We conducted numerical simulations of the dust heating in accretion shocks induced by the interaction between the infalling envelope and the Keplerian disk surrounding a protostar, in order to investigate the thermal desorption of molecules from the dust-grain surfaces. It is thought that the surfaces of the amorphous dust grains are inhomogeneous; various adsorption sites with different binding energies should therefore exist. We assumed that the desorption energy has a Gaussian distribution and investigated the effect of the desorption energy distribution on the desorption-efficiency evaluation. We calculated the desorption fractions of the grain-surface species for wide ranges of input parameters and summarized our results in a shock diagram. The resulting shock diagram suggests that the enhanced line emissions around protostars observed using the Atacama Large Millimeter Array cannot be explained by the thermal desorption in an accretion shock if typical interstellar dust-grain sizes (∼0.1 μm) and a single desorption energy are considered. On the other hand, if significantly smaller dust grains are the main grain-surface species carriers and the desorption energy has a Gaussian distribution, the origin of the enhanced line emission can be explained by the accretion shock heating scenario for all of the three protostars examined in this study: IRAS 04368+2557, IRAS 04365+2535, and IRAS 162932422. The small-grain-carrier supposition is quite reasonable when the dust grains have a power-law size distribution because the smaller grains primarily contribute to the dust-grain surface area.
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