Modelling Deuterated Isotopologues of Methanol toward the Pre-Stellar Core L1544

Aims. We aim to improve a previous model for the prediction of column densities and deuterium fractions of non- and singly deuterated methanol. Thereby, we try to identify crucial chemical and physical parameters, for which the study of deuteration could provide valuable additional constraints. Methods. We employed a gas-grain chemical code to devise a model that is in agreement with the observed column density and deuterium fraction profiles of the innermost region of the pre-stellar core L1544. For that purpose, we developed a new treatment of reactive desorption, deriving an individual reactive desorption efficiency for every product species in a chemical reaction, that depends on the reaction enthalpy and type of underlying surface. Furthermore, we explored several options to promote the diffusion of hydrogen and deuterium atoms over the surface of interstellar dust grains, in order to increase methanol formation. Results. Our fiducial model employs diffusion by quantum tunneling of hydrogen and deuterium atoms, resulting in CH$_3$OH and CH$_2$DOH column densities that are approximately an order of magnitude lower than the observed values, which improves the results compared to the previous model by a factor 10. The $N$(CH$_2$DOH)/$N$(CH$_3$OH) ratio is reproduced within a factor of 1.2 for the centre and 1.8 for the position of the methanol peak. Given the large uncertainties that chemical models typically have, we consider our predictions to be in agreement with the observations. In general, we conclude that a diffusion process with a high diffusion rate needs to be employed to obtain methanol column densities that are in accordance with the observed values. Also, we find that the introduction of abstraction reactions into the methanol formation scheme suppresses deuteration, when used in combination with a high diffusion rate.