Astron. Astrophys. 358, 1058-1068 (2000)

3. Method of analysis

3.1. Computer codes

For each nebula in the sample, a photo-ionization model was constructed to fit the published intensities of the emission lines. The nebula is approximated as a spherical shell defined by an inner and outer radius, a total mass, a radial density distribution and a radial velocity field. The central star is modelled using a black-body atmosphere. The free parameters of the photo ionization model are adjusted to obtain a good agreement between the observed and calculated H flux and the emission line ratios.

Subsequently, a radial velocity profile is assumed, with the velocity varying smoothly with radius. The predicted profile for the emission line is calculated by integrating the velocity field over the ionization equilibrium and resulting line emission coefficients. The slit parameters and seeing are used as input parameters for this calculation. Comparing the predicted profile with the observations allows one to correct the assumed velocity field. The photoionization model requires knowledge of stellar and nebular parameters as initial parameters. Distances and radii are mostly taken from Van de Steene & Zijlstra (1994), however other values are sometimes adopted. The effective stellar temperature and luminosity are taken from the literature, but are often slightly changed during the fit in order to reproduce the observed H flux and the line ratios (CGPN). The adopted nebular and stellar parameters are presented in Table 2. They are in succession: PN designation according to galactic coordinates, PN common name, central star's effective temperature and luminosity, assumed distance, outer radius, and the ionized mass.

The computer codes as well as the recipe for modelling the nebulae are described elsewhere. Below we list the most important limitations: for further details we refer to Paper I.

3.2. Model simplifications

The model fits as performed in this paper allow more freedom than those in papers I and II, firstly because less is known for most objects from our sample, and secondly because only one spectral line was observed.

Most of the PNe in our sample have small angular diameters and high-resolution images are not available. Therefore the density distribution (especially the inner radius) cannot be determined from imaging data. For all objects, we used a density distribution in the shape of an inverted parabola, with the central density twice as large the density at the edge. This is closer to predictions from hydrodynamical models than a constant density (e.g. Marten & Schönberner 1991). The inner nebular radius is taken as 0.4 of the outer radius.

Because we only have data for the [OIII ] line which mainly probes the inner nebular region, the fitted velocity field was limited to a linearly increasing velocity. Whenever possible we assumed a constant velocity. There are some indications of a steeper velocity gradient in the outer regions (as shown by wings on the line profile), but these can not be accurately modelled without information from lines formed in the outer regions, e.g. [NII ].

Our calculated spectral lines are affected by thermal and instrumental broadening. The thermal broadening is calculated from the electron temperature indicated by the photoionization calculations and is different for each PN. The instrumental broadening corresponds to the spectral resolution of the instrument and is adopted as 0.17 Å. Additional turbulent motions may be present in a few nebulae, especially in those with [WC]-type nuclei (see Gesicki & Acker 1996), but this can be proven only by comparing the line widths from different ions (especially hydrogen is desirable). We therefore neglect the effects of possible turbulence.

3.3. Uniform velocities

The models relate the line profile to a `true' expansion velocity. Previously, expansion velocities were determined directly from the line profiles. For the case of a constant velocity throughout the nebula, we can quantify the difference between the expansion velocity as obtained from the line profile, and the `true' model velocity. (For a non-uniform velocity the relations are very complicated.)

The observed line profile will depend on whether the slit resolves the nebula, e.g. line splitting is only seen for a spherically symmetric nebula if the slit width is smaller than the object. The effect of the relative size of the aperture on the derived expansion velocity is illustrated in Fig. 1, showing the calculated emission line profile for an artificial nebula with all parameters constant with radius. The line profiles in the figure correspond to circular apertures of different sizes. Table 1, lists, for different apertures, the ratio between the true expansion velocity, and the velocity derived from (i ) line splitting, (ii ) half width at half maximum (HWHM), (iii ) half width at ten per cent of maximum (HWTP).

Fig. 1. Modelled [OIII ] lines for a spherical nebula with constant velocity and emissivity. This nebula is observed with a centered circular aperture: the profiles are given for aperture size relative to the PN diameter of: 1.0, 0.8, 0.5, 0.2. The vertical dashed line indicates the true expansion velocity of 40 km s-1.

Table 1. The expansion velocities defined for the models shown in Fig. 1, determined from line splitting (e.g. Weinberger 1989), half width at half maximum (Robinson et al. 1982) and the half width at ten per cent (Dopita et al. 1985), divided by the true expansion velocity.

Most catalogued PNe expansion velocities are derived from the line splitting as . This can significantly underestimate the true expansion velocity: the peak-to-peak separation is a good measure of the expansion velocity only if the nebula is fully resolved (e.g. Robinson et al. 1982). The half-width-at-half-maximum is the most accurate approximation to the true expansion velocity for spatially unresolved or marginally resolved nebulae. However, it overestimates the expansion velocity for well-resolved objects. Thirdly, the half-width at 10 percent of the peak (as used by Dopita et al. 1985) is almost independent of the size of the aperture but overestimates the true value by about 10 per cent. We note that the precise values depend on the relative effect of thermal broadening which increases for lower expansion velocities.

3.4. Velocity gradients

Fig. 1 illustrates another important feature: the emission profile from a shell with uniform emissivity and constant expansion velocity is flat-topped when the nebula is unresolved. It would be rectangular in shape without line broadening (thermal, turbulent or instrumental): the flat top is visible when the expansion velocity is larger than the broadening. This can be seen as follows. When the spherical nebula expands with a constant velocity the Doppler-shifted contribution to the spectral line from a small volume element will depend on the angle () between the line connecting the nebular center with the element and the line from the center to the distant observer. The total contribution to the profile at the given velocity is obtained by integrating all volume elements at the appropriate . Apply a spherical coordinate system, centered on the nebula and with the main axis pointing towards the observer. In such a system the volume element is:

After performing the integration over (around the main axis) and over (from 0 to R):

Consider the uniformly spaced intervals in the velocity dv. Because

equal intervals dv in velocity correspond to equal volumes dV. Therefore the fluxes emitted in each velocity interval are equal and the spectral line is rectangular in shape. Such flat-topped lines are not observed, indicating that the velocity in our objects is more complicated. This is clear even from only one [OIII ] line formed near the nebular center.

The velocity fields presented in papers I and II, based on two or three lines, mostly show two components with a slowly increasing velocity in the inner nebula but steeply increasing in the outer parts. In Fig. 2 we present an example of this. For the nebula H 1-35 spectral lines of both [NII ] 6584 Å (Acker, priv. comm.) and [OIII ] 5007 Å are shown. The bold line indicates a fit using only the [OIII ] line assuming a linear velocity field (this case is exceptional in that a constant velocity gives a reasonable fit). This fit fails for [NII ]: instead the thin line shows a velocity profile which fits both lines. The nitrogen line shows evidence for a fast acceleration in the outer nebula. If only the [OIII ] line is available, the inner region of the nebula is well determined but the outermost velocities may be poorly constrained. This limitation should be kept in mind for the remainder of the paper.

Fig. 2. [NII ] 6584 Å and [OIII ] 5007 Å lines for object H 1-35 (PN G 355.7-03.5). The circles present the observed profiles, and the lines the modelled ones. Thick and thin lines correspond to two different velocity fields shown in the lowest box as a function of the relative nebular radius.

© European Southern Observatory (ESO) 2000

Online publication: June 20, 2000
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