We selected 27244 nearby, red, giant early-type galaxies (RGEs) from the Sloan Digital Sky Survey (SDSS). In a large fraction (53%) of their spectra the [O III] emission line is detected, with an equivalent width (EW) distribution strongly clustered around ~0.75 A. The vast majority of those RGEs for which it is possible to derive emission line ratios (amounting to about half of the sample) show values characteristic of LINERs. The close connection between emission lines and stellar continuum points to stellar processes as the most likely source of the bulk of the ionizing photons in RGEs, rather than active nuclei. In particular, the observed EW and optical line ratios are consistent with the predictions of models in which the photoionization comes from to hot evolved stars. Shocks driven by supernovae or stellar ejecta might also contribute to the ionization budget. A minority, ~4%, of the galaxies show emission lines with an equivalent that is width a factor of ~2 greater than the sample median. Only among them are Seyfert-like spectra found. Furthermore, 40% of this subgroup have a radio counterpart, compared to ~6% of the rest of the sample. These characteristics argue in favor of an AGN origin for their emission lines. Emission lines diagnostic diagrams do not reveal a distinction between the AGN subset and the other members of the sample, and consequently they are not a useful tool for establishing the dominant source of the ionizing photons, which is better predicted by the EW of the emission lines.Comment: Accepted for publication in A&
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