The Herschel-PACS legacy of low-mass protostars: Properties of warm and hot gas and its origin in far-UV illuminated shocks

Recent observations from Herschel allow the identification of important mechanisms responsible for the heating of gas surrounding low-mass protostars and its subsequent cooling in the far-infrared (FIR). Shocks are routinely invoked to reproduce some properties of the far-IR spectra, but standard models fail to reproduce the emission from key molecules, e.g. H$_2$O. Here, we present the Herschel-PACS far-IR spectroscopy of 90 embedded low-mass protostars (Class 0/I). The Herschel-PACS spectral maps covering $\sim55-210$ $\mu$m with a field-of-view of $\sim$50'' are used to quantify the gas excitation conditions and spatial extent using rotational transitions of H$_{2}$O, high-$J$ CO, and OH, as well as [O I] and [C II]. We confirm that a warm ($\sim$300 K) CO reservoir is ubiquitous and that a hotter component ($760\pm170$ K) is frequently detected around protostars. The line emission is extended beyond $\sim$1000 AU spatial scales in 40/90 objects, typically in molecular tracers in Class 0 and atomic tracers in Class I objects. High-velocity emission ($\gtrsim90$ km s$^{-1}$) is detected in only 10 sources in the [O I] line, suggesting that the bulk of [O I] arises from gas that is moving slower than typical jets. Line flux ratios show an excellent agreement with models of $C$-shocks illuminated by UV photons for pre-shock densities of $\sim$$10^5$ cm$^{-3}$ and UV fields 0.1-10 times the interstellar value. The far-IR molecular and atomic lines are a unique diagnostic of feedback from UV emission and shocks in envelopes of deeply embedded protostars.