# On the universal validity of Case B recombination theory

C. Scarlata1 M. Hayes2 N. Panagia3,4 V. Mehta1 F. Haardt5,6 M. Bagley1
1Minnesota Institute for Astrophysics, University of Minnesota, Minneapolis,
MN 55455, USA.  2Stockholm Observatory, Department of Astronomy, Stockholm
University, 106 91 Stockholm, Sweden  3Space Telescope Science Institute, 3700
San Martin Drive, Baltimore MD 21218, USA  4Supernova Limited, OYV #131,
Northsound Rd., Virgin Gorda VG1150, Virgin Islands, UK  5DiSAT, Università
dell’Insubria, via Valleggio 11, 22100 Como, Italy  6 INFN, Sezione di Milano-
Bicocca, Piazza delle Scienze 3, 20123 Milano, Italy  Corresponding Author:
<EMAIL_ADDRESS>
###### Abstract

In an ongoing search for low-mass extreme emission line galaxies, we
identified a galaxy with a H$\alpha$/H$\beta$ Balmer line ratio of $2.620\pm
0.078$. H$\alpha$/H$\beta$ Balmer ratios lower than the dust-free Case B value
appear relatively frequently in extreme emission line galaxies. These low
values suggest that the Case B assumption may not be valid in these objects.
After ruling out the possibility that the low H$\alpha$/H$\beta$ ratio is due
to systematic errors introduced by observational effects, we use constraints
from the total H$\beta$ luminosity, the [O iii]/[O ii] line ratio and the
Balmer line equivalent widths, to suggest that the gas is optically thick to
both Balmer H$\alpha$ and Ly$\alpha$ photons, and the geometry and orientation
of the scattering gas causes H$\alpha$ photons to be preferentially removed
from the line of sight with respect to higher order Balmer series photons.
Finally, we use data from the SDSS survey to show that Balmer self–absorption
may be more important than previously assumed in high excitation emission line
galaxies, where Ly$\alpha$ pumping of the hydrogen excited state can be
effective. If not recognized, Balmer self-absorption could lead to inaccurate
estimates of galaxy physical properties. As an example, the effect of dust
extinction could be over-estimated, for spherically symmetric scattering
medium, or under-estimated, for a not spherically-symmetric distribution.

##  1 Introduction

Over the past decade, it has become clear that the galaxies we are able to
observe in, or close to, the reionization epoch are characterized by young
starbursts with strong nebular emission lines, compact sizes, low dust
content, and high ionization parameters (e.g., Schaerer & de Barros, 2009;
Finkelstein et al., 2012; Dunlop et al., 2013; Capak et al., 2015; Bouwens et
al., 2017; De Barros et al., 2019; Endsley et al., 2020). These results have
now been confirmed with extensive spectroscopic campaigns conducted with the
James Webb Space Telescope, which push observations of rest-frame optical
lines out to redshifts $\approx 9$ (e.g., Topping et al., 2024; Pirzkal et
al., 2023).

Wide field optical surveys like the Sloan Digital Sky Survey (SDSS Blanton et
al., 2017) have shown that galaxies can be identified at low-$z$ that have
properties, comparable to these reionization era systems (Schaerer et al.,
2022). These analogs are characterized by compact morphologies and extreme
nebular spectra, with high [O iii]/[O ii] line ratios (typically above 5,
Izotov et al., 2017) and extreme Balmer line equivalent widths (EWs, over
several hundreds Å for H$\alpha$, and [O iii]). Additionally, UV spectra of
these objects show a high incidence of Ly$\alpha$ in emission (very close to
100%), much higher than in normal star forming galaxies with similar stellar
masses (e.g. Henry et al., 2015; Jaskot et al., 2019). These properties are
indicative of low-metallicity (Z$\lesssim$10% Z⊙, Cardamone et al., 2009;
Amorín et al., 2010; Yang et al., 2017b; Senchyna et al., 2017), highly
excited gas, ionized by young and hot starbursts, and with typically small
amount of dust (e.g., Henry et al., 2018; Jaskot et al., 2017, 2019). Finally,
direct measurements of their ionizing spectra (below rest–frame 912Å) proved
that a substantial fraction of their ionizing radiation is able to escape the
interstellar medium, with escape fraction up to 70%, (Izotov et al., 2016b;
Izotov et al., 2016a; Schaerer et al., 2016; Verhamme et al., 2017; Izotov et
al., 2018; Flury et al., 2022b, a).

Many of the physical properties derived from the emission line spectra of
these galaxies are based on one fundamental assumption: that the Case B
recombination limit is valid and can be used to interpret their spectra.
Although this assumption is verified in the vast majority of star-forming
galaxies, there has been some evidence that the spectra of extreme emission
line galaxies (EELGs) may deviate from Case B: 32 out of 41 of the EELGs in
Yang et al. (2017b) have H$\alpha$/H$\beta$ ratio smaller than
(H$\alpha$/H$\beta$)CaseB. Similarly, Atek et al. (2009) find that a number of
$z\sim 0.3$ Ly$\alpha$ emitters have comparably low Balmer line ratios. Early
results based on JWST spectra show similar trends: Pirzkal et al. (2023) find
a significant number of $1\lesssim z\lesssim 3.5$ sources in the NGDEEP-NISS
survey which appear to violate Case B recombination. This result seems to hold
at even higher redshifts, where examples of H$\alpha$/H$\beta$ line ratios
smaller than Case B are found at $z\approx 6$ by, e.g., Cameron et al. (2023)
and Topping et al. (2024). As part of an ongoing search aimed at pushing the
selection of local analogs to lower masses and lower metallicities, described
in Lin et al. (2022), we discovered one object (hereafter, SXDF308) that shows
an emission line spectrum that could dramatically change the physical
interpretation of EELG spectra. The Balmer H$\alpha$/H$\beta$ line ratio
measured in this object suggests that the universally accepted Case B limit is
in fact not a good assumption for this galaxy. We argue in Section 4 that the
physical conditions specific to this galaxy may in fact be more common than
previously assumed.

The implications of this finding are important: if Case B is not a valid
assumptions for some high excitation emission line galaxies, then physical
properties such as dust extinction, intrinsic ionizing power, and consequently
Lyman continuum escape fractions, computed from the intensity of the Balmer
lines will be incorrect. In this paper we present a thorough investigation of
the emission line spectrum of SXDF308, including Balmer and Oxygen line
emission, to determine the properties of the nebular gas and fully explore the
reasons for the departure from the Case B scenario.

The paper is organized as follows: in Section 2 we present the observations
and the data analysis. Different physical interpretations for the anomalous
line ratios are addressed in Section 3 and discussed in Section 4. We present
our conclusions in Section 5.

Figure 1: Main panel: MMT spectrum of SXDF308. The full wavelength and flux
range is shown to emphasize the line ratios between the strongest emission
lines. In particular, note the large [O iii]5007/H$\beta$ line ratio,
indicating the presence of a hard ionizing radiation field. Insets: zooms in
around faint spectral features, indicating very low abundance (e.g., low [N
ii]/H$\alpha$ ratio), high electron temperature (from the strong [O
iii]$\lambda 4363$) and hard ionizing radiation field (He ii,[Ar iv]). The
blue shaded region indicates the $1\sigma$ error on the spectrum.

##  2 Observations and data reduction

SXDF308 (RA=02:19:12.857, DEC=–04:30:17.62, J2000) was discovered as an
extreme emission line candidate using the $u$, $g$, $r$ broad band images
covering the 2 deg2 Subaru Deep Field (Mehta et al., 2018). Extreme emission
line candidates are selected as objects showing a significant $g-$band flux
excess, that we assume is a result of a strong emission line with observed
equivalent width larger than few hundreds Å. This filter combination
identifies galaxies in the 0.05$\lesssim z\lesssim 0.1$ and 2.3$\lesssim
z\lesssim 3.1$ redshift ranges, where the emission line(s) in question may be
[O iii]+H$\beta$ and Ly$\alpha$, respectively. SXDF308 was spectroscopically
confirmed to be a low redshift galaxy in the spectroscopic followup and has a
redshift of $z=0.0695$.

The spectroscopic data for SXDF308 were obtained in queue mode on Sept. 26,
2017 using the Hectospec Multifiber Spectrograph at the MMT 6.5m telescope in
Arizona. Hectospec uses the $f/5$ secondary mirror, which incorporates
atmospheric dispersion compensation prisms to correct for atmospheric
dispersion between airmass of 1 and 2. We used the 270 mm-1 grating, blazed at
5200Å to provide a spectral coverage of 3650–9000Å (Fabricant et al., 2005),
at a spectral resolution, for the used fiber diameter (15), of $\approx 6$Å.
Second-order blocking filters are not available to use with Hectospec.
However, as we discuss below, the reddest line of interest ([S ii] $\lambda
6731$ Å) falls at 7200 Å, where the second order contamination from blue light
is negligible (Gezari et al., 2012). The total integration time was 4,500
seconds, split in 5 exposures to remove cosmic ray hits. The seeing full-
width-half-maximum during the observations was stable at 10, conditions were
clear, and the average airmass of the observations was 1.4. A large number
(103) of fibers were placed on empty sky111Empty sky regions were identified
using SPLASH deep imaging regions, resulting in an accurate determination of
the sky background, with the sky fiber-to-fiber variation of $\approx$0.2%.

The data were reduced using the HSRED 2.0222http://archive.mmto.org/node/536
which was originally written by R. Cool. The pipeline performs bias, flat-
field, illumination, and wavelength calibrations, background sky subtraction,
and extracts one-dimensional spectra. The flux calibration was performed using
3 standard stars (BD+28-4211, HD-217086, HD-192281) observed on Oct. 1–3,
2017, at airmasses between 1 and 1.4, along with a set of 4 stars with
available broad–band photometry from SPLASH-SXDF, observed together with
SXDF308. We computed the wavelength dependent sensitivity function as the
average of the sensitivity functions obtained from all the standard stars
exposures. We verified that second-order contamination does not affect the
results of the flux calibration. First, at the wavelength of interest (7020Å
for H$\alpha$) the contamination of the stellar continuum due to radiation at
3500Å is minimal (Fabricant, private communication). Second, as the amount of
second-order contamination from blue light depends on the color of the source,
we verified that the computed sensitivity curve does not depend on the color
of the calibration star, demonstrating that second-order contamination is not
an issue bluewards of 7200 Å. At the high signal–to–noise ratio of the
available data, the major uncertainty in the flux measurements comes from
systematic uncertainties introduced during the flux calibration step (e.g.,
Berg et al., 2015). We quantify this systematic contribution in Appendix A.2.

The extracted spectrum was corrected for Milky Way foreground extinction using
the Cardelli et al. (1989) extinction law, assuming $R_{V}=3.1$ and an
$E(B-V)=0.0182\pm 0.0002$ computed from the Schlafly & Finkbeiner (2011)
recalibration of the Schlegel et al. (1998) extinction map [corresponding to
$A_{V}=0.0561$(mag)]. Figure 1 shows the extracted spectrum over the full
wavelength and flux range. The three insets show zooms around faint spectral
features.

Fluxes of isolated emission line were computed via direct integration of the
continuum–subtracted line profiles. The continuum was estimated by computing
the median flux over 5Å on each side of the emission line. This approach was
chosen because the line profiles, when measured at high signal-to-noise, were
found to deviate from perfect Gaussian functions in the core region. This
discrepancy is shown in Figure 2, where we present the full line profiles for
both the H$\alpha$ and H$\beta$ emission lines together with the best fit
Gaussian functions333The fits were performed using the optimize.curve_fit
algorithm within the SciPy libraries (Jones et al., 2001) (solid curves).
However even when computing the line ratios using Gaussian integration, the
result remains unchanged. The inset zooms in on the continuum to allow the
reader a visual assessment of the data quality. Given the large EW of the
emission lines, the error budget on the total line flux is dominated by the
flux calibration uncertainty. The flux measurements, relative to H$\beta$, and
the corresponding errors are listed in Table 1. We measure a Balmer
H$\alpha$/H$\beta$ of $2.620\pm 0.078$, significantly smaller than the
canonical Case B recombination theory limit, commonly assumed in the analysis
of H ii regions. For typical electron temperatures ($T_{e}=10^{4}$K) and
electron densities ($n_{e}=10^{2}$ cm-3), a completely dust free H ii region
would have (H$\alpha$/H$\beta$)${}_{\rm CaseB}=2.863$ (e.g., Osterbrock,
1989).

Figure 2: Full emission line profiles for both the H$\alpha$ and H$\beta$
lines. The inset zooms in on the continuum around the lines. The solid curve
indicate the best-fit Gaussian profiles to the emission lines.

###  2.1 Impact of observational effects

We explored (and rejected) the possibility that the low Balmer ratio is a
result of observational effects producing a wavelength dependent bias in the
measured fluxes.

We first consider atmospheric differential refraction (Filippenko, 1982) and
chromatic seeing (Fried, 1966; Boyd, 1978). Both these effects can introduce a
wavelength dependent biases. However, they cause a larger flux loss at the
blue end of the spectral range compared to the red end and thus, corrections
for these effects would make the H$\alpha$/H$\beta$ ratio smaller. The
differential atmospheric refraction is corrected for via the atmospheric
dispersion corrector installed at the secondary mirror of the MMT (Fabricant
et al., 2008).

We estimate the contribution of chromatic seeing following the recent work of
Herenz et al. (2017). These authors use Integral Field Spectroscopic (IFU)
data to characterize the wavelength dependency of the seeing FWHM. Using their
best fit linear fit (Table 2 in Herenz et al., 2017), we find that the seeing
is $\approx$ 9% larger at 5200Å compared to 7000Å (the observed wavelengths of
H$\beta$ and H$\alpha$ respectively). Accordingly, given the 10 seeing during
the observations, the energy enclosed by the 15 Hectospec fibers is $\approx$
7% smaller at 5200Å compared to 7000Å. These calculations are valid for point
sources. Given the uncertainty on the real galaxy profile, we do not apply
this correction to the observed line ratio. We however caution the reader that
the real H$\alpha$/H$\beta$ could be as low as 2.43, as a consequence of
chromatic seeing.

As mentioned above, our observations were conducted without an order-sorting
filter to remove second order blue light. This phenomenon only affects
spectrophotometry of the stellar continuum radiation, unless by chance a blue
emission line would fall at exactly the same detector coordinate as H$\alpha$,
and we have verified that this is not the case. At the observed wavelength of
the H$\alpha$ emission line ($\sim 7020$Å) the second order contamination of
stellar continuum is negligible, due the combination of the second order blaze
throughput curve falling off and the fiber throughput also decreasing at 3500Å
(Fabricant, private communication). To test this, we compared the sensitivity
functions inferred from the three standard stars, observed over the course of
three nights. These stars span a broad range of colors, with
m${}_{3500}-$m7000 varying from $-$1.4 to $+$1.4, AB magnitudes. If second
order contamination at 7000Å were important, we would expect to see a
correlation between relative differences in the sensitivity curves and the
stellar colors. We do not see a measurable trend with stellar color,
confirming that the effect is not important at the accuracy of the
calibration.

We also examine the possible contribution of underlying stellar absorption at
the wavelength of the Balmer H$\alpha$ and H$\beta$ lines. We see no
indications of absorption. Given the large EWs of the Balmer emission lines
(i.e., EW[H$\beta$]$=165\pm 14$Å), and the extremely hard ionizing radiation
field implying very hot stars (Figure 1), this is not surprising. The
underlying absorption correction for a stellar age of a few Myrs is smaller
than 3Å at H$\beta$ (González Delgado et al., 1999) and it is larger at
H$\beta$ than H$\alpha$. Thus, similarly to the effects of atmospheric
refraction, more flux would be lost to H$\beta$ than H$\alpha$. Correcting for
stellar absorption would then result in an even lower H$\alpha$/H$\beta$
ratio.

Finally, we applied a correction for Milky Way reddening, using the Schlafly &
Finkbeiner (2011) recalibration of Schlegel et al. (1998) extinction maps.
Given the low resolution of these maps, we consider the possibility that this
correction is overestimated. The most conservative assumption would be no MW
extinction at the position of SXDF308. An E(B-V)=0, however, would increase
the H$\alpha$/H$\beta$ ratio to 2.649$\pm$0.078, still inconsistent with the
typical Case B recombination theory value of 2.86 at $\approx$ 2.7$\sigma$.

To conclude, observational effects are not able to explain the low value of
the H$\alpha$/H$\beta$ ratio.

##  3 Physical explanation of the low H$\alpha$/H$\beta$ ratio

In order to assess the implications of the measured Balmer ratio, we need to
first derive the physical conditions of the nebula. This is done in Section
3.1. We then review the values expected in textbook Case B and Case A limits
(Section 3.2), and move on to explore other mechanisms that can impact the
emissivity of the Balmer lines. The Case C limit is discussed in section 3.3,
and Balmer self absorption in Section 3.4. As we argue below, we conclude that
Balmer self absorption in non-spherically symmetric gas is the favored
interpretation.

###  3.1 Estimate of $T_{e}$, $n_{e}$

The electron temperature ($T_{e}$) in the $O^{2+}$ region can be estimated
directly using the ratio between the intensity of the
${}^{1}S_{0}\rightarrow\,^{1}D_{2}$ 4363Å emission line and the combined
intensities of the ${}^{1}D_{2}\rightarrow\,^{2}P_{2}$ 4959Å and
${}^{1}D_{2}\rightarrow\,^{2}P_{1}$ 5007Å lines. The energies required to
collisionally excite the ${}^{1}S_{0}$ and ${}^{1}D_{2}$ states in the O2+ ion
are 5.4 and 2.5eV respectively. Thus, in the limit of low electron density
($n_{e}<10^{3}$cm-3), as $T_{e}$ increases, the relative population of ions in
the ${}^{1}S_{0}$ state increases, resulting in a relative brightening of the
4363Å line. We use the PyNeb software (Luridiana et al., 2015) to compute the
electron temperature, using the atomic data of Froese Fischer & Rubin (2004).
We compute an electron temperature of $T_{e}(O^{2+})=15380\pm 850$K. This
electron temperature is higher than typical values observed in galactic H ii
regions, but is consistent with the temperatures measured in the most metal
poor galaxies (e.g., Skillman et al., 2013). We cannot compute the temperature
of the low ionization zone directly, as the S/N of the spectrum is not
sufficient to measure lines past rest–frame 7100Å, and the [N ii]$\lambda$5755
is not detected, likely because of the low nitrogen abundance. Thus, we follow
Skillman et al. (2013) and use the Pagel et al. (1992) relation between
$T_{e}(O^{+})$ and $T_{e}(O^{2+})$444Defining
$t_{3}=\frac{10^{4}}{T_{e}(O^{2+})}$ and $t_{2}=\frac{10^{4}}{T_{e}(O^{+})}$,
then $t_{2}^{-1}=0.5(t_{3}^{-1}+0.8)$. This relation depends on the
photoionization modeling, and Pagel et al. (1992) used the calculations by
Stasińska (1982) . Similarly, we used the relationship derived by Garnett
(1992, see footnote 555$t(S^{2+})=0.83t_{3}+0.17$) to derive the temperature
of the S2+ region. The computed temperatures are tabulated in Table 2.

Finally, we estimated an electron density of
$n_{e}(S^{+})=10^{+200}_{-10}$cm-3 using the [S ii]$\lambda\lambda$6717,6731
doublet. In addition to the [S ii] doublet, we also detect the [Ar
iv]$\lambda\lambda$4712,4740 doublet, originating in high excitation gas. The
ratio between the two emission lines ($R1\equiv I_{4712}/I_{4740}$) can be
used to derive the electron density, following Keenan et al. (1997). At the
resolution of the MMT spectrum, the [Ar iv]$\lambda$4712 line is contaminated
by the He i 4713 line. We estimated the He i 4713 line flux from the He i 4471
line using the emission line ratios presented in Porter et al. (2012). The
corrected ratio $R1$=0.7$\pm$0.4 only allows us to place an upper limit on the
$n_{e}(Ar^{3+})<10^{3}$cm-3.

###  3.2 Case B and Case A limit

Emission line spectra from star-forming galaxies are interpreted within the
successful Case B recombination limit theory, which describes the limit in
which all recombination photons in the Lyman series are scattered in a
spherically symmetric medium often enough to be eventually degraded into
Balmer and higher series photons and either two-photon continuum or
Ly$\alpha$. We used PyNeb to compute the expected Case B Balmer ratio, given
the gas temperature and density measured in Section 3.1. We find
(H$\alpha$/H$\beta$)${}_{\rm CaseB}=2.79$, 2.2$\sigma$ higher than the
observed value. The opposite limit from Case B is Case A, where the H i column
density is low enough for the gas to be optically thin in all recombination
lines, including those of the Lyman series. In this limit, the emissivities of
Balmer photons are reduced, and the predicted H$\alpha$/H$\beta$ ratio is
lower than the corresponding Case B value. Figure 3 shows the variation of the
Balmer ratio with electron temperature, computed for $n_{e}=10$ cm-3 in the
Case A and Case B limits. The curve does not change appreciably for
$n_{e}=100$ cm-3, at the measured electron temperature. Although the
(H$\alpha$/H$\beta$)CaseA decreases more rapidly with temperature than
(H$\alpha$/H$\beta$)CaseB, it never reaches the value observed in SXDF308,
although it is consistent (within 1.5$\sigma$) with it.

Figure 3: H$\alpha$/H$\beta$ ratio as a function of electron temperature,
computed for two case limits (A, B), assuming a gas electron density of 10
cm-3.

###  3.3 Case-C limit

It is instructive to consider other physical mechanisms that can affect the
intrinsic value of the H$\alpha$/H$\beta$ ratios and change it from the Case B
limit. One possible mechanism, important for optically thin gas, is the
contribution to the Balmer line emission from reprocessed continuum photons.
This mechanism, known as Case C, was first discussed by Baker & Menzel (1938)
and more recently by Ferland (1999) and Luridiana et al. (2009). Case C
relaxes the assumption, implicit in the Case A and Case B calculations, that
excited levels can only be populated by radiative recombinations and cascades
from higher levels, and includes the possibility that they can be effectively
excited via fluorescence, i.e., through the absorption of continuum photons at
the wavelength of the Lyman lines with subsequent decay into n$\geq 3$ levels.
Under Case C, the resulting emission line spectrum from the nebula depends not
only on its physical conditions (i.e., electron temperature and density), but
also on the detailed shape of the continuum at energies between 1 Ryd and
Ly$\alpha$. Ferland (1999) and Luridiana et al. (2009) show that continuum
pumping contributes substantially to the H$\alpha$/H$\beta$ ratio (compared to
Case A) in the most internal regions of ionized clouds.

In order to explore the viability of this interpretation we computed
photoionization models using Cloudy version 17.01 (Ferland et al., 2017). The
photoionization models presented in Figure 4 assume a dustless spherical cloud
with a inner radius of 1pc and a constant hydrogen density of $n_{\rm H}$=10
cm-3. We modeled the ionizing source using a single burst stellar population
models created with Starburst 99 (Version 7.0.0 Leitherer et al., 2014). We
run the calculations of the stellar population synthesis models using the
Geneva stellar tracks (both with and without rotation), assuming a single
stellar metallicity of $Z=0.008$, and a Kroupa (2001) initial mass function.

In addition to the H$\alpha$/H$\beta$ ratio, we also computed the expected O32
ratio and H$\beta$ luminosity as a function of the cloud’s depth. We know that
SXDF308 shows an unusually high O32 ratio (O${}_{32}=11.80\pm 0.74$), so any
photoionization model needs to be able to reproduce the high excitation state
of oxygen as well. The varying line ratios as a function of cloud’s depth
(computed using Cloudy’s cumulative intensities) are presented in Figure 4 for
a burst age of three million years. Older bursts were not able to reproduce
the observed ranges of line ratios, indicated in the Figure with horizontal
grey areas. Lines of different color show models with varying total gas
metallicity between Zgas=1 and 0.001Z⊙, as indicated in the Figure. All models
show similar trends in the H$\alpha$/H$\beta$ ratio with depth, which is due
to the varying contribution of fluorescent emission in the two emission lines.
H$\alpha$ and H$\beta$ fluorescent photons are mainly produced after the
absorption of Ly$\beta$ and Ly$\gamma$ continuum photons, which are abundant
in the illuminated part of the nebula. Because of the larger Ly$\beta$ optical
depth compared to Ly$\gamma$, the H$\alpha$ emissivity peaks at smaller depth
than H$\beta$, resulting in the observed variation with depth. Moving deeper
into the cloud, Lyman line photons are consumed and the H$\alpha$/H$\beta$
ratio approaches the Case B limit. The thin grey lines in the top panel of
Figure 4 show, for reference, a set of identical calculations where the
continuum radiative pumping of the hydrogen Lyman lines was turned off. All
line ratios converge at large depth. Decreasing the gas metallicity pushes the
curves to the right, i.e., the same value of H$\alpha$/H$\beta$ is observed
deeper in the cloud. The gas metallicity mainly affects the cloud electron
temperature, going from Te of 12kK to 15kK for Zgas of 0.1 and 0.025Z⊙,
respectively, resulting in a change of the recombination-only line ratio.
Indeed, the curves are shifted to the right even when continuum pumping is
turned off.

All explored models are able to reproduce the observed value of the hydrogen
emission line ratio in the depth range between approximately 100 and 500 pc,
depending on the gas metallicity. The situation is, however, different when
the ratio of the oxygen lines is considered. The O32 ratio as a function of
cloud depth is shown in the bottom panel of Figure 4. The lines represent the
same models as in the top panel, although we do not show the models with
continuum pumping turned off, since it does not affect the O32 values. As
expected, the oxygen ionization state and the resulting O32 line ratio is very
sensitive to the position in the cloud, and depends very little on the gas
metallicity. The combined constraints from the Balmer and oxygen line ratios
limit the viable range of gas metallicities to $Z<0.1Z_{\odot}$.

In addition to line ratios, the model needs to reproduce the absolute
luminosity of the Balmer lines. In order to produce line ratios within the
observed ranges, the 3 Myr starburst was normalized to a total number of
ionizing photons (Q) of $\log{Q}=52$. The Cloudy total H$\beta$ line
luminosity for this optically thin cloud is, however, a factor of $\approx 10$
smaller than the observed 3$\times 10^{40}$ erg s-1. Although it is possible
to obtain a H$\beta$ luminosity consistent with the observed value, these
models would require a near UV flux many orders of magnitudes larger than the
observed one. Invoking absorption of dust is not a viable option given the
observed line ratios. Thus, although promising, the fluorescent mechanism
cannot be uniquely responsible for the observed line ratios and luminosities.

Figure 4: Top panel: Predicted H$\alpha$/H$\beta$ ratio as a function of
depth, computed for a 3Myr burst with different metallicities (colored lines,
see legend for details). Also shown are models in which the continuum
radiative pumping of the hydrogen Lyman lines was turned off (thin grey
lines). Bottom panel: same as top panel, but for the [O iii]/[O ii] ratio
(curves with metallicity smaller than $Z_{\odot}$ lie very close to each
other).

###  3.4 Balmer self–absorption

Another mechanism that can affect the relative intensity of the Balmer lines
is self-absorption of Balmer photons. This effect is important if the gas’
optical depth to Balmer photons is not negligible and requires a large
population of hydrogen atoms in the $n=2$ energy level.

Figure 6 shows the position of SXDF308 in the H$\alpha$/H$\beta$ versus
H$\gamma$/H$\beta$ ratio diagram, together with the values of the emission
line ratios predicted for Balmer self-absorption (from Cox & Mathews, 1969).
Specifically, each track shows the calculations for constant Ly$\alpha$
optical depth (labeled in each curve) and varying H$\alpha$ optical depth (on
each track we mark the positions for $\tau_{H\alpha}=5$ and 10). The values of
the ratios expected for dust-free Case B recombination theory are also
indicated (thin black lines). When Balmer self-absorption is important,
H$\alpha$/H$\beta$ smaller than the Case B limit is predicted for a medium
with a low optical depth to Ly$\alpha$ photons ($\log{\tau_{Ly\alpha}}\sim
2.6$). This is because in the limit of low optical depth to the Ly$\alpha$
line, H$\alpha$ photons scattered in the neutral gas are effectively converted
into Ly$\beta$ photons and escape. In this (unrealistic) situation, however,
the medium would be very thin to ionizing radiation, with a $\tau_{LyC}\sim
0.04$, making the nebula substantially fainter in recombination lines.

In the more realistic scenario of a gas is optically thick to Lyman line
photons, as the H$\alpha$ optical depth increases the H$\alpha$/H$\beta$ ratio
also increases, because H$\alpha$ photons are simply scattered in the medium,
while H$\beta$ photons get converted into H$\alpha$$+$Pa$\alpha$. In this
regime, however, the geometry of the scattering medium becomes important, and
the effects of radiative transport of Balmer photons cannot be neglected. If
the scattering gas is asymmetric and preferential directions exist that offer
lower total optical depth to Balmer photons, the resulting H$\alpha$/H$\beta$
ratio depends on the specific line of sight into the nebula and low values of
the H$\alpha$/H$\beta$ ratio can be allowed. This would be the case if, for
example, the gas were organized in a flattened geometry, oriented edge–on with
respect to the line of sight. In this case, H$\alpha$ photons would
preferentially escape along the direction perpendicular to the disk (i.e.,
perpendicular to the line of sight), thus effectively reducing the H$\alpha$
output toward the observer and decreasing the H$\alpha$/H$\beta$ ratio.
Detailed radiative transfer calculations are being performed to quantify this
effect and will be presented in a forthcoming paper.

##  4 Discussion

Section 3 presents evidence that the canonical Case B limit interpretation of
H ii region emission line spectra fails for SXDF308. We suggest that Balmer
self-absorption and scattering of Balmer line photons may be important
mechanisms to regulate the intensity of the Balmer emission lines. For
SXDF308, the low Balmer H$\alpha$/H$\beta$ ratio can be explained if the
geometry of the medium responsible for scattering the Balmer photons is not
spherically symmetric, but rather is similar to a disk oriented edge-on with
respect to the line of sight. In this configuration, H$\alpha$ photons would
preferentially be scattered off the line of sight, explaining the observed
values. Figure 5 shows a schematic diagram of the proposed scenario. Galaxies
with low H$\alpha$/H$\beta$ Balmer line ratios are the result of preferential
scattering of H$\alpha$ photons out of the line of sight, in galaxies where
the scattering gas is characterized by an asymmetric geometry (panel b in
Figure 5). Galaxies with symmetric gas distribution would show both
H$\alpha$/H$\beta$ and H$\gamma$/H$\beta$ ratios larger than Case B, as a
result of Balmer self-absorption (panel a).

Figure 5: Schematic diagram of the scenario proposed in this paper.
H$\alpha$/H$\beta$ Balmer line ratios smaller than predicted for Case B
recombination theory are the result of preferential scattering of H$\alpha$
photons out of the line of sight, in galaxies where the scattering gas shows
an asymmetric geometry (panel b). In galaxies with a symmetric gas
distribution both the H$\alpha$/H$\beta$ and H$\gamma$/H$\beta$ ratios would
be larger than Case B, indicating the effect of Balmer self-absorption.

If our interpretation is correct we would expect the effects of Balmer self-
absorption to be at play in other galaxies. However, given only the H$\alpha$,
H$\beta$, H$\gamma$ observations as we have here, these galaxies would only be
identifiable under certain conditions: that they have a polar asymmetry in the
column densities, are oriented edge-on, and have very small/negligible nebular
reddening. Conditions other than this would all result in line ratios
consistent with reddened gas, and they would mostly go unnoticed. Therefore we
need to explore whether or not evidence exists in other galaxies that Balmer
self-absorption is not negligible. In general, we would expect that radiative
transport effects would not always work in favor of H$\beta$ as in the case of
SXDF308. Consequently, we should see a population of galaxies with both the
H$\alpha$/H$\beta$ and the H$\gamma$/H$\beta$ ratios larger than Case B.
Clearly, a large value of the H$\alpha$/H$\beta$ alone cannot be uniquely
interpreted as the result of Balmer self-absorption, as dust extinction would
also go in the same direction. However, the two mechanisms have a different
effect on the H$\gamma$/H$\beta$ ratio: while dust extinction decreases the
H$\gamma$/H$\beta$, Balmer self-absorption increases it (Osterbrock, 1989).

Figure 6:  H$\alpha$/H$\beta$ as a function of the H$\gamma$/H$\beta$ ratio
for star-forming galaxies in the SDSS survey. The contours indicated the
number of galaxies per bin of H$\alpha$/H$\beta$ and H$\gamma$/H$\beta$
ratios, starting from a density of 100 galaxies. We plot individual points for
lower densities of galaxies. The dashed lines show the predicted ratios for
varying Ly$\alpha$ optical depth (labeling each track), and H$\alpha$ optical
depth (increasing along each track). Along each track, we mark the position of
$\tau_{H\alpha}=5$ and 10, with plus signs and triangles respectively.
Galaxies in the upper left quadrant of the plot can easily be explained by
dust extinction, while galaxies in other parts of the diagram cannot. The
figure shows that there is a population of galaxies with line ratios
inconsistent with extinguished Case B values, but that can be reproduced with
Balmer self-absorption and a range of optical depth to H$\alpha$ and
Ly$\alpha$ photons.

To test if such a population of galaxies exists we explore the ratios of
Balmer lines for star-forming galaxies in the SDSS survey (see Figure 6). We
use the publicly available666Available from
https://www.sdss.org/dr14/spectro/galaxy_mpajhu/ MPA-JHU Data Release 8
catalogue, developed by the Max Planck Institute for Astrophysics and the
Johns Hopkins University groups (Kauffmann et al., 2003a; Brinchmann et al.,
2004). The measurement of the nebular emission lines accounts for the galaxy
stellar continuum and stellar absorption at the Balmer lines as explained in,
e.g.,Tremonti et al. (2004). To realistically account for uncertainties in the
stellar-continuum subtraction, we increased the emission line errors by
factors determined by comparisons of duplicate observations within the SDSS
sample. These factors, listed on the catalog website777http://www.mpa-
garching. mpg.de/SDSS/, amount to 1.882 and 2.473 for the H$\beta$ and
H$\alpha$ lines, respectively. Additionally, Groves et al. (2012) report that
the stellar absorption EW of H$\beta$ is systematically underestimated by
0.35Å in the MPA-JHU catalog. We apply the recommended correction to the
H$\beta$ emission line fluxes888$F_{\rm H\beta}^{\rm corrected}=F_{\rm
H\beta}^{\rm SDSS}+0.35/{\rm EW(H_{\beta})}F_{\rm H\beta}^{\rm SDSS}$.

Figure 7: Cumulative distributions of H$\alpha$/H$\beta$ (left) and
H$\gamma$/H$\beta$ (right) line ratios binned according to the O32 line ratio.
The fraction of galaxies with H$\alpha$/H$\beta$(H$\gamma$/H$\beta$) lower
(higher) than Case B recombination theory increases with the O32 ratio.

In Figure 6 we only include SDSS galaxies with emission line fluxes measured
with a signal-to-noise ratio greater than 15 in all lines, and exclude Active
Galactic Nuclei (AGN) using the spectral classification based on the Baldwin
et al. (1981, BPT) diagram, with the separation line between AGN and star-
forming galaxies of Kauffmann et al. (2003b). The black point in the lower
left of the panel shows the median uncertainty as shown by the error bars. To
avoid saturation of the density of point, we show isodensity contours where
the density of SDSS galaxies is larger than 100 galaxies per bin of
H$\alpha$/H$\beta$ and H$\gamma$/H$\beta$ (corresponding to the outermost
contour), and individual points where the density is lower.

The arrow in Figure 6 shows the direction in which a point would move for an
extinction of one magnitude in the $V$ band, for a Calzetti et al. (2000)
extinction law and the assumption of a uniform screen geometry in front of a
point source. Changing the dust distribution to a clumpy screen $+$ extended
source would still not help explaining galaxies with H$\alpha$/H$\beta$ and
H$\gamma$/H$\beta$ smaller and larger, respectively, than those predicted for
Case B recombination theory (e.g., Scarlata et al., 2009). Clearly, there
exist a population of galaxies on the right of the Case B line at
H$\gamma$/H$\beta$$=0.466$ with line ratios that cannot be explained with dust
extinction. Instead, these line ratios are consistent with those predicted by
calculations including Balmer self-absorption. The scenario that emerges from
these observations is that the effects of self-absorption of the Balmer lines
in galaxies may not be negligible as commonly assumed.

The consequences of this conclusion are important:

First, analysis of emission line spectra that proceeds from the assumption
that Case B recombination limit holds will be erroneous. Dust extinction
correction to galaxy spectra are usually computed using only the Balmer
H$\alpha$/H$\beta$ ratio, under the assumption that a value greater than Case
B is uniquely due to dust. This assumption will cause an over-correction of
the observed spectra, and thus an overestimate of quantities such as the
intrinsic star-formation rate, galaxy ionizing power, etc. Adding the
intensities of Balmer lines originating from $n>3$, as demonstrated in Figure
6, can help in identifying those objects where absorption of Balmer line
photons is important.

Second, if, in addition to Balmer self-absorption, dust is also present in a
galaxy, then the line ratios will move to the region of the diagram allowed by
Case B $+$ dust, and these galaxies could only be identified with the addition
of other lines, such as, e.g., those of the Paschen series. Emission lines of
helium could also be used to identify departure from Case B due to radiative
transfer effects in hydrogen. At the signal-to-noise of our spectrum, the He i
lines are consistent with no absorption by dust. It is also worth mentioning
that radiative transfer of Balmer photons will also alter other ratios
involving Balmer lines, like the R23 Pagel’s index (Pagel et al., 1992), and
the indices used in the BPT diagrams. For large H$\alpha$ optical depths,
galaxies will move upward on the BPT, toward high [O iii]/H$\beta$ ratios.

Third, the possibility that in some galaxies hydrogen atoms are in the excited
state implies that it is possible to ionize them with photons more energetic
than the Balmer ionization energy limit of 3.4eV (including Ly$\alpha$
photons), rather than Lyman continuum photons. This possibility, in turn,
implies that the luminosity of the Balmer lines is not a direct tracer of
Lyman continuum photons, as assumed in the Case B limit. More in general, all
calculations relying on the intrinsic intensity of the Balmer lines will be
affected, including, e.g., element abundances.

If Balmer self-absorption is the correct explanation for the anomalous Balmer
line ratios, there has to be a population of hydrogen atoms in the excited
$n=2$ state. Purcell (1952) and Pottasch (1960) suggest that trapping of
Ly$\alpha$ photons can be responsible of producing a sizable population of
excited H i. Although Ly$\alpha$ photons can be easily destroyed by dust
(e.g., Ershov, 1987; Dennison et al., 2005; Hayes, 2015; Dijkstra et al.,
2016), SXDF308 in particular, and extreme emission line galaxies in general
(Henry et al., 2015), have a modest dust content, implying a low destruction
rate of Ly$\alpha$ photons. Detailed calculations are beyond the scope of this
paper, however, an order of magnitude analysis shows that the proposed
mechanism is indeed possible. In order to decrease the H$\alpha$/H$\beta$
ratio by scattering $\sim$10% of H$\alpha$ photons off the line of sight, we
need $\tau_{H\alpha}\sim 0.1$. According to Ferland & Netzer (1979)
$\tau_{H\alpha}\sim 4\times 10^{-7}\langle N_{Ly\alpha}\rangle
Q(H)_{52}R^{-2}$, where $\langle N_{Ly\alpha}\rangle$ is the average number of
Ly$\alpha$ scattering, $Q(H)_{52}$ is the number of hydrogen ionizing photons
per second in units of $10^{52}$ s-1 and $R$ is the distance of the scattering
medium to the source in pc. The observed H$\alpha$ luminosity of $\approx
10^{41}$ erg s-1 translates into a $Q(H)_{52}\gtrsim 10$. Assuming a size of
$\sim 10$pc, $\tau_{H\alpha}\sim 0.1$ implies $\langle N_{Ly\alpha}\rangle\sim
10^{6}$, ensuring that Ly$\alpha$ pumping of the excited state is effective.

Extreme emission line galaxies such as SXDF308 and the Green Peas are the only
galaxy population to consistently show Ly$\alpha$ in emission (i.e., more than
90%, Henry et al., 2015; Yang et al., 2017a). They are therefore an obvious
population in which Ly$\alpha$ pumping may be at play. To further examine this
idea we explore the dependency of the Balmer ratios on the $O_{32}$ line
ratio: the most extreme values of O32, similar to SXDF308, are a common
characteristic of extreme emission line galaxies. In Figure 7 we show the
distributions of H$\alpha$/H$\beta$ (left) and H$\gamma$/H$\beta$ (right) line
ratios binned according to O32. These panels show a clear trend with a larger
fraction of galaxies having lower(higher)
H$\alpha$/H$\beta$(H$\gamma$/H$\beta$) value than the Case B.

It is important to exclude the possibility that the trends observed in Figure
7 are a consequence of systematic biases introduced by the SDSS data reduction
pipeline. Such effects can be important as demonstrated in, e.g., Lan et al.
(2018) and Rodríguez (2020). Specifically, Lan et al. (2018) shows that the
flux calibration of the SDSS spectra can result in three main spurious
spectral features: $i)$ wiggles in the $4500<\lambda<7000$Å wavelength range
(due to mismatches between stellar templates and observations) that can
introduce additional random uncertainty to the flux calibration at the level
of at most 1%; $ii)$ a strong absorption at the wavelength of Calcium H&K
($\lambda\lambda$3934,3969) and Sodium D ($\lambda\lambda$5891,5897), due to
absorption by the Milky Way interstellar medium; and finally $iii)$ a strong
absorption line at the wavelength of H$\alpha$ due to a problem in the masking
of the calibration spectra around this wavelength. The first effect is not an
issue because we consider galaxies at a range of redshifts. This effect
therefore introduces random rather than systematic uncertainties in the flux
calibration (i.e., for some galaxies a line will fall on a minimum of the
wiggle, and for others it will fall on a maximum). The latter two points,
however, could be an issue if the sample of SDSS galaxies we considered has
any of the lines of interests (H$\alpha$, H$\beta$, H$\gamma$, [O iii], and [O
ii]) falling close to some of the absorption features. We therefore computed
Figure 7 excluding all objects at those redshifts where any of the lines could
be compromised.

The trends presented in Figure 7 strongly suggests that there is a link
between the galaxies ionization state and the observed Balmer ratios, possibly
indicating that a significant population of galaxies with strong Balmer self
absorption could exist and hide among the ostensibly dust–reddened population.

##  5 Conclusions

We presented the emission line spectrum of SXDF308, an extreme emission line
galaxy showing a H$\alpha$/H$\beta$ Balmer line ratio significantly lower that
the value predicted for dust free Case B recombination theory. We ruled out
the possibility that this ratio is due to systematic errors introduced by
observational effects (e.g., wavelength dependent seeing, atmospheric
dispersion). We discussed various mechanisms that can alter the intrinsic
ratios of the Balmer emission lines, and exclude both Case A and Case C as
explanation of the observed spectrum. The mechanisms that best reproduces the
observations of SXDF308 invokes resonant scattering of H$\alpha$ photons, in a
non spherically-symmetric geometry that would preferentially scatter H$\alpha$
photons away from the line of sight. Basic calculations show that some extreme
starburst galaxies produce sufficient Ly$\alpha$ to effectively populate the
n=2 level of hydrogen.

Using data from the Sloan Digital Sky Survey, we show that Balmer self-
absorption may be relevant more generally than just in this galaxy, with
important implications for the interpretation of galaxy spectra. If the Case B
limit is not valid, galaxies’ physical properties computed from the intensity
of the Balmer lines will be incorrect, if Case B is assumed. These properties
include, but are not limited to, the dust extinction, the intrinsic ionizing
power, and the intrinsic star-formation rate.

SXDF308, together with the other examples among extreme emission line galaxies
stand out and are identified as problematic because of their "unphysical" low
Balmer H$\alpha$/H$\beta$ ratio, which in our explanation, is likely due to
the particular geometry of the scattering gas. In general, though, galaxies
for which radiative transfer effects of Balmer photons are important cannot be
identified on the basis of this ratio alone, as, in the more general
predictions, their values overlaps with the range expected for Case B plus
extinction by dust. The addition of higher order Balmer lines and Paschen
lines can help identifying these cases. More detailed modeling and additional
observations are required to understand the impact of Balmer self-absorption
to the interpretation of galaxies’ spectra.

## Acknowledgements

CS and VM acknowledge the support from Jet Propulsion Laboratory under the
grant award #RSA-1516084. VM also acknowledges support from the University of
Minnesota Doctoral Dissertation Fellowship 2016-17. M.H. acknowledges the
support of the Swedish Research Council, Vetenskapsrådet and the Swedish
National Space Board (SNSB), and is fellow of the Knut and Alice Wallenberg
Foundation. This work made use of the following software packages: ipython
(Pérez & Granger, 2007), numpy (van der Walt et al., 2011), matplotlib
(Hunter, 2007), astropy (and A. M. Price-Whelan et al., 2018).

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##  Appendix A Assessing systematic uncertainties

A number of factors can limit the accuracy to which line ratios can be
measured. In this Appendix we assess the impact of the most important
contributors to the flux uncertainty.

###  A.1 Atmospheric absorption

Atmospheric absorption bands become increasingly important for wavelengths
above 6500Å, and contribute to add substantial uncertainty to the flux of
narrow emission lines Stevenson (1994); Fabricant et al. (2008). These bands
are formed by dozens of intrinsically narrow absorption lines that get blurred
together in low resolution spectroscopy, masking the exact nature of the
absorption. Stevenson (1994) caution that when the intrinsic profile of an
object’s spectroscopic feature has a width comparable or smaller than the
width of the telluric features ($\sigma<40$ km s-1), the standard correction
procedure to divide by an observed telluric reference spectrum may fail,
depending on the exact relative position of the feature of interest and the
narrow atmospheric absorption lines.

We checked for this problem by comparing the wavelengths of the SXDF308
H$\alpha$ line, after all appropriate velocity corrections, with the
wavelengths of nearby strong telluric absorption lines. The results of this
test are presented in Figure 8, from where we can see that at the redshift of
SXDF308, the H$\alpha$ line falls in a region clear of strong telluric
absorption. In the inset we compare the atmospheric telluric correction
computed for the observed spectral resolution, and what we would obtain
assuming a galaxy’s Gaussian intrinsic H$\alpha$ profile with a velocity
dispersion of $\sigma=35$ km s-1. We find that at the wavelength of the
H$\alpha$ emission line, the difference is negligible (below 0.2%). Given the
integrated nature of the observations, we do not expect the intrinsic line
profile to be narrower than $\sigma=35$ km s-1 (e.g., Amorín et al., 2012).

Figure 8: The H$\alpha$ line falls in a spectral region clean of strong
telluric absorption features. The atmospheric transmission is shown at the
intrinsic resolution of the (Noll et al., 2012) library (0.1 Å) and at the
resolution of the observations (6.2 Å, or 265 km s-1). The inset zooms in
around the position of the H$\alpha$ emission line, to show the difference
between the correction calculated for the observed spectral resolution and
assuming an intrinsic Gaussian emission line profile with $\sigma=35$ km s-1.
This difference is insignificant at the position of H$\alpha$. Given the
integrated nature of the observations, we do not expect the intrinsic profile
to be narrower than $\sigma=35$km s-1 (e.g., Amorín et al., 2012).

###  A.2 Flux calibration

The accuracy with which line ratios can be measured in high signal-to-noise
spectra is limited by the uncertainty introduced by the flux calibration step
(e.g., Skillman et al., 2013). When only one exposure of one standard star is
available, as in most cases, the flux calibration step introduces a
conservative $\approx 2$% systematic uncertainty to the flux measurement, due
to the inherent uncertainty in the spectra of the standard stars (Oke, 1990;
Berg et al., 2015).

Our observing strategy included observations of three standard stars observed
multiple times over the course of three nights. With these observations we
were able to constrain the relative uncertainty in the H$\alpha$/H$\beta$ line
ratio directly. As explained in Section 2 the sensitivity function is derived
as the average of the sensitivity functions obtained from all the standard
stars. In Figure 9 we show the fractional variation between the individual
sensitivity functions, normalized at the wavelength of the observed H$\beta$
emission line. Curves of the same colors refer to observations of the same
stars. The standard deviation at the wavelength of the observed H$\alpha$ line
is a direct measurement of the uncertainty in the H$\alpha$/H$\beta$ line
ratio due to the flux calibration step.

Figure 9: Fractional variation among the sensitivity functions obtained from
individual standard stars observed multiple times over the course of the
observing run. Curves of the same color show sensitivity functions for the
same star. The curves are normalized to the wavelength of the H$\beta$
emission line, so that the standard deviation at the wavelength of the
observed H$\alpha$ emission line is a direct measure of the uncertainty in the
H$\alpha$/H$\beta$ line ratio due to the flux calibration step.

Table 1: MW extinction-corrected emission-line fluxes  Line | I($\lambda$)/I(H$\beta$)
---|---
[S ii]$\lambda$6731 | 0.030 $\pm$ 0.014
[S ii]$\lambda$6716 | 0.041 $\pm$ 0.015
H$\alpha$$\lambda$6563 | 2.620 $\pm$ 0.078
[O iii]$\lambda$5007 | 6.763 $\pm$ 0.141
[O iii]$\lambda$4959 | 2.261 $\pm$ 0.051
[Ar iv]$\lambda$4740 | 0.024 $\pm$ 0.008
[Ar iv]+He i$\lambda$4712 | 0.022 $\pm$ 0.009
[O iii]$\lambda$4363 | 0.136 $\pm$ 0.014
H$\gamma$$\lambda$4340 | 0.468 $\pm$ 0.021
H$\delta$$\lambda$4102 | 0.245 $\pm$ 0.015
[O ii]$\lambda$3727 | 0.573 $\pm$ 0.034
Table 2: Physical properties for the nebular gas in SXDF308.  Parameter | Value
---|---
$T_{e}(O^{2+})$ | 15370 $\pm$ 850 K
$T_{e}(O^{+})$ (inferred) | 13940 $\pm$ 425 K
$T_{e}(S^{2+})$ (inferred) | 14100 $\pm$ 590 K
$n_{e}(S^{+})$ | 10${}^{+200}_{-10}$ cm-3