5 The physical conditions

5.1 General considerations

We can obtain the $T{\rm e}$ radial profile from the diagnostic line ratios of ions in p² or p⁴ configurations (like [O III] and [N II]), and the $N{\rm e}$ radial distribution from both the diagnostics of ions in p³ configuration (like [S II]) and the absolute H$\alpha$ flux. According to Paper IV, the zvpc, which is independent on the expansion velocity field, must be used.

In the specific case of NGC 6818 the large H$\alpha$ broadening (due to thermal motions, fine structure and expansion velocity gradient) prevents the accurate determination of F(H$\alpha$) $_{\rm zvpc}$ (and then of $N{\rm e}$(H$\alpha$)). Thus, in this section $T{\rm e}$[O III], $T{\rm e}$[N II] and $N{\rm e}$[S II] are derived from the corresponding line intensity ratios. Later on (Sect. 6.3) we will illustrate the adopted escamotage providing F(H$\alpha$) $_{\rm zvpc}$ and $N{\rm e}$(H$\alpha$) from the observed radial ionization structure and the assumption $\rm O/H=$constant across the nebula.

5.2 Interstellar absorption

First of all the observed line intensities must be corrected for interstellar absorption according to:

$$\log \frac{{I}(\lambda)_{\rm corr}}{{I}(\lambda)_{\rm obs}}=f_{\lambda}~ $c$ ({\rm H}\beta)$$ (2)

where $f_{\lambda}$ is the interstellar extinction coefficient given by Seaton (1979). The logarithmic extinction at H$\beta$, c(H$\beta$), is normally obtained by comparing the observed Balmer decrement (in particular H$\alpha$/H$\beta$) to the intrinsic value given by Brocklehurst (1971) and Hummer & Storey (1987). The estimates of c(H$\beta$) (from the Balmer ratio) reported in the literature for NGC 6818 span the range 0.25 (Aller & Czyzak 1983, Liu & Danziger 1993) to 0.41 (Collins et al. 1961). Moreover, Tylenda et al. (1992) and Condon et al. (1999) obtained $c(H\beta)=0.33$ and 0.40, respectively, from the radio to H$\beta$ fluxes.

Thanks to the excellent spatial and spectral accuracies achieved by the superposition technique used ($\pm$0.15 arcsec and $\pm$1.0 km s^-1, respectively) we can extend the H$\alpha$/H$\beta$ analysis to the whole spectral image, as recently introduced (Paper IV) in the study of NGC 6565, a compact, dust embedded PN exhibiting a complex c(H$\beta$) profile. The results are less dramatic for NGC 6818, since the blurred appearance of both H$\alpha$ and H$\beta$ limits the resolution: besides some indications of a soft decline in the innermost regions (likely caused by a local increase of $T{\rm e}$), the spectral maps appear quite homogeneous at $\rm H\alpha /H\beta=3.73 (\pm 0.06$), corresponding to $c(H\beta)=0.37 (\pm 0.03$) (for the case B of Baker & Menzel 1938, $T{\rm e}=12~000$ K and log  $N{\rm e}= 3.00$; Brocklehurst 1971; Aller 1984; Hummer & Storey 1987).

Figure 4: Position-velocity maps at the nine observed PA for the high (He II, blue), medium ([O III], green) and low ([N II], red) ionization regions of NGC 6818, scaled according to the relation $V{\rm exp}~({\rm km~s}^{-1})=3.5\times r''$. The orientation of these tomographic maps is the same of Fig. 2.

5.3 Te[O III], Te[N II] and Ne[S II]

First we obtain $T{\rm e}$[O III] from $\lambda$5007 Å/$\lambda$4363 Å, the line ratio being almost independent on $N{\rm e}$ for $N{\rm e} <10^4$ cm^-3. $N{\rm e}$[S II] is then derived from $\lambda$6717 Å/$\lambda$6731 Å (using $T{\rm e}$[O III] to take into account the weak dependence of the ratio on $T{\rm e}$). Last, $T{\rm e}$[N II] comes from $\lambda$6584 Å/$\lambda$5755 Å (adopting $N{\rm e}$[S II] for its weak dependence on the electron density).

Although the resulting $T{\rm e}$[O III], $T{\rm e}$[N II] and $N{\rm e}$[S II] profiles rapidly change with PA, as expected of the chaotic structure of NGC 6818, nevertheless there are some common features. In order to highlight both the differences and the analogies we have selected two PA close to the apparent major axis (PA $=10\hbox {$^\circ $ }$ and PA $=30\hbox{$^\circ$ }$) and two PA close to the apparent minor axis (PA $=90\hbox {$^\circ $ }$ and PA $=110\hbox{$^\circ$ }$) as representative of the whole nebular phenomenology.

The results are shown in Fig. 5; their main limitation is evident: due to the weakness of the [N II] auroral line and of the [S II] doublet, $T{\rm e}$[N II] and $N{\rm e}$[S II] can be obtained only at the corresponding intensity peaks (in the best case).

$T{\rm e}$[O III] presents a well-defined radial profile common to all the PA: it is $\ge$15 000 K in the tenuous, innermost regions, it gradually decreases outward down to 12 000-12 500 K in the densest layers, and later it remains more or less constant (the last statement is weakened by the auroral line faintness). Such a radial trend is in quantitative agreement with the results by Rubin et al. (1998), based on $\lambda$4363 Å and $\lambda$5007 Å HST/WFPC2 imagery. The presence of a $T{\rm e}$ gradient across the nebula is also suggested by Hyung et al. (1999). Previous ground-based $T{\rm e}$[O III] determinations are mean values for the brightest (i.e. densest) regions and span the range 11400 K (de Freitas Pacheco et al. 1991) to 12 770 K (Mathis et al. 1998).

In Fig. 5 $T{\rm e}$[N II] refers to the intensity peaks of the low ionization regions and is systematically below $T{\rm e}$[O III], in agreement with both the previous reports for NGC 6818 (9500 K, Hyung et al. 1999, to 11 280 K, McKenna et al. 1996) and the general results for PNe (see Aller 1990; Gruenwald & Viegas 1995; Mathis et al. 1998). More $T{\rm e}$ values for NGC 6818 are: 11 500 K (C III], Kaler 1986), 12 700 K and 13 800 K ([Ne V] and [O IV], respectively, Rowlands et al. 1989), 14 700 K (Balmer discontinuity, Liu & Danziger 1993), 12 500 K (C III], Mathis et al. 1998), 13 000 K ([Cl IV], Hyung et al. 1999) and 19 000 K ([O II], Keenan et al. 1999).

In summary, the mean kinetic energy of the free electrons in NGC 6818 is quite large. This on the one hand explains both the H$\alpha$ broadening and the strength of the forbidden lines (in particular, $\lambda$5007 Å of [O III]), on the other hand is indicative of a very hot central star ( $T_*\ge 150~000$ K, also supported by the presence of high excitation emissions, up to [Ne V], IP =97.1 eV; see Table 1).

Concerning the [S II] electron densities in the zvpc (Fig. 5), they are limited to the brightest parts of the external, low ionization layers and show peaks up to $2000(\pm 200$) cm^-3. Previously Hyung et al. (1999) obtained $N{\rm e} \rm [S~II]\simeq 2000$ cm^-3 (they also report $N{\rm e}\rm [S~II]\simeq 3000$ cm^-3 from the improved calculations for 3 equivalent p-electrons by Keenan et al. 1996).

Besides the zvpc, we have extended the $\lambda$6717 Å/$\lambda$6731 Å analysis to the prominent knots of the entire [S II] spectral images. The resulting $N{\rm e}$[S II] values span the range $1500(\pm 200$) to $2800(\pm 200$) cm^-3, the equatorial moustaches being the densest regions of NGC 6818.

In the next section we will derive the local filling factor, $\epsilon_{\rm l}$, by combining $N{\rm e}$[S II] and $N{\rm e}$(H $\alpha )_{\rm zvpc}$, since $N{\rm e}$[S II]$\times$ $\epsilon_{\rm l}^{0.5} \ \simeq\ N{\rm e}$(H $\alpha )_{\rm zvpc}$ (Aller 1984; Osterbrock 1989).

Figure 5: The diagnostics radial profile 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). Left ordinate scale: $T{\rm e}$[O III] (continuous line) and $T{\rm e}$[N II] (squares). Right ordinate scale: $N{\rm e}$[S II] (circles). $T{\rm e}$[N II] is lacking at PA $=90\hbox {$^\circ $ }$ and 110 $\hbox {$^\circ $ }$, because of the auroral line weakness.