Motivated by recent observations with Herschel/PACS, and the availability of new rate coefficients for the collisional excitation of CO, the excitation of warm astrophysical CO is revisited with the use of numerical and analytic methods. For the case of an isothermal medium, results have been obtained for a wide range of gas temperatures (100-5000?K) and H2 densities (103-109 cm–3), and presented in the form of rotational diagrams, in which the logarithm of the column density per magnetic substate, log (NJ /gJ ), is plotted for each state, as a function of its energy, EJ . For rotational transitions in the wavelength range accessible to Herschel/PACS, such diagrams are nearly linear when n(H2) ≥ 108 cm–3. When n(H2) ~ 106.8-108?cm–3, they exhibit significant negative curvature, whereas when n(H2) ≤ 104.8?cm–3, the curvature is uniformly positive throughout the PACS-accessible range. Thus, the observation of a positively curved CO rotational diagram does not necessarily require the presence of multiple temperature components. Indeed, for some sources observed with Herschel/PACS, the CO rotational diagrams show a modest positive curvature that can be explained by a single isothermal component. Typically, the required physical parameters are densities in the 104-105?cm–3 range and temperatures close to the maximum at which CO can survive. Other sources exhibit rotational diagrams with more curvature than can be accounted for by a single temperature component. For the case of a medium with a power-law distribution of gas temperatures, dN/dT∝T –b , results have been obtained for H2 densities 103-109 cm–3 and power-law indices, b, in the range 1-5; such a medium can account for a CO rotational diagram that is more positively curved than any resulting from an isothermal medium.
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