Observations of the high-mass star forming region AFGL 2591 reveal a large abundance of CO+, a molecule known to be enhanced by far UV (FUV) and X-ray irradiation. In chemical models assuming a spherically symmetric envelope, the volume of gas irradiated by protostellar FUV radiation is very small due to the high extinction by dust. The abundance of CO+ is thus underpredicted by orders of magnitude. In a more realistic model, FUV photons can escape through an outflow region and irradiate gas at the border to the envelope. Thus, we introduce the first 2D axi-symmetric chemical model of the envelope of a high-mass star forming region to explain the CO+ observations as a prototypical FUV tracer. The model assumes an axi-symmetric power-law density structure with a cavity due to the outflow. The local FUV flux is calculated by a Monte Carlo radiative transfer code taking scattering on dust into account. A grid of precalculated chemical abundances, introduced in the first part of this series of papers, is used to quickly interpolate chemical abundances. This approach allows to calculate the temperature structure of the FUV heated outflow walls self-consistently with the chemistry. Synthetic maps of the line flux are calculated using a raytracer code. Single-dish and interferometric observations are simulated and the model results are compared to published and new JCMT and SMA observations. The two-dimensional model of AFGL 2591 is able to reproduce the JCMT single-dish observations and also explains the non-detection by the SMA. We conclude that the observed CO+ line flux and its narrow width can be interpreted by emission from the warm and dense outflow walls irradiated by protostellar FUV radiation.
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