9 The photo-ionization model

9.1 General approach

In this section the photo-ionization code CLOUDY (Ferland et al. 1998) is applied to a PN having the same distance, matter distribution and chemical composition of NGC 6818.

We immediately point out that CLOUDY is a "steady-state'' model (a constant UV flux hits the nebula), whereas we argued that the ionization and thermal structure of NGC 6818 are out of equilibrium, because of the fast luminosity drop of the star. A similar, but less dramatic situation occurs in the recombining PN NGC 6565, which is in quasi-equilibrium, since the stellar evolution slows down and the luminosity gradient rapidly decreases while approaching the white dwarf domain (Paper IV).

In absence of an "evolving'' photo-ionization code, which takes into account the gas reactions to a fast changing UV flux, at the moment we apply CLOUDY to the present nebula, (NGC 6818)(t₀), as well to (NGC 6818)(t₀-100 yr) and (NGC 6818)(t₀+100 yr). Both the past and the future "snapshots'' come from a homologous expansion (i.e. $r\propto t$ and N(H) $\propto t^{-3}$), being the dynamical time t₀= 2300 yr and $\Delta t<<t_0$. Moreover, the evolutionary track of the 0.625 $M_\odot$ H-burning post-AGB star by Blöcker (1995) was adopted for the stellar parameters. Last, we have selected the quite thin East sector at PA $=90\hbox {$^\circ $ }$ as "mean'' density radial profile of (NGC 6818)(t₀). The input data are contained in Table 5. The results, shown in Fig. 9, can be synthesized as follows:
- at (t₀-100 yr) (upper row) the intense UV flux of the luminous and hot star completely ionizes the nebula, which appears as a very high excitation object at large $T{\rm e}$, optically thin in all the directions;
- at (t₀) (central row) the stellar luminosity is hardly sufficient for ionizing the gas, and the high excitation nebula is almost thick in some directions, where the low ionization emissions emerge;
- at (t₀+100 yr) (bottom row) the UV flux is far inadequate: the innermost plasma maintains a high degree of excitation, but the external layers are thick, with prominent [O I], [O II], [N II] and [S II] lines. Note that the outermost parts of the "steady'' photo-ionization model are neutral, whereas are recombining in a "true'' evolving nebula (they create a faint halo embedding the main object).

Although these results call for caution due to the photo-ionization model limitations, nevertheless they support the evolutive scenario of the previous sections: NGC 6818 is a PN at the very beginning of the recombination phase.

A direct confirmation is expected of the detailed analysis of the equatorial moustaches, representing the densest and brightest (i.e. thickest) regions of NGC 6818 (Sect. 5.3). To this end we have first de-projected the spectral images through Eq. (1), and then repeated the complete procedure already applied to the zvpc.

9.2 Equatorial moustaches

Figure 9: "Steady'' photo-ionization model of NGC 6818, East sector in PA $=90\hbox {$^\circ $ }$, at three epochs: (t0-100 yr) (upper row), t0 (central row), and (t0+100 yr) (bottom row), t0=2300 yr being the dynamical nebular age. Left panels: radial profiles of the absolute line fluxes. Right panels: the diagnostics; left ordinate scale: $T{\rm e}$ (dotted line); right ordinate scale: $N{\rm e}$ (thick continuous line), N(H $_{\rm tot}$) (dotted-dashed thin line), N(H+) (long-dashed thin line) and N(H0) (short-dashed thin line).

Let's consider the southern, approaching moustache in PA $=10\hbox {$^\circ $ }$. The de-projected radial profiles of the main emissions, the diagnostics and the $\frac{{\rm X}^i}{{\rm O}^{++}}$ ionic abundances are shown in Fig. 10.

We are particularly interested to $\lambda$6300 Å of [O I], marking the neutral (or almost neutral) nebular layers, where the efficiency of the resonant charge-exchange reaction $\rm O^+ + H^0\getsto O^0+ H^+$ largely increases (Williams 1973). Note that on the one hand the [O I] line is very sensitive to the physical conditions, on the other hand no precise information is yet available for the electron temperature of the external, neutral (or almost neutral) gas dominated by the recombination and cooling processes. We arbitrarily adopt $T{\rm e}\rm (almost \ neutral \ gas)=8000$ K, which is lower than $T{\rm e}$(ionized gas), but not enough low to compromise the [O I] emissivity (a choice supported by the large kinetic energy of the free electrons in NGC 6818; see Sect. 5.3).

Moreover, referring to the ionization correcting factor $icf({\rm O}^{++})_{\rm outer}$ of Sect. 6.3, it includes H⁰, whose contribution comes from the equilibrium condition $\rm O^+/O^0 \simeq 0.82\times(H^+/H^0$), valid for $\rm O^0>O^+$, i.e. in the nebular regions affected by the $\rm O^+ + H^0\getsto O^0+ H^+$ reaction (Williams 1973; Stancil et al. 1999).

The observational results of Fig. 10 must be compared with the "steady'' photo-ionization model of the moustache presented in Fig. 11 (the stellar parameters obviously refer to time t₀ of Table 5).

Besides the general features common to all the directions (i.e. the decreasing $T{\rm e}$[O III] radial profile, $T{\rm e}{\rm [N~II]} < T{\rm e}$[O III], ${\rm He}^{++}/{\rm O}^{++}={\rm He}^+/{\rm O}^{++}\simeq 120$, $N{\rm e}({\rm H}\alpha)= {N}{\rm e}$[S II] for ( $\epsilon_{\rm l}\times r_{\rm cspl}\times D)\simeq$ 9.5 arcsec kpc, and so on), the moustache of NGC 6818 (Fig. 10) exhibits a remarkable peculiarity: the external layers are partially neutral, and the outward ionization decline is very smooth, as expected of a recombining region. On the contrary, the "steady'' photo-ionization model in Fig. 11 predicts an abrupt ionization fall (to be noticed: the photo-ionization model at (t₀-100 yr), not shown here, indicates that a century ago the moustache was optically thin to the UV stellar flux).

The $N{\rm e}$ depletion rate for recombination is given by:

$${\rm d}N{\rm e}/{\rm d}t = -\alpha_{\rm B}\times N{\rm e}\times N{\rm (H^+) }$$ (13)

where $\alpha_{\rm B}$ is the recombination coefficient (Storey & Hummer 1995). Integrating Eq. (13) and assuming $N{\rm e}=1.15\times N({\rm H}^+$), we obtain:

$$t =1.15\times\frac{N{\rm e}(t_0-t)-N{\rm e}(t_0)}{\alpha_{\rm B}\times N{\rm e}(t_0-t)\times N{\rm e}(t_0)}$$ (14)

which provides the time t elapsed from the beginning of the recombination once are known $N{\rm e}(t_0)$, the present electron density, and $N{\rm e}(t_0-t)$, the electron density at the start of the process.

The application of Eq. (14) to the external parts of the "true'' moustache (Fig. 10) furnishes t=30-60 yr (t = 60-120 yr and 20-40 yr for $T{\rm e}$(almost neutral gas) = 5000 K and 12 000 K, respectively). In spite of the heavy assumptions, this agrees with all the previous evidences suggesting that the recombination phase has just begun in NGC 6818.