We explore the minimal conditions which enable the formation of metal-enriched solar and subsolar-mass stars. Using a one-zone semi-analytical model, we accurately follow the chemical and thermal evolution of the gas with the aim of understanding how the initial metal and dust content alters the cooling and fragmentation properties, hence the characteristic stellar mass. We find that in the absence of dust grains, gas fragmentation occurs at densities nH∼ [104-105]cm-3 when the metallicity exceeds Z∼ 10-4Z⊙. The resulting fragmentation masses are ≥10M⊙. The inclusion of Fe and Si cooling does not affect the thermal evolution as this is dominated by molecular (mostly OH, H2O and CO) cooling even for metallicities as large as Z= 10-2Z⊙. The presence of dust is the key driver for the formation of low-mass stars. We focus on three representative core-collapse supernova (SN) progenitors (a Z= 0 star with 20M⊙ and two Z= 10-4Z⊙ stars with 20 and 35M⊙), and consider the effects of reverse shocks of increasing strength: these reduce the depletion factors, fdep=Mdust/(Mdust+Mmet), alter the shape of the grain size distribution function and modify the relative abundances of grain species and metal species in the gas phase. We find that the lowest metallicity at which fragmentation occurs is Z= 10-6Z⊙ for gas pre-enriched by the explosion of a 20M⊙ primordial SN (fdep≥ 0.22) and/or by a 35M⊙, Z= 10-4Z⊙ SN (fdep≥ 0.26); it is ∼1dex larger, when the gas is pre-enriched by a Z= 10-4Z⊙, 20 M⊙ SN (fdep≥ 0.04). Cloud fragmentation depends on the depletion factor and it is suppressed when the reverse shock leads to a too large destruction of dust grains. These features are all consistent with the existence of a minimum dust-to-gas ratio, above which fragmentation is activated. We derive a simple analytic expression for, which depends on the total grain cross-section per unit mass of dust; for grain composition and properties explored in the present study, When the dust-to-gas ratio of star-forming clouds exceeds this value, the fragmentation masses range between 0.01 and 1M⊙, thus enabling the formation of the first low-mass stars.
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