RESOLVING THE ELECTRON TEMPERATURE DISCREPANCIES IN H ii REGIONS AND PLANETARY NEBULAE: κ-DISTRIBUTED ELECTRONS

Initially, κ-distributions were used as an empirical fit to directly measure electron energies in solar system plasmas, and were criticized as lacking a theoretical basis. More recently, the distribution has been shown to arise from entropic considerations. See, for example, Tsallis et al. (1995), Treumann (1999), Leubner (2002), and the comprehensive analysis by Livadiotis & McComas (2009). They explored the so-called q-nonextensive entropy statistics in which the entropies of adjacent samples of plasma are not simply additive, and have shown that κ energy distributions arise naturally in such plasmas. The requirement for this to occur is that there be long-range interactions between particles, in addition to the short-range Coulombic forces that give rise to M-B equilibration. Although there is ongoing debate over whether the Tsallis statistical mechanics is the best generalization of Boltzmann–Gibbs statistics, it provides a sound physical basis for the overtly successful use of the κ-distribution in plasma physics.

There are a number of slightly differing expressions for the energy distributions that can arise from q-nonextensive entropy, but the forms are generally similar. Here, we adopt the Vasyliunas form of the distribution as representative of the possible variants. This is referred to by Livadiotis & McComas (2009) as a κ-distribution of the "second kind." The successful use of the κ-distributions to describe physical phenomena in many disciplines, and especially in solar system physics, provides ample justification for exploring their use in H ii regions and PNe.

The κ-velocity distribution can be expressed (after Vasyliunas 1968) as

$$\begin{equation} n(v) {d}v\!=\!\frac{4 N}{\sqrt{\pi }w_0^3}\!\left(\!\frac{\Gamma(\kappa +1)}{\kappa ^{3/2} \Gamma \big(\kappa -\frac{1}{2}\big)} \!\right)\!\!\frac{v^2}{\big(1 + v^2/ \left[(\kappa - \frac{3}{2})w_0^2 \right]\big)^{\kappa + 1}} {d}v, \end{equation} \tag{ 1 }$$

where n(v) is the number of electrons with speeds between v and v + dv. The velocity w₀ is related to w_mp, the most probable speed (i.e., the velocity value at the distribution peak) by

$$\begin{equation} w_0 = w_{{\rm mp}} \left({\frac{\kappa }{ \big(\kappa - {\frac{3}{2}}\big)}}\right)^{1/2} \end{equation} \tag{ 2 }$$

and related to the "physical temperature" of the system, T_U, as defined in Livadiotis & McComas (2009):

$$\begin{equation} w_0 = \left({\frac{2 k_BT_U}{m_e}}\right)^{1/2}, \end{equation} \tag{ 3 }$$

where m_e is the electron mass and k_B is the Boltzmann constant.

κ is a parameter that describes the extent to which the distribution departs from an equilibrium distribution. In the Vasyliunas form, (3/2) < κ ⩽ ∞. In the limit as κ → ∞, the velocity distribution reverts to the standard M-B form:

$$\begin{equation} n(v) {d}v=\frac{4 N}{\sqrt{\pi }w_0^3} v^2 \exp \left[-\frac{v^2}{w_0^2}\right] {d}v. \end{equation} \tag{ 4 }$$

The physical temperature, T_U, also referred to as the kinetic temperature, is a generalization of the M-B equilibrium temperature, and is related to the energy density (system kinetic energy), U, which for a monatomic gas is the internal energy of the system:

$$\begin{equation} U = \frac{3}{2}k_BT_U. \end{equation} \tag{ 5 }$$

Figure 1 shows a family of κ velocity distributions with an M-B distribution.

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Expressed in energy terms, the κ-distribution becomes

$$\begin{eqnarray} n(E) {d}E&=&\frac{2 N}{\sqrt{\pi }} \left(\frac{\Gamma (\kappa +1)}{\big(\kappa -\frac{3}{2}\big)^{3/2} \Gamma \big(\kappa -\frac{1}{2}\big)} \right)\nonumber\\ &&\times\frac{\sqrt{E}}{(k_BT_U)^{3/2} \big(1 + E/ \left[\big(\kappa -\frac{3}{2}\big) k_BT_U \right]\big)^{\kappa + 1}} {d}E.\nonumber\\ \end{eqnarray} \tag{ 6 }$$

Again, the κ energy distribution tends in the limit as κ → ∞ to the M-B:

$$\begin{equation} n(E) {d}E=\frac{2 N}{\sqrt{\pi }} \frac{\sqrt{E} \ \exp \left[-E/k_BT\right]}{(k_BT)^{3/2}} {d}E. \end{equation} \tag{ 7 }$$

A family of normalized κ energy distributions is shown in Figure 2, together with an M-B equilibrium distribution. Note that the peak of the M-B energy distribution occurs at E = (1/2)k_BT_U, whereas κ-distributions with the same internal energy peak at E = (1/2)k_BT_U (2κ − 3)/(2κ + 1).

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The same distributions are shown in Figure 3 but plotted on a log scale. The high-energy power-law tail of the κ-distributions is clearly shown.

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Figures 1–3 illustrate the key characteristics of the κ-distribution: the peak of the distribution moves to lower energies; at intermediate energies there is a population deficit relative to the M-B distribution; and at higher energies the "hot tail" again provides a population excess over the M-B. The κ-distribution behaves as an M-B distribution at a lower temperature, but with a significant high-energy excess.

This can be seen in Figure 4, where an M-B distribution is peak-fitted to a κ = 2 distribution. The peak-fitted "core" M-B distribution is in fact at a lower physical temperature than the κ-distribution to which it is fitted. The low-energy "core" occurs because the equilibration timescale is a strong function of energy (∝exp E^3/2), allowing the low-energy electrons to approach an equilibrium distribution. The relationship between the physical temperature of the κ-distribution, T_U, and the equilibrium temperature of the "core" M-B, T_core, as implied by Equation (3), is

$$\begin{equation} T_{{\rm core}}=T_U\left(\frac{\kappa -\frac{3}{2}}{\kappa }\right). \end{equation} \tag{ 8 }$$

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For physical processes that involve low-energy electrons such as recombination line excitation, reactions "see" the cool M-B core distribution. In other words, any physical property sensitive to the region of the electron energy distribution around or below the distribution peak will interact with an effective M-B electron energy distribution at a lower temperature than the M-B distribution with the same total internal energy as the κ-distribution.

Conversely, in processes that depend on the high energy end of the distribution, such as collisional line excitation, the κ-distribution behaves like an M-B distribution at a higher temperature than one with the same total internal energy. In other words, physical properties sensitive to energies above the peak will behave as if they were interacting with a hotter distribution. The κ-distribution thus exhibits a "split personality," depending on the physical process involved. We show below that this behavior resolves many of the discrepancies between different nebular temperature measurement techniques.

In addition to the energy injection mechanisms capable of maintaining the excitation of suprathermal distributions, several authors (Livadiotis & McComas 2011 and references therein; Shizgal 2007; Treumann 2001) have investigated the possibility that the κ-distribution may remain stable against thermalization longer than conventional thermalization considerations would suggest (e.g., Spitzer 1962). In particular, distributions with 2.5 ≳ κ > 1.5 appear to have the capacity, through increasing entropy, of moving to values of lower κ (Livadiotis & McComas 2011) i.e., away from (M-B) equilibrium. While the physical application of this aspect of κ-distributions remains to be explored fully, it suggests that where q-nonextensive entropy conditions operate, the suprathermal energy distributions produced exist in "stationary states" where the behavior is, at least in the short term, time-invariant (Livadiotis & McComas 2010). These states may have longer lifetimes than expected classically. This is fully consistent with the numerous observations that in solar system plasmas, κ electron and proton energy distributions are the norm. It seems reasonable to expect that such conditions will also be present in H ii regions and PNe.

Consider the collisional excitation of an atomic species from energy level 1 to energy level 2. The rate of collisional population of the upper energy level per unit volume is given in terms of the collision cross-section, σ₁₂(E), by

$$\begin{equation} R_{12}=n_e N_1 \int _{E_{12}}^\infty \sigma _{12}(E) \sqrt{E} f(E) {d}E. \end{equation} \tag{ 9 }$$

We separate out the strong energy dependence of the collision cross-section σ₁₂(E), by expressing it in terms of the collision strength, Ω₁₂ and the energy, E:

$$\begin{equation} \sigma _{12}(E)=\left(\frac{h^2}{8\pi m_e E}\right)\frac{\Omega _{12}}{g_1}, \end{equation} \tag{ 10 }$$

where h is the Planck Constant, m_e is the electron mass, and g₁ is the statistical weight of the lower energy state.

The collisional population rate now becomes

$$\begin{equation} R_{12}=n_e N_1 \frac{h^2}{8 \pi m_e g_1}\int_{E_{12}}^\infty \frac{\Omega _{12}}{\sqrt{E}}\ f(E) {d}E, \end{equation} \tag{ 11 }$$

where the appropriate form of the distribution is substituted, giving the well-known collisional population rate formula for an M-B distribution:

$$\begin{eqnarray} R_{12}(\mathrm{M-B})&=&n_e N_1 \frac{h^2 }{4 \pi ^{3/2} m_e g_1} \left(k_B T_U \right)^{-3/2}\nonumber\\ &&\times\int_{E_{12}}^\infty \Omega _{12} \ {\rm exp}\left[-\frac{E}{k_BT_U}\right] {d}E. \end{eqnarray} \tag{ 12 }$$

For a κ-distribution, the corresponding rate is

$$\begin{eqnarray} R_{12}(\kappa)&=&n_e N_1 \frac{h^2}{4 \pi ^{3/2} m_e g_1} \frac{\Gamma (\kappa +1)}{\big(\kappa -\frac{3}{2}\big)^{3/2}\Gamma \big(\kappa -\frac{1}{2}\big)} \left({k_BT_U}\right)^{-3/2}\nonumber\\ &&\times\int_{E_{12}}^\infty \frac{\Omega _{12}}{\big(1 + E\big/\big[\big(\kappa -\frac{3}{2}\big) k_BT_U\big]\big)^{\kappa + 1}} {d}E. \end{eqnarray} \tag{ 13 }$$

Adopting the approximation that Ω₁₂ is independent of energy, the integral parts of the above equations (after taking a k_BT_U factor outside the integrals) reduce to

$$\begin{equation} \exp \left[-\frac{E_{12}}{k_BT_U}\right] \end{equation} \tag{ 14 }$$

and

$$\begin{equation} \left(1-\frac{3}{2\kappa }\right) \left(1+\frac{E_{12}}{\big(\kappa -\frac{3}{2}\big)k_BT_U} \right) ^{-\kappa }, \end{equation} \tag{ 15 }$$

respectively. Note that in the limit as κ → ∞, Equation (15) transforms into Equation (14), as it should.

It is useful to compare the relative rates of population to an upper state for M-B and κ-distributions. Taking Equations (12) and (13) and assuming constant Ωs, we can derive an analytical equation that expresses the ratio of the population rates:

$$\begin{eqnarray} \frac{R_{12}(\kappa)}{R_{12}(\mathrm{M-B})}&=& \frac{\Gamma (\kappa +1)}{\big(\kappa -\frac{3}{2}\big)^{3/2}\Gamma \big(\kappa -\frac{1}{2}\big)}\left(1-\frac{3}{2\kappa } \right)\nonumber\\ &&\times\exp \left[\frac{E_{12}}{k_BT_U}\right] \left(1+\frac{E_{12}}{\big(\kappa -\frac{3}{2}\big)k_BT_U} \right) ^{- \kappa }.\qquad \end{eqnarray} \tag{ 16 }$$

Figure 5 shows the effect of Equation (16) in the enhancement of the collisional excitation rate for κ: 2 → 100 compared to the M-B rate for an electron distribution having the same internal energy. Very similar curves were obtained by Owoki & Scudder (1983) in the context of collisional ionization rates in the solar corona. Data for Figure 5 for an extended energy ratio range are given in Table 1.

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The implication is that collisionally excited UV lines—with higher excitation energies—should show strong enhancements compared to standard theory, while lines in the optical and IR would be relatively little changed, unless the region has a low kinetic temperature. Collisionally excited lines of non-ionized species such as [O i] λλ6300, 63 or [N i] λλ5198, 5200 will be differently affected, since the collision strength has a strong energy dependence, increasing above threshold. For these ions, the degree of enhancement in the collisional excitation rate compared with the M-B case will be much larger than shown in Figure 6. This could explain a long-standing problem in H ii region models, which tend to systematically underestimate the strength of these lines, compared to observations. New data on the collision strengths of O i and N i, in the context of the κ-distribution, have implications for using the lines of these neutral species as low-temperature diagnostics in partially ionized regions. We will explore this in a future paper.

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The direct method of estimating electron temperature in H ii regions relies on measuring the flux ratios of different excited states from the same atomic species. Most often used are collisional excitation to the ¹D₂ and ¹S₀ states of [O iii], which give rise to the forbidden nebular lines at 5007 Å and 4959 Å, and the "auroral" line at 4363 Å. Lines of [O ii], [N ii], [S ii], [Ne iii], [Ar iii], and [S iii] are also used, when the relevant lines can be observed. The threshold excitation energies for the upper states in these species are different, so the degree of enhancement in a κ-distribution of electrons will differ from one ion to the next. This fact provides a possible resolution of the discrepancies encountered when measuring temperatures using these different atomic species.

As a specific and important example, we consider here the excitation of the O iii ion. For reference, the configuration of the lower states which give rise to the forbidden lines used in temperature determinations is illustrated in Figure 6. The threshold excitation temperatures of the excited states are 29,130 K (¹D₂) and 62,094 K (¹S₀). For a κ-distribution at temperature T_U = 10, 000 K from Figure 5, the excitation rate to the lower state will be little changed, while the collision excitation rate to the upper state will be strongly enhanced, leading to an overestimate of the true electron temperature computed by formulae such as given by Osterbrock & Ferland (2005). We now proceed to quantify this remark.

For an equilibrium electron energy distribution, the relative population rates can be calculated using Equation (11) for the upper and lower excited states. If one assumes for simplicity that the Ω values are energy-independent, then the ratio of the two population rates in the M-B case is simply

$$\begin{equation} \frac{R_{13}}{R_{12}}=\frac{\Omega _{13}}{\Omega _{12}} \exp \left[-\frac{E_{23}}{k_BT_U}\right]. \end{equation} \tag{ 17 }$$

In a more accurate analysis, the collision strength Ω is integrated over the energy range E₁₂ to ∞ as per Equation (11), to give the effective (temperature averaged) collision strength, ϒ.

Allowing for other transitions from the ¹S₀ state via the branching ratio and correcting for the energies of the two lower state transition photons, the ratio of the fluxes of the (λ5007 + λ4959) to λ4363 give a direct measure of the population rates to the two excited states, and therefore of the electron temperature of the energy distribution. Following Osterbrock & Ferland (2005), we can derive a simple equation for the electron temperature, T_e, in terms of the line fluxes, for low density plasmas:

$$\begin{equation} \frac{j5007+j4959}{j4363}=7.90 \exp \left[\frac{32900}{T_e}\right]. \end{equation} \tag{ 18 }$$

This formula is, of course, an approximation, as it assumes the values Ω₁₂ and Ω₁₃ (and therefore of ϒ₁₂ and ϒ₁₃) do not depend on temperature. A more accurate iterative formula was given by Izotov et al. (2006). Values of the effective collision strength ϒ, assuming constant Ω, differ slightly from those computed numerically, taking into account the detailed collisional cross-section resonances. To identify what effect this would have on determining the electron temperature, we compared electron temperatures derived using constant Ω, the Izotov iterated formula, and the values computed using the detailed collision strengths from Lennon & Burke (1994) and Aggarwal (1993).

Figure 7 shows the relationship between the flux ratio and the electron temperature, using the Osterbrock equation, the Izotov iteration, and computed numerically from the Lennon & Burke (1994) and Aggarwal (1993) Ω data. (The values in most common use for ϒ₁₂ are those published by Lennon & Burke 1994, available online via TIPbase (Hummer et al. 1993); or the data from Aggarwal & Keenan 1999, but the latter are only available in abbreviated tabular form.)

For M-B energy distributions, the use of a constant Ω gives a slightly higher T_e than is obtained using the Lennon & Burke (1994) data, and higher still compared to the data from Aggarwal (1993). At an equilibrium temperature of 20,000 K, the difference between the Izotov value and the L&B data is 130 K and for the Aggarwal data, 285 K.

These differences are of minor import. An altogether different result occurs when we use a κ-distribution instead of the M-B to calculate the line ratio versus T_U graph. The results are shown in Figure 8. For κ = 100, 50, 20, and 10 the kinetic temperature differences from the M-B equilibrium value at T_U = 20, 000 K are 180 K, 385 K, 980 K, and 2100 K. Data for the extended temperature range 5000–20,000 K are given in Table 2. The data are calculated using the detailed collision strengths (from Lennon & Burke 1994), but assuming constant Ωs makes only a small difference for κ > 6.

As can be seen in Figures 2 and 3, these values of κ are visually relatively minor deviations from the M-B distribution, but they have a considerable effect. This implies that even κ-distributions which diverge slightly from equilibrium can have a significant effect on electron temperatures measured using the [O iii] lines. The same result is true for other collisionally excited lines, [O ii], [S ii], and [N ii], but to differing extents, owing to the different collisional excitation energy thresholds for the upper and lower states. The differences in apparent electron temperatures calculated using different collisionally excited species allow us to obtain an estimate of the effective value of κ for remote H ii regions, as we show in Section 6.

As noted earlier (Equation (8)), a κ-distribution with kinetic temperature T_U can be characterized at low electron energies below the peak of the distribution by a "core" Boltzmann distribution with an effective temperature T_core = (κ − 3/2)T_U/κ. Since T_core is systematically lower than T_U, and as it is the energy distribution of these low-energy electrons that determines the recombination temperature (T_rec ≡ T_core), it is clear that in a κ-distribution the recombination rate is systematically enhanced. These same low-energy electrons also determine the size of the Balmer and Paschen discontinuities of Hydrogen, so that the measured Balmer and Paschen break temperatures will reflect T_B, rather than T_U.

At the same time, lines with excitation temperatures comparable to T_U may be either mildly enhanced or suppressed in a κ-distribution, but lines with excitation temperatures well above T_U are strongly enhanced by the power-law tail of high-energy electrons present in the κ-distribution. This has the effect of enhancing the apparent electron temperature inferred using well-known temperature-sensitive line ratios such as [O iii] 4363 Å/5007,4959 Å, [S ii] 4069,76 Å/6717,31 Å, [N ii] 5755 Å/6548,84 Å, or [O ii] 7318,24 Å/3726,29 Å. The degree of enhancement of the inferred temperature, compared to T_U, is strongly dependent on T_ex, the excitation temperature of the upper levels involved in these transitions. Those ions in which T_ex is much greater than T_U will have their inferred collisional excitation temperature, T_CEL, very strongly enhanced over T_U, while those for which T_ex is comparable to T_U will show little change (T_CEL ∼ T_U).

These properties of the κ-distribution can be used as the basis for the determination of κ from spectrophotometric observations of both H ii regions and PNe.

We can test the κ-distribution hypothesis using the excellent echelle spectrophotometry which has been gathered by a number of authors in recent years. For Galactic H ii regions, we have data in M 42 (Esteban et al. 2004), NGC 3576, S311 (García-Rojas et al. 2004), M20 & NGC 3603 (García-Rojas et al. 2006), and M 8 & M 17 (García-Rojas & Esteban 2007). For the extragalactic H ii regions, we have data for 30 Dor (Peimbert 2003), NGC 5253 (López-Sánchez et al. 2007), NGC 595, NGC 604, VS 24, VS 44, NGC 2365, and K 932 (Esteban et al. 2009).

One of the strongest tests of the validity of the κ-distribution is the comparison of the [O ii] 7318,24 Å/3726,29 Å and [S ii] 4069,76 Å/6717,31 Å temperatures. These ions have very similar ionization potentials and are therefore distributed in a very similar way in the nebula. The photoionization models used by López-Sánchez et al. (2012) have line emission weighted temperatures in the O ii and S ii zones which differ from each other by less than 400 K over the abundance range 0.3–3.0 times solar. However, the excitation temperatures of the lines are quite different. For the [S ii] 4069,76 Å lines, T_ex = 35, 320 K, and for the [S ii] 6717,31 Å lines, T_ex = 21, 390 K. In the case of [O ii] 7318,24 Å, T_ex = 58,220 K, while for the 3726,29 Å lines, T_ex = 38, 590 K. Thus, the [O ii] lines are much more sensitive to the high-energy tail of the κ-distribution than are the [S ii] lines. On this basis, one would expect higher temperatures to be derived from the [O ii] line ratio than from the [S ii] line ratios in the presence of a κ-distribution.

We have computed for the observed data the effect that a κ-distribution has on the inferred collisional excitation temperature, T_CEL, of these ions. This is shown in Figure 9(a). The observations suggest that the [O ii] temperatures are indeed higher than the [S ii] temperatures, and that this difference can be accounted for in most of the H ii regions by 20 ≳ κ ≳ 10.

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As an alterative approach, we show in Figure 9(b) the average of the inferred collisional excitation temperatures for the [O ii] and [N ii] ions compared with that of the [S ii] ion. For the [O ii] and [N ii] ions, the sensitivity to κ is not as great as for the [O ii] lines alone, but the errors in the temperature determination produced by assumptions about the dust extinction curve are reduced. Again, the [S ii] temperatures are lower. The inferred range of κ is somewhat wider, but most H ii regions are still consistent with 20 ≳ κ ≳ 10.

Because the recombination temperature, T_B, is appreciably lower than the kinetic temperature T_U, which is itself lower than the excitation temperature for collisionally excited line ratios, T_ex, a comparison of recombination temperatures and inferred line temperatures is strongly sensitive to the choice of κ. The recombination temperatures, T_REC, should be significantly lower than the collisionally excited line temperatures, T_CEL, in the presence of a κ-distribution. Again, this is clearly shown in the analysis of the observed H ii region spectra.

In Figures 9(c) and (d), we show this comparison for the H ii regions, using the excitation temperature given by the average of the [O ii] and [N ii] line ratios and by the [O iii] line ratio, respectively. How well these determine κ depends on which of the ionic species is the dominant ionization stage in the nebula. For cooler exciting stars, [O ii] and [N ii] line ratios are the better ones to use, but when the central star(s) is hot enough to ionize helium to He ii in the bulk of the nebula, then the [O iii] line ratio is the better one to use. From Figures 9(c) and (d), it would appear that κ ∼ 20, or even higher. Because the upper state of [O iii] has the highest excitation temperature, it is the most sensitive of all the ions to the high-energy electrons, and therefore lower values of κ have greater effects.

High-quality spectrophotometry also exists for many PNe, although published values of electron temperatures for many ions is rather sparse. However, rather complete data exist for temperatures determined from the Balmer discontinuity and from the [O iii] line ratio. The direct comparison of these should be a fairly reliable means of estimating κ, since the O iii ion is most often the dominant ionization stage in PNe.

We have taken data from Tsamis et al. (2003), Liu et al. (2004), Zhang et al. (2004), Wesson et al. (2005), Wang & Liu (2007), and Fang & Liu (2011). These data are plotted in Figure 10, along with the expectation for κ = 6, 10, and 20. The points with error bars are from Zhang et al. (2004). Again, the results are consistent with κ > 10 in most cases, i.e., a mild departure from an equilibrium electron energy distribution. A few points lie above the line of equality of temperatures shown by the dotted line, corresponding to κ = ∞. However, many of these are consistent with no departure from an M-B distribution, within the observational errors. Of course, it is likely that local temperature and density fluctuations in the PNe contribute to the measured results, and it is not obvious what the contributions are from the various effects.

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There are two apparent trends in Figure 10: both cooler objects and objects with high [O iii] electron temperatures tend to show greater deviations from equilibrium (smaller inferred κ). For high electron temperature objects, the energetic stellar winds from the hot central star are likely to lead to significant suprathermal heating of the ionized plasma, leading to lower values of κ. Indeed, some of the points identified by name (NGC 2440 and NGC 6302) are famous Peimbert Type I PNe, known to contain the most luminous and massive central stars. Among the rest, IC 4997 is a young PNe which also contains a Wolf–Rayet type central star, subject to rapid mass-loss.

The coolest objects tend to be more metal rich, and/or of lower excitation class. As a result, the excitation is dominated by the high-energy tail of the energy distribution, as illustrated in Figure 5: κ effects for [O iii] become more apparent. These and other characteristics of the κ-distribution will be the subject of detailed modeling in future papers.

The results for both PNe and H ii regions are consistent with mild departures from equilibrium (κ ≳ 10). It is interesting to note that in the analysis of the observed spectra, extreme values of κ are not required. The implied departures from an M-B distribution are quite small, but their effect on the nebular diagnostics is quite gross, with serious implications for the chemical abundance determinations in both classes of object. Large departures from the M-B distribution in either PNe or H ii regions (lower values of κ) would in any case not be expected, since at the peak of the energy distribution the mean collision time between electrons is short.