We propose an extended Schmidt law with explicit dependence of the star formation efficiency (SFE=SFR/Mgas) on the stellar mass surface density. This relation has a power-law index of 0.48+-0.04 and an 1-sigma observed scatter on the SFE of 0.4 dex, which holds over 5 orders of magnitude in the stellar density for individual global galaxies including various types especially the low-surface-brightness (LSB) galaxies that deviate significantly from the Kennicutt-Schmidt law. When applying it to regions at sub-kpc resolution of a sample of 12 spiral galaxies, the extended Schmidt law not only holds for LSB regions but also shows significantly smaller scatters both within and across galaxies compared to the Kennicutt-Schmidt law. We argue that this new relation points to the role of existing stars in regulating the SFE, thus encoding better the star formation physics. Comparison with physical models of star formation recipes shows that the extended Schmidt law can be reproduced by some models including gas free-fall in a stellar-gravitational potential and pressure-supported star formation. By implementing this new law into the analytic model of gas accretion in Lambda CDM, we show that it can re-produce the observed main sequence of star-forming galaxies (a relation between the SFR and stellar mass) from z=0 up to z=2.
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