X > X ∗ ) = C (3 − p ) / X − (3 − p ) / ∗ ≈ X − (3 − p ) / ∗ . (46)The treatment thus far has neglected projection effects. Since circumstellar disks arenot, in general, aligned toward the supernova ejecta, the distribution derived above must beconvolved with the distribution of orientation angles. In practice, we have the probabilitydistribution dP/dX for the variable X that specifies the SLR masses delivered to individualsolar systems (where M = M X ). We need to find the corresponding probability distribution dP/dZ for the composite variable Z ≡ µX , where µ = cos θ is distributed uniformly on [0,1].The cummulative probability is then given by the integral P ( Z > Z ∗ ) = Z dµ Z ∞ Z ∗ /µ dPdX dX . (47)Since we are interested in the regime where X, Z ≫
1, we can use the limiting form givenby equation (45) to specify dP/dX ; as a result, the expression becomes P ( Z > Z ∗ ) = Z dµ Z ∞ Z ∗ /µ CX − (5 − p ) / dX , (48)which can be evaluated to obtain P ( Z > Z ∗ ) = 25 − p Z − (3 − p ) / ∗ . (49)As a result, the fraction of solar systems that are enriched at a given level is reduced bya factor of 2 / (5 − p ) ≈ . − .
67 due to projection effects. The corresponding differentialprobability for the variable Z then becomes dPdZ = 3 − p − p Z − (5 − p ) / . (50)To fix ideas, we use the index p = 3 / Z = µX > P ( Z > ≈ . p = 1 (2), we obtain P ( Z >
Z > ∼
10 percent for clusters with p = 3 / ∼
21 percent for clusters with p = 2. Forcomparison, after taking into account the projection factor, the expectation value is about h Z i ∼ Al that is delivered to constituent solarsystems in clusters with membership size N = 3000. The solid curve shows the distributionobtained from numerical sampling, whereas the dashed curve shows the result obtained fromequations (43) and (33). To obtain the numerical curve, we construct a theoretical sample ofone million clusters with N = 3000. For each cluster, the stellar IMF is sampled N = 3000times; for the massive stars, the yield of Al is then determined using the WW models. Withthe total supply of Al specified for the cluster, the amount delivered to individual solarsystems is determined from equation (25) after sampling the distribution of radial positions,as given by equation (8). For both distributions, the cluster radius is given by equation (7)and the index of the cluster density distribution is p = 3/2. The semi-analytical result is inexcellent agreement with the numerically determined distribution, provided that the sameparameters are used.The distributions shown in Figure 13 are subject to additional uncertainties. For ex-ample, we have used the cluster radius from equation (7), which implies R ≈ .
16 pc. Someclusters could have slightly smaller radii (say, R ∼ ∼ M given by equation (33) for the preferred choice ofexpectation value, as well as specify the width of the distribution λ using equation (38).Similarly, we can find the distributions for other SLR species (e.g., Fe) by using the ap-propriate values of M and λ . Finally we note that the SLR abundances quoted here donot include radioactive decay, so they represent the starting abundances, immediately afterinjection. The subsequent decay results in a reduction factor, e.g., as shown in Figure 8 asa function of time since the supernova explosion (see also Section 3.3 and the discussion ofSection 7). 38 – Fig. 13.— Distribution of total mass in Al delivered to solar systems living in clusters withstellar membership size N = 3000. Solid curve shows the composite distribution calculatedby numerical sampling both the stellar IMF and the radial positions within the cluster (usingdensity profile with index p = 3/2). The dashed curve shows the analytic distribution givenby equation (43) with benchmark mass scale given by equation (33). 39 –
6. Global Considerations
The radioactive yields considered here produce observable consquences on larger scales.We note that production of SLRs has been considered previously on galactic scales (e.g.,Timmes et al. 1995a,b; Diehl et al. 2006) and in molecular clouds (e.g., Gounelle & Meibom2008). This section briefly revisits the issue to see how observations on these larger scalescan constrain the distributions of SLR enrichment considered in previous sections.Let Γ SF be the star formation rate of a well-defined system. Here we consider the“system” to be either the entire Galaxy or, on a smaller scale, a molecular cloud complex.For a given system, the time evolution of the supply of radioactive elements, of species A k ,is given by dM [ A k ] dt = 1 h m i Γ SF h M [ A k ] i ∗ − ln 2 τ / M [ A k ] , (51)where τ / is the half-life; note that the star formation rate is given in units of mass per unittime (rather than the number of stars per unit time). The steady-state condition, which setsthe equilibrium abundance, can then be written in the form M [ A k ] = τ / Γ SF h M [ A k ] i ∗ (ln 2) h m i . (52)This simple treatment assumes that the radioactive nuclei decay, but cannot leave the systemby other channels. Since the half-lives of interest are short, of order 1 Myr, this assumptionshould be valid for the Galaxy as a whole. For molecular clouds, however, supernovae maynot inject all of their SLRs into the cloud, so that losses can occur. For the Milky Way galaxy, the current star formation rate is estimated to be Γ SF ≈ . ± . M ⊙ yr − , where this result (and error estimate) takes into account many independentlines of evidence (Chomiuk & Povich 2011). By definition, the corresponding supernova rateΓ SN is then given by Γ SN = F SN h m i Γ SF . (53)For our nominal values h m i = 0.5 and F SN = 0.005, we thus obtain a supernova rate of0.019 explosions per year, or, one supernova every 53 years.Using the star formation rate discussed above (Chomiuk & Povich 2011), or the cor-responding supernova rate (equation [53]), we can estimate the abundances of the nuclear 40 –species of interest for the Galaxy. The equilibrium abundance of Al is given by M [ Al] =0.78 M ⊙ (1.20 M ⊙ ) for the WW (LC) stellar evolution calculations, where we haved used γ = 1.5 as the index of the stellar IMF. Similarly, for Fe abundances, we obtain M [ Fe] =2.77 M ⊙ (2.64 M ⊙ ) for the WW (LC) models. The Fe abundances are nearly the same forthe two models, whereas the LC model predicts ∼
50% more Al; these results are a directreflection of the yields shown in Figure 3.Observations of gamma ray emission, in particular the 1808.65 keV line from Al,indicate that the current abundance of Al in the Galaxy is 2.8 ± M ⊙ (Diehl et al.2006). This value is larger than the estimate found above by a factor of ∼ . Feindicate that the line ratio of Fe to Al is R (Fe/Al) = 0.148 ± Fe to Al by the ratio 26/60 of their atomic weights. For IMF index γ = 1.5, the ratio of Fe to Al by mass is 0.67 (using LC models) so that the predicted line ratio becomes R (Fe/Al) =0.29, larger than the observed line ratio by a factor of about two. A similar conclusion wasreached by Prantzos (2004); the line ratio obtained in that work depends on the maximumstellar mass included in the analysis, as well as the possible contribution from Wolf-Rayetstars.Given the masses of Al and Fe inferred for the Galaxy, we can divide by the totalmass in gas (about 10 M ⊙ , e.g., Stahler & Palla 2004) and thereby obtain Galaxy-averagedabundances. The resulting mass fractions are X Al ∼ × − and X Fe ∼ − . As expected,these mass fractions are smaller than those inferred for the early Solar Nebula by an orderof magnitude for both Al and Fe (hence the need for radioactive enrichment).To summarize: We find that the predicted values for both the overall production rateof Al and the ratio of Fe to Al agree with observations at the factor of two level. Thisapparent discrepancy can be interpreted in two ways: We can use this level of agreement asan estimate for the uncertainties inherent in the results of the rest of this work (the calculateddistributions of nuclear yields, etc.), i.e., our results would be uncertain by a factor of two.On the other hand, both the overall abundances and the line ratio can be understood (tohigher precision) if twice as much Al is produced while keeping the production rate of Fethe same.Possible uncertainties in the theoretical formulation that affect this discrepancy includethe following:[1] The yield of Al calculated here assumes a value of F SN = 0 . F SN ). However,increasing F SN would increase the yields of both Al and Fe by the same factor, so thatthe line ratio would still be in disagreement.[2] We have assumed that the index of the IMF γ = 1 .
5. Smaller values of the indexresult in more high-mass stars in the tail of the distribution and hence larger yields (seeFigure 3). However, smaller values of γ increase the ratio of Fe to Al, which works in thewrong direction (see Figure 6). If, instead, we increase the value of the index to γ = 2, themass ratio decreases to 0.46, which implies a corresponding line ratio R (Fe/Al) = 0.20. Thisvalue is consistent (within the error bars) with the observed value of R (Fe/Al) = 0.148 ± F SN ) to produce the correct overall yields.[3] The star formation rate for the Galaxy could be underestimated. However, the esti-mate used here includes multiple constraints (see Chomiuk & Povich 2011), so that changingthe star formation rate could result in disagreement with other observations. In addition,a larger star formation rate would increase the abundance of Al, but would leave the lineratio unexplained.[4] The radioactive yields per star, as determined from the stellar evolution models, couldrequire modification. Yields from the current stellar models most likely contain uncertaintiesat (approximately) the factor of two level, as indicated by the differences between the models.[5] The galaxy could have an additional source of Al. In this scenario, supernovaewould provide all of the Fe and half of the Al, thereby allowing the star formation rateand stellar IMF parameters to have their canonical values. In this case, the other half mustcome from other sources, which could include winds from massive stars (Prantzos 2004), TypeIa supernovae (from white dwarfs), and spallation sources (including X-winds; see Shu et al.2001; Gounelle et al. 2006). The presence of spallation sources are indicated for our ownSolar System by evidence for short-lived Be (McKeegan et al. 2000), because this SLRcannot be produced via stellar nucleosynthesis. Spallation from X-winds can also producethe isotopes Cl, Mn, and Ca at the levels inferred for the early Solar nebula, but Alis underproduced by factors of a few (Shu et al. 2001; see also Chaussidon & Gounelle 2006;Desch et al. 2010 for additional discussion).
Next we find the equilibrium abundances of the SLRs on the smaller scale of molecularclouds using equation (52). In this setting, the star formation rate Γ SF is that of the cloud. 42 –Star formation is notoriously inefficient (Shu et al. 1987; McKee & Ostriker 2007), whereonly a small fraction ǫ sfe of the mass of the cloud is turned into stars during a free-fall time τ ff . As a result, the star formation rate can be written in the formΓ SF = ǫ sfe M cloud τ ff , (54)where M cloud is the mass of the entire cloud. The free-fall time τ ff takes the form τ ff = (cid:18) π Gρ (cid:19) / . (55)The appropriate mass density ρ is that corresponding to number density n ≈ cm − , sothat the free-fall time τ ff ≈ . τ ff , then equation (54) isexact if we use it as the definition of the star formation efficiency ǫ sfe . The equilibrium massfraction of a given SLR can be determined by combining equations (52) and (54), i.e., X k = ǫ sfe ln 2 τ / τ ff h M [ A k ] i ∗ h m i . (56)Inserting typical values, we obtain X k ≈ . × − (cid:16) ǫ sfe . (cid:17) (cid:18) τ / (cid:19) (cid:18) h M [ A k ] i ∗ . µM ⊙ (cid:19) . (57)This result indicates that the mass fraction of Al is expected to be fall in the range X Al ≈ . − . × − , where the lower (upper) end of range arises from the WW (LC) stellarevolution yields (again using γ = 1.5 for the stellar IMF). For both models, the mass fractionof Fe is expected to be X Fe ≈ − . For comparison, the inferred mass fraction of Al forthe Solar Nebula is X Al ⊙ ≈ . × − , whereas the mass fraction of Fe is X Fe ⊙ ≈ − .For both Al and Fe, these mass fractions are an order of magnitude larger than thosemeasured for the Galaxy as a whole (see Section 6.1).These results have three important implications: [1] For this simple estimate of theequilibrium abundances of SLRs in molecular clouds, the mass fractions delivered to star/disksystems are roughly comparable to those delivered via direction injection (Section 5). [2]The enrichment values are high enough to account for the estimated abundances of SLRs inthe early Solar Nebula. As a result, a distributed enrichment scenario is viable for our SolarSystem (Gounelle & Meibom 2008; Gounelle et al. 2009). [3] The supernova yields predictlarger mass fractions for Fe, whereas the meteoritic data from the Solar System indicatehigher mass fractions for Al. This discrepancy, once again, points toward an additionalsource for Al (see the discussion of this issue in Section 6.1). 43 –This estimate for the distributed contribution to SLR abundances assumes no losses, i.e.,all of the SLRs are captured by the molecular cloud. In practice, however, supernovae oftenexplode near the cloud edges, so that some fraction of the SLRs could escape. In addition,stars near the lower end of the mass range (for supernovae) live for nearly 30 Myr, therebyallowing them time to leave the clouds before detonation. Recent numerical simulationsprovide some guidance on this issue. If the supernova explosion has clumpy ejecta, it canbe readily mixed with the surrounding cloud (Pan et al. 2012); specifically, the metals froman individual supernova will mix with ∼ × M ⊙ of cloud material. Since the masses ofboth Al and Fe are of order M k ∼ − M ⊙ (see Figure 1), the mass fractions for theseSLRs are predicted to be X k ∼ − . This result is comparable to, but somewhat smallerthan, the estimate of equation (57). However, some regions could have higher mass fractionsif they are enriched by multiple supernovae. Simulations of nuclear enrichment for entirecloud complexes have also been carried out (Vasileiadis et al. 2013). These calculations arealso able to reproduce the levels of nuclear enrichment inferred for our early solar system;however, these computations are done using a periodic box and do not include losses fromthe cloud.Observations of SLR emission in star forming regions are in their infancy. A recentreview (Diehl 2013) discusses the current experimental status for mesaurements of Al linesin Sco-Cen, Carina, and Orion. Emission from Al is detected in these regions, but thefluxes (and hence the inferred abundances) are somewhat lower than expected (given thestar formation rate and the expected nuclear yields). This discrepancy could be interpretedas evidence for some SLRs being lost from the cloud. In addition, the observed emission fromOrion originates from a region that is much larger than the molecular cloud itself; here again,the standard interpretation is that some fraction of the radioactive material has escapedfrom the cloud. Although a full assessment of the probability for substantial SLR losses isbeyond the scope of this work, it seems likely that most solar systems will not experiencethe maximum levels of enrichment considered here. Nonetheless, distributed enrichment ofSLRs from the background cloud will compete with direct injection into circumstellar disks.One important role played by SLRs is their contribution to the ionization rate. Wecan illustrate the contribution of distributed populations of SLRs as follows: The ionizationrate ζ k due to a given SLR is given by equation (10), although we can neglect the decayingexponential factor since we are using the equilibrium abundances. Using the result (56) forthe mass fraction, the ionization rate can be written in the form ζ k = E k ω I ǫ sfe A k τ ff h M [ A k ] i ∗ h m i , (58)where A k is the atomic number of the nuclear species, E k is the energy of the decay products,and ω I is the energy required for ionization. Equation (58) specifies the ionization rate due 44 –to only one SLR; the total ionization rate is determined by a sum over all species. Againinserting typical values, we obtain the estimate ζ SLR = X k ζ k ≈ × − sec − (cid:16) ǫ sfe . (cid:17) X k A k (cid:18) E k (cid:19) (cid:18) h M [ A k ] i ∗ . µM ⊙ (cid:19) . (59)For typical parameter values, the sum in the above equation is about 0.165, so that thebenchmark ionization rate due to SLRs in molecular clouds ζ SLR ≈ × − sec − . This valueis comparable to that expected for SLRs in the early Solar Nebula (Umebayashi & Nakano2009; Cleeves et al. 2013b), but smaller than the ionization rate due to cosmic rays in theinterstellar medium, ζ CR ≈ − × − (van der Tak & van Dishoek 2000).
7. Conclusion
This paper explores the degree to which young stellar clusters can influence their con-stituent solar systems by providing enhanced abundances of short-lived radionuclides. TheseSLRs, in turn, affect disk evolution, disk chemistry, and planet formation by providing heat-ing and ionization. Previous work has focused on the possible enrichment of our own SolarSystem, and on the total galactic supply of SLRs. This work generalizes previous treatmentsby considering SLR enrichment for typical solar systems residing in a range of cluster envi-ronments. This section presents a summary of our specific results (Section 7.1) and providesa discussion of their implications (Section 7.2).
Using results from two different sets of stellar evolution calculations (the WW and LCmodels), we have calculated the expectation values of the SLR yields per star h M [ A k ] i ∗ ,along with the widths σ ∗ of the individual distributions (Table 1). Although previous workhas considered these expectation values, little focus has been given to the widths of thedistributions; these widths affect the distributions of the radioactive yields provided byclusters as well as the distributions of SLRs delivered to individual solar systems. Theseexpectation values for the yields per star have been calculated as a function of the index γ ofthe stellar IMF (Figure 3), the minimum mass of the progenitor included in the distribution(Figure 4), and the time span included in the treatment (Figure 5). For both Al and Fe,the two species of greatest importance, the expectation values for the SLR mass per startypically fall in the range h M [ A k ] i ∗ = 0 . − . µM ⊙ (where 1 µM ⊙ = 10 − M ⊙ ). Althoughthe expectation values calculated from the WW and LC models are roughly comparable, thedistributions are significantly wider for the LC results (Table 1). 45 –For clusters with fixed stellar membership size N , we have calculated the distributions ofthe SLR yields (see Figures 9 – 12). In the limit of large N , the distributions become nearlygaussian, with expectation values given semi-analytically via equation (12) and with widthsgiven by equation (13). Clusters with N = 3000 show nearly gaussian distributions, whereassomewhat smaller clusters with N = 1000 show significant departures from gaussianity. Thetransition to the “large- N limit” thus takes place near N ≈ N = 5000, for example, the typical mass of short-livedradioactive material provided by the cluster is of order 10 − M ⊙ = 1 mM ⊙ , and this valuescales linearly with increasing stellar membership size N .For individual solar systems residing in clusters, we have determined the distribution ofSLR masses provided to their circumstellar disks (Section 5 and Figure 13). For clusters withlarge N , where the distribution of (total) radioactive yields per cluster is nearly gaussian, wehave derived a semi-analytic expression for the distribution of SLR masses delivered to solarsystems (see equation [43]). This function is in good agreement with the results found bynumerical sampling. We have also found the corresponding cummulative distribution, whichprovides the fraction of solar systems that are exposed to radioactive material above a giventhreshold. The fraction of solar systems that could be enriched at the levels found for theSolar Nebula falls in the range 0.01 – 0.10.In addition to finding the distributions (of total cluster yields and masses deliveredto solar systems) for clusters with a given size N , we have also estimated the expectationvalues integrated over the entire range of cluster membership sizes (Section 4.3). The “typicalcluster”, in terms of the number of clusters, is quite small and has a correspondingly smallradioactive yield (a few µM ⊙ ). However, the “typical star” is predicted to reside in a largercluster (which has more stars and more supernovae), so that the cluster yield that a typicalsolar system would experience is much larger, about 0.015 M ⊙ = 15 mM ⊙ . Only a smallfraction of the total mass in SLRs produced by a cluster impinges upon any given solarsystem, so the typical mass enrichment is of order 10 pM ⊙ = 10 − M ⊙ .We have also considered the connection between the radioactive yields produced byembedded stellar clusters and the supply of SLRs on larger scales represented by molecularclouds and the galaxy (Section 6). Observations of gamma ray lines indicate that the galaxy-wide supply of Al and Fe, and their abundance ratio, is consistent at (only) the factor oftwo level. More specifically, the observed line emission from Al is too strong by a factor of ∼
2, which could indicate another source (in addition to supernovae). On the scale of themolecular cloud, we find that supernovae can enrich the entire cloud at levels comparable tothose inferred for the early Solar Nebula, provided that all of the SLRs are confined to thecloud. In general, losses of SLRs from the cloud will reduce the abundances below those of 46 –the early Solar Nebula. Nonetheless, distributed enrichment of SLRs will compete with thedirect injection of SLRs considered in the rest of this paper.Finally, we have shown that the ionization rate due to SLRs falls in the range ζ SLR ∼ − × − sec − for both direct nuclear enrichment in clusters and for distributed enrichment inmolecular clouds. This ionization rate is smaller than the canonical value usually attributedcosmic rays in the interstellar medium, ζ CR ∼ − sec − . Nonetheless, the CR flux is oftensuppressed in young stellar objects (Cleeves et al. 2013a), so that SLR ionization will beimportant those systems (see Umebayashi & Nakano 2009; Cleeves et al. 2013b). The mass scales in this problem can be summarized as follows: For the most abundantisotopes, the yields of SLRs produced by individual supernovae have masses in the range10 – 100 µM ⊙ (where 1 µM ⊙ = 10 − M ⊙ ). The corresponding yields per star, obtained byaveraging over the stellar IMF, have masses corresponding to fractions of µM ⊙ . The yieldsof SLRs per cluster are much larger and have masses measured in mM ⊙ (where 1 mM ⊙ =10 − M ⊙ ). Within cluster environments, the typical mass of SLRs delivered to circumstellardisks is of order 10 pM ⊙ = 10 − M ⊙ , but the range extends up to 100 pM ⊙ (but only forseveral percent of the solar systems). For comparison, the most abundant SLRs found inour own Solar Nebula are thought to have masses in the range 20 – 200 pM ⊙ (see Table 2).One implication of this work is that both distributed enrichment in molecular clouds(Section 6.2) and direct enrichment within stellar clusters (Sections 4 and 5) can providesignificant abundances of SLRs. The enrichment levels are large enough to contribute to theionization rates and — under favorable circumstances — large enough to explain the inferredabundances of SLRs in the early Solar Nebula. Nonetheless, the manner in which these twoenrichment scenarios compete with each other remains an open question. In both cases, theamount of radioactive material delivered to a given star/disk system will be drawn from awide distribution. For supernova enrichment in clusters, these distributions are constructedin this paper (see Section 5). For distributed enrichment, however, more work must becarried out to define the distributions. A number of questions remain, including the amountof nuclear material that is not seeded into the parental molecular cloud, the amount of cloudmaterial that is mixed with the supernova ejecta, and the number of high mass stars thatescape the cloud before exploding. All of these quantities have a range of values that varyfrom cloud to cloud, and will contribute to the distribution of possible enrichment levels.The uncertainties in the nuclear yields from supernovae remain an important unresolved 47 –issue. The level of agreement between the WW and LC nuclear models (as considered herein)suggest that the yields are uncertain by a factor of ∼ Fe and Al are in apparent disagreementwith the stellar nucleosynthesis calculations at the same factor of ∼ Al in addition to supernovae(which provide the only source of Fe), but the uncertainties in the nuclear yields are largeenough that this possibility remains inconclusive (see Section 6.1 and references therein). Animportant task for the future is thus to determine the expected nuclear yields with greaterspecificity.Orthogonal to the uncertainties in nuclear yields, a number of other issues should beexplored in greater depth. This paper shows that the expected yields only reach their fullvalues after ∼
30 Myr, when the smallest progenitor stars have exploded as supernovae.Nonetheless, the nuclear yields reach a healthy fraction of their asymptotic values after ∼ −
10 Myr (Hern´andez et al.2007), or even shorter times (Cizea et al. 2007; Williams et al. 2013), not every disk willexperience full enrichment. The resulting timing issues are thus important and should bestudied in the future. The direct injection scenario can suffer from additional inefficienciesdue to supernova fallback, inhomogeneities in the supernova ejecta, and incomplete captureby the circumstellar disks; all of these issues should be examined further. With the massesof SLRs that are delivered to solar systems specified, the implications should also be studied,including ionization of the gas and heating of planetesimals. These processes, in turn, willinfluence disk accretion, planet formation, and the chemical content of the disk gas. Finally,the long-term properties of forming planets depend not only on SLRs, but also on radioactivenuclei with longer half-lives. The abundances of these isotopes will be more affected bydistributed enrichment, but will be augmented by direct injection as considered herein.
Acknowledgments:
We thank Ilse Cleeves and Frank Timmes for useful comments anddiscussions; we also thank an anonymous referee for a prompt and constructive report thatimproved the manuscript. This work was supported at the University of Michigan throughthe Michigan Center for Theoretical Physics and at Xavier University through the HauckFoundation. 48 –
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