We explore the minimal conditions which enable the formation of metal–enriched solar and sub–solar 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 n _ { H } \sim [ 10 ^ { 4 } -10 ^ { 5 } ] cm ^ { -3 } when the metallicity exceeds Z \sim 10 ^ { -4 } Z _ { \odot } .
The resulting fragmentation masses are \geq 10 M _ { \odot } .
The inclusion of Fe and Si cooling does not affect the thermal evolution as this is dominated by molecular ( mostly OH , H _ { 2 } O and CO ) cooling even for metallicities as large as Z = 10 ^ { -2 } Z _ { \odot } .
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 20 M _ { \odot } and two Z = 10 ^ { -4 } Z _ { \odot } stars with 20 M _ { \odot } and 35 M _ { \odot } ) , and consider the effects of reverse shocks of increasing strength : these reduce the depletion factors , f _ { dep } = M _ { dust } / ( M _ { dust } + M _ { met } ) , alter the shape of the grain size distribution function and modify the relative abundances of grain species and of metal species in the gas phase .
We find that the lowest metallicity at which fragmentation occurs is Z = 10 ^ { -6 } Z _ { \odot } for gas pre–enriched by the explosion of a 20 M _ { \odot } primordial SN ( f _ { dep } \geq 0.22 ) and/or by a 35 M _ { \odot } , Z = 10 ^ { -4 } Z _ { \odot } SN ( f _ { dep } \geq 0.26 ) ; it is \sim 1 dex larger , when the gas is pre–enriched by a Z = 10 ^ { -4 } Z _ { \odot } , 20 M _ { \odot } SN ( f _ { dep } \geq 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 , { \cal D } _ { cr } , above which fragmentation is activated .
We derive a simple analytic expression for { \cal D } _ { cr } which depends on the total grain cross–section per unit mass of dust ; for grain composition and properties explored in the present study , { \cal D } _ { cr } = [ 2.6 - 6.3 ] \times 10 ^ { -9 } .
When the dust–to–gas ratio of star forming clouds exceeds this value , the fragmentation masses range between 0.01 M _ { \odot } and 1 M _ { \odot } , thus enabling the formation of the first low–mass stars .