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 E_u/k = 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 power-law density structure and a bipolar outflow cavity. Three heating mechanisms are considered: passive heating by the protostellar luminosity, ultraviolet irradiation of the outflow cavity walls, and small-scale C-type shocks along the cavity walls. Most of the model parameters are constrained from independent observations; the two remaining free parameters considered here are the protostellar UV luminosity and the shock velocity. Line fluxes are calculated for CO and H₂O and compared to Herschel data and complementary ground-based data for the protostars NGC 1333 IRAS2A, HH 46 and DK Cha. The three sources are selected to span a range of evolutionary phases (early Stage 0 to late Stage I) and physical characteristics such as luminosity and envelope mass. Results. The bulk of the gas in the envelope, heated by the protostellar luminosity, 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 and within the context of our model, both UV-heated gas and C-type shocks are needed to reproduce the emission in far-infrared rotational lines of CO and H₂O. [ABSTRACT FROM AUTHOR]