Connecting Primordial Star-forming Regions and Second-generation Star Formation in the Phoenix Simulations

We introduce the Phoenix Simulations, a suite of highly resolved cosmological simulations featuring hydrodynamics, primordial gas chemistry, primordial and enriched star formation and feedback, UV radiative transfer, and saved outputs with Delta t = 200 kyr. We observe 73,523 individual primordial stars within 3313 distinct regions forming 2110 second-generation enriched star clusters by z >= 12 within a combined 177.25 Mpc(3) volume across three simulations. The regions that lead to enriched star formation can contain greater than or similar to 150 primordial stars, with 80% of regions having experienced combinations of primordial Type II, hypernovae, and/or pair-instability supernovae. Primordial supernovae enriched 0.8% of the volume, with 2% of enriched gas enriched by later-generation stars. We determine the extent of a primordial stellar region by its metal-rich or ionized hydrogen surrounding cloud; the metal-rich and ionized regions have time-dependent average radii r less than or similar to 3 kpc. 7 and 17% of regions have r > 7 kpc for metal-rich and ionized radii, respectively. We find that the metallicity distribution function of second-generation stars overlaps that of subsequent Population II star formation, spanning metal-deficient (similar to 7.94 x 10(-8) Z (circle dot)) to supersolar (similar to 3.71 Z (circle dot)), and that 30.5% of second-generation stars have Z > 10(-2) Z (circle dot). We find that the metallicity of second-generation stars depends on progenitor configuration, with metals from pair-instability supernovae contributing to the most metal-rich clusters; these clusters form promptly after the supernova event. Finally, we create an interpretable regression model to predict the radius of the metal-rich influence of Population III star systems within the first 7-18 Myr after the first Population III stars form in the region.