We discuss the detection of 14 rovibrational lines of CH$^+$, obtained with the iSHELL spectrograph on NASAs Infrared Telescope Facility (IRTF) on Maunakea. Our observations in the 3.49 - 4.13 $mu$m spectral region, obtained with a 0.375 slit width that provided a spectral resolving power $lambda/Delta lambda sim 80,000$, have resulted in the unequivocal detection of the $R(0) - R(3)$ and $P(1)-P(10)$ transitions within the $v=1-0$ band of CH$^+$. The $R$-branch transitions are anomalously weak relative to the $P$-branch transitions, a behavior that is explained accurately by rovibronic calculations of the transition dipole moment reported in a companion paper (Changala et al. 2021). Nine infrared transitions of H$_2$ were also detected in these observations, comprising the $S(8)$, $S(9)$, $S(13)$ and $S(15)$ pure rotational lines; the $v=1-0$ $O(4) - O(7)$ lines, and the $v=2-1$ $O(5)$ line. We present a photodissociation region model, constrained by the CH$^+$ and H$_2$ line fluxes that we measured, that includes a detailed treatment of the excitation of CH$^+$ by inelastic collisions, optical pumping, and chemical (formation) pumping. The latter process is found to dominate the excitation of the observed rovibrational lines of CH$^+$, and the model is remarkably successful in explaining both the absolute and relative strengths of the CH$^+$ and H$_2$ lines.
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