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In order to make a comparison between the models and data, we have considered line ratios, involving the following most intense emission lines observed in the spectra of Seyfert 2's and LINERs: [O II] lambda3727, [O III] lambdalambda4959, 5007, [N II] lambdalambda6548, 6584, and [S II] lambdalambda6717, 6731.

To begin, we have used the ratio I (lambda6717) / (lambda6731) in order to obtain an estimate for the gas density. As is well-known (e.g., Osterbrock 1974), this ratio is sensitive to the density between the limits 102 and 104-105 cm-3. So we have plotted, in Figure 1, this ratio for the observational data and the above-described photoionization models, with the slope of the ionizing continuum alpha = -1.5 (see below) and gas density in the above range. From the distribution of points in Figure 1 it can be concluded that, in most objects, the gas producing [S II] has densities in the range 102-104 cm-3.

Figure 1

Figure 1. The density-sensitive ratio I (lambda6717) / I (lambda6731) for the sample galaxies (dots) and photoionization models (lines) for gas densities 102, 103, and 104 cm-3.

Using this range of densities, we have calculated models with the slope alpha having three values: -1, -1.5, and -2. The best agreement with the data was obtained for alpha = -1.5. The behavior of the models for alpha = -1 and -2 is discussed at the end of this section. We remark that a slope alpha = -1.5 was also used by previous authors (e.g., Ferland and Netzer 1983) for nonthermal photoionizing sources. Even for the alternative scenario of WARMERs (Terlevich and Melnick 1985), it is shown that the integrated ionizing spectrum is similar, in the ultraviolet region, to a power-law spectrum with slope alpha = -1.5.

We have constructed four diagrams, involving the strongest emission lines: [O III] / Hbeta x [O II]/[O III], [N II] / Halpha x [O II]/[O III], [S II]/Halpha x [O II] / [O III], and [N II] / Halpha x [S II] / Halpha. We have plotted in Figures 2a-d the data and models with alpha = -1.5, densities 102, 103, and 104 cm-3 and ionization parameter (U) varying in the range 10-2.5 to 10-4. This range of U is necessary in order to reproduce the [O II] / [O III] values presented by the data. The abundances are solar for all the elements. We show also the shock models mentioned in the previous section. From the diagrams in Figure 2 we obtain the following results.

Figure 2

Figure 2a-d. Sample galaxies (dots), shock models (broken lines), and photoionization models (continuous lines) for gas densities 102, 103, and 104 cm-3 in diagrams involving the stronger emission lines, in part "a" models for density 105 cm-3 are also plotted (in the others they lie over the previous models), and typical photoionization parameters are indicated.

(1) The shock models do not reproduce most of the data; they could only reproduce the data with an [O II]/[O III] ratio larger than 2.5, but most observational points present smaller values for this ratio.

(2) In order to cover the data in the diagram [O III] / Hbeta x [O II] / [O III] it is necessary to consider the presence of clouds of densities higher than 104 cm-3. It can be seen that models with densities between 104 and 105 cm-3 reproduce most of the data. This higher density for the clouds producing [O III] can be understood considering that emission is enhanced in the higher density clouds, as long as the density remains lower than the critical value (7.0e5 cm-3 for [O III]). In fact, the observed correlations between the FWHM of the emission lines and their critical densities (e.g., Storchi-Bergmann, Bica, and Pastoriza 1990; Filippenko and Sargent 1988) indicate the presence of clouds of different densities in the narrow-line region of active galactic nuclei.

(3) The [O II] / [O III] and [O III] / Hbeta ratios are strongly dependent on the ionization parameter U.

(4) The ratios [N II] / Halpha and [S II] / Halpha depend only weakly on the density and U. Model sequences for N = 10-5 cm-3 were not plotted in the corresponding diagrams for the sake of clarity; The values become lower due to collisional deexcitation and superpose with some of the other values. It is not possible, by varying only U and Ne to reproduce the range of values presented by the ratios [N II] / Halpha and [S II] / Halpha. Figure 1d shows that the [N II] / Halpha x [S II] / Halpha relation can be in part reproduced by varying U; nevertheless, we have plotted, for the different densities, the U range which reproduces the range of the [O II] / [O III] ratios. Higher U values would give [O II] / [O III] ratios which do not correspond to the data. So it is necessary to vary another parameter to reproduce the range of [N II] / Halpha and [S II] / Halpha values.

Figure 3

Figure 3a-d. Sample galaxies (dots) and photoionization models (lines) for gas density 104 cm-3 and varying metallicity in the same diagrams as Figure 2.

In order to obtain a larger range in the model values of [N II] / Halpha and [S II] / Halpha, we computed models with metallicities half-solar and three and five times solar. These models, together with the solar abundance model and the data points, are plotted in Figures 3a-d. For clarity we have fixed the density as N = 104 cm-3. The results which can be obtained from this set of models are:

(1) Comparing Figure 3a to Figure 2a, it can be seen that the effect of a higher metallicity in the gas in the diagram [O III] / Hbeta is equivalent to the effect of a higher density. A higher metallicity lowers the temperature-sensitive ratio [O III] / Hbeta due to the cooling of the gas produced by the forbidden line emission. It can be seen that the data on this diagram are compatible with a higher metallicity in the nuclear gas.

(2) In the diagram [N II] / Halpha x [O II] / [O III] it can be seen that the models can reproduce part of the vertical dispersion of the data, but not all. In particular, we note that the sequence for metallicity three times solar is an upper limit for the [N II] / Halpha ratio; The sequence for metallicity five times solar gives lower values than these. So, a varying abundance for all the elements does not reproduce the range of[N II] / Halpha ratios presented by the data.

(3) A varying metallicity also does not reproduce the range presented by the [S II] / Halpha ratios nor the relation [N II] / Halpha x [S II] / Halpha.

Figure 4

Figure 4a-d. Same diagrams as Figures 2 and 3 for model with simultaneous variation of sulfur and nitrogen abundances from 0.5 to 5 times solar. The other elements have solar abundance.

The next step we have tried was to vary only the abundances of nitrogen and sulfur, keeping the abundance of the other elements at the solar value. We have computed models for nitrogen and sulfur with abundances half-solar and two, three, and five times solar. These models, for density N = 104 cm-3, are plotted in Figures 4a-d. It can be seen that:

(1) The result regarding the [O III] / Hbeta x [O II] / [O III] diagram is similar to that obtained with the enhancement of all the metals; Due to the cooling produced by the forbidden nitrogen and sulfur lines, the [O III] / Hbeta ratio is lowered. So it is possible to reproduce the data with clouds of density 104 to 105 cm-3 and nitrogen and sulfur abundances from half-solar to five times solar.

(2) The data points in the [N II] / Halpha x [O II] / [O III] diagram are well covered by these models. The lowest points can be reproduced by models with the same abundances and densities 102-103 cm-3 (see Fig. 2b).

(3) The data in the [S II] / Halpha x [O II] / [O III] and [N II] / Halpha x [O II]/[O III] diagrams are also much better reproduced by this set of models. It is not possible to reproduce the data on the last diagram by varying only the abundance of N or S; it is necessary to vary the abundances of both elements together.

A similar analysis to that described above was also done for slopes of the ionizing continuum alpha = -1 and alpha = -2. Both values led to a worse overall agreement between the model values and the data. In particular, models for alpha = -2 do not reproduce the observed [O III] / Hbeta x [O II]/[O III] diagram except for densities lower than 102 cm-3. And, for these densities, the [N II] / Halpha and [S II] / Halpha values can only be reproduced with higher N and S abundances than that obtained for alpha = -1.5. Models with alpha = -1 give higher [S II] / Halpha ratios than those observed, requiring unacceptably low S abundances (< 0.5 solar) in order to reproduce the data.

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