The Excitation of N$_2$H$^+$ in Interstellar Molecular Clouds. I - Models

We present LVG and non-local radiative transfer calculations involving the rotational and hyperfine structure of the spectrum of N$_2$H$^+$ with collisional rate coefficients recently derived by us. The goal of this study is to check the validity of the assumptions made to treat the hyperfine structure and to study the physical mechanisms leading to the observed hyperfine anomalies. We find that the usual hypothesis of identical excitation temperatures for all hyperfine components of the $J$=1-0 transition is not correct within the range of densities existing in cold dense cores, i.e., a few 10$^4$ $\textless$ n(H$_2$) $\textless$ a few 10$^6$ cm$^{-3}$. This is due to different radiative trapping effects in the hyperfine components. Moreover, within this range of densities and considering the typical abundance of N$_2$H$^+$, the total opacity of rotational lines has to be derived taking into account the hyperfine structure. The error made when only considering the rotational energy structure can be as large as 100%. Using non-local models we find that, due to saturation, hyperfine anomalies appear as soon as the total opacity of the $J$=1-0 transition becomes larger than $\simeq$ 20. Radiative scattering in less dense regions enhance these anomalies, and particularly, induce a differential increase of the excitation temperatures of the hyperfine components. This process is more effective for the transitions with the highest opacities for which emerging intensities are also reduced by self-absorption effects. These effects are not as critical as in HCO$^+$ or HCN, but should be taken into account when interpreting the spatial extent of the N$_2$H$^+$ emission in dark clouds.