We investigate the process of metal-free star formation in the first galaxies with a high-resolution cosmological simulation. We consider the cosmologically motivated scenario in which a strong molecule-destroying Lyman-Werner (LW) background inhibits effective cooling in low-mass haloes, delaying star formation until the collapse or more massive haloes. Only when molecular hydrogen (H2) can self-shield from LW radiation, which requires a halo capable of cooling by atomic line emission, will star formation be possible. To follow the formation of multiple gravitationally bound objects, at high gas densities we introduce sink particles which accrete gas directly from the computational grid. We find that in a 1 Mpc^3 (comoving) box, runaway collapse first occurs in a 3x10^7 M_sun dark matter halo at z~12 assuming a background intensity of J21=100. Due to a runaway increase in the H2 abundance and cooling rate, a self-shielding, supersonically turbulent core develops abruptly with ~10^4 M_sun in cold gas available for star formation. We analyze the formation of this self-shielding core, the character of turbulence, and the prospects for star formation. Due to a lack of fragmentation on scales we resolve, we argue that LW-delayed metal-free star formation in atomic cooling haloes is very similar to star formation in primordial minihaloes, although in making this conclusion we ignore internal stellar feedback. Finally, we briefly discuss the detectability of metal-free stellar clusters with the James Webb Space Telescope.
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