6 The radial ionization structure

6.1 General considerations

All the main ionic species are present in the echellograms, with the usual, dreadful handicap for hydrogen: due to the large H$\alpha$ broadening, the deconvolution for instrumental resolution plus thermal motions plus fine structure appears inadequate, and the detailed F(H $\alpha )_{\rm zvpc}$ profile cannot be determined. Moreover:

- the principal O⁺ emissions ($\lambda$3726 $\AA\/$ and $\lambda$3729 Å) fall outside our spectral range. We have considered the much weaker $\lambda$7319.87 Å  the strongest line of the red O⁺ quartet;

- $\lambda$3967 Å of [Ne III] being at the extreme blue edge of the frame, the quantitative Ne⁺⁺ analysis appears quite uncertain.

In absence of H$\alpha$ we are forced to adopt $\lambda$5007 Å as reference line, thus obtaining the radial ionization structure relative to O⁺⁺ according to:

$$\frac{{\rm X}^{+a}}{{\rm O}^{++}} = \frac{F(\lambda({\rm X}^{... ...pc}}{F(\lambda 5007~{\rm\AA})_{\rm zvpc}} f(T{\rm e},N{\rm e})$$ (3)

where a=b for the forbidden lines, and a=b+1 for the recombination ones. Note that the solution of the equations of statistical equilibrium requires the detailed knowledge of the plasma diagnostics, whereas $N{\rm e}$[S II] has been derived only at the intensity peaks of the outermost, low ionization layers. Luckily, in the $N{\rm e}$ range here considered $f(T{\rm e},N{\rm e})$ reduces to  $f(T{\rm e})$ for most ionic species. Moreover all the emissions, with the exception of $\lambda$5876 Å of He I and $\lambda$4686 Å of He II, are forbidden lines, whose emissivity is a direct function of $T{\rm e}$. Thus, in first approximation we can put $N{\rm e}=1.5\times 10^3$ cm^-3, constant across the nebula; this introduces uncertainties up to $\pm$10$\%$ (for O⁺/O⁺⁺ and Ar⁺⁴/O⁺⁺), which do not modify the results here obtained.

The radial profiles of $\frac{{\rm X}^{+i}}{{\rm O}^{++}}$ at the four selected PA of NGC 6818, shown in Fig. 6, contain a number of interesting features:
- the weakness of the low excitation emissions in PA $=10\hbox {$^\circ $ }$ northern sector, PA $=90\hbox {$^\circ $ }$ both sectors and PA $=110\hbox{$^\circ$ }$ both sectors indicates that the nebula is optically thin in these directions. It is almost thick in PA $=10\hbox {$^\circ $ }$ southern sector and PA $=30\hbox{$^\circ$ }$ both sectors;
- as expected, He⁺⁺/O⁺⁺ decreases outward and He⁺/ O⁺⁺ increases. They cross for $\rm He^{++}/O^{++}=He^+/O^{++}\simeq 120$, implying that $\rm He_{\rm tot}/O^{++}\simeq 240$. Since in these internal regions $\rm O_{\rm tot}\simeq 1.30\times O^{++}$ (see Eqs. (10) and (11) in Sect. 6.3) we derive $\rm He/O\simeq 185$;
- from similar considerations we infer $\rm N/O\simeq 0.23$, $\rm N{\rm e}/O\simeq 0.15$, $\rm S/O\simeq 0.012$ and $\rm Ar/O\simeq 0.008$.

These chemical abundances (relative to oxygen) must be compared with the corresponding values (relative to hydrogen) obtained from the conventional method, as illustrated in the next Section.

6.2 Total chemical abundances

According to the critical analysis by Alexander & Balick (1997) we consider the total line fluxes (i.e. integrated over the whole spatial profile and the expansion velocity field). The resulting ionic abundances must be multiplied for the corresponding ICFs, the correcting factors for the unobserved ionic stages. These were obtained both empirically (Barker 1983, 1986) and from interpolation of theoretical nebular models (Shields et al. 1981; Aller & Czyzak 1983; Aller 1984; Osterbrock 1989).

The final mean chemical abundances of NGC 6818, presented in Table 3 (last column), are in reasonable agreement with the previous estimates reported in the literature (also listed in the Table), and in excellent agreement with the indications of Sect. 6.1.

Figure 6: The radial ionization structure (relative to O++) at PA $=10\hbox {$^\circ $ }$ and 30 $\hbox {$^\circ $ }$ (close to the major axis of NGC 6818), and at PA $=90\hbox {$^\circ $ }$ and 110 $\hbox {$^\circ $ }$ (close to the minor axis). The orientation is as in Fig. 5.

6.3 F(H$\alpha$)_zvpc and Ne(H$\alpha$)

As emphasized in Paper IV, the H$\alpha$ flux distribution in the zvpc, F(H$\alpha$) $_{\rm zvpc}$, is the fundamental parameter linking $N{\rm e}$ with both the spatial and the kinematical properties of the expanding plasma through the relation:

$$N{\rm e} = \frac{1.19\times 10^9}{T{\rm e}^{-0.47}} \times (... ...m zvpc}}{\epsilon_{\rm l} \times r_{\rm cspl} \times D})^{1/2}$$ (4)

where:
- D is the nebular distance;
- $r_{\rm cspl}$ is the angular radius of the cspl (i.e. the nebular size in the radial direction);
- $\epsilon_{\rm l}$ is the "local filling factor'', representing the fraction of the local volume actually filled by matter with density $N{\rm e}$.

In order to recover F(H$\alpha$) $_{\rm zvpc}$ we start assuming $\rm O/H= 5.5\times 10^{-4}$, constant across the nebula. At each radial position:

$$\frac{\rm O}{\rm H} =\frac{\sum_{i=0}^8 {\rm O}^{0+i}}{{\rm H}^0 + {\rm H}^+}\cdot$$ (5)

It can be written in the form:

$$\frac{{\rm O}}{{\rm H}} =\frac{{\rm O}^{++}}{{\rm H}^+}\times icf({\rm O}^{++}).$$ (6)

That is:

$$\frac{{\rm O}}{{\rm H}} =\frac{F(\lambda 5007~{\rm\AA})_{\rm ... ...\rm zvpc}}\times f(T{\rm e},N{\rm e})\times icf({\rm O}^{++}).$$ (7)

Thus obtaining:

$$F({\rm H}\alpha)_{\rm zvpc} = 1.9\times10^3 F(\lambda 5007~{\... ...{\rm zvpc}\times f(T{\rm e},N{\rm e})\times icf({\rm O}^{++}).$$ (8)

Also in this case $f(T{\rm e},N{\rm e})$ essentially reduces to $f(T{\rm e})$, with opposite trends for the emissivity of the forbidden and the recombination line.

Equation (8) provides the H$\alpha$ flux distribution in the zvpc (and $N{\rm e}$(H$\alpha$) through Eq. (4)) once the ionization correcting factor $icf({\rm O}^{++})$ is known. The complex structure of NGC 6818 implies that $icf({\rm O}^{++})$ strongly changes across the nebula. In the innermost regions we have: O⁰/O⁺⁺<<1, O⁺/O⁺⁺<<1 and H⁰/H⁺<<1. Thus $icf({\rm O}^{++})$ becomes:

$$icf({\rm O}^{++})_{\rm inner} = 1 + \frac{\sum_{i=3}^8 {\rm O}^{+i}}{{\rm O}^{++}}\cdot$$ (9)

According to Seaton (1968), $\frac{\sum_{i=3}^8 {\rm O}^{+i}}{{\rm O}^{++}}$ can be derived from the ionization structure of helium thanks to the closeness of the O⁺⁺ and He⁺ ionization potentials (54.9 and 54.4 eV, respectively). To this end we have performed a number of photo-ionization simulations (see also Alexander & Balick 1997) obtaining the fairly good relation:

$$\frac{\sum_{i=3}^8 {\rm O}^{+i}}{{\rm O}^{++}}\simeq 0.30\times\frac{{\rm He}^{++}}{{\rm He}^+} \cdot$$ (10)

Therefore we will adopt:

$$icf({\rm O}^{++})_{\rm inner} = 1+ 0.30\times\frac{{\rm He}^{++}}{{\rm He}^+} \cdot$$ (11)

Concerning the outermost nebula, the contribution of $\sum_{i=3}^8 {\rm O}^{+i}$ can be neglected. Moreover in these regions we have O⁰/O⁺<1 (see Fig. 6), implying a low efficiency of the charge-exchange reaction $\rm O^+ + H^0\getsto O^0+ H^+$ (Williams 1973; Aller 1984; Osterbrock 1989), i.e. $\rm H^0<<H^+$. Thus:

$$icf({\rm O}^{++})_{\rm outer} = 1 + \frac{{\rm O}^0}{{\rm O}^{++}} + \frac{{\rm O}^+}{{\rm O}^{++}} \cdot$$ (12)

All this is summarized in Fig. 7, showing the $N{\rm e}$ radial profile (at the four selected PA of NGC 6818) obtained from F(H$\alpha$) $_{\rm zvpc}$ for some representative values of $\epsilon _{\rm l}\times r_{\rm cspl}\times D$ (in arcsec kpc), superimposed to $N{\rm e}$[S II] (taken from Fig. 5).

Figure 7 evidences the basic advantage of using $F(\rm H\alpha)_{\rm zvpc}$ in the determination of the electron density radial distribution: $N{\rm e}$(H$\alpha$) extends all over the nebular image, whereas $N{\rm e}$[S II] is limited to the peaks of the low excitation regions.

 

Table 3: Total chemical abundances (relative to hydrogen).

Element	Aller & Czyzak	de Freitas Pacheco et al.	Liu & Danziger	Hyung et al.	This paper
	(1983)	(1991)	(1993)	(1999)
He	0.107	0.126	0.114	0.105	0.106($\pm$0.003)
C	4.47$\times$10-4	-	-	8.0$\times$10-4	-
N	1.41$\times$10-4	1.1$\times$10-4	-	4.0$\times$10-4	1.4($\pm$0.2)$\times$10-4
O	5.50$\times$10-4	5.25$\times$10-4	6.37$\times$10-4	7.0$\times$10-4	5.5($\pm$0.5)$\times$10-4
Ne	1.23$\times$10-4	-	-	1.0$\times$10-4	9.0($\pm$2.0)$\times$10-5
Na	3.09$\times$10-6	-	-	3.0$\times$10-6	-
Mg	-	-	-	3.0$\times$10-5	-
Si	-	-	-	9.0$\times$10-6	-
S	8.9$\times$10-6	1.2$\times$10-5	-	7.0$\times$10-6	6.2($\pm$1.4)$\times$10-6
Cl	1.9$\times$10-7	-	-	3.0$\times$10-7	-
Ar	3.8$\times$10-6	-	-	4.0$\times$10-6	4.0($\pm$0.8)$\times$10-6
K	1.02$\times$10-7	-	-	2.0$\times$10-7	-
Ca	1.15$\times$10-7	-	-	1.5$\times$10-7	-

In detail:
- at PA $=10\hbox {$^\circ $ }$ (along the apparent major axis), $N{\rm e}$(H$\alpha$) presents a single, broad and asymmetric (i.e. steeper outwards) bell-shaped profile;
- the double-peak structure is very subtle at PA $=30\hbox{$^\circ$ }$ (close to the apparent major axis), the inner peaks being predominant;
- at both PA $=90\hbox {$^\circ $ }$ and 110 $\hbox {$^\circ $ }$ (close to the apparent minor axis) the two-shell distribution clearly appears; the $N{\rm e}$(H$\alpha$) top corresponds to the inner peaks at PA $=110\hbox{$^\circ$ }$, whereas at PA $=90\hbox {$^\circ $ }$ the peaks are almost equivalent;
- an outward, low density tail is present at all directions, extending up to about 15 arcsec from the star.

Note in Fig. 7 the match between $N{\rm e}$(H$\alpha$) and $N{\rm e}$[S II] for $\epsilon_{\rm l}\times r_{\rm cspl}\times D\simeq$ 9.5($\pm$1) arcsec kpc, implying that $\epsilon_{\rm l}\times D\simeq 0.88 (\pm 0.10$) kpc (see Table 2 and Sect. 4). Although the assumption $\epsilon _{\rm l}=1$ provides a lower limit of 0.9 kpc for the nebular distance, we can no longer delay a better quantification of this fundamental parameter.

Figure 7: The $N{\rm e}$(H $\alpha )_{\rm zvpc}$ radial profile at the four selected PA of NGC 6818 for some representative values of $\epsilon _{\rm l}\times r_{\rm cspl}\times D$ (dotted line = 5 arcsec kpc, short-dashed line = 10 arcsec kpc, long-dashed line = 15 arcsec kpc), superimposed to $N{\rm e}$[S II] (thick line; same as Fig. 5). The orientation is as in Figs. 5 and 6.