Understanding star formation is problematic as it originates in the large scale dynamics of a galaxy but occurs on the small scale of an individual star forming event. This paper presents the first numerical simulations to resolve the star formation process on sub-parsec scales, whilst also following the dynamics of the interstellar medium (ISM) on galactic scales. In these models, the warm low density ISM gas flows into the spiral arms where orbit crowding produces the shock formation of dense clouds, held together temporarily by their external pressure. Cooling allows the gas to be compressed to sufficiently high densities that local regions collapse under their own gravity and form stars. The star formation rates follow a Schmidt-Kennicutt Sigma_{SFR} ~ Sigma_{gas}^{1.4} type relation with the local surface density of gas while following a linear relation with the cold and dense gas. Cooling is the primary driver of star formation and the star formation rates as it determines the amount of cold gas available for gravitational collapse. The star formation rates found in the simulations are offset to higher values relative to the extragalactic values, implying a constant reduction, such as from feedback or magnetic fields, is likely to be required. Intriguingly, it appears that a spiral or other convergent shock and the accompanying thermal instability can explain how star formation is triggered, generate the physical conditions of molecular clouds and explain why star formation rates are tightly correlated to the gas properties of galaxies.
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