THE CRAB NEBULA'S COMPOSITION AND PRECURSOR STAR MASS

We begin by discussing details of our analysis for Domain 2, because it involves the most prevalent nebular gas. In addition, the relevant photoionization models also help us to understand the other domains.

First, we developed a representative "median observed spectrum" for Domain 2. This involved a subset of 19 spectra for which we have both optical and NIR measurements (see Paper 1) and for which the Hα-normalized [N ii] λ6583 emission lies in the range 1–2.5. For each emission-line measurement, we identified a median value and combined those, as given in Table 1. Then we adjusted photoionization model input parameters and chemical abundances in order to obtain a satisfactory fit to the observed median data.

Following considerable experimentation, all models presented here employ the ionizing synchrotron radiation spectrum given by Davidson & Fesen (1985), with added points across the unobserved ultraviolet range as suggested in the documentation that accompanies the Cloudy code. In addition, we employed a constant hydrogen density of 3000 cm⁻³ and log ionization parameter = −3.2. The ionization parameter is defined as U = Φ/cN, where Φ is the flux of photons more energetic than 1 ryd striking the face of a cloud, c is the speed of light, and N is the total density of hydrogen. The ionizing spectrum did a good job of producing typically observed He ii/He i line ratios (e.g., MacAlpine et al. 1989), the adopted hydrogen density led to calculated electron densities in S⁺ zones comparable with those measured by Davidson (1979) and Fesen & Kirshner (1982), and the log U value was necessary to obtain a consistent overall emission-line spectrum that matches the observations, in particular the [S iii] λ9069/[S ii] λ6731 ratio.

The ionization parameter employed here is almost a factor of 10 lower than that which is necessary to account for observed ultraviolet C iv and [C iii] lines (see Davidson et al. 1982; Blair et al. 1992). Henry & MacAlpine (1982) noted similar ionization parameter differences when comparing photoionization calculations that could match optical and ultraviolet observations separately. We do not consider this to be a problem. As discussed by Davidson (1978), Davidson et al. (1982), and Blair et al. (1992), much of the ultraviolet line emission probably comes from lower density, more diffuse gas, compared with the gas in bright optical-line-emitting filaments analyzed here.

For our Domain 2 model, the derived helium abundance was 89% by mass fraction, consistent with some stellar model expectations (Woosley & Weaver 1995) for a region with partial helium burning and also with numerous observations of He i λ5876 emission (e.g., Uomoto & MacAlpine 1987; MacAlpine et al. 1989; Paper 1).

Figure 2 presents the computed Domain 2 hydrogen and helium ionization fractions as a function of depth into an emitting-gas cloud, with ionizing radiation entering from the left. We note that the He²⁺ and He⁺ zones do not extend far into the cloud. Because of the high helium abundance, ionizing photons more energetic than 54.4 eV (the ionization potential of He⁺) and 24.6 eV (the ionization potential of He⁰) are absorbed before they can penetrate a substantial distance. The hydrogen-ionized depth is about 10¹⁶ cm, which is consistent with observations for Crab Nebula filaments (see Davidson & Fesen 1985).

Zoom In Zoom Out Reset image size

Figure 3 shows the computed electron temperature and density, plotted as a function of the distance into the cloud. The electron density is roughly 8000 cm⁻³ where helium is ionized, drops to 3000 cm⁻³ through the H⁺ zone, and then falls to 400–800 cm⁻³. The temperature peaks near 1.2 × 10⁴ K, is about 6–8 × 10³ K through the H⁺ zone, and then remains above about 6000 K to the depth plotted.

Zoom In Zoom Out Reset image size

Figure 4 gives the modeled carbon and nitrogen ionization. Because the C⁺ ionization potential is comparable with that for He⁰, and the C⁰ ionization potential is below that for H⁰, the dominant ionization stage for carbon is C⁺, all the way from the He⁺ zone to where the calculations were stopped at a depth of approximately seven times the hydrogen ionization zone thickness. In addition, because the N⁺ and N⁰ ionization potentials are slightly above those for He⁰ and H⁰, respectively, N⁺ occupies an extensive, very proficient "[N ii] emitting zone" along with the neutral helium and ionized hydrogen, in which the temperature and electron density effectively support collisional excitation.

Zoom In Zoom Out Reset image size

Figure 5 presents an expanded view of the lower 10% of the vertical scale from Figure 4. This illustrates how a relatively small, but significant amount of C⁰ exists in the zones where helium and hydrogen are ionized. Progressing from left to right in the figure, there is a small peak in the C⁰ fraction where the electron density is highest due to ionized helium. Then there is a gradual C⁰ increase as the temperature declines, and finally there is essentially no C⁰ beyond the hydrogen ionization edge where the electron density becomes very low. These distribution characteristics for C⁰ are consistent with what would be expected from C⁺→ C⁰ recombination.

Zoom In Zoom Out Reset image size

Figure 6 shows selected, computed line emissivities for this model, as a function of depth into the cloud. First, we note that [N ii] λ6583 is "off the chart" because of electron collisional excitation in the warm, high-electron-density [N ii] emitting zone discussed above. H i λ6563 illustrates what would be expected from recombination, and [C i] λ9850 shows a similar distribution since it results from C⁺→ C⁰ recombination followed by electron collisional excitation. The [O i] λ6300 line arises from collisional excitation of O⁰, and it is primarily important in the neutral hydrogen region because of charge exchange processes between hydrogen and oxygen.

Zoom In Zoom Out Reset image size

Previously, high nitrogen mass fractions for many locations were deduced from observations of strong [N ii] λλ6548, 6583 emission and models using the Cloudy code in MacAlpine et al. (1996). However, those models did not adequately take into account high helium abundance and its effect on broadening the [N ii] emitting zone. The helium mass fraction over most of the nebula ranges from about 75% to greater than 90% (Uomoto & MacAlpine 1987). For such helium-rich regions, strong [N ii] lines would not necessarily imply nitrogen mass fractions greater than solar.

It has been known for decades that [C i] λ9850 emission is very strong in parts of the Crab Nebula (Dennefeld & Andrillat 1981; Henry et al. 1984). Pequignot & Dennefeld (1983) tried to match the observed strengths of this line by postulating a high dielectronic recombination rate, and Henry et al. (1984) postulated that collisional excitation by H⁰ could play an important role. Neither group was successful in using [C i] λ9850 for reliable carbon abundance estimates. Therefore, difficult measurements of the ultraviolet [C iii] and C iv emission were attempted, as noted previously. However, production of those lines may not be well understood either, in the sense that model predictions can be wrong by as much as a factor of 2 depending on how He ii Lα absorption is treated (Eastman et al. 1985). In the present calculations, we do understand the production of modeled [C i] λ9850 emission as being due to thermal electron collisional excitation. This ¹D₂→ ³P₂ transition is an analog of the [O iii] λ5007 line. Making use of modeled ionization fractions, electron densities, temperatures, and the relevant effective collision strengths given by Pequignot & Aldrovandi (1976), we manually calculated λ9850 emissivities at various locations and matched the model-computed emissivities exactly.

Predicted relative emission-line intensities from this model are shown in Table 1, where it may be seen that there is a reasonable match with the Domain 2 median spectrum for element mass fractions given in Table 2. The deduced Domain 2 nitrogen mass fraction is about a third of its solar value. We also note that the carbon mass fraction required to produce the observed median [C i] λ9850 emission in Domain 2 is roughly six times solar, and the deduced oxygen mass fraction is close to solar. Finally, we note that the sulfur and argon mass fractions necessary to account for their observed, median emission lines are about a fourth and half of their solar values, respectively.

The most suitable photoionization calculations for our defined Domain 1 gas employed the same ionizing spectrum, hydrogen density, and ionization parameter as discussed above; and the model characteristics are similar to those in Figures 2–6. The median measured spectrum and computed line intensities are shown in Table 1. Measured Hα-normalized [N ii] λ6583 values for Domain 1 range from 3.1 to 5.6, and the adopted median is 4.5. Unfortunately, the subset of NIR data with measured [C i] λ9850 did not include any nebular locations for [N ii] λ6583/Hα > 2.5 (see Figure 10 of Paper 1). Therefore, no [S iii] or [C i] line values are given for this domain in Table 1.

As may be seen in Table 2, the helium mass fraction required to match the observed He i λ5876 line is again taken to be 89% for Domain 1, which is reasonably consistent with expectations for a zone containing largely CNO-processed material. However, we note that the inferred nitrogen mass fraction is only solar, because [N ii] λ6583 arises so effectively in the [N ii] emitting zone discussed above and illustrated in Figures 4 and 6. We could expect the nitrogen mass fraction to be several times solar for gas in which processing has ended with the CNO cycle, so this result suggests significant gas mixing or helium burning to match our Domain 1 median spectrum. The most extreme [N ii] emission would imply a somewhat larger nitrogen mass fraction, but that would involve only a very small fraction of the overall nebular gas. The deduced Domain 1 oxygen mass fraction is solar, while the data suggest that sulfur and argon are significantly less than solar, even more so than in Domain 2. Because [C i] λ9850 was not measured (suggesting weakness) in this domain, the carbon mass fraction input to the Domain 1 model was assumed to be 0.5 solar.

Our defined Domain 3 for the [N ii]:[S ii] plane is illustrated in Figure 1. It involves extreme cases with measured [S ii] λ6731/Hα over the range 3–5.9 and with [N ii] λ6583/Hα < 1.

The measured median spectrum and computed line intensities for Domain 3 are also shown in Table 1. The best-fit photoionization model was similar to that for Domains 1 and 2, except it was necessary in this case to lower the value of log U to −3.5 to reproduce the measured [S iii] λ9069/[S ii] λ6731 ratio.

As seen in Table 2, the adopted helium mass fraction to match observed median He i λ5876 is still 89% for Domain 3, which would seem to imply gas mixing for regions in which processing beyond helium burning has taken place. As expected, the nitrogen mass fraction is now well below solar. The deduced carbon and oxygen mass fractions are both about ten times solar. We note that the [O i] λ6300 line emissivity plotted in Figure 6 illustrates that it arises in a different region from [C i] λ9850, so the inferred, correlated carbon and oxygen mass fractions are not the result of high-optical-depth regions in the emitting gas.

The sulfur mass fraction is roughly four times solar in this domain where it is hypothesized that oxygen burning has taken place. Further support for oxygen burning comes from the even higher inferred argon mass fraction in this domain, as well as from Figure 6 of Paper 1, which shows a very tight correlation between [Ar iii] and [S ii] emission, reinforcing the idea that sulfur and argon are produced together by the same process. Finally, a correlation shown in Figure 8 of Paper 1 is consistent with the related hypothesis that some iron-peak nuclei like nickel are produced by "alpha-rich freezeout" (see Woosley & Weaver 1995; Jordan et al. 2003) in Domain 3 regions that contain enhanced silicon-group elements from oxygen burning.

It is important that we consider whether the NIR emission-line measurements presented in Paper 1, and used here, may be biased toward high [C i] λ9850 emission. As discussed in Paper 1, the NIR data were obtained during nonoptimal observation conditions, and [C i] measurements were made only where potentially useful [S iii] or [C i] line-emission could be identified in the two-dimensional, NIR spectral images. To investigate the possibility of consequent biasing, we compared the current measurements with [C i] λλ9823, 9850 emission reported by Henry et al. (1984). The ranges of measured λ9850 from that study and this one are comparable, with marginally lower and higher values in the larger data set used here. However, we still suspect that the current median spectrum for Domains 2 and 3 may be biased toward stronger [C i] in view of how the measured positions were selected. That having been said, we note that the primary reasons for the current suggestions of high carbon mass fractions are new photoionization calculations whereby we can derive potentially useful carbon abundances from the measured [C i] emission. Furthermore, even though our λ9850 median values may bias the inferred carbon mass fractions for our domains, the individual measurements alone imply that carbon-rich gas exists at numerous locations in the nebula. In any case, a more extensive set of NIR spectra would be useful for understanding the overall nebular carbon.

Here we discuss the fact that the Cloudy photoionization code gave results for plane-parallel geometrical situations, whereas line-emitting gas in the Crab Nebula may be immersed in ionizing synchrotron radiation from all directions (see, e.g., Davidson 1973 and Figure 1 therein). We developed algorithms for converting the plane-parallel results to what would be expected for convex cylindrical and spherical clouds. For a computed plane-parallel model, the line emissivities (erg cm⁻³ s⁻¹) as a function of depth into the cloud were extracted. Then, considering the end point of the calculations (at greatest cloud depth) to be the central axis (or center) of a cylindrical (or spherical) configuration, the data were integrated outward, weighting by appropriately enhanced volume factors, in order to obtain total computed line emission (erg s⁻¹) for each geometry. Potentially noteworthy changes in Hα-normalized lines occurred for only [S ii] λ6731 and [O i] λ6300 emission. In going from the plane-parallel to spherical geometry, the computed λ6731/Hα could be reduced by as much as 30%, and λ6300/Hα could be reduced by about 40%.

By employing convex cylindrical or spherical emitting clouds, we could increase the computed oxygen or sulfur mass fractions by perhaps as much as 30–40%. The deduced oxygen would still be roughly solar or higher in the various domains, and sulfur (as well as argon) would still be low in Domains 1 and 2. We note that Henry & MacAlpine (1982) also found lower-than-solar sulfur mass fractions for particular filaments.

Furthermore, because adopted ionization parameters for the photoionization calculations involved matching the observed [S iii] λ9069/[S ii] λ6731 ratio, an employed ionization parameter could be slightly lower for a convex spherical geometry. However, we investigated this possibility and found that it would probably not have a significant effect on the conclusions presented here. The most significant emission lines for supporting the primary conclusions of this investigation are [N ii] λ6583 and [C i] λ9850, both of which arise mainly within the ionized regions of the models and are not significantly affected by geometry. Therefore, we believe our primary conclusions that nitrogen abundances are not strongly enhanced and that carbon is enhanced throughout much of the Crab Nebula gas would not be changed by more elaborate geometrical considerations.

The photoionization calculations employed here assume that the emitting gas is in ionization and thermal equilibrium. However, as first pointed out by Davidson & Tucker (1970), low-density emitting gas that has not had time to achieve equilibrium since the Crab Nebula formed may exist. This could impact the computation of some low-ionization lines produced in neutral gas. For the present calculations, the line with the greatest potential to be impacted is [O i] λ6300, which is produced in gas where recombination times can be about a sixth of the observed age of the nebula. All other relevant lines develop in gas with higher electron density. In particular, the [C i] and [N ii] lines, which are of primary importance for the conclusions of this work, are produced within the hydrogen-ionized gas (see Figure 6) where recombination times are more like a thirtieth the age of the nebula and where the gas is likely to be in ionization equilibrium.