A cautionary note about composite Galactic star formation relations

We explore the pitfalls which affect the comparison of the star-formation (SF) relation for nearby molecular clouds with that for distant compact molecular clumps. We show that both relations behave differently in the ($Sigma_{gas}$, $Sigma_{SFR}$) space, where $Sigma_{gas}$ and $Sigma_{SFR}$ are, respectively, the gas and SF rate surface densities, even when the physics of star formation is the same. This is because the SF relation of nearby clouds relates gas and star surface densities measured locally, that is, within a given interval of gas surface density, or at a given protostar location. We refer to such measurements as local measurements, and the corresponding SF relation as the local relation. In contrast, the stellar content of a distant molecular clump remains unresolved. Only the mean SF rate can be obtained from e.g. the clump infrared luminosity. One clump therefore provides one single point to the ($Sigma_{gas}$, $Sigma_{SFR}$) space, that is, its mean gas surface density and SF rate surface density. We refer to this SF relation as a global relation since it builds on the global properties of molecular clumps. Its definition therefore requires an ensemble of cluster-forming clumps. We show that, although the local and global relations have different slopes, this per se cannot be taken as evidence for a change in the physics of SF with gas surface density. It therefore appears that great caution should be taken when physically interpreting a composite SF relation, that is, a relation combining together local and global measurements.