Far-Infrared Line Diagnostics: Improving N/O Abundance Estimates for Dusty Galaxies

The [N iii] 57 $\mu$m and [O iii] 52 $\mu$m lines arise from the ground state term of the N⁺⁺ and O⁺⁺ fine structure configurations. Their emitting levels are collisionally excited by electrons in H ii regions so that the line ratio is affected by the H ii region physical properties and ionization structure in the following way:

The first term is the gas phase N/O abundance ratio; the second term is the ratio of the fractions of N and O that are doubly ionized; the last term is the ratio of emissivity, defined here as the power emitted in the line per ion. The emissivity is primarily a function of electron density, and more weakly dependent on electron temperature. The line emission is assumed to be optically thin.

The [N iii]/[O iii]52 line ratio is a good tracer for the N/O abundance ratio for several reasons. First, the energies required to produce both ions are very close (Table 1), so that the N⁺⁺ and O⁺⁺ ions occupy very similar regions in ISM, typically H ii regions of young stars or the ionized regions surrounding Active Galactic Nuclei (AGNs). Notice that their formation potentials are just above that of Helium (24.6 eV) which means they share nearly the same volume as He⁺ (Lester et al., 1983), the ion which dominates the ionization structure in the H ii regions with hard radiation field. Secondly, their critical densities are very similar so that the [N iii] to [O iii] 52 $\mu$m emissivity ratio changes only by a factor of 5 from the low density to high density limit. This is much smaller than the variation if the [O iii] 88 $\mu$m line is used. Thus in Equ. 1, the ionization and emissivity ratios nearly cancel out, so that the [N iii]/[O iii]52 line ratio is a good first order proxy for the N/O abundance ratio. Furthermore, on large scales in galaxies, this doubly ionized line flux probe can be interpreted as arising primarily from a few H ii regions with high excitation. This is different from lower ionization state lines which come from a much larger collection of H ii region excitation environments (e.g. the diffuse ionized ISM; see Díaz-Santos et al., 2017), and thus are more difficult to interpret.

Our first estimate of the N/O abundance ratio uses only the [N iii]/[O iii]52 line flux ratio:

The index is dubbed as $N3O3$ parameter for clarity. The numerical factor is what one obtains in the low density limit ($n_{e}\ll n_{\mathrm{crit}}$) for Equ. 1. For extragalactic work, this is a good approximation since electron densities in starburst galaxies are usually of the order $10^{2}\ \mathrm{cm^{-3}}$ (Inami et al., 2013), whereas $n_{\mathrm{crit}}$ of both lines are above $2\times 10^{3}\ \mathrm{cm^{-3}}$. We have also assumed an electron temperature of $T_{e}=10^{4}$ K, which is typical for H ii regions. The exact value chosen has only a small effect on far-IR lines. More description of the parameters and details of the calculation can be found in Appendix A.2.

This method is advantageous in theoretical and practical aspects. It is based on simple arguments and is more physically robust than other N/O diagnostics, especially those involving the O⁺⁺ to N⁺ ion ratio, since these ions occupy different ionization zones. It also benefits from the weak dependence on temperature of far-IR lines over optical methods. The simplicity of using only two lines lends great applicability when studying high redshift galaxies, where observation are often difficult or impossible due to telluric absorption, and data is therefore sparse. An additional benefit is that as one of the brightest spectral lines from star forming galaxies, the [O iii] 52 $\mu$m line encodes additional scientific value that is linked to star-formation rates, and that when combined with the equally bright [O iii] 88 $\mu$m line, the line pair constrains electron density and gas mass.

Though the low density limit works well in most cases and the simplicity of using only two lines is attractive, it is desirable to correct for the electron density dependence when the density can be estimated, for example by the [O iii]52/[O iii]88 ratio or the [N ii]122/[N ii]205 ratio. Therefore we also provide here a more precise diagnostic tool that is corrected for electron density and temperature. The index is denoted $N3O3_{n_{e}}$ and is defined as

where $n_{e}$ is the electron density value in unit $\mathrm{cm^{-3}}$ and electron temperature $T_{e}$ in Kelvin. The correction factor starts to have a non-negligible effect at $n_{e}/T_{e}^{1/2}>1$, and rises until the value $\sim 4.7$ at the high density end.

Clearly the densities derived from [O iii]52/88 are the best since it reuses one of the same lines in $N3O3$ thereby measuring densities in the same regions. For convenience, we also provide an expression of $N3O3_{n_{e}}$ that uses the [O iii]52/88 ratio:

in which $R_{52/88}=\frac{F_{\mathrm{[O\,III]}52}}{F_{\mathrm{[O\,III]}88}}$ is the [O iii] line flux ratio. The detailed derivation of $N3O3$ and $N3O3_{n_{e}}$ as well as the parameters used for collisional excitation calculation are given in Appendix A.

As stated in the previous section, one of the major benefits of using [O iii] and [N iii] is the near co-spatial nature of N⁺⁺ and O⁺⁺ ions in H ii regions. But in actuality the O⁺⁺ volume is usually smaller than the N⁺⁺ volume: the volumes are only closely matched when the radiation field is hard enough such that the Helium Strömgren sphere fills the whole H ii region. Only in this case can the line ratio trace the abundance ratio at high accuracy without corrections for ionization structure. Furthermore, if the radiation field is too hard, our $N3O3$ diagnositic would fail as N⁺⁺ gets ionized into N⁺⁺⁺ and the [N iii] flux would decrease. Therefore an indicator for the radiation hardness is essential for calibrating the $N3O3$ diagnostic to higher precision.

We use the [Ne iii] $15.5\ \mathrm{\mu m}$ to [Ne ii] $12.8\ \mathrm{\mu m}$ line ratio as a tracer for radiation hardness. Because their ionization potentials are 40.96 eV and 21.56 eV (Table 1), with one higher and the other lower than that of Helium, the Ne⁺⁺ to Ne⁺ ion ratio closely follows the fraction of volume inside and outside the Helium Strömgren sphere in H ii region. In addition, the critical densities of both lines are $\sim 10^{5}\ \mathrm{cm}^{-3}$, much higher than the typical densities in H ii regions, so that their emissivity ratio is almost density invariant. This is a major advantage over the [N iii]/[N ii] ratio, which is another commonly used hardness tracer (c.f. Unger et al., 2000): because the [N ii] lines have relatively low critical densities ($\leq 200$ cm^-3), the line emissivity ratio is much more sensitive to the electron density at $\sim 10^{2}\ \mathrm{cm}^{-3}$ values. Furthermore, since the energy to ionize the third electron off from Neon is 63.45 eV, higher than that energy for Nitrogen (47.4 eV) or Oxygen (54.9 eV), so that the Neon line ratio can still function as hardness tracer in the environments where N⁺⁺ ions are further ionized and the [N iii]/[N ii] ratio would fail.

Other factors that favor the [Ne iii]15/[Ne ii]12 ratio as a radiation hardness tracer include that the emitting levels are roughly 1000 K above ground so that they have reduced sensitivity to gas temperature in H ii regions compared with optical lines; they lie in the mid-IR range so that they are affected less by extinction, and they lie close in wavelength so the differential extinction correction is small; there is ample data available from mid-IR observatories (Spitzer, ISO, etc); and both Neon lines are often strong in star-forming galaxies. But one caveat of the [Ne iii] line is the possible contamination from AGN. Though it is not a concern for nearby galaxies, as spectral classification can identify AGNs, and spatial resolution is often enough to separate active nuclei from star-forming regions, AGN contamination might undermine its application on high-redshift objects. Fortunately, the nearby [Ne V] 14.3 $\mu$m line is only bright from AGN on galactic scales and there is a strong correlation between the [Ne iii] lines and the [Ne V] line emission in nearby AGN dominated systems, so that the fraction of the [Ne iii] line that arises from the AGN is well determined by measuring the [Ne V] line (Gorjian et al., 2007).

In order to validate the $N3O3$ parameter, and correct it for ionization hardness with the Neon line ratio, we compare our results against photoionization models that account for different stellar populations and include detailed physical calculations on ionization structure and radiative transfer. These models can be used to study the uncertainty of this diagnostic as well because they cover a large volume in the parameter space, representative of the diversity of galaxies.

The grids of photoionization models used in this paper are drawn from the CALIFA project (Cid Fernandes et al., 2014) and BOND project (Vale Asari et al., 2016). These models are run and hosted by the Mexican Million Models Database (3MDB; Morisset et al., 2015), available for public use¹¹1https://sites.google.com/site/mexicanmillionmodels/. All the models result from running Cloudy photoionization code v17.01 (Ferland et al., 2017) through pyCloudy (Morisset, 2013).

The CALIFA Project is a grid of photoionization models which feature a broad diversity in input stellar populations, with ages ranging from 1 Myr to 14 Gyr. The N/O abundance ratio is an independent parameter in the input configuration, spanning from $\log$ N/O = -1.36 to -0.36 in 5 steps. These photoionization models were initially run for analyzing IFU observation in the CALIFA survey (Sánchez et al., 2012).

BOND is a grid of models that also do not assume an N/O-O/H relation, and which cover a broad range in abundance of the two elements as well as radiation field strength. These models only use a fixed density of 100 cm^-3, in either a filled sphere or a thin shell geometry. The input starburst ages are 1, 2, 3, 4, 5 and 6 Myr. The spectral energy distribution (SED) of the ionizing radiation is obtained from stellar population synthesis models accounting for the appropriate metallicity.

It is essential to use both BOND and CALIFA models, because the former are run on electron density $100\ \mathrm{cm^{-3}}$ while the latter use $10\ \mathrm{cm^{-3}}$. They complement each other on electron density, which is the parameter that has the largest effect on far-IR fine-structure line emissivity. Our calibration of the $N3O3$ parameter below justifies the need to combine the two grids, and the usefulness of the density corrected $N3O3_{n_{e}}$ parameter.

Because BOND and CALIFA cover slightly different regions in the parameter space, only a subset of parameters common for both models is used for consistency. The selections of parameters are as follows:

• •

  Only filled sphere geometry is used for both grids, corresponding to geometric fraction = 0.03, meaning the inner radius is 3$\%$ that of Strömgren sphere radius. Although 3MDB contains results of partial cuts of radiation bounded models, only fully radiation bounded results (H_β depth $>$ 95%) are used;

• •

  CALIFA has a smaller step size in ionization parameter U than BOND, but covers a slightly smaller range. $\log$U = -4 to -1.5 with step size = 0.5 used for both grids;

• •

  BOND only runs on input SEDs with star burst ages from 1 to 6 Myr with step size 1 Myr, while the CALIFA SEDs range from 1 Myr to 14 Gyr, with a more coarse sampling. Age = 1, 3, 4, 6 Myr are used in BOND, and 1, 3, 4, 5.6 Myr are used in CALIFA in the comparison;

• •

  BOND is run on a wider range in $\log$ O/H, but only models with log O/H = -3.2, -3.4, -3.8, -4.0 are used to overlap with CALIFA’s range of log O/H = -3.09, -3.31, -3.71, -4.02;

• •

  Again only BOND models with $\log$ N/O = -1.25, -1.0, -0.75, -0.5, -0.25 are used, because CALIFA only has log N/O = -1.36, -1.11, -0.86, -0.61, -0.36.

These selection cuts result in 480 models in each project, and a combined model count of 960. These photoionization models cover the most commonly discussed parameter space in metallicity, the N/O ratio, and the ionization parameter, so that they form a representative collection for calibrating and diagnosing chemical abundance. It also samples the relatively low metallicities and young stellar populations that are relevant to the dwarf galaxies and LIRGs tested in this paper, and are usesful for the prospective application to high-redshift counterparts.

The $N3O3$ parameter divided by the N/O ratio of each photoionization model are computed and plotted against the Neon line ratio in the upper left panel in Fig. 1. BOND data points are marked by square and CALIFA as circle symbols. All the $N3O3$ are divided and color-coded by the N/O ratio in the models to better illustrate that most of the residual dispersion is due to factors other than the abundance ratio. Only data points with $\log$ [Ne iii]/[Ne ii] between -2 and 2 are shown in the figure, which reflects the dynamical range of currently available spectral observations. As a result, only 820 out of 960 models are shown in the figure. The mean and standard deviation are calculated in bins of size $\Delta(\log\mathrm{[Ne\,{\sc iii}]/[Ne\,{\sc ii}]})=0.5$, plotted in the black solid lines and enveloping cyan shades. The standard deviation in each bin is interpreted as the error of this diagnostic. As the [Ne iii]/[Ne ii] ratio increases along the x-axis, $\log\ N3O3$ gradually converges and flattens to $0$, the expected value, but it diverges below $0$ at higher Neon ratios. This behavior is consistent with the discussion on soft and very hard radiation fields at the beginning of Sec. 3.1. It shows that the original $N3O3$ ratio works reasonably well in the regime $-0.5<\log\mathrm{[Ne\,{\sc iii}]/[Ne\,{\sc ii}]}<1.5$, and can be calibrated to function in a wider range of radiation field with the help of Neon line ratio.

The data points are color-coded by the N/O ratio, and many points that show a continuous gradient of N/O often cluster together very closely. This indicates that as expected, the N/O ratio has a negligible effect on ionizing structure and emissivity, and the scatter seen in the left figure is mainly due to differences in electron density, metallicity and radiation strength, and not due to changes in the N/O ratio. This relation is split into abundance groups in the right panels by multiplying with the N/O abundance ratio ranging from $\log$ N/O = 0 to -2. This illustrates how the N/O ratio can be determined with the $N3O3$ to Neon line ratio diagram.

One obvious issue with the upper left figure is that the BOND data points settle below CALIFA by about 0.1 dex, especially in the high [Ne iii]/[Ne ii] ratio region where $N3O3$ has the highest accuracy. This is because BOND is run on electron density $n_{e}=100\ \mathrm{cm^{-3}}$, resulting in lower $\mathrm{\varepsilon_{[N\,III]}}/\mathrm{\varepsilon_{[O\,III]52}}$ ratio.

The calibration using photoionization models validates the effectiveness of $N3O3$, as well as a more precise and reliable relation with an error estimation created with the addition of the Neon line-based radiation hardness tracer. However, this calibration also displays some limitations: the $N3O3$ diagnostic works best in hard radiation environments where the $\log$ [Ne iii]/[Ne ii] ratio is higher than -0.5 since this is the regime where the N⁺⁺ and O⁺⁺ ionization zones are co-spatial exactly; the $N3O3$ to Neon line ratio relation still has a dispersion of at least 0.03 dex, which largely comes from small variations in the ionization structure caused by metallicity differences; and, the models used for calibration only cover a limited and sparse sample in large parameter space.

To demonstrate this new diagnostic, several starburst and low metallicity galaxies in the local universe are selected for application. The choice is primarily subject to data and observation scheduling availability, but also various factors including the representativeness of the galaxies and their similarity to the supposedly low-metallicity high redshift galaxies. The galaxy sample and their basic properties are summarized in Table 2.

Arp 299 is an interacting system comprising two galaxies. The nuclear regions of the galaxies are named A (eastern component) and B (southwestern component), separated by about 20”. The component C located at $\sim$10” north of B is part of the overlapping/external star-forming region. (see Neff et al., 2004). The A source is the brightest IR continuum source in the system and hosts a young starburst estimated to have peaked 6 - 8 Myr ago (Alonso-Herrero et al., 2000). B is the second brightest in the far-IR continuum, but interestingly the [O iii] line coming from B is significantly weaker than its close neighbor C. While C shows the opposite feature, its [O iii] emission is almost as bright as A, but the continuum flux density is an order of magnitude lower than B. Because B and C are only separated by $\sim 10$ arcsec and both appear extended, they are blended in both SOFIA/FIFI-LS and Herschel observations, thus they are treated as one source “B&C” in the rest of the paper. But the majority of the line emission is actually from component C. Both A and B&C are observed in our SOFIA/FIFI-LS mapping. It is in the Great Observatories All-Sky LIRG Survey (GOALS; Armus et al., 2009) sample, too.

Haro 3 is a Blue Compact Dwarf galaxy (BCD) with two very young starburst regions. One of the two regions hosts a starburst younger than 5 Myr, and the other is $8\sim 10$ Myr old (Johnson et al., 2004). The stellar mass in Haro 3 is $\sim 10^{6}\ \mathrm{M_{\odot}}$, and SFR $\sim 0.8\ \mathrm{M_{\odot}/yr}$ measured by combining FUV and MIR photometry (De Looze et al., 2014). It is included in the Herschel Dwarf Galaxy Survey sample (DGS; Cormier et al., 2015).

II Zw 40 is also a Blue Compact Dwarf galaxy with a young starburst. The stellar mass is estimated to be $\sim 9\times 10^{5}\ \mathrm{M_{\odot}}$, and SFR $\sim 0.8\ \mathrm{M_{\odot}/yr}$(De Looze et al., 2014). The age of the starburst is estimated to be $\sim 3\ \mathrm{Myr}$ (Leitherer et al., 2018). II Zw 40 is also among the DGS sample.

MCG+12-02-001 is a LIRG and an interacting system. The stellar mass is about $8\times 10^{10}M_{\odot}$ and SFR is 30 to 55$\ \mathrm{M_{\odot}/yr}$ (De Looze et al., 2014; Howell et al., 2010). The starburst age is found to be 40–200 Myr from SED fitting (Pereira-Santaella et al., 2015). It is in the GOALS sample, but this source has very little optical observational data available.

NGC 2146 is a barred spiral galaxy, and one of the nearest luminous infrared galaxies. The star formation rate is estimated to be 7.9 $\mathrm{M_{\odot}/yr}$, dominated by the nuclear starburst. It is in the GOALS sample. NGC 2146 has Infrared Space Observatory (ISO) detection of the [N iii] line as well as both of the [O iii] lines. Besides, it is the only source not classified as “extended,, in Brauher et al. (2008) and with all three lines detected, plus it has ample optical and mid-infrared data available to aide our N/O measurement. While the star formation activities in NGC 2146 are extended over 2’ scales (Stacey et al., 1991), the ISO beam is uniform at about 70” for these lines and therefore provides good measurements of the inner 4.9 kpc region of this galaxy.

NGC 4194 is a LIRG and a minor merger. Estimates for its SFR are not in agreement, with SFR ranging from $\sim 46\ \mathrm{M_{\odot}/yr}$ by $H_{\alpha}$ observation (Hancock et al., 2006) to $\sim 13\ \mathrm{M_{\odot}/yr}$ in far-IR (De Looze et al., 2014). Most of the star formation happens in the super star clusters thought to be 5 to 15 Myr old (Konig et al., 2014; Pellerin & Robert, 2007). It is in the GOALS samples.

NGC 4214 is a Blue Compact Dwarf galaxy with 2 major components: region I is the larger component centered on the nucleus and is brighter in far-IR; region II is a smaller source located about 30 arcsecond to the southeast. Because only region I is within the field of view in our SOFIA/FIFI-LS observations, the discussion in this paper focuses on this component. The SFR in NGC 4214 as a whole is $0.063\ \mathrm{M_{\odot}/yr}$ (De Looze et al., 2014). Its stellar population age is estimated at only 4 to 5 Myr through FUV photometry (Pellerin & Robert, 2007) and UV point source observations (Williams et al., 2011). It is in the DGS sample.

M83 is a nearby spiral galaxy with a prominent bar. At present, only the central regions are mapped in the [O iii] 52 $\mu$m line by SOFIA/FIFI-LS, so only the nuclear region of M83 is discussed here. M83 hosts a circumnuclear starburst, which displays a burst age gradient along the star-forming arc spanning from 6 to 8 Myr (Houghton & Thatte, 2008; Knapen et al., 2010).

One major limitation of the application of the far-IR diagnostics we advocate here is the limited availability of the [O iii]52 and [N iii] line measurements. During its lifetime, the Herschel Space Observatory (Pilbratt et al., 2010) provided far-IR spectroscopy at unprecedented sensitivity and spatial resolution. However, the 52 $\mu$m [O iii] line is outside the 55 to 210 $\mu$m spectral range of PACS (Poglitsch et al., 2010) for z $<$ 0.06 objects. Furthermore the [N iii] line, which is typically $\sim 5$ times weaker than the [O iii] line was not often observed so that Herschel measurements of both the [O iii]52 and [N iii] lines are very rare. Herschel was decommissioned in 2013. Luckily NASA’s SOFIA observatory (Temi et al., 2018) is in operation and can provide the spectroscopy we need for the science presented here.

FIFI-LS (Fischer et al., 2018) is a Far-IR integral field unit onboard SOFIA, with the design and capabilities similar to Herschel/PACS instrument, but with a shorter wavelength spectral cut-off. In the 51 - 125 $\ \mathrm{\mu m}$ “blue channel”, it has $5\times 5$ spatial pixels (spaxels) of size $6"/\mathrm{pixel}$. Observations are dithered, so that our maps have finer spatial sampling. The velocity resolution is about 300 km/s at 50 - 60 $\mathrm{\mu m}$.

We obtained new far-IR spectroscopic observations for all of our sources except NGC 2146 over a period from February 2017 to May 2019 using FIFI-LS on SOFIA. The newly acquired spectral observations are listed in Table 3. All the SOFIA data have gone through the level 4 pipeline reduction, which is chopped and corrected for atmospheric transmission. However, for NGC 4214 and MCG+12-02-001 the atmospheric transmission around the [N iii] line is problematic, and to produce the [N iii] spectra, we manually corrected the raw data for the atmosphere transmission using the information in the spectral cube. This reduction shows a line flux similar to the pipeline reduction, but with a better line shape and improved signal-to-noise ratio (SNR). For other data sets which were pipeline processed we also manually restored data channels blanked out by the pipeline, since the FIFI-LS pipeline blanks channels with transmission $<$ 0.6. We find that this puts too harsh a standard in accepting data, and restoring the blanked channels can improve the baseline determination.

The SOFIA/FIFI-LS data are delivered as spectral cubes, and the process of measuring the spectra is as follows: first, the raw data is manually corrected for transmission to fill in the missing channels in the pipeline corrected data; second, weights ($w$) are calculated as $w=\mathrm{N}\times\tau$ for each spaxel, where $N$ is the number of scans in each observation and $\tau$ is the transmission. The error is calculated based on the rms in spectral dimension, taking into account that $\sigma\propto 1/\sqrt{w}$. The continuum map is then computed at each spatial position as the weighted average of the channels free of the spectral line. The continuum map is later subtracted from the whole spectral cube to produce a continuum subtracted cube, which is collapsed in spectral dimension to produce a line moment 0 map. Now an ellipse is defined as the emitting region of the spectral line, and integrated in each channel to get the spectrum. The error estimation takes into account that the noise is correlated across the $5.5"$ beam. Finally the line channels in the spectrum are added together as the line flux measurement.

The integrated line flux map and spectra for all but NGC 4194 [N iii] data are presented in Appendix B. The NGC 4194 [N iii] observation is not used in this paper, because it lies right beside a deep and wide telluric feature, and another narrow feature is at +200 km/s with respect to the expected source line center. The ISO [N iii] flux is used instead. For each source we include the line flux map and the continuum subtracted spectrum. On the line map we plot an ellipse showing the region from which the spectrum is extracted.

There are several caveats on the spectra. Instead of matching the same areas across different lines, we choose to pick the integration region such that it encloses most of the emission. This is justified by two reasons: first it would give us the highest SNR for our flux measurement, which is essential for [N iii] observations that in some cases only have tentative detections; second, the Herschel/PACS line fluxes used in this work are mostly integrated over a 5 by 5 map, so maximizing the flux in SOFIA/FIFI-LS observation ensures to include any extended emission contained in the PACS data. This strategy works well when we compare the SOFIA extracted fluxes with available ISO observations which has an even larger beam so that it could include more extended emission, and find they agree within $1\sigma$ error. Other SOFIA data, especially the [N iii] observations, also suffer from the low and/or highly variable transmission. In addition, many spectra have excess noise and sometimes systematic trends in their long and/or short wavelength edge channels. This is because FIFI-LS can saturate and enter a non-linear regime when the foreground (sky + telescope) emission is too strong, and the integration times on the edge channels are much less than those on the central channels as the instantaneous FIFI-LS bandwidth is swept across the line to increase spectral coverage (private communication with the SOFIA help desk). These effects complicate baseline determination. The [N iii] line from II Zw 40 is also very close to a deep absorption feature. Its baseline turns downwards at $v_{\mathrm{rest}}>150\ \mathrm{km/s}$ and the error rapidly increases as the telluric transmission declines, making the absolute line flux measurement unreliable. Though the line channels add up to flux = $5.51\pm 1.76\times 10^{-16}\ \mathrm{W/m^{2}}$, the asymmetric line shape indicates that up to $30\%$ of the flux could be missing. Thus we assess the uncertainty of the measurement should be $4\times 10^{-16}\ \mathrm{W/m^{2}}$, which is used in the paper.

Herschel/PACS (Poglitsch et al., 2010) provides the largest archive of [N iii] and [O iii] spectral observations to date. For the sample galaxies, all but NGC 4194 and MCG+12-02-001 have [N iii] spectra taken by PACS. Most of the data used here are already processed in Cormier et al. (2015) and Fernández-Ontiveros et al. (2016). Because PACS has a similar sized of field of view as our FIFI-LS maps, the flux measurements by FIFI-LS secure any loss of extended emission compared with that of PACS.

Infrared Space Observatory (Kessler et al., 1996) supplies the vital far-IR data for MCG+12-02-001, NGC 2146 and NGC 4194. The data were obtained with the ISO/LWS (Clegg et al., 1996) and are spatially unresolved. The line fluxes are taken from Brauher et al. (2008), and the explanation of data reduction details can be found in that paper.

The mid-IR [Ne ii] 12.8 $\mu$m and [Ne iii] 15.5 $\mu$m lines are originally observed with Spitzer/IRS in high-resolution mode (Werner et al., 2004; Houck et al., 2004). The flux data are directly taken from Inami et al. (2013) and Cormier et al. (2015), where a description of data processing can be found.

Arp 299 and NGC 4214 are two exceptions as they contain multiple components. The far-IR data of Arp 299 are from Hodis et al. (2020, in prep.), where fluxes are measured for A and B&C components separately within a Herschel/PACS map. For NGC 4214, we use both the far-IR and the mid-IR line flux values reported in Dimaratos et al. (2015) which measures two regions individually.

In addition to the Neon lines used for calibration, all sources except NGC 4214 have [Ne V] 14.3 $\mu$m flux upper limits reported in the same references. The upper limits are at least one order of magnitude lower that their [Ne iii] 15.5 $\mu$m fluxes, indicating negligible AGN contribution. For NGC 4214, the resolved photometric study in Williams et al. (2011) and PDR modeling in Dimaratos et al. (2015) show the central region is dominated by starburst activity with no trace of AGN. Thus all the sources in our sample have little to none AGN contamination in their [Ne iii]15 and [Ne ii]12 line fluxes, and the photoionization model calibration based on stellar population synthesis is applicable to this sample.

All the data we use in our application of the $N3O3$ diagnostic are summarized in Table 4.

The $N3O3$ parameter is calculated for each galaxy and then corrected for density by using the [O iii]52/88 line ratio to get $N3O3_{n_{e}}$. With these strong line parameters in hand, we computed the [Ne iii]15/[Ne ii]12 line ratio, and compared it with the calibrated diagnostic as in Sec. 3.3 to derive the high-precision N/O abundance ratio. The original and density corrected $N3O3$ parameters are plotted against our model calibration in Fig. 2. Their positions in the diagram represent the N/O ratio estimates. The values of strong line parameters and the calibrated N/O ratio are listed in Table 5.

The derived N/O ratio covers a large range from -1.6 to -0.5 in Table 5. The N/O of different types of galaxies also cluster together: those of dwarf galaxies are systematically lower than LIRGs, and M83 nucleus has the highest value. This is consistent with our understanding of the N/O evolution along with the starburst age and metallicity. The errors also change in an increasing trend as more spectral lines are used and more statistical uncertainty is introduced, but the systematic uncertainty in the calibration would decrease. And the final error budget is often dominated by the [O iii]52 and [N iii]57 line flux errors.

Notice that in the case where the noise is dominated by these two lines, the strong line methods are good estimations of the N/O abundance ratio. Comparing the $N3O3$ in Column (2), the most simplistic estimation of N/O using only two spectral lines, to the value in Column (5) which is the most precise and reliable diagnostic utilizing 5 lines, the value changes by $<0.15$ dex for all targets except Haro 3 and II Zw 40. This small change is because the effect of density correction which would lift N/O ratio result, is offset by the model calibration for the $\log$[Ne iii]/[Ne ii] in the -0.5 to 1.5 range which would decrease the N/O estimate. For the exceptions Haro 3 and II Zw 40, they have moderately high electron densities as well as hard ionization conditions, resulting in noticeable differences between $N3O3$ and the model calibration, but $N3O3_{n_{e}}$ parameter instead shows great agreement with the model calibrated results. This indicates that the $N3O3$ parameter is pretty good an estimator for the N/O abundance when the availability of data is limited, or when the uncertainty is dominated by observational error instead of the intrinsic dispersion of diagnostic, like in the case of samples used here. But for galaxies with moderately high electron density and hard radiation field, such as some dwarf galaxies and high-redshift galaxies, $N3O3_{n_{e}}$ could yield a better N/O estimate than $N3O3$.

However, using multiple lines and model calibration is still advantageous as it can mitigate the effects of density or radiation hardness in a way that is physically sound. The statistical errors may increase with more lines used, but systematic errors will decrease, so that the values obtained with the density and radiation field correcting lines are more reliable. Therefore the diagnostic method using more spectral lines is always recommended.

Of the nine galaxies that we have showcased here, only 6 have optical line-based N/O abundance measurements in the literature. Haro 3, II Zw 40, and NGC 4214 are in the DGS paper (Cormier et al., 2019), while the NGC 4214 N/O measurement is quoted from Kobulnicky & Skillman (1996) based on [N II]$\lambda 6584$/[O II]$\lambda 3727$ strong line ratio at various positions in long slit observation. In addition, Kobulnicky & Skillman (1996) also calculates N/O for II Zw 40. Haro 3, M83, NGC 2146 and NGC 4214 are measured in De Vis et al. (2019) using a new calibration in Pilyugin & Grebel (2016), denoted as PG16 hereafter. To get a global abundance ratio De Vis et al. (2019) combines observations of various scales including fiber, IFU and drift scan, then tries to model the N/O gradient and use the value at $R_{25}$. One major concern on comparing optical and far-IR N/O measurement is that the optical observations often resolve galaxies and potentially probe smaller regions than the far-IR lines, which use the integrated fluxes of the whole galaxy and measure N/O averaged over a larger aperture. Hence we calculate N/O using the integrated optical spectroscopic observation in Moustakas & Kennicutt (2006), and the $N2S2$ index defined in Pérez-Montero & Contini (2009). This ensures the optical line fluxes also cover the whole galaxy, and the [N II]$\lambda 6584$ to [S II]$\lambda 6717,6731$ ratio is insensitive to extinction. The values taken from these papers and re-calculated with $N2S2$ are listed in Table 6 along with those derived from the $N3O3_{n_{e}}$ model calibration as our best optical estimate of N/O. Fig. 3 shows a comparison of these estimates.

For NGC 4214, NGC 2146 and M83 nucleus, the far-IR derived and optical derived N/O abundance ratios agree within the $1\sigma$ error. But for Arp 299 A and Arp 299 B&C, the optically derived values are 1.5 to 2 $\sigma$ higher than our $N3O3_{n_{e}}$ calibrations. In the case of Haro 3 and NGC 4194, there are more than one optical measurements. For Haro 3, the N/O ratios quoted in Cormier et al. (2015) and Shi et al. (2005) are higher but within $\sim 2\sigma$ away from the one-to-one agreement (diagonal line), while the values in the other two sources are more than $4\sigma$ higher than our far-IR result. Similar for NGC 4194, the optical results are higher and at 2 to 2.5 $\sigma$ away from the diagonal line. As for II Zw 40, the various optical measurements are off the far-IR estimate but within the $1\sigma$ errorbar, which is understandable given its highly unreliable measurement of [N iii] flux.

It is worth noting that the optically derived N/O ratios are overall $\sim$0.2 dex higher than the far-IR results, and N/O measured by different optical probes do not agree with each other. Therefore, we also compare the relationship of optical and far-IR N/O diagnostics by computing the N/O ratios of photoionization models using the optical methods against those by our far-IR approach. The photoionization models are the ones selected in Sec. 3.2 for consistency. We adopted two optical measurements, the $N2S2$ index and PG16, because they are used in our optical N/O calculation and De Vis et al. (2019) respectively. The optical N/O ratio of models are plotted against far-IR N/O in Fig.3 as blue ($N2S2$) and red (PG16) translucent points. The model N/O ratios calculated by $N2S2$ show a similar trend as the far-IR measurements, and the least mean square fitting relation (shown as the blue dashed line in Fig. 3) suggests a close match between the two methods. However, the N/O ratio estimated by PG16 systematically deviate from the far-IR method, and follows a different trend as shown by the fitted line (red dashed line) in Fig. 3. In the figure, PG16 overestimates the N/O ratio in the low N/O regime, then transits to underestimation at N/O beyond -0.9. This indicates a significant discrepancy of PG16 calibration with our far-IR method and $N2S2$. When compared with the fitted lines, data points of both optical diagnostics show dispersion of 0.4 dex. Because $N2S2$ and PG16 are derived through empirical calibration on H ii regions, the discrepancy and large scatter seen in the comparison using photoionization models imply systematic differences in the empirical and model based calibrations. It might be because either the sample of H ii regions used for calibrating these diagnostics don’t have enough coverage in the physical parameter space, or the photoionization models explore too large a region in parameter space so that they include physical conditions not present in actual galaxies. Calibrations based on individual H ii regions could also be at odds with model calibrations that utilize stellar population synthesis, as the latter is more likely to capture the global properties of a galaxy. It requires samples of H ii regions and photoionization model grids that have finer and wider parameter coverage to further study the relationship and reliability of the two kinds calibrations, but that goes beyond the topic of this work. Even though the $N2S2$ shows good agreement with the model calibrated $N3O3_{n_{e}}$ index, the N/O of our galaxy sample measured by $N2S2$ are still distributed at upper left to the fitted line in Fig. 3. Hence the difference or the large scatter in the optical to far-IR diagnostics model comparison are not enough to explain the offset in Fig. 3.

There are other factors that may affect the result of far-IR N/O estimate and its comparison to optical techniques. The first to consider is the different aperture size for far-IR data. We do not expect beam effect between SOFIA/FIFI-LS and Herschel/PACS for the reasons stated in Sec.4.2. In the case of NGC 4194, ISO [N iii] data is used which has a 75 arcsec aperture and could contain more extended flux than in the SOFIA field. But that would only contribute to $<30\%$ [N iii] flux in ISO data given the small size of NGC 4194, and would lead to overestimation of N/O instead of underestimation. As for NGC 2146, it is not affected by aperture size because all the far-IR data are taken from ISO observations. Besides, in order to increase the far-IR N/O estimates by 0.2 dex to match with optical results, it would need either $60\%$ more [N iii] flux, or 36% less [O iii]52 flux, or the [Ne iii]15/[Ne ii]12 line ratio to increase by at least ten folds. The beam effect can not account for such difference. Hence we conclude that the aperture size difference of far-IR data can affect the precision of line ratio, but do not cause the lower value of N/O probed by far-IR lines.

Although the optical N/O results quoted in the previous studies resolve the galaxies, hence they may probe different and much smaller regions than our far-IR measurements, this is not the case for our $N2S2$ calculation which uses spectroscopic observations integrated over the whole galaxies. Use of $N2S2$ also avoids the impact of dust extinction. So the spatial mismatch of regions and extinction may not be responsible for the discrepancy.

However, the optical diagnostics may still probe regions different from far-IR in terms of physical conditions. Because the optical forbidden lines are enhanced in high temperature regions, while the far-IR fine-structure lines are enhanced in high density regions, we suspect that the optical lines probe hotter and more diffuse ISM than the far-IR lines. This means the optical lines are more susceptible to diffuse ionized gas (DIG). All the optical abundance probes including PG16 and $N2S2$ rely on the low ionized [N II]$\lambda 6584$ line, or other low ionized species such as [O II] and [S II] doublets, for which up to 30% of the line emission may arise from DIG. Although the study by Zhang et al. (2017) shows that [N II]/[S II] ratio is less affected by DIG emission than the indexes using Hydrogen recombination lines, we suspect the N/O ratio measured by $N2S2$ is still affected by the abundance in DIG that is not probed by the doubly ionized lines used in $N3O3$. This effect can be non-negligible for massive star-forming galaxies, which host most of the recent star formation activities around the nuclei or in a few compact regions, while the DIG across most of the galaxies are enriched by the relatively old population of stars. It also indicates an interesting approach to study and dissect the effect of DIG.

Another factor to consider is that the low and highly ionized lines could come from H ii regions with different physical conditions. As optical low ionized lines emerge largely from the population of H ii regions that host less massive stars with softer radiation, the ISM probed by the low ionized lines have longer life time and could be more enriched with secondary Nitrogen. In contrast, far-IR [N iii] and [O iii] lines are dominated by the dense ISM surrounding young, massive stars with hard radiation fields. These effects combined can also lead to lower N/O ratio probed by highly ionized lines, accounting for the offset shown in the comparison. Detailed chemical evolution models combined with photoionization grids would be needed to test this hypothesis.

We can not yet draw a conclusion for what causes the discrepancy in the optical and far-IR derived N/O ratios. But we suggest this could be related to the systematic differences between the empirical and photoionization model based calibration, and that optical and far-IR methods probe ISM of different physical conditions. Both questions touch the fundamental question of the reliability and nature of those abundance diagnostics. To further validate this new N/O diagnostic in nearby galaxies and study the probable difference in optical and far-IR derived N/O, it is essential to obtain more high-quality [O iii] 52 $\mu$m and [N iii]57 $\mu$m spectra with SOFIA/FIFI-LS.