Fragmentation and the formation of primordial protostars: the possible role of collision‐induced emission

The mechanisms which could lead to chemo‐thermal instabilities and fragmentation during the formation of primordial protostars are investigated analytically. We introduce new analytic approximations for H 2 cooling rates bridging the optically thin and thick regimes. These allow us to discuss chemo‐thermal instabilities up to densities when protostars become optically thick to continuum radiation ( n ≡ρ/ m H ≲ 10 16 –10 17 cm −3 ) . During the protostellar collapse, instabilities are active in two different density regimes. In the well‐known ‘low‐density’ regime ( n ∼ 10 8 –10 10 cm −3 ) , instability is due to three‐body reactions quickly converting atomic hydrogen into H 2 . In the ‘high‐density’ regime ( n ≳ 10 14 cm −3 ) , another instability is triggered by the strong increase in the cooling rate due to H 2 collision‐induced emission (CIE). In agreement with the three‐dimensional simulations, we find that the low‐density instabilities cannot lead to fragmentation, both because fluctuations are too small to survive turbulent mixing, and because their growth times are too slow. The situation for the newly found high‐density instability is analytically similar. This continuum cooling instability is as weak as low‐density instability, with similar ratios of growth and dynamical time‐scales, as well as allowing for the necessary fragmentation condition t cool ≲ t dyn . Because the instability growth time‐scale is always longer than the free‐fall time‐scale, it seems unlikely that fragmentation could occur in this high‐density regime. Consequently, one expects the first stars to be very massive, not to form binaries nor harbour planets. Nevertheless, full three‐dimensional simulations are required to be certain. Such 3D calculations could become possible using simplified approaches to approximate the effects of radiative transfer, which we show to work very well in 1D calculations, giving virtually indistinguishable results from calculations employing full line transfer. This indicates that the effects of radiative transfer during the initial stages of formation of primordial protostars are local corrections to cooling rather than influencing the energetics of distant regions of the flow.