We present a coherent database of spectroscopic observations of far-IR fine-structure lines from the Herschel/PACS archive for a sample of 170 local AGN, plus a comparison sample of 20 starburst galaxies and 43 dwarf galaxies. Published Spitzer/IRS and Herschel/SPIRE line fluxes are included to extend our database to the full 10-600 $\mu m$ spectral range. The observations are compared to a set of CLOUDY photoionisation models to estimate the above physical quantities through different diagnostic diagrams. We confirm the presence of a stratification of gas density in the emission regions of the galaxies, which increases with the ionisation potential of the emission lines. The new [OIV]25.9$\mu m$/[OIII]88$\mu m$ vs [NeIII]15.6$\mu m$/[NeII]12.8$\mu m$ diagram is proposed as the best diagnostic to separate: $i)$ AGN activity from any kind of star formation; and $ii)$ low-metallicity dwarf galaxies from starburst galaxies. Current stellar atmosphere models fail to reproduce the observed [OIV]25.9$\mu m$/[OIII]88$\mu m$ ratios, which are much higher when compared to the predicted values. Finally, the ([NeIII]15.6$\mu m$ + [NeII]12.8$\mu m$)/([SIV]10.5$\mu m$ + [SIII]18.7$\mu m$) ratio is proposed as a promising metallicity tracer to be used in obscured objects, where optical lines fail to accurately measure the metallicity. The diagnostic power of mid- to far-infrared spectroscopy shown here for local galaxies will be of crucial importance to study galaxy evolution during the dust-obscured phase at the peak of the star formation and black-hole accretion activity ($1 < z < 4$). This study will be addressed by future deep spectroscopic surveys with present and forthcoming facilities such as JWST, ALMA, and SPICA.

We observed the far-IR fine-structure lines of 26 Seyfert galaxies with the Herschel-PACS spectrometer. These observations are complemented with Spitzer Infrared Spectrograph and Herschel SPIRE spectroscopy. We used the ionic lines to determine electron densities in the ionized gas and the [C I] lines, observed with SPIRE, to measure the neutral gas densities, while the [O I] lines measure the gas temperature, at densities below ∼104 cm-3. Using the [O I]145 μm/63 μm and [S III]33/18 μm line ratios, we find an anti-correlation of the temperature with the gas density. Various fine-structure line ratios show density stratifications in these active galaxies. On average, electron densities increase with the ionization potential of the ions. The infrared lines arise partly in the narrow line region, photoionized by the active galactic nucleus (AGN), partly in H II regions photoionized by hot stars, and partly in photo-dissociated regions. We attempt to separate the contributions to the line emission produced in these different regions by comparing our observed emission line ratios to theoretical values. In particular, we tried to separate the contribution of AGNs and star formation by using a combination of Spitzer and Herschel lines, and we found that besides the well-known mid-IR line ratios, the line ratio of [O III]88 μm/[O IV]26 μm can reliably discriminate the two emission regions, while the far-IR line ratio of [C II]157 μm/[O I]63 μm is only able to mildly separate the two regimes. By comparing the observed [C II]157 μm/[N II]205 μm ratio with photoionization models, we also found that most of the [C II] emission in the galaxies we examined is due to photodissociation regions.