# Rest-frame UV and optical emission line diagnostics of ionised gas
properties: a test case in a star-forming knot of a lensed galaxy at $z\sim
1.7$

Ayan Acharyya1,2, Lisa J. Kewley1,2, Jane R. Rigby3, Matthew Bayliss4, Fuyan
Bian1,5, David Nicholls1, Christoph Federrath1,2, Melanie Kaasinen6, Michael
Florian3, and Guillermo A. Blanc7,8
1Research School of Astronomy and Astrophysics, Australian National
University, Canberra, ACT 2611, Australia
2ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D)
3Astrophysics Science Division, Goddard Space Flight Center, 8800 Greenbelt
Rd., Greenbelt, MD 20771, USA
4MIT Kavli Institute for Astrophysics and Space Research, 77 Massachusetts
Ave., Cambridge, MA 02139, USA
5European Southern Observatory, Alonso de Córdova 3107, Casilla 19001,
Vitacura, Santiago 19, Chile
6Max Planck Institute für Astronomie, Königstuhl 17, 69117, Heidelberg,
Germanyy
7Observatories of the Carnegie Institution for Science, 813 Santa Barbara
Street, Pasadena, CA 91101, USA
8Departamento de Astronomía, Universidad de Chile, Camino del Observatorio
1515, Las Condes, Santiago, Chile E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We examine the diagnostic power of rest-frame ultraviolet (UV) nebular
emission lines, and compare them to more commonly used rest-frame optical
emission lines, using the test case of a single star-forming knot of the
bright lensed galaxy RCSGA 032727-132609 at redshift $z\sim 1.7$. This galaxy
has complete coverage of all the major rest-frame UV and optical emission
lines from Magellan/MagE and Keck/NIRSPEC. Using the full suite of diagnostic
lines, we infer the physical properties: nebular electron temperature (Te),
electron density (ne), oxygen abundance ($\log{(\mathrm{O/H})}$), ionisation
parameter ($\log{(q)}$) and interstellar medium (ISM) pressure
($\log{(P/k)}$). We examine the effectiveness of the different UV, optical and
joint UV-optical spectra in constraining the physical conditions. Using UV
lines alone we can reliably estimate $\log{(q)}$, but the same is difficult
for $\log{(\mathrm{O/H})}$. UV lines yield a higher ($\sim$1.5 dex)
$\log{(P/k)}$ than the optical lines, as the former probes a further inner
nebular region than the latter. For this comparison, we extend the existing
Bayesian inference code IZI, adding to it the capability to infer ISM pressure
simultaneously with metallicity and ionisation parameter. This work
anticipates future rest-frame UV spectral datasets from the James Webb Space
Telescope (JWST) at high redshift and from the Extremely Large Telescope (ELT)
at moderate redshift.

###### keywords:

ISM: evolution, galaxies: abundances, galaxies: emission lines, ultraviolet:
ISM

††pubyear: 2017††pagerange: Rest-frame UV and optical emission line
diagnostics of ionised gas properties: a test case in a star-forming knot of a
lensed galaxy at $z\sim 1.7$–A

##  1 Introduction

Understanding the evolution of inter-stellar medium (ISM) properties requires
measuring galaxies at both lower redshifts ($z\leqslant 1$) where the star
formation has been quenched due to feedback from several physical processes
(e.g. Cicone et al., 2014; Bluck et al., 2014; Schwamb et al., 2016; Leslie et
al., 2016) including turbulence (e.g. Federrath & Klessen, 2012; Federrath et
al., 2017; Zhou et al., 2017), and at higher redshifts ($z\sim 2$) where the
star-formation rate (SFR) is at its peak (e.g. Uzgil et al., 2016; Kewley et
al., 2016; Kulas et al., 2010).

The star formation history (e.g. Schulte-Ladbeck et al., 2003), ISM conditions
(e.g. Tremonti et al., 2004; Lamareille et al., 2004; Gallazzi et al., 2005;
Contursi et al., 2017) and kinematics (e.g. Cicone et al., 2016) of low-
redshift galaxies have been studied extensively thanks to large, targeted
surveys (e.g. Ho et al., 2016; Medling et al., 2018; Ellison et al., 2018;
Belfiore et al., 2017). Not only are low-$z$ galaxies brighter then the
distant ones, but in many cases we have spatially resolved properties. In
contrast, the physical conditions of high-$z$ ($z$>1) galaxies are still
poorly understood, because these galaxies are fainter and harder to observe.
Spatially resolved studies of high-$z$ galaxies are difficult with current
telescopes and the resolution is poorer than for the local samples (e.g.
Förster Schreiber et al., 2009; Wisnioski et al., 2015). The advent of the
next generation of large ground-based (Giant Magellan Telescope, GMT (Johns,
2004); Extremely Large Telescope, ELT (Sanders, 2013); Thirty Metre Tescope,
TMT (Hook, 2009)) and space-based telescopes (James Webb Space Telescope, JWST
(Gardner et al., 2006)) will facilitate spectroscopic observations of galaxies
with unprecedented spatial resolution ($\sim 0.1^{\prime\prime}$), out to very
high redshifts ($z\sim 6-10$) and detection of the first generation of
galaxies ($z\sim 15$).

JWST will be able to observe the rest-frame optical emission lines in a
multiplexed way with its NIRSpec instrument for galaxies at $z\leqslant 6.4$.
Moreover, for $z\geqslant 3$ JWST can additionally capture diagnostic rest-
frame UV lines. At moderate redshifts ($z\geqslant 3$) JWST /NIRSpec will be
able to capture both rest-frame optical and rest-frame UV diagnostics. At
higher redshifts ($z\geqslant 6.4$), the rest-frame optical lines redshift out
of the NIRSpec bandpass and are no longer accessible for multiplexed
spectroscopy, though they can be captured singly by the MIRI (Mid-Infrared
Instrument Instrument). For the highest redshift (7 < $z$ < 10) galaxies, all
JWST may spectroscopically detect are the rest-frame UV lines. This motivates
the development of a suite of UV emission line diagnostics, as well as a full
UV$+$optical suite of diagnostics to be used at moderate redshifts.

Even prior to the onset of the next generation telescopes, we can preview the
measurements that will be possible with such facilities by studying galaxies
that are strongly magnified through lensing. Gravitational lensing has been
used by astronomers to quantify the ISM conditions in galaxies at $z\geqslant$
1.5 (e.g. Yuan & Kewley, 2009; Bian et al., 2010; Jones et al., 2013; Yuan et
al., 2013; Cañameras et al., 2015; Leethochawalit et al., 2016; James et al.,
2018; Rigby, 2009; Rigby et al., 2011; Bayliss et al., 2014; Jones et al.,
2015; Wang et al., 2017). In addition to magnifying background galaxies,
gravitational lensing stretches the images of the galaxies into extended arcs,
making it possible to spatially resolve individual star-forming knots (Figure
1 of Bayliss et al., 2014; Cañameras et al., 2015; Swinbank et al., 2015;
Sharda et al., 2018).

Diagnostics using rest-frame optical nebular lines have been extensively
applied to H ii regions and galaxies (e.g. Kewley & Dopita, 2002; Pagel et
al., 1979; McGaugh, 1991; Zaritsky et al., 1994; Kewley et al., 2001;
Kobulnicky & Kewley, 2004; Pettini & Pagel, 2004; Sanders et al., 2016a,
henceforth KD02, KK04, PP04). UV diagnostics appear promising (e.g. Jaskot &
Ravindranath, 2016; Feltre et al., 2016; Stark et al., 2014; Byler et al.,
2018; Izotov et al., 2006b; Garnett et al., 1995a; Garnett et al., 1995b,
henceforth I06, G95a and G95b respectively) but have not yet been extensively
tested. Moreover, the atomic data and stellar atmospheric libraries used for
the calibrations of the earlier works have now been updated. These updates
affects the suite of UV and optical diagnostics developed to date.

Recent studies have presented improvements to the rest-frame UV and optical
diagnostics. Kewley et al. (2019a, hereafter K19a) present a set of rest-frame
optical and UV line diagnostics for electron density and ISM pressure using
the updated version of the MAPPINGS photoionzation models (Sutherland et al.,
2013) (explained in section 5.2). Using these models, Kewley et al. (2019b,
hereafter K19b) further present a set of optical and UV diagnostics for
ionisation parameter and metallicity. Nicholls et al. (in prep; N18 hereafter)
propose new methods for determining the nebular electron temperature using
rest-frame UV oxygen emission lines combined with improved theoretical models.
Dopita et al. (2016, hereafter D16) present a robust technique to isolate the
dependence of oxygen abundance on a set of nebular optical lines (independent
of ionisation parameter and ISM pressure) applicable for high-$z$ ($z\sim 2$)
galaxies. These new diagnostics, together with the existing diagnostics,
constitute a comprehensive suite that can be employed to determine ISM
properties of high-$z$ galaxies. However, the new diagnostics have not yet
been tested using observations of galaxies.

In addition to strong emission line diagnostics (SEL), Bayesian techniques are
also becoming increasingly important in inferring ionised gas properties due
to their ability to probe asymmetry and non-trivial topography in the
probability distributions of the properties. Recent Bayesian estimation tools
like IZI (Blanc et al., 2015), BOND (Vale Asari et al., 2016), HII-CHI-mistry
(Pérez-Montero, 2015) and NebulaBayes (Thomas et al., 2018) have proven useful
in inferring nebular gas properties. However, in light of the recent
development of SEL diagnostics, particularly the rest-frame UV diagnostics, it
is necessary to test the agreement between the Bayesian and SEL techniques.
Hence, in this work we compare the SEL diagnostics with a new, extended
version of IZI, as described later in Section 6.

The purpose of this paper is to test these new diagnostics by applying them to
a star-forming knot of a $z$$\sim$2 galaxy with full coverage of these
diagnostic lines. Herein we exploit the advantage that spatially resolved
spectroscopy has over other high-z samples. Choosing a single star-forming
knot of the lensed galaxy RCS0327 for the testing allows us to probe a small
($\sim$100 pc) spatial region, which can be expected to have fairly
homogeneous ISM properties with high fraction of H ii region relative to
diffuse gas.

This paper is organized as follows. In Section 2 we justify the selection of a
single star-forming knot of RCS0327 as the test case and describe the
selection of the local galaxies used for comparison. Section 3 describes the
observation and data reduction for both the Keck/NIRSPEC and Magellan/MagE
data. We explain the tools used for our analysis, including line fitting
algorithms, various diagnostics used and the results obtained from them in
Section 5. The comparison among various results are discussed in Section 7
followed by a summary of our work in Section 8.

We use a solar oxygen abundance 12+log(O/H) = 8.72 (Asplund et al., 2009)
throughout the paper. We assume a standard flat $\Lambda$ cold dark matter
cosmology with $H_{0}$ = 70 km s-1 Mpc-1 and matter density $\Omega_{M}$ =
0.27. For the emission lines, we adopt the sign convention of negative
equivalent width and use the wavelengths from the NIST database, the Leitherer
et al. (2011) atlas, and MAPPINGS v5.

##  2 Sample selection

The primary limitation in the development of rest-frame UV spectral
diagnostics has been the lack of high-quality spectra. Project MEGaSaURA
(Rigby et al., 2018) has obtained high signal to noise, moderate spectral
resolution (R $\sim$ 3000) spectra for 15 bright gravitationally lensed
galaxies. From that sample, we select the spectrum of knot-E of RCS0327
(henceforth referred to as RCS0327-E) for our pilot study, for the following
reasons.

  * •

RCSGA 032727-132609 (henceforth RCS0327) is a very bright lensed galaxy with
an r-band magnitude of 19.1 (Wuyts et al., 2010; Sharon et al., 2012).

  * •

The fact that the lensed galaxy appears as a very extended (38") arc makes it
possible to resolve and target individual star-forming knots. Spectra of four
knots have been published; we select knot-E (see Sharon et al., 2012, Figure
1) for our analysis in this paper because it has the highest signal-to-noise
ratio (SNR).

  * •

The spectrum of RCS0327 has a SNR per resolution element $\sim$ 20 at
$\lambda_{\mathrm{obs}}$ = 5000 Å, sufficient to clearly detect the rest-frame
UV emission lines.

  * •

Rest-frame optical spectra for this object have been obtained from
Keck/NIRSPEC (Rigby et al., 2011, henceforth R11), Keck/OSIRIS (Wuyts et al.,
2014b) and HST/WFC3 G141 grism (Whitaker et al., 2014, hereafter W14). Wuyts
et al. (2014b) also report optical fluxes for knots B, U from Magellan/FIRE
observations and that for knot U from re-extracted Keck/NIRSPEC observations.
W14 report grism fluxes from several other star-forming knots too. However,
the largest number of optical$+$UV emission lines are detected in Knot E,
which allow us to compare the extensively used optical diagnostics to the new
UV diagnostics.

RCS0327 was first discovered (Wuyts et al., 2010) in a dedicated search for
highly magnified giant arcs (Bayliss, 2012) in the Red Sequence Cluster Survey
2 (RCS2; Gilbank et al., 2011). Sharon et al. (2012) performed a source plane
reconstruction of RCS0327, based on HST/WFC3 imaging data, down to a scale of
$\sim 100$ pc. Due to its apparent brightness, this galaxy has been subjected
to extensive subsequent spectroscopic analyses. R11 constrained the nebular
physical properties of RCS0327 using the spatially summed Keck/NIRSPEC (rest-
frame optical) spectra. Wuyts et al. (2014a) studied the stellar populations
and concluded that RCS0327 is a starburst galaxy (SFR $30-50$ M⊙yr-1; Wuyts et
al., 2012a) with a young ($5-100$ Myr; Wuyts et al., 2014a) stellar population
that has just started to build up its stellar mass ($10^{7}-5\times 10^{8}$ M⊙
; Wuyts et al., 2014a). W14 analyzed spatial variations in HST/WFC3 grism
spectra of RCS0327 and reported no appreciable knot-to-knot variation in
reddening, and an enhanced star formation rate ($\sim 2$ dex above the star-
formation main sequence) due to an ongoing interaction. They also found
spectroscopic evidence of the presence of O stars in most knots (except knots
E and F which have lower He i/H$\beta$ ratios), which is consistent with the
young stellar population scenario. Bordoloi et al. (2016) examined the
galactic outflows for this galaxy as traced by the Mg ii and Fe ii emission
and its spatial variation, finding large outflow velocities ($\sim$ 170-250 km
s-1) and mass outflow rates ($\gtrsim$ 30-50 M⊙yr-1). However, Rigby et al.
(2014) report a lack of correlation between the Mg ii and Ly-$\alpha$
emissions, which implies that the source of Mg ii emission is not nebular, but
may instead be resonantly scattered continuum. Overall, the picture that has
emerged is that RCS0327 hosts a young stellar population that is driving a
large-scale outflow.

In this work, we analyze the spectra of one particular knot of star-formation
(knot E) within RCS0327. A unique feature of this analysis is that the spectra
cover a $\sim$100 pc (Sharon et al., 2012) star-forming region (classified
using the Baldwin, Phillips & Terlevich (BPT) diagram and KD02 line). Our work
has the distinct advantage in that the physical conditions can be expected to
be fairly homogeneous within the small ($\sim$100 pc) spatial region that the
spectra probe, rather than being averaged across several kiloparsecs, which
ensures fair comparison to photoionisation models. Moreover, the relative
contribution from H ii regions with respect to diffuse ionised gas is expected
to be very high because we integrate over a small star-forming knot, rather
than the whole galaxy.

We compare the rest-frame UV and rest-frame optical emission line fluxes to
newly-developed diagnostics (K19a,b, N18), to constrain the ionisation
parameter ($\log{(q)}$), electron density (ne), ISM pressure ($\log{(P/k)}$),
electron temperature (Te) and oxygen abundance (12+log(O/H)). We also consider
how the results would differ if we had only the UV dataset, or only the
optical dataset, using a Bayesian approach (detailed in Section 6). Moreover,
given that the [O ii]$\lambda\lambda$3727,9 doublet will be within the
wavelength coverage of JWST up to $z\sim$ 12, we also investigate the effect
of including the [O ii]$\lambda\lambda$3727,9 lines with the set of rest-frame
UV emission lines.

##  3 Observations

We use the rest-frame optical spectra from the NIRSPEC instrument on Keck and
rest-frame ultraviolet spectra from the MagE instrument on Magellan.

###  3.1 Rest-frame optical spectroscopy from NIRSPEC on Keck

Near-infrared spectra of RCS0327-E, covering the rest-frame optical, were
obtained on UT 2010 Feb. 4 with the NIRSPEC spectrograph (McLean et al., 1998)
on the Keck II telescope. The spectra were originally published in R11, along
with a detailed description of the observation and data reduction procedures.
Since only a basic lensing model was available at that time, the spectrum in
the long-slit was summed across the spatial direction. Subsequently, high-
resolution images with the Hubble Space Telescope (HST) revealed that the
NIRSPEC observations had captured multiple physically distinct knots of star
formation (see Sharon et al., 2012, Figure 6), the brightest of which they
labeled Knot E and Knot U. Guided by the HST–enabled lensing model, Wuyts et
al. (2014b) re-extracted the spectra, producing spectra for these two physical
regions. Subsequently, W14 improved the measurement of the reddening for
RCS0327 . Therefore, we take the NIRSPEC line fluxes reported for Knot E by
Wuyts et al. (2014b), and apply the reddening of E(B-V)${}_{\rm gas}=0.40\pm
0.07$ measured by W14, to compute updated cross-filter scaling factors, as we
describe in more detail in Section 4.1.

###  3.2 Rest-frame UV spectroscopy from MagE on Magellan

Optical spectra of RCS0327-E (see Sharon et al., 2012, Figure 1), covering the
rest-frame ultraviolet, were obtained on multiple nights in the range UT
2008-07-31 to UT 2010-12-10 with the MagE instrument (Marshall et al., 2008)
on the Magellan Clay telescope. The observations and data reduction are
described in Rigby et al. (2018). The data were reduced using the MagE
pipeline, which is part of the Carnegie Python
Distribution111http://code.obs.carnegiescience.edu. The full spectra from each
observation were obtained by combining the weighted average of different
spectral orders. Observations from different nights were then combined via a
weighted average to obtain the resulting rest-frame UV spectra of RCS0327-E
used in this paper. The spectra were flux-calibrated by comparing to
spectrophotometric standard stars, as described in Rigby et al. (2018). The
spectra were corrected for Milky Way reddening. The effective spectral
resolution of the final combined spectrum, measured from the widths of night
sky lines, is $R=3650\pm 120$ (median and absolute median deviation).

##  4 Flux measurements

Table 1: Keck/NIRSPEC line flux measurements. These measurements include updated (relative to Wuyts et al. (2014b)) tweak factors to take into account the cross-filter flux calibration. fluxobs and $\delta$ fluxobs denote the observed flux and uncertainty respectively. fluxdereddened is the dereddened flux using E(B-V) = 0.4 $\pm$ 0.07. All fluxes and uncertainties are in units of 10-17 ergs/s/cm2 Line ID |  $\lambda_{\mathrm{rest}}$ (Å) | fluxobs |  $\delta$ fluxobs | fluxdereddened
---|---|---|---|---
Filter N1
[O II] 3727,9 | 3728.483 | 165.3 | 7.3 | 958.36
[O II] 3727 | 3727.092 | 87.8 | 3 | 509.23
[O II] 3729 | 3729.875 | 76.6 | 4 | 443.94
[Ne III] 3869 | 3869.860 | 25.4 | 2.6 | 141.33
Filter N2
H$\zeta$ | 3890.166 | 17.8 | 2 | 98.408
H$\delta$ | 3971.198 | 16 | 4.1 | 86.09
H$\epsilon$ | 4102.892 | 7 | 3 | 35.882
Filter N3
H$\gamma$ | 4341.692 | 49.8 | 2 | 231.64
O III 4363 | 4364.435 | 5.5 | 2 | 25.34
[Ar IV] 4741 | 4741.449 | 0 | 0 | 0
H$\beta$ | 4862.691 | 119 | 1.6 | 449.53
[O III] 4959 | 4959.895 | 190 | 1.4 | 693.78
[O III] 5007 | 5008.239 | 613 | 3 | 2202.5
H$\alpha$ | 6564.632 | 465.2 | 23.2 | 1183.4
[N II] 6584 | 6585.273 | 55 | 3.9 | 139.4
[S II] 6717 | 6718.294 | 26.3 | 2.3 | 65.114
S II 6731 | 6732.674 | 19.4 | 6.2 | 47.91
[Ar III] 7136 | 7137.770 | 10.8 | 3.1 | 24.824

###  4.1 Keck/NIRSPEC spectral line fits

For each NIRSPEC filter setting (N1, N3, and N6, roughly corresponding to
J-band, H-band, and Ks-band), all emission lines were fit simultaneously with
Gaussian profiles by Wuyts et al. (2014b, see their Sections 2.3 and 2.4).
Relative flux offsets are expected in the NIRSPEC spectra across the three
grating settings, due to slit losses associated with variable seeing. As a
result, the fluxes of the NIRSPEC-1 and NIRSPEC-6 spectra had to be adjusted
with respect to the flux calibration in NIRSPEC-3. Section 3.2 of R11
describes how the Balmer lines were used to perform this process, using the
at-the-time best measurement of reddening, $E(B-V)_{gas}=0.23\pm 0.23$. This
procedure produced offset factors of 1.15 for N1 and 0.61 for N6, both with
respect to N3.

Table 2: MagE/Magellan line flux measurements. fluxobs and $\delta$ fluxobs denote the observed flux and uncertainty respectively. fluxdereddened is the dereddened flux using E(B-V) = 0.4 $\pm$ 0.07. Wr,fit and $\delta$ Wr,fit denote the rest-frame equivalent width measured and the corresponding uncertainty in Å respectively. For cases of non-detection (i.e. < 3 $\sigma$ detection), the 3 $\sigma$ upper limit on equivalent widths and fluxes are quoted. Uncertainty estimates for these entries are not quoted because they do not provide any meaningful information. Line ID | $\lambda_{\mathrm{rest}}$ | Wr,fit |  $\delta$ Wr,fit | Wr,signi | fluxobs |  $\delta$ fluxobs | fluxdereddened
---|---|---|---|---|---|---|---
| (Å) | (Å) | (Å) | (Å) | (10-17 ergs/s/cm2) | (10-17 ergs/s/cm2) | (10-17 ergs/s/cm2)
Ly$\alpha$ | 1215.6700 | >-0.5706 | .. | .. |  $<$0.43 | .. |  $<$23.32
O I 1304 | 1304.8576 | >-0.2609 | .. | .. |  $<$0.25 | .. |  $<$5.26
O I 1306 | 1306.0286 | >-0.2589 | .. | .. |  $<$0.24 | .. |  $<$5.22
Si II 1309 | 1309.2757 | -0.6397 | 0.15 | 7.37 | 1.62 | 0.37 | 34.82
C II 1335a | 1334.5770 | >-0.1547 | .. | .. |  $<$0.15 | .. |  $<$3.16
C II 1335b | 1335.6630 | >-0.1575 | .. | .. |  $<$0.15 | .. |  $<$3.22
C II 1335c | 1335.7080 | >-0.1575 | .. | .. |  $<$0.15 | .. |  $<$3.22
N II] 1430 | 1430.4100 | >-0.1131 | .. | .. |  $<$0.10 | .. |  $<$2.21
N II] 1431 | 1430.9730 | >-0.116 | .. | .. |  $<$0.11 | .. |  $<$2.26
N IV] 1486 | 1486.5000 | -0.2242 | 0.09 | 5.92 | 0.55 | 0.23 | 11.17
Si II 1533 | 1533.4312 | -0.3645 | 0.05 | 10.83 | 0.86 | 0.12 | 16.84
He II 1640 | 1640.4170 | -0.4451 | 0.07 | 15.06 | 1.03 | 0.16 | 18.56
O III] 1660 | 1660.8090 | -0.1209 | 0.07 | 4.24 | 0.28 | 0.16 | 4.94
O III] 1666 | 1666.1500 | -0.4817 | 0.07 | 15.88 | 1.10 | 0.16 | 19.52
N III] 1750 | 1749.7000 | -0.3396 | 0.05 | 13.45 | 0.73 | 0.11 | 12.82
[Si II] 1808 | 1808.0130 | >-0.0896 | .. | .. |  $<$0.07 | .. |  $<$1.22
[Si II] 1816 | 1816.9280 | -0.3036 | 0.06 | 9.83 | 0.62 | 0.13 | 11.14
[Si III] 1882 | 1882.7070 | -0.3104 | 0.05 | 11.59 | 0.61 | 0.10 | 11.75
Si III] 1892 | 1892.0290 | -0.4239 | 0.06 | 15.22 | 0.82 | 0.12 | 16.10
[C III] 1906 | 1906.6800 | -1.5011 | 0.32 | 52.55 | 2.89 | 0.61 | 57.99
C III] 1908 | 1908.7300 | -1.1148 | 0.11 | 38.19 | 2.14 | 0.21 | 43.16
N II] 2140 | 2139.6800 | >-0.1038 | .. | .. |  $<$0.06 | .. |  $<$2.24
[O III] 2320 | 2321.6640 | >-0.1132 | .. | .. |  $<$0.06 | .. |  $<$1.39
C II] 2323 | 2324.2140 | >-0.1023 | .. | .. |  $<$0.05 | .. |  $<$1.24
C II] 2325c | 2326.1130 | -0.7885 | 0.20 | 13.72 | 1.08 | 0.27 | 25.58
C II] 2325d | 2327.6450 | -0.3731 | 0.10 | 10.39 | 0.51 | 0.13 | 12.01
C II] 2328 | 2328.8380 | -0.1435 | 0.08 | 4.10 | 0.20 | 0.11 | 4.59
Si II] 2335a | 2335.1230 | >-0.1064 | .. | .. |  $<$0.05 | .. |  $<$1.22
Si II] 2335b | 2335.3210 | >-0.1064 | .. | .. |  $<$0.05 | .. |  $<$1.22
Fe II 2365 | 2365.5520 | -0.5465 | 0.07 | 14.59 | 0.71 | 0.10 | 14.68
Fe II 2396a | 2396.1497 | >-0.1234 | .. | .. |  $<$0.06 | .. |  $<$1.07
Fe II 2396b | 2396.3559 | >-0.1279 | .. | .. |  $<$0.06 | .. |  $<$1.11
[O II] 2470 | 2471.0270 | -0.9819 | 0.07 | 25.09 | 1.20 | 0.08 | 18.01
Fe II 2599 | 2599.1465 | >-0.1064 | .. | .. |  $<$0.04 | .. |  $<$0.50
Fe II 2607 | 2607.8664 | >-0.1694 | .. | .. |  $<$0.07 | .. |  $<$0.77
Fe II 2612 | 2612.6542 | -0.732 | 0.20 | 16.06 | 0.77 | 0.21 | 8.89
Fe II 2614 | 2614.6051 | >-0.1453 | .. | .. |  $<$0.06 | .. |  $<$0.65
Fe II 2618 | 2618.3991 | >-0.1384 | .. | .. |  $<$0.05 | .. |  $<$0.61
Fe II 2621 | 2621.1912 | >-0.1351 | .. | .. |  $<$0.05 | .. |  $<$0.59
Fe II 2622 | 2622.4518 | >-0.1569 | .. | .. |  $<$0.06 | .. |  $<$0.69
Fe II 2626 | 2626.4511 | -0.9937 | 0.10 | 21.09 | 1.04 | 0.11 | 11.67
Fe II 2629 | 2629.0777 | >-0.1376 | .. | .. |  $<$0.05 | .. |  $<$0.59
Fe II 2631 | 2631.8321 | >-0.1259 | .. | .. |  $<$0.05 | .. |  $<$0.54
Fe II 2632 | 2632.1081 | >-0.1245 | .. | .. |  $<$0.05 | .. |  $<$0.53
Mg II 2797b | 2798.7550 | -1.089 | 0.07 | 23.33 | 1.08 | 0.07 | 10.00
Mg II 2797d | 2803.5310 | -0.4567 | 0.06 | 10.97 | 0.45 | 0.06 | 4.19
He I 2945 | 2945.1030 | >-0.1515 | .. | .. |  $<$0.07 | .. |  $<$0.55
Table 3: Intervening absorption lines in RCS0327-EMagellan/MagE spectrum. Wr,fit denotes the rest-frame equivalent width measured, in Å. $z$ an $\Delta z$ are the redshift and corresponding uncertainty respectively, as measured from our line fitting code. Line ID | $\lambda_{\mathrm{rest}}$ | Wr,fit |  $\delta$ Wr,fit | zz | zzu
---|---|---|---|---|---
| (Å) | (Å) | (Å) |  |
Fe II 2344 | 2344.2140 | 0.4791 | 0.0612 | 0.98285 | 0.00006
Fe II 2383 | 2382.7650 | 0.5997 | 0.0578 | 0.98298 | 0.00004
Al II 1670 | 1670.7874 | 0.0572 | 0.0609 | 1.87880 | 0.00150
Fe II 2586 | 2586.6500 | 0.3040 | 0.0456 | 0.98290 | 0.00004
Fe II 2600 | 2600.1729 | 0.8633 | 0.3090 | 0.98295 | 0.00003
Mg II 2796 | 2796.3520 | 1.0939 | 0.0671 | 0.98293 | 0.00003
Mg II 2803 | 2803.5310 | 1.2589 | 0.0637 | 0.98295 | 0.00003
Mg I 2853 | 2852.9640 | 0.2608 | 0.0519 | 0.98299 | 0.00006
Figure 1:  Gaussian fits to the MagE spectra for individual lines. The black
line is the observed spectrum, whereas grey and yellow denote the 1 $\sigma$
uncertainty spectrum and the continuum respectively. Brown solid lines
represent the Gaussian fit of individual lines. Thick dark cyan lines are the
sum of the Gaussian fits, wherever there are multiple lines. The fitting
routine works on a portion of the spectrum as shown bounded by blue dashed
lines. The fitted value of the central wavelength is denoted by a vertical
brown solid line, if the line is detected (i.e. > 3 $\sigma$) and by a black
line if not (i.e. < 3 $\sigma$). A single Gaussian is fit to a single line or
multiple Gaussians are simultaneously fit to a group of lines, depending on
the separation between neighboring lines. This is visible, for example, in the
C iii] 1906-1908 doublet fitting.  Figure 2:  Our fits of the C ii] 2323 (top)
and FeII 2600 (bottom) complex. The colour coding is as in Figure 1. There are
>3 $\sigma$ detections for only some of the lines (brown vertical lines) in
this complex. For the rest (black vertical lines) we quote upper limits in
Table 2. Blue solid lines indicate the initial guess/es of the line center/s
provided to the fitting code. For each line/group, only the portion bounded by
blue dashed lines is fit.

###  4.2 Magellan/MagE spectral line fits

We fit the continuum with an automatic routine that masks the positions of all
expected spectral features, including from known intervening absorption
systems, and then apply boxcar smoothing. The result is almost identical to
the hand-fit spline continuum described in Rigby et al. (2018). We
subsequently fit the emission lines in the combined, continuum normalized
spectrum using a Python-based, automated line fitting code, explained below.

We simultaneously fit all neighboring spectral lines with one Gaussian profile
per line employing the python tool scipy.optimize.curve_fit that implements
the non-linear least-squares method. A neighbor is defined as follows. Each
line centroid is assumed to have a window of $\pm 5$ spectral resolution
elements, both blueward and redward. If the windows of any two adjacent lines
overlap they are neighbors. A neighbor of a neighbor is considered in the same
group of lines, which are fit simultaneously. As an example, if there are 4
lines with separation between each adjacent pair of line centroids less than
$\pm 10$ resolution elements (5 resolution elements from the window of each
line), the lines are considered as a single group. This group is fit with a
quadruple Gaussian function with $4\times 3=12$ parameters, where the
parameters are height, width and centroid of the Gaussians for each line in
the group. We set the continuum value to unity222The code is capable of
fitting the continuum value, for each group, as an additional parameter if
desired. because, in this case, we use a continuum normalized spectrum.
Initially, the nebular redshift is measured by fitting the strongest emission
lines (for RCS0327-E it is the $[$C iii$]\lambda\lambda$1906,8 doublet) with
sufficiently large allowance for the fitted redshift. Subsequently, an initial
guess for each line centroid $\lambda_{in}$ is provided to curve_fit() by
redshifting the rest-frame vacuum wavelengths by the nebular redshift. The
centroids are forced to be fit within a window of $\lambda_{in}\pm 3\sigma$
wavelength interval corresponding to the uncertainties in the nebular
redshift. The line width is fit within upper and lower bounds of 300 km s-1
and one spectral resolution element respectively. The amplitude of the
Gaussians were allowed to vary freely. We measure rest-frame equivalent widths
($W_{r}$) from both the fitted Gaussian parameters and by direct summation. In
this paper, we quote the $W_{r}$ values derived using the former method.

In order to determine the significance of the $W_{r}$ measurements
($W_{\mathrm{r,signi}}$), we define a quantity $W_{\mathrm{r,Schneider}}$ as
the $W_{r}$ derived by interpolating a rolling average of the error weighted
$W_{r}$ at every point throughout the spectrum, following Schneider et al.
(1993, Section 6.2). The $W_{\mathrm{r,signi}}$ is then defined as the ratio
of the measured $W_{r}$ of a line to the $W_{\mathrm{r,Schneider}}$ computed
at the line center. The rolling average technique gives us a quantifiable
estimate of the spectral noise. We consider all lines that meet the criteria
(a) $W_{\mathrm{r,signi}}>$3, and (b) SNR > 1, to be detected333There are
cases where only criteria (a) is satisfied, especially when the code fits a
broad emission in the noisy part of the spectrum.. Features not satisfying
these criteria are considered to be non-detections, for which we quote
$3\times W_{\mathrm{r,Schneider}}$ as the $3\sigma$ upper limits of $W_{r}$s.
We repeat the same operation on measured line flux values ($f$) to derive
$f_{\mathrm{Schneider}}$, $f_{\mathrm{signi}}$ and $f_{\mathrm{uplim}}$. In
the absence of flux ($f$), we translate $f_{\mathrm{uplim}}$ to lower/upper
limits on line ratios and consequently to limits on the ISM properties.

We present the emission line fluxes and upper limits of nebular lines in the
MagE spectrum in Table 2 and show the Gaussian fits to the MagE data in
Figures 1 and 2. Some of the emission lines of interest in our spectrum are
affected by intervening absorption lines. We fit the intervening lines
simultaneously with the emission lines, to properly account for the missing
(absorbed) emission line fluxes (e.g. see bottom-left panel of Figure 1). The
intervening absorption features, along with their strengths and redshifts are
presented in Table 3. These intervening lines have been studied by Lopez et
al. (2018).

Table 4: Complete list of all UV and optical diagnostics and the line ratios involved. Name of diagnostic | Inferred quantity | Line ratio used | Reference
---|---|---|---
Te_O3a_I06 | Te | [O iii]$\lambda\lambda$4959,5007/[O iii]$\lambda$4363 | Izotov et al. (2006b)
Te_O3a_N18 | " | [O iii]$\lambda\lambda$5007/[O iii]$\lambda$4363 | Nicholls et al. (in prep)
Te_O3b | " | [O iii]$\lambda$5007/O iii]$\lambda\lambda$ 1660,6 | "
Te_O3a_O32 | 12+log(O/H) | Te_O3a_I06 ratios (as above) and [O iii]$\lambda\lambda$4959,5007/H$\beta$ & | Izotov et al. (2006b)
(Direct method) |  | [O ii]$\lambda\lambda$3727,9/H$\beta$ |
Te_O3b_O32 | " | Te_O3b ratios (as above) and [O iii]$\lambda\lambda$4959,5007/H$\beta$ & | "
(Direct method) |  | [O ii]$\lambda\lambda$3727,9/H$\beta$ |
KD02N2O2 | " | [N ii]$\lambda$6584/[O ii]$\lambda\lambda$3727,3729 | Kewley & Dopita (2002)
PP03N2 | " | ([O iii]$\lambda$5007/H$\beta$)/([N ii]$\lambda$6584/H$\alpha$) | Pettini & Pagel (2004)
PPN2 | " | [N ii]$\lambda$6584/H$\alpha$ | "
S07 | " | [Ne iii]$\lambda$ 3869/[O ii]$\lambda\lambda$3727,3729 | Shi et al. (2007)
BKD18 | " | [Ne iii]$\lambda$ 3869/[O ii]$\lambda\lambda$3727,3729 | Bian et al. (2018)
D16 | " | [N ii]$\lambda$6584/[S ii]$\lambda\lambda$6717,31 & | Dopita et al. (2016)
|  | [N ii]/H$\alpha$ |
KK04 | 12+log(O/H), $\log{(q)}$ | ([O ii]$\lambda$3727+[O iii]$\lambda\lambda$4959,5007)/H$\beta$ & | Kobulnicky & Kewley (2004)
LR14Ne3O2 | $\log{(q)}$ | [Ne iii]$\lambda$ 3869/[O ii]$\lambda\lambda$3727,3729 | Levesque & Richardson (2014)
LR14O3O2 | " | [O iii]$\lambda\lambda$4959,5007/[O ii]$\lambda\lambda$3727,9 | "
KD02O32 | " | [O iii]$\lambda$5007/[O ii]$\lambda\lambda$3727,9 | Kewley & Dopita (2002)
S18O32 | " | [O iii]]$\lambda\lambda$4959,5007/[O ii]$\lambda\lambda$3727,9 | Strom et al. (2018)
S18Ne3O2 | " | [Ne iii]$\lambda$ 3869/[O ii]$\lambda\lambda$3727,9 | "
S18O3Hb | " | [O iii]$\lambda$5007/H$\beta$ | "
K19bNe3O2 | " | [Ne iii]$\lambda$ 3869/[O ii]$\lambda\lambda$3727,9 | Kewley et al. (2019b)
K19bO3Hb | " | [O iii]$\lambda$5007/H$\beta$ | "
K19bO32a | " | [O iii]$\lambda$5007/[O ii]$\lambda$3727 | "
K19bO32b | " | [O iii]$\lambda\lambda$1660,6/[O ii]$\lambda$2470a,b | "
K19bC32a | " | [C iii]$\lambda\lambda$1906,8/[C ii]$\lambda$1335 | "
K19bC32b | " | [C iii]$\lambda\lambda$1906,8,8/[C ii]$\lambda$2323-8 | "
Ost_O2 | ne | [O ii]$\lambda$3729/[O ii]$\lambda$3727 | Osterbrock (1989)
S16O2 | " | [O ii]$\lambda$3729/[O ii]$\lambda$3727 | Sanders et al. (2016a)
S16S2 | " | [S ii]$\lambda$6731/[S ii]$\lambda$6717 | "
K19aO2 | ne, $\log{(P/k)}$ | [O ii]$\lambda$3729/[O ii]$\lambda$3727 | Kewley et al. (2019a)
K19aS2 | " | [S ii]$\lambda$6731/[S ii]$\lambda$6717 | "
K19aC3 | " | [C iii]$\lambda$1908/[C iii]$\lambda$1906 | "
K19aSi3 | " | [Si iii]$\lambda$1892/[C iii]$\lambda$1882 | "
Figure 3:  Comparison of individual diagnostics for the physical parameters – electron temperature, ionisation parameter, ISM pressure, electron density and gas phase oxygen abundance. For each panel the curves indicate the normalized, scaled (to unity) probability density function (PDF) of the measured physical quantity generated by performing every diagnostic 104 times. For each realization we randomly draw the line fluxes from a Gaussian distribution with mean and width equal to the measured flux and corresponding uncertainty, respectively (see Section 5.1). Different colours denote different diagnostics. The median of each PDF is shown with a filled circle of the corresponding colour. PDFs with dashed lines denote rest-frame UV diagnostics whereas solid lines denote optical diagnostics. The line ratios used for each diagnostic can be looked up in Table 4. For diagnostics involving lines for which we only have upper limits, we do not plot the PDF. Instead, we show the median by thick dashed vertical lines. Whether these are the upper or lower limits of the physical parameter, are denoted by arrows (right arrow: lower limit, left arrow: upper limit). We demonstrate that the UV and optical diagnostics for Te, $\log{(q)}$ and $\log{(\rm O/H)}$ broadly agree, with some exceptions where the diagnostics either could not be transformed to the common reference frame (for $\log{(\rm O/H)}$) or used the Ne3O2 index (for $\log{(q)}$) (detailed in Section 5). For $\log{(P/k)}$ and ne, however, we find that the UV diagnostics probe different (denser, higher pressure) physical nebular regions than their optical counterparts (detailed in Section 7.5).  Table 5: Inferred physical parameters. The diagnostics (see Table 4 for a list) are discussed in Section 5.2.5 & 5.3.5. For all further analysis, we consider log(O/H)+12 values in column 3 as the final values, if available. Otherwise we use the values from column 2. We use the 12+log(O/H)$=$8.23 and $\log{(P/k)}$$=$8.0 branch of the K19b calibrations, for all the K19b diagnostics. K18C32a,b methods have zero uncertainty because they are lower limits on $\log{(q)}$. The values quoted for the IZIP methods correspond to Figure 4, where a [N ii]$\lambda$6584/Hα based prior on $Z$ has been used. Oxygen abundance | Ionization parameter | ISM pressure | Elecron density | Electron temperature
---|---|---|---|---
Name of | 12+log(O/H) | 12+log(O/H) | Name of | $\log{(q(cm/s))}$ | Name of | $\log{(P/k(K/cm^{3}))}$ | Name of | $\log{(n_{e}(cm^{-3}))}$ | Name of | Te ($\times$104 K)
diagnostic |  | in KK04 frame | diagnostic |  | diagnostic |  | diagnostic |  | diagnostic |
Optical diagnostics
Direct (Te_O3a_O32) | 8.19 ${}^{+0.16}_{-0.10}$ | - | KK04 | 8.19 ${}^{+0.05}_{-0.05}$ |  K19aO2 | 7.31 ${}^{+0.09}_{-0.09}$ | Osterbrock O2 | 3.21 ${}^{+0.10}_{-0.10}$ | Te_O3a_I06 | 1.22 ${}^{+0.16}_{-0.17}$
KK04 | 8.65 ${}^{+0.02}_{-0.02}$ | 8.65 ${}^{+0.02}_{-0.02}$ | KD02O32 | 8.06 ${}^{+0.04}_{-0.04}$ |  K19aS2 | 7.59 ${}^{+0.51}_{-0.44}$ |  K19aO2 | 2.90 ${}^{+0.09}_{-0.09}$ | Te_O3a_N18 | 1.22 ${}^{+0.16}_{-0.17}$
PPN2 | 8.32 ${}^{+0.02}_{-0.02}$ | 8.70 ${}^{+0.02}_{-0.02}$ | LR14Ne3O2 | 7.21 ${}^{+0.06}_{-0.06}$ | - | - |  K19aS2 | 3.22 ${}^{+0.55}_{-0.46}$ | - | -
PPO3N2 | 8.21 ${}^{+0.01}_{-0.01}$ | 8.65 ${}^{+0.01}_{-0.01}$ | LR14O32 | 7.88 ${}^{+0.04}_{-0.03}$ | - | - | S16O2 | 2.88 ${}^{+0.09}_{-0.09}$ | - | -
KD02N2O2 | 8.71 ${}^{+0.05}_{-0.06}$ | 8.80 ${}^{+0.05}_{-0.05}$ |  K19bO32a | 7.78 ${}^{+0.03}_{-0.03}$ | - | - | S16S2 | 3.1 ${}^{+0.5}_{-0.41}$ | - | -
S07 | 8.26 ${}^{+0.25}_{-0.26}$ | - |  K19bNe3O2 | 7.36 ${}^{+0.05}_{-0.06}$ | - | - | - | - | - | -
D16 | 8.62 ${}^{+0.13}_{-0.14}$ | - |  K19bO3Hb | 7.75 ${}^{+0.02}_{-0.01}$ | - | - | - |  | - | -
BKD18 | 8.15 ${}^{+0.03}_{-0.03}$ | - | S18O32 | 7.91 ${}^{+0.04}_{-0.03}$ | - | - | - |  | - | -
- | - | - | S18Ne3O2 | 7.73 ${}^{+0.03}_{-0.03}$ | - | - | - |  | - | -
- | - | - | S18O3Hb | 7.85 ${}^{+0.01}_{-0.01}$ | - | - | - |  | - | -
UV diagnostic
Direct(Te_O3b_O32) | 8.29 ${}^{+0.07}_{-0.06}$ | - | K18C32a |  $\geq$ 6.94 |  K19aC3 | 8.31 ${}^{+0.37}_{-0.56}$ |  K19aC3 | 3.91 ${}^{+0.36}_{-0.56}$ | Te_O3b | 1.10 ${}^{+0.07}_{-0.06}$
- | - | - | K18C32b |  $\geq$ 7.45 |  K19aSi3 | 8.82 ${}^{+0.12}_{-0.19}$ |  K19aSi3 | 4.45 ${}^{+0.19}_{-0.21}$ | - | -
- | - | - | K18O32b | 7.99 ${}^{+0.07}_{-0.07}$ | - | - | - | - | - | -
IZIP diagnostic
All lines | 8.56 ${}^{+0.07}_{-0.03}$ | - | - | 8.21 ${}^{+0.12}_{-0.12}$ | - | 6.55 ${}^{+0.31}_{-0.31}$ | - | - | - | -
Only optical lines | 8.53 ${}^{+0.03}_{-0.03}$ | - | - | 8.01 ${}^{+0.12}_{-0.12}$ | - | 7.06 ${}^{+0.31}_{-0.51}$ | - | - | - | -
Only UV lines | 8.04 ${}^{+0.13}_{-0.26}$ | - | - | 8.01 ${}^{+0.29}_{-0.29}$ | - | unconstrained | - | - | - | -
UV+[O ii]3727,9 | 8.11 ${}^{+0.13}_{-0.26}$ | - | - | unconstrained | - | 7.88 ${}^{+0.31}_{-0.20}$ | - | - | - | -

##  5 Strong Emission Line Diagnostics

W14 measured a luminosity-weighted average reddening value of E(B-V) = $0.4\pm
0.07$. We use the W14 E(B-V) value and Cardelli et al. (1989) reddening law to
correct the NIRSPEC and MagE fluxes for extinction. The quoted uncertainities
in the measured fluxes are inferred directly from the Gaussian fits to the
emission lines. To obtain the dereddened flux uncertainty, we scale the
measured flux uncertainty with the same dereddening factor as applied to the
flux measurements. We take into account the uncertainties in both the
dereddened flux and the E(B-V) via a Monte Carlo approach, as described in
Section 5.1.

We list the emission line ratios used and corresponding labels for all the
diagnostics used in Table 4. The following sections describe the diagnostics
used. Figure 3 shows the results and Table 5 quotes the corresponding values.

###  5.1 Deriving uncertainties

For each emission line involved in a particular diagnostic, we randomly draw a
flux value from a Gaussian distribution which has a mean equal to the measured
(non-dereddened) flux and a width equal to the 1$\sigma$ uncertainty in the
measurement. We also randomly draw a value of E(B-V) from the measured range
of 0.4$\pm$0.07. Then we deredden the fluxes relative to the reddest line
involved in that particular diagnostic, using the randomly drawn E(B-V) for
that particular iteration. For instance, if the diagnostic involves the [O
iii]$\lambda\lambda$ 4959,5007/[O iii]$\lambda$ 4363 ratio, we de-redden the
[O iii]$\lambda$ 4363 and [O iii]$\lambda$ 4959 fluxes to bring them to the
reference frame of the [O iii]$\lambda$ 5007 line. Because only the line
ratios are relevant for our purposes, and not the overall shape of the
spectrum, the relative de-reddening is performed to ensure that we are not
over-estimating the de-reddening uncertainties. We calculate all the
diagnostics, explained in the preceding sections, with this set of de-reddened
line fluxes and repeat the process 104 times. This leaves us with 104
different values of each parameter we are trying to estimate, which we convert
to a probability distribution function (PDF). We quote the median values of
each distribution, along with the 16th and 84th percentiles as the 1$\sigma$
lower and upper limits of that parameter. To calculate the mean value and
uncertainties of a quantity from multiple SEL diagnostics, we simply add the
PDFs from the concerned diagnostics and then quote the 50th, 16th and 84th
percentiles of the summed PDF. We adopt this approach to account for asymmetry
(non-Gaussianity) in the uncertainties of physical parameters, which indeed is
the case for some of the properties.

###  5.2 Rest-frame optical diagnostics

We use the nebular [O ii]$\lambda\lambda$ 3727,3729, [O iii]$\lambda\lambda$
4959,5007, [O iii]$\lambda$ 4363, [N ii]$\lambda$ 6583 and [S
ii]$\lambda\lambda$ 6716, 6731 lines from the rest-frame optical NIRSPEC
spectrum (Table 1) to determine the physical quantities in RCS0327-E.

####  5.2.1 Electron temperature

We measure the electron temperature (Te), in the [O iii] nebular zone, using
the [O iii]$\lambda\lambda$4959,5007/[O iii]$\lambda$4363 ratio, following
equations 1 and 2 of I06, iteratively. This method has almost no dependence on
the ne, determined using [S ii] line fluxes, and can therefore constrain Te
even in absence of the [S ii] lines. The [O iii]$\lambda\lambda$ 4959,5007 and
[O iii]$\lambda$4363 emission lines originate from the ${}^{1}S\rightarrow\
^{1}D$ and ${}^{1}D\rightarrow\ ^{3}P$ transitions, which can be completely
constrained by atomic physics from a given Te. Therefore, the [O
iii]$\lambda\lambda$ 4959,5007/[O iii]$\lambda$4363 ratio is an excellent Te
diagnostic in the low-density (ne < 105 cm-3) regime. Using the I06 diagnostic
we derive Te = 1.22 ${}^{+0.16}_{-0.17}$ $\times$ 104 K.

We also use the [O iii]$\lambda$4363/[O iii]$\lambda$5007 ratio calibration
from N18 and which also yields Te = 1.22 ${}^{+0.16}_{-0.17}$ $\times$ 104 K.
The N18 calibrations are based on the latest version of MAPPINGS v5.1 (see
Sutherland & Dopita, 1993; Dopita et al., 2013) photoionisation code. The
MAPPINGS photoionisation code self-consistently computes the ionisation
structure of the nebulae, accounting for dust absorption, radiation pressure,
grain charging, and photoelectric heating of small grains (Groves et al.,
2004).

N18 is based on the latest atomic data for an ensemble of atoms at constant
temperature and density. The I06 diagnostic (from Aller, 1984) is based on the
same model, but uses older atomic data. As a result, both are emission-
weighted average temperatures, despite the existence of temperature gradients
in real nebulae. The remarkable agreement between Te derived from I06 and N18
methods is expected because the oxygen atomic data for the relevant lines has
not changed substantially.

####  5.2.2 Ionization parameter

The ionisation parameter $q$ is defined as the ratio of incident ionising
photon flux to the hydrogen density at the inner boundary of the ionised
shell. It is a measure of the hardness of the ionising radiation and
bolometric luminosity of the ionising source. We quote $q$ in units of cm s-1
throughout this paper. The ionization parameter can also be represented as a
dimensionless quantity $U$, by dividing $q$ by the speed of light.

We measure $\log{(q)}$ using ten different diagnostics (Table 4): four employ
the O32 ([O iii]$\lambda\lambda$4959,5007/[O ii]$\lambda\lambda$3727,9) index
from the calibrations of KK04, KD02, Levesque & Richardson (2014, henceforth
LR14), Strom et al. (2018, henceforth S18) and new photoionisation models of
K19b, two are based on the Ne3O2 ([Ne iii]$\lambda$3869/[O ii]$\lambda$3727)
calibrations by LR14, S18 and K19b, and another using the [O iii]$\lambda$
5007/H-$\beta$ ratio diagnostic from S18 and K19b. Averaging over all the
optical SEL diagnostics, we derive a weighted mean $\log{(q)}$ =
7.8${}^{+0.2}_{-0.5}$. We compare the different diagnostics in the top-right
panel of Figure 3 and quote the values in Table 5.

Reddening is a concern for the methods involving the O32 ratio, because the [O
ii] and [O iii] wavelengths are widely separated, and hence the dereddened
flux obtained for these lines greatly depends on the E(B-V) value and
extinction law assumed. As a possible solution, LR14 proposed a new
$\log{(q)}$ diagnostic using the Ne3O2 index, which uses lines with smaller
wavelength separation and makes for a more powerful diagnostic in spite of
reddening concerns. K19b adds to the LR14 calibrations by using updated
photoionisation models. We use both methods and compare the results.

The O32 and Ne3O2 line ratios are sensitive to metallicity. We use the mean
metallicity obtained from all the abundance diagnostics to specify the
metallicity branch of the $\log{(q)}$ calibrations in the LR14 and K19b
methods. K19b point out that the Ne3O2 index should only be used when reliable
estimates of ISM pressure and metallicity are available. We use the mean ISM
pressure derived using our pressure diagnostics, to define the ISM pressure
for use in the K18 methods.

The Ne3O2 diagnostic consistently yields lower ($\approx$0.7 dex) $\log{(q)}$
values than the other diagnostics, for both K19b and LR14 calibrations. This
is because the Ne3O2 ratio is extremely sensitive to ISM pressure and
metallicity. RCS0327-E yields $\log{(P/k)}$ $\approx$ 7.5 and 8.5 for the low
and high ionisation zones, respectively. The K19b calibrations quoted in this
paper correspond to a mean pressure branch of $\log{(P/k)}$ = 8.0. However,
the $\log{(P/k)}$ = 7.5 branch of the K19b calibrations yields a $\approx$0.2
dex higher $\log{(q)}$ and the $\log{(P/k)}$ = 8.5 branch yields $\approx$0.2
dex lower $\log{(q)}$, compared to the $\log{(P/k)}$ = 8.0 branch. Thus, using
$\log{(P/k)}$ = 7.5, brings the Ne3O2 ratio into agreement with the mean
$\log{(q)}$ value derived from other diagnostics. The LR14 diagnostic assumes
fixed ne = 100 cm-3 ($\log{(P/k)}$ $\approx$6), which is not a good
approximation for RCS0327-E. S16, on the other hand, include all galaxies,
irrespective of their metallicty and pressure, while fitting $\log{(q)}$ as a
function of Ne3O2. This leads to a $\sim 0.2$ dex intrinsic scatter in the
Ne3O2 calibration of S16. Consequently, the $\log{(q)}$ thus derived, agrees
well (within $<$0.1 dex) with the mean $\log{(q)}$, but has a large intrinsic
scatter which is not reflected in the quoted uncertainty. The example of
RCS0327-E demonstrates the drawbacks of the assumptions in individual emission
line diagnostics. It is in such cases that Bayesian inference methods (Section
6) can be more useful, provided the relevant emission lines are available.

####  5.2.3 Electron density

We compute the electron density ne in RCS0327-E using the O2 ([O
ii]$\lambda$3729/[O ii]$\lambda$3727) and S2 ([S ii]$\lambda$6731/[S
ii]$\lambda$6717) calibrations from Sanders et al. (2016a, hereafter S16) and
from the constant density models of K19a. The K19a models cover a range of
$\log{(n_{e}/\mathrm{cm^{-3}})}$ = 0 to 5, in increments of 0.5 dex. The S16
diagnostics are based on a 5-level atom approximation of the O ii and S ii
ions which yield $\log{(n_{e})}$ = 2.9${}^{+0.1}_{-0.1}$ and
3.1${}^{+0.5}_{-0.4}$ from the O2 and S2 ratios respectively, where ne is in
units of cm-3.

The $\log{(n_{e})}$ we obtain using the K19a O2 and S2 ratios, are in
excellent agreement with the corresponding S16 values. K19a provide a 3D
(metallicity, ionisation parameter and pressure/density) grid of models with
predicted emission line fluxes for every combination of the three parameters.
As such, obtaining electron density (or pressure) given a line ratio, requires
the metallicity ($Z$) and ionisation parameter ($q$) as inputs. We use the
mean of metallicity and $\log{(q)}$ values, measured using the different SEL
diagnostics, to constrain the $Z$ and $\log{(q)}$. Thereafter, we interpolate
the line ratios as a function of the electron density (or pressure) to obtain
our desired physical quantity.

For comparison, we also use the theoretical O2 vs ne curves from Osterbrock
(1989) that are based on single atom models. We infer a weighted mean
$\log{(n_{e})}$ = 3.0${}^{+0.4}_{-0.2}$ using rest-optical diagnostics. The
middle-right panel in Figure 3 shows our ne measurements and Table 5 quotes
the corresponding values.

The electron density ne is fundamentally related to the ISM pressure (e.g. see
Dopita et al., 2006; Kewley et al., 2019a, for discussion). Assuming a
constant electron temperature Te, the ISM pressure $P$ is related to the total
density $n=P/T_{e}k$ and the total density $n$ is related to the electron
density through $n=n_{e}(1+(4X+Y)/(2X+2))$ where $X$ and $Y$ are the mass
fractions of Hydrogen and Helium respectively. Thus, for a fixed Te the ISM
pressure is directly proportional to ne. In reality, however, neither the
electron temperature nor the density is constant. The ISM is often clumpy and
has fluctuations and/or gradients in temperature and density. The constant
density models of K19a allow for the temperature structure within the H ii
region but they do not allow the electron density to vary. Hence, K19a point
out that the constant density models are likely to be less realistic than the
constant pressure models because typical H ii regions have shorter sound
crossing timescale than cooling/heating timescale, allowing the pressure to
equalise throughout the nebula.

####  5.2.4 ISM pressure

We use the O2 ([O ii]$\lambda$3729/[O ii]$\lambda$3727) and S2 ([S
ii]$\lambda$6731/[S ii]$\lambda$6717) calibrations of K19a to measure the ISM
pressure, given in terms of $\log{(P/k)}$ where $k$ is the Boltzmann constant.
We derive a weighted average $\log{(P/k)}$ = 7.4${}^{+0.6}_{-0.2}$ for
RCS0327-E from the rest-frame optical diagnostics, where $P/k$ is in units of
$\mathrm{K\ cm^{-3}}$. The middle-left panel in Figure 3 shows the PDFs for
all diagnostics.

Reddening is not a concern for either of these sets of closely spaced lines.
Nevertheless, we performed reddening corrections (as described in Section 5)
for consistency. K19a point out that the S2 ratio changes by $\sim 1$ dex
within a range of 5.5 $\leq\log{(P/k)}\leq$ 9.0 and the O2 index drops by
$\sim 1$ dex within 5.5 $\leq\log{(P/k)}\leq$ 8.0, demonstrating that these
diagnostics are extremely sensitive to the pressure for this range.

K19a used plane-parallel MAPPINGS v5.1 H ii region models at constant
pressure, with $\log{(P/k)}$ ranging from 4.0 to 9.0 in increments of 0.5.
MAPPINGS calculates detailed electron temperature and density structure within
the H ii region for each of these models at a fixed ISM pressure. K19a point
out that the constant pressure models are more realistic than the constant
density models used for the electron density calibrations (Section 5.2.3) and
recommend using the former. We refer the reader to K19a for a detailed
description of the models.

####  5.2.5 Gas phase oxygen abundance

We measure the gas phase oxygen abundance from the available set of optical
lines in eight different ways (Table 4). We use the combined method of KD02
(Section 6 KD02), which, for our abundance regime, uses the [N
ii]$\lambda$6584/[O ii]$\lambda\lambda$3727,3729 ratio. We also employ the
iterative method of KK04 which uses the R23 (([O ii]$\lambda$3727+[O
iii]$\lambda\lambda$4959,5007)/H$\beta$) and O32 indices to solve for both
12+log(O/H) and $\log{(q)}$. Both these works stem from MAPPINGS
photoionisation models of H ii regions. We additionally use the N2 ([N
ii]$\lambda$6584/H$\alpha$) and O3N2 (([O iii]$\lambda$5007/H$\beta$)/([N
ii]$\lambda$6584/H$\alpha$)) calibrations from PP04 to compare the abundance
values derived using different calibration methods. We also use the empirical
Ne3O2 (Ne iii]$\lambda$ 3869/[O ii]$\lambda\lambda$3727,9) calibrations from
Shi et al. (2007, hereafter S07) and Bian et al. (2018, hereafter BKD18), and
the theoretical [N ii]/[S ii] and [N ii]/H$\alpha$ ratios, following D16. Both
the BKD18 and D16 methods are suitable for high redshift ($z\gtrsim$2)
galaxies. In addition to these SEL methods, which depend on photoionisation
models (with the exception of S07 and BKD18), we also use the direct
estimation of the abundance from the electron temperature Te, following the
I06 procedure.

Each method has its own drawbacks. The $T_{e}$ method, KK04 and KD02 methods
are sensitive to reddening corrections because they involve lines with widely
spaced wavelengths. The KK04 R23 diagnostic is double valued and requires an
initial guess of abundance, which we provide by using the [N
ii]$\lambda$6584/H$\alpha$ ratio. The R23 index is also sensitive to
ionisation parameter $\log{(q)}$. We use R23 in conjunction with the O32 index
to iteratively solve for both log(O/H)+12 and $\log{(q)}$. The S07, BKD18, D16
and both the PP04 diagnostics, do not suffer from reddening issues because
they use lines that are closely spaced in wavelength.

In addition to the above shortcomings, all the methods have systematics
offsets, relative to one another, on their zero-points (Kewley & Ellison,
2008; Bian et al., 2017). This is because of the different photoionisation
models and samples of H ii regions used to derive the diagnostics. The
discrepancy between the strong line diagnostics and the Te method are well
known and are mainly attributed to the assumption of a constant temperature in
the Te methods. (e.g. Stasińska, 2002; López-Sánchez et al., 2012).
Additionally, the existence of a temperature gradient within the H ii regions
may lead the Te method to systematically underestimate the oxygen abundance
because of the assumption of a one or two-zone temperature model (Stasińska,
2005). Therefore, it is sensible to compare among these methods, only after we
have corrected for the relative offsets.

We correct for this offset following KE08 prescription which was developed
using local SDSS galaxies. We convert the 12+log(O/H) values from the
empirical and theoretical calibrations to the reference frame of the KK04
method. The choice for this common reference frame was motivated by the fact
that KK04 take into account the dependence of the metallicity sensitive lines
on the ionisation parameter. However, Kewley & Ellison (2008, hereafter KE08)
do not prescribe a conversion scheme from the direct Te method to the other
SEL diagnostics. Therefore, we quote the Te metallicity as it is, without any
conversion. We also transform the R11 abundance value to the KK04 frame, in
order to facilitate comparison between our work and R11. Note that we cannot
account for any potential relative offsets in the S07, BKD18 and D16 methods
following the KE08 prescription because KE08 predates both.

Table 5 lists the values of 12+log(O/H) computed using various diagnostics and
the bottom-left panel in Figure 3 shows the corresponding PDFs. Wuyts et al.
(2014b) inferred a 12+log(O/H) = 8.28 $\pm$ 0.02 for RCS0327-E, which
transforms to 12+log(O/H) = 8.65 $\pm$ 0.03 in the KK04 frame. We plot the
Wuyts et al. (2014b) value as a dotted line in Figure 3 for comparison.
Averaging over all diagnostics, we derive a weighted average 12+log(O/H) =
8.6${}^{+0.1}_{-0.4}$ for RCS0327-E in the KK04 frame.

The optical SEL diagnostics yield a weighted mean 12+log(O/H) =
8.6${}^{+0.1}_{-0.4}$. Both the Te methods and the Ne3O2 methods (S07 and
BKD18) result in lower ($\sim$0.4 dex) abundance. With the exception of BKD18,
the other three measurements have large uncertainties associated with them
(broad PDFs in Figure 3). As discussed in Section 5.2.5, the discrepancy
between the Te and strong line methods is well known and can be attributed to
the temperature fluctuations and gradients in the H ii region, which the Te
methods do not take into account. S07 and BKD18 use the Te method to calibrate
their diagnostic and hence, suffer from the same discrepancy. The higher
uncertainties of the Te and S07 methods is because the [O iii]$\lambda$4363
line is often weak or undetected. Moreover, the strong line methods use
additional information in terms of $\log{(q)}$ or a specific branch of
metallicity, leading to lower uncertainties.

###  5.3 Rest-frame UV diagnostics

The rest-frame UV diagnostics use the O iii]$\lambda\lambda$1660,6, C
iii]$\lambda$1907, [C iii]$\lambda$ 1909 and Si ii]$\lambda\lambda$1883,92
line fluxes measured from the Magellan/MagE spectra (Table 2). The majority of
these diagnostics (refer Table 4) are from recent works of K19a,b and N18,
which use the latest, improved version 5.1 of MAPPINGS. The new MAPPINGS v5.1
uses the latest available atomic data from the CHIANTI8 atomic database (Del
Zanna et al., 2015), which is a prime factor that governs the nebular emission
line strengths. As described in detail in Section 5.2, the plane parallel
isobaric H ii region models have been used for both N18 and K18 (except for
density diagnostics, where constant density models have been used). Moreover,
reddening corrections are not important for the diagnostics involving only
rest-frame UV lines, because these pairs of lines have closely spaced
wavelengths.

####  5.3.1 Electron temperature

We derive electron temperature (Te) using the [O iii]$\lambda$ 5007/O
iii]$\lambda\lambda$ 1660,6 ratio from the theoretical calibrations of N18.
This method relies on the fact that the ${}^{5}S\rightarrow\ ^{3}P$ ($\lambda$
5007) and ${}^{1}D\rightarrow\ ^{3}P$ ($\lambda\lambda$ 1660,6) transition
rate ratio depends only on one physical parameter, the Te. The other
dependencies of the ratio are constants that can be derived from atomic
physics. The UV N18 calibrations yield a Te = 1.10 ${}^{+0.07}_{-0.06}$
$\times$ 104 K which is $\sim 1000$ K lower than that derived from the optical
Te diagnostics. The optical and UV Te measurements agree to within $1\sigma$.

The large wavelength baseline renders the [O iii]$\lambda$ 5007/O
iii]$\lambda\lambda$ 1660,6 ratio highly susceptible to uncertainties in
reddening corrections. An uncertainty of $\pm$0.07 in E(B-V) leads to
$\sim$+33%, -25% uncertainty in this ratio. Moreover, discrepancies in the
relative flux calibration from the rest-frame UV to the rest-frame optical
spectra can contribute to uncertainties in Te.

Although we refer to the [O iii]$\lambda$5007/O iii]$\lambda\lambda$1660,6
method as a UV diagnostic, it still requires [O iii]$\lambda$5007\. Therefore,
no Te diagnostic used in this paper is completely independent of rest-optical
spectra. However, the method involving only the optical lines requires the [O
iii]$\lambda$4363 line, which is barely detected (SNR = 2.75) in RCS0327-E.
Consequently, the [O iii]$\lambda$5007/O iii]$\lambda\lambda$1660,6 ratio
provides a better constraint on Te. Taking a weighted average of all UV-
optical methods, we find a mean Te = 1.2${}^{+0.2}_{-0.1}$ $\times 10^{4}$ K
for RCS0327-E.

####  5.3.2 Ionization parameter

K19b outline many rest-frame UV emission line ratios that can potentially be
used as ionisation parameter ($\log{(q)}$) diagnostics. We choose to use three
line ratios: [C iii]]$\lambda\lambda$1907,9/[C ii]$\lambda$1335 (blend of
1334.58 Å, 1335.66 Å and 1335.71 Å), [C iii]]$\lambda\lambda$1907,9/[C
ii]$\lambda$2323-8 (2323.50 Å, 2324.69 Å, 2325.40 Å, 2326.93 Å, and 2328.12 Å)
and [O iii]]$\lambda\lambda$1660,6/[O ii]$\lambda$2470\. We derive $\log{(q)}$
entirely from UV lines, for the first time, by using these three diagnostics.
The top-right panel of Figure 3 shows the PDFs of our $\log{(q)}$
measurements. The [C ii]$\lambda$1335 and [C ii]$\lambda$2323 group of lines
are not detected in RCS0327-E. We therefore use the 3$\sigma$ upper limits for
the [C ii] line fluxes to estimate a lower limit for $\log{(q)}$, wherever
applicable. Lower limits are represented in Figure 3 as dashed vertical lines
with horizontal arrows.

The [C iii]/[C ii] ratios are very effective measures of the ionisation
parameter, especially in the low metallicity regime (12+log(O/H) $\leq$ 8.5),
because they have negligible sensitivity to ISM pressure. Moreover, the [C
iii]$\lambda\lambda$1907,9/[C ii]$\lambda$1335 ratio does not vary with
metallicity for low metallicities, making it an ideal $\log{(q)}$ diagnostic.
The [O iii]$\lambda\lambda$1660,6/[O ii]$\lambda$2470 ratio is analogous to
the O32 index in optical. K19b advise against the use of the [O iii]/[O ii]
diagnostic in the high pressure ($\log{(P/k)}$ $\geq$7) and low metallicity
(12+log(O/H) $\leq$ 8.23) regime owing to the high sensitivity of these lines
(varies $>$ 0.5 dex) to ISM pressure. We derive the ISM pressure of RCS0327-E,
and use it to interpolate between the pressure grid, thereby minimising the
sensitivity issue.

The $\log{(q)}$ estimated from all rest-frame UV and optical diagnostics
(except the Ne3O2 ratio) broadly agree within $\sim$0.7 dex. K19bO32b is the
only UV diagnostic which is not a lower limit, and it yields $\log{(q)}$ =
8.0${}^{+0.1}_{-0.1}$ which agrees with the mean optical result within
1$\sigma$ uncertainty, as do all the individual K19b diagnostics. The KD02 and
KK04 diagnostics were based on an earlier version of MAPPINGS (v3) and as
such, yield slightly ($\sim$0.2 dex) higher $\log{(q)}$ than the K19b methods,
while agreeing to within 1$\sigma$ with each other. The lower limits on
$\log{(q)}$ obtained from the undetected [C ii] emission lines are consistent
with the other diagnostics. Although the scatter in ionization parameter is
large ($\sim$0.7 dex), the UV estimates are not systematically offset from the
optical estimates within the uncertainties, which is encouraging. This clearly
demonstrates that it is indeed possible to determine $\log{(q)}$ using only
rest-frame UV spectra, provided at least one of C
iii]$\lambda\lambda$1906,8/[C ii]$\lambda$1335, C
iii]$\lambda\lambda$1906,8/[C ii]$\lambda$2325, or [O
iii]$\lambda\lambda$1660,6/[O ii]$\lambda$2470 ratios is available.

####  5.3.3 ISM Pressure

We determine the ISM pressure ($\log{(P/k)}$) using the K19a calibrations of
rest-frame UV line ratios C iii]$\lambda$1907/[C iii]$\lambda$1909 (C3) and Si
iii]$\lambda$1883/Si iii]$\lambda$1892 (Si3). These calibrations are derived
using the plane parallel, isobaric model grid from MAPPINGS, as described
Section 5.2. The middle-left panel in Figure 3 shows the PDFs of our ISM
pressure measurements and the corresponding values are quoted in Table 5. We
derive a weighted average $\log{(P/k)}$ = 8.8${}^{+0.2}_{-0.6}$ from the UV
diagnostics.

In general, the UV diagnostics (dashed lines) yield considerably higher
($\sim$1.4 dex) ISM pressures than the optical diagnostics (solid lines),
though there is significant ovelap between the UV and optical PDFs. The
discrepancy is the result of the K19a C3 and Si3 diagnostics probing the high
pressure ($\log{(P/k)}$ > 7.5) regime. We further discuss this difference in
Section 7.5.

The Si iii and C iii ratios approach the high pressure limit at $\sim 1.4$ and
cease to be sensitive to pressure. Both the pressure diagnostics are sensitive
to $\log{(q)}$ and metallicity. The Si iii ratio is more sensitive to the ISM
pressure, varying by over an order of magnitude in the range 7.5 <
$\log{(P/k)}$ < 9.0, as compared to the C iii ratio (which varies by almost an
order of magnitude in the same range of $\log{(P/k)}$).

####  5.3.4 Electron density

Due to the interdependency between the electron density (ne) and the ISM
pressure, the line ratios sensitive to one property are sensitive to the other
as well. We derive ne using the same C3 and Si3 ratios, as in the case of
pressure, from the K19a calibrations. However, in this case, the H ii region
models used assume constant density throughout the nebula, which may not be a
valid assumption. We infer a weighted mean $\log{(n_{e})}$ =
4.3${}^{+0.3}_{-0.6}$ using rest-UV diagnostics.

Note that the K19a C3 and Si3 diagnostics are sensitive only in the high
density (ne > 1000 cm-3) regime, which holds for RCS0327-E. Both the Si iii
and C iii ratios have almost no dependence on $\log{(q)}$ (Figures 1 and 2 in
K19a) but are slightly sensitive to the metallicity, which has to be provided
in order to obtain ne.

Similar to the ISM pressure case, we derive consistently higher ne ($\sim 1.5$
dex) with the UV lines than the optical lines, which is reflected as a
bimodality in the PDFs in Figure 3. Again, this discrepancy is the result of
different emission line species tracing different physical regions of the
nebula, as described in Section 7.5. Thus, it is possible to infer ne using
only the rest-frame UV lines, although the values inferred do not represent
the same physical region as the optical diagnostics.

####  5.3.5 Oxygen abundance

We measure 12+log(O/H) by the direct method following equations 2 and 3 of
I06. Here we use the Te obtained from the [O iii]$\lambda$5007/O
iii]$\lambda\lambda$1660,6 ratio. Because this abundance estimate makes use of
the UV lines O iii]$\lambda\lambda$ 1660,6 we classify this as a rest-frame UV
abundance diagnostic. The UV Te diagnostic yields 12+log(O/H) = 8.3 $\pm$ 0.1,
which agrees with the optical Te abundance within $1\sigma$ uncertainty. The
bottom-left panel in Figure 3 compares the abundance diagnostics.

##  6 Joint Bayesian Diagnostics

IZI is an IDL-based software developed by Blanc et al. (2015, hereafter B15)
that uses Bayesian inference to simultaneously infer metallicity ($Z$) and
ionisation parameter ($\log{(q)}$) of the ionised nebular gas. IZI requires a
set of emission line fluxes observed from the nebulae and a 2D ($Z$ and
$\log{(q)}$) grid of models, as its inputs.

We extend the publicly available version of IZI to 3D – to enable the
metallicity, ionisation parameter and ISM pressure to be inferred
simultaneously. This 3D Bayesian method avoids the need for assumptions about
$\log{(P/k)}$ and hence constrains the physical properties in a self-
consistent way. The 3D IZI, referred to as IZIP (Inferring metallicities (Z),
Ionization, and Pressure) henceforth, requires a 3D grid of models ($Z$,
$\log{(q)}$ & $\log{(P/k)}$) as an input to interpolate. We emphasize that
IZIP is simply an extension of IZI to include an extra dimension, but
otherwise preserves the functionality of the original IZI algorithm. We use
MAPPINGS-V photoionisation models to produce a grid of $Z$, $\log{(q)}$,
$\log{(P/k)}$ and emission line fluxes as inputs to IZIP. In the hope that it
would benefit the community, we make IZIP publicly available at [URL to be
added at proofs stage] and ask future users to consider this paper as the
appropriate reference for IZIP.

The input to the MAPPINGS-V models, which in turn is input to IZIP, accounts
for the primary and secondary nucleosynthetic components of the N/O and C/O
ratios. The application assumes that the nitrogen and carbon in galaxies have
a primary and a secondary origin and that there is no N/O or C/O excess. We
are assuming that RCS0327-E lies along the local relation for the above
nucleosynthetic origin. Thus, in effect, IZIP accounts for variations in N/O
and C/O as a function of metallicity, when used with the MAPPINGS models as
the input.

Most individual emission line diagnostics suffer from a major drawback: they
are simultaneously sensitive to metallicity, ionisation parameter and
pressure. IZIP simultaneously computes the likelihood of each of 12+log(O/H),
$\log{(q)}$ and $\log{(P/k)}$ without any assumption about the others.
Moreover, IZIP makes use of all the available emission line information
simultaneously as opposed to using a specific pair of lines to derive $Z$ and
$\log{(q)}$. The Bayesian approach allows us to calculate joint and
marginalised posterior probability distribution functions (PDFs). PDFs allow
for multiple peaks and/or asymmetry, which reflect degeneracies in the
relation between line fluxes and nebular properties. These degeneracies are
harder to deal with while using a specific emission line pair. IZIP also takes
into account upper limits for lines that are not formally detected and
translates them to a limit on the derived $Z$, $\log{(q)}$, and $\log{(P/k)}$.
This is extremely useful for high-redshift spectroscopic studies for which,
often, only upper limits on emission lines are available.

A potential disadvantage of IZIP is that it assumes equal weights on all the
available emission lines. In other words, lines that are potentially not
sensitive to the concerned nebular property or are also sensitive to other
nebular properties are weighted the same as the lines that are only sensitive
to the concerned property. Consider, for example, the metallicity. Some
metallicity sensitive emission lines are also sensitive to ionisation
parameter and/or ISM pressure. Individual nebular diagnostics involving these
lines would be considered less robust as they may not be ideal probes of
metallicity if $\log{(q)}$ and $\log{(P/k)}$ are not accurately known. IZIP,
however, is unable to make such informed decisions and would treat these lines
with the same weights as other lines that are sensitive to metallicity only
(and thus are ideal metallicity indicators). This might lead to poor
constraints in the derived metallicity because IZIP includes emission lines
that are dominated by other properties (.e.g. $\log{(q)}$) which may be
dependent on, but not necessarily positively correlated with, metallicity. One
way to remedy this would be to provide only those lines to IZIP that are
sensitive only to the particular physical parameter. We carry out such tests
for different physical quantities and present the results in Appendix A. The
issue of using less sensitive emission lines is especially relevant for
high-$z$ galaxies because high-$z$ spectra often contain only a few emission
lines above a desired SNR threshold, all of which may not necessarily be
exclusively sensitive to $Z$, $\log{(q)}$ or $\log{(P/k)}$. Additionally, not
all emission lines have a nebular origin. For instance, Prochaska et al.
(2011) suggest that the Mg ii and Fe ii lines arise due to scattering in the
galactic wind. Hence, we remove all the Mg ii and Fe ii lines, before
providing the set of emission lines to IZIP.

Another way to force IZIP to give different weights to different lines is by
providing user defined priors to IZIP. For instance, to break the degeneracy
of the double valued metallicity branches (e.g. R23), we use a top-hat prior
on $Z$ such that 12+log(O/H) > 8.52 if $[$N ii$]\lambda$6584/H$\alpha$ >
0.0776 and vice versa, wherever $[$N ii$]$/H$\alpha$ ratio is avaiable. The
choice of this prior is motivated by the fact that $[$N ii$]$/H$\alpha$ varies
monotonically with $Z$. We chose $[$N ii$]$/H$\alpha$ over the $[$N ii$]$/$[$O
ii$]$ ratio because the latter could be affected by relative flux calibration
issues as the $[$N ii$]$ and $[$O ii$]$ lines are captured by different
NIRSPEC filters. We also investigate the impact of using a flat prior (i.e. no
user-defined prior) on the inferred physical quantities in Section 7.4.

Table 6: List of emission lines provided to IZIP for Bayesian analysis. We provide different combinations of the rest-frame optical and UV emission line fluxes for different cases, as described in Section 6. The third column denotes whether the emission line is undetected and hence upper limits are used by IZIP, or if it is a blended doublet. Line ID |  $\lambda_{\mathrm{rest}}$ (Å) | Comments
---|---|---
Rest-frame UV
C II 1335a | 1334.5770 | Upper limit
C II 1335b | 1335.6630 | Upper limit
C II 1335c | 1335.7080 | Upper limit
Si II 1533 | 1533.4312 | -
He II 1640 | 1640.4170 | -
O III] 1660 | 1660.8090 | -
O III] 1666 | 1666.1500 | -
N III] 1750 | 1749.7000 | -
[Si III] 1882 | 1882.7070 | -
Si III] 1892 | 1892.0290 | -
[C III] 1906 | 1906.6800 | -
C III] 1908 | 1908.7300 | -
N II] 2140 | 2139.6800 | -
[O III] 2320 | 2321.6640 | -
C II] 2323 | 2324.2140 | Upper limit
C II] 2325c | 2326.1130 | -
C II] 2325d | 2327.6450 | -
C II] 2328 | 2328.8380 | -
Si II] 2335a | 2335.1230 | Upper limit
Si II] 2335b | 2335.3210 | Upper limit
[O II] 2470 | 2471.0270 | Unresolved doublet
He I 2945 | 2945.1030 | Upper limit
Rest-frame optical
[O II] 3727,9 | 3727.092, 3729.875 | Unresolved doublet
[Ne III] 3869 | 3869.860 | -
H$\zeta$ | 3890.166 | -
H$\delta$ | 3971.198 | -
H$\epsilon$ | 4102.892 | -
H$\gamma$ | 4341.692 | -
O III 4363 | 4364.435 | -
H$\beta$ | 4862.691 | -
[O III] 4959 | 4959.895 | -
[O III] 5007 | 5008.239 | -
H$\alpha$ | 6564.632 | -
[N II] 6584 | 6585.273 | -
[S II] 6717 | 6718.294 | -
S II 6731 | 6732.674 | -
[Ar III] 7136 | 7137.770 | -

To investigate how the ISM properties depend on the amount of spectral
information available, we supply IZIP with four different sets of emission
lines tailored to mimic observations with different wavelength coverage, as
follows.

  1. 1.

All the UV and optical emission lines (in Table 6) as input: Using all the
nebular spectral information available would help us understand by how much
the constraints on the physical properties of RCS0327-E are improved on
inclusion of the UV information in addition to the existing optical spectra.

  2. 2.

Only the rest-frame optical emission lines of Table 6: This combination
emulates the scenario when the rest-frame UV is not within the observed
wavelength coverage and all the information we have is from the rest-frame
optical spectra.

  3. 3.

Only the rest-frame UV emission lines of Table 6: We use this combination to
study how well the nebular properties can be constrained in a scenario where
only the rest-frame UV spectra are available, the redder part of the spectra
having been redshifted out of the observed wavelength range.

  4. 4.

The rest-frame UV emission lines along with the [O ii]$\lambda\lambda$3727,9
doublet: The [O ii] doublet would be within wavelength coverage until $z\sim
12$ with JWST. Hence, combined information from both the [O ii] doublet and
the UV spectra could be used to infer the physical properties for high-$z$
galaxies.

  5. 5.

All the UV and optical emission lines except [Si iii]$\lambda\lambda$1882,92:
Excluding the [Si iii] doublet allows us to investigate its effect on the
inferred nebular properties.

  6. 6.

Only the UV emission lines except [Si iii]$\lambda\lambda$1882,92. This
combination of emission lines is one of the many combinations we tested,
including or excluding certain emission lines each time. Excluding [Si iii]
produced a considerable impact, as we discuss in Section 6.1. We discuss all
the other additional tests in Appendix A.

IZIP does not account for extinction by dust and requires the user to provide
extinction-corrected flux values. We propagate the uncertainties in the
reddening value via a Monte Carlo (MC) technique. The MC approach is better
than the analytic error propagation because the latter is only applicable up
to first order expansion in Taylor series whereas, given enough iterations,
the MC approach better samples the parameter space. We randomly draw from a
normal distribution of the measured value of E(B-V) = 0.4 $\pm$ 0.07 (W14).
After correcting for the reddening using this randomly drawn E(B-V) and
Cardelli et al. (1989) reddening law, we supply the dereddened fluxes to IZIP.
We repeat this process 100 times, adding and normalizing the resulting PDFs
for each iteration, to give the final marginalised and joint PDFs presented in
Section 6.1. The results converge well before the 100 iterations used here.

(a) All emission lines used

(b) Only optical emission lines used

(c) Only UV emission lines used

(d) Only UV and [O ii]$\lambda\lambda$3727,9 emission lines used

Figure 4:  Results from 100 iterations of IZIP (Section 6) with uniform priors
on $\log{(q)}$ and $\log{(P/k)}$. Wherever both [N ii]$\lambda$6584 and
H$\alpha$ are available, we use a top-hat prior on metallicity (black dashed
line) such that 12+log(O/H) > 8.52 if [N ii]$\lambda$6584/H$\alpha$ > 0.0776
and vice versa. For cases where at least one of the lines is unavailable we
use a uniform prior on $Z$. In each group of plots: the bottom-left plot shows
the 2D joint PDF for ISM pressure on the y-axis and metallicity on the x-axis,
the middle-left plot shows the 2D joint PDF for ionisation parameter (y-axis)
and metallicity (x-axis) and bottom-middle plot denotes that for ISM pressure
(y-axis) and ionisation parameter (x-axis). The remaining plots show the 1D
marginalized posterior PDFs for metallicity (top), ionisation parameter
(middle) and ISM pressure (bottom). The blue circle represents the peak of the
marginalised distribution, with error bars being the 16th and 84th
percentiles. The top-left group of plots denote results when all rest-frame UV
and optical emission lines are used by IZIP. The top-right and bottom-left
groups of plots show the results when IZIP works on only the rest-frame
optical and only rest-frame UV line measurements, respectively. The bottom-
right group of plots corresponds to the scenario where only rest-frame UV and
[O ii]$\lambda\lambda$3727,9 doublet were provided to IZIP. Fe or Mg emission
lines have not been included in the IZIP analysis, as discussed in Section 6.
The black contours and dotted histograms denote the PDFs (2D and 1D
respectively) of the corresponding fiducial cases i.e. when all the emission
lines are used. By definition, the fiducial case for Figure 4(b) is Figure
4(a). For Figures 4(c) and 4(d), we consider the configuration using all the
lines but not the [N ii]/H$\alpha$ prior, as the fiducial case (Figure 5(a)).
We demonstrate that the ISM properties are well constrained on using all the
lines or only the rest-frame optical lines. Using UV lines in addition to
opitcal lines generally improves the constraints but the UV lines alone find
it difficult to constrain metallicity and pressure. Using the [O ii] doublet
in addition to the UV lines helps constrain the ISM pressure.

(a) All emission lines used

(b) Only optical emission lines used

Figure 5:  Same as in Figure 4 but here we use a flat prior on $Z$ even if we
include the rest-frame optical emission lines. Figure 5(a) is shown on Figure
5(b) with black contours and dotted histograms for visual aid. On using only
the optical lines, we infer a lower ($\sim$2 dex), poorly constrained
$\log{(P/k)}$ (right) with a flat prior on $Z$ as compared to using a top-hat
prior on $Z$ (Figure 4(b)).

###  6.1 Results from Bayesian methods

We describe our results from the Bayesian method, including the cases where
different sets of emission lines were provided to IZIP. Figures 4 and 5 show
the marginalised PDFs for the different physical parameters and the last
section of Table 5 quotes the corresponding peak values. Figure 5(a)
represents the case where all available emission lines were provided to IZIP.
For panels Figures 5(b) and 4(c) only rest-frame optical and only rest-frame
UV (Table 6) nebular emission lines were used, respectively. The shaded plots
denote the 2D PDF (marginalised over the third parameter) and the solid curves
denote the 1D PDF for the corresponding physical parameter. The filled circles
at the base of each 1D PDF show the peak value of the PDF, with errorbars
being the 16th and 84th perecentiles.

####  6.1.1 Ionization parameter

The ionisation parameter, $\log{(q)}$ = 8.21${}^{+0.12}_{-0.12}$ derived by
IZIP using all the emission lines (Figure 5(a)), is $\sim$0.4 dex higher than
the mean $\log{(q)}$ = 7.77${}^{+0.01}_{-0.01}$ derived using the emission
line diagnostics. The mean $\log{(q)}$ from the rest-frame optical strong line
diagnostics $\log{(q)}$ = 7.76${}^{+0.01}_{-0.01}$ agrees at 1$\sigma$ level
of the peak $\log{(q)}$ = 8.01${}^{+0.12}_{-0.12}$ inferred by IZIP using only
the rest-frame optical spectra. The $\log{(q)}$ = 8.01${}^{+0.29}_{-0.29}$
inferred by IZIP while using only the UV spectra agrees with the $\log{(q)}$ =
7.99${}^{+0.07}_{-0.07}$ derived using the rest-frame UV diagnostic of K18,
within 1$\sigma$ uncertainties. This agreement demonstrates that, at least for
the test-case of RCS0327-E, the Bayesian approach can be used to reliably
determine the ionisation parameter when only the UV lines are available.

####  6.1.2 ISM pressure

IZIP yields a low $\log{(P/k)}$ $\sim$7 (Figure 4(b)) when only the optical
emission lines are provided. However, using only the UV emission lines results
in the $\log{(P/k)}$ PDF saturating at the edge of the model parameter space
(Figure 4(c)). Hence, UV lines alone fail to constrain $\log{(P/k)}$ in the
case of RCS0327-E. However, the inclusion of the [O ii]$\lambda\lambda$3727,9
doublet with the UV lines (Figure 4(d)), constrains the ISM pressure fairly
well ($\sigma<$ 0.4) at a value $\log{(P/k)}$ = $\sim$7 in spite of the
inclusion of the high ionisation UV species. This is because the [O
ii]$\lambda\lambda$3727,9 doublet has much higher SNR than the UV lines and
hence gets higher weight during the Bayesian analysis. In contrast, when only
the rest-optical lines are used, models predict that the [O ii] lines have
comparable strength as the [S ii]$\lambda\lambda$6717,31 doublet in the high-
metallicity regime (such as in RCS0327-E) and hence it is not obvious that [O
ii] would dominate the Bayesian inference. Moreover, on using the [O ii] lines
along with the UV lines (Figure 4(d)), the PDF is shifted down by $\sim$1.6
dex, implying that the [O ii] lines probe a lower ISM pressure than the higher
ionised UV species (see Section 7.5). Therefore, the inclusion of [O ii] helps
to constrain the pressure but also biases $\log{(P/k)}$ towards lower values
than probed by the UV lines.

####  6.1.3 Oxygen abundance

IZIP infers 12+log(O/H) = 8.56${}^{+0.07}_{-0.03}$ when the full suite of UV
and optical spectra is used (top panel, Figure 4(a)), which agrees within
1$\sigma$ of the mean abundance of all the individual diagnostics. The oxygen
abundance is poorly constrained by IZIP when only the rest-frame UV spectra is
used, suggesting that it is difficult to constrain the abundance using only UV
emission lines. The uncertainties in the Extreme Ultraviolet (EUV) radiation
field that is used to produce the H ii region models is a potential cause for
this difficulty. Providing IZIP with only the UV lines and the Balmer lines
does not help improve constraints on $\log{(\mathrm{O/H})}$ (see Appendix A).
Our work implies that rest-frame optical emission lines are necessary to
determine the metallicity. It is difficult to obtain reliable metallicity
estimates if only rest-frame UV spectra are present. However, we will further
investigate the UV sensitivity to metallicity in our future papers (Acharyya
et al., Kewley et al., Byler et al. in prep).

##  7 Discussion

In this section we compare our results based on the availability of different
parts of the spectra – UV, optical, UV-optical – in regards to the Bayesian
approach. We also discuss our results in the context of previous studies, the
implications of our work on upcoming telescopes, and potential caveats.

###  7.1 Comparison Between UV, Optical or UV-optical Bayesian Results

##### Ionization parameter:

Ionization parameter values are well constrained when only optical emission
lines are provided to IZIP, along with an user defined prior on $Z$ based on
[N ii]$\lambda$6584/Hα (Figure 4(b)). However, inferring $\log{(q)}$ from only
the optical lines or only the UV lines, in absence of a prior, show a broad
tail (small likelihood) of high $\log{(q)}$ values and hence put a poorer
constraint on $\log{(q)}$. The bias towards high $\log{(q)}$ is more
pronounced when the [O ii]$\lambda$3727 doublet is included in addition to the
UV lines, leading to an unconstrained PDF. The $\sim 0.2$ dex higher
$\log{(q)}$ inferred by using all the emission lines, compared to when only
the UV or optical lines are used, is not concerning given that the values
still agree within 1$\sigma$. The blue and black dotted histograms for
$\log{(q)}$ in Figures 4(b), 4(c) and 5(b) have considerable overlap to
demonstrate the agreement within the uncertainties. We conclude that adding
the rest-frame UV information to the existing optical spectra, puts a better
($\sigma\sim$0.2) constraint on $\log{(q)}$ overall, than when only the
optical spectra are used. The UV lines by themselves barely constrain
$\log{(q)}$, albeit with a larger uncertainty than when all lines are used.

##### ISM Pressure:

In the event of only the UV lines being available, the Bayesian approach would
give an ISM pressure value of $\log{(P/k)}$ = 8.90${}^{+0.10}_{-1.22}$
whereas, inclusion of the [O ii]$\lambda\lambda$3727,9 lines would yield a
lower $\log{(P/k)}$ = 7.88${}^{+0.31}_{-0.20}$). Both these values agree
(within 1$\sigma$ uncertainties) with $\log{(P/k)}$ = 8.8${}^{+0.2}_{-0.6}$
and $\log{(P/k)}$ = 7.4${}^{+0.6}_{-0.2}$ derived from the individual UV and
optical emission line diagnostics, respectively. This demonstrates that with a
rest-frame UV coverage up to the [O ii]$\lambda\lambda$3727,9 lines, it is
possible to effectively probe the different physical regions in the nebula
using the Bayesian approach. Suitably designed future surveys with the JWST
should take advantage of this fact.

##### Oxygen abundance:

When only optical lines are used, IZIP shows a double peaked PDF (Figure
5(b)), with the stronger peak being at 12+log(O/H) = 8.20${}^{+0.16}_{-0.10}$,
reflecting the fact that the optical collisionally excited emission lines are
doubled valued with metallicity. For the case where only UV lines are used
(Figure 4(c), 4(d)), the abundance PDF has a broad low metallicity tail, with
a peak at 12+log(O/H) = 8.04${}^{+0.13}_{-0.26}$. Providing IZIP with the [O
ii]$\lambda$3727 doublet along with the UV lines (Figure 4(d)) helps constrain
the oxygen abundance to a slightly ($\sim$0.07 dex) higher value 12+log(O/H) =
8.11${}^{+0.13}_{-0.26}$ but does not lead to better (narrower) constraints.
The broader 12+log(O/H) PDFs when only UV lines are used, imply that it is
difficult to infer the oxygen abundance with only the rest-frame UV lines
using a Bayesian approach.

###  7.2 Comparison With Other Work

In this section we compare the physical properties of RCS0327-E with those of
other galaxies from the literature, over a wide range of redshifts. However,
one should bear in mind that RCS0327-E is a $\sim$100 pc region of vigorous
star-formation whereas most of the literature data correspond to spatially
integrated spectra of entire galaxies that include both star-forming regions
as well as passively evolving stellar populations. As such, some ISM
parameters e.g. $\log{(q)}$ are expected to be higher in RCS0327-E than the
spatially averaged properties of other galaxies.

##### Electron temperature:

van Zee & Haynes (2006) studied a sample of nearby dwarf galaxies and reported
the ISM Te $\sim$ 1.3$\times$104 K. Jones et al. (2015) obtained Te
$\leq$2$\times$104 K for a sample of 32 $z\sim$0.8 galaxies. On the other
hand, Yuan & Kewley (2009) reported a Te = 2.3$\times$104 K for a lensed
galaxy at a redshift $z\sim$1.7. Christensen et al. (2012) estimated
$1.3\times$104 K $\gtrsim$ Te$\gtrsim 2.7\times$104 K for three lensed
galaxies in the redshift range $2\gtrsim z\gtrsim 3.5$. Further adding to the
sample of lensed galaxies, Stark et al. (2013) and James et al. (2014)
estimated Te $\sim 1.5$ and $1.7\times$104 K for two galaxies at $z\sim 1.4$.
Steidel et al. (2014) reported a mean Te of $\sim$ 1.3$\times$104 K for 3
KBSS-MOSFIRE galaxies at redshift $\sim 2$. Sanders et al. (2016b) estimated
an [O iii] Te = 1.4${}^{+0.20}_{-0.14}$ $\times$ 104 K, and Bayliss et al.
(2014) derived an upper limit on Te $\leq$1.4$\times$ 104 K for two galaxies
at $z\sim$3 and 3.6, respectively. Thus, RCS0327-E has a [O iii] Te
(1.2${}^{+0.2}_{-0.1}$ $\times 10^{4}$ K) similar to that of local galaxies,
and marginally lower than that of $z\sim 2-3$ galaxies.

##### Ionization parameter:

We derive a weighted mean $\log{(q)}$ = 7.77 $\pm$ 0.01 for RCS0327-E,
averaging over all the diagnostics. The mean $\log{(q)}$ agrees with the study
of $z$ $\sim$2-3 galaxies by Steidel et al. (2014) where they reported
$\log{(q)}$ values between 7.6 and 8.7 using CLOUDY photoionisation models.
Moreover, Kaasinen et al. (2018) analyse spectra from $\sim$200,000 SDSS
galaxies and derive a mean $\log{(q)}$ = 7.4 (using both KK04 and IZI),
further supporting the scenario that ionisation parameter at high ($z\sim 2$)
redshift is higher than that in the local universe (Steidel et al., 2014;
Strom et al., 2017; Kewley et al., 2015).

##### Electron density:

S16 measured a mean $\log{(n_{e}/\mathrm{cm}^{-3})}$ $\approx$ 2.4 for 225
star-forming galaxies at $z\sim 2.3$ with median stellar mass of M∗ = 1010 M⊙
and median SFR = 21.6 M⊙yr-1. In agreement with S16, Strom et al. (2017)
reported a mean $\log{(n_{e}/\mathrm{cm}^{-3})}$ $\approx$ 2.44 for another
high-redshift ($z\simeq 2-3$) sample of $\sim$ 380 galaxies, whereas Kennicutt
(1984) measured electron densities between 2
$\leq\log{(n_{e}/\mathrm{cm}^{-3})}\leq$ 3 for H ii regions in nearby
galaxies. The Strom et al. (2017) sample consists of star-forming galaxies
with M∗ = 109 – 1011.5 M⊙ and SFR = 3 – 1000 M⊙yr-1, which encompasses the
properties of RCS0327 (M∗ = 1010 M⊙ and SFR = 60 M⊙yr-1). The local galaxies
of the Kennicutt (1984) sample, however, are quiescent galaxies spanning M∗ =
108 – 1011 M⊙ and SFR = $\leq$ 1 M⊙yr-1. All of these studies used rest-frame
optical spectra to determine ne. RCS0327-E has a slightly higher electron
density ($\log{(n_{e})}\simeq$ 3.04) than both the local and $z\simeq 2-3$
galaxies. Kaasinen et al. (2017) study a sample of z$\sim$1.5 galaxies with
median M∗ = 107.5 M⊙ and SFR = 15 M⊙yr-1 and report $\log{(n_{e})}\simeq$
2.05. Their M∗\- and SFR-matched local analogs yield $\log{(n_{e})}\simeq$
1.99. RCS0327 being more massive and more rapidly star-forming than the
z$\sim$1.5 sample, has a higher ($\sim$1 dex) electron density.

Figure 6:  Redshift evolution of oxygen abundance in the stellar mass bin 109
M⊙ < M∗ < 1010 M⊙ , based on Yuan et al. (2013). Yuan et al. (2013) obtained
the SDSS (black) and DEEP2 (brown) data from Zahid et al. (2011) and the green
point from the UV-selected Erb et al. (2006) galaxies. The ‘Lensed’ (blue)
data denotes the mean abundance for the lensed sample in Yuan et al. (2013),
which also includes measurements from Yuan et al. (2011), Wuyts et al. (2012b)
and Richard et al. (2011). The 12+log(O/H) we derived using the [N
ii]$\lambda$6584/H$\alpha$ diagnostic of PP04 (PPN2) is shown as a star, while
all the other mean abundance values have been taken from Table 2 of Yuan et
al. (2013) and are denoted by circles. We show only the PPN2 diagnostic in
this plot in order to be consistent with Yuan et al. (2013). The uncertainties
quoted for the SDSS and DEEP2 samples by Yuan et al. (2013) is the 1$\sigma$
standard deviation of the mean from bootstrapping. Uncertainties thus derived
are too small to be visible on this scale, because the surveys comprise of a
large number of galaxies. In addition to Table 2 of Yuan et al. (2013), we
also include 12+log(O/H) measurement of the composite spectrum from Steidel et
al. (2016) as a purple circle.

##### Oxygen abundance:

Figure 6 shows the evolution of oxygen abundance as a function of redshift and
where, in that evolutionary track, RCS0327 lies with its stellar mass content
of $<$1010 M⊙ (Wuyts et al., 2014a). We adopt the mean abundance values in
Table 2 of Yuan et al. (2013) to compare with our work. The 12+log(O/H) = 8.32
$\pm$ 0.02 for RCS0327-E derived from the [N ii]$\lambda$6584/H$\alpha$
diagnostic of PP04 (PPN2) is shown Figure 6 (red star) in order to be
consistent with Yuan et al. (2013). The abundance of RCS0327-E (z$\sim$1.7),
based on PPN2, is also comparable to those of other high-redshift studies like
Jones et al. (2010, z$\sim$2.0), Shapley et al. (2004, z$\sim$2.0) and Steidel
et al. (2014, z$\sim$2.3) that report 12+log(O/H) $\simeq$8.4-8.5 using PPN2.
The metallicity of RCS0327-E is $\sim$0.2 dex lower than that of the local
galaxies of KINGFISH survey, which are reported to have 12+log(O/H) $\sim$ 8.5
(converted to PPN2 frame using KE08) by Kennicutt et al. (2011). Using
Bayesian methods, Kaasinen et al. (2018) find 12+log(O/H) = 8.7 for local
galaxies and 12+log(O/H) = 8.0 for redshift z$\sim$1.5 galaxies, in the mass
bin 9 < $\log{(M_{*}/M_{\bigodot})}$ < 10\. Bayesian (using IZIP) abundance
estimates for RCS0327-E are $\sim$0.2 dex lower and $\sim$0.5 dex higher than
the low and high redshift measurements of Kaasinen et al., respectively. Thus,
RCS0327-E is consistent with the trend of decreasing oxygen abundance with
redshift.

###  7.3 Implications for JWST

In this work, we have used the O iii]$\lambda\lambda$1660,6 lines in the UV
(along with [O iii]$\lambda$5007) to directly determine the oxygen abundance
from Te. We have also inferred the abundance using a Bayesian approach by
providing the UV line and [O iii]$\lambda$5007 to IZIP (see Appendix A). From
both these tests we conclude that it is difficult to reliably estimate
metallicity with only the rest-frame UV lines – the [O iii]$\lambda$5007 line
is required to break the degeneracy. JWST/NIRSpec will be able to
simultaneously capture the [O iii]$\lambda$5007 line and the [O
iii]$\lambda\lambda$1660,6 doublet within redshift $3.5\lesssim z\lesssim 9$,
thereby making it possible to obtain reliable estimates of oxygen abundance at
such high redshifts. Currently, rest-frame UV nebular emission line
diagnostics for oxygen abundance are scarce. However, K19b and Byler et al.
(in prep) investigate the diagnostic power of the UV lines, in detail.

We demonstrate that $\log{(q)}$ can be determined using only rest-frame UV
spectra, provided at least one of [C iii]$\lambda\lambda$1906,8/[C
ii]$\lambda$1335, [C iii]$\lambda\lambda$1906,8/[C ii]$\lambda$2325, or [O
iii]$\lambda\lambda$1660,6/[O ii]$\lambda$2470 ratios is available (Section
5.3.2). These ratios will be within the wavelength coverage of JWST/NIRSpec
for redshifts above $z\sim$ 3.5, 1.6, and 2.6, respectively. However, the [C
ii]$\lambda$1335 group is very weak ($\sim$10-3 \- 10${}^{-4}\times$ H$\beta$
flux for $\log{(q)}$=8.0 and $Z$=0.4 $Z_{\bigodot}$). The unresolved [C
ii]$\lambda$2325 group of lines is very closely spaced in wavelength, thereby
making it difficult to estimate the fluxes for moderate resolution spectra.
Large uncertainties stemming from line blending could potentially be
translated into uncertainties in $\log{(q)}$.

###  7.4 IZIP with and without priors

A point to note about the Bayesian inference method is that collisionally
excited emission lines are double valued with metallicity. IZIP, by default,
is unable to make an informed choice between the two branches based on
specific emission line ratios, and yields bimodal PDFs in many such cases. For
instance, one could not use the [N ii]$\lambda$6584/H$\alpha$ ratio to break
the degeneracy while using the R23 metallicity indicator. Instead, IZIP would
use the R23 line flux information and weigh them same as all the other lines,
thus leading to a double peaked metallicity PDF. This is not wrong from a
Bayesian perspective, because the models indeed predict two probable values of
$\log{(\mathrm{O/H})}$ given the emission line ratios. However, the observed H
ii region can only have one or the other abundance; this is where an observer
would use other emission line information to decide between the two branches.
To facilitate this, IZIP takes into account user defined priors on the
physical parameters while computing the posterior distribution. Providing
user-defined priors to IZIP based on certain line ratios can help break the
degeneracy.

We impose a top-hat prior on the [N ii]$\lambda$6584/H$\alpha$ ratio, where
available (discussed in Section 6), to help select the relevant metallicity
branch. In the absence of a prior, we infer 12+log(O/H) =
8.53${}^{+0.1}_{-0.07}$ when the full suite of UV and optical spectra is used
(Figure 5(a)); and 12+log(O/H) = 8.20${}^{+0.16}_{-0.1}$ when only the rest-
frame optical lines are used (Figure 5(b)). This discrepancy in
$\log{(\mathrm{O/H})}$ based on whether UV lines are included could be
potentially due to inconsistencies in photoionzation models, as discussed in
Section 7.6. On using the prior on [N ii]/H$\alpha$, the oxygen abundance is
well constrained to a peak value of 12+log(O/H) = 8.56${}^{+0.06}_{-0.03}$
(Figure 4(a)) and 12+log(O/H) = 8.53${}^{+0.03}_{-0.03}$ (Figure 4(b)) by
using UV-optical and only optical lines, respectively. In Figure 4(b), the
top-hat prior used to derive the posterior probability is shown by a black
dashed line in the 1D PDF of the metallicity (top) panel. Using the [N
ii]/H$\alpha$ prior leads to narrow constraints, as expected, and closer
agreement between the abundances measured with different set of emission
lines. Comparing Figure 4(a) with 5(a) and Figure 4(b) with 5(b) reveals that
usage of a prior on abundance, also has considerable impact on $\log{(q)}$ and
$\log{(P/k)}$. The high $\log{(q)}$ and low $\log{(P/k)}$ solution is
eliminated on using the prior, leading to much tighter constraints on both
these physical properties. We therefore recommend employing priors where
possible (e.g. [N ii]/H$\alpha$ available), to break the degeneracy while
using IZIP.

###  7.5 ISM pressure – UV vs optical

Figure 7:  Normalised strength of different emission line species as a
function of normalised shell thickness. The high ionisation species C iii and
Si iii originate from inner, denser zones of the nebula whereas the lower
ionisation O ii and S ii ions trace the less dense peripheral part of the
ionised nebular shell. The curves correspond to a MAPPINGS photoionisation H
ii region model, assuming plane parallel geometry, for 12+log(O/H) = 8.23 and
$\log{(q)}$ = 8.0.

ISM pressures measured using the new UV diagnostics are $\sim 1.5$ dex higher
than those determined using optical line ratios. This discrepancy can be
attributed to the fact that the different emission lines originate from
different regions of the ionised ISM. As K19a point out, the regions closer to
the ionising source receive a greater proportion of the UV photon flux
resulting in a greater fraction of higher ionisation species. Consequently,
the higher ionisation emission lines in the rest-frame UV probe the inner,
high pressure region of the nebula, whereas the lower ionisation lines in the
optical probe the outer, lower pressure zone. This is due to the different
sensitivity of various emission species to different density regimes depending
on critical density. It is therefore unsurprising that we would derive a
higher pressure and higher density with the UV diagnostics.

Figure 7 shows the emissivity of various line species as a function of the
depth into the nebula. Higher ionisation species (rest-frame UV diagnostics)
used in this work – [C iii] and [Si iii] are produced throughout the H ii
region, and concentrated in the inner regions ($\lesssim$70% of nebular shell
radius). In contrast, the lower ionisation species (rest-frame optical
diagnostics) used in this work – [O ii] and [S ii] are primarily produced in
the outer parts of the nebula, specifically, from $\sim$80% and $\sim$100% of
the nebular shell radius. We might expect that the [C iii] and [Si iii]
species represent the average physical conditions within $\lesssim$70% of
nebular shell radius, whereas the [O ii] and [S ii] species probe the H ii
region at 80% radius and extreme outskirts, respectively.

Our work demonstrates that, given only the rest-frame UV spectra, it is
possible to derive $\log{(P/k)}$ (or ne) using the SEL diagnostics. Whether
the Bayesian methods can reliably constrain $\log{(P/k)}$ using only the UV
lines remains unclear. However, one has to bear in mind that the $\log{(P/k)}$
(or ne) derived from the SEL diagnostics would be representative of the inner
regions of the nebula and consequently would be biased towards higher values.
There is no clear choice for a “better" diagnostic between the different UV
and optical diagnostics. It is simply a case of different emission line
species probing different physical regions in the nebulae.

Rest-frame optical spectra, when available, probe the outskirts of the nebula.
Additionally including rest-frame UV spectra probes the inner physical regions
of the nebula as well. K19a suggest using the Si iii$\lambda$33$\mu m$ and Cl
iii$\lambda$5518 Å lines as diagnostics that are more representative of the
entire nebula because these emissions originate fairly uniformly throughout
the nebula. However, we do not detect either of these emission lines in
RCS0327-E.

The measured electron densities are not representative of the entire nebula
either. Different emission lines have different critical densities, making
them sensitive to different density regimes. K19a point out that the density
structure of an H ii region can be quite complex. Moreover, clumpy star
formation knots have been observed in RCS0327. As such, it is more sensible to
measure the ISM pressure in this case than measuring a constant density which
may not be representative of the entire ionised ISM.

The above discussion naturally raises a general (not specific to ISM pressure)
concern regarding the combined usage of the emission lines originating in
inner nebular regions with those emitted from the outer regions. While this is
a valid concern, the extent to which such combined usage would make a
difference depends on the particular emission line ratio and the ISM property
involved. In case of $\log{(P/k)}$ and ne, all the lines involved in a
particular diagnostic are of the same ionisation species which originate at
similar physical regions, but different diagnostics have lines originating
from different regions, indicating that each diagnostic clearly probes a
different nebular region. As such, it is not sensible to combine the two
groups – UV and optical diagnostics – for pressure and density measurements
and should be considered as probes of distinctly different nebular regions.
For abundance and ionisation parameter however, some diagnostics involve
emission lines of different ionisation species e.g. the R23, O32 and Ne3O2
ratios (see Table 4). The [O iii] lines originate throughout the nebula,
implying that the R23 and O32 diagnostics are representative of the entire gas
cloud. Thus, it is sensible to compare diagnostics involving [O iii] with
other diagnostics for 12+log(O/H) or $\log{(q)}$. It is difficult to compare
Ne3O2 index with other diagnostics as the former yields systematically
different values of 12+log(O/H) or $\log{(q)}$, due to reasons discussed in
Section 5. Moreover, these diagnostics are based on models for a single H ii
region. whereas in reality, a star-forming knot would comprise of several such
H ii regions. Thus, we probe the emission-weighted average properties of an
ensemble of H ii regions. We conclude that, it is possible to combine rest-
frame UV and optical diagnostics; modulo the fact that they probe different
physical regions in the case of pressure and density. We refer the reader to
K19bfor a review of ionisation parameter and metallicity diagnostics.

###  7.6 Caveats

One potential reason for the discrepancy between the ISM pressure ($\sim$1.5
dex) and oxygen abundance ($\sim$0.5 dex) derived from UV and optical emission
lines, is inconsistencies in the inputs to the H ii model grid. The currently
available stellar atmosphere libraries and stellar evolutionary tracks are
based on different abundance standards and do not agree. The O-star models are
only sparsely represented in the available model sets. Moreover, the stellar
population synthesis model Starburst99 (Leitherer et al., 1999) linearly
interpolates between these sparsely sampled libraries.

A second, but less concerning, source of uncertainty comes from the scatter in
the available atomic data, which leads to a variation in the ionising energies
of crucial nebular emission lines (up to a factor of 2). Thus, there is an
intrinsic uncertainty in the input stellar spectra. For instance, the stellar
spectra obtained from Starburst99 in different spectral resolution modes do
not agree in the rest-frame UV regime. Nevertheless, in the absence of more
extensive stellar model sets, the Starburst99 spectra are used by the
photoniozation code MAPPINGS v5.1 (Dopita et al., 2013) as the driving source
of the radiation field leading to all the high ionisation emission lines. The
inherent discrepancy in Starburst99 translates to uncertainties in output UV
and optical emission line fluxes in the H ii region model grids of MAPPINGS.
The MAPPINGS model grids, in turn, have been used to calibrate the K19a
diagnostics. Thus, there are inherent discrepancies in the rest-frame UV
diagnostics. Moreover, our Bayesian analysis is also based on the MAPPINGS
model grids. This could potentially lead to discrepancies in the metallicity
determined using only UV or only optical emission lines.

Different stellar population synthesis (SPS) models incorporating different
physics could potentially have an impact on the ionising stellar spectra used
as an input to the photoionisation models, but such a comparative study is
beyond the scope of this paper. However, D’Agostino et al. (2019, MNRAS, in
press) have recently compared different SPS models and concluded that for most
cases the emission line ratios do not change considerably. The ionising
spectra were found to be somewhat senstive to the cluster age and SPS codes,
extremely sensitive ($\sim 8$ orders of magnitude) to stellar evolutionary
tracks, with very little ($\lesssim$ 2%) dependence on the stellar atmospheres
and the inclusion of binaries. D’Agostino et al. however, have not
investigated the impact of different functional forms for the initial mass
function (IMF) on the spectra.

Work is currently in progress to combine stellar atmosphere models and
evolutionary tracks with improved, stochastic stellar population synthesis
codes. Once complete, we will have state-of-the-art diagnostics with self-
consistent inputs and will be able to determine how much of the observed
discrepancy in physical parameters stems from the observed target itself.
Given all uncertainties in the models, we can constrain the physical
properties remarkably well within the observed level of agreement. Moreover,
the caveats discussed in this section are likely to have a smaller effect on
the $\log{(P/k)}$ measurements than the fact that UV and optical lines probe
different physical nebular regions.

##  8 Summary and Conclusions

We measure equivalent widths and fluxes of the emission lines in the rest-
frame UV and optical spectra of RCS0327-E. By applying the full suite of new
and existing UV and optical strong emission line (SEL) diagnostics on the
dereddened fluxes, we determine the ISM properties of RCS0327-E. We show that
it is possible to infer some of the ISM properties – ionisation parameter,
electron density and ISM pressure in the inner nebular region – with only the
rest-frame UV emission lines. The rest-frame optical spectra better constrain
the abundance and probe the pressure and density at the outer nebular regions.

We develop a new extension of IZI, called IZIP, which uses Bayesian inference
method to simultaneously infer 12+log(O/H), $\log{(q)}$, and $\log{(P/k)}$
values. Given a theoretical model grid of emission line fluxes and a set of
observed emission lines, IZIP constrains the three physical parameters
simultaneously. We run IZIP with four different sets of emission lines – all
available emission lines, only the rest-frame UV lines, only the optical lines
and UV+[O ii]$\lambda\lambda$3727,9 lines – to mimic observations with
different rest-frame wavelength coverage.

By comparing the individual emission line diagnostics and the results from the
four different configurations of IZIP, we draw the following conclusions:

  1. 1.

The rest-frame UV emission lines infer $\sim$1.5 dex higher ISM pressures than
the optical emission lines, because they probe different physical regions. The
latter probe the outskirts of the nebula whereas the higher ionisation UV
species probe the inner, denser regions. Because it is directly related to ISM
pressure, the electron density also exhibits the same behavior.

  2. 2.

The rest-frame UV emission lines used in this work (see Table 6) are
insufficient to accurately constrain the oxygen abundance for RCS0327-E. The
[O iii]$\lambda$5007 emission line, used along with O iii]$\lambda\lambda$
1660,6, [O ii]$\lambda\lambda$3727,9 and H$\beta$, helps constrain the oxygen
abundance through the direct (Te) method, and is within range of JWST/NIRSpec
wavelength coverage for redshifts $z\,\lesssim$9.

  3. 3.

If only rest-frame UV spectra are available, it is possible to derive the
ionisation parameter $\log{(q)}$ as long as at least one of the ([C
iii]$\lambda$1906 + $\lambda$1908)/[C ii]$\lambda$1335, ([C iii]$\lambda$1906
+ $\lambda$1908)/[C ii]$\lambda$2325 or ([O iii]$\lambda$1660 +
$\lambda$1666)/[O ii]$\lambda$2470 ratios are available. JWST/NIRSpec will be
able to capture these ratios in the redshift ranges
3.5$\lesssim\,z\,\lesssim$27, 2.1$\lesssim\,z\,\lesssim$22 and
2.6$\lesssim\,z\,\lesssim$20, respectively.

  4. 4.

Joint Bayesian analysis is useful to determine $\log{(q)}$ and $\log{(P/k)}$
when the rest-frame optical lines are available and yield results comparable
to SEL diagnostics. Bayesian techniques have the capability to explore non-
trivial topology in the PDFs of the inferred parameters e.g. multiple peaks
and assymetry. However, when only the UV lines listed in Table 6 are
available, it is difficult to constrain the oxygen abundance using Bayesian
methods and currently available photoioniation grids. Inclusion of [O
ii]$\lambda\lambda$3727,9 with the UV lines does not make a noticeable
difference either. This is a potential problem for JWST at very high redshifts
if the [O iii]$\lambda$5007 line is not available to break the degeneracy.

  5. 5.

Given rest-frame UV coverage and the optical [O ii]$\lambda\lambda$3727,9
doublet, it is possible to effectively probe the ISM pressure in different
physical regions in the nebula using the Bayesian approach. Future surveys
with the JWST will be designed to take advantage of this fact.

In summary, we have demonstrated the power of rest-frame UV emission line
diagnostics used in conjunction with rest-frame optical diagnostics, for
inferring the ionised gas properties of moderate to high redshift galaxies.
The ionisation parameter can be determined with only UV lines, whereas the
electron density and ISM pressure additionally require the [O
ii]$\lambda\lambda$3727,9 doublet. We find the UV diagnostics used in this
work alone cannot reliably constrain the electron temperature and oxygen
abundance. This work paves the way for upcoming large telescopes (e.g. JWST,
GMT, TMT, ELT) which will carry out rest-frame UV spectroscopic studies of
galaxies out to redshifts exceeding 10.

## Acknowledgements

We thank the anonymous referee for their thorough review and thoughtful
comments that greatly improved this paper. This work includes data gathered
with the 6.5 meter Magellan Telescopes located at Las Campanas Observatory,
Chile. Some of the data presented herein were obtained at the W.M. Keck
Observatory, which is operated as a scientific partnership among the
California Institute of Technology, the University of California and the
National Aeronautics and Space Administration. The Observatory was made
possible by the generous financial support of the W.M. Keck Foundation. We
acknowledge the very significant cultural role and reverence that the summit
of Mauna Kea has always had within the indigenous Hawaiian community. We are
most fortunate to have the opportunity to conduct observations from this
mountain. Parts of this research were conducted by the Australian Research
Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO
3D), through project number CE170100013. L.J.K. gratefully acknowledges the
support of an ARC Laureate Fellowship (FL150100113). C. F. acknowledges
funding provided by the Australian Research Council (Discovery Projects
DP150104329 and DP170100603, and Future Fellowship FT180100495), and the
Australia-Germany Joint Research Cooperation Scheme (UA-DAAD). G. B. is
supported by CONICYT/ FONDECYT, Programa de Iniciación, Folio 11150220. This
research has made use of the following: NASA’s Astrophysics Data System
Bibliographic Services, the NASA/IPAC Extragalactic Database (NED), Astropy –
a community-developed core Python package for Astronomy (The Astropy
Collaboration et al., 2013), scipy (Jones et al., 01), and pandas (Mckinney,
2010) packages.

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##  Appendix A Additional IZIP analyses

In this section we present the additional tests we conducted with IZIP. No
user defined priors are used for these analyses in order to isolate the
effects of the individual lines on the Bayesian estimates. Moreover, for
computational efficiency, only 10 realisations (refer to Section 6) were
performed for each of the cases discussed here, unlike the 100 realisations