We introduce non-equilibrium molecular hydrogen chemistry into the radiation- hydrodynamics code Ramses-RT. The H2 physics that we include are formation on dust grains, gas phase formation, formation by three-body collisions, collisional destruction, photodissociation, photoionisation, cosmic ray ionisation, and self-shielding. In particular, we implement the first model for H2 self-shielding that is tied locally to moment-based radiative transfer by enhancing photo-destruction. This self-shielding from Lyman-Werner line overlap is critical to H2 formation and gas cooling. We can now track the non-equilibrium evolution of molecular, atomic, and ionised hydrogen species with their corresponding dissociating and ionising photon groups. We use this recipe in an isolated, Milk Way-like galaxy. The disc profile exhibits a spike in H2 towards the galactic centre, and a flat HI profile similar to observations. Fountains of HI gas extend above the disc. With a star formation recipe that is tied to the total gas density and temperature, and not H2 content, we reproduce Kennicutt-Schmidt relation for both neutral gas vs. SFR and H2 vs. SFR. This suggests that the relationship between H2 and star formation is more of a correlation caused by the H2 cooling of the gas.
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