Tracking the distribution of $^{26}$Al and $^{60}$Fe during the early phases of star and disk evolution

The short-lived $^{26}$Al and $^{60}$Fe radionuclides are synthesized and expelled in the interstellar medium by core-collapse supernova events. The solar system's first solids, calcium-aluminium refractory inclusions (CAIs), contain evidence for the former presence of the $^{26}$ Al nuclide defining the canonical $^{26}$Al/$^{27}$ Al ratio of $\sim5 \times10^{-5}$. A different class of objects temporally related to canonical CAIs are CAIs with fractionation and unidentified nuclear effects (FUN CAIs), which record a low initial $^{26}$Al/$^{27}$Al of $10^{-6}$. The contrasting level of $^{26}$Al between these objects is often interpreted as reflecting the admixing of the $^{26}$Al nuclide during the early formative phase of the Sun. We use giant molecular cloud (GMC) scale adaptive mesh-refinement numerical simulations to trace the abundance of $^{26}$Al and $^{60}$Fe in star-forming gas during the early stages of accretion of individual low mass protostars. We find that the $^{26}$Al/$^{27}$Al and $^{60}$Fe/$^{56}$Fe ratios of accreting gas within a vicinity of 1000 AU of the stars follow the predicted decay curves of the initial abundances at time of star formation without evidence of spatial or temporal heterogeneities for the first 100 kyr of star formation. Therefore, the observed differences in $^{26}$Al/$^{27}$Al ratios between FUN and canonical CAIs are likely not caused by admixing of supernova material during the early evolution of the proto-Sun. Selective thermal processing of dust grains is a more viable scenario to account for the heterogeneity in $^{26}$Al/$^{27}$Al ratios at the time of solar system formation.