10 The 3-D morpho-kinematical structure

10.1 General

The reconstruction of the gas distribution in the nebular slices covered by the slit was introduced in Papers I and II. In the case of NGC 6818 we have selected $\lambda$4686 Å of He II, $\lambda$5007 Å of [O III] and $\lambda$6584 Å of [N II] as representative of the high, mean and low ionization regions, respectively. Note that H$\alpha$, the marker of the whole ionized gas distribution, cannot be utilized because of the blurred appearance.

The spectral images of the forbidden lines have been de-convolved for seeing, spectral resolution and thermal motions, while also fine structure has been taken into account for $\lambda$4686 Å of He II. They all were de-projected through Eq. (1), and assembled by means of the 3-D rendering procedure described in Paper III.

In order to reproduce the spatial structure on the paper we adopt the usual method: a series of opaque reconstructions of the nebula seen from different directions, separated by 15 $\hbox {$^\circ $ }$. Each couple forms a stereoscopic pair providing a 3-D view of NGC 6818.

The novelties are represented by the multicolor projection and the movies.

10.2 Opaque reconstruction

For reasons of space we only present the opaque reconstruction in He II, [O III] and [N II] for a rotation of 180 $\hbox {$^\circ $ }$ around the East-West axis (close to the minor axis). This is shown in Figs. 12 to 16, where the upper-right panel corresponds to the nebula seen from the Earth (West is up and North to the left, to allow the reader the stereo-view).

The high excitation layers of NGC 6818 (Fig. 12) form an inhomogeneous shell seen almost equatorial on, open-ended at North and South; it merges in a closed ellipsoid at lower $\lambda$4686 Å fluxes.

The opaque reconstruction in [O III] is given for two absolute flux cuts: log $E(\lambda 5007~{\rm\AA})=-17.32$ erg s^-1 cm^-3 (Fig. 13), and -17.75 erg s^-1 cm^-3 (Fig. 14). Since ${\rm O/H}=5.5\times 10^{-4}\simeq {\rm O}^{++}/{\rm H}^+$ and $N{\rm e}=1.15\times N{\rm (H}^+$), they correspond to $N{\rm e}\simeq 1500$ cm^-3 (Fig. 13), and $N{\rm e} \simeq 900$ cm^-3 (Fig. 14) (for $T{\rm e}=12~000$ K and $\epsilon _{\rm l}=1$). The [O III] high cut represents the densest regions of the inner shell, mainly constituted by the equatorial moustaches and by an extended cup in the southern part, whereas the external, inhomogeneous shell appears in the low cut frames, characterized by the large hole at North (along the major axis).

 

Table 5: The CLOUDY photo-ionization code: input parameters of the model nebula.

Radial density profile	right panels of Fig. 9 (cf. Sects. 6.3 and 9)
Chemical abundances:
C, Na, Mg, Si, Cl, K, Ca	Hyung et al. (1999)
He, N, O, Ne, S, Ar	this paper
other elements	PN (CLOUDY default)
Dust	PN (CLOUDY default)
Filling factor	0.5
	at time (t0-100 yr): blackbody with T*=170 000 K and $\log L_*/L_\odot= 3.45$ (*)
Exciting star	at time (t0): blackbody with T*=160 000 K and $\log L_*/L_\odot= 3.10$ (see Sect. 8)
	at time (t0+100 yr): blackbody with T*=140 000 K and $\log L_*/L_\odot= 2.60$ (*)

(*) according to Blöcker (1995).

The brightest and the faintest [N II] regions refer to $\lambda$6584 Å at the cuts $\log E(\lambda6584~{\rm\AA})=-18.20$ erg s^-1 cm^-3 (Fig. 15), and -18.80 erg s^-1 cm^-3 (Fig. 16). The [N II] (high cut) distribution mimics the corresponding [O III] one, whereas at lower fluxes the outermost, knotty structure emerges. Note the general weakness of the low ionization layers for $-135\hbox{$^\circ$ }<\psi<-45\hbox{$^\circ$ }$.

These stereo-reconstructions confirm that NGC 6818 consists of a double shell projected almost equatorial on; the external one is spherical ( $r\simeq 0.090$ pc), faint and patchy. It circumscribes a dense and inhomogeneous tri-axial ellipsoid ( $a/2\simeq 0.077$ pc, $a/b\simeq 1.25$, $b/c \simeq 1.15$) characterized by a large hole along the major axis and a couple of thick equatorial regions (the moustaches).

10.3 Multicolor projection

A representative sample of the different morphologies assumed by NGC 6818 when changing the line of view is given in Figs. 17 and 18. They show the assembled, multicolor projection at high (He II, blue), medium ([O III], green) and low ([N II], red) ionization for a rotation around the N-S axis (close to the major axis; Fig. 17), and around the E-W axis (close to the minor axis; Fig. 18). Also in this case the upper-right panel corresponds to the nebula seen from the Earth (North is up and East to the left), to be compared with Fig. 1, and with the HST multicolor image by A. Hajian & Y. Terzian at http://ad.usno.navy.mil/pne/gallery.html.

These "almost'' true color reproductions highlight the variety of looks exhibited by the Little Gem: roundish to elliptical, to quasi-bipolar (according to the current morphological classifications; Greig 1972; Stanghellini et al. 1993; Corradi & Schwarz 1995; Gorny et al. 1997). NGC 6818 resembles NGC 6153, Hu 1-1 and NGC 4071 when seen from (0,60), K 3-57 and M 2-51 from (0,90), A 70 from (90,0), IC 4663 from (0,30) and NGC 7354 from (150,0) (see the imagery catalogues of PNe by Acker et al. 1992; Schwarz et al. 1992; Manchado et al. 1996; Gorny et al. 1999).

10.4 Movies

A number of limitations are implicit when showing the 3-D structure on the paper. They are both objective (space, choice of the rotation axes, the nebular parameters, the cuts etc.), and subjective (difficulty in the stereo view). In order to overcome all these handicaps, we have decided to introduce a series of movies as integral and functional part of each PN analysis.

Such an "in fieri'' film library, providing the multicolor projection, the opaque recovery in different ions and at various cuts, some slices and radial profiles etc. (suggestions are welcome), can be found at http://web.pd. astro.it/sabbadin (where the acronym "sabbadin'' stands for "stratigraphy and best boundary analytic determination in nebulae'', sic!).

Figure 10: The observed radial properties of the southern, approaching moustache of NGC 6818 in PA $=10\hbox {$^\circ $ }$. Top panel: the absolute flux (in erg cm-3 s-1) of the main emissions; same symbols as Figs. 6 and 9. Middle panel: the diagnostics; left ordinate scale: dotted line = $T{\rm e}$[O III], short-dashed line = $T{\rm e}$[N II]; right ordinate scale: thick continuous line = $N{\rm e}$[S II], long-dashed line = N(H+) for ( $\epsilon _{\rm l}\times r_{\rm cspl} \times D)= 9.5$ arcsec kpc, dotted-dashed line = N(H $_{\rm tot}$). Bottom panel: the $\frac{{\rm X}^i}{{\rm O}^{++}}$ ionic abundances (same symbols as the top panel).

Figure 11: "Steady'' photo-ionization model for the southern, approaching moustache of NGC 6818 in PA $=10\hbox {$^\circ $ }$. Top panel: absolute radial flux distribution (erg cm-3 s-1) in the different emissions; same symbols as Figs. 6, 9 and 10. Bottom panel: diagnostics radial profile; left ordinate scale: dotted line = $T{\rm e}$; right ordinate scale: thick continuous line= $N{\rm e}$, long-dashed line =  N(H+), short-dashed line = N(H0), dotted-dashed line = N(H $_{\rm tot}$).

We are also exploring new graphical solutions, rendering at best the (present) spatial reconstruction and the (forthcoming) spatio-temporal one. In particular, we are sounding the wide potential of the virtual reality.