Modelling Herschel observations of hot molecular gas emission from embedded low-mass protostars

Aims. Young stars interact vigorously with their surroundings, as evident from the highly rotationally excited CO (up to Eup=4000 K) and H2O emission (up to 600 K) detected by the Herschel Space Observatory in embedded low-mass protostars. Our aim is to construct a model that reproduces the observations quantitatively, to investigate the origin of the emission, and to use the lines as probes of the various heating mechanisms. Methods. The model consists of a spherical envelope with a bipolar outflow cavity. Three heating mechanisms are considered: passive heating by the protostellar luminosity, UV irradiation of the outflow cavity walls, and C-type shocks along the cavity walls. Line fluxes are calculated for CO and H2O and compared to Herschel data and complementary ground-based data for the protostars NGC1333 IRAS2A, HH 46 and DK Cha. The three sources are selected to span a range of evolutionary phases and physical characteristics. Results. The passively heated gas in the envelope accounts for 3-10% of the CO luminosity summed over all rotational lines up to J=40-39; it is best probed by low-J CO isotopologue lines such as C18O 2-1 and 3-2. The UV-heated gas and the C-type shocks, probed by 12CO 10-9 and higher-J lines, contribute 20-80% each. The model fits show a tentative evolutionary trend: the CO emission is dominated by shocks in the youngest source and by UV-heated gas in the oldest one. This trend is mainly driven by the lower envelope density in more evolved sources. The total H2O line luminosity in all cases is dominated by shocks (>99%). The exact percentages for both species are uncertain by at least a factor of 2 due to uncertainties in the gas temperature as function of the incident UV flux. However, on a qualitative level, both UV-heated gas and C-type shocks are needed to reproduce the emission in far-infrared rotational lines of CO and H2O.