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.
