We study the gravitational fragmentation of cold accretion streams flowing into a typical first galaxy. We use a one-zone hydrodynamical model to examine the thermal evolution of the gas entering a 10^8 M_sun DM halo at z=10. The goal is to find the expected fragmentation mass scale and thus a characteristic mass of the first population of stars to form by shock fragmentation at high redshift. Our model accurately describes the chemical and thermal evolution of the gas as we are specifically concerned with how the cooling of the gas alters its fragmentation properties. We find there to be a sharp drop in the fragmentation mass at a metallicity of ~10^-4 Z_sun when a strong molecule destroying, LW background is present. However, If molecules can efficiently form, they dominate the cooling at T < 10^4 K, demonstrating no critical metallicity. Dust grains are not included in our chemical model, but we argue their inclusion would not significantly the results. We also find that this physical scenario allows for the formation of a cluster of solar mass fragments, or a single 10^4 M_sun fragment, possibly the precursors to primeval clusters and SMBHs. Lastly, we conclude that the usual assumption of isobaricity for galactic shocks breaks down in gas of sufficiently high metallicity, suggesting that metal cooling may lead to thermal instabilities.
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