The ultraviolet CII lines as a diagnostic of kappa-distributed electrons in planetary nebulae

Non-Maxwellian $\kappa$ electron energy distributions (EEDs) have been proposed in recent years to resolve the so-called ``electron temperature and abundance discrepancy problem'' in the study of planetary nebulae (PNe). Thus the need to develop diagnostic tools to determine from observations the EED of PNe is raised. Arising from high energy levels, the ultraviolet (UV) emission lines from PNe present intensities that depend sensitively on the high-energy tail of the EED. In this work, we investigate the feasibility of using the \ion{C}{2}]$\lambda$2326/\ion{C}{2}$\lambda$1335 intensity ratios as a diagnostic of the deviation of the EED from the Maxwellian distribution (as represented by the $\kappa$ index). We use a Maxwellian decomposition approach to derive the theoretical $\kappa$-EED-based collisionally excited coefficients of \ion{C}{2}, and then compute the \ion{C}{2} UV intensity ratio as a function of the $\kappa$ index. We analyze the archival spectra acquired by the {\it International Ultraviolet Explorer} and measure the intensities of \ion{C}{2} UV lines from 12 PNe. By comparing the observed line ratios and the theoretical predictions, we can infer their $\kappa$ values. With the Maxwellian-EED hypothesis, the observed \ion{C}{2}]$\lambda$2326/\ion{C}{2}$\lambda$1335 ratios are found to be generally lower than those predicted from the observed optical spectra. This discrepancy can be explained in terms of the $\kappa$ EED. Our results show that the $\kappa$ values inferred range from 15 to infinity, suggesting a mild or modest deviation from the Maxwellian distribution. However, the $\kappa$-distributed electrons are unlikely to exist throughout the whole nebulae. A toy model shows that if just about 1--5 percent of the free electrons in a PN had a $\kappa$-EED as small as $\kappa=3$, it would be sufficient to account for the observations.


INTRODUCTION
Photoionized gaseous nebulae, including planetary nebulae (PNe) and H II regions, serve as a vital probe of the chemical evolution of galaxies and as a natural laboratory for understanding the plasma physics found in extreme physical environments. Extensive observational studies of nebulae have revealed stark differences between the electron temperatures and elemental abundances derived from collisionally excited lines (hereafter CELs) and those derived from recombination spectra, which is often called the "electron temperature and abundance discrepancy problem" (e.g., Peimbert 1967;Liu et al. 2000;Zhang et al. 2004;García-Rojas et al. 2017). Several alternative solutions have been previously proposed and continue to be hotly debated (see e.g., Danehkar 2021;Hajduk et al. 2021;García-Rojas et al. 2022;Ruiz-Escobedo & Peña 2022, as examples of the recent contributions to this debate). Two mechanisms often cited are temperature fluctuations (characterized by the t 2 temperature fluctuation parameter) and the two-component model consisting of hydrogen-deficient clumps distributed within a diffuse nebula.
Recently, a new promising scenario has been presented for solving the problem. The determinations of nebular temperatures and abundances in most published literature are based on a fundamental but insufficiently demonstrated assumption that the electron energy distribution (hereafter EED) necessarily follows a Maxwell-Boltzmann (hereafter MB) energy distribution. If the EED was in-deed non-thermal, the inconsistencies of current plasma diagnostics would be essentially resolved (Nicholls et al. 2012Dopita et al. 2013), in which case the EED becomes characterized by a κ function where a lower κ index corresponds to a larger departure from the MB distribution.
Despite the plausibility of this solution, the mechanism for energizing non-thermal electrons is so far unknown. Much of the criticism to the κ-EED hypothesis is from a theoretical perspective. Given the relatively short relaxation timescale of non-thermal electrons, the photoionization process of gaseous nebulae is expected not to generate a κ EED (Ferland et al. 2016;Draine & Kreisch 2018), casting serious doubt on this hypothesis. Nevertheless, it is premature to conclude that the currently proposed alternative scenarios are necessarily valid. For instance, if the hypothesis of temperature fluctuations held, the t 2 values required to explain the observations would be too large to be reproduced by canonical photoionization models. Regarding the two-component model, the proposed but unconfirmed hydrogen-deficient inclusions are also of unknown origin.
The κ distribution was first introduced to provide an empirical fit to the energy distribution of the earth's magnetospheric plasma (Vasyliunas 1968), and was then found to be a natural consequence of the Tsallis statistical mechanics (Livadiotis & McComas 2009). It has been widely found to be present in the solar wind (Pierrard & Lazar 2010), in the heliosheath (Livadiotis & McComas 2011), and the solar corona (Nicolaou et al. 2018). The proposed origin for particle acceleration, such as magnetic reconnection, is a hot topic in space plasma research. It is not unreasonable to speculate that such processes may also affect the EED of PNe, where the "electron temperature and abundance discrepancy problem" is more prominent than observed in H II regions.
Spectral observations may allow us to confirm or refute the presence of the κ EED in PNe. The C II dielectronic recombination lines and the continuum emission spectra near the Balmer jump (hereafter BJ) have been proposed to determine the EED (Storey & Sochi 2013Zhang et al. 2014). However, primarily due to large uncertainties, these results are inconclusive or even conflicting. Zhang et al. (2016) determined the κ index for a sample of PNe by comparing the CEL ratios and BJ intensities, which appears to not correlate with the other properties of PNe. Lin & Zhang (2020) found that the introduction of the κ EED can bring in line the intensity ratios of the O II recombination lines and those predicted by [O III] CEL observations, but the inferred κ values are not in agreement with those obtained from comparison of the CEL and BJ intensities. Such studies based on optical spectroscopy suggest that the non-thermal electrons are unlikely to be uniformly distributed across the whole nebulae, but they cannot exclude their presence on a small spatial scale as is found to be the case for the solar system plasma.
The non-thermal high-energy tail of the κ EED would cause a stronger impact on the excitation of the ultraviolet (UV) emission lines than on the optical lines. Therefore, UV line fluxes can in principle serve as an excellent probe of the EED in PNe. Based on the observed C II UV line ratios, Humphrey & Binette (2014) and Morais et al. (2021) found evidences supporting the existence of the κ EED in Type 2 Active Galactic Nuclei (AGNs), the high-energy analogue of PNe.
In this paper, we investigate the impact of the κ EED on the C II]λ2326/C IIλ1335 line ratios, aiming at examining whether the UV observations indicate the presence of κ-EED in PNe. Section 2 describes the archival data we will be using. Section 3 describes the method used to derive the C II UV line ratio under a κ EED. The results are presented in Section 4 and are discussed in Section 5.
Section 6 summarizes the conclusions.

SAMPLE AND OBSERVATIONAL DATA
The UV observations of PNe have been obtained from the International Ultraviolet Explorer (IUE).
We extracted the low-resolution spectra obtained with the Short Wavelength Prime (SWP) and Long Wavelength Prime/Long Wavelength Redundant (LWP/LWR) cameras from the INES (IUE Newly Extracted Spectra) Archive 1 . The SWP and LWP/LWR spectra cover the following wavelength ranges: 1150-1980 and 1850-3350Å, which therefore include the C IIλ1335 and C II]λ2326 lines, respectively. For the sample selection, we focused on the PNe studied by Zhang et al. (2016) for the sake of comparison. Due to the severe extinction at UV wavelengths, The C II UV lines for most of the PNe are faint or invisible. Discarding those nebulae with only one of the C II lines being detected, we selected the PNe which possess a reasonably good signal-to-noise ratios. In order to ascertain whether the observed features are indeed C II lines rather than glitches or other contaminations, we carefully compared the widths and peak velocities of the two features in each PN, and excluded those with apparent inconsistencies. Finally, we ended with the 12 PNe listed in Table 1. These PNe are ionized bounded and show relatively high electron temperatures, which are the physical conditions favoring the emission of C II UV CELs. All IUE observations were taken with a large aperture of 10 ′′ × 23 ′′ except for two PNe. For NGC 6572, the LWR spectrum obtained with the small aperture (3 ′′ diameter) was used. We did not attempt to make an aperture correction since NGC 6572 is a compact young PN. For NGC 3918, both spectra were obtained using the smaller aperture.
We subtracted the continuum emission from the IUE spectra by performing a linear fit to the linefree spectral regions. The fluxes of the two C II lines were measured using the Gaussian line profile fitting. Figure 1 shows the IUE spectra and the line fittings for the 12 PNe. The integrated fluxes F (λ) were then deredened to derive the intensities by I(λ) = 10 c(Hβ)f (λ) F (λ), where c(Hβ) is the logarithmic extinction constant taken from Cahn et al. (1992), and f (λ) is the standard Galactic extinction law with R V = 3.1 (Howarth 1983). See Table 1 for the details.
where E, k B and Γ are the electron energy, Boltzmann constant, and the Gamma function, respectively. The dimensionless κ index ranges from 1.5 to ∞. T U is the nonequilibrium temperature representing the mean kinetic energy of free electrons, which satisfies the relation When κ → ∞, the formula decays to a MB form with T U = T c .
The theoretical intensity ratios of C II CELs can be computed by solving the ten energy level balance equation. For that purpose, the collisionally excited rate coefficients (q) between any two energy levels are derived from the energy-dependent collision cross sections, σ(E), integrated over the EED function. This means under the MB-EED hypothesis (hereafter, we use the subscripts 'M' and 'κ' to distinguish between the cases under the MB or the κ EED assumption, respectively). The tabulated q M (T M ) values are available in Blum & Pradhan (1992). In principle, q κ (T U , κ) could be obtained by substituting 3, but such procedure is impractical because numerical evaluation of σ(E) is not available from the literature. Hahn & Savin (2015) proposed an easier approach to determine q κ (T U , κ) using the tabulated q M (T M ). The κ function can be mathematically decomposed into several weighted MB functions, with T M,j = a j T U , where a j and the weighting factor c j are independent of T U . Equs. 3 and 4 indicate that Therefore, we can obtain q κ (T U , κ) by using the above formula and the a j , c j , and q M (T M ) values provided by Hahn & Savin (2015) and Blum & Pradhan (1992). This approach for instance has been successfully employed to develop diagnostics for the solar plasma ). Dzifčáková et al. (2021) noted that this approach provides highly precise results with an uncertainty comparable to the typical uncertainty inherent in the atomic calculations.
The lowest seven levels of C + relevant to the C II UV lines are sketched in Figure 2. As shown in this figure, the λ1335 and λ2326 features are respectively composed of three and five blended lines that cannot be resolved with IUE spectra. In this work, we will focus on the intensity ratios between the 4 P 5/2 -2 P 3/2 transition at 2326.12Å and the 2 D 3/2 -2 P 1/2 transition at 1334.52Å, which is hereafter called r(C II), in short. Figure 3 shows r(C II) as a function of both T U and electron density (N e ) at various κ values. It is conceivable that r(C II) decreases with increasing T U and decreasing κ, which means that higher-energy electrons enhance the λ1334.52 line to a larger degree than the λ2326.12 line. An inspection of this figure also suggests that r(C II) is essentially independent of N e . This is because the two lines have critical densities much higher than the typical density found in PNe.
At a temperature of 10 4 K, the λ2326.12 and λ1334.52 lines show a critical density of 7.3 × 10 8 cm −3 and 2.2 × 10 15 cm −3 , respectively. Consequently, once T U is known, we can infer the κ index from the observed r(C II). At lower T U , r(C II) is more sensitive to κ.
The observed r(C II) can be derived by considering the theoretical relative intensities between the blended lines. The λ1334.52 and λ1335.66 lines arise from the same upper level, and thus their intensity ratio is strictly equal to 5.1, while I(λ1335.71) is about 1.8 × I(λ1334.52). It follows that the λ1334.52 line has an intensity of 0.34 times that of the blended feature. The λ2326.12 line is the strongest among the five blended lines, with an intensity of 0.50 times that of the whole blended feature. These relative intensity ratios are extremely insensitive to the nebular physical conditions in the sense that the blended lines arise from the upper levels with very similar excitation energies and high critical densities (> 10 7 cm −3 ). After deblending the C II λ1335 and λ2326 features, the observed r(C II) are derived and their values are listed in the 7th column of Table 1.

RESULTS
The theoretical and observed r(C II) values are compared in Figure 4, where we simply assume N e = 10 3 cm −3 . For that we have derived the T U based on the BJ temperatures, T M (BJ), given in the literature (see the 9th column of Table 1). Because the BJ mostly 'sees' the cold MB core of the κ EED, it is a good approximate to assume T c = T M (BJ). Then κ and T U can be iteratively determined from r(C II) and Equ. 2. Their values are presented in the 11th and 13th columns of Table 1. The uncertainties of κ are found to increase with increasing T U and κ.
If the EED obeyed the MB distribution, the data points would overlay the κ = ∞ curve in Figure 4.
On the contrary, if the free elections got closer to the κ EED with a smaller κ index, the CEL with a higher excitation energy would be relatively enhanced, and thus r(C II) would decrease and T M (BJ) would be lower than T U , meaning that the data points would shift towards the left and down in this figure. We found this to be the case for most of the PNe of our sample, thus providing us a favorable evidence for the κ EED in PNe. We could derive well constrained κ values for six PNe. NGC 6572 shows a low κ index of 17. However, the small-aperture of its LWR spectrum may have caused an underestimation of the C II]λ2326 line flux, and thus likely resulted in an underestimation of the κ index. The medium κ value for the PN sample turns out to be larger than 50, suggesting that most PNe exhibit only mild to modest deviations from the MB distribution.
Humphrey & Binette (2014) found that the r(C II) ratio in their AGN sample suggests the presence of a κ EED. We reexamined their results in Figure 4 using the observational data retrieved from Vernet et al. (2001), where we have assumed a typical electron temperature of 1.5 × 10 4 K (Osterbrock & Ferland 2005). Most AGNs exhibit a prominent κ EED unless they have an abnormally high temperature, in agreement with the conclusion of Humphrey & Binette (2014). Figure 4 shows that the medium κ value inferred from the AGN spectra is around 20. Recently, through photoionization model fitting to a large sample of type 2 AGNs, Morais et al. (2021) found that, for most of the objects, a κ EED with κ = 5 provides a better fit to the observed spectra than the MB EEDs. Such a low κ value suggests a generally larger departure from the MB distribution in AGNs than in PNe. This could probably be attributed to their high-energy physical condition.
Humphrey & Binette (2014) also indicated that shocks can decrease r(C II) to < 5 and therefore account for the measured values of their AGN sample except for one object. As can be inferred from Figure 4, however, shock models cannot reproduce the PN observations.
An effect we did not consider is the continuum fluorescence through the C IIλ1335 resonance line, which may either enhance or suppress the r(C II) ratio, depending on the nebular geometry (see discussion in Humphrey & Binette 2014). However, almost all our PNe show a measured r(C II) ratio lower relative to the MB values ( Figure 4). Moreover, the stellar UV continuum luminosity of the PNe is presumably low within the C + regions due to geometrical dilution. This would hinder continuum fluorescence as being the cause of the r(C II) discrepancy.
In order to facilitate the evaluation of plasma diagnostics using κ-EED calculations appropriate to photoionized gaseous nebulae, we present in Table 2 the theoretical r(C II) calculated for various κ and T U values. Although we assumed N e = 1000 cm −3 in the calculations, the results are essentially insensitive to the electron density. Once T c using Equ. 2 is derived from the BJ, the κ and T U values can be derived from the observed r(C II) ratio through iteration. Alternatively, one could use the CELs instead of the BJ for the determination of κ and T U , in which case the relations between T U and T M (CELs) proposed by  can be used.

DISCUSSION
We have investigated the feasibility of invoking the κ EED to solve the "electron temperature and abundance discrepancy problem" encountered in PN studies. The electron temperature of the singly ionized C + regions can be evaluated using the temperature inferred from the [N II] lines, T M (N II).
We have retrieved T M (N II) from the literature (Table 1), which is generally found to be higher than T M (BJ). Within the scenario of κ EED, T M (N II) strongly depends on the electronic highenergy tail, while the BJ depends mostly on the cold core of the distribution. It is interesting to note that the derived T U values consistently lie between T M (N II) and T M (BJ), suggesting that the temperature discrepancy can indeed be alleviated under the κ EED hypothesis. The ratio between the O 2+ /H + abundance derived from recombination lines and that derived from CELs, the so-called abundance discrepancy factor (ADF), is introduced to quantify the degree of abundance discrepancy.
As listed in Table 1, the ADFs of our sample PNe range from 1.2 to 5.4. It is unfortunate that those nebulae with extremely large ADFs do not have C II UV lines detected. In Figure 5, we show that the κ index negatively correlates with both T M (N II)−T M (BJ) and ADFs. This is compatible with the expectation that a larger departure from the MB EED should lead to larger temperature and abundance discrepancies. However, the correlation coefficients are only moderate, suggesting that the situation is not as simple as described here.
If the PN is homogeneous, that is, is presenting a constant EED, the κ indices deduced from different diagnostic line ratios should have an identical value. For the purpose of comparison, Table 1 lists the κ indices obtained by Zhang et al. (2016)  with the κ EED hypothesis is the lack so far of a plausible mechanism for pumping the non-thermal electrons in PN environments. This might be the main reason for the fact that the κ EED has been generally ignored in mainstream PN studies. However, we should be aware that the κ distribution is a cutting edge research area in the field of solar plasma physics although its origin still remains unclear (e.g. Dzifčáková et al. 2021). Furthermore, the other proposed solutions to the "electron temperature and abundance discrepancy problem" of PNe, such as temperature fluctuations or two-component models, are facing a similar challenge, namely, how to generate a sufficiently large t 2 or the hydrogen-deficient clumps? We therefore see no legitimate reason for the study of κ distribution to be overlooked and undervalued by the PN research community. Recent studies reveal that high ADFs might be associated to the evolution of close binary central stars (Wesson et al. 2018) and, at this point, we may tentatively conjecture that a fast magnetosonic wave excited by the interaction with a strong stellar magnetic fields could instantly inject energetic electrons into the free-electron pool, thereby forming a κ EED within the central regions of PNe.
We have built a toy two-EED model in an attempt to reproduce the observations, in which, besides a regular nebular component with a MB EED, we added a κ-EED component. In this scenario, the predicted C II line ratio is given by where ω is the volume filling factor of the κ-EED gas, and the emissivities of the κ-and MB-EED We have assumed that the κ-EED nebular component present values that are similar to those of solar plasmas, and calculate r(C II) for four preselected κ values (3, 5, 10, and 20); the results are presented in Figure 6. It shows that with increasing filling factor (ω) of the κ-EED component, the r(C II) versus T M relation changes from that of a uniform MB-EED (ω = 0) up to the case of a complete κ-EED (ω = 1). For a given ω, there is a trend that, along with decreasing temperature, the r(C II) versus T M curves get progressively closer to the complete κ-EED case. This is because at such a low temperature, the MB-EED hardly excites the C II CELs unless non-thermal high-energy electrons exist, and thus in the low temperature case, the r(C II) ratio from Equ. 6 is essentially dominated by the κ-EED component.
The observed PN data are plotted in Figure 6. We can see that a very small amount of κ-EED gas with an extremely low κ index is sufficient to explain the observations. For instance, we would require only 1-5% of the nebular gas to have κ = 3 in order to reproduce the observed r(C II). If the non-thermal gas had a κ value of 5 or 10, the required filling factor would need to increase up to 10% or 50%. On the other hand, if one assumes κ = 20, we could no longer reproduce the lowest r(C II) ratios. Alternatively, we could relax the assumption of a uniform temperature for the two components since the ejection of non-thermal electrons into the regular nebular gas is likely to cause T U to become > T M . This would enhance the ǫ κ value relative to ǫ M . As a result, the inferred ω would be further reduced (see Equ. 6). It therefore appears that the problem of the origin of κ-electrons pumping is not a very crucial issue.

CONCLUSION
The current work shows that the C IIλλ1335,2326 UV lines can serve as a useful tool to investigate the EED of PNe. The observed C II λ2326/λ1335 intensity ratios are systematically lower than those expected with the MB-EED hypothesis. This would be satisfactorily explained if we assume the κ-EED hypothesis. We would suggest that the deviation from the MB-EED in PNe is lower than for AGNs. However, the κ-EED is unlikely to extend uniformly across the whole nebulae. Using a toy model of two-EED components, we have demonstrated that only a few percent of nebular gas in κ-EED is sufficient to reproduce the observations.
By no means can we state that the evidence of the existence of κ-EED in PNe is conclusive.
Instead, we have shown that the generation of non-thermal electrons can definitely improve, rather than worsen, the agreement with the observations of C II UV lines as well as other emission lines. We are grateful to the anonymous referee for constructive comments that contributed to improve the manuscript. This work was partially based on INES data from the IUE satellite. We thank Bao-Zhi Lin for fruitful discussions. We are grateful for financial supports from the National Science     Note that the F (λ) and I(λ) are those of the blended features.
Objects F(λ) (10 −13 erg cm −2 s −1 ) c(Hβ) a I(λ) (10 −13 erg cm −2 s −1 )  et al. (2003, 2004),  Under the MB−EED assumption, the relative contribution of DR and collision excitation (CE) to the C IIλ1335 line intensity can be estimated using R M = C ++ Neα eff (Te) C + Pu(Te,Ne)A ul , where the electron density N e is assumed to be 10 3 cm −3 , the effective DR coefficient α eff (T e ) is taken from Storey (1981), P u (T e , N e ) is the fractional population of the upper level of C + that is obtained by solving the level population equations assuming only collisional processes, and A ul is the Einstein coefficient for the transition from the upper level to the lower level. If assuming the electron temperature T e = 10 4 K and the C ++ /C + abundance ratio of 1, we obtain R M = 2.6, suggesting that the DR effect is not negligible under the MB−EED assumption.
Under the κ-EED assumption, as the κ EED has a superthermal tail that significantly enhances the CE, the DR effect should be less significant than that under the MB−EED assumption. For a κ-EED with a kinetic temperature T U , the DR is mainly from the electrons in the low-temperature MB core (with a temperature of T c = κ−1.5 κ T U ) of the κ EED. The relative contribution of DR and CE can be written as R κ = C ++ N eff e α eff (Tc) C + Pu(T U ,Ne,κ)A ul , where N eff e is the effective electron density obtained by applying a correction for N e (see , for the reason and the correction formula).
In order to correct for the effects of DR and resonance scattering, we multiply the observed C II]λ2326/C IIλ1335 line intensity ratio by a factor of (1 + R κ )f e to obtain r(C II), where f e is the escape probability of the λ1335 photons. It is reasonable to assume that the λ2326 line is free from these effects. An accurate estimation of f e requires a sophisticated radiative-transfer computation, which is beyond the scope of the paper. As a first approximation, we simply assume f e = 0.5, as obtained for IC 418 by Clavel et al. (1981). Because no κ-EED-based C ++ /C + ratio has been derived for planetary nebulae (PNs), we only consider two cases where C ++ /C + = 1 and 5.
The new r(C II) versus T U plot is shown in Figure 4 (also Figure 4 in Yao & Zhang 2022), where we compare the results before and after correcting for the effects of DR and resonance scattering.
When the κ index is low or the C ++ /C + ratio is small, the DR is weak compared to the CE. When C ++ /C + = 1, the change of the resultant κ values is slight, and we obtain the corrected κ indices of NGC 2867, NGC 6543, NGC 6565, NGC 6572, NGC 6720, and IC 4406 to be 16 +2 −1 , 9 +2 −1 , 40 +10 −6 , 10 +7 −4 , 23 +6 −5 , and 70 +∞ −40 , respectively. However, when C ++ /C + = 5, most of the data points move above the κ = ∞ curve in Figure 4 due to a substantial correction of the DR, and thus the observations would not support the κ-EED hypothesis in our PN sample In summary, the results in Yao & Zhang (2022) are barely changed if C ++ /C + ≤ 1 or κ ≤ 10.
Otherwise, accurate treatments of photoionization and radiative transfer are required for using r(C II) as a diagnostic of the κ EED.