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Project Progress

In 2013, we carried out studies of the self-enrichment of the earliest star clusters, building on the turbulence simulations in Pan & Scannapieco 2010 and Pan et al. (2011). Population III (PopIII) stars are thought to have had very different properties than later stellar generations, as they were formed from a “pristine” gas that was free of metals. Normal star formation took place only after the first stars polluted the surrounding turbulent interstellar gas, increasing its metallicity beyond a critical value, Zc, between 10^-6 and ∼ 10^-3 Z⊙. The critical value is much smaller than the typical overall average metallicity, and therefore the mixing efficiency of the pristine gas in the interstellar medium plays a crucial role in determining the transition from Pop III to a normal star formation.

With the goal of developing a sub-grid model for tracking the evolution of the first stars in, we carried out a detailed numerical and theoretical investigation of the fundamental physics of the pollution of pristine fluid elements in isotropic compressible turbulence, such as it occurs in star forming molecular clouds. While the evolution of the metallicity distribution function in a turbulent medium cannot be solved exactly, in Pan, Scannapieco, & Scalo (2012), we derived predictions of the evolution of the metal-free fraction for several closure models from the literature, and showed that a class of closure models, called self-convolution models (Venaille & Sommeria 2007, Duplat & Villermaux 2008), provided successful fitting functions that matched our numerical results (Refer to Figure 1).

These models are based on the physical picture of turbulent stretching causing a cascade of metallicity structures toward the small scales at which molecular diffusivity can efficiently operate as a “convolution” between polluted and unpolluted material. The models are dependent only on two major parameters: τcon, which sets the characteristic timescale for convolution of the metal abundance PDF through turbulent stretching of concentration structures, and n, which quantifies the degree of spatial locality of the PDF convolution process.

Using a suite of numerical simulations to expand these results we were not only able measure τcon and n for all pollutant conditions relevant for primordial star formation, but also derive an extremely simple equation for the evolution of the pristine fraction throughout a simulation:

where, r is the density of the gas, P is the primordial fraction, v is the velocity field, ej is the rate at which SN ejecta are being released, g is the turbulent diffusion rate, which is proportional to the velocity times the length scale of the turbulence, and we have fit n and τcon with simple functions of the local turbulent Mach number and average metallicity (Pan, Scannapieco, & Scalo, 2013). This makes it just as easy to track the fraction of primordial gas in each zone in a simulation, as it is to track its average metallicity. With this tool, future studies will be equipped to consider the evolution of the unmixed fraction of the earliest star clusters.