TRACKING THE DISTRIBUTION OF ²⁶Al AND ⁶⁰Fe DURING THE EARLY PHASES OF STAR AND DISK EVOLUTION

The current simulation is a partial rerun, with much higher numerical fidelity, of the model in Vasileiadis et al. (2013), which used a ${(40\mathrm{pc})}^{3}$ periodic box with a total mass of 9 × 10⁴M${}_{\odot }$, and a mean magnetic field of 3.5 μG. The initial evolution was simulated using the unigrid stagger-code (Kritsuk et al. 2011; Padoan & Nordlund 2011). That model was started up by driving turbulence with a typical root mean square velocity of 6–7 km s⁻¹ (Padoan & Nordlund 2011), consistent with Larson's velocity dispersion-size relation $\sigma {(\mathrm{km}{\rm{s}})}^{-1}\propto {L}_{\mathrm{pc}}^{0.38}$ (Larson 1969, 1981). Self-gravity was then turned on and, subsequently, sufficiently dense gas was converted to a distributions of sink particles, sampled according to a Salpeter initial mass function (IMF) (Padoan & Nordlund 2002). When the kinetic energy feedback due to supernova explosions became significant, the turbulence driving was turned off, and the evolution was continued with the ramses code, with a root grid of 128³ and 16 levels of refinement relative to the box size, i.e., with a minimum cell size of about 126 au. Essentially the same star formation recipe as described below was used to self-consistently follow the formation of massive stars, which gradually took over driving from the generation of massive stars originating from IMF sampling. The result is a realistic, evolving GMC model, polluted through supernova explosions with SLRs, and having a mature and diverse population of massive stars (cf. Vasileiadis et al. 2013, for details). Here we repeat parts of that run with 4–6 additional levels of refinement, using an updated version of ramses that also provides higher fidelity at a given grid size than the version used by Vasileiadis et al. (2013), by allowing more aggressive choices of Riemann solver and slope limiter. In the current paper we define t = 0 as the time of birth of the first massive star formed in the Vasileiadis et al. (2013) ramses run.

Sink particles are used as a subgrid model for stars. Sink particles form from cold gas on the highest refinement level, when it exceeds a certain threshold density, has a convergent velocity flow, is at a local minimum of the potential, and is at least 30 cells (≈3777 au) from any already formed sink particle. In addition, the temperature of the gas has to be below 2000 K. Over time, it is evident that stars form clustered in filaments of high density, in contradiction to the classical model of stars forming isolation due to gravitational collapse (Shu 1977). This is consistent with numerical simulations by other groups (e.g., Hennebelle 2013; Myers et al. 2014; Banerjee & Körtgen 2015) and as seen in observations (Lada & Lada 2003; Bressert et al. 2010). It demonstrates the necessity of using our more complex large-scale zoom-in model, instead of using idealized spherical core-collapse models. The sink particles move through the molecular cloud, accreting gas from surrounding cells within a radius of eight cells from the sink if the total energy in the gas in a nearby cell is negative. The rate of accretion increases gradually from zero at the edge to a fraction of $\approx 0.01$ per orbital time near the sink particle, similar to the prescription given in (Padoan et al. 2014). Galilean transformations to the rest frame of a single sink particle make it possible to turn on the built-in ramses geometric refinement and keep the particle centered, allowing us to explicitly zoom to the environment around the sinks of interest. The geometric refinement in ramses does not force refinement. Instead, it constrains potential refinement only to cells that are located close enough to the sink in order to avoid unnecessary computational costs. We set up geometric of refinement in such a way that with decreasing distance from the sink the allowed maximum level of refinement gradually increases. Here we allow refinement in concentric spherical regions with radii of at least 40 cells at each refined level.

Self-gravity in the simulation is accounted for in three steps: first, we compute the potential from only the gas, using it to compute the gravitational force from the gas on the sink particles. Second, we deposit the sink particle masses to the grid using a triangular-shaped cloud method and use the combined potential from the gas and sink particles to compute the gravitational forces on the gas. The gravitational forces between sink particles are accounted for by explicitly using Newton's law, with a smoothed gravitational potential using a piece-wise polynomial with a softening length of $0.3{\rm{\Delta }}x$ (Federrath et al. 2010). Particles are evolved with a symplectic kick–drift–kick leap-frog integrator, identical to the one used for dark matter particles in ramses. This ensures that close encounters between particles are properly accounted for, and close binaries settle in orbits of $\sim {\rm{\Delta }}x$. To make sure sink particles move in stable orbits we have added two Courant conditions related to the maximum velocity and acceleration of the sink particles

$$\begin{eqnarray}&&{\rm{\Delta }}{t}_{v}={C}_{{\rm{\Delta }}t}\displaystyle \frac{\mathrm{min}({r}_{{ss}^{\prime} },{\rm{\Delta }}x)}{v}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{t}_{a}={\left[{C}_{{\rm{\Delta }}t}2\displaystyle \frac{\mathrm{min}({r}_{{ss}^{\prime} },{\rm{\Delta }}x)}{a}\right]}^{1/2},\end{eqnarray} \tag{ 2 }$$

where ${C}_{{\rm{\Delta }}t}$ is the Courant number (typically 0.5), ${r}_{{ss}^{\prime} }$ is the minimum distance between two sink particles, ${\rm{\Delta }}x$ is the cell size, v is the speed of each sink, and a is the size of the acceleration.

To follow the accretion history of a number of individual stars in detail, while simultaneously retaining realistic initial and boundary conditions of the surrounding medium in the model, we use the method of zoom-in introduced in Nordlund et al. (2014), proceeding in two steps: first, we simulate the evolution of the entire box for approximately 4 Myr with a maximum resolution of 126 au. During this step, several hundred stars of different masses are formed, from which eleven stars are selected. Ten of these stars accrete to from 1 to 2 ${M}_{\odot }$ and one (star 4) accretes to a final mass of about 2.8 ${M}_{\odot }$. In Table 1, we provide an overview of the different sinks. The selected stars are formed—at different points in time and in different environments—from collapsing pre-stellar cores generally located in filamentary structures of the GMC. Our choice of selecting stars that accrete to more than 1 ${M}_{\odot }$ is motivated by the fact that young stars eject a fraction of the accreting mass in strong outflows, which we do not resolve with a resolution of 126 au but (partly) resolve when zooming in. Consequently, in order to model the formation and evolution of what become solar-mass stars, we need to select stars that accrete more than 1 ${M}_{\odot }$ in the first step, i.e., the low resolution run.

In the second step, we rerun the simulation with higher resolution around the selected stars to follow the accretion onto these individual stars in as much detail as we can afford. A compromise between resolution and time coverage allows using up to 20–22 AMR levels of refinement relative to the box size, instead of the 16 levels used in the first step. This corresponds to a minimum cell size of either 8 au (20 levels) or 2 au (22 levels), which still allows following the accretion over about 100 kyr (during which we use output file cadences between 0.2–1 kyr). This is sufficient to cover the periods of time when most of the mass of the stars accretes. To ensure proper coverage of the early phase of star formation, we start most of the second step simulations more than ten thousand years before formation of the selected star, while imposing a "geometric refinement" zoom-in region centered on the pre-stellar core of the star.³ Simultaneously, we also insert about 10 million tracer particles in a cubic region of about $1.28\times {10}^{5}$ au (0.62 pc) in diameter for eight of the zoom-in runs, namely for stars 1, 5, 6, 7, 8, 9, 10, and 11. The tracer particles are distributed with a probability density proportional to mass density, and are passively advected with the gas motion.

Overview of the eleven stars selected for zoom-in. First column: number of star, second column: cell size at the highest resolution, third column: time of formation of the star in the parental run, fourth to sixth column: x, y and z-coordinate of the star at the time of formation.

During the roughly 4 Myr of GMC evolution considered for this paper, nine of the massive stars adopted from the previous stagger and ramses runs explode as supernovae after their mass-dependent life-times and admix ²⁶Al as well as ⁶⁰Fe at different locations in the GMC.

In Figure 1, we show the average mass-weighted abundance of ²⁶Al (left panel) and ⁶⁰Fe (right panel) as green horizontal lines together with the abundances of the individual stars of masses from 0.2 to 0.5 ${M}_{\odot }$ (purple asterisks), 0.5 to 2.5 ${M}_{\odot }$ (black asterisks), and 2.5 to 8 ${M}_{\odot }$ (yellow asterisks) at their times of formation. The initial abundance of ²⁶Al and ⁶⁰Fe in the cloud prior to the first supernova explosion reflect contributions from earlier supernova events that occurred prior to our t = 0. It is clear from Figure 1 that the average ²⁶Al and ⁶⁰Fe abundances in the cloud are highly modulated by supernova events (illustrated by the blue stars on top of the plots), followed by a gradual decrease due to radioactive decay. The first supernovae corresponds to a star of 13 ${M}_{\odot }$ and result in a significant enhancement of the ²⁶Al abundance relative to ⁶⁰Fe. There are two reasons for this. First, ²⁶Al (${\tau }_{1/2{,}^{26}\mathrm{Al}}\approx 717$ kyr) decays about three times faster than ⁶⁰Fe (${\tau }_{1/2{,}^{60}\mathrm{Fe}}\approx 2.6\;{\rm{Myr}}$; Rugel et al. 2009) and, hence, the ⁶⁰Fe abundance is depleted less than ²⁶Al before the first supernova event. Second, the first supernova event is not a very massive star, which produces ⁶⁰Fe per unit less mass relative to more massive supernovae. The second supernova explodes with a mass of 22 ${M}_{\odot }$ less than 200 kyr later and significantly enriches the cloud in ⁶⁰Fe as it is more efficient in producing ⁶⁰Fe than the first supernova. During the time until the next supernova explosion, one can clearly recognize the characteristic decay of both SLRs before the box becomes efficiently enriched in SLRs by the third supernova. This supernova is the most massive one during the entire evolution of the GMC with 75 ${M}_{\odot }$ and is thus particularly responsible for the enhancement in ⁶⁰Fe. The subsequent supernovae, with masses of $15\;{M}_{\odot }$, $29\;{M}_{\odot }$, $40\;{M}_{\odot }$, $13\;{M}_{\odot }$, $22\;{M}_{\odot }$, and $29\;{M}_{\odot }$ do not enhance the average abundance as much, partly because enrichments of the already enhanced GMC appear less significant on the logarithmic scale. In general, we can see an overall increase of SLR abundances (ranging from about $2.5\times {10}^{-6}$ to about $1\times {10}^{-4}$ for ²⁶Al/²⁷Al and from about 5 × 10⁻⁷ to about 7 × 10⁻⁶ in ⁶⁰Fe) over time, consistent with earlier results of Vasileiadis et al. (2013).

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Enrichment in SLR abundances is also reflected in the spatial distributions of SLRs at different times. The two left panels of Figure 2 show the distribution of ²⁶Al/²⁷Al in the cloud for all cells with respect to density and temperature for two different times. The left panel illustrates the distribution just at the end of the quiescent period at $t=2.2\;{\rm{Myr}}$, while the middle panel corresponds to the end of the simulation. In both diagrams one can see that the temperature of the gas decreases with increasing density. Also, the spread in ²⁶Al/²⁷Al is wider for low density gas than for high density gas. Both diagrams reveal that the highest abundances occur for highest temperatures but, due to several recent supernova enrichments, this property is much more evident at the end of the simulation (middle panel) than after the supernova quiet phase (left panel). The significant number of cells with high temperatures and low density reveals the admixing of enriched gas from supernova explosions into the GMC. To illustrate the spatial distribution of the SLRs, in particular ²⁶Al/²⁷Al, we present the distribution of ²⁶Al/²⁷Al inside our entire box with the visualization software vapor (Clyne et al. 2007). The color scheme represents ²⁶Al/²⁷Al ratios from a lowest value of about $3.9\times {10}^{-8}$ to highest values of about $5.6\times {10}^{-2}$. For clarity, we set the floor values to 10⁻² (10⁻⁸) and color all values above (below) this ratio in white (violet). However, because the high ²⁶Al/²⁷Al values are typically associated with hot and, hence, very low density gas, the gas enriched in SLRa does not contribute significantly to the overall mass distribution in the GMC. The variability in ²⁶Al/²⁷Al ratios present in dense and cold star-forming gas is much more limited relative to that of the entire GMC.

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To quantify the distribution of SLR abundances in our GMC at different times more accurately, we plot the ²⁶Al/²⁷Al and ⁶⁰Fe/⁵⁶Fe distribution in the form of histograms in Figure 3. The distributions clearly show that large spatial heterogeneities in ²⁶Al/²⁷Al abundance exist throughout the entire GMC due to supernova enrichments. The SLR ratios cover a range of up to six orders of magnitude for ²⁶Al/²⁷Al and up to about four orders of magnitude for ⁶⁰Fe/⁵⁶Fe at times not too long after recent supernova events. Furthermore, the distribution is narrower and the GMC lacks very high values of SLRs at the end of more quiescent periods (blue solid line, red dotted line). Comparing the histogram for $t=2.229\;{\rm{Myr}}$, corresponding to the end of a quiescent period, with the distribution shortly after the two first supernova enrichments at $t=0.679\;{\rm{Myr}}$ shows that the maximum SLR values are about two orders of magnitude lower as seen in Figure 2. This significant decrease in ²⁶Al and ⁶⁰Fe abundance observed at the end of the quiescent period cannot be explained by radioactive decay and, instead, must reflect progressive admixing of the SLR enriched high density gas with older, lower density gas present in the GMC. Considering the range of abundances found in our model, both the canonical value measured in bulk CAIs from Carbonaceous chondrites of Vigarano type (CV CAIs) as well as lower values measured in FUN CAIs are well represented within the range found in our simulation, although the majority of star-forming gas is of lower abundance.

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Similarly to Figure 1, we illustrate in Figure 4 (left panel) the evolution of the average ⁶⁰Fe/²⁶Al ratio in the GMC together with the abundances of the individual stars at their time of formation (asterisks) and the exploding supernovae. Although the average ratios vary due to the different decay times of ⁶⁰Fe and ²⁶Al as well as the different supernova enrichments during the evolution of the GMC, the value generally decreases from about 0.3 to about 0.16 at the end of the simulation due to the larger amount of supernovae that admix more ²⁶Al than ⁶⁰Fe into the GMC. Again, this value is in agreement with Vasileiadis et al. (2013), who found an average value of about 0.2. Furthermore, the value at later times is also consistent with the galactic value of 0.15 ± 0.06 (Diehl et al. 2006; Wang et al. 2007). Our average value is higher than the galactic value throughout the entire evolution of about 4 Myr, but could eventually have become lower, if we had continued the simulation for a longer time. Despite natural fluctuations of the ⁶⁰Fe/²⁶Al value, the overall trend of a decrease is expected, since supernovae generally admix less Fe relative to Al for decreasing masses of the progenitor, and therefore the ⁶⁰Fe/²⁶Al in GMCs will tend to decrease over time. This argument is also supported by the changing distribution of ⁶⁰Fe/²⁶Al ratios inside our cloud Figure 4 (right panel). Since our GMC has already evolved long enough before the start of our simulation, the less massive supernova already occurs at the beginning of our simulation. Although somewhat counter-intuitive, the associated distribution (black solid line) reveals that the lowest ⁶⁰Fe/²⁶Al ratios occur after a low-mass supernova. In general, however, the early SLR abundances are predominantly dominated by enrichments of short-lived heavy supernovae, while longer-lived low-mass supernovae mostly occur at later times and cause new enrichment. At this point, we emphasize the difficulties of measuring and estimating one single ⁶⁰Fe/²⁶Al value for a GMC considering the large range of ⁶⁰Fe/²⁶Al ratios reflected in Figures 3 and 4 (right panel) present throughout the entire GMC. Although mixing occurs inside the GMC, fluctuations are still significant indicating that the process of mixing occurs on longer timescales relative to that depicted in our simulations.

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To better understand the heterogeneities inside the GMC, we compare the timescales that are relevant for mixing of the SLRs. The dynamical crossing time is

$$\begin{eqnarray}&&{t}_{{\rm{cross}}}=\displaystyle \frac{l}{{ \mathcal M }{c}_{s}},\end{eqnarray} \tag{ 3 }$$

where $l={L}_{{\rm{box}}}/2$ is a characteristic length and ${ \mathcal M }{c}_{s}=6\mbox{--}7$ $\mathrm{km}\;{{\rm{s}}}^{-1}$ is the typical turbulent rms speed. This gives a mixing time of ${t}_{{\rm{cross}}}\sim 3\;\mathrm{Myr}$. Another relevant timescale is the cooling time of the hot medium

$$\begin{eqnarray}&&{t}_{{\rm{cool}}}=\displaystyle \frac{3{k}_{b}T}{2n{\rm{\Lambda }}}\end{eqnarray} \tag{ 4 }$$

where Λ is the cooling rate. If we assume approximate pressure equilibrium between different phases, then hot gas has a vastly different density compared to cold gas, and it has to be cooled down and compact to efficiently mix with the cold gas. In our case the cooling time can be calculated to be $\sim 1\;\mathrm{Myr}$ for 10⁶ K gas. The average time between different supernovae ${t}_{{\rm{\Delta }}\mathrm{SN}}$ provides the time between supernova enrichments. Given that nine supernova explosions occurred in roughly 4 Myr during our simulation, we set ${t}_{{\rm{\Delta }}\mathrm{SN}}$ to 450 kyr. This is significantly lower than either the crossing time or the cooling time, which explains the heterogeneous SLR abundance in the gas of the GMC.

After having shown that SLR abundances of the gas are heterogeneous, we investigate to what extent the heterogeneity is present in the stars (represented by the asterisks in Figure 1). Altogether 252 stars of masses higher than 0.2 ${M}_{\odot }$ form between t = 0 and the end of the simulation, of which 46 evolve to masses higher than 8 ${M}_{\odot }$ and will end their lives in supernova explosions. As mentioned earlier, none of these stars has exploded in a supernova event by the end of the simulations. The GMC also contains lower mass stars, but we exclude stars of masses lower than 0.2 ${M}_{\odot }$ because the minimum cell size of 126 au is not sufficient to properly sample the tail of the turbulence and resolve the cores of lower mass stars. As we are mostly interested in the evolution of solar-mass stars and due to the lack of radiative transfer, we also exclude the high mass stars for our analysis and only focus on the 206 stars in the range of 0.2 ${M}_{\odot }$–8 ${M}_{\odot }$. In agreement with Vasileiadis et al. (2013) and Gounelle et al. (2009), the stars show different relative abundances in ²⁶Al (varying from about 1 × 10⁻⁷ up to about 1 × 10⁻⁵), as well as in ⁶⁰Fe (varying from about 1 × 10⁻⁷ up to 3 × 10⁻⁶) at their time of formation. Moreover, similar to the distribution of all the gas in the cloud, the initial abundances are on average lower and show a narrower spread in abundance than seen at later times.

However, there are significant differences in the overall distributions of SLR abundances between the gas and the stars. The stars show a smaller range of abundances than the gas and the stars have abundances that always lie below the average abundance in the gas at that time. Moreover, a significant number of stars show abundances that follow the decay curve of the average value of the gas at the very beginning of the simulation (the barely visible small green lines). We interpret this result such that these stars formed from a first rather old gas reservoir. The parental run does not include tracer particles, which would have allowed us to track the history of the gas from at least one supernova enrichment during the evolution of the GMC. Nevertheless, we can use the decay time of the SLRs as a clock to draw some qualitative conclusions about the origin of the gas in stars. In agreement with the delay of enriched SLR abundances for the stars in our box, we suggest that although highly SLR enriched gas from supernovae is present in the GMC, it does not contribute to the formation of stars until at least several 100 kyr later. This is in agreement with results from Vasileiadis et al. (2013), who followed the motion of gas injected by supernovae by using tracer particles and estimated that it takes of the order of $1\;\mathrm{Myr}$ until such gas is incorporated in star-forming cores. Observations show that stars form in regions of cold, dense gas. Hence it is obvious that the gas in supernova ejecta needs time to cool before it can take part in star formation. This is in agreement with the results seen in Figure 2 (left and middle panels), as well as in Figure 3. In general, the gas covers a large range of ratios and densities. However, ²⁶Al/²⁷Al abundances higher than 10⁻³ only occur for densities that are lower than ${10}^{-13}\;{\rm{g}}\;{\mathrm{cm}}^{-3}$ and, thus, cannot yet contribute to star formation. The plot shows that only a small faction of the cells have densities higher than ${10}^{-16}\;{\rm{g}}\;{\mathrm{cm}}^{-3}$; these cells correspond to potential star-forming cores. Also, the gas does not show such large spreads in the ²⁶Al/²⁷Al ratio as for lower densities. Considering that the densities of the star-forming cores are several orders of magnitude higher than the SLR enriched gas in the vicinity of recent supernovae, this indicates the difficulty of contaminating the star-forming cores with new gas of different abundance. Given that high abundances must be associated with recent supernova activities, we conclude that the gas needs time to cool sufficiently before it is able to clump and subsequently to form stars only from the gas with lower SLR ratios.

With respect to the measured differences in ²⁶Al/²⁷Al between canonical CAIs and FUN CAIs of more than one order of magnitude, it is of particular interest to investigate whether such differences occur during the accretion process. It is generally accepted that the formation of CAIs (both CV and FUN types) is restricted to the very early phase of star formation and very close to the star, probably only to the first few ten thousand years and the inner astronomical units (Krot et al. 2009; Holst et al. 2013). Since the resolution around the stars is 126 au and the time between snapshots in the parental run already is 50 kyr, we cannot resolve the surrounding to this level for all the stars in our simulation. However, late-stage contamination of an accreting star by freshly synthesized supernova material requires that the differences in abundance originate at distances far beyond the sizes of star-forming cores. Therefore, we test whether significant differences in abundance occur within 5000 au of the star 50 kyr after its formation. In Figure 5, we show the mass-weighted distribution around the stars in the mass range of 0.2 ${M}_{\odot }$–8 ${M}_{\odot }$ that show ten times higher ²⁶Al/²⁷Al ratios within a distance of 5000 au and less than 50 kyr after the birth of the star with respect to the abundance in the corresponding star. Altogether only nine of the 206 stars show contaminations of more than a factor of ten, among them only two stars with masses higher than 1 ${M}_{\odot }$. We emphasize that all of the cells with very different abundance are at least 1500 au away from the star and belong to times already up to 50 kyr after the star has formed. Hence, we do not see any contaminations at early times relevant for CAI formation ($t\lt 10$ kyr) that can account for the measured differences of ²⁶Al/²⁷Al in FUN and canonical CAIs.

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To ensure that this result is robust, we selected a few stars to follow their formation phase with higher resolution. The eleven stars selected for zoom-in are marked with red circles in Figure 1. Ten of the eleven stars accreted between 1 and 2 ${M}_{\odot }$. Additionally, we also modeled one star that accreted to more than $2.8{M}_{\odot }$. Figures 6 and 7 show the temporal evolution of the average ²⁶Al/²⁷Al ratio and ⁶⁰Fe/⁵⁶Fe in spherical shells at distances of 10 and 1000 au as well as their standard deviations for the selected stars. These have different relative abundances of ²⁶Al/²⁷Al between 10⁻⁷ and 2 × 10⁻⁶, while ⁶⁰Fe/⁵⁶Fe ratios are between 10⁻⁷ and 8 × 10⁻⁷. Although the stars selected for zoom-ins show abundances below the canonical value of $5\times {10}^{-5}$, we emphasize that our selection is valid to test the hypothesis of ²⁶Al enrichment in the solar system through supernovae. Considering that bulk CV CAIs are supposed to reflect ²⁶Al enrichment after solar birth, the initial abundances in our selected stars are consistent with values less than $3\times {10}^{-6}$ as measured in FUN CAIs. Since these values are considered to reflect the original abundance in the collapsing pre-solar core, and due to the fact that higher abundances are available in the GMC, our selection provides an adequate sample to test the enrichment hypothesis.

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As expected from the results obtained using the parental run, the average ²⁶Al/²⁷Al and ⁶⁰Fe/⁵⁶Fe ratios are almost indistinguishable at different distances from the parent stars (left and right upper panel of Figures 6 and 7). This suggests a spatially homogeneous distribution of ²⁶Al and ⁶⁰Fe within 1000 au during the accretion process for the first 100 kyr of evolution. We note that the apparent time integrated variability is consistent with the typical decay curve for ²⁶Al and ⁶⁰Fe. In principle, inflow of gas with different SLR abundances at two different locations and identical in-fall speeds could nevertheless cause spatial heterogeneities without affecting the average SLR value, which would be reflected in large deviations from the mean value. However, the ratios deviate only marginally from the average values in the shells as illustrated by plotting the standard deviation from the mean value at distances of 50 au and 1000 au in the lower panels of Figures 6 and 7. Generally, the different stars all have deviations that range from less than 1 ‰ to at most 20% of the mean value. Importantly, the fluctuation observed in the star with the largest fluctuation in ²⁶Al/²⁷Al ratios (star 9) is still lower by more than one order of magnitude relative to the difference between canonical and FUN CAIs.

To better understand the reason for the spatial homogeneity of ²⁶Al during the accretion, we investigated the origin of the gas and compared it with the SLR distribution at time t = 0. The right panels of Figures 8 and 9 illustrate the ²⁶Al/²⁷Al distribution of all cells within a distance of 100 kau around the stars at time t = 0 for the eight zoom runs including tracer particles. As indicated by Figure 6, the gas does not show spreads of more than a factor of five in ²⁶Al/²⁷Al abundance within about 10 kau at the time of stellar birth. In contrast, the gas distribution beyond $\sim {10}^{4}\;{\rm{au}}$ can be of very different abundance, but it is also of much lower density. When following the motion of the gas with tracer particles, we find that—for at least the first 50 kyr—all of the gas within the inner 100 au was located less than 10 kau away from the star at stellar birth. Considering the narrow spread in SLR abundances for the inner several thousand astronomical units at the time of star formation, this explains why we only see small differences in SLR distribution during the early accretion process of the stars. In Figure 10, we show the same phase diagrams as in the right panels of Figures 8 and 9, but coloring the temperature instead of the density of the gas. Densities decrease with increasing distance from the star and moreover the gas close to the star is of low temperature. This in agreement with observations that stars form from cores of cold gas of about ∼10 kau in size (Enoch et al. 2007).

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Our results indicate that heterogeneous accretion processes cannot account for the variability in the ²⁶Al/²⁷Al ratios between canonical and FUN CAIs during the early stages of star formation, which may appear counter-intuitive considering the variability of several orders of magnitude in SLR abundances present within the entire GMC. We consider below the cascade of events leading to star formation in a GMC to better understand our results. Turbulent motions inside GMCs predominantly cause the formation of filaments of parsec-size inside GMCs. Inside these filaments gas becomes further compressed to form pre-stellar cores of sizes of 5–10 kau consistent with theoretical predictions of a Bonnor–Ebert sphere. Eventually, the cores become dense enough to overcome the threshold value for gravitational collapse and they collapse to stars. Compared to typical sizes of GMCs, pre-stellar cores are about three orders of magnitude smaller and fill only a small part of the GMC (Figure 2). In order to contaminate a pre-stellar core during formation, it has to be near the boundary between two regions with different SLR abundances. Nevertheless, we consider the hypothetical case of a pre-stellar core that is located close to such a large difference boundary. Then we are likely to have the following two regimes: on the one hand, densities inside pre-stellar cores are very high compared to the average in the rest of the GMC; on the other hand, SLR enriched gas is associated with recent supernova events and therefore located in regions of warm gas, and particularly of low density (right panels of Figures 8–10). Thus, even if the gas that is in the vicinity of the pre-stellar core has a significantly different abundance, it is difficult for that gas to penetrate the core because of the large density contrast.

Although our simulations have not identified the existence of appreciable spatial and/or temporal heterogeneity in SLR abundances during the early accretion phases, we consider also the possibility that a pre-stellar core is contaminated by enriched gas at the beginning of its existence when the densities are still rather low. Similarly to our analysis in the previous section for the mixing on GMC scales, we compare the relevant timescales for the mixing at the scale of pre-stellar cores, which is the life-time of a pre-stellar core with the crossing time of the gas. Observational constraints suggest that the life-times of pre-stellar cores range from 100 kyr up to 500 kyr (Enoch et al. 2008). As stars form in regions of cold gas of mostly molecular hydrogen, we adopt a value of 10 K for the temperature of the sound speed. Considering radii of pre-stellar cores of about 5–10 kau for a solar-mass star, we obtain crossing times of about 100–250 kyr, similar to the life-times of pre-stellar cores, which could in principle allow for insufficient mixing inside the core. However, our results show only modest variations in SLR ratios and thus we conclude that the cold gas is already well mixed before the formation of the star-forming cores. Even if a potential contamination occurs, it only contributes slightly to the abundance per mass and only penetrates the outer edge of the core, from where it takes often more than 100 kyr for the gas to fall in toward the star (left panels of Figures 8–10).

Although we do not detect any significant differences in ²⁶Al/²⁷Al at early times, the fact that gas from initial distances beyond 10 kau accretes to the star over timescales >100 kyr allows for the possibility of contaminations at later times. Therefore, we consider the evolution of the stellar surroundings, including the protoplanetary disks, at a phase later than the initial ∼100 kyr, which we refer to as the "late phase" for simplicity. Due to high computational costs, only data for three stars (star 2, 3, and 4) were acquired for the late phase and none of these runs include tracer particles. One of these stars shows enhancements of up to a factor of 2 in ²⁶Al/²⁷Al within a distance of only 10 au from the star after about 160 kyr (Figure 11). Such contaminations at later times are possible due to massive accretion of mass onto the young star during the early phase, which causes a decrease in density around the star. Hence, it becomes possible for material that was not initially bound to the protostar to approach the vicinity of the star at later times. Unfortunately, we do not have data from simulations including tracer particles for this run, with which we could analyze the origin of the gas causing the enrichments. We point out that this late pollution is different from the idea of contaminating the collapsing pre-stellar core or a specific local injection into the protoplanetary disk from a supernova (Ouellette et al. 2007). Finally, we emphasize that such contaminations occurring at late stages in our run probably cannot account for differences in ²⁶Al abundance in CAIs, considering that the temperatures and pressures are too low to form CAIs directly out of the gas phase at this later stage.

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In contrast to the early phase, our results suggest the presence of gas with variable ²⁶Al/²⁷Al ratios in the stellar surroundings during later phases of accretion, namely $\gt 100$ kyr after star formation. As such, we evaluate the possibility that FUN CAIs represent objects formed around other ²⁶Al-poor solar-mass stars located in the vicinity of the proto-Sun and thereafter transported to the Sun via stellar outflows (MacPherson & Boss 2011). Two lines of evidence are in apparent support of such a scenario. First, most stars form in clusters often separated by less than 1000 au (Lada & Lada 2003). Second, approximately 1 solar mass of material is lost to outflows during the accretion of a solar-mass type star (Offner et al. 2014, p. 53). Thus, the cross contamination of nearby nascent systems through stellar outflows appears likely to be the outcome of star formation in clusters. We assess this possibility by investigating the spatial distribution of stars with contrasting ²⁶Al/²⁷Al values in our simulations. We restrict our analysis to stars more massive than 0.2 ${M}_{\odot }$ that formed within a timeframe of 1.6 Myr, as this represent the maximum age difference between canonical and FUN CAIs inferred from the ¹⁸²Hf–¹⁸²W system (Holst et al. 2013). In Figure 12, we show that all pair of stars formed within a distance of 50 kau have initial ²⁶Al/²⁷Al values within one order of magnitude, although greater variability is observed at greater distances. Therefore, our analysis requires the transport of CAI material in the ISM over distances greater than 50 kau to explain the contrasting initial ²⁶Al/²⁷Al ratios observed between the FUN and canonical CAIs, which appears unrealistic.

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One might argue that CAIs formed instead around stars lower than 0.2 ${M}_{\odot }$ in mass, which are closer to the star. Even with our state-of-the-art zoom-run, we do not resolve the formation of these very low-mass stars properly and we do not have a reliable statistics for these stars. However, we are confident that the SLR to distance relation would not be very different for low-mass stars, considering that these stars formed from the same gas reservoir as the higher mass stars. Taking additionally into account the uncertainty whether CAIs can travel through the ISM and accrete onto foreign star–disk systems, we consider this scenario to be unlikely.

Instead, we argue that measured differences in CAIs and chondrules are most likely caused by physical processes neglected in our simulation. In our model, we only considered the motion of the gas to evaluate the influence from GMC scales down to protoplanetary disk scales. However, observations show that GMCs and protoplanetary disks consist of about 1% of dust. Therefore, we suggest that the contrasting initial ²⁶Al/²⁷Al values recorded by canonical and FUN CAIs reflect unmixing of the ²⁶Al carrier by thermal processing during the early stage of solar system formation. This is supported by the observation that FUN CAIs plot on the solar system nucleosynthetic correlation line defined by inner solar system solids, asteroids, and planets (Schiller et al. 2015a). Although a detailed study of thermal processing is beyond the scope of this paper, we discuss the basics of the process here.

Consider the evaporation time

$$\begin{eqnarray}&&{t}_{{\rm{ev}}}\propto {\nu }^{-1}{e}^{\tfrac{{E}_{{\rm{B}}}}{{k}_{{\rm{B}}}T}}\end{eqnarray} \tag{ 5 }$$

(Tielens & Allamandola 1987; Boogert et al. 2015), where ν is the vibrational frequency, E_B the binding energy of the dust grain, k_B the Boltzmann constant, and T the temperature. Approximating dust grains to be perfectly spherical, the energy of one photon $h\nu$, where h is the Planck constant, is assumed to scale with the inverse of the volume of the grains. Thus, we obtain

$$\begin{eqnarray}&&{t}_{{\rm{ev}}}\propto {a}^{3}{e}^{\tfrac{{E}_{b}}{{k}_{{\rm{B}}}T}}.\end{eqnarray} \tag{ 6 }$$

The dust composition in the ISM consists of grains of different size (up to micrometer size) and different age. Due to radioactive decay, older dust grains are considered to show lower SLR abundances than younger grains. Taking additionally into account that older grains were exposed to potentially destructive radiation for a longer time, the surviving grains have higher binding energies and/or larger grain sizes than the younger grains. Furthermore, older components and thus SLR depleted components are more likely to be in the central layers of the grains, whereas younger components are more likely to accumulate on the surface of the grains. Thus the younger grain components generally

• 1.  
  vaporize at lower temperatures,
• 2.  
  are considered to be in smaller grains and,
• 3.  
  shield the older components from radiation.

During the star-forming process, the well mixed dust grains in the pre-stellar core fall in toward the star, where they are exposed to stellar radiation. Since the younger, SLR enriched components vaporize more easily, the gas phase can become locally enriched in SLRs. Depending on the temporally changing strength of irradiation, more or fewer layers of the dust grains are vaporized, eventually causing a continuous distribution of different ²⁶Al/²⁷Al ratios in the gas around the star. Due to the short timescales of this collapsing phase, of the order of kyr or less, the gas cannot mix to a homogeneous reservoir before CAIs are formed by condensation out of the gas phase. In this way, CAIs inherit the thermally induced, local heterogeneities of ²⁶Al/²⁷Al ratios in the early gas phase.

Progressive thermal processing of in-falling ²⁶Al-rich molecular cloud material in the inner solar system has also been invoked to account for the large-scale heterogeneity in ²⁶Al that existed at the time of accretion of most asteroidal bodies (Trinquier et al. 2009; Larsen et al. 2011; Paton et al. 2013; Schiller et al. 2015a, 2015b; van Kooten et al. 2016). We note that the initial ⁶⁰Fe abundance for stars in our simulations is much higher than that inferred for the early solar system based on differentiated meteorites and chondritic components (Tang & Dauphas 2012, 2015). Similarly to Vasileiadis et al. (2013), we interpret this discrepancy as reflecting efficient removal of ⁶⁰Fe from disk solids via thermal processing of their precursors, which requires that the carrier phase of ⁶⁰Fe was significantly more volatile than the ²⁶Al carrier. As such, inferred initial ⁶⁰Fe estimates based on meteoritic material such as differentiated meteorites and/or chondrules may not be representative of that of the bulk solar system.