Radiative and mechanical energies in galaxies I. Contributions of molecular shocks and PDRs in 3C 326 N

Context: Atomic and molecular lines in galaxies offer insights into energy budgets and feedback mechanisms. Aims: This study establishes a new framework for interpreting these lines and deducing energy budgets from observations. Methods: Atomic and molecular lines detected in a given object are assumed to result from the combination of distributions of shocks and photo-dissociation regions (PDR). Using the Paris-Durham shock code and the Meudon PDR code, emissions are computed over a wide range of parameters. Total emissions are calculated using probability distribution functions, with a defined distance metric based on observed and predicted intensity ratios. Results: We analyze the radio galaxy 3C 326 N, finding both shocks and PDRs necessary to explain the line fluxes. Viable solutions occur only at low densities ($\rm n_H < 100 cm^{-3}$), indicating emission from diffuse interstellar matter. The optimal solution involves low-velocity shocks (5-20 km/s) in PDRs illuminated by UV radiation ten times stronger than in the solar neighborhood. The H$2$ 0-0 S(0) $28 \mu$m, [CII] $158 \mu$m, and [OI] $63 \mu$m lines originate from PDRs, while other H$2$ lines mostly come from shocks. The reprocessed radiative and mechanical energies are $\rm {L_{UV} = 6.3\times10^9 L\odot}$ and $\rm {L_K = 3.9\times10^8 L_\odot}$, respectively, in agreement with 3C 326 N's infrared luminosity, and consistent with 1% of the AGN jet kinetic power dissipated in the interstellar medium. Conclusions: This study demonstrates that the radiative and mechanical energy budgets of galaxies can be derived from observations of atomic and molecular lines alone. It highlights the unexpected importance of the diffuse medium for 3C 326 N. Comparison with new JWST data for 3C 326 N shows striking agreement, opening new prospects for predicting and interpreting extragalactic observations.