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

We have conducted interferometric observations with the Combined Array for Research in Millimeter Astronomy (CARMA) and an on-the-fly mapping with the 45 m telescope at Nobeyama Radio Observatory (NRO45) in the CO (J = 1-0) emission line of the nearby spiral galaxy NGC 3521. Using the new combined CARMA + NRO45 data of NGC 3521, together with similar data for NGC 5194 (M51a) and archival SINGS H Delta *a, 24 Delta *mm THINGS H I, and Galaxy Evolution Explorer/Far-UV (FUV) data for these two galaxies, we investigate the empirical scaling law that connects the surface density of star formation rate (SFR) and cold gas (known as the Schmidt-Kennicutt law or S-K law) on a spatially resolved basis and find a super-linear slope for the S-K law when carefully subtracting the background emissions in the SFR image. We argue that plausibly deriving SFR maps of nearby galaxies requires the diffuse stellar and dust background emission to be subtracted carefully (especially in the mid-infrared and to a lesser extent in the FUV). Applying this approach, we perform a pixel-by-pixel analysis on both galaxies and quantitatively show that the controversial result whether the molecular S-K law (expressed as ) 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 super-linear () at the highest available resolution (~230 pc) and decreases monotonically for decreasing resolution. We also find in both galaxies that the scatter of the molecular S-K law () monotonically increases as the resolution becomes higher, indicating a trend for which the S-K law breaks down below some scale. Both and are systematically larger in M51a than in NGC 3521, but when plotted against the de-projected scale ( Delta *ddp), both quantities become highly consistent for the two galaxies, tentatively suggesting that the sub-kpc molecular S-K law in spiral galaxies depends only on the scale being considered, without varying among spiral galaxies. A logarithmic function and a linear relation 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 sensitivity, resolution, and broader field of view are required to test the general applicability of these relations.