Conditions for HD Cooling in the First Galaxies Revisited: Interplay
  between Far-Ultraviolet and Cosmic Ray Feedback in Population III Star
  Formation
  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.
