We model the star formation relation of molecular clumps in dependence of their dense-gas mass when their volume density profile is that of an isothermal sphere, i.e. $rho_{clump}(r) propto r^{-2}$. Dense gas is defined as gas whose volume density is higher than a threshold $rho_{th}=700,M_{odot}.pc^{-3}$, i.e. HCN(1-0)-mapped gas. We divide the clump into two regions: a dense inner region (where $rho_{clump}(r) geq rho_{th}$), and low-density outskirts (where $rho_{clump}(r) < rho_{th}$). We find that the total star formation rate of clumps scales linearly with the mass of their dense inner region, even when more than half of the clump star formation activity takes place in the low-density outskirts. We therefore emphasize that a linear star formation relation does not necessarily imply that star formation takes place exclusively in the gas whose mass is given by the star formation relation. The linearity of the star formation relation is strengthened when we account for the mass of dense fragments (e.g. cores, fibers) seeding star formation in the low-density outskirts, and which our adopted clump density profile $rho_{clump}(r)$ does not resolve. We also find that the star formation relation is significantly tighter when considering the dense gas than when considering all the clump gas, as observed for molecular clouds of the Galactic plane. When the clumps have no low-density outskirts (i.e. they consist of dense gas only), the star formation relation becomes superlinear and progressively wider.
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