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It is well-established that a gas density gradient inside molecular cloudsand clumps raises their star formation rate compared to what they wouldexperience from a gas reservoir of uniform density. This effect should beobserved in the relation between dense-gas mass $M_{dg}$ and star formationrate $SFR$ of molecular clouds and clumps, with steeper gas density gradientsyielding higher $SFR/M_{dg}$ ratios. The content of this paper is two-fold.Firstly, we build on the notion of magnification factor introduced byParmentier (2019) to redefine the dense-gas relation (i.e. the relation between$M_{dg}$ and $SFR$). Not only does the $SFR/M_{dg}$ ratio depend on the meanfree-fall time of the gas and on its (intrinsic) star formation efficiency perfree-fall time, it also depends on the logarithmic slope $-p$ of the gasdensity profile and on the relative extent of the constant-density region atthe clump center. Secondly, we show that nearby molecular clouds follow thenewly-defined dense-gas relation, provided that their dense-gas mass is definedbased on a volume density criterion. We also find the same trend for the densemolecular clouds of the Central Molecular Zone (CMZ) of the Galaxy, althoughthis one is scaled down by a factor of $10$ compared to nearby clouds. Therespective locii of both nearby and CMZ clouds in the $(p, SFR/M_{dg})$parameter space is discussed.

A new parameterization of the star formation rate-dense gas mass relation: embracing gas density gradients