The Super-linear Slope of the Spatially-Resolved Star Formation Law in NGC 3521 and NGC 5194 (M51a)

We have conducted interferometric observations with CARMA and an OTF mapping with the 45-m telescope at NRO in the CO (J=1-0) emission line of NGC 3521. Combining these new data, together with CARMA+NRO45 data for M51a and archival SINGS H$alpha$, 24$mu$m, THINGS H I and GALEX FUV data for both galaxies, we investigate the empirical scaling law that connects the surface density of SFR and cold gas (the Schmidt-Kennicutt law) on a spatially-resolved basis. We argue that plausibly deriving SFR maps of nearby galaxies requires the diffuse stellar/dust background emission to be carefully subtracted. An approach to complete this task is presented and applied in our pixel-by-pixel analysis on both galaxies, showing that the controversial results whether the molecular S-K law is super-linear or basically linear is a result of removing or preserving the local background. In both galaxies, the power index of the molecular S-K law is 1.5-1.9 at the highest available resolution (230 pc), and decreases monotonically for decreasing resolution; while the scatter (mainly intrinsic) increases as the resolution becomes higher, indicating a trend for which the S-K law breaks down below some scale. Both quantities are systematically larger in M51a than in NGC 3521, but when plotted against the de-projected scale (delta_{dp}), they become highly consistent between the two galaxies, tentatively suggesting that the sub-kpc molecular S-K law in spiral galaxies depends only on the considered scale, without varying amongst spiral galaxies. A logarithmic function gamma_{H_2}=-1.1 log[delta_{dp}/kpc]+1.4 and a linear relation sigma_{H_2}=-0.2 [delta_{dp}/kpc]+0.7 are obtained through fitting to the M51a data, which describes both galaxies impressively well on sub-kpc scales. A larger sample of galaxies with better quality data are required to test the general applicability of these relations.