3 Spectral trends in the general population

3.1 Characteristics of the [OI] transition

It is apparent from both narrow and broad-band imaging of planetary nebulae that shell structures are often extremely complex, whilst fine-scale components (whether clumpy or filamentary) may be associated with anomalous enhancements in [OI] (see references cited in Sect. 1). The present results suggest that some such deviation from normal trends must apply for the large majority of sources, with [OI] in particular (Figs. 1c, d, f, h) lying substantially outside the range defined through the modelling of Gruenwald & Viegas (1992): observed values are in almost all cases appreciably in excess of simulated results. It therefore appears that the present results confirm for the generality of nebulae what has long been known for individual sources.

A diagram representing the excess in line ratios for [OI] and other transitions is illustrated in Fig. 3 (see the trend designated "OBSERVED"), wherein we indicate a parameter R(GEN) = $\langle \log (I_{\rm G})\rangle - \langle \log (I_{\rm M})\rangle$ for the general population of sources; where $\langle \log (I\rm _M)\rangle$ represents the mean logarithmic line ratio for the radiative models combined, $\langle \log (I\rm _G)\rangle$ is the corresponding parameter for the observed nebular sample, and bipolar nebulae have been excluded (see the further discussion in Sect. 4.1). We have also corrected R([OIII]) for use of inappropriately low model central star temperatures, as discussed in Sect. 4. Broadly speaking, therefore, R(GEN) is an indication of the proportionate excess in line ratios over those anticipated through radiative modelling.

  

Figure 3: Observed variation of R(GEN) as a function of transition, together with comparative predictions for selected planar shock models

It is apparent that excesses in [OI] represent by far the most serious departures from modelling of any of the transitions considered here. This, in part, is likely to be the result of low [OI] model line ratios, such that small excesses in all lines would lead to disproportionately large effects in this particular transition. Indeed, the evaluation of a median excess parameter $\langle \Delta I_{\rm D}\rangle = \langle I_{\rm G}\rangle - \langle I\rm _M\rangle$ suggests that the trends illustrated in Fig. 3 are consistent with a similar excess in all transitions - a species of "veiling" over and above that arising through normal radiative excitation. How could such a component arise?

One possible indication is afforded through the work Reay et al. (1988) and Phillips et al. (1992), where it is noted that [OI] line intensities are proportional to the intensity of shocked $\lambda = 2.1~\mu$m H₂ S(1) emission. One plausible explanation, therefore, is that we are witnessing a small shock excess at the HI/HII interface of optically thick sources, or over the surfaces of neutral condensations within the primary ionised mass.

Under these circumstances, and given that $I_{\rm M} = ^{\rm R}I_{\rm L}/^{\rm R}I_\beta$, and $I_{\rm G} = (^{\rm R}I_{\rm L}+^{\rm R}I_{\rm S})/(^{\rm R}I_\beta +^{\rm S}I_\beta )$, then $\langle \Delta I_{\rm D}\rangle$ may be equated with $(^{\rm S}I_{\rm L} - ^{\rm R}I_{\rm L}(^{\rm S}I_\beta /^{\rm R}I_\beta ))(^{\rm R}I_\beta + ^{\rm S}I_\beta )^{-1}$; where $I_\beta$ is the H$\beta$ emission intensity, $I_{\rm L}$ is the intensity of the transition under investigation, and superscripts R and S refer respectively to the radiative and shock terms. Where $^{\rm S}I_\beta \ll ^{\rm R}I_\beta$ and $^{\rm R}I_{\rm L}\leq ^{\rm S}I_\beta$ then $\langle \Delta I_{\rm D}\rangle \approx ^{\rm S}I_{\rm L}/ ^{\rm R}I_\beta$. In other words, $\langle \Delta I_{\rm D}\rangle$ for [OI] will be broadly proportional to the shocked line intensity. Note, in this respect, that force-fitting of observed [OI] line ratios to a mix of shock and radiative modelling would imply values $^{\rm S}I_\beta/ ^{\rm R}I_\beta \sim 0.06$ for shock velocities $V_{\rm s} = 45$ km s^-1; a ratio which decreases still further as $V_{\rm s}$ increases.

Inspection of the shock results of Raymond et al. (1988) and Hartigan et al. (1987) reveals that a variety of specifications might satisfy required excess characteristics, including models I20$\Rightarrow$I80, A100, D100 and bow shock models 3 and 4. These are characterised by fairly modest velocities $V_{\rm S}\sim 20\Rightarrow 100$ km s^-1; comparable to [OIII]/HI expansion velocities $V_{\rm exp}\sim 25$ km s^-1 in normal PN (e.g. Phillips 1989; although note that [OII]/[NII] velocities (characteristic of the nebular peripheries) are typically $\sim 60\%$ larger).

Although the excess trends, observed expansion velocities and shock modelling appear therefore to offer a consistent scenario, care must be taken in the interpretation of such results; in particular, the synthetic data used to assess $\langle I_{\rm M}\rangle$ are far from representing an appropriate balance of models. Thus, although the radiative modelling appears to simulate observed trends tolerably well (see later), slight errors in $\langle I_{\rm M}\rangle$would have a disproportionate effect upon $\langle \Delta I_{\rm D}\rangle$in all excepting the [OI] results.

A more interesting question, under these circumstances, is whether the very much more greatly enhanced [OI] line ratios are consistent with plausible shock modelling. To investigate this, we note that for a shock extending uniformly over the surface of a spherical nebula

$$\frac{^{\rm s}I({\rm OI})}{^{\rm R}I({\rm H}\beta )}=\frac{7... ...rgs~cm^{-2}s^{-1}}}\right]}{n^2_{\rm e}(R/{\rm pc})\varepsilon}$$

for case B radiative conditions, where $\varepsilon$ is the nebular filling factor, $n_{\rm e}$ is the electron density, and $^{\rm s}F({\rm H}\beta )$ is the H$\beta$ flux emerging from the shock front. Given a proton number density

$$n_{\rm p}=\frac{3~M}{4\pi R ^3 m_{\rm H}\varepsilon \mu}$$

where M is the shell mass, and $\mu$ is the mean atomic mass per proton, and taking $^{\rm s}F({\rm H}\beta ) = U(n_{\rm e}/10^{2}~{\rm cm}^{-3})$then gives

$$\frac{^{\rm s}I({\rm OI})}{^{\rm R}I({\rm H}\beta )}=\frac{3... ...}I({\rm OI})/^{\rm s}I({\rm H}\beta )] [R/{\rm pc}]^2 U}{M/M_0}$$

where U is a steeply varying function of shock velocity $V_{\rm S}$ ($U \propto V_{\rm S}^{3.5})$, and we assume pre-shock densities to be comparable to $n_{\rm e}$. For representative values $M \sim 0.1\ M_\odot$, $R \sim 0.1$ pc and employing a value $U \cong 0.22$ appropriate for $V_{\rm s} \sim 40$ km s^-1 then yields $^{\rm s}I$(OI)/$^{\rm R}I$(H$\beta$) = .038; a ratio which increases to .053 for $V_{\rm s} \sim 50$ km s^-1 ($U \cong 0.54$). For comparison, the observed value of $\Delta I_{\rm D}$([OI]) based on logarithmic mean estimates (i.e. log $\langle \Delta I_{\rm D}([{\rm OI}]) \rangle = \langle \log (I_{\rm M}$([OI])$\rangle \log (10^{\rm R([OI])} - 1))$ is of order 0.052. It is therefore clear that much of the observed emission may indeed be explicable through such a mechanism. Indeed, the viability of such a model may be even greater than supposed above, since a variation in nebular mass $M \propto n_{\rm e} ^\gamma$, $\gamma = -1\Rightarrow -0.7$ (e.g. Boffi & Stanghellini 1994; Pottasch 1984) would tend to proportionately enhance [OI] in smaller nebulae.

In contrast, the trend towards smaller values of $V_{\rm exp}(R)$ with decreasing R (e.g. Phillips 1989) might be expected to work in the reverse direction, and lead to corresponding decreases in [OI] shock excesses; a factor which may be responsible for the shallow secular variations in [OI]/H$\beta$ noted in Sect. 3.2.

Finally, the theoretical trends for a range of transitions and velocities are presented in Fig. 3. It can be seen that shock modelling predictions are more than adequate, and accommodate a good proportion of the excess in [SII] which, on this basis, would appear to be shock enhanced by a factor $\sim 2$. Such a result may have severe consequences for our understanding of nebular densities, since it is apparent that [SII] line ratios would be to some degree representative of compressed post-shock regimes, and imply higher densities than are appropriate for the primary nebular shells.

A further possible source for such trends may arise through UV shadowing, appreciable ionisation stratification, and charge exchange reactions in zones of partial ionisation. Such mechanisms may not, in fact, be entirely divorced from the process of shock excitation considered above, since Rayleigh-Taylor and Kelvin-Helmholtz instabilities would be expected to lead to frontal irregularities, and the possible development of globular neutral condensations within the primary ionised zones (cf. Capriotti 1973). Whether such features could account for the "veiling" excesses noted above is, however, far from clear, and requires further analysis.

Finally, we have noted that low-excitation emission appears often to be associated with small-scale condensations and filamentary structures (see for instance the spectro-morphological studies of Phillips & Reay (1980) and Boeshaar (1974), high resolution imaging of nearby nebulae (e.g. NGC 7293, NGC 6543; e.g. Harrington 1995; O'Dell & Handron 1996), and the line modelling analyses of Hyung et al. (1994, 1995), Köppen (1979), Boeshaar (1974) and Hyung & Aller (1995). It is therefore pertinent to ask whether the strength of the [OI] transition may be related to the degree of fragmentation of the primary shell. More specifically, is there a correlation between [OI] line strengths and the nebular filling factor $\varepsilon$? To investigate this question, we have plotted the variation of I([OI]) against values of $\varepsilon$ derived from Boffi & Stanghellini (1994), Kingsburgh & Barlow (1992), Kingsburgh & English (1992) and Mallik & Peimbert (1988) (Fig. 4; where we have adopted averaged values of $\varepsilon$ where multiple estimates are available). Cursory inspection suggests that there is little correlation between the parameters. Whilst [OI] excitation may be associated with nebular condensations, therefore, it would appear that such emission is only superficially related to primary shell fragmentation.

  

Figure 4: Variation of [OI]/H$\beta$ with respect to the nebular filling factor $\varepsilon$

3.2 Radiative excitation of ionic transitions

In contrast to [OI], the other transitions investigated here appear to follow radiative modelling trends tolerably closely. We briefly review the characteristic line ratio trends for the remaining transitions:
1) Trends in [SII], [OII] and HeI line ratios appear broadly consistent with radiative modelling (Figs. 1a, f; although note comments concerning [SII] in Sect. 3.1).
2) Power-law trends exist between the transitions [NI], [OI], [OII], and [SII] (see Figs. 1a, d, g, h and j for representative examples, and Table 1 for a summary of various least-squares fits; where Y=MX+C, and r ² is the correlation coefficient). In comparison, trends between the radiative modelling results for [SII] and [OII], and HeI and [OII] appear closely similar in terms of absolute intensities and gradients

  

Table 1: Least squares regression analysis for spectral line ratios

Given that many of these line intensities increase with increasing source radius (see Sect. 4 below), the trends also correspond to an evolutionary sequence in which younger nebulae (high densities, lower central star temperatures) are located to the lower left of Figs. 1, and older nebulae are to the upper right.
3) Comparison between theoretical and model results for [NI], [OI], [OII], and [SII] reveals that the levels of scatter are comparable. Much of the theoretical scatter derives from the range of parameters $T\rm _e$, $n\rm _e$, and Z employed for the radiative modelling, as well as through the contributions of differing lines-of- sight through the nebular shells. This, in turn, underlines the necessity of employing multiple line-of-sight analyses for any radiative investigation of nebular line strengths; a factor not always appreciated in previous analyses.

Note that the size of scatter exceeds probable errors (Sect. 2) by a factor $\sim 5$.
4) Although correlation coefficients are relatively low, there appears to be evidence for statistically significant variations in [OI], [SII], and [NI] line intensity with nebular radius (Fig. 2). This does not apply for higher excitation transitions such as [SIII], HeII, [OIII] and He I. The radial gradient for [NI] appears significantly greater than for any other ion, whilst gradients for [OI] and [SII] are comparable.

Such trends would be at variance with radiative modelling of ionisation-bound nebulae, given likely evolutionary trends in T_* and $n\rm _e$. On the other hand, careful perusal of these figures suggests that much of the apparent variation may arise from a jump in low-excitation line-strengths close to R = 0.1 pc, by typical factors of between 0.5 and 0.75 dex; that is, as a result of the transition from radiatively-bound to density-bound ionisation structures.
5) [SIII] intensities were not modelled by Gruenwald & Viegas (1992), although they appear to be broadly comparable to those expected through shocks (Fig. 1i).
6) The variation of HeI with HeII differs from most other trends in revealing a relatively constant ratio I(HeI)/I(H$\beta$) up to the limiting value log(10² I(HeII)/I(H$\beta$)) $\sim 2$, after which there is some down-turn in ratios. Such trends are broadly consistent with radiative analyses, as noted from the model results illustrated in Fig. 1b; where we have assumed model nebulae to be optically thick, spherical and homogeneous, with typical densities $n\rm _e = 10^4~cm^{-3}$, and electron temperatures $T\rm _e = 10^4$ K (the relevant emission parameters are insensitive to these parameters). We have also assumed n(He)/n(H) = 0.115, case B conditions, a range of stellar temperatures 3 10⁴ K $\leq T_* \leq 1.5\ 10^5$ K, and adopted a blackbody approximation to the stellar continuum. Ratios refer to lines of sight through the nebular cores.

The downturn in I(HeI)/I(H$\beta$) for large values of I(HeII)/I(H$\beta$) occurs for high central star temperatures, and applies (in particular) to density-bound structures.

Finally, note that the similarity of HeI and H$\beta$ emission regimes for large ranges of central star temperature, and the relative insensitivity of emission coefficients to $n\rm _e$ and $T\rm _e$ would lead to comparable HeI line ratios irrespective of the mode of spectral sampling. This, in large part, is probably responsible for the reduced scatter in HeI compared to [SII] and other intermediate excitation transitions.
7) The low scatter, invariant trend of [OIII] line ratios (Figs. 1c, e) is also broadly consistent with the distribution expected for radiative line excitation. For very much the same reasons as were cited for HeI above, [OIII]/H$\beta$ratios are expected to be reasonably consistent over a broad range of central star temperatures, and as a function of projected shell location.

The failure of the radiative modelling to reproduce these trends appears to arise, in part, from a sensitivity in [OIII]/H$\beta$ ratios to low central star and nebular temperatures; both sets of parameters are smaller (in certain models) than would be appropriate for the spectral data base investigated here.