Results from the UV satellite GALEX revealed large extensions of disks in some nearby spiral galaxies, extending out to 3 to 4 times the isophotal radius, r25. M63 is a remarkable example of a spiral galaxy with one of the most extended UV disks, so it offers the opportunity to search for the molecular gas and characterize the star formation in outer disk regions as revealed by the UV emission. We obtained deep CO(1-0) and CO(2-1) observations on the IRAM 30 m telescope along the major axis of the M63 disk from the center out to the galactocentric radius rgal = 1.6 r25 and over a bright UV region at rgal = 1.36 r25. CO(1-0) is detected all along the M63 major axis out to r25, and CO(2-1) is confined to rgal = 0.68 r25, which may betray lower excitation temperatures in the outer disk. CO(1-0) is also detected in the external bright UV region of M63. The radial profiles of the CO emission and of the Halpha, 24 micron, NUV and FUV star formation tracers and HI taken from the literature show a severe drop with the galactocentric radius, such that beyond r25 they are all absent with the exception of a faint UV emission and HI. The CO emission detection in the external UV region, where the UV flux is higher than the UV flux observed beyond r25, highlights a tight correlation between the CO and UV fluxes, namely the amount of molecular gas and the intensity of star formation. This external UV region is dominated by the atomic gas, suggesting that HI is more likely the precursor of H2 rather than the product of UV photodissociation. A broken power law needs to be invoked to describe the Kennicutt-Schmidt (K-S) relation of M63 from the center of the galaxy out to rgal = 1.36 r25. While all along the major axis out to r25 the K-S relation is almost linear, in the external UV region the SFR regime is highly nonlinear and characterized by a steep K-S relation and very low star formation efficiency.
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