The mass of dusty clumps with temperature and density structure
  We consider a dusty clump in the two cases of spherical and cylindrical
symmetry to investigate the effect of temperature and density gradients on the
observed flux density. Conversely, we evaluate how the presence of such
gradients affects the calculation of the clump mass from the observed flux. We
provide the reader with approximate expressions relating flux density and mass
in the optically thick and thin limits, in the Rayleigh-Jeans regime, and
discuss the reliability of these expressions by comparing them to the outcome
of a numerical code. Finally, we present an application of our calculations to
three examples taken from the literature, which shows how the correction
introduced after taking into account temperature and density gradients may
affect our conclusions on the stability of the clumps.
