We present Herschel SPIRE Fourier Transform Spectrometer (FTS) observations of N159W, an active star-forming region in the Large Magellanic Cloud (LMC). In our observations, a number of far-infrared cooling lines including CO(4-3) to CO(12-11), [CI] 609 and 370 micron, and [NII] 205 micron are clearly detected. With an aim of investigating the physical conditions and excitation processes of molecular gas, we first construct CO spectral line energy distributions (SLEDs) on 10 pc scales by combining the FTS CO transitions with ground-based low-J CO data and analyze the observed CO SLEDs using non-LTE radiative transfer models. We find that the CO-traced molecular gas in N159W is warm (kinetic temperature of 153-754 K) and moderately dense (H2 number density of (1.1-4.5)e3 cm-3). To assess the impact of the energetic processes in the interstellar medium on the physical conditions of the CO-emitting gas, we then compare the observed CO line intensities with the models of photodissociation regions (PDRs) and shocks. We first constrain the properties of PDRs by modelling Herschel observations of [OI] 145, [CII] 158, and [CI] 370 micron fine-structure lines and find that the constrained PDR components emit very weak CO emission. X-rays and cosmic-rays are also found to provide a negligible contribution to the CO emission, essentially ruling out ionizing sources (ultraviolet photons, X-rays, and cosmic-rays) as the dominant heating source for CO in N159W. On the other hand, mechanical heating by low-velocity C-type shocks with ~10 km/s appears sufficient enough to reproduce the observed warm CO.
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