State-to-state chemistry and rotational excitation of CH$^+$ in photon-dominated regions

We present a detailed theoretical study of the rotational excitation of CH$^+$ due to reactive and nonreactive collisions involving C$^+(^2P)$, H$_2$, CH$^+$, H and free electrons. Specifically, the formation of CH$^+$ proceeds through the reaction between C$^+(^2P)$ and H$_2(\nu_{\rm H_2}=1, 2)$, while the collisional (de)excitation and destruction of CH$^+$ is due to collisions with hydrogen atoms and free electrons. State-to-state and initial-state-specific rate coefficients are computed in the kinetic temperature range 10-3000~K for the inelastic, exchange, abstraction and dissociative recombination processes using accurate potential energy surfaces and the best scattering methods. Good agreement, within a factor of 2, is found between the experimental and theoretical thermal rate coefficients, except for the reaction of CH$^+$ with H atoms at kinetic temperatures below 50~K. The full set of collisional and chemical data are then implemented in a radiative transfer model. Our Non-LTE calculations confirm that the formation pumping due to vibrationally excited H$_2$ has a substantial effect on the excitation of CH$^+$ in photon-dominated regions. In addition, we are able to reproduce, within error bars, the far-infrared observations of CH$^+$ toward the Orion Bar and the planetary nebula NGC~7027. Our results further suggest that the population of $\nu_{\rm H_2}=2$ might be significant in the photon-dominated region of NGC~7027.