Chemical abundances in Seyfert galaxies - VI. Empirical abundance calibration

We derived a bi-dimensional calibration between the emission-line ratios $R_{23}= ([\mathrm{ O}\, {\small {\rm II}}]\,\lambda 3726 +\lambda 3729 +[\mathrm{ O}\mathrm{ }\, {\small {\rm III}}]\,\lambda 4959 + \lambda 5007)/\rm H\,\beta$, ${\it P}=[([\mathrm{ O}\, {\small {\rm III}}]\,\lambda 4959+\lambda 5007)/{\rm H}\,\beta ]/R_{23}$ and the oxygen abundance relative to hydrogen (O/H) in the gas phase of Seyfert 1 and 2 nuclei. In view of this, emission-line intensity ratios for a sample of objects taken from the Sloan Digital Sky Survey Data Release 7 measured by the MPA/JHU group and direct estimates of O/H based on T_e-method, adapted for active galactic nuclei (AGNs), are considered. We find no variation of R₂₃ observed along the radii of AGNs, which shows that this line ratio is a good oxygen abundance (O/H) indicator for the class of objects considered in this work. The derived O/H = f(R₂₃, P) relation produces O/H values similar to estimations via T_e-method in a wide range of metallicities [$\rm 8.0 \: \lesssim \: 12+\log \,(O/H) \: \lesssim \: 9.2$]. Conversely to star-forming regions in the high-metallicity regime, R₂₃ shows a positive correlation trend with O/H in AGNs. This indicates that the hardness of ionizing radiation is not affected by the metallicities in these objects or narrow-line regions are not significantly modified by changes in the spectral energy distribution due to metallicity variations.