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We aim at determining the spatial distribution of the gas and dust instar-forming regions and address their relative abundances in quantitativeterms. We also examine the dust opacity exponent beta for spatial and/ortemporal variations. Using mapping observations of the very dense rho Oph Acore, we examined standard 1D and non-standard 3D methods to analyse data offar-infrared and submillimeter (submm) continuum radiation. The resulting dustsurface density distribution can be compared to that of the gas. The latter wasderived from the analysis of accompanying molecular line emission, observedwith Herschel from space and with APEX from the ground. As a gas tracer we usedN2H+, which is believed to be much less sensitive to freeze-out than CO and itsisotopologues. Radiative transfer modelling of the N2H+(J=3-2) and (J=6-5)lines with their hyperfine structure explicitly taken into account providessolutions for the spatial distribution of the column density N(H2), hence thesurface density distribution of the gas. The gas-to-dust mass ratio is varyingacross the map, with very low values in the central regions around the core SM1. The global average, =88, is not far from the canonical value of 100,however. In rho Oph A, the exponent beta of the power-law description for thedust opacity exhibits a clear dependence on time, with high values of 2 for theenvelope-dominated emission in starless Class -1 sources to low values close to0 for the disk-dominated emission in Class III objects. beta assumesintermediate values for evolutionary classes in between. Since beta isprimarily controlled by grain size, grain growth mostly occurs in circumstellardisks. The spatial segregation of gas and dust, seen in projection toward thecore centre, probably implies that, like C18O, also N2H+ is frozen onto thegrains.

Gas and dust in the star-forming region rho Oph A: The dust opacity exponent beta and the gas-to-dust mass ratio g2d