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Astron. Astrophys. 327, 909-920 (1997)

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2. The model

2.1. The computer code

The multipurpose code MAPPINGS IC was used to compute the photoionization models. As noted in Ferruit et al. (1997), the new version IC maintained by one of us (LB) now includes Fe and allows calculations of all ionization stages up to bare nuclei. The atomic data is taken from a compilation of Ralph Sutherland (see Appendix in Ferruit et al. 1997). The uncertainties in collision strengths of many coronal lines are often large (see review by Oliva 1997) and may dominate the errors in the predicted line strengths.

2.2. Two powerlaws to represent the EUV and the X-ray domaines

Given the success of Paper I in accounting for a broad range of line excitation with a simple power-law with index [FORMULA] ([FORMULA]), we adopt a similar energy distribution. We recall that our choice of [FORMULA] was motivated by the desire to produce matter-bounded (MB) clouds with high enough temperature (T [FORMULA]), as indicated by the ratio [O III ] [FORMULA] 4363/[O III ] [FORMULA] 5007 but with the index close to the canonical values proposed in the literature (i.e., [FORMULA] or -1.5; e.g., Ferland & Osterbrock 1986). The energy distribution starts at 0.1 eV. Above 500 eV we introduce a flatter component with canonical index [FORMULA]. We impose a high energy cut-off to this X-ray component at 100 keV (see Mathews & Ferland 1987). The resultant ionizing energy distribution is shown in Fig. 1 (dash line). It is different from that adopted by Mo96. In particular, it does not contain the pronounced UV bump at 70 eV (in a plot of [FORMULA] versus [FORMULA]) invoked by Mo96. By joining the UV and X-ray power-laws at 4 000 eV instead of 500 eV (so as to produce a smaller [FORMULA]), we have verified that a weaker X-ray component produces very similar results.

[FIGURE] Fig. 1. Ionizing energy distributions, plotted as [FORMULA] versus [FORMULA]. The X-ray cut-off is at 100 keV. The long dashed line represents the direct nuclear source radiation impinging on the MB component. The solid line represents the spectrum escaping from the back of the MB slab (which contains 65% of the original number of ionizing photons). This radiation (after further geometrical dilution) impinges on the IB component. The dotted line represents the ionizing distribution escaping the back of the IB slab (only 0.9% of the number of ionizing photons but 60% of the ionizing energy flux). The energy between [FORMULA] keV and the cut-off at 100 keV escapes the IB slab unabsorbed.

2.3. Gas pressure vs radiation pressure

The ionization parameter [FORMULA] is customarily defined as the ratio between the density of impinging ionizing photons and the gas density at the face of the cloud:


where [FORMULA] is the monochromatic energy flux impinging on the cloud, [FORMULA] the Lyman frequency, q0 the number of ionizing photons incident on the slab per cm [FORMULA] per second, c the speed of light and [FORMULA] the total gas density at the irradiated surface of the slab (all quantities relevant to the irradiated surface carry 0 as a superscript or subscript).

A different definition of the ionization parameter is the following (Krolik et al. 1981):


where T0 is the temperature at the irradiated surface, Fion is the incident flux of ionizing radiation and k is Boltzmann's constant. Since F [FORMULA] is the pressure of the ionizing radiation (assuming the radiation is confined to a small solid angle and is normally incident on the cloud), [FORMULA] gives us the ratio of the pressure of ionizing radiation to the gas pressure at the irradiated face : [FORMULA] (cf. Krolik et al. 1981). Radiation pressure exerts a force only if the ionizing radiation is absorbed. Even in the case of an ionization-bounded slab, which would absorb, say, 99% of the number of ionizing photons (see dotted line in Fig. 1), of order 60% of the ionizing flux (i.e. that between [FORMULA] keV and the cut-off at 100 keV) escapes unabsorbed and therefore does not exert any force on the gas. For the matter-bounded slab discussed below, as much as 75% of the ionizing flux will not be absorbed. Therefore the actual radiation pressure absorbed by the matter-bounded slab is reduced to [FORMULA]. This implies that for [FORMULA] radiation pressure exceeds gas pressure. To determine the corresponding value of [FORMULA], we plot in Fig. 2 the temperature of a thin gas layer photoionized by the distribution described in Sect. 2.2 (Fig. 1, dashed line) as a function of both ionization parameters - [FORMULA] (solid line, to be read along the top axis) and [FORMULA] (dotted line, to be read along the bottom axis). We see that [FORMULA] is 0.25 when [FORMULA] is 10. In Sect. 2.5 we propose a density prescription in which the strong force exerted by radiation sets up an internal density gradient. The suggestion that clouds might be compressed by radiation pressure at high U was first made by Davidson (1972).

[FIGURE] Fig. 2. Equilibrium gas temperature as a function of the pressure ionization parameter [FORMULA] (solid line, top axis) and as a function of the number density ionization parameter [FORMULA] (dotted line, bottom axis). The position of the three models (L, M, H) of Table 1 which have [FORMULA], 0.05 and 0.5, respectively, appear as stars on both curves.

Because it corresponds to an integral of [FORMULA], the detailed shape of the [FORMULA] curve up to [FORMULA] is quite insentive to the high energy cut-off and is therefore more appropriate for determining the point where radiation pressure exceeds gas pressure (i.e. at [FORMULA] for the matter-bounded slab). At the highest end of [FORMULA] and [FORMULA] in Fig. 2, both curves join at the corresponding Compton temperature which is, however, a strong function of the X-ray cut-off. The sharp structure seen in the [FORMULA] curve around T [FORMULA] was not found in earlier work (e.g. Krolik et al. 1981; Mathews & Ferland 1987). A similar feature is nevertheless present in the more recent papers of Komossa & Fink (1997a, b) who used the code Cloudy.

2.4. Matter-bounded vs ionization-bounded clouds

The excitation mechanism of the NLR and the ENLR is generally agreed to be photoionization. However, despite broad success in fitting the strongest optical lines, there are still significant problems with photoionization models. For instance, one has the temperature problem, in which the temperature sensitive ratio [O III ] [FORMULA] 4363/[O III ] [FORMULA] 5007 is predicted to be smaller than is observed (Tadhunter et al. 1989; Paper I; Wilson, Binette & Storchi-Bergmann 1997). Another significant problem is that of the high excitation lines which are predicted to be much weaker than observed (e.g. [Ne V ] /H [FORMULA]) (cf. Stasi[FORMULA]ska 1984; Viegas & Gruenwald 1988; Paper I). Shock excitation or a mixture of shock and photoionization have been proposed to resolve these discrepancies (cf., Contini & Viegas-Aldrovandi 1989, Dopita & Sutherland 1996).

In Paper I, we have shown how inclusion of photoionized matter-bounded (MB) clouds of sufficiently high excitation ([FORMULA] =0.04) to reproduce the high excitation [Ne V ] [FORMULA] 3426 lines can solve the above problems. These MB clouds were considered to be sufficiently thick to reprocess [FORMULA] % of the ionizing photons to which they were exposed. The low excitation lines were accounted for by a population of low ionization parameter, ionization-bounded (IB) clouds exposed to the ionizing radiation spectrum which exits the MB clouds (solid line, Fig. 1). Varying the relative proportion of the two types of clouds was found to have a similar effect on the combined emission line spectrum to varying [FORMULA] in a traditional ionization parameter sequence. We adopt here a similar scheme by considering that all the coronal lines are produced within MB clouds, some of which must evidently be much more highly excited than the clouds invoked in Paper I. The extensive set of coronal lines measured by Mo96 will be used as constraints on the physical parameters of the matter-bounded photoionized gas. The IB clouds, which account for the low excitation lines, will be discussed later in Sect. 5.

2.5. Strong density gradient as a result of radiation pressure

Binette & Raga (1989) computed the emission-line spectra emanating from one-dimensional clouds with fixed internal density gradients of arbitrary steepness. They showed that the [FORMULA] case (density decreasing outward from the center of the cloud as [FORMULA]) was particularly interesting since the overall excitation of the spectrum (i.e. the emissivity averaged [FORMULA]) and the line ratios tended asymptotically towards constant values, independent of the strength of the ionization radiation. This surprising result originates from the fact that a model with [FORMULA] generates a `frozen-in' excitation structure provided [FORMULA] is sufficiently large. Further increases in q0 produce over-ionized zones with low emissivity at the face of the cloud; such zones do not contribute to the lines of interest. Although they did not explore how such gradients might come about, we propose here that they result from radiation pressure. In effect, for [FORMULA], the radiative force is strong enough to set up a density gradient within the cloud, which approaches the case [FORMULA] as demonstrated in Appendix.

Sect. A.3 in Appendix describes our implementation of the effects of radiation pressure in MAPPINGS IC. Fig. 3 shows the results of calculations with this code which confirm the asymptotic behaviour of the line ratios by comparing successive calculations with increasing ionizing photon number, q0 ([FORMULA] being kept constant). For instance, the [O III ] [FORMULA] 5007, [Ne VI ]7.66 [FORMULA] m and all the C, Si lines tend at very large [FORMULA] towards constant values relative to H [FORMULA]. The notable exceptions are, of course, lines with low critical densities which are progressively suppressed as the increasing radiation compresses further the slab, pushing the relevant ionized zones deeper into the cloud and causing [FORMULA] to increase with increasing [FORMULA]. Alternatively, if we generate a sequence of models in which the increase in [FORMULA] is obtained by progressively reducing the density [FORMULA] (with q0 held constant), as shown in Fig. 4, then the mean density [FORMULA] (dotted line) as well as all line ratios tend towards a constant value at very large [FORMULA]. In both figures, the mean ionization parameter [FORMULA], as defined in the Appendix (Sect. A.2), tends toward a constant value.

[FIGURE] Fig. 3. High excitation line ratios relative to H [FORMULA] emitted by a matter-bounded slab which reproceses 35% of the incident ionizing radiation. The calculations were performed for a range of ionization parameter ([FORMULA]) obtained by increasing q0 and taking into account the variation of density with distance into the cloud resulting from radiation pressure (Sect. 2.5). The lines are labelled with their wavelength in Å or [FORMULA] m. a oxygen (solid line), iron (dotted-dashed line) and carbon (long dash line). The short dash line indicates the relative importance of the radiative pressure exerted within the MB slab. b neon (long dash line) and silicon (solid line). The dotted line represents the average density relative to the density [FORMULA] ([FORMULA] cm-3) at the irradiated surface of the slab. For [FORMULA] [FORMULA], line ratios of density stratified models tend asymptotically towards constant values except for the density sensitive lines like [O IV ]25.90 [FORMULA] m, [Ne V ]24.31 [FORMULA] m and [Ne V ]14.32 [FORMULA] m.
[FIGURE] Fig. 4. Same as in Fig. 3 except that the increase in [FORMULA] was obtained by decreasing [FORMULA]. For [FORMULA] [FORMULA], the mean density [FORMULA] /1000 cm-3 and all line ratios tend asymptotically towards a constant value.

The behaviour of the average density [FORMULA] as one increases q0 in Fig. 3b illustrates the magnitude of the density gradient across the MB structure : first, there is a gradual increase of [FORMULA] with increasing [FORMULA] (for [FORMULA]) then a more rapid change (for [FORMULA]). When [FORMULA] and [FORMULA] become proportional, [FORMULA] is constant (see eqs. A.12-A.14) indicating that the excitation structure relative to the Strömgren boundary is `frozen in'. In Fig. 3a and 4a, the short dash line describes the relative importance of the radiation pressure absorbed within the MB slab, which exceeds [FORMULA] for [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998