  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.
