PUMPING UP THE [N i] NEBULAR LINES

The upper panel of Figure 3 shows the temperature structure of the cloud along our ray. The H⁺ region has a temperature of around 10⁴ K while the gas kinetic temperature falls to around 300 K in the H⁰ region or PDR. The deeper H₂ region is also colder. The H⁺ region is thicker than was found in Baldwin et al. (1991) due to magnetic support.

Zoom In Zoom Out Reset image size

The lower panel of Figure 3 shows the volume emissivity of the λ5199⁺ lines along this ray. For reference the lower panel also shows the emissivity of some well-observed H₂, CO, and [O i] lines. We see that both [O i] and [N i] lines form near the H⁺–H⁰ ionization front, while the H₂ and CO lines form near the H⁰–H₂ dissociation front.

The H₂ line is mainly formed by continuum fluorescent excitation for a PDR near an H ii region (Tielens & Hollenbach 1985). This formation processes is very similar to that forming the [N i] lines. The H₂ and [N i] lines form at either edge of the PDR due to a combination of abundance and FUV line optical depth effects.

Figure 4 shows the volume emissivity of the [N i] line as a function of gas kinetic temperature. This is a convenient way to visualize the rapid changes in emissivity that occur near the H⁺–H⁰ ionization front, where both emissivity and temperature change rapidly. This does not indicate the total contribution of various processes to the observed line since the surface brightness is the integral of the emissivity over the emitting volume. The size of each volume element changes dramatically as the conditions change.

Zoom In Zoom Out Reset image size

The peak emissivity occurs at a gas kinetic temperature of ∼4000 K, with contributions from continuum pumping and collisions. The emissivity increases as the temperature decreases due to the increasing N⁰ abundance in cooler regions. Continuum pumping increases with increasing N⁰ abundance. The emissivity decreases at lower temperatures due to increasing optical depths in the FUV lines. The rise in emissivity at temperatures ∼300 K is due to formation by molecular dissociation, using the estimates outlined in Appendix A.4.

Given these assumptions the only free parameter is the turbulent contribution to the line width. Figure 5 shows the predicted [N i] 5199⁺/Hβ intensity ratio as function of the turbulence. We have determined the broadening of the [N i] lines from spectra made with the HIRES spectrograph as part of a program studying mass loss from proplyds in the Huygens Region (Henney & O'Dell 1999). In the proplyd-free portions of these long-slit spectra, the average observed FWHM was 13.65 ± 1.91 km s⁻¹. The instrumental FWHM of the comparison lines was 8.28 ± 0.40 km s⁻¹. If the [N i] emission arises from the region with T = 1000–3000 K, then the thermal component of the broadening would be 2–3 km s⁻¹. After quadratic subtraction of the instrumental and thermal widths from the observed FWHM, there is a residual non-thermal broadening component of 10.6 ± 1.9 km s⁻¹. The four proplyds in the sample (150–353, 170–337, 177–341, 182–413) have an average distance from θ¹ Ori C of 0.56 ± 029. After examination of panel (A) of Figure 1, we see that the expected ratio I([N i])/I(Hβ) would be 0.0034 ± 0.0007. These values are indicated in Figure 5 for comparison with the predictions of our model. The line width is given as an upper limit, since the observed emission profile will in general include contributions from macroscopic and microscopic broadening processes (see discussion in Appendix C) whereas only the latter will contribute to the pumping efficiency.

Zoom In Zoom Out Reset image size

In Appendix A, we show that in the fluorescent scenario the strength of the [N i] emission lines with respect to Hβ is close to linearly proportional to the degree of line broadening in the region in which the lines are pumped. In order to reproduce the observed brightness, our model requires an FWHM for the broadening (assuming a Gaussian line profile) of ≃ 10 km s⁻¹. If this broadening were to be thermal, then a temperature of >10, 000 K in the pumping region would be required, which is much larger than the ≈2000 K predicted by our Cloudy model. Instead, it is likely that the majority of the broadening is non-thermal in nature. Significant non-thermal line widths have been reported in the spectra of Orion Nebula emission lines (O'Dell 2001; O'Dell et al. 2003; García-Díaz et al. 2008). The nature of the processes producing this broadening is not known, but it must be important as its magnitude indicates that as much energy is contained there as is contained in the components explained by basic photoionization physics.

In Section 3, we established a model of an essentially substellar point in the Orion Nebula that adequately explains the observed I([N i])/I(Hβ) line ratios. However, we see in Figure 1(A) that this ratio varies across the Huygens Region and its near vicinity, rising monotonically with increasing distance from θ¹ Ori C. A similar increase is seen for the line ratios in M43 with increasing distance from NU Ori. On the other hand, Figure 6 shows that the corrected equivalent width of the [N i] lines is essentially constant at 2 ± 1 Å and shows no detectable variation either within M42, nor between M42 and M43. Since the PDR is optically thick to the irradiating stellar continuum, its visual scattered light is a measure of the FUV continuum that pumps the upper states of 5199+. The quantitative relation is determined by the scattering properties of the grains. These are elaborated in Appendix B.

Zoom In Zoom Out Reset image size

Appendix B shows that the line intensity ratios and equivalent widths can be expressed in terms of the ratios of "scattering" efficiencies (albedos) and the ratios of the stellar continuum luminosities in different wavelength bands. The pumping contribution to 5199+ depends on the intensity of the SED around 1000 Å while the intensity of Hβ, which forms by recombination, is proportional to the continuum intensity at hydrogen-ionizing energies. These are denoted by FUV and EUV in Table 2. In the simplest case we expect the 5199+/Hβ intensity ratio to scale with FUV/EUV, while the 5199+ equivalent width should scale with FUV/visual.

Table 2 lists the ratios of the average value of the SED λL_λ, calculated for the visual, FUV, and ionizing EUV bands, and for three different OB stellar populations characteristic of the inner Orion Nebula (Trapezium region), the outer Orion Nebula, and M43 (we will consider the Crab and Ring Nebulae, objects very different from Orion and with strong [N i] emission, in Section 4.5 below). It can be seen from the table that the FUV/visual luminosity ratio is approximately constant between the three stellar populations in Orion (variation <10%), which arises because the spectral shape in all cases approximately follows the Rayleigh–Jeans behavior of L_λ ∼ λ⁻². On the other hand, the relative strength of the EUV band with respect to the FUV and optical bands shows significant variation, being roughly five times greater for the Trapezium stars than for the exciting star of M43. We suggest that it is the presence or otherwise of variations in the broadband illuminating spectrum that is the principal determinant of the observed spatial variations in intensity ratios.

The observed constant value of EW([N i], Corr), coupled with the lack of variation in $\langle \lambda L_\lambda \rangle _{\scriptscriptstyle \mathrm{FUV}}/\langle \lambda L_\lambda \rangle _\mathrm{\scriptscriptstyle vis}$ implies (Equation (B5)) that the ratio of [N i] albedo to dust-scattering albedo is also constant within and between the nebulae: $\varpi _{\scriptscriptstyle 5199}/\varpi _{\mathrm{\scriptscriptstyle dust}}= (4\pm 2) \times 10^{-4}$. The Cloudy model of Section 3 implies that $\varpi _{\scriptscriptstyle 5199}\simeq 2 \times 10^{-4}$ for a non-thermal broadening of 5 km s⁻¹ if the illumination and viewing angles are both close to face-on (see Figure 11(b)). The dust-scattering effective albedo is therefore constrained to be 0.5 ± 0.3, which is consistent with the expectations for back-scattering from the background molecular cloud (see Appendix B.2.3 and Figure 12) so long as the single-scattering albedo is relatively high.

The ratio I([N i])/I(Hβ) (Figure 1(A)) increases by roughly a factor of three between the inner and outer regions of M42 and by a factor of 4–8 between M42 and M43. The different values of $\langle \lambda L_\lambda \rangle _{\scriptscriptstyle \mathrm{FUV}}/\langle \lambda L_\lambda \rangle _{\scriptscriptstyle \mathrm{EUV}}$ for the stellar populations (Table 2) can fully explain the difference between M42 and M43 but can only account for half of the variation within M42 and cannot explain any of the variation within M43 (where the single dominant star means that the illuminating spectrum should be constant). This implies that a small systematic increase with radius within each nebula of the ratio of albedos $\varpi _{\scriptscriptstyle 5199}/\varpi _{\mathrm{H}\beta }$ may also play a role. The analysis of Appendix B.2.1 shows that ϖ_Hβ ≃ 0.1 when the illumination and viewing angles are face-on, which, combined with the above value of $\varpi _{\scriptscriptstyle 5199}$ and using Equation (B3), implies I([N i])/I(Hβ) ≃ 0.03 for illumination by the Trapezium spectrum, as is observed for the innermost regions of M42. Inspection of Figure 11 shows that, as long as the plane-parallel approximation is valid, variations in the viewing angle cannot account for the inferred increase in $\varpi _{\scriptscriptstyle 5199}/\varpi _{\mathrm{H}\beta }$ with radius since the two albedos depend on angle in a similar way, except for close to edge-on orientations where $\varpi _{\scriptscriptstyle 5199}/\varpi _{\mathrm{H}\beta }$ is predicted to decrease. However, as discussed in Appendix B.4, a finite curvature of the scattering layer has the effect of limiting the limb brightening for edge-on viewing angles, and this effect is much greater for Hβ, where the scattering layer is much thicker than for [N i]. This effect may explain the increase in $\varpi _{\scriptscriptstyle 5199}/\varpi _{\mathrm{H}\beta }$ if the average viewing angle became increasingly edge-on in the outskirts of the nebula (see Appendix B.3 for further discussion). An alternative explanation could be an increase with radius of the turbulent broadening within the fluorescent [N i] layer, but there is no independent evidence for such an increase.

The original motivation for this work was to calibrate models of the formation of [N i] lines in the relatively quiescent Orion environment, as a step toward understanding what these lines indicate in the more exotic environments where they are unusually strong. Although continuum fluorescent excitation does account for the [N i] lines in Orion, the process cannot produce the stronger [N i] emission seen in planetary nebulae or the Crab supernova remnant.

The last two rows of Table 2 show the continuum intensity ratios produced by the SEDs of the Ring Nebula (a T = 1.2 × 10⁵ K Rauch stellar atmosphere; O'Dell et al. 2007) and the Crab Nebula (Davidson & Fesen 1985). The [N i]/Hβ intensity ratio scales with the FUV/EUV continuum ratio. Table 2 shows that this ratio is 7 and three times smaller for the Ring and Crab Nebulae than in Orion. Accordingly, the continuum fluorescent excitation contribution to the [N i]/Hβ intensity ratio produced by fluorescence will be of order I([N i])/I(Hβ) ∼ 10⁻³. The observed line ratio is several orders of magnitude larger, showing that other processes must be at work. Thermal excitation by warm gas, perhaps produced by penetrating energetic photons or ionizing particles, seems to be needed. High-resolution observations of the lines given in Figure 9 could test whether thermal processes do account for the observed spectrum.

In order to optimize the speed of the model, we have chosen to model the N i atom using a five-level atom for the metastable levels. The fluorescence processes discussed in this paper (as well as the recombination pumping) are added as rates populating the various excited metastable levels. The level energies were obtained from Moore (1975) and the lowest five levels are listed in Table 3 and shown in Figure 7. An additional 10 FUV lines can absorb photons in the 951–1161 Å range. These lines drive the fluorescence process and will be discussed in more detail below.

Zoom In Zoom Out Reset image size

Transition probabilities have been computed by Butler & Zeippen (1984), Godefroid & Fischer (1984), Hibbert et al. (1991), Tachiev & Froese Fischer (2002), and Froese Fischer & Tachiev (2004).

Table 4 compares the Butler & Zeippen (1984) and Godefroid & Fischer (1984) rates for the forbidden transitions, referred to as the "1984" rates, with the more recent calculation of Froese Fischer & Tachiev (2004), referred to as the "2004" rates. The latter rates are used. Hibbert et al. (1991) do not give transition probabilities for the forbidden transitions.

The electron collision rates for [N i] have been the subject of a number of studies. Berrington et al. (1975) computed electron collision cross sections which Dopita et al. (1976) converted into collision strengths. These were later summarized by Berrington & Burke (1981). Dopita et al. (1976) found some discrepancies with existing observations and speculated that the disagreement was due to uncertainties in the collision strengths. Tayal (2000) presented close-coupling calculation of the effective collision strengths while Tayal (2006) redid the calculation with significantly different results.

Table 5 gives the history of these electron collision strengths. The ⁴S^o − ²D^o collision strength affects the intensity of the collisionally excited contribution to the [N i] λ5199⁺ lines, while the ²D^o_3/2 − ²D_1/2^o collision strength affects the density diagnostic. The Berrington & Burke (1981) and Tayal (2006) results are in reasonable agreement suggesting that the theoretical calculations have converged onto a stable value.

We know of no rates for collisions with hydrogen atoms. This should be included since we expect that [N i] may form in shallow regions of the PDR, where n(H⁰) ≫ n_e. Cloudy does include a general correction for H⁰ collisions based on the n_e rate. The effective electron density, with this correction, is n_e + 1.7 × 10⁻⁴n(H⁰) based on the discussion by Drawin (1969).

Cloudy includes many other line formation processes in addition to thermal collisional impact excitation (Ferland & Rees 1988). Continuum pumping of the FUV lines around 1000 Å will be very important. We treat this as described in Ferland (1992) and Shaw et al. (2005). Fluorescent excitation is mainly produced by stellar FUV photons. Pumping will be efficient until the N i lines become optically thick At this point they will have absorbed stellar photons over the Doppler width of the line. Below we explore how the pumping efficiency depends on the turbulent contributor to the line width. Other opacity sources will affect the strength of the pumped contributor to N i by removing FUV photons before they are absorbed by N i. The two most important opacity sources are extinction of the stellar radiation field by grains within the H ii region and PDR, and shielding by the forest of overlapping H₂ lines in deeper parts of the PDR. These processes are all included self-consistently in our calculations.

In strict LS coupling, transitions that change the total spin of the atom (called intercombination transitions) are forbidden and FUV pumping out of the quartet ground term could not eventually populate the doublet excited terms that produce the observed lines. However, deviations from strict LS coupling make it possible that a significant fraction of excitations by FUV photons will eventually populate the excited doublets. There are two possible routes: either direct excitation by an intercombination line from the ground state or indirect excitation by a resonance line followed by de-excitation through an intercombination line. In the case of N i both routes contribute. A complete list of the driving lines can be found in Table 6. For each of the driving lines we calculate a two-level atom giving us the excitation rate for each of these transitions. We also calculated branching ratios for the cascade down from each of the upper levels. In these calculations, we exclude the transition straight back to the ground state as this does not destroy the photon. Instead it can be absorbed over and over again until finally a different cascade from the upper level occurs (this neglects background opacities which will be discussed further down). Intercombination lines from the doublet system back to the quartet system are included in the cascade, but not tracked any further after that. This implies that routes quartet → doublet → quartet → doublet are not included in the pumping rates. We expect the error introduced by this approximation to be negligible. The branching ratios were calculated using transition probabilities from Froese Fischer & Tachiev (2004). By combining all different routes in the cascade we could calculate a probability that an excitation of a given driving line would result in populating any of the metastable levels. The results of these calculations are shown in Table 7. The list of intercombination lines populating the doublet metastable levels after an excitation in the Ind1 driving line is given in Table 8.

In the previous discussion, we mentioned that transitions in any of the driving lines straight back to the ground level were not counted because these photons would simply be re-absorbed until a different cascade occurs. This assumption is not entirely correct as there is a finite probability P_dest that the photon is destroyed before it can be absorbed again (e.g., due to background opacities such as the grain opacity or bound-free opacity of elements with sufficiently low-ionization potentials). Additionally, there is a probability P_esc that the photon escapes from the cloud before it can be absorbed again. In order to account for these processes, we modify the excitation rate j₂ in s⁻¹ obtained from the two-level atom as follows:

$$\begin{equation} j_{\rm c} = j_2 \times \frac{1 - \beta }{1 - \beta (1 - P_{\rm dest} - P_{\rm esc})} , \end{equation} \tag{ A1 }$$

where β is a constant that gives the fraction of excitations in a driving line that is followed directly by a de-excitation back to the ground level. For Ind1 β = 0.7955 and for Dir1 β = 0.1384. For all other driving lines β < 0.01 and is assumed to be zero. Given this formula we can the write the total pump rate for each of the metastable levels as

$$\begin{equation} j_{\rm meta} = \sum _i P_{\rm pump}^{\, i} \, j_{\rm c}^{\, i} , \end{equation} \tag{ A2 }$$

where the summation runs over all the driving lines and the constants Pⁱ_pump are given in Table 7 for each of the metastable levels and driving lines.

For completeness we should also mention that pumping of the metastable states through recombination from N⁺ is also included in our modeling. We use the formulae given in Pequignot et al. (1991). These only give the rates to the full ²D and ²P metastable terms. In our modeling we split up these rates for each level according to statistical weight. This pumping mechanism will of course only be effective inside the ionized region as nitrogen has a slightly higher ionization potential than hydrogen.

From the data in Table 6 it is clear that all driving lines have wavelengths longward of the Lyman limit. This implies that the fluorescence mechanism is effective beyond the ionization front in the PDR. Since the temperature in the PDR is generally too low to collisionally excite the metastable doublet states, fluorescence can even become the dominant excitation mechanism for the forbidden N i lines in the PDR. It should also be noted that even very weak direct excitation lines can have a significant contribution to the fluorescence mechanism. If the PDR has sufficient column density, then all driving photons will eventually be absorbed. A low transition probability in the driving line only means that the effect is spread over a larger area.

The fluorescence mechanism will produce permitted N i emission lines that are observable in deep spectra. The cascade routes that populate the metastable levels will produce lines in the doublet system with wavelengths ranging between 8567 Å and 5.382 μm, as well as UV lines that cannot be observed from the ground. The shortest wavelength lines will tend to be the strongest since they come from the lowest levels where there are only a few alternative routes the cascade can take. In the ionized region these lines can also be produced by recombination from N⁺ → N⁰, but in the PDR these lines can only be produced by the fluorescence mechanism described here. So if these lines are observed in the PDR, it is conclusive proof for continuum pumping of the [N i] lines. In Table 9 we list all optical cascade lines with wavelengths shorter than 1 μm and in Table 10 we list the branching probabilities for each of the driving lines. Excitations by the Dir8 and Dir9 driving lines only produce UV cascade lines and are therefore not included in Table 10. Each driving line has its own characteristic spectrum. The relative strength of the contribution for each driving line depends on the incident spectrum, the optical depth in each of the driving lines, and the escape and destruction probability for the Ind1 and Dir1 driving lines. So no generic prediction for the spectrum can be made. However, a straight average indicates that the λλ9387, 9029, 9060, and 8629 lines will be the strongest, with the λ9387 line having about 1.8% of the flux of the λ5199⁺ doublet. It should be noted that the cascade lines shown in Table 9 are not predicted by Cloudy.

At this point we should discuss the accuracy of the transition probabilities of the intercombination lines. Accurate values for such lines are hard to obtain since they are quite sensitive to the details of the calculation. However, for direct excitation lines, accurate values for the transition probability are not crucial. Using the argument from the previous paragraph it becomes clear that an error in the transition probability would only imply that the absorption of the driving photons would happen over a smaller or larger area, but the total amount of pumping would remain the same when integrated over the entire PDR. This of course assumes that the PDR is optically thick. If that is not the case, then an error in the transition probability would alter the escape probability of the driving line. In such circumstances accurate transition probabilities are needed. For indirect excitation lines, accurate transition probabilities are always needed (even when the PDR is optically thick) since the intercombination line has to compete with stronger, fully allowed transitions in the cascade down from the upper level of the driving line. However, this problem is mitigated by the fact that there is only one indirect driving line versus nine direct driving lines. So, an error in this component would only have a limited effect on the total pumping. In Table 11, we compare the transition probabilities of the lines involved in the cascade down from the indirect excitation using data from Hibbert et al. (1991, length form) and Froese Fischer & Tachiev (2004). It is apparent that discrepancies up to 1 dex and more can occur, indicating the difficulty in calculating these data.

If the lines were collisionally excited then the electron density could be determined from the ratios of the intensities of two lines of the same ion, emitted by different levels with nearly the same excitation energy (AGN3). Temperature is indicated by emission from levels with different excitation energies. Together, the gas pressure could be directly measured. This can test whether the lines are thermally excited, and is useful for reference by future studies which will look into the formation of [N i] lines in planetary nebulae, the Crab Nebula, and cool-core cluster filaments.

We show several emission line diagnostics for the collisionally dominated case, using our updated atomic data and model atom. Line pairs such as the ratio of lines of [N i]

$$\begin{equation} R_n = {I(^2D_{5/2} \rightarrow \,^4S_{3/2})} / {I(^2D_{3/2} \rightarrow \,^4S_{3/2})} = \lambda 5200/\lambda 5198 \vspace*{4pt} \end{equation} \tag{ A3 }$$

indicate the electron density in gaseous nebulae, as shown by Seaton & Osterbrock (1957) and Saraph & Seaton (1970) for [O ii]. Note that the energy order of the J levels within the ²D term depends on both the charge and electronic configuration. The line ratio is defined so that it decreases as density increases.

Every collisional excitation is followed by the emission of a photon in the low-density limit. Since the relative excitation rates of the ²D_5/2 and ²D_3/2 levels are proportional to their collision strengths, the ratio is

$$\begin{equation} R_n (n_e \rightarrow 0) = \frac{ \Upsilon (^2D_{5/2} - ^4S_{3/2})}{\Upsilon (^2D_{3/2} - ^4S_{3/2}) } = \frac{0.337}{0.224} = 1.5. \end{equation} \tag{ A4 }$$

This is valid when kT ≫ δ, where δ is the difference in energies of the upper levels. This holds for all temperatures where the optical lines emit due to the small energy difference of the upper levels. In the high-density limit collisional processes dominate and set up a Boltzmann level population distribution. The relative populations of the ²D_5/2 and ²D_3/2 levels are in the ratio of their statistical weights, and the relative intensities of the two lines are in the ratio

$$\begin{equation} R_n \left({n_e \rightarrow \infty } \right) = \frac{\omega (^2D_{5/2})A_{\lambda 5200} }{{\omega ({}^2D_{3/2})A_{\lambda 5198} }} = \frac{3}{2}\frac{7.57 \times 10^{ - 6} }{2.03 \times 10^{ - 5} } = 0.60. \vspace*{4pt} \end{equation} \tag{ A5 }$$

The line ratio varies between these intensity limits as the density varies. The critical density, the density where the collisional and radiative de-excitation rates are equal, is n_crit ∼ 10³ cm⁻³ at ∼10⁴ K.

Figure 8 compares the [N i] R_n with the more commonly used [O ii], [S ii], and [Cl iii] density indicators using data from B2000. The [O ii] collision strengths computed by Kisielius et al. (2009) were used. The behavior of these curves is qualitatively similar, going to the ratio of statistical weights at low densities and a ratio that depends on the radiative transition probabilities at high densities.

Zoom In Zoom Out Reset image size

The gas kinetic temperature can be determined from ratios of intensities from two levels with considerably different excitation energies. For [N i] we have the ratio

$$\begin{eqnarray} R_T &=& {I(^2P \rightarrow \,^4S)} / {I(^2D \rightarrow \,^4S)} \nonumber\\ &=& (\rm 3466.49 + 3466.54)/(5198 + 5200)\nonumber\\ & =& \lambda 3467^+/\lambda 5199^+ \, . \end{eqnarray} \tag{ A6 }$$

These density–temperature indicators can be combined to form a unified diagnostic diagram, as has long been done for [O iii] (AGN3 Figure 5.12). Figure 9 shows calculated curves of the values of the two [N i] intensity ratios for various values of T and n_e.

Zoom In Zoom Out Reset image size

We used the line intensities measured in bright central regions (B2000; Esteban et al. 2004, hereafter E2004) to estimate n_e and T. B2000 presented high-resolution spectrophotometric observations of the Orion Nebula in the 3500–7060 Å range. Their slit position was 37'' west of θ¹ Ori C. This is close to the position modeled by Baldwin et al. (1991). E2004 covered the 3100–10400 Å range. Their slit was oriented east–west and centered at 15 arcsec south and 10 arcsec west of θ¹ Ori C.

R_n was 0.60 ± 0.02 for the average of the blue and red spectra in B2000 and 0.59 for E2004. This is plotted in both Figures 8 and 9. The results are surprising—the electron density indicated by the [N i] lines, which should form in partially ionized gas giving lower electron densities, is 0.2–0.4 dex larger than densities indicated by lines which form in highly ionized regions.

It is not now possible to measure the temperature using the λ3467⁺ line. We know of no detection of this line in the Orion environment. However Esteban et al. (1999) reported the upper limit of I(3467⁺)/I(Hβ) < 10⁻³ from the data presented by Osterbrock et al. (1992). This corresponds to λ3467⁺/λ5199⁺ < 0.22.

These values of the line ratios are shown in Figure 9. The temperature limit indicated by the line is consistent with formation in a photoionized environment. The high density would be surprising if true. Actually this can be taken as independent evidence that the lines do not form by collisional excitation.

Störzer & Hollenbach (2000) show that significant optical [O i] emission can result from dissociation of oxygen-bearing molecules. Could an analogous process contribute to the [N i] emission we observe in Orion?

Störzer & Hollenbach (2000) consider OH photodissociation and subsequent [O i] 6300+ emission in detail. The intensity of the line that is produced depends on the photodissociation rate, the branching ratio for populating the excited level producing a particular line, and the extinction between the point where the emission is produced and the surface of the cloud.

We include molecular photodissociation by the processes included in a modified version of the UMIST (Le Teuff et al. 2000) database (Röllig et al. 2007). Our original treatment, described in Abel et al. (2005), considered each reaction on an ad hoc basis. In our upcoming release we will generalize our treatment of the chemistry to more systemically consider reactions, their inverses, and maintain an accounting of the consistency (Williams et al., in preparation). This is a step toward treating the chemistry as a coupled system that is driven by external databases.

Using the results from the chemistry network we can identify all photodissociation processes. The N-bearing molecules NH, CN, N₂, NO, and NS produce N⁰ following photodissociation. We save this photodissociation rate per unit volume at each point in the cloud and assume that each dissociation produces N⁰ in the ²D^o level. Using that we can estimate the contribution of this pumping process to the production of the λ5199⁺ lines. The observed emission is predicted by attenuating the local emission by the absorption optical depth from the creation point to either side of the cloud. Grains are the dominant opacity source at optical wavelengths for conditions similar to the Orion Nebula. This produces an upper limit to the emission because of the assumption that 100% of photodissociations produce N⁰ in the excited state producing [N i] 5199⁺. This upper limit is added to the flux of the λ5199⁺ lines.

Figure 4 shows that there are regions of the cloud where photodissociation could make [N i] emission. However, this is deep enough within the cloud that the process makes no significant contribution to the observed flux. The process may be important in other environments, however.

This physics is included in the current release of Cloudy (C10.00).

In this Appendix, we develop a simple framework for evaluating the intensity of radiatively driven continuum and line processes in a photoionized nebula, which will elucidate the dependence of line ratios and equivalent widths on the shape of the exciting stellar spectrum and on geometric factors. All three emission mechanisms, dust continuum, [N i], and Hβ lines, can be thought of as diffuse reflection or scattering processes in the broadest sense, with each being driven by a different wavelength band of the stellar continuum⁶.

• 1.  
  Dust continuum is coherent scattering in the optical sense of visual band photons (∼5000 Å).
• 2.  
  Fluorescent [N i] is highly incoherent scattering of FUV pumping photons (∼1000 Å) into visual photons.
• 3.  
  To the degree that static photoionization equilibrium holds, then Hβ emission is scattering (albeit in a statistical and indirect way) of ionizing EUV photons (<912 Å) into visual photons.

In each case, one can define an effective albedo ϖ, which is an efficiency factor that relates the intensity of scattered or emitted photons to the intensity of incident photons (see Figure 10 and Appendix B.1 below). Therefore, any variation in the observed intensity ratios must be due to either (1) variations in the SED of the stellar radiation field, or (2) variation in the effective albedos, or (3) a breakdown of the simplifying assumption of a single infinite plane-parallel scattering layer. In the remainder of this appendix we discuss in detail the contributions of (2) and (3), while the role of (1) is explored in Section 4.1 above.

Zoom In Zoom Out Reset image size

B.1. Formal Calculation of Intensity Ratios and Equivalent Widths

Under this black-box "scattering" or "reprocessing" description, the efficiency of the scattering can be described by an effective albedo ϖ_eff (see Figure 10), so that the photon intensity I for each line or continuum process is proportional to the local continuum flux F₀ in the spectral band that excites the scattering process:

$$\begin{equation} I = \frac{F_0 \varpi _{\mathrm{eff}}}{4 \pi } \ \mathrm{photons\ s^{-1}\ cm^{-2}\ sr^{-1}}, \end{equation} \tag{ B1 }$$

where

$$\begin{equation} F_0 = \frac{\langle \lambda L_\lambda \rangle _{\mathrm{band}}(\Delta \lambda)_{\mathrm{band}}}{4 \pi R^2 h c} \ \mathrm{photons\ s^{-1}\ cm^{-2}}. \end{equation} \tag{ B2 }$$

In this expression, R is the distance from the star to the scattering layer, and 〈λL_λ〉_band and (Δλ)_band are respectively the mean SED and wavelength width of the continuum band that excite the process. In general, the albedo will be a function of the illumination angle and viewing angle (Figure 10). The particular values of the albedo for the production of the Hβ recombination line, the [N i] fluorescent lines, and dust-scattered continuum in the nebula are calculated in Appendix B.2 below.

The line ratios and equivalent widths measured in Section 2 will then be given by

$$\begin{equation} \frac{I([{{\rm N}\,\mathsc {i}}])}{I(\mathrm{H}\beta)} = \frac{\varpi _{\scriptscriptstyle 5199}\, \langle \lambda L_\lambda \rangle _{\scriptscriptstyle \mathrm{FUV}}\, (\Delta \lambda)_{\scriptscriptstyle \mathrm{FUV}}}{\varpi _{\mathrm{H}\beta }\, \langle \lambda L_\lambda \rangle _{\scriptscriptstyle \mathrm{EUV}}\, (\Delta \lambda)_{\scriptscriptstyle \mathrm{EUV}}} \end{equation} \tag{ B3 }$$

$$\begin{equation} \mathrm{EW}(\mathrm{H}\beta , \mathrm{Corr})= \frac{\varpi _{\mathrm{H}\beta }\, \langle \lambda L_\lambda \rangle _{\scriptscriptstyle \mathrm{EUV}}\, (\Delta \lambda)_{\scriptscriptstyle \mathrm{EUV}}}{\varpi _{\mathrm{\scriptscriptstyle dust}}\, \langle \lambda L_\lambda \rangle _\mathrm{\scriptscriptstyle vis}} \end{equation} \tag{ B4 }$$

$$\begin{equation} \mathrm{EW}([{{\rm N}\,\mathsc {i}}], \mathrm{Corr})= \frac{\varpi _{\scriptscriptstyle 5199}\, \langle \lambda L_\lambda \rangle _{\scriptscriptstyle \mathrm{FUV}}\, (\Delta \lambda)_{\scriptscriptstyle \mathrm{FUV}}}{\varpi _{\mathrm{\scriptscriptstyle dust}}\, \langle \lambda L_\lambda \rangle _\mathrm{\scriptscriptstyle vis}}. \end{equation} \tag{ B5 }$$

A particularly simple limiting case is provided by the situation in the extreme outskirts of the Extended Orion Nebula. For these regions studies have shown that, except for the lowest ionization lines, essentially all the radiation, including the emission lines, is scattered by dust rather than being emitted locally (O'Dell & Goss 2009; O'Dell & Harris 2010). For positions far outside the bright core of the nebula, it is reasonable to make the additional assumption that the angular distribution of the incident radiation (as seen by the scatterers) is on average similar for the continuum (which comes from the star cluster) and for the emission lines (which come from the nebular gas). That being the case, all geometrical factors will cancel out and the effective albedo for an emission line will be the same as that for the adjacent continuum, so that the equivalent width of the line will be simply EW(Int) = L_line/L_λ, where L_line is the total intrinsic line luminosity of the nebula and L_λ is the total intrinsic continuum luminosity of the star cluster.⁷ One would therefore expect that the observed corrected equivalent widths should tend to a constant value of EW(Int) in the extreme outskirts of the nebula. Just such a behavior is seen in the observed Hβ equivalent width (O'Dell & Harris 2010, Figure 8), which around the outer rims of the nebulae tends to a value of ∼150 Å for M42 and ∼100 Å for M43. Comparison with the intrinsic Hβ equivalent widths in Table 2 shows good agreement in the case of the M43, although for M42 the predicted value of 213 Å is rather higher than is observed.

In Table 2 we show SED ratios between different wavebands for OB stellar populations characteristic of the inner Orion Nebula (Trapezium region), the outer Orion Nebula, and M43. These can be compared with different observed emission ratios shown in Figures 1 and 6: FUV/visual corresponds to EW([N i], Corr), EUV/visual corresponds to EW(Hβ, Corr), and FUV/EUV corresponds to I([N i])/I(Hβ).⁸

B.2. Estimation of Effective Albedos for Particular Processes

B.2.1. Hβ Recombination Line

Assuming a thin, plane-parallel, ionization-bounded layer of dust-free hydrogen that is illuminated by ionizing photons with a flux $F_{\scriptscriptstyle \mathrm{EUV}}$ (photons s⁻¹ cm⁻²) incident from a direction μ₀, then the condition of global static photoionization equilibrium is given by

$$\begin{equation} \mu _0 F_{\scriptscriptstyle \mathrm{EUV}}= \int \alpha _{\mathrm{B}}\, n_{\mathrm{p}}n_{\mathrm{e}}\, dz, \end{equation} \tag{ B6 }$$

in which α_B is the "Case B" recombination coefficient and n_p, n_e are the proton and electron densities. At the same time, the emergent intensity I (photons s⁻¹ cm⁻² sr⁻¹) of the recombination line Hβ is given by

$$\begin{equation} I = \frac{1}{4\pi \mu } \int \alpha _{\mathrm{H}\beta } \, n_{\mathrm{p}}n_{\mathrm{e}}\, dz, \end{equation} \tag{ B7 }$$

where α_Hβ is an effective recombination coefficient that only includes those recombinations that give rise to the emission of an Hβ photon. Combining these, the effective albedo (see Figure 10) is found to be

$$\begin{equation} \varpi _{\mathrm{H}\beta } = \left\langle \!\frac{\alpha _{\mathrm{H}\beta }}{\alpha _{\mathrm{B}}}\!\right\rangle \, \frac{\mu _0}{\mu }, \end{equation} \tag{ B8 }$$

where 〈α_Hβ/α_B〉 ≃ 0.12 for typical H ii region conditions (Osterbrock & Ferland 2006).

The correction to this result for the presence of helium will be small, but the presence of dust in the ionized gas may have a much larger effect. Both the incident ionizing radiation and the emergent emission line will be affected by dust absorption. The fraction $f_{\mathrm{\scriptscriptstyle dust}}$ of ionizing photons that are absorbed by dust is an increasing fraction of the ionization parameter (Aannestad 1989; Arthur et al. 2004), but for the conditions found in Orion reaches a maximum value of about 30% so long as the illumination is close to face-on. The effect is greater for edge-on illumination, but such cases, with μ₀ ≪ 1, have already a small albedo and so will contribute little to the observed emission so long as a variety of illumination angles is present (see discussion in Appendix B.3). The absorption of emergent Hβ photons is more important since this is largest for precisely those cases μ ≪ 1 which would give the highest albedo in the dust-free case (Equation (B8)). If the dust absorption optical depth of the scattering layer at the wavelength of Hβ is τ, then in the approximation that the ionized density is constant Equation (B8) becomes

$$\begin{equation} \varpi _{\mathrm{H}\beta } = \left(1 - \left\langle f_{\mathrm{\scriptscriptstyle dust}}\right\rangle \right) \left\langle \!\frac{\alpha _{\mathrm{H}\beta }}{\alpha _{\mathrm{B}}}\!\right\rangle \, \frac{\mu _0}{\tau } (1 - e^{-\tau /\mu } ). \end{equation} \tag{ B9 }$$

The maximum relative boost in the albedo due to limb brightening as μ → 0, which is infinite in the dust-free case, is now limited to 1/(1 − e^−τ), which is a factor of 3–5 for the values of τ ≃ 0.1–0.3 expected in Orion. Note that scattering by dust of the Hβ photons is ignored in this approximation.

B.2.2. Fluorescent [N i]

The FUV fluorescent pumping of the optical [N i] lines is only efficient at wavelengths where the opacity of the pumping line exceeds the background continuum opacity, which at FUV wavelengths is dominated by dust. This gives a limit δλ = λδv/c to the wavelength interval that contributes to the pumping, where δv is of order the Doppler width of the line.⁹ If there are a number N_line pumping lines, each of effective width δv, then the fraction of the total FUV continuum that contributes to the pumping is ${\simeq } N_{\mathrm{line}}(\delta v / c) (\langle \lambda \rangle _{\scriptscriptstyle \mathrm{FUV}}/ (\Delta \lambda)_{\scriptscriptstyle \mathrm{FUV}})$, where $\langle \lambda \rangle _{\scriptscriptstyle \mathrm{FUV}}$ is the average wavelength of the pumping lines and $(\Delta \lambda)_{\scriptscriptstyle \mathrm{FUV}}$ is the wavelength width of the FUV band.¹⁰ If a fraction $f_{\scriptscriptstyle 5199}$ of all pumps results in the emission of a line in the optical [N i] λλ5198, 5200 doublet, then the effective albedo for "scattering" of FUV continuum into these lines is

$$\begin{eqnarray} \varpi _{\scriptscriptstyle 5199} &=& \frac{ 4\pi I_{\scriptscriptstyle 5199}}{ F_{\scriptscriptstyle \mathrm{FUV}}} = f_{\scriptscriptstyle 5199}N_{\mathrm{line}}\left(\frac{\delta v}{c}\right) \nonumber\\ && \times \left(\frac{\langle \lambda \rangle _{\scriptscriptstyle \mathrm{FUV}}}{(\Delta \lambda)_{\scriptscriptstyle \mathrm{FUV}}}\right) \left(\frac{\mu _0}{\mu }\right)\, e^{-\tau _{\scriptscriptstyle \mathrm{FUV}}/\mu _0} \, e^{-\tau _{\scriptscriptstyle 5199}/\mu }, \quad \end{eqnarray} \tag{ B10 }$$

where $\tau _{\scriptscriptstyle \mathrm{FUV}}$ and $\tau _{\scriptscriptstyle 5199}$ are the continuum absorption optical depths between the star and the pumping layer, measured perpendicular to the layer, and at the wavelengths of the pumping FUV lines and the emerging optical lines, respectively.

The opacity of the FUV pumping lines τ_pump is proportional to the abundance of N⁰, and so is very low inside the ionized gas, rising suddenly at the ionization front. Therefore, for strong pumping lines, the fluorescent excitation (which peaks at τ_pump ≃ μ₀), is concentrated in a thin layer just behind the ionization front so that $\tau _{\scriptscriptstyle \mathrm{FUV}}$ and $\tau _{\scriptscriptstyle 5199}$ are insensitive to variations in the illumination cosine μ₀. For the weakest pumping lines, on the other hand, the pumping layer extends deeper into the neutral PDR and so $\tau _{\scriptscriptstyle \mathrm{FUV}}$ and $\tau _{\scriptscriptstyle 5199}$ are generally larger and become roughly proportional to μ₀.

Figure 11(b) shows results for $\varpi _{\scriptscriptstyle 5199}$ for the case of perpendicular illumination μ₀ = 1 and assuming N_line = 10, δv = 10 km s⁻¹, $f_{\scriptscriptstyle 5199}= 0.1$, $\langle \lambda \rangle _{\scriptscriptstyle \mathrm{FUV}}/ (\Delta \lambda)_{\scriptscriptstyle \mathrm{FUV}}= 4.3$, and $\tau _{\scriptscriptstyle \mathrm{FUV}}= 1.5 \tau _{\scriptscriptstyle 5199}$. It can be seen that the same optical depth of dust has a considerably larger effect on the [N i] albedo than on the Hβ albedo, particularly for oblique viewing angles (small μ). This is because the dust absorption layer completely overlies the fluorescent scattering layer in the [N i] case, whereas in the case of Hβ the dust is mixed in with the line-emitting gas. As a result, whereas the Hβ albedo ϖ_Hβ simply saturates at small μ, the [N i] albedo $\varpi _{\scriptscriptstyle 5199}$ has a maximum at $\mu = \tau _{\scriptscriptstyle 5199}$ and then drops to zero as μ → 0.

Zoom In Zoom Out Reset image size

B.2.3. Scattered Starlight

The dust scattering of starlight in the nebula can be divided into two parts: (1) back-scattering by dust in the PDR and molecular cloud located behind the nebula, which has a high optical depth, and (2) small-angle scattering by dust located in the diffuse clouds in front of the nebula (the neutral veil, Abel et al. 2004, 2006), which has a smaller optical depth (τ = 0.1–1; O'Dell & Yusef-Zadeh 2000). In both cases, the results will be sensitive to the optical properties of the dust grains, which at the simplest level can be characterized by the single-scattering albedo ϖ₀, which is the probability that a photon interacting with a grain is scattered rather than absorbed, and the asymmetry parameter g, which is the mean cosine of the scattering angle (g = 0 for isotropic scattering). Dust in Orion is found to have a high value of the total/selective extinction ratio R_V ≃ 5, possibly due to grain coagulation (Cardelli & Clayton 1988). Theoretical calculations of the optical properties of a grain population with this value of R_V (Figure 4 of Draine 2003) imply that at optical wavelengths (∼5000 Å) the albedo is relatively high (ϖ₀ ≃ 0.8) and the scattering is moderately forward-throwing (g ≃ 0.6), whereas at FUV wavelengths (∼1000 Å) the albedo is lower (ϖ₀ ≃ 0.4) and the scattering is extremely forward-throwing (g ≃ 0.8). Observations in Orion of scattered FUV continuum (Shalima et al. 2006) and scattered optical emission lines (Section 3.1 of Henney 1998) are consistent with these values, although in both cases it is only a combination of ϖ₀ and g that is constrained. Earlier studies (Schiffer & Mathis 1974; Mathis et al. 1981; Patriarchi & Perinotto 1985) have found different values, and even evidence that the dust properties vary with position, but the results are very sensitive to the assumed geometry of the scattering. In the following we present results for both high-albedo and low-albedo grains, which can be taken as representative of the range of possible optical grain properties at visual and FUV wavelengths.

The problem of back-scattering by the molecular cloud has a well-known solution in the case of isotropic scattering (Chandrasekhar 1960), giving an effective albedo of

$$\begin{equation} \varpi _{\mathrm{\scriptscriptstyle dust}}= \varpi _0 \,\frac{\mu _0}{\mu + \mu _0}\, H(\mu)\, H(\mu _0) \end{equation} \tag{ B11 }$$

where H(μ) is the Chandrasekhar H-function. An approximate analytic form for the H-function (Henney 1998), accurate to <5% for ϖ₀ ⩽ 0.9, is

$$\begin{equation} H(\mu) = 1 + 0.5 \varpi _0\, \mu \big(1 + 1.8 \mu ^{0.4} \varpi _0^2 \big) \ln (1 + \mu ^{-1}). \end{equation} \tag{ B12 }$$

For the more relevant case of asymmetric scattering (g ≠ 0), the problem is more difficult to solve, and is no longer axially symmetric unless μ₀ = 1. However, for illumination angles that are not far from face-on, a good approximation is found by simply multiplying the isotropic results by a factor of (1 − g)^3/2 (Henney 1998). The results of this approximation are shown in Figure 12(a), where it is seen that typical values of $\varpi _{\mathrm{\scriptscriptstyle dust}}= 0.2$–0.3 are obtained at visual wavelengths, but much smaller values ($\varpi _{\mathrm{\scriptscriptstyle dust}}< 0.05$) are seen at FUV wavelengths. In both cases, the scattering is approximately Lambertian (brightness independent of viewing angle) when the illumination is close to face on. Note however, that this approximation ignores the fact that as μ₀ is decreased, then the forward-throwing part of the phase function begins to be sampled at small μ for favorable viewing azimuths ϕ, which would tend to increase the limb brightening for μ₀ < 1.

Zoom In Zoom Out Reset image size

For the case of forward scattering, one can use the results for diffuse transmission through a homogeneous plane-parallel layer of optical thickness τ (Chandrasekhar 1960), where the effective albedo can be expressed in terms of Chandrasekhar's X and Y functions:

$$\begin{eqnarray} \varpi _{\mathrm{\scriptscriptstyle dust}} &=& \frac{\mu _0}{\mu - \mu _0} \varpi _0\, \Phi (\mu , \mu _0, \phi) [ Y(\mu , \tau) X(\mu _0, \tau)\nonumber\\ && - X(\mu , \tau) Y(\mu _0, \tau) ]. \end{eqnarray} \tag{ B13 }$$

where Φ is the scattering phase function and X and Y depend implicitly on Φ and ϖ₀. In the limit of small τ, it is sufficient to include only single scattering, which yields the approximation X⁽¹⁾(μ, τ) = 1, Y⁽¹⁾(μ, τ) = e^−τ/μ. For multiple scattering in the isotropic case, extensive tables have been published for X and Y (e.g., Mayers 1962) and we find that an acceptable approximation to these results is given by

$$\begin{equation} X(\mu , \tau) \simeq 1 + 0.75 A, \quad \quad Y(\mu , \tau) \simeq (1 + 1.5 A)\, e^{-\tau /\mu } , \end{equation} \tag{ B14 }$$

where

$$A = \varpi _0^2 \,\frac{2 \tau }{1 + \tau }\, \mu ^{\tau /(1+\tau)}.$$

This approximation is accurate to <10% for all the cases covered by Mayers (1962) (τ = 0.1–5, ϖ₀ = 0.5–1). To extend this result to forward-throwing phase functions, we assume that the anisotropy can be neglected for all orders of scattering higher than the first, so that Equations (B14) and (B13) may be directly combined.

Example results for scattering from foreground dust in this approximation are shown in Figure 12(b), assuming ϕ = 45° and a Henyey–Greenstein form for the scattering phase function:

$$\begin{eqnarray} \Phi (\mu , \mu _0, \phi) &=& \frac{1-g^2}{(1 + g^2 - 2 g \mu _\mathrm{s})^{3/2}} \nonumber\\ \mathrm{where} \quad \mu _\mathrm{s} &=& \mu \mu _0 + (1-\mu ^2)^{1/2} (1-\mu _0^2)^{1/2} \cos \phi. \quad\quad \end{eqnarray} \tag{ B15 }$$

The results are shown for a fixed value of the optical depth measured along the line of sight, τ/μ, since it is this quantity that is constrained by observations of the extinction in the neutral veil. Therefore, the actual thickness of the layer τ goes to zero as μ → 0 and no limb brightening is seen. Instead, the albedo tends to have a maximum when μ ≃ μ₀ since this maximizes Φ for the small values of ϕ considered here. It can be seen that the effective albedo is generally of order ϖ₀τ/μ, although it can be several times larger than this for favorable combinations of μ, μ₀, and ϕ that give sufficiently small scattering angles (μ_s > 0.8 for the optical-band grain properties, or μ_s > 0.9 for the FUV-band grain properties).

B.3. Variations within the Nebula of the Illumination and Viewing Angles

The effective scattering albedos derived in the previous sections are strong functions of the angle of illumination of the scattering layer μ₀ and of the observer's viewing angle μ, with the albedo generally being highest when the illumination is close to face-on (μ₀ ≃ 1) and the view is close to edge-on (μ ≃ 0). It is therefore important to consider whether these angles vary systematically between the core and the outskirts of the nebula. This depends critically on the large-scale geometry of the scattering layers within the nebula. For instance, if the nebula were a simple hemispherical shell centered on the Trapezium stars (illustrated in Figure 13(a)), then the illumination angle would be constant at μ₀ = 1 while the viewing angle would vary from μ = 1 in the center to μ = 0 at the edge. On the other hand, if the nebula were a plane-parallel layer (as in the models of Henney et al. 2005a and illustrated in Figure 13(b)), then μ would be constant, whereas μ₀ would vary from ≃ 1 at the center to ≃ 0 toward the edges. In reality, neither of these simple geometries works well as model for the nebula. In particular, the hemispherical-shell model would predict a constant ionized gas density and a surface brightness that increases with radius, both in violent disagreement with observations. On the other hand, the plane-layer model fails to explain the fine-scale structure seen in many emission lines (e.g., O'Dell & Yusef-Zadeh 2000; García-Díaz & Henney 2007), as well as the sharp edge of the EON. For an observational aperture that is larger than the angular size of the individual emission structures, the observed emission will be biased toward face-on illumination angles and edge-on viewing angles, simply because those are the cases that give the highest effective albedo.

Zoom In Zoom Out Reset image size

B.4. Breakdown of the Infinite Plane-parallel Layer Approximation