2 Observational and model data

For the purposes of this analysis we have compiled in excess of 2700 line intensities for the transitions of [OII] $\lambda 3729/6$ Å, [OIII] $\lambda 4959/5007$ Å, [NI] $\lambda 5200$ Å, [OI] $\lambda 6300$ Å, [SII] $\lambda 6716/31$ Å, [SIII] $\lambda 9532$ Å, HeI $\lambda 5876$ Å, and HeII $\lambda 4686$ Å, using 807 spectra derived from Kaler et al. (1996; a catalogue which is in turn based upon data from in excess of 450 referenced sources); where following standard convention, the various line intensities I(X) are quoted in terms of the normalised function 10² I(X)/I(H$\beta$). Since shocks may be confined to isolated regimes within the nebular shells (see later), we have also included spectra taken at multiple locations within the same source.

The spectra have subsequently been dereddened using the extinction measurements by Tylenda et al. (1992) in combination with the extinction curve of Savage & Mathis (1979). Note, in this latter case, that extinctions based upon radio/H$\beta$ measurements appear to be particularly susceptible to uncertainties, and have not been included in the present analysis. Similarly, results acquired at the Observatoire de Haute Provence appear to be prone to error, and have been employed sparingly.

From comparison with independent extinction measurements (Tylenda et al. 1992) it appears that typical internal errors do not exceed $\Delta C$ $\sim$ 0.6 mag, which would generate corresponding logarithmic errors in de-reddened line ratios $\Delta$log[OII] $\sim 0.17$ and $\Delta$log[SII] $\sim 0.19$ (errors in [OIII] would be significantly less, and in [SIII] somewhat larger). Similarly, errors in line ratio measurements are unlikely to exceed 30%, implying logarithmic errors $\leq 0.1$.

Finally, the resulting corrected line strengths have been correlated with nebular radii from Cahn et al. (1992) with the aim of evaluating evolutionary variations in line strength. Similarly, the observed trends are compared to bow- and planar-shock results deriving from Hartigan et al. (1987) and Shull & McKee (1979), and a variety of radiative modeling results from Gruenwald & Viegas (1992).

The latter analysis includes line ratio estimates for various lines-of-sight through the model nebular shells; a procedure which differs from most prior evaluations, and parallels the procedures employed in acquiring the present observations. We have included modeling for central star temperatures 3.09 10⁴ K $\leq T_*\leq 5~10^4$ K, densities 10² cm$^{-3}\leq n_{\rm e}\leq 10^3$ cm^-3, and abundances $1/30 \leq Z/Z_0 \leq 1$. Taking account of the evolutionary tracks of Schonberner et al. (1979, 1981, 1983), and the mean shell expansion velocities of Phillips (1989), it is apparent that model ratios conform most closely to radiatively-limited shells with radii R<0.1 pc.

A selection of figures resulting from this analysis is illustrated in Figs. 1 and 2.

  

Figure 1: a) Logarithmic variation of [SII]/H$\beta$ with respect to [OII]/H$\beta$, where we have indicated separately the results corresponding to "normal" planetary nebulae (denoted "OBSERV." in the internal caption), bipolar sources, shock and radiative models, and sources containing FLIERs. The primary regimes corresponding to shock modeling and FLIERs are also indicated separately. Multiple measures of individual FLIER sources are connected by dashed lines, whence it is apparent that NGC 7009 (- - - - -) and NGC 6543 (-- - -- - -- -) are associated, in particular, with a broad range of line ratios. The single solid diagonal line corresponds to the least squares regression trend for non-bipolar sources

 

Figure 1: b) Logarithmic variation of HeI/H$\beta$ with respect to HeII/H$\beta$, where we have included the trend expected for radiative line excitation in radiatively-bound sources (see text for details). Commencing from the extreme left, model points correspond to central star temperatures of 30000 K, 40000 K, 50000 K, 60000 K, 70000 K, 80000 K, 90000 K, 100000 K, and 150000 K. The range in scatter of the results is significantly less than for the transitions illustrated in Fig. 1a

 

Figure 1: c-j) Comparative logarithmic trends for a selection of nebular transitions, where the solid diagonal lines again correspond to least-squares regressions for non-BPN sources

  

Figure 2: a-c) Line ratio variations as a function of nebular radius for [NI], [SII] and [OI], where the solid lines correspond to least squares regression trends. Note that most of these variations may be attributed to a quantal increase in line strength for density bound sources ($R \geq 0.1$ pc)