Importance of source structure on complex organics emission - II. Do disks explain lack of methanol emission from low-mass protostars?

Context. The protostellar stage is known to be the richest star formation phase in emission from gaseous complex organic molecules. However, some protostellar systems show little or no millimetre (mm) line emission of such species. This can be interpreted as a low abundance of complex organic molecules. Alternatively, complex species could be present in the system, but are not seen in the gas.Aims. The goal is to investigate the second hypothesis for methanol as the most abundant complex organic molecule in protostellar systems. This work aims to determine how effective dust optical depth is in hiding methanol in the gas, and whether methanol can mainly reside in the ice due to the presence of a disk that lowers the temperatures. Hence, we attempt to answer the question whether the presence of a disk and optically thick dust reduce methanol emission even if methanol and other complex species are abundant in the ices and gas.Methods. Using the radiative transfer code RADMC-3D, we calculated methanol emission lines from an envelope-only model and from an envelope-plus-disk model. We compared the results with each other and with the observations. Methanol gas and ice abundances were parametrised inside and outside of the snow surfaces based on values from observations. Both models included either dust grains with low mm opacity or high mm opacity, and their physical parameters such as envelope mass and disk radius were varied.Results. Methanol emission from the envelope-only model is always stronger than from the envelope-plus-disk model by at least a factor ∼2 as long as the disk radius is larger than ∼30 au (for L = 8 L⊙). In most cases, this is due to lower temperatures (disk shadowing), which causes the smaller amount of warm (≳70 K) methanol inside the snow surface of the envelope-plus-disk model. The intensities drop by more than an order of magnitude for models including high mm opacity dust grains and disk radii of at least ∼50 au (for L = 8 L⊙) due to continuum over-subtraction.Conclusions. The line intensities from the envelope-only models match the observations moderately well when methanol emission is strong, but they overproduce the observations of protostars with lower methanol emission even with large dust optical depth effects. The envelope-plus-disk models can explain the bulk of the observations. However, they can only reproduce the observations of sources with high luminosities and very low methanol emission when the dust optical depth is significant in the envelope and continuum over-subtraction becomes effective in the disk (high mm opacity dust grains are used). Therefore, both the effects of disk and dust optical depth should be considered to explain the observations. In conclusion, it is important to take physical structure into account in future chemical studies of low-mass protostars: absence of gas-phase methanol emission does not imply absence of methanol molecules in either gas or ice