The Origin of Cosmic Rays: How Their Composition Defines Their Sources and Sites and the Processes of Their Mixing, Injection, and Acceleration

"Any theory of the origin of cosmic rays cannot expect serious success unless it rests on a detailed analysis of the observed composition of primary cosmic radiation" (Ginzburg & Syrovatskii 1964).

The quest for the origin of cosmic rays was best charted in a pair of truly prescient papers by Baade & Zwicky (1934a, 1934b). There they first identified a powerful new class of novae, which they called "supernovae," and suggested that these were the gravitational collapse and explosion of massive stars into hypothetical, highly compact remnants, which they dubbed "neutron stars." They also suggested that these powerful supernova explosions were the source of the cosmic rays, and as a test of the idea, they further predicted that the cosmic rays would include heavy elements with nuclear charge Z > 2.

They were right on all counts, for as observations have shown, such core-collapse supernovae (CCSNe; SN II and Ib/c) make up the bulk of Galactic supernovae, producing not only all of the neutron stars, but most of the supernova shock energy, making them the principal source of the Galactic cosmic rays.

Supernovae produce more explosive power than any other known sources in the Galaxy and an order of magnitude more power than that required to generate the Galactic cosmic rays. Observations of supernovae in surrounding galaxies of various types suggest that the supernova rate in our Galaxy is 2.8 ± 0.6 per century (Li et al. 2011; Shivvers et al. 2017). Of these, ∼81% are core-collapse supernovae, from the core collapse of young (∼3–35 Myr), massive (∼8–120 M_⊙) stars, whose winds and explosions blow off most of their mass, leaving only a roughly ∼2 M_⊙ neutron star remnant, though some simply collapse into black holes without a supernova explosion. The remaining ∼19% of the supernovae are the SN Ia explosions of smaller, much older stars that have evolved into white dwarfs and are accreting from binary companions and eventually undergo a thermonuclear explosion, blowing off all of their mass and leaving no remnant.

Both types of supernovae release roughly the same ∼10⁵¹ erg in ejecta and shocks (e.g., Nomoto et al. 1984; Woosley & Weaver 1995; Branch & Wheeler 2017), generating a mean explosive power of ∼10⁴² erg s⁻¹. The cosmic rays with a mean interstellar energy density of w ∼ 10⁻¹² erg cm⁻³ (Cummings et al. 2016) filling a Galactic volume, V, and a mean Galactic residence time, τ, require (e.g., Lingenfelter 2013) a sustaining power Q ∼ wV/τ, or ∼wcM/x ∼ 10⁴¹ erg s⁻¹, where x ∼ ρτc ∼ τcM/V, with the velocity of light, c, the mass of Galactic gas M ∼ 10⁴³ g, or about 10% of the mass of the Galaxy, and the mean cosmic-ray path length, x ∼ 5 g cm⁻², as determined by local cosmic-ray elemental abundance measurements of the energy-dependent secondary/primary ratios, such as Be/CNO and B/CNO, of secondary nuclei produced by spallation of primary nuclei during propagation. Thus the Galactic cosmic rays can be produced by supernova shocks if they gain ∼10% of the shock energy.

The most efficient cosmic-ray acceleration, as Axford (1981) emphasized, occurs in strong shocks expanding in low-density, hot, fully ionized H ii regions, and supernova remnants can indeed generate cosmic rays with such an efficiency. Strong shocks with compression ratios s ∼ 3 to 4 can also produce power-law spectra at relativistic energies with an index γ ∼ (s+2)/(s − 1) of −2 to −2.5. This range is quite consistent with the cosmic-ray proton spectral indexes of around −2.2, implied by cosmic-ray-produced, high-energy pion-decay gamma-rays in both isolated supernova remnants in the interstellar medium (ISM; Acero et al. 2016) and those collectively clustered in superbubbles (Ackermann et al. 2011; Aharonian et al. 2019).

Moreover, that source spectral index is in excellent agreement with the values of −2.2 to −2.4 determined from the local-equilibrium cosmic-ray index of −2.7 (e.g., Engelmann et al. 1990; Obermeier et al. 2012) with a −0.33 to −0.5 energy-dependent cosmic-ray escape lifetime in the Galaxy. That spectral index shift reflects the steepening of the source spectrum during propagation, measured (Obermeier et al. 2012; Adriani et al. 2014; Aguilar et al. 2016) from the spallation production ratio of secondary B to parent CNO nuclei as a function of energy.

The heavy elements in the cosmic rays have now been measured over the energy range from about ∼10⁸ eV to at least 10¹⁴ eV, and perhaps even ∼10¹⁸ eV. They all appear to have essentially the same power-law energy spectrum of about −2.7, and their relative abundances in the local cosmic rays are found to be effectively independent of energy, as seen in Figure 1.

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Detailed elemental and isotopic abundances have now been measured from H to U (Binns et al. 1989; Engelmann et al. 1990; Rauch et al. 2009; Ahn et al. 2010; Donnelly et al. 2012; Cummings et al. 2016; Murphy et al. 2016). These measurements not only further confirmed Baade and Zwicky's predictions, but revealed broader sources, mixing, and injection processes. As Ginzburg & Syrovatskii (1964) foresaw, cosmic-ray abundances have provided a wealth of data whose diagnostic value has surpassed all expectation.

The cosmic-ray abundance ratios of the metals, Z > 2, probe critical nuclear, atomic, and even solid-state processes in addition to plasma processes that are not accessible from the study of the H and He ratio alone, which is actually dominated by the interstellar medium, unlike the metals. In particular, the metals give multiple measures of the mass mixing ratio of the ejecta and the ambient ISM in the cosmic-ray acceleration region, together with the nucleosynthetic yields of the ejecta that can identify the type of supernova or other source. They also probe major atomic and mineralogical processes, including differential ionization and grain condensation fractions that determine the relative elemental ion injection factors that dominate both the cosmic-ray abundances and their overall acceleration efficiencies. Thus the metals measure major processes not accessible from plasma studies alone.

First, we find that these cosmic-ray abundances Z/H are not simply solar or local ISM, but are enriched by factors of ∼20 to 200 times solar system values (Lodders 2003, predominantly C1 chondritic meteorites). Further, we show that these measurements quantitatively define the driving processes of supernova source mixing and injection that can determine the cosmic-ray elemental abundances to within ±35% with no free parameters. Moreover, we find that the spread of the cosmic-ray bulk mass mixing ratio of swept-up ISM to CCSN ejecta $\sim {4.3}_{-2.0}^{+2.4}$, based on the best-measured cosmic-ray source ejecta mass fraction ${F}_{\mathrm{EJ}}\sim {19}_{-6}^{+11} \%$ (Murphy et al. 2016), matches very well the early range of the intrinsic, homologous ratio calculated (e.g., McKee & Truelove 1995; Truelove & McKee 1999, 2000). Basically, this is the growing accumulation and mixing of the fast ejecta and shocked ISM swept up by it, in supernova remnants during the period right after the onset of the turbulent, strong shock-generating, Sedov–Taylor stage of supernova expansion and mixing. The need for grain injection further constrains any metal-accelerating remnants to the hot (∼10⁶ K), tenuous (∼0.001–0.01 H cm⁻³) superbubbles, where the grains will not be wholly destroyed by the reverse shock. Altogether, these abundance measurements clearly establish the major sources of cosmic-ray injection and acceleration, and also delineate the lesser contributions of thermonuclear SN Ia supernovae, Wolf–Rayet winds, asymptotic giant branch (AGB) stars, and r-process elements potentially in both CCSNe and binary neutron star mergers.

As a handy reference, definitions of the major notations, used here in developing the transformation of the mix of the supernova ejecta and swept-up interstellar abundances into that of cosmic rays, are given in the Appendix in Table 1. Quantitative values of the major steps along the way, shown in Figures 2 and 4–6 and 8, are also listed in Tables 2 and 3 of the Appendix.

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Independent of the specific site of cosmic-ray acceleration, a number of other processes operate both on the generation of their composition and energy spectrum at their source and on the subsequent modification of these properties to what we now observe. The various elemental and isotopic abundance ratios of the cosmic rays are the most extensive and diverse tracers of all and place powerful constraints on these processes. In order to determine the cosmic-ray spectrum and composition generated by the cosmic-ray sources, the experimental groups have first corrected their measurements for detector sensitivity and biases, the effects of nuclear spallation in the Earth's atmosphere, if needed, and both geomagnetic and heliospheric modulation, to determine the local interstellar cosmic-ray abundances, and the Voyager Mission (Cummings et al. 2016) has now directly measured the interstellar cosmic rays. All then model the effects of their propagation and spallation in the ISM en route from their sources in order to derive the cosmic-ray source composition, (Z/H)_GCRS.

Although various cosmic-ray acceleration processes have been proposed, the most likely and most efficient appears to be diffusive shock acceleration (DSA) of suprathermal ions injected into supernova blast waves. This process, developed by Axford et al. (1977), Bell (1978a, 1978b), Blandford & Ostriker (1978), Axford (1981), Jokipii (1982), and Ellison et al. (1997), shows that such ions in the shock-generated turbulent medium both behind and ahead of the forward and reverse shocks can diffuse back and forth through the shock and be repeatedly accelerated with a high overall efficiency of ∼10% or more.

The similar spectral indices of the various elements in the cosmic-ray source and their constant abundance ratios (Figure 1) are also quite consistent with DSA, which does not differentiate between elements at the same ultrarelativistic rigidity, which is proportional to their kinetic energy divided by their nuclear charge. Thus the observed differences in the elemental source abundance ratios with respect to solar system and interstellar values are not expected to result from plasma acceleration processes (e.g., Ellison et al. 1997; Ohira et al. 2016), except perhaps for small spectral variations if shock injection is significant (Hanusch et al. 2019).

Thus we explore in detail just what the measured cosmic-ray elemental abundances and other observations reveal about the nature of their sources and their mixing and injection.

This all develops in the powerful, homologous mass mixing of elements in the slowing of the supernova ejecta plasma by the reverse shock of the Sedov–Taylor phase of the expansion once the mass swept up from the surrounding ISM becomes comparable to that of the ejecta. These two components are each divided into gases and solids: fast ejecta grains that condensed out in the early free expansion and fast but previously condensed older ISM grains, shock accelerated (Ellison et al. 1997) by the blast wave that swept them up. The fast grains in turn swept up and implanted both heavy refractories and volatiles (e.g., Bibring et al. 1974; Audouze et al. 1976; Deneault et al. 2003 during the reverse shocks). Simultaneously, there is selective injection of these mixed elements into the accelerating shocks as suprathermal ions by sputtering of the condensed and implanted elements in the fast grains, colliding primarily with H and He in the turbulent remnant.

The fresh supernova ejecta and swept-up, older interstellar material are the two main components of the cosmic-ray source mix. The bulk compositions of the ejecta are determined from nucleosynthesis calculations for various stellar progenitor masses and supernova types. Those of the ISM come from optical observations of interstellar gas and dust and laboratory measurements of meteoritic dust. The relative fractions of refractory and volatile elements that were either condensed or implanted into grains are also particularly important factors. They determine any added abundance enrichment of each element from differential injection into the supernova shocks as suprathermal ions from sputtered refractories and volatiles in fast ejecta grains (Cesarsky & Bibring 1981; Lingenfelter et al. 1998) and in accelerated ISM grains (Epstein 1980; Ellison et al. 1997; Meyer et al. 1997). This is as opposed to the common alternative shock acceleration of the most easily ionized of the highly volatile elements of the ejecta and ISM that remained in the gas phase, especially H, which sets the baseline abundance.

Cosmic-ray ejecta mixing is the key to cosmic-ray acceleration, for it is in fact the measure of the shocked supernova remnant mix during the most effective period of such acceleration, and it defines the time frame in which that acceleration occurs in the remnant expansion.

The basic model of supernova remnant mixing of ejecta and ISM is drawn from the pioneering explosion studies of Taylor (1946) and Sedov (1959). This mixing has long been recognized as a result of the slowing down of freely expanding young supernova remnants after they have swept up a mass of ISM greater than that of the supernova ejecta (e.g., Shklovskii 1962). More recently, McKee & Truelove (1995) and Truelove & McKee (1999) have shown that this mixing is a basic process in the homologous transformation of remnants from free expansion to adiabatic, or Sedov–Taylor expansion, tracing the growth of both forward and reverse shocks, turbulent mixing of swept-up ISM and expanding ejecta, and its increasing ISM/ejecta mass ratio, before finally sweeping up enough gas to enter the radiative or "snowplow" stage, where it radiates away most of its remaining kinetic energy and becomes subsonic.

The cosmic-ray source ISM/ejecta mass mixing ratio has also recently been found to be a rather broad universal constant of ∼4 ± 2 both in individual elemental cosmic-ray abundances and in the Galactic evolution of cosmic-ray spallation products Be and B in old halo stars. As we discuss in detail below (Section 5), this range of cosmic-ray source mixes strongly suggests that the cosmic rays are in fact primarily accelerated from the shocked ejecta/ISM mix in the remnants within a couple doubling times following the nominal onset of the Sedov–Taylor stage at an ISM/ejecta ratio of ∼1.6 (McKee & Truelove 1995; Truelove & McKee 1999).

The earliest measures of supernova mixing and acceleration of fresh ejecta in the cosmic rays have come from extensive observations of old halo star abundances of Be and B, which are produced (Reeves et al. 1970) almost entirely by cosmic-ray spallation. The measurements of their evolving enrichment throughout the Galaxy from the earliest times also define the evolving cosmic-ray metallicity and have directly shown (e.g., Tatischeff & Gabici 2018, and references therein) that the cosmic rays cannot be accelerated from the ISM alone, but must include a major contribution from highly enriched supernova ejecta mixed with swept-up ISM, as will be discussed in detail in Section 2. Comparison (Figure 2) of the solar system composition, (Z/H)_SS, with the Galactic cosmic-ray source composition, (Z/H)_GCRS, determined from extensive recent measurements of the local cosmic-ray source abundances of the best-measured elements, extending all the way up to Pb, clearly shows that they do not match with solar system values. Instead, the cosmic rays are greatly enriched by factors of ∼20 to ∼200 times solar system and local interplanetary medium values. Most individual cosmic-ray abundances also differ from one another by factors of as much as ∼10.

But, noting that in a subset of the cosmic-ray abundance ratios, several of the more refractory elements (Mg, Al, Si, Ca, Fe, Ni, Sr, Zr, and Ba) with ratios from 90 to 160 times solar are roughly the same as one another to within a factor of 2, some have suggested (e.g., Meyer et al. 1997) that these cosmic rays are accelerated almost entirely out of the ISM. However, such similarity is also expected from the CCSN s-process ejecta, since that is the primary source of the ISM (e.g., Timmes et al. 1995). Moreover, the highly refractory, most fully condensed, grain-forming, and most strongly injected elements show the strongest ejecta mixing effect, while all the other less refractory, only partially condensed elements do not. So only these refractories show the ejecta composition similarity with the ISM.

Still, both these large differences and selected similarities between the cosmic-ray source and the solar system compositions offer clear clues as to their causes. As noted, detailed analyses of these source abundances relative to those of the ISM show that these differences can be almost entirely resolved without any free parameters by assuming that the source abundances seen by the accelerating supernova shocks result from just the two processes: bulk mixing of ejecta with the ISM and selective injection of these mixed elements into the accelerating shocks by grain sputtering.

In Sections 2 and 3 we apply the two basic mixing and sputtering processes sequentially to solar system elemental abundances (Lodders 2003) to show, as Rauch et al. (2009) and Murphy et al. (2016) have done with the isotopic abundances, just how remarkably revealing a comparison of the cosmic-ray source abundances with the mix of the supernova ejecta and the ISM can be. We see particularly how it organizes those abundance ratios to clearly define the injection process in terms of the fraction of each refractory element that condensed into and volatile element that was implanted into high-velocity grains, and was then sputtered off as a suprathermal ion. Taking that procedure further, we find that next applying the elemental charge Z^2/3-dependent Coulomb sputtering cross section or relative enrichment and then the grain condensation fraction moves the fit of the abundances closer to and finally into agreement with the cosmic-ray observations. Thus the procedure demonstrates both the need for such processes and their effectiveness and clearly reveals the remaining process.

In Section 4, using the ability of these processes to explain the bulk of the cosmic-ray metal abundances primarily in terms of core-collapse supernova nucleosynthesis, we define the contribution of thermonuclear SN Ia supernovae, Wolf–Rayet winds, AGB stars, and the relative production of r-process elements by core-collapse supernovae and neutron star merger kilonovae.

Finally, in Section 5, we explore the major implications of these processes in the context of the other constraints on the supernova sources and sites of cosmic-ray source mass mixing, grain injection, and DSA. Most importantly, we find that this uniform Galactic range of cosmic-ray bulk mass mixing, ISM/CCSN ∼4.3_−2.0^+2.4 (Murphy et al. 2016), matches that of the fundamental, homologous range of that mixing ratio calculated (McKee & Truelove 1995; Truelove & McKee 1999, 2000) in evolving young supernova remnants at the onset of turbulent, shock-generating Sedov–Taylor expansion, outlining the interconnected spatial and temporal structures of the cosmic-ray source mixing, injection, and acceleration.

Moreover, the occurrence of strong Z-dependent enrichment of cosmic rays from sputtering interactions between fast supernova grains and swept-up gas also requires that these grains not be >90% destroyed by grain–grain collisions in reverse shocks, which is expected if supernovae are expanding in interstellar gas denser than n > 0.1 H cm⁻³ (Bianchi & Schneider 2007), and, unlike ion injection by only limited but not complete sputtering, does not result in any Z-dependent enrichment.

We now consider the evidence for mixing of heavy elements in the supernova ejecta and swept-up ISM in the cosmic-ray source material, the fundamental nature of the supernova remnant mixing by mass, and finally what a comparison of the cosmic-ray source composition with that source mix reveals about the additional preferential injection of these elements as suprathermal ions sputtered from fast refractory grains into the accelerating shocks. As observations have shown, nearly all of the refractory elements condense into fast dust grains very early in the freely expanding ejecta, as it rapidly cooled to as low as 20 K in the moving frame. With the transition to Sedov–Taylor expansion came massive ejecta/ISM mixing and heating of the grains back up to ∼1000 K (Bianchi & Schneider 2007) and their partial sublimation in the reverse shock. Those grains that survived destruction in that shock were decoupled from the slowing, shocked plasma and moved on ahead at suprathermal velocities close to their expansion velocity of ∼3000 km s⁻¹, or energies of several MeV per atom for the major refractory elements, into the much slower, swept-up, compressed and mixing ISM. There, in interactions with turbulent H, He, and other volatiles, they were sputtered off as suprathermal ions at similar velocities, or ∼10 times 150 km s⁻¹ that of the 10⁶ K H gas, injecting the metals into the DSA process to be selectively carried to cosmic-ray energies.

The earliest and most far-reaching evidence for accelerated supernova ejecta in the cosmic rays, however, came not from measurements of the various current, local cosmic-ray abundances, but instead from quite different studies of the long-term evolution of the cosmic-ray abundances of just the CNO group, determined from abundances of Be and B in the atmospheres of old halo stars over the last ∼10 Gyr. These two exceedingly rare elements, (Be/H)_SS ∼ 3 × 10⁻¹¹ and (B/H)_SS ∼ 7 × 10⁻¹⁰ (Lodders 2003), are unique in that they cannot be produced by standard stellar nucleosynthesis, and, as Reeves et al. (1970) first suggested, they are produced instead by spallation of CNO in cosmic-ray interactions in the ISM. They are produced by two reciprocal processes, the spallation of interstellar CNO atoms primarily by cosmic-ray protons and helium and inversely by spallation of cosmic-ray CNO nuclei colliding primarily with interstellar H and He atoms. Currently the contribution of spallation of interstellar CNO nuclei is about twice that of cosmic-ray nuclei (Ramaty et al. 1997).

Measurements (Duncan et al. 1992; Gilmore et al. 1992; and others, see review by Tatischeff & Gabici 2018) of Be and B in old halo stars born in the early Galaxy with low interstellar gas metallicity, however, showed that at that time the Be and B resulted just from spallation of cosmic-ray CNO, for these Be and B abundance ratios were constant relative to both O and Fe, and within a factor of ∼2 of the present values, while the interstellar (Fe/H)_ISM grew by orders of magnitude from 10⁻³ to 10⁻¹ (Fe/H)_SS. Thus the Be and B production was proportional to the Galactic supernova rate and ejecta production and had to be produced by spallation of some fraction of their ejecta in the cosmic rays, since the elemental yields of the dominant core-collapse supernovae are independent of their initial metallicity in this range (e.g., Woosley & Heger 2007). There is no significant measurable evidence of any component of the Be and B production that was proportional to the interstellar metallicity, which varied by a factor of >10², as would have been expected if they were produced solely by cosmic-ray spallation of the ISM (Duncan et al. 1992, 1997; Ramaty et al. 1997, 2000; Alibes et al. 2002).

The estimated fraction by mass, F_EJ, of the highly enriched supernova ejecta in a mix with the very low metallicity ISM in the cosmic-ray source material, necessary to satisfy the Be and B observations, was at least 15% (Ramaty et al. 1997), or a bulk cosmic-ray mass mixing fraction of swept-up ISM/SN ejecta <6. Subsequent calculations (Alibes et al. 2002) of the evolution of the spallation-produced Be and B versus Fe/H, as a function of the supernova ejecta mass fraction in the cosmic-ray source, compared to Be and B abundances measured in halo stars over the range of Galactic metallicities, (Fe/H)_ISM from 10⁻³ to 1 (Fe/H)_SS, gave a best-fit supernova ejecta mass fraction F_EJ ∼ 25 ± 15%, or a mass mixing fraction of ∼3. But this ejecta fraction is not the elemental abundance ratio of the source mix, which is dominated by the highly enriched supernova ejecta. With such an ejecta mass fraction, this mixing increases the elemental cosmic-ray source abundances by factors of ∼2 to ∼10 times those of the ISM.

Moreover, these studies show that this supernova ejecta mass fraction in the cosmic-ray source mix, which essentially describes the initial mixing of the ejecta with the ISM, is not just a local or transient value, but a global property of cosmic-ray source abundances, since these stars sample the whole Galaxy for over ∼10 Gyr.

Although alternative models for Be and B production without cosmic-ray acceleration of supernova ejecta have been explored, none are consistent with other constraints (e.g., Tatischeff & Gabici 2018 and references therein), and all fall short by more than an order of magnitude. Thus, the observed Galactic evolution of Be and B abundances requires that a constant fraction of the supernova ejecta CNO be accelerated to cosmic-ray energies. The simplest acceleration process would be for the ejecta of a supernova to be accelerated by its own shocks (Lingenfelter et al. 1998). This is also quite consistent with what is expected (McKee & Truelove 1995; Truelove & McKee 1999, 2000) in homologous, expanding supernova remnants at the onset of the Sedov–Taylor stage of expansion (Section 5.2) and a core feature of cosmic-ray acceleration that easily explains its universality.

A constraint on such acceleration, however, is set by limits (Wiedenbeck et al. 1999) on the abundance of cosmic-ray radioactive ⁵⁹Ni, which decays to ⁵⁹Co with a half-life of 7.6 × 10⁴ yr only by bound electron capture, and cannot significantly decay once the ⁵⁹Ni is accelerated and its electrons are stripped off (Casse & Soutoul 1975; Soutoul et al. 1978). The standard nonrotating star nucleosynthesis models of CCSNe (Woosley & Weaver 1995, Table 5A) predict that radioactive Ni makes up ∼60% of the combined, freshly produced mass-59 Co and Ni nuclei, or ⁵⁹Ni/⁵⁹Co ∼ 1.5, although recent models (Limongi & Chieffi 2018, Table 30) of both rotating and nonrotating stars predict only 37%–35%, or ⁵⁹Ni/⁵⁹Co ∼ 0.56, instead.

Thus, since the measured (Wiedenbeck et al. 1999) cosmic-ray abundance of stable ⁵⁹Co/⁶⁰Ni nuclei is ∼0.182 ± 0.021, we would expect the fresh, undecayed cosmic-ray ⁵⁹Ni/⁶⁰Ni to lie between 0.56 and 1.5 times that, or ∼(0.10 and 0.27) ± 0.021. However, the measured cosmic-ray ⁵⁹Ni/⁶⁰Ni is <0.055, which is consistent with the 0.049 ± 0.012 abundance that might be expected purely from secondary production by spallation of heavier cosmic rays during subsequent propagation with no significant evidence of ⁵⁹Ni remaining. If supernova ejecta alone were being accelerated to cosmic-ray energies, they could not have been so in <10⁵ yr.

But as the Galactic Be and B evolution alone shows, that cannot be the case. The mass mixing of old (≫10⁵ yr), well-decayed, swept-up ISM into the fresh CCSN ejecta greatly cuts the relative abundance of ⁵⁹Ni in the remnant itself and its significance in the cosmic rays. Mixed with a mean F_EJ ∼ 20% of ejecta ⁵⁹Ni plus 80% of none in the ISM, the expected ⁵⁹Ni/⁶⁰Ni ∼ (0.10 to 0.27) ± 0.021 is cut by 5 to just (0.020 to 0.054) ± 0.021. But that is offset by the factor of ∼10 Ni metallicity of the ejecta, for a net cut of a factor of 0.6 to (0.06 to 0.16) ± 0.021. However, with most of the 10 to 20 M_⊙ CCSN remnants clustered in low-density (∼0.001H cm⁻³) superbubbles around OB associations, the cosmic-ray accelerating phase of the Sedov–Taylor expansion is stretched from relatively short (<10⁴ yr) scales observed in isolated radio and high-energy gamma-ray remnants in the typical ∼1H cm⁻³ ISM to as much as the ⁵⁹Ni decay half-life. There, radioactive decay by bound electron capture can also significantly cut the expected ⁵⁹Ni/⁶⁰Ni by roughly an additional one-half to only ∼(0.03 to 0.08) ± 0.021, less than 2σ statistical errors. Thus, the combined effects of decay, mixing dilution, and rotation yields can now account for the absence of ⁵⁹Ni in cosmic rays.

The general mixing in the cosmic-ray source of fresh, high-metallicity supernova ejecta with the swept-up interstellar gas and dust has been better quantified by a number of different measurements of local cosmic-ray elemental and isotopic ratios. These studies began (Higdon & Lingenfelter 2003) with the measured (Binns et al. 2001) cosmic-ray ²²Ne/²⁰Ne ratio of 0.366 ± 0.015, which is fully 5.0 ± 0.2 times that of the solar system ratio. Using the core-collapse supernova s-process yields (Woosley & Weaver 1995) and allowing for large uncertainties in the added yields of these isotopes in Wolf–Rayet winds (Maeder & Meynet 2000), we find this isotope ratio implies a mass fraction F_EJ of 18 ± 5%. Similar mass fractions of about 20% have since been found (Binns et al. 2005, 2006; Wiedenbeck et al. 2007), using later Wolf–Rayet models, for ²²Ne/²⁰Ne and the other three largest cosmic-ray isotopic deviations from solar system values, ¹²C/¹⁶O, ⁵⁸Fe/⁵⁶Fe, and <3 Myr old radioactive ⁶⁰Fe/⁵⁶Fe from CCSNe (Binns et al. 2016, Figure 3), in addition to the ⁵⁹Ni (Wiedenbeck et al. 1999; Binns et al. 2008b).

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Calculations (Lingenfelter et al. 2003) of cosmic-ray elemental enrichment ratios of the ThU/Pt group from supernovae also suggest a similar ejecta mixing ratio F_EJ close to 20% for the products of explosive r-process supernova nucleosynthesis. These calculations based on Kratz et al. (1993) of the radioactive actinide/Pt group abundance ratio predicted a cosmic-ray source value of 0.027 ± 0.005, which is quite consistent with the latest balloon measurements (Combet et al. 2005; Donnelly et al. 2012) of ∼0.032.

Subsequent studies (Wiedenbeck et al. 2007; Binns et al. 2008a, 2008b; Lingenfelter & Higdon 2007) of the origin of the cosmic-ray composition explored not only the CCSN/ISM mixing ratio, but also the grain condensation and sputtering enrichment. Recent higher energy measurements (Ahn et al. 2010) of the abundance ratios of cosmic-ray He, C, N, Mg, Si, and Fe relative to O up to ∼4 TeV/nucleon also found a good fit with a 20% ejecta mixing ratio.

In the most extensive study, including new cosmic-ray isotopic abundance measurements of heavier elements of Zn to Zr on the Antarctic long-duration balloon flights TIGER and Super-TIGER, Rauch et al. (2009) and Murphy et al. (2016) have found best-fit mixing mass fractions of around 19% newly synthesized, high-metallicity massive star ejecta and 81% old interstellar gas and dust. Moreover, comparing the cosmic-ray source composition with this source mix, they also quantified the subsequent elemental injection biases that further enrich the cosmic-ray composition, as did Ahn et al. (2010), which we discuss further in Section 3.

Thus, the local cosmic-ray measurements all suggest an essentially constant cosmic-ray source component of F_EJ ∼ 20% high-metallicity ejecta of newly synthesized nuclei over a wide energy range of at least four decades of energy from ∼100 MeV/nucleon up to ∼4 TeV/nucleon for the nominal ∼20 Myr lifetime of the current local <1 kpc cosmic rays. As we also saw, analyses of old halo star abundance ratios of Be, produced primarily by the spallation of cosmic-ray CNO, give a similar constant ratio of ∼25% in the far older and much more distant cosmic-ray position extending out to ∼10 kpc and back over >10 Gyr, when the average Fe metallicity of the ISM was as low as (Fe/H)_ISM ∼ 10⁻³ (Fe/H)_⊙.

All of these measurements show that the individual elemental abundance ratios in the core-collapse supernova source mix can be simply defined by a single mixing mass fraction, F_EJ, of highly enriched supernova ejecta mixed with a swept-up interstellar mass fraction, F_ISM = 1 − F_EJ, or a bulk ISM/CCSN mass ratio of ∼(1 − F_EJ)/F_EJ. The resulting elemental abundance ratios of the supernova source mix relative to solar system abundances are

$$\begin{eqnarray}\begin{array}{rcl}{({\rm{Z}}/{\rm{H}})}_{\mathrm{SM}/\mathrm{SS}} & = & {F}_{\mathrm{EJ}}{({\rm{Z}}/{\rm{H}})}_{\mathrm{EJ}/\mathrm{SS}}+{F}_{\mathrm{ISM}}{({\rm{Z}}/{\rm{H}})}_{\mathrm{ISM}/\mathrm{SS}}\\ & = & {F}_{\mathrm{EJ}}{({\rm{Z}}/{\rm{H}})}_{\mathrm{EJ}/\mathrm{SS}}+{(1-{F}_{\mathrm{EJ}})({\rm{Z}}/{\rm{H}})}_{\mathrm{ISM}/\mathrm{SS}.}\end{array}\end{eqnarray} \tag{ 1 }$$

The source mix abundances, resulting from this ubiquitous supernova ejecta mass fraction F_EJ of 20% mixed with an ISM mass fraction F_ISM of 80%, or a bulk ISM/CCSN mass mixing ratio of ∼4, are very different from those of the solar system (Lodders 2003). Since the present local ISM (Z/H)_ISM ∼ 1.32(Z/H)_SS (Timmes et al. 1995), the source mix reduces to simple ratios of ejecta abundances to solar system values, (Z/H)_SM/(Z/H)_SS ∼ 0.2(Z/H)_EJ/(Z/H)_SS + 1.06, which gives individual abundance ratios varying by a factor of 10, as can be seen in Figure 4. Even though the swept-up ISM dominates the source mix mass by a factor of ∼4, the freshly synthesized, high-metallicity ejecta still greatly dominates the elemental source mix abundances (Z/H)_SM by factors of ∼2 to 6.

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The individual elemental abundances, (Z/H)_EJ/SS, of the supernova ejecta combined with the presupernova winds are the Salpeter IMF weighted yields calculated relative to solar system values for the dominant core-collapse supernovae, SN II and Ib/c, that make up the bulk (81%, Li et al. 2011) of Galactic supernovae. These yields for both the standard nonrotating star (e.g., Woosley & Weaver 1995; Woosley & Heger 2007) and recent rotating and nonrotating star (Limongi & Chieffi 2018) models of the slow neutron capture s-process nucleosynthesis are the primary process in main-sequence stellar evolution. Both of these supernova yields, based on an extensive network of nuclear reactions, have been shown (Timmes et al. 1995; Prantzos et al. 2018) to be consistent with Galactic chemical evolution.

For the rotating star models, Limongi & Chieffi (2018, Table 30) calculated the CCSN elemental nucleosynthetic yields for three fiducial surface rotation velocities, v = 0, 150, and 300 km s⁻¹, and defined an initial distribution of rotational velocities (IDROV), paralleling the initial mass function (IMF). The relative contributions of these three fiducials of the velocity distribution were then calibrated (Limongi & Chieffi 2018; Prantzos et al. 2018; Figure 4) by the measured Galactic chemical evolution as a function of Galactic metallicity [Fe/H], shifting from a relative weight of 75% for v = 300 km s⁻¹ yields and 25% for v = 150 km s⁻¹ from early times at metallicity −3, to 67% v = 0 and 33% v = 150 km s⁻¹ at solar metallicity zero. Using the latter weights for their fiducial yields, we calculated the current rotating-star CCSN source mix abundances shown in Figure 4(b), and for the standard nonrotating-star model yields (Woosley & Heger 2007, Figure 7), we calculated the nonrotating CCSN source mix abundances shown in Figure 4(a).

These CCSN abundances for the two models, 4(a) and (b), appear to be quite similar and robust for the major elements measured in the cosmic rays. The elemental differences between these models are generally much less than one-half, while their CCSN supernova mix abundance ratios relative to solar system and interstellar values each differed much more widely by ∼2 to ∼8 times.

Even the CCSN supernova ejecta abundances, which are dominated (∼70%) by the more numerous and productive SN II progenitors below ∼25 M_⊙, are only very weakly affected by whether the more massive stars above ∼25 M_⊙ are able to explode as supernovae contributing to the mix, or whether they fail to explode and collapse in a black hole, taking most of their nucleosynthetic products with them. Various studies (Heger et al. 2003; Smartt 2015; Sukhbold et al. 2016; Mirabel 2017) suggest such masses for the onset of stellar black hole formation, and calculations (Woosley & Heger 2007, Figure 6) have shown that even if these more massive stars did explode as CCSNe, they would make "little difference except for the iron group," and even there they would only increase Fe/Si by 10% to 20%. The other massive stars up to ∼120 M_⊙ are thought (Woosley & Heger 2007; Branch & Wheeler 2017) to be stripped of H envelopes by mass transfer to binary companions or blown off in stellar winds to become He-rich stars of <6 M_⊙ and SN Ib or Ic, accounting for ∼20% and 10% respectively of CCSNe (Shivvers et al. 2017).

The s-process yields, together with important, but limited, contributions to C and N from AGB stars and Wolf–Rayet winds, and to the Fe peak from thermonuclear SN Ia supernovae of accreting white dwarfs (e.g., Nomoto et al. 1984, model W7), can account for the solar elemental abundances (e.g., Anders & Grevesse 1989; Lodders 2003) of the bulk of the elements up through Zn at Z = 30. For heavier nuclei all the way up through the actinides, the fast neutron capture r-process of explosive nucleosynthesis (e.g., Kappeler et al. 1989, 2011) competes with the core-collapse s-process, with each contributing about one-half overall, but their relative contributions vary from isotope to isotope. The core-collapse supernovae are a significant source of the r-process nucleosynthesis, but the newly detected neutron star mergers are also thought to be important sources, and the cosmic-ray abundances can provide a measure of their relative strength.

The major differences between the heavy elemental cosmic-ray mix and the ISM is that the latter is essentially the unbiased mix of core-collapse SN II/Ibc and thermonuclear SN Ia, while the cosmic-ray source mix is a highly biased one. This mix, which is dominated by the enriched core-collapse supernova ejecta and the refractory element grain condensation in it, plus those in the swept-up ISM, is even further enriched by grain sputtering injection into the accelerating supernova shocks. On the other hand, the cosmic-ray source abundances also show that only a negligible fraction of the cosmic-ray metals come from the very highly enriched ejecta of SN Ia supernovae, as we discuss in detail in Section 4.

First, however, we explore what the cosmic-ray source abundances reveal about refractory supernova grains and their sputtered suprathermal ion injection in core-collapse supernovae. We present a new analysis in Figure 5, comparing the most recent elemental cosmic-ray source abundances, (Z/H)_GCRS, from H to Pb with those of the best-fit source mix, (Z/H)_SM.

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Hints of a grain injection were first recognized (Reeves 1975) from the fact that the relative cosmic-ray source abundances of refractory elements, which are the first to condense into grains in a cooling medium, depleting the gas, and the last to sublime in a warming or hot medium, were rather uniformly larger than those of volatiles. Such fast grains are condensed in the rapidly expanding supernova ejecta (Cesarsky & Bibring 1981; Lingenfelter et al. 1998), but they can also be accelerated by supernova shocks out of the older, swept-up ISM (Epstein 1980; Meyer et al. 1997).

A grain connection was further supported (Lingenfelter & Higdon 2007) by measurements (Engelmann et al. 1990) of the cosmic-ray source abundances, most clearly of the volatiles, showing an atomic charge Z dependence close to that of the grain sputtering Coulomb cross section Z^2/3 (Sigmund 1969, 1981; Draine & Salpeter 1979; Tielens et al. 1994). This sputtering also appears as an A^2/3-dependence for relatively low mass elements A < 40, where the elemental mass A is simply equal to 2Z.

Such sputtering of fast grains can inject both the refractory elements originally condensed in the grains and also the refractories and volatiles implanted in them (e.g., Bibring et al. 1974; Audouze et al. 1976; Deneault et al. 2003) as suprathermal ions. Thus, as the mean cosmic-ray bulk mass mixing ratio of swept-up ISM to CCSN ejecta $\sim {4.3}_{-2.0}^{+2.4}$ (Murphy et al. 2016) indicates, both the peak shock acceleration of the cosmic rays and the ${Z}^{2/3}$-dependent sputtering enrichment of the suprathermal ions that feed it occur in the turbulent region of the remnant behind both the forward and reverse shocks during the period right after the onset of the turbulent, strong shock-generating, Sedov–Taylor stage of expansion (McKee & Truelove 1995; Truelove & McKee 1999, 2000).

This basic elemental Z-dependent Coulomb sputtering F_CS from interactions between two nuclei of charges z and Z is the sputtering cross section ${zZ}/{\left({z}^{2/3}+{Z}^{2/3}\right)}^{1/2}$ (Sigmund 1969, 1981). Since fast grain–element interactions are predominantly with ambient H, the enrichment factor for refractories, for example Z > 6, reduces to F_CS ∼ Z/(1+Z^2/3)^1/2. Although the enrichment for interactions with He is roughly a constant ∼2 times that of H, the relative abundance of He/H in the swept-up medium is only ∼0.1, so He contributes only about one-fifth as much as H and only about one-sixth of their total with virtually the same Z dependence. The He contribution becomes just a small addition to the general normalization constant.

Thus the calculated ratio of Coulomb sputtering cross sections between any two cosmic-ray elements would be expected to define their relative H sputtering enrichment with respect to their bulk mixing abundances solely as a function of their different charges Z, if their sputtered elements, Z and H, are directly injected as suprathermal ions into the accelerating shocks. So we should not be surprised to see in Figure 5 that the measured ratios (Z/H)_GCRS/(Z/H)_SM of the cosmic-ray source abundances divided simply by the uniform 4:1 bulk mass mix from Figure 4 match very well the nominal ${Z}^{2/3}$ enrichment ratios (dashed lines) expected (Sigmund 1969, 1981) from sputtering enrichment by near MeV/atom interactions injecting sputtered suprathermal ions into cosmic-ray accelerating shocks. This is also confirmed in the ratio (Z/H)_GCRS/(Z/H)_SMCSI and the maximum refractory grain enrichment constant relative to volatiles (Figure 6).

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These cosmic-ray injection ratios, however, are quite distinct from the standard integral "sputtering yield" (Sigmund 1969, 1981) that includes, and is dominated by, a far larger cascade of much lower energy sputtering interactions down to just eV/atom binding energy thresholds. These also result from some initial interactions. But only a very small fraction of these can pass the high-energy interaction threshold, required to produce suprathermal ions that seed cosmic-ray acceleration. Moreover, the relative abundances of all these with low energy thresholds are shifted by factors of ∼1 to ∼10 in their binding energies in addition to their Coulomb sputtering ratios. These standard yields also match very well with laboratory measurements (e.g., Andersen & Bay 1981) and supernova grain survival calculations (e.g., Micelotta et al. 2016).

A seemingly attractive alternative to the refractory versus volatile bias was preferential acceleration of elements with the lowest first-ionization potential (Casse & Goret 1978), which would be most easily and highly ionized in the warm neutral or partially ionized gas and thus the most easily accelerated. Indeed, such an effect was clearly observed in energetic solar flare particles. But, as Axford (1981) first pointed out, such injection would not operate in the hot, fully ionized plasma, where shock acceleration is most efficient, and instead would limit the acceleration to cooler, denser gas where it is least efficient, because of much higher competing ionization losses, rapid Alfvén wave damping, and radiative losses. Since many of the most favored elements were the same, it was at first hard to discriminate between the two alternatives observationally. Meyer et al. (1997), however, found several elements where the two differed significantly, and all favored a grain bias. They also noted a strong mass-dependent, A^0.8±0.2, enrichment of the highly volatile cosmic-ray elements, N, Ne, S, Ar, Se, and Kr, which as later pointed out (Lingenfelter & Higdon 2007) is also quite consistent with the classical Z^2/3 enrichment expected from grain sputtering injection (Sigmund 1969, 1981) of cosmic rays.

Both the ejecta mixing ratio and the grain connections have now been greatly strengthened by new cosmic-ray source composition measurements and analyses of Murphy et al. (2016) from their SuperTIGER and the earlier TIGER (Rauch et al. 2009). They not only found best-fit mixing mass fractions of around 19% fresh CCSN ejecta and 81% old interstellar gas and dust, but on comparing the cosmic-ray source composition with this source mix, rather than that of the solar system, they were able to quantify the elemental injection biases.

They found that those more refractory cosmic-ray elements Mg, Al, Si, Ca, Fe, Ni, Sr, and Zr were all roughly ∼4 to ∼4.5 times more abundant with respect to solar system values than the most volatile ones, Ne, S, Ar, Cu, Ge, Se, and Kr. They further found that the best-fit atomic mass dependent enrichment of the highly volatile cosmic-ray elements was A^0.632±0.119, which is quite consistent with the ∼A^2/3 expected from the grain Coulomb sputtering and scattering cross section for A/Z = 2. However, for the more abundant, highly refractory cosmic-ray elements with tighter uncertainties, the best-fit value was just A^0.583±0.072, so the authors concluded only that both the refractory and volatile elements show a mass-dependent enrichment with similar slope.

But A^2/3 is just approximately the A-dependence for the Z^2/3 Coulomb sputtering dependence, since A/Z = 2 only for the limited range of refractories in 20 < A < 40, while for all of the neutron-rich elements above that, A grows relative to Z with what can be approximated as a power-law dependence, where $Z\sim A/2{A}^{y}\sim 0.5{A}^{(1-y)}$. Thus the A-dependent Coulomb sputtering and scattering yield enrichment of such heavy elements that is equivalent to Z^2/3 is $\sim {A}^{2(1-y)/3}$.

For the cosmic-ray source abundances analyzed by Murphy et al. (2016), the refractories ranged between ²⁴Mg and ⁹¹Zr, or a relative atomic mass range $({A}_{\mathrm{MAX}}/{A}_{\mathrm{MIN}})$ of 91/24 = 3.79, with A/Z growing from 2.0 to a maximum of 2.28. This gives a nominal exponent y = log (2.28/2)/log (91/24) = 0.10 and the expected Z^2/3 equivalent A-dependent index 2(1–y)/3 = 0.60. That value is quite consistent within 0.24σ of the best-fit value of 0.583 ± 0.072 (Murphy et al. 2016). Similarly, the volatiles, which ranged from ²⁰Ne to ⁸⁰Br with A/Z rising from 2.0 to 2.32, give a nominal exponent y = log (2.32/2)/log (80/20) = 0.11 and the expected Z^2/3 equivalent A-dependent index 2(1–y)/3 = 0.59, and that too is in very good agreement with the best-fit value of 0.632 ± 0.119. Thus, the cosmic-ray source abundance analyses of Murphy et al. (2016) would actually provide strong evidence for a Z^2/3 fast grain sputtering injection of the cosmic-ray source mix of core-collapse ejecta (Woosley & Heger 2007).

3.1. Grain Condensation Fraction

3.2. Refractory/Volatile Constant

The cosmic-ray refractory/volatile enrichment constant, C_R/V = 5.5F_GC+1.0F_V, brackets the range of partially condensed refractory grains, where F_GC is the partially condensed grain fraction. From the Coulomb sputtering dependence of the noble gases, the remaining volatile fraction ${F}_{V}=(1-{F}_{\mathrm{GC}})$ may also be interpreted as essentially the mean saturated implantation fraction for noble gases, or ∼1/5.5 or ∼0.18. That would also appear to be consistent with the base abundance fractions of such volatiles in C3 chondrules indicated in Figure 7, and with extensive calculations (Marhas & Sharda 2018) of relative implantation fractions versus condensation of <0.25 for both Fe and Ni for the most probable models of 15–25 M_⊙ SN II CCSN grain formation.

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Thus, ${C}_{R/V}=5.5{F}_{\mathrm{GC}}+1(1-{F}_{\mathrm{GC}})=1+4.5{F}_{\mathrm{GC}}$. The elemental grain condensation fractions, F_GC, have been determined both indirectly from optical determinations (e.g., Savage & Sembach 1996; Jenkins 2009) of the average depletion of elements in the interstellar gas observed along various lines of sight through the ISM compared to their solar system abundances, and directly from analysis (e.g., Wasson & Kallemeyn 1988) of early solar system and presolar grains, or type C chondrules, in carbonaceous chondritic meteorites. For each element, these fractions show a clear correlation with the condensation/sublimation temperature of its refractory compounds (Kallemeyn & Wasson 1981; Wasson & Kallemeyn 1988; Savage & Sembach 1996; Lodders 2003; Jenkins 2009), as can be seen in Figure 7. The refractory elements in the most common carbonaceous chondrules, type C1, are thought to be fully condensed, so F_GC = 1 for condensation temperatures T_C greater than a few 100 K. These relative abundances are consistent with solar photospheric values, and together they are taken (e.g., Anders & Grevesse 1989; Lodders 2003) as the two best measures of solar system abundances, as well as local ISM abundances.

Detailed calculations (Bianchi & Schneider 2007; Nozawa et al. 2007; Sarangi & Cherchneff 2015; Bocchio et al. 2016; Biscaro & Cherchneff 2016; Micelotta et al. 2016) of dust formation and survival in core-collapse supernova ejecta model the condensation of ∼0.1 to 0.6 M_⊙ in refractory grains at ∼20 K, consistent with observations mentioned above. They also suggest a surviving mass of the order of >0.1 M_⊙ in refractory grains after the destructive passage of reverse shocks, expanding in ambient densities n < 0.1 H cm⁻³, running roughly proportional to $\sim {n}^{-1/2}$. During that passage, grains of radius 10 to 200 A are calculated (Bianchi & Schneider 2007) to reach temperatures between about 100 and 1000 K.

Such a grain temperature range is consistent with recent observations of grains in supernova remnants (Matsuura et al. 2015; Micelotta et al. 2018; Sarangi et al. 2018; and others noted above), as well as the ISM (Savage & Sembach 1996). These temperatures are also quite consistent with the range of grain exposure temperatures between ∼400 K and ∼1400 K (see Figure 7) inferred from the partial condensation or sublimation fractions of carbonaceous chondrules of types C1 through C3 (Kallemeyn & Wasson 1981; Wasson & Kallemeyn 1988). We might expect that the highest temperatures that determined the partial grain sublimation fractions for both the ejecta and swept-up dust would be that in the hot, turbulent, supernova-shocked regions where the ejecta and swept-up interstellar grains are mixed and accelerated, and could therefore be essentially the same F_GC for both.

Not only are core-collapse supernovae thought (Dwek 1998) to be the main source of silicate grains in the ISM, but there is also clear evidence of a direct connection between the supernova grains and some of the presolar grain (Wasson 2017) inclusions in the C3 chondrules, particularly in calcium–aluminum inclusions (CAIs), which are the oldest known solids in the solar system. Isotopic anomalies in these inclusions reveal the presence of short-lived radioactive nuclei with half-lives <15 Myr: ²⁶Al, ³⁶Cl, ⁴¹Ca, ⁵³Mn, ⁶⁰Fe, ¹⁰⁷Pd, ¹²⁹I, ¹⁸²Hf, and ²⁴⁴Pu, all of which are attributed to core-collapse supernovae and their Wolf–Rayet winds (Adams 2010; Fujimoto et al. 2018, and references therein). Although these radionuclei were first thought to have resulted from a rare chance occurrence of a nearby supernova, they are now recognized as the natural result of the fact that the Sun, like nearly all stars, was formed in a highly spatially and temporally clustered star formation region. Moreover, the injection of these freshly synthesized nuclei into the protosolar nebula is thought (e.g., Connelly et al. 2008; Goodson et al. 2016) to be via grains like the CAIs.

We suggest, therefore, that the temperature-dependent fraction of each refractory element that condensed and survived sublimation in the grains measured in the most heated sample, the class C3 of meteoritic chondrules, may also provide a reasonable first estimate of F_GC for both the fast ejecta grains and the swept-up interstellar grains, since their different relative elemental abundances do not appear to have a major effect on the principal condensates. Nonetheless, the meteoritic fractions can also be significantly modified by subsequent heating or cooling in the protosolar nebula, particularly those of the chemically active and more volatile elements, such as Pb and Cd with condensation temperatures below ∼900 K, which show (Figure 7) fluctuations of as much as a factor of ∼2 above the relatively tight thermal correlation of those at higher temperatures and may result from subsequent recondensation or accretion (e.g., Snow 1975).

For O, in fact, the fraction in grains appears to be primarily determined (Meyer et al. 1997; Lingenfelter et al. 1998) by the sum of the fractions of O bound in highly refractory SiO₂, MgO, Fe₃O₄, Al₂O₃, and CaO weighted by their relative source mix abundances, 1.48:1.67:1.21:0.21:0.07, which gives a grain F_R ∼ 0.15 relative to the O abundance.

If the temperature and corresponding elemental grain condensation fractions F_R for both fast ejecta grains and swept-up interstellar grains are indeed the same, then the grain condensation/sublimation constant, ${C}_{R/V}=5.5\,{F}_{\mathrm{GC}}\,+1(1-{F}_{\mathrm{GC}})=1+4.5\,{F}_{\mathrm{GC}}$, times the grain Coulomb sputtering factor, F_CS = Z/(1+Z^2/3)^1/2, defines the overall elemental grain injection factor, F_GI = C_R/V F_CS. This factor operates on the supernova source-mix abundances to produce the final grain-injected source-mix elemental abundances,

$$\begin{eqnarray}\begin{array}{rcl}{({\rm{Z}}/{\rm{H}})}_{\mathrm{SMGI}} & = & {F}_{\mathrm{GI}}{({\rm{Z}}/{\rm{H}})}_{\mathrm{SM}}\\ & = & (1+4.5{F}_{\mathrm{GC}})[Z/{\left(1+{Z}^{2/3}\right)}^{1/2}]{\left({\rm{Z}}/{\rm{H}}\right)}_{\mathrm{SM}},\end{array}\end{eqnarray} \tag{ 2 }$$

which can be directly compared with the observationally determined Galactic cosmic-ray source elemental abundances, (Z/H)_GCRS.

3.3. Galactic Cosmic-Ray Source Composition/Grain-injected Supernova Mix

With the core-collapse supernova mix of the ejecta and the ISM, (Z/H)_SM, we can define the overall supernova source mix and grain enrichment, ${({\rm{Z}}/{\rm{H}})}_{\mathrm{SMGI}}={F}_{\mathrm{GI}}{({\rm{Z}}/{\rm{H}})}_{\mathrm{SM}}$, which is the expected supernova cosmic-ray abundance. Comparing this model distribution with the cosmic-ray source distribution determined from local cosmic-ray abundance measurements (Z/H)_GCRS/(Z/H)_SMGI, we see from Figure 8 that the model is in good agreement, ∼±35%, with the cosmic-ray source abundances. This agreement also clearly suggests that the refractory grain condensation/sublimation fraction F_R from the C3/C1 chondrules (Figure 7) gives, in fact, a reasonable estimate of both the ejecta and swept-up grain fractions.

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Thus, we see that the extensive measurements of the local cosmic-ray elemental and isotopic abundances from H to U have now shown very clearly that they differ radically from those of the solar system and local ISM by factors varying from ∼20 to ∼200. But the nature of these differences have also revealed their primary causes: the mixing of supernova ejecta with the ISM, and the injection of these nuclei as suprathermal ions sputtered from refractory grains into the supernova shocks, where DSA carries them to cosmic-ray energies.

Moreover, these new measurements have at last quantified these two basic processes in the generation of the cosmic-ray composition. In particular, they show that the cosmic-ray source elemental abundance ratios can be simply defined by a single mixing ratio by mass of high-metallicity supernova ejecta and swept-up ISM, coupled with the calculated s-process nucleosynthesis yields of the dominant core-collapse supernovae. This mixing increases the elemental cosmic-ray source abundances by factors of ∼2 to ∼10 over those of the ISM.

The accompanying grain injection, which provides even larger enrichment, can be defined by the fractions of individual refractory elements that condensed into fast grains in the freely expanding ejecta, as estimated from measurements of meteoritic chondrules, which can account for additional factors of as much as ∼5.5, and combined with their sputtering and scattering that inject them as suprathermal ions into the cosmic-ray accelerating shocks explains their strong elemental charge-dependent enrichment of ∼4 to ∼20, proportional to Z^2/3, which reflects their relative Coulomb sputtering rates. We have shown that applying these two basic processes of mixing and injection to interstellar values can thus give source abundances of the major cosmic-ray elements to within ±35% at ±1σ uncertainty. This vastly reduces the net differences of cosmic-ray source abundances compared to solar system abundances by factors of ∼20 to ∼200, and all with no free parameters.

3.4. CCSN Ejecta and ISM Mixing, Grain Condensation, and Sputtering Injection Equations

In summary, we have shown that the elemental (Z) abundances of the cosmic-ray metals at their source, (Z/H)_GCRS, determined from local flux measurements after correcting for the effects of differential Z-dependent propagation and spallation, can be well modeled by two factors, a supernova source mix, (Z/H)_SM/SS, and a grain sputtering injection, F_GI, applied to the solar system abundances (Z/H)_SS, namely

$$\begin{eqnarray}&&{\left({\rm{Z}}/{\rm{H}}\right)}_{\mathrm{SMGI}}={F}_{\mathrm{GI}}{\left({\rm{Z}}/{\rm{H}}\right)}_{\mathrm{SM}/\mathrm{SS}}{\left({\rm{Z}}/{\rm{H}}\right)}_{\mathrm{SS}},\end{eqnarray} \tag{ 3 }$$

where the source mixing factor is

$$\begin{eqnarray}\begin{array}{rcl}{\left({\rm{Z}}/{\rm{H}}\right)}_{\mathrm{SM}} & = & ({F}_{\mathrm{EJ}}{\left({\rm{Z}}/{\rm{H}}\right)}_{\mathrm{EJ}/\mathrm{SS}}\\ & & +\,[1-{F}_{\mathrm{EJ}}]{\left({\rm{Z}}/{\rm{H}}\right)}_{\mathrm{ISM}/\mathrm{SS}}){\left({\rm{Z}}/{\rm{H}}\right)}_{\mathrm{SS}}.\end{array}\end{eqnarray} \tag{ 4 }$$

This enriches the individual elemental cosmic-ray abundances by ∼2 to ∼10 times that of the solar system values and depends both on the fluid dynamical process of the bulk mass mixing of the ejecta and swept-up interstellar material and on the underlying individual nuclear reaction processes generating the relative abundances of each element.

Here, F_EJ ∼ 0.2 is the best-fit, single mixing mass fraction of highly enriched supernova ejecta mixed with a swept-up interstellar mass fraction, and F_ISM = (1 − F_EJ), determined both from Be and B measurements of old halo stars (Duncan et al. 1992; Ramaty et al. 1997; Alibes et al. 2002; Tatischeff & Gabici 2018) and cosmic-ray source abundances (Higdon & Lingenfelter 2003; Binns et al. 2005; Lingenfelter & Higdon 2007; Rauch et al. 2009; Ahn et al. 2010; Murphy et al. 2016). Also, ${({\rm{Z}}/{\rm{H}})}_{\mathrm{EJ}/\mathrm{SS}}$ are the individual elemental abundances of the combined presupernova winds and supernova ejecta relative to solar system values, integrated over a Salpeter initial mass function, calculated from s-process yields (Woosley & Heger 2007; Limongi & Chieffi 2018) for core-collapse supernovae that make up the bulk (81%, Li et al. 2011) of Galactic supernovae; ${({\rm{Z}}/{\rm{H}})}_{\mathrm{ISM}/\mathrm{SS}}=1.32{({\rm{Z}}/{\rm{H}})}_{\mathrm{SS}}$ are determined from Galactic chemical evolution models (Timmes et al. 1995); and (Z/H)_SS are the elemental abundances in the solar system (Lodders 2003), primarily from fully condensed C1 chondritic meteorites. The highly effective suprathermal ion injection factor,

$$\begin{eqnarray}&&{F}_{\mathrm{GI}}=(1+{[({C}_{R/V})-1]{F}_{\mathrm{GC}})(Z/[1+{Z}^{2/3}]}^{1/2}),\end{eqnarray} \tag{ 5 }$$

further enriches the individual elemental cosmic-ray abundances by even greater factors of ∼4 to ∼100 and depends on both the individual elemental refractory grain condensation and sublimation temperature and the elemental charge, Z, dependent grain sputtering and scattering yields. Here, C_R/V is a constant equal to the enrichment of the most refractory elements compared to the most volatile, equal to ∼5.5 as shown above, while F_GC are the grain temperature-dependent fractions of each element that condensed from the ejecta gas into grains, determined from the elemental abundance ratios C3/C1 of the most heated and sublimed versus the most cooled and condensed chondrule measurements (Wasson & Kallemeyn 1988). Finally, $Z/{[1+{Z}^{2/3}]}^{1/2}$ is the classic Coulomb sputtering cross section calculated from interactions between H and refractory grain nuclei of charge Z (Sigmund 1969, 1981).

The ability of these processes to explain within ±35% the bulk of the major cosmic-ray metal abundances in terms of s-process nucleosynthesis in the dominant core-collapse supernovae also provides a powerful new tool for determining the contributions of the Wolf–Rayet winds, the rarer thermonuclear SN Ia supernovae, the AGB stars, and the relative production of r-process elements by core-collapse supernovae and neutron star mergers.

Last, we discuss the broader implications of these results on the cosmic-ray source and sites and their injection and acceleration, and we show that they can provide a new, self-consistent framework that ties together the Galactic sources, injection, acceleration, and propagation of the cosmic rays.

The comparison of the observationally determined cosmic-ray source compositions (Z/H)_GCRS with that of cosmic rays generated by the s-process elements in the core-collapse supernovae ejecta, (Z/H)_SMGI, shows that this grain-injected ejecta-mix model can account for most of the cosmic-ray elemental compositions to within ±35% at ±1σ. In Figures 8(a) and (b), however, there are several elemental abundances, Na, Mg, Fe, Cu, Ge, Se, Br, Hf, and W, among the ∼1/3 of the values that could be expected to randomly lie above or below ±1σ.

But there are also a few very conspicuous outliers in this comparison that appear to signal contributions from other sources. As noted above, those outliers that lie well above unity suggest significant underproduction of cosmic rays by the core-collapse s-process model and indicate possible contributions from extra sources of (Z/H)_XMI = [{(Z/H)_GCRS/(Z/H)_SMGI} − 1] (Z/H)_SMGI. In particular, the cosmic-ray C and N overabundances compared to expected CCSN yields suggest an extra contribution from Wolf–Rayet winds or AGB stars. Other scattered outliers above Z > 30, particularly Pt, may result from extra contributions of r-process elements from CCSNe not calculated along with the s-process by Woosley & Heger (2007). Perhaps most important, there is a surprising lack of an Fe outlier expected from SN Ia supernovae, which shows a clear lack of acceleration of cosmic-ray metals by these supernovae.

4.1. C and N

4.2. Missing SN Ia Cosmic Rays

4.3. CCSN r-process

Here we focus on the newly demonstrated link between the ubiquitously measured cosmic-ray source mass mixing ratio of supernova ejecta to swept-up ISM, and the homologously evolving supernova remnant shocked mix at the onset of adiabatic, Sedov–Taylor expansion. This uniquely defines at last the source, time, and place of peak cosmic-ray acceleration within that evolution. We also consider the dominance of grain sputtering and ion injection on the cosmic-ray elemental composition variations and the overall efficiency of DSA in supernova remnants. This places critical constraints on the source environments and sites of such cosmic-ray acceleration because of the major destruction of such grains by reverse shocks at that same time in the Sedov–Taylor expansion.

But first we briefly review the nature of the sources and distribution of supernovae in our Galaxy and the basic nature of their evolution. Most supernovae do not occur randomly throughout the ISM. Instead, their progenitors, which account for just ∼0.1% of all stars, are born in episodic starbursts in the densest parts of giant molecular clouds that contain up to $\sim 3\times {10}^{6}{M}_{\odot }$ of gas and dust (e.g., Pudritz 2002; Branch & Wheeler 2017). There, many thousands of stars are formed within a megayear in tight clusters only a few parsecs in diameter. The most massive and most rapidly evolving of these stars, the 15–120 M_⊙ O stars (Martins et al. 2005) and the 8–15 M_⊙ brightest B stars, the red supergiants (RSG), end their lives as core-collapse supernovae. Moreover, they pass on much of their natal stellar clustering from their compact OB associations. With a mean radial dispersion velocity of only ∼2 km s⁻¹, or ∼2 pc Myr⁻¹ (de Zeeuw et al. 1999), most of these supernova progenitors disperse <50 pc (Higdon et al. 1998; Higdon & Lingenfelter 2005) during their relatively short lives of just 3–35 Myr before they become supernovae (Schaller et al. 1992; Chieffi & Limongi 2013). Thus, the core-collapse supernovae are also highly correlated in both space and time.

Such core-collapse supernovae account (Li et al. 2011; Shivvers et al. 2017) for ∼81% of the Galactic total at a rate of one CCSN every ∼43 yr. That leaves the thermonuclear SN Ia from the far more slowly evolving (a gigayear or more, Greggio 2005), and hence quite widely scattered, accreting white dwarf binaries to produce the bulk of single supernova remnants in the ISM at one SN Ia every ∼180 yr. There are on the order of 100 supernova progenitors in a typical OB association (McKee & Williams 1997; Williams & McKee 1997; Higdon & Lingenfelter 2005). Observations (e.g., Blaauw 1991) show that these are generally produced as part of a series of several bursts of star formation separated in space and time by ∼50 pc and ∼4 Myr, as their parent molecular cloud complexes form multiple OB associations.

These OB clusters are surrounded by self-generated, giant H II superbubbles of basically the local ISM, a prominent feature of Galactic star-formation regions. Blown by the powerful UV radiation and the Wolf–Rayet winds of the most massive O stars and the culminating supernovae, they formed cavities in the molecular clouds, blowing out most of the gas and magnetic field into the surrounding warm (∼10⁴ K) and dense (∼10 H cm⁻³) H i supershells or cocoons. These enclose their merged, supernova-remnant superbubble cavities of >10⁵ pc³ filled with hot (>10⁶ K) and tenuous (∼0.001 to <0.01 H cm⁻³) gas (e.g., Cox & Smith 1974; McKee & Ostriker 1977; Weaver et al. 1977; McCray & Snow 1979; Heiles 1987; Mac Low & McCray 1988; Shull & Saken 1995; Kim & Ostriker 2017).

Because of the asymmetry of the swept-up, compressed magnetic field pressure, the superbubbles tend to be roughly tubular in shape, extending along the magnetic field (Tomisaka 1992) until their radii approach the scale height of the Galactic disk. Then they vent into the halo (Kim & Ostriker 2018), forming what Heiles (1979, 1994) dubbed "worms" and "chimneys" of hot superbubble gas, enclosed by supershells of warm ionized gas.

As few as five supernovae in a cluster appear to be all that are needed to generate a superbubble (Higdon & Lingenfelter 2005). So from the size distribution of the young star clusters, ∼85% of the core-collapse stellar progenitors are expected to be born in superbubble-generating clusters. In that crowded environment, as many as one-half of those smaller OB clusters may also be engulfed by neighboring superbubbles.

In addition, it appears that essentially all of the core-collapse supernova progenitors that missed being engulfed by superbubbles still explode very close to some part of the parent molecular cloud complex, and can thus account (Lingenfelter 2018) for essentially all of the single gamma-ray supernova remnants observed by Fermi to be interacting with filaments of molecular clouds. For, assuming a detectable lifetime (Acero et al. 2016) of anywhere from ∼10 to 50 kyr, those 30 to 44 gamma-ray remnants interacting with molecular gas give an occurrence rate of ∼1 to 4 SN/kyr out of ∼23 SN/kyr for all Galactic core-collapse supernovae. Independent of the Fermi sky coverage, this amounts to ∼4% to 17% of core-collapse supernovae, and from one-quarter to all of the estimated ∼15% that were born in some of the smaller clusters that were unable to create superbubbles of their own.

Core-collapse supernovae in hot (∼10⁶ K) superbubbles likely account (Higdon & Lingenfelter 2005) for ∼75% of all Galactic supernovae, occurring at an overall rate of one CCSN every ∼43 yr, with an average of ∼300 supernova occurring in each of an estimated ∼3000 merged superbubbles over their ∼50 Myr lifetimes. The supernovae in these low-density (n < 0.01 H cm⁻³) superbubbles do not produce many small (∼20 pc), scattered individual supernova remnants like those in the cooler, denser (>0.1 H cm⁻³) phases of the ISM that radiate away most of their shock energy in <50 kyr. Instead they can grow to ∼100 pc. Moreover, these supernovae occurring in the bursts of highly clustered stars in the larger, more tenuous superbubble gas (∼0.001 H cm⁻³) can interact with older, nearby supernova remnants, and their blastwave shocks may extend as much as a few hundred parsecs as the superbubbles of nearby associations merge and collectively (e.g., Bykov & Fleishman 1992; Bykov & Toptygin 2001) continue accelerating cosmic rays for ∼50 Myr or more until their last supernova explodes.

Since ∼300 such spatially and temporally clustered CCSNe occur in a superbubble at rate of about one CCSN every ∼150 kyr in its typical lifetime of ∼50 Myr, not only do they each serially accelerate fresh ejecta and swept-up ISM by DSA, these shocks also further accelerate some of the cosmic rays from previous supernovae.

So the extensive superbubble acceleration processes also appear (Bykov & Toptygin 2001; Ferrand & Marcowith 2010) to be able to account for both the acceleration and the escape spectra of the locally measured cosmic rays with the simple −2.7 power-law energy spectra with constant nuclear abundance ratios up to the so-called "knee" above 10⁶ GeV. There the proton spectra break (Horandel 2013) at E_p ∼ 4 × 10⁶ GeV, followed successively at higher energies by the less-abundant helium and heavier nuclei breaking at E_Z ∼ E_pZ, where Z is their nuclear charge, all the way up to ∼4 × 10⁸ GeV (Z ∼ 92) as the mean nuclear mass of the remaining cosmic rays increases. At the same time, the further acceleration by the multiple supernova shocks helps to smooth the overall spectral index to about −3.1 above the knee on up to the "ankle" at ∼10⁹ GeV.

The thermal X-ray emission from the hot, tenuous cores of these superbubbles is generally too faint and diffuse to be resolved in all but some of the closest and largest bubbles (e.g., Cash et al. 1980). The surrounding, much denser H i supershells nonetheless are readily seen in 21 cm emission, and these and their enclosed voids led to their discovery (Heiles 1979). The youngest superbubbles are also revealed by much shorter lived (∼7 Myr) and bright Hα-emitting H ii shells on the inner walls of the cocoons, ionized by the intense far-ultraviolet emission of the Wolf–Rayet stars before they blow off most of their mass, collapse, and explode as SN Ic (Van Dyk et al. 1996; Anderson et al. 2012).

Most recently, observations of high-energy gamma-rays from decay of π^o, which are produced by relativistic pp-interactions and are a signature of cosmic-ray proton interactions, have also shown that cosmic rays are being accelerated with a spectrum of ∼E^−2.3 up to the "knee" at a PeV in supernova remnants of spatially and temporally clustered CCSNe in such massive superbubbles as the Cygnus Cocoon, the Galactic Central Molecular Zone (CMZ), and both Westerlund 1 and 2 (Abdo et al. 2009; Ackermann et al. 2011; Aharonian et al. 2019).

The Cygnus Superbubble (Figure 9), the closest and best studied of these, is fed by at least 10 OB associations and smaller clusters (Wright et al. 2015), and it has been growing steadily for at least 20 Myr (Comeron et al. 2016, and references therein). In Cygnus OB2, the youngest and biggest of these associations, some 55 O stars of 15 M_⊙ or more have currently been identified (Wright et al. 2015), plus at least 88 early B0–B2 star RSG supernova progenitors of 15–8 M_⊙. Most of the O stars in Cyg OB2 were produced between 1 and 7 Myr ago, and possibly peaked around 4 to 5 Myr ago (Wright et al. 2015). Since the lifetimes of the most massive of these stars are only ∼3 Myr (e.g., Schaller et al. 1992; Chieffi & Limongi 2013), a number of these stars have already exploded and others have lost significant mass in their winds, so their mass distribution has evolved considerably. In addition, there are all the other older OB associations within the superbubble with ages of as much as ∼20 Myr (Comeron et al. 2016). Together these suggest (Lingenfelter 2018) a mean supernova rate of ∼6–9 CCSN/Myr that will produce some ∼300 to ∼500 core-collapse supernovae over the expected ∼50 Myr lifetime of just the Cygnus Superbubble alone.

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High-energy π^o-decay γ-rays have also shown (Morlino & Caprioli 2012) proton acceleration up to nearly the same energy in one of the three youngest SN Ia supernovae in our Galaxy (Tycho Brahe's of 1572), as well as from about 30 individual CCSN remnants, such as W44 and IC433, scattered throughout the ISM (Albert et al. 2007; Ackermann et al. 2013). These latter supernova remnants, though not in superbubbles, all belonged to a small, complex class of "mixed-morphology" remnants (Rho & Petre 1998) that were interacting in dense (>10 H cm⁻³), filamentary molecular clouds, which made them much brighter than others in the average ∼0.1–1 H cm⁻³ ISM. But, because of their much higher accompanying ionization losses, they were far less efficient, and their inferred cosmic-ray proton spectra broke (Albert et al. 2007; Acciari et al. 2011) below E^−2/3 at barely 20–200 GeV, which is only ∼0.01 to 1% of the PeV break energy of the cosmic rays in CCSN remnants in hot, low-density (<0.01 H cm⁻³) superbubbles (Aharonian et al. 2019; Lingenfelter 2018).

5.1. Supernova Remnant Evolution, Grains, and Cosmic Rays

Tracing the evolution of the ejecta mixing and grain injection processes, we follow the general three-stage evolutionary approximations of supernova remnant expansion from free expansion to the Sedov–Taylor or adiabatic stage, when after a time depending on the surrounding interstellar gas density, the ejecta sweeps up a comparable mass of interstellar matter, driving a strong blastwave or forward shock into the ISM and in reaction a reverse shock into the ejecta, both of which can efficiently accelerate cosmic rays. The remnant continues in that stage until the radiative or "snowplow" stage, when so much more mass has been swept up that most of the remaining energy is radiated away and the expansion becomes subsonic.

In particular, we adopt McKee & Truelove's (1995) treatment of supernova remnant expansion, which demonstrates that all uniform nonradiative SNRs moving into uniform, homogeneous media can be described by a single, universal solution in terms of a characteristic remnant age, t_ST = 209 M_EJ^5/6 E₅₁ ^−1/2 n_o ^−1/3 yr, and radius, ${R}_{\mathrm{ST}}=2.23{M}_{\mathrm{EJ}}^{1/3}{n}_{o}^{-1/3}$ pc, at the effective onset of the Sedov–Taylor stage of expansion. These analytic expressions for the remnant radius versus age, and that for the velocity below, define the homologous evolution of the supernova remnant from the explosion to the end of the Sedov–Taylor expansion at the onset of the pressure-driven snowplow (PDS) stage, ${t}_{{\rm{PDS}}}\,\sim 13300\,{E}_{51}^{3/14}\,{n}_{o}^{-4/7}$ yr and ${R}_{\mathrm{PDS}}\sim 14.0\,{E}_{51}^{-2/7}\,{n}_{o}^{-3/7}$ pc. The supernova remnant evolutionary tracks drawn from these expressions are shown in Figure 10.

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Immediately after a supernova explosion, its high-temperature ejecta remnant approaches a free expansion stage and quickly cools to <100 K in the moving frame, and the refractory elements condense into grains. Assuming that all supernovae produce the same blast energy of E₅₁ ∼ 10⁵¹ erg, independent of their ejecta mass, M_EJ, the remnants are calculated (McKee & Truelove 1995) to be expanding freely at a velocity V_FE = R_ST/t_ST ∼ 10400 E₅₁ ^1/2 M_EJ ^−1/2 km s⁻¹, or ∼pc Myr⁻¹. For the CCSN remnants, whose ejecta masses lie primarily between ∼10 and 20 M_⊙, their free expansion velocities range between about 3300 and 2300 km s⁻¹. Thus, for the majority of supernovae, the CCSNe whose progenitor O and B stars are both born and die in the uniform hot, tenuous superbubbles, with densities ranging from n ∼ 0.01 to 0.001 H cm⁻³ (e.g., Mac Low & McCray 1988; Tomisaka 1992; Higdon et al. 1998; Higdon & Lingenfelter 2005), their remnants typically take about 10–15 kyr to expand to about ∼30 to 50 pc radius and enter the Sedov–Taylor stage of expansion. A fraction of the CCSN remnants (Lingenfelter 2018), such as those isolated Fermi high-energy gamma-ray SNRs that encounter denser (>0.1 H cm⁻³) interstellar gas, can even sweep up enough to enter the Sedov–Taylor expansion phase in <10 kyr. In all, the bulk mixing of the CCSN ejecta and swept-up ISM drives further Rayleigh–Taylor turbulent mixing, leading to gas–grain sputtering interactions and suprathermal ion injection into the DSA of a small fraction of the ejecta/ISM mix to cosmic-ray energies.

5.2. Swept-up ISM to CCSN Ejecta Mass Mix

Here we discuss the telling implications of the ubiquitous ∼4 to 1 mixing ratio of swept-up ISM to CCSN ejecta mass, defined by analyses of the cosmic-ray source composition in Section 2 and Figure 5.

We (Higdon et al. 1998) originally suggested that the cosmic-ray mass mixing ratio might result from the build-up of an essentially constant ejecta-enriched interstellar material from the clustered CCSN in the cores of superbubbles. But as we now see in Figures 10 and 11, there is no constant ejecta fraction in the CCSN during the reverse shock passage. In every remnant, that fraction continuously decreases by ∼t^−6/5, diluted by more swept-up ISM, and reaching just a few percent as its radius approaches that of the galactic disk scale height of ∼100 pc at which superbubbles themselves vent their contents through "worms" and "chimneys" (e.g., Heiles 1987; Kim & Ostriker 2017).

Yet, as we also see in Figure 11 particularly, the OB-clustered CCSN explosions in the hot, low-density superbubbles can, in fact, produce the uniform cosmic-ray mass mixing ratio simply through the standard homologous evolution of supernova remnants. What is critical about superbubbles is their low density, <0.01 H cm⁻³. That allows the bulk of the grains to survive the passage of the reverse shock through the mixing ejecta during the early Sedov–Taylor phase of grand sputtering, injection, and acceleration, and it delays the major fraction of that acceleration until the expected cosmic-ray ⁵⁹Ni abundance is consistent with observed limits.

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The latest analysis (Murphy et al. 2016) of the major cosmic-ray source abundances from N through Zr gives a best-fit 1σ value of the CCSN ejecta mass fraction in the cosmic-ray source F_EJ of ${19}_{-6}^{+11} \%$, or a mean swept-up ISM to CCSN ejecta mass mixing ratio ISM/CCSN = (1 − F_EJ)/F_EJ of ${4.3}_{-2.0}^{+2.4}$, spanning a broad t*/t_ST range from 2.3 to 6.7 at peak acceleration.

At the onset of the Sedov–Taylor stage, CCSNe building strong forward and reverse shocks had swept up and mixed with ∼1.6 times as much mass of ISM as that of their own ejecta (McKee & Truelove 1995; Truelove & McKee 1999, 2000). This homologous, evolving remnant mass mixing ratio of shocked ISM mass to SN ejecta would seem quite naturally to be roughly that of the relative mix of the source material that is accelerated to cosmic-ray energies by those shocks.

The bulk mixing mass ratio of this growing swept-up ambient ISM to SNR ejecta is simply M_SISM/M_EJ = (4π/3)R_TS ³ρ_o/M_EJ, and ρ_o is the ISM mass density. Since at the onset of the Sedov–Taylor expansion stage the characteristic mass ratio Ms_ISM/M_EJ = 1.6 (McKee & Truelove 1995), and the homologous relative shock radius R*/R_ST ∼ (t*/t_ST) ^2/5, the age-dependent mixing mass ratio is simply M_SISM/M_EJ ∼ 1.6 (t*/t_ST)^6/5 during Sedov–Taylor expansion.

Thus, the measured mean cosmic-ray mixing mass ratio of 4.3 (Murphy et al. 2016) is equal to the homologous mixing value calculated in supernova remnants around ∼2.3 times the Sedov–Taylor onset time, t*/t_ST, on the reverse-shock evolutionary track (McKee & Truelove 1995). For the principal M_EJ ∼ 10 to 20 M_⊙ CCSNe in superbubbles of ∼0.001 H cm⁻³, the peak acceleration occurs at ages of ∼30 to 60 kyr (Figure 10). Also as noted before, even with the ISM dominating the mass by a factor of ∼4:1, the still-larger factor of ∼10 metallicity of the ejecta still dominates ∼2:1 by element over the ISM.

The composition of the cosmic rays is once again the key that graphically ties together the ensemble of homologous and concurrent processes of cosmic-ray acceleration driven by supernova shock and turbulence generation. The reverse shock decouples the fast ejecta grains from the slowing ejecta plasma, which they leave behind and move forward to mix with the swept-up and accelerated ISM. In this growing, turbulent mix between the forward and reverse shocks, these grains sputter off enriched suprathermal ions, injecting them into the DSA, which carries them to cosmic-ray energies. Through numerical simulations of the blastwave and reverse-shock trajectories versus time, McKee & Truelove (1995) map out the homologous temporal and spatial structure within the expanding supernova remnants during that peak period of cosmic-ray generation (Figures 10 and 11), around the mean cosmic-ray mixing mass ratio of ∼4.3, at ∼2.3 times the Sedov–Taylor onset time t*/t_ST.

All of the shocked, swept-up ISM and CCSN ejecta at any instance lies in the growing shell of the remnant between the blastwave and reverse shocks, and between both shocks are the highly turbulent regions (e.g., Chevalier & Fransson 2003), where most grain–gas interactions that inject suprathermal ions into those shocks occur. Diffusing back and forth into these confining shocks, these fast ions are differentially accelerated to cosmic-ray energies.

The peak cosmic-ray acceleration occurs quite early, at just ∼5%, or ∼20 kyr out of the full ∼400 kyr Sedov–Taylor stage. This peak also occurs together with the strong shocks and hard compression ratios of s ∼ 3 to 4 that can accelerate cosmic rays to power-law energy spectra of γ = (s + 2)/(s −1) and are seen in cosmic-ray-produced p⁰-decay γ's from both supernovae in the ISM (Acero et al. 2016) and in superbubbles (Ackermann et al. 2011; Aharonian et al. 2019).

5.3. Dust Grain Survival in Supernova Ejecta

Far exceeding the expectations of Baade & Zwicky (1934a, 1934b) and Ginzburg & Syrovatskii (1964), measurements of the abundances of the Galactic cosmic-ray metals (Z/H)_GCRS have defined a self-consistent framework of their sources, sites, and processes of mixing, injection, and acceleration. Their elemental composition is basically that of core-collapse supernova ejecta, mixed and partially diluted with swept-up ISM, differentially enriched by grain condensation and sputtering injection into DSA. Their composition has in fact defined these two basic processes so clearly that they permit the construction of the elemental cosmic-ray abundances to within a 1σ uncertainty of ±35% with no free parameters. Moreover, the measured array of elemental cosmic-ray source abundances determines the ubiquitous bulk mass mixing ratio of swept-up interstellar gas to CCSN ejecta to such a degree, ∼4:1, that it identifies the time span within the homologous remnant expansion, when this mixing, injection, and acceleration occur shortly after the onset of the decelerating, shock-generating, turbulent Sedov–Taylor stage of expansion and mixing.

Core-collapse supernovae are also by far, ∼81%, the most frequent in our Galaxy and the major sources of both its supernova ejecta and shock accelerating power. The cosmic-ray metals are not accelerated by all of the core-collapse supernovae, however, only those that occur in the hot (∼10⁶ K) and low-density (0.01–0.001 H cm⁻³) superbubble gas. There the reverse shocks of remnants expanding into such low densities are not strong enough to destroy more than a small fraction of the crucial refractory grains necessary to produce the observed Z^2/3-dependent sputtering injection yields. Also in these low densities, cosmic-ray accelerating shocks enjoy much lower ambient ionization losses and are far more efficient (e.g., Axford 1981).

Thus, the robust success of both the nonrotating and rotating models (Woosley & Heger 2007; Limongi & Chieffi 2018) of the CCSN ejecta compositions, as mixed with swept-up ISM, in clearly organizing and revealing the refractory grain injection processes (Section 3, Figures 5–8) provides strong evidence that CCSNe are not just the primary source of Galactic supernova shock energy and cosmic-ray protons and He, whose ratio still involves all supernovae, but nearly the sole source of cosmic-ray metals, Z > 2.

This is particularly reinforced by the fact that the cosmic-ray composition measurements have shown no significant evidence of the dominant Fe-group contribution expected (Section 4.2) from SNe Ia, based on their measured luminosity and frequency. These cosmic-ray measurements do, nonetheless, strongly show additional C and N contributions, expected (Section 4.1) from Wolf–Rayet winds of some of the most massive (>40 M_⊙) CCSN progenitors.

We have also seen that extensive astronomical observations of star-forming regions and detailed measurements of the unusual cosmic-ray composition have provided compelling evidence that at a minimum over 75% of the Galactic cosmic rays from about 0.1 GeV up to the "ankle" at around 10⁹ GeV are accelerated by supernova shocks out of highly ejecta-enriched dust and gas by spatially and temporally clustered bursts of core-collapse supernovae, SN II and SN Ib/c, in superbubbles formed around massive OB associations, while most of the remainder of the cosmic rays are accelerated by the shocks of single thermonuclear supernovae, SN Ia, scattered randomly throughout the ISM.

Not all is explained, of course, because the source of the very highest energy cosmic rays above the ankle at ∼10⁹ GeV is still unknown (e.g., Olinto 2013).

But these abundance measurements have explained the origin of by far the great bulk of the Galactic cosmic rays. For we see that applying these basic processes of mixing, condensation, and injection to solar system abundances can thus match major cosmic-ray abundances to within ±35% uncertainty with no free parameters.

Thus we have seen just how truly revealing the cosmic-ray abundance ratios of the metals, Z > 2, can be in probing critical nuclear, atomic, and even solid-state processes, in addition to plasma processes, that are not accessible from the study of the H and He ratio alone, which is actually dominated by the ISM, unlike the metals. In particular, the metals give multiple measures of the mass mixing ratio of the ejecta and the ambient ISM in the cosmic-ray acceleration region, together with the nucleosynthetic yields of the ejecta that can identify the type of supernova or other source. They also probe major atomic and mineralogical processes, including differential ionization and grain condensation fractions that determine the relative elemental ion injection factors that dominate both the cosmic-ray abundances and their overall acceleration efficiencies. Thus the metals measure major processes not accessible from plasma studies alone.

There is obviously still much to be done. What we need to explore next is the impact and interplay of these cosmic-ray compositional source constraints on the details of the DSA of the cosmic rays. In particular, it would appear that simulations are most needed of DSA by both blastwave and reverse shocks in both the swept-up ISM and the high-metallicity, grain-loaded ejecta, respectively, of core-collapse supernovae exploding in the hot (∼10⁶ K), tenuous (∼0.001 H cm⁻³), turbulent gas and magnetic fields of superbubbles surrounding OB-association star-formation regions. These are the particular source conditions required by the cosmic-ray compositional constraints, but they have been commonly overlooked in favor of the more common warm, denser phases of the ISM, which we now find are not consistent with the measured cosmic-ray composition.

Such cosmic-ray acceleration simulations need to be made in connection with homologous models of supernova remnant evolution and (1) grain freeze-out fractionization during free expansion at >2000 km s ⁻¹, (2) early Sedov–Taylor mixing of ejecta and swept-up ISM that produces the ubiquitous cosmic-ray mass ratio, (3) high-speed grain sputtering with the classical Z^2/3 Coulomb cross section (e.g., Sigmund 1969) to produce suprathermal ions, and (4) injection of enriched metals into the cosmic rays, which are dictated by both the measured cosmic-ray composition and supernova hydrodynamics, rather than a simple grain–grain collision estimate, or by the commonly assumed partially ionized ISM alone injected by first-ionization potential, which are also not at all consistent with cosmic-ray measurements. Such are the new simulations that can help put together a general integrated program to explore how these interact to generate the bulk of the local cosmic rays in measurable detail.

As we have also seen, the cosmic-ray source composition is quite consistent with a mix of the swept-up ISM and the ejecta of the most common, ∼81%, supernovae, the CCSNe or SN II and Ib/c. The bulk (∼70%; Shivvers et al. 2017) of these are the SN II with ejecta masses of roughly ∼10 M_⊙ to ∼20 M_⊙ and main-sequence lifetimes of ∼12–35 Myr before they explode. But only a fraction (<1/3) of the SN II come from the prominent O-stars, all of which explode by the first ∼13 Myr of an OB association's life, leaving all of the remaining SN II progenitors uncounted in the surveys of OB associations, which are based only on the O-star counts. Recently, however, Comeron et al. (2016) have successfully surveyed the early B-star red supergiants in the Cygnus OB2 association, significantly increasing the potential CCSN count in that association, and because these are roughly one-half of the cosmic-ray sources, more such surveys need to be made.

Finally, in order to determine the sources of the explosive nucleosynthetic r-process, it is particularly important to measure the cosmic-ray source abundances of the major r-process elements and particularly the actinides to higher precision, and this should be pursued with high priority.

I am particularly indebted to my old friend James C. Higdon of Claremont Colleges with whom I collaborated on much of this early work, to my close colleague Richard E. Rothschild of UCSD for many stimulating discussions, to W. Robert Binns of Washington University for very valuable comments on cosmic-ray measurements and modeling, and especially to the anonymous reviewer's very thoughtful, incisive comments and criticisms that have greatly helped to focus and improve this paper.

Definitions of the major notations, used here in tracing the transformation of the mixing of the supernova ejecta and swept-up interstellar abundances and its grain sputtering enrichment and injection as suprathermal ions into diffusive shock acceleration to that of cosmic rays, are given in the in Table 1. Quantitative values of cosmic-ray elemental abundances at the major steps along the way, shown in Figures 2, 4–6, and 8, are also listed with source citations in Tables 2 and 3 for both nonrotating and rotating star core collapse supernova nucleosynthetic ejecta yield calculations.

In this appendix, we present a comparison of galactic cosmic-ray source abundances with a source mix of massive star winds, core collapse supernova ejecta, and swept-upism, plus further enrichment by fast refractory ejecta grain condensation, sputtering, and suprathermal ion injection into diffusive shock acceleration.

Table 1.  List of Notations

${C}_{R/V}$	Cosmic-ray refractory/volatile enrichment constant
E51	Supernova blast energy, 1051 erg
FCS	Coulomb sputtering and scattering injection
FEJ	Supernova ejecta bulk mixing mass fraction
FGC	Grain condensation fraction
FGI	Elemental grain injection factor
FISM	Swept-up ISM bulk mixing mass fraction
FV	Volatile implantation fraction
FX	Unknown source fractions
MEJ	Ejecta mass
MSISM	Swept-up ISM mass
nO	Density, H cm−3
RST	Sedov–Taylor radius
tST	Sedov–Taylor onset time
VFE	Free expansion velocity, km s−1
(Z/H)CCSN	Core-collapse supernova ejecta abundances
(Z/H)EJ	Supernova ejecta abundances
(Z/H)GCRS	Galactic cosmic-ray source abundances
(Z/H)ISM	ISM abundances, ∼1.32 (Z/H)SS
(Z/H)SM	Source-mix model abundances
(Z/H)SMCSI	Coulomb sputtering-injected, source-mix model abundances
(Z/H)SMGI	Grain-injected, source-mix model abundances
(Z/H)SS	Solar system abundances
(Z/H)XMI	Potential extra source abundances

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Table 2.  For Nonrotating Star Core Collapse Supernova Ejecta Yields

	Z	(Z/H)GCRS1	(Z/H)SS2	GCRS/SS	(Z/H)EJ/SS3	(Z/H)EJ	(Z/H)ISM	(Z/H)SM	SM/SS	GCRS/SM	FCS	GCRS/SMCSI	${{F}_{\mathrm{GE}}}^{4}$	GCRS/SMGI
				Figure 2					Figure 4	Figure 5		Figure 6		Figure 8
H	1	10000	10000	1.00	1.00	10000	10000	10000	1.00	1.00	10000	1.00	1.00	1.00
He	2	4016	963	4.17	2.15	2068	963	1184	1.23	3.39	1468	2.74	1.00	2.74
C	6	166 ± 5	2.91	57 ± 2	28.0	81.6	3.78	19.3	6.63	8.6 ± 0.3	55.8	2.97 ± 0.09	1.39	2.14 ± 0.06
N	7	14.9 ± 0.8	0.80	19 ± 1	6.02	4.81	1.04	1.79	2.24	8.3 ± 0.5	5.80	2.57 ± 0.14	1.14	2.25 ± 0.12
O	8	208 ± 4	5.81	36 ± 1	30.8	179	7.55	41.8	7.19	5.0 ± 0.1	149	1.40 ± 0.03	1.66	0.84 ± 0.02
Ne	10	23.6 ± 0.3	0.88	27 ± 2	28.4	25.0	1.14	5.91	6.72	4.0 ± 0.05	24.8	0.95 ± 0.01	1.00	0.95 ± 0.01
Na	11	1.63 ± 0.11	0.024	68 ± 5	22.5	0.54	0.031	0.13	5.42	12.5 ± 0.9	0.59	2.76 ± 0.19	2.84	0.97 ± 0.07
Mg	12	45.1 ± 0.5	0.42	107 ± 1	14.6	6.13	0.55	1.67	3.98	27.0 ± 0.3	8.02	5.62 ± 0.06	4.64	1.21 ± 0.01
Al	13	3.93 ± 0.34	0.035	112 ± 10	15.3	0.54	0.046	0.14	4.00	28.1 ± 2.4	0.71	5.54 ± 0.48	5.50	1.00 ± 0.08
Si	14	40.6 ± 0.1	0.412	98.5 ± 0.2	12.7	5.24	0.54	1.48	3.59	27.4 ± 0.1	7.92	5.13 ± 0.01	4.65	1.10 ± 0.01
P	15	0.29 ± 0.08	0.0034	85 ± 24	16.9	0.057	0.0045	0.015	4.41	19 ± 5	0.085	3.4 ± 0.9	3.43	0.99 ± 0.26
S	16	5.33 ± 0.08	0.18	30 ± 1	15.3	2.76	0.23	0.74	4.11	7.2 ± 0.1	4.37	1.22 ± 0.02	1.86	0.66 ± 0.01
Ar	18	0.76 ± 0.07	0.042	18 ± 2	9.84	0.41	0.055	0.13	3.10	5.9 ± 0.5	0.83	0.92 ± 0.08	1.00	0.92 ± 0.08
Ca	20	2.55 ± 0.04	0.026	98 ± 2	8.41	0.22	0.034	0.071	2.73	35.9 ± 0.6	0.49	5.20 ± 0.08	5.60	0.93 ± 0.01
Fe	26	45.2 ± 1.5	0.345	131.0 ± 4.4	7.93	2.74	0.449	0.91	2.64	49.7 ± 1.7	7.53	6.00 ± 0.20	4.22	1.42 ± 0.05
Co	27	0.10 ± 0.04	0.00096	104 ± 42	14.3	0.014	0.0013	0.0038	4.00	26.3 ± 10.5	0.032	3.13 ± 1.25	4.21	0.74 ± 0.30
Ni	28	2.44 ± 0.04	0.020	122 ± 2	9.77	0.20	0.026	0.061	3.05	40.0 ± 0.7	0.53	4.60 ± 0.08	4.07	1.13 ± 0.02
Cu	29	225 ± 37	2.17	104 ± 17	41.0	88.9	2.86	20.1	9.26	11.2 ± 1.8	180	1.25 ± 0.21	3.26	0.38 ± 0.06
Zn	30	314 ± 19	5.05	62.2 ± 3.8	14.6	73.7	6.57	20.0	3.96	15.7 ± 1.0	184	1.71 ± 0.10	1.84	0.93 ± 0.05
Ga	31	27 ± 3	0.148	182 ± 20	49.2	7.28	0.19	1.61	10.9	16.8 ± 1.9	15.1	1.78 ± 0.20	2.62	0.68 ± 0.08
Ge	32	40 ± 4	0.496	81 ± 8	30.3	15.0	0.64	3.51	7.08	11.4 ± 1.1	33.7	1.19 ± 0.12	2.44	0.49 ± 0.05
As	33	5.5 ± 1.5	0.025	220 ± 60	36.6	0.92	0.033	0.21	8.40	26.2 ± 7.1	2.06	2.67 ± 0.73	3.34	0.80 ± 0.22
Se	34	17.6 ± 2.4	0.271	65 ± 9	21.8	5.91	0.35	1.46	5.39	12.1 ± 1.6	14.6	1.20 ± 0.16	1.99	0.60 ± 0.08
Br	35	5.0 ± 1.4	0.047	106 ± 30	28.4	1.33	0.061	0.31	6.60	16.1 ± 4.5	3.17	1.58 ± 0.44	1.95	0.81 ± 0.23
Kr	36	7.8 ± 1.4	0.227	34 ± 6	14.9	3.38	0.30	0.92	4.05	8.5 ± 1.5	9.59	0.81 ± 0.15	1.00	0.81 ± 0.15
Rb	37	5.3 ± 1.4	0.027	196 ± 52	30.3	0.82	0.035	0.19	7.04	27.9 ± 7.4	2.02	2.62 ± 0.69	2.95	0.89 ± 0.24
Sr	38	15.2 ± 2.1	0.097	157 ± 22	8.4	0.81	0.13	0.27	2.78	56.3 ± 7.8	2.92	5.20 ± 0.72	5.30	0.98 ± 0.14
Zr	40	6.2 ± 1.3	0.047	132 ± 28	3.9	0.18	0.06	0.085	1.81	73 ± 15	1.00	6.5 ± 1.4	6.15	1.06 ± 0.23
Ba	56	2.1 ± 0.3	0.0179	117 ± 17	2.4	0.043	0.023	0.027	1.51	78 ± 11	0.40	5.3 ± 0.8	5.86	0.90 ± 0.14
Hf	72	0.14 ±	0.00070	200 ±	1.85	0.0013	0.00091	0.0010	1.43	140 ±	0.017	8.1 ±	4.77	1.70±
W	74	0.11 ±	0.00053	208 ±	1.87	0.0010	0.00069	0.00075	1.42	147 ±	0.013	8.3 ±	5.26	1.58±
Pt	78	0.97 ± 0.10	0.0056	173 ± 18	1.41	0.0079	0.0073	0.0074	1.32	131 ± 14	0.14	7.2 ± 0.7	4.01	1.80 ± 0.17s
Pb	82	0.42 ± 0.08	0.0134	31 ± 6	2.00	0.027	0.0174	0.0193	1.44	22 ± 4	0.36	1.15 ± 0.23	1.00	1.15 ± 0.23

References. (1) Engelmann et al. (1990)/Cummings et al. (2016) (mean 1 < Z < 28), Rauch et al. (2009), Murphy et al. (2016) (30 < Z < 40), Binns et al. (1989) (40 < Z < 70), Donnelly et al. (2012) (Z > 70). (2) Lodders (2003). (3) Woosley & Heger (2007), Figure 4. (4) Wasson & Kallemeyn (1988) (C3/C1), normalized to 1.00 for mean of Al, Ca, Sr, Ba, and W.

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Table 3.  For Rotating Star Core Collapse Supernova Ejecta Yields

	Z	(Z/H)GCRS1	(Z/H)SS2	GCRS/SS	(Z/H)EJ/SS3	(Z/H)EJ	(Z/H)ISM	(Z/H)SM	SM/SS	CRS/SM	FCS	CRS/SMCSI	${{F}_{\mathrm{GE}}}^{4}$	CRS/SMGI
				Figure 2					Figure 4	Figure 5		Figure 6		Figure 8
H	1	10000	10000	1.00	1.00	10000	10000	10000	1.00	1.00	10000	1.00	1.00	1.00
He	2	4016	963	4.17	2.12	2038	963	1178	1.22	3.41	1465	2.73	1.00	2.73
C	6	166 ± 5	2.91	57 ± 2	20.4	50.3	3.78	14.9	5.11	11.2 ± 0.3	43.1	3.86 ± 0.12	1.39	2.78 ± 0.08
N	7	14.9 ± 0.8	0.80	19 ± 1	6.06	4.95	1.04	1.80	2.25	8.3 ± 0.5	5.83	2.55 ± 0.14	1.14	2.24 ± 0.12
O	8	208 ± 4	5.81	36 ± 1	23.6	137	7.55	33.4	5.76	6.2 ± 0.1	120	1.73 ± 0.03	1.66	1.04 ± 0.02
Ne	10	23.6 ± 0.3	0.88	27 ± 2	19.8	17.4	1.14	4.40	5.00	5.37 ± 0.07	18.5	1.28 ± 0.02	1.00	1.28 ± 0.02
Na	11	1.63 ± 0.11	0.024	68 ± 5	11.9	0.29	0.031	0.082	3.41	19.9 ± 1.4	0.37	4.38 ± 0.31	2.84	1.54 ± 0.10
Mg	12	45.1 ± 0.5	0.42	07 ± 1	10.3	4.33	0.55	1.31	3.12	34.5 ± 0.4	6.29	7.17 ± 0.08	4.64	1.55 ± 0.01
Al	13	3.93 ± 0.34	0.035	12 ± 10	10.4	0.36	0.046	0.11	3.13	35.9 ± 3.1	0.56	7.05 ± 0.61	5.50	1.28 ± 0.11
Si	14	40.6 ± 0.1	0.412	98.5 ± 0.2	18.8	7.75	0.54	1.98	4.81	20.5 ± 0.1	10.62	3.82 ± 0.01	4.65	0.82 ± 0.01
P	15	0.29 ± 0.08	0.0034	85 ± 24	17.0	0.058	0.0045	0.015	4.41	19 ± 5	0.085	3.4 ± 0.9	3.43	0.99 ± 0.26
S	16	5.33 ± 0.08	0.18	30 ± 1	13.2	2.38	0.23	0.66	3.66	8.1 ± 0.2	3.89	1.37 ± 0.02	1.86	0.74 ± 0.01
Ar	18	0.76 ± 0.07	0.042	18 ± 2	9.58	0.40	0.055	0.12	3.13	6.1 ± 0.5	0.80	0.95 ± 0.08	1.00	0.95 ± 0.08
Ca	20	2.55 ± 0.04	0.026	98 ± 2	9.11	0.24	0.034	0.074	2.84	34.5 ± 0.6	0.51	5.00 ± 0.08	5.60	0.89 ± 0.01
Fe	26	45.2 ± 1.5	0.345	31.0 ± 4.4	7.42	2.56	0.449	0.87	2.53	51.8 ± 1.7	7.24	6.25 ± 0.21	4.22	1.48 ± 0.05
Co	27	0.10 ± 0.04	0.00096	04 ± 42	10.9	0.0105	0.0013	0.0031	3.27	31.9 ± 12.8	0.027	3.73 ± 1.49	4.21	0.90 ± 0.36
Ni	28	2.44 ± 0.04	0.020	22 ± 2	13.6	0.27	0.026	0.075	3.76	32.5 ± 0.5	0.66	3.72 ± 0.06	4.07	0.91 ± 0.02
Cu	29	225 ± 37	2.17	04 ± 17	17.0	36.9	2.86	9.67	4.45	23.3 ± 3.8	86.7	2.59 ± 0.43	3.26	0.79 ± 0.13
Zn	30	314 ± 19	5.05	62.2 ± 3.8	32.8	166	6.57	38.4	7.61	8.16 ± 0.5	125	2.50 ± 0.15	1.84	1.36 ± 0.06
Ga	31	27 ± 3	0.148	82 ± 20	45.2	6.69	0.19	1.49	10.1	18.1 ± 1.9	14.0	1.93 ± 0.20	2.62	0.74 ± 0.07
Ge	32	40 ± 4	0.496	81 ± 8	27.3	13.5	0.64	3.22	6.49	12.4 ± 1.2	30.9	1.29 ± 0.13	2.44	0.53 ± 0.05
As	33	5.5 ± 1.5	0.025	20 ± 60	39.2	0.98	0.033	0.22	8.90	24.7 ± 7	2.09	2.63 ± 0.75	3.34	0.79 ± 0.22
Se	34	17.6 ± 2.4	0.271	65 ± 9	17.6	4.78	0.35	1.24	4.56	14.3 ± 2.0	12.4	1.42 ± 0.20	1.99	0.71 ± 0.10
Br	35	5.0 ± 1.4	0.047	06 ± 30	7.9	0.37	0.061	0.12	2.63	40.5 ± 11.3	1.23	4.07 ± 1.1	1.95	2.09 ± 0.59
Kr	36	7.8 ± 1.4	0.227	34 ± 6	15.2	3.45	0.30	0.93	4.10	8.4 ± 1.5	9.69	0.80 ± 0.15	1.00	0.80 ± 0.15
Rb	37	5.3 ± 1.4	0.027	96 ± 52	17.7	0.48	0.035	0.12	4.58	42 ± 12	1.32	4.02 ± 1.06	2.95	1.36 ± 0.36
Sr	38	15.2 ± 2.1	0.097	57 ± 22	13.5	1.30	0.13	0.36	3.75	42.2 ± 5.8	4.02	3.78 ± 0.56	5.30	0.71 ± 0.10
Zr	40	6.2 ± 1.3	0.047	32 ± 28	4.11	0.19	0.06	0.087	1.89	71.6 ± 15.0	0.98	6.4 ± 1.3	6.15	1.03 ± 0.22
Ba	56	2.1 ± 0.3	0.0179	17 ± 17	1.9	0.034	0.023	0.025	1.40	84 ± 12	0.35	5.9 ± 0.8	5.86	1.01 ± 0.14
Pb	82	0.42 ± 0.08	0.0134	31 ± 6	0.72	0.027	0.0096	0.016	1.19	26 ± 5	0.29	1.43 ± 0.28	1.00	1.43 ± 0.28

References. (1) Engelmann et al. (1990)/Cummings et al. (2016) (mean 1 < Z < 28), Rauch et al. (2009), Murphy et al. (2016) (30 < Z < 40), Binns et al. (1989) (40 < Z < 70), Donnelly et al. (2012) (Z > 70). (2) Lodders (2003). (3) Limongi & Chieffi (2018) (CCSN; see the text). (4) Wasson & Kallemeyn (1988) (C3/C1), normalized to 1.00 for mean of Al, Ca, Sr, Ba, and W.

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