Role of the H⁺₂ channel in the primordial star formation under strong radiation field and the critical intensity for the supermassive star formation

We investigate the role of the H⁺₂ channel on H₂ molecule formation during the collapse of primordial gas clouds immersed in strong radiation fields which are assumed to have the shape of a diluted blackbody spectra with temperature Trad. Since the photodissociation rate of H⁺₂ depends on its level population, we take full account of the vibrationally resolved H+2 kinetics.We find that in clouds under soft but intense radiation fields with spectral temperature T_rad ≲ 7000 K, the H⁺₂ channel is the dominant H₂ formation process. On the other hand, for harder spectra with T^rad ≳ 7000 K, the H^- channel takes over H⁺₂ in the production of molecular hydrogen.We calculate the critical radiation intensity needed for supermassive star formation by direct collapse and examine its dependence on the H⁺₂ level population. Under the assumption of local thermodynamic equilibrium (LTE) level population, the critical intensity is underestimated by a factor of a few for soft spectra with T_rad ≲ 7000 K. For harder spectra, the value of the critical intensity is not affected by the level population of H⁺₂ . This result justifies previous estimates of the critical intensity assuming LTE populations since radiation sources like young and/or metal-poor galaxies are predicted to have rather hard spectra.