Context. While [O III] narrow-band imaging is commonly used to measure the size of the narrow-line regions (NLRs) in active galactic nuclei (AGNs), it can be contaminated by emission from surrounding starbursts. Recently, we have shown that long-slit spectroscopy provides a valuable alternative approach to probe the size in terms of AGN photoionisation. Moreover, several parameters of the NLR can be directly accessed. Aims. We here apply the same methods developed and described for the Seyfert-2 galaxy NGC 1386 to study the NLR of five other Seyfert-2 galaxies by using high-sensitivity spatially-resolved optical spectroscopy obtained at the VLT and the NTT. Methods. We probe the AGN-photoionisation of the NLR and thus, its "real'' size using diagnostic line-ratio diagrams. We derive physical properties of the NLR such as reddening, ionisation parameter, electron density, and velocity as a function of distance from the nucleus. Results. For NGC 5643, the diagnostic diagrams unveil a similar transition between line ratios falling in the AGN regime and those typical for H II regions as found for NGC 1386, thus determining the size of the NLR. For the other four objects, all measured line ratios fall in the AGN regime. In almost all cases, both electron density and ionisation parameter decrease with radius. Deviations from this general behaviour (such as a secondary peak) seen in both the ionisation parameter and electron density can be interpreted as signs of shocks from the interaction of a radio jet and the NLR gas. In several objects, the gaseous velocity distribution is characteristic for rotational motion in an (inclined) emission-line disk in the centre. We compare our results to those of NGC 1386 and show that the latter can be considered as prototypical also for this larger sample. We discuss our findings in detail for each object.



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