We analyze the collisional excitation of the 158 micron (1900.5 GHz) fine structure transition of ionized carbon (C+) in terms of line intensities produced by simple cloud models. The single C+ fine structure transition is a very important coolant of the atomic interstellar medium and of photon dominated regions in which carbon is partially or completely in ionized form. The [CII] line is widely used as a tracer of star formation in the Milky Way and other galaxies. Excitation of the [CII] fine structure transition can be via collisions with hydrogen molecules, atoms, and electrons. Velocity-resolved observations of [CII] have become possible with the HIFI instrument on Herschel and the GREAT instrument on SOFIA. Analysis of these observations is complicated by the fact that it is difficult to determine the optical depth of the [CII] line due to the relative weakness and blending of the components of the analogous transition of 13C$+. We discuss the excitation and radiative transition of the [CII] line, deriving analytic results for several limiting cases and carry out numerical solutions using a large velocity gradient model for a more inclusive analysis. We show that for antenna temperatures up to 1/3 of the brightness temperature of the gas kinetic temperature, the antenna temperature is linearly proportional to the column density of C+ irrespective of the optical depth of the transition, which can be referred to as the effectively optically thin (EOT) approximation. We review the critical densities for excitation of the [CII] line by various collision partners. We briefly analyze C+ absorption and conclude with a discussion of C+ cooling and how the considerations for line intensities affect the behavior of this important coolant of the ISM.
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