The Dust You Cannot See: Millimetre Optical Depths in Protoplanetary Discs and the Resulting Mass Underestimate

Millimetre continuum fluxes are routinely converted into dust masses on the assumption that the emission is optically thin, so that every grain contributes. In the inner regions of a disc of ordinary mass that assumption fails badly. With the standard 1.3 mm opacity of 2.3 cm^2 per gram of dust, the face-on optical depth on the reference profile is 39 at 1 AU, 7.5 at 3 AU and 1.24 at 10 AU, falling below unity only outside 12 AU. Integrating the true profile against the thin-inferred profile over 0.1 to 100 AU, the true dust mass is 78 Earth masses while the optically thin conversion returns 53 Earth masses — an underestimate by a factor of 1.5, before any allowance for grain growth changing the opacity itself. The missing mass hides exactly where planet formation is fastest, which biases disc demographic conclusions: discs are more massive in solids, and planet formation less efficient per unit observed dust, than the thin conversion suggests. The effect is a radiative-transfer triviality with demographic consequences, and it is computed here from one opacity and one profile. Computed from stated assumptions, not measured.