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.ρclump(r) ∝ r-2. Dense gas is defined as gas whose volume density is higher than a threshold ρth=700 M☉.pc-3, i.e. HCN(1-0)-mapped gas.
We divide the clump into two regions: a dense inner region (where ρclump(r) ≥ ρth), and low-density outskirts (where ρclump(r) < ρth). 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 ρclump(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.