  The influence of the first stars on the formation of second-generation
objects at high redshift may be determined, in part, by their metal enrichment
of surrounding gas. At a critical metallicity, Zcrit, primordial gas cools more
efficiently by fine-structure lines of CII (157.74 um), OI (63.18 um, 145.5
um), SiII (34.8 um), and FeII (25.99 um, 35.35 um) than by HI or H2 emission.
This cooling may alter the process of fragmentation into smaller units. We
study the time-dependent cooling of primordial gas enriched by heavy elements
from early massive stars, particularly O, Si, and Fe. We define Zcrit as the
point when the total cooling rate by metals plus H2 equals the adiabatic
compressional heating. We explore two metallicity scenarios: (1) a single
metallicity for all heavy elements; (2) individual metallicities (Z_C, Z_O,
Z_Si, Z_Fe) from theoretical supernova yields. For dense gas [n > 10^3 cm^(-3)]
with metals in relative solar abundances, fragmentation occurs at Zcrit ~
10^(-3.5) Z_sun. However, for lower density gas, [n = 1-100 cm^(-3)],
particularly in halos enriched in Si, O, and Fe, we find Zcrit = 0.1-1% Zsun.
The critical metallicity approaches a minimum value at high-n set by efficient
LTE cooling, with thermalized level populations of fine-structure states and H2
rotational states (J = 2 and J = 3). Primordial clouds of 10^8 Msun at 200K are
detectable in redshifted fine-structure lines, with far-infrared fluxes between
10^-22 and 10^-21 W/m^2. For metallicities Z_O = 10^(-3) and molecular
fractions f(H2) = 10^(-3) the fine-structure emission lines of OI, SiII, and
FeII could be 10^2 - 10^3 times stronger than the H2 rotational lines at 28.22
um (J = 2-0) and 17.03 um (J = 3-1).
