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 similar to 2 as long as the disk radius is larger than similar to 30 au (for L = 8 L-circle dot). In most cases, this is due to lower temperatures (disk shadowing), which causes the smaller amount of warm (greater than or similar to 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 d