HD dominates the cooling of primordial clouds with enhanced ionization, e.g. shock-heated clouds in structure formation or supernova remnants, relic HII regions of Pop III stars, and clouds with cosmic-ray (CR) irradiation. There, the temperature decreases to several 10 K and the characteristic stellar mass decreases to $sim 10 {rm M}_{odot}$, in contrast with first stars formed from undisturbed pristine clouds ($sim 100 {rm M}_{odot}$). However, without CR irradiation, even weak far ultra-violet (FUV) irradiation suppresses HD formation/cooling. Here, we examine conditions for HD cooling in primordial clouds including both FUV and CR feedback. At the beginning of collapse, the shock-compressed gas cools with its density increasing, while the relic HII region gas cools at a constant density. Moreover, shocks tend to occur in denser environments than HII regions. Owing to the higher column density and the more effective shielding, the critical FUV intensity for HD cooling in a shock-compressed gas becomes $sim 10$ times higher than in relic HII regions. Consequently, in the shock-compressed gas, the critical FUV intensity exceeds the background level for most of the redshift we consider ($6 lesssim z lesssim 15$), while in relic HII regions, HD cooling becomes effective after the CR intensity increases enough at $z lesssim 10$. Our result suggests that less massive ($sim 10 {rm M}_{odot}$) Pop III stars may be more common than previously considered and could be the dominant population of Pop III stars.
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