Physical conditions in the gas phases of the giant H？II region LMC-N 11 - II. Origin of [C？II] and fraction of CO-dark gas

Context. The ambiguous origin of the [C？II] 158μm line in the interstellar medium complicates its use for diagnostics concerning the star-formation rate and physical conditions in photodissociation regions.Aims. We investigate the origin of [C？II] in order to measure the total molecular gas content, the fraction of CO-dark H_{2 gas, and how these parameters are impacted by environmental effects such as stellar feedback.Methods. We observed the giant H？II region N 11 in the Large Magellanic Cloud with SOFIA/GREAT. The [C？II] line is resolved in velocity and compared to H？I and CO, using a Bayesian approach to decompose the line profiles. A simple model accounting for collisions in the neutral atomic and molecular gas was used in order to derive the H2 column density traced by C^{+.Results. The profile of [C？II] most closely resembles that of CO, but the integrated [C？II] line width lies between that of CO and that of H？I. Using various methods, we find that [C？II] mostly originates from the neutral gas. We show that [C？II] mostly traces the CO-dark H2 gas but there is evidence of a weak contribution from neutral atomic gas preferentially in the faintest components (as opposed to components with low [C？II]/CO or low CO column density). Most of the molecular gas is CO-dark. The CO-dark H2 gas, whose density is typically a few 100s cm？3 and thermal pressure in the range 103.5？5 K cm？3, is not always in pressure equilibrium with the neutral atomic gas. The fraction of CO-dark H2 gas decreases with increasing CO column density, with a slope that seems to depend on the impinging radiation field from nearby massive stars. Finally we extend previous measurements of the photoelectric-effect heating efficiency, which we find is constant across regions probed with Herschel, with [C？II] and [O？I] being the main coolants in faint and diffuse, and bright and compact regions, respectively, and with polycyclic aromatic hydrocarbon emission tracing the CO-dark H2 gas heating where [C？II] and [O？I] emit.Conclusions. We present an innovative spectral decomposition method that allows statistical trends to be derived for the molecular gas content using CO, [C？II], and H？I profiles. Our study highlights the importance of velocity-resolved photodissociation region (PDR) diagnostics and higher spatial resolution for H？I observations as future steps}}