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The influence of the first stars on the formation of second-generationobjects at high redshift may be determined, in part, by their metal enrichmentof surrounding gas. At a critical metallicity, Zcrit, primordial gas cools moreefficiently by fine-structure lines of CII (157.74 um), OI (63.18 um, 145.5um), 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. Westudy the time-dependent cooling of primordial gas enriched by heavy elementsfrom early massive stars, particularly O, Si, and Fe. We define Zcrit as thepoint when the total cooling rate by metals plus H2 equals the adiabaticcompressional heating. We explore two metallicity scenarios: (1) a singlemetallicity 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 efficientLTE cooling, with thermalized level populations of fine-structure states and H2rotational states (J = 2 and J = 3). Primordial clouds of 10^8 Msun at 200K aredetectable in redshifted fine-structure lines, with far-infrared fluxes between10^-22 and 10^-21 W/m^2. For metallicities Z_O = 10^(-3) and molecularfractions f(H2) = 10^(-3) the fine-structure emission lines of OI, SiII, andFeII could be 10^2 - 10^3 times stronger than the H2 rotational lines at 28.22um (J = 2-0) and 17.03 um (J = 3-1).Comment: submitted to ApJ (13 pages, 10 figs, 5 tables

Critical Metallicity and Fine‐Structure Emission of Primordial Gas Enriched by the First Stars