11institutetext: Department of Astronomy, Kyoto University, Kitashirakawa-
Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan
11email<EMAIL_ADDRESS>22institutetext: Academia Sinica
Institute of Astronomy and Astrophysics, 11F of Astronomy-Mathematics
Building, AS/NTU, No.1, Section 4, Roosevelt Road, Taipei 10617, Taiwan
33institutetext: Research Center for Space and Cosmic Evolution, Ehime
University, 2-5 Bunkyo-cho, Matsuyama, Ehime 790-8577, Japan 44institutetext:
Dipartimento di Fisica e Astronomia, Università di Bologna, via Gobetti 93/2,
40129 Bologna, Italy 55institutetext: INAF- Osservatorio di Astrofisica e
Scienza dello Spazio di Bologna, via Gobetti 93/3, 40129 Bologna, Italy
66institutetext: Max-Planck-Institut für Extraterrestrische Physik (MPE),
Giessenbachstrasse 1, 85748 Garching bei München, Germany 77institutetext:
Leibniz-Institut für Astrophysik, Potsdam (AIP), An der Sternwarte 16, 14482
Potsdam, Germany 88institutetext: Astronomical Institute, Tohoku University,
6-3 Aramaki, Aoba-ku, Sendai, Miyagi 980-8578, Japan 99institutetext:
Department of Astrophysical Sciences, Princeton University, Princeton, NJ
08540, USA 1010institutetext: Faculty of Science and Engineering, Kindai
University, Higashi-Osaka, 577-8502, Japan 1111institutetext: Department of
Economics, Management and Information Science, Onomichi City University,
Hisayamada 1600-2, Onomichi, Hiroshima 722-8506, Japan

# The eROSITA Final Equatorial-Depth Survey (eFEDS):

An X-ray bright, extremely luminous infrared galaxy at $z$ = 1.87

Yoshiki Toba 112233 Marcella Brusa 4455 Teng Liu 66 Johannes Buchner 66 Yuichi
Terashima 33 Tanya Urrutia 77 Mara Salvato 66 Masayuki Akiyama 88 Riccardo
Arcodia 66 Andy D. Goulding 99 Yuichi Higuchi 1010 Kaiki T. Inoue 1010
Toshihiro Kawaguchi 1111 Georg Lamer 77 Andrea Merloni 66 Tohru Nagao 33
Yoshihiro Ueda 11 Kirpal Nandra 66

In this study, we investigate the X-ray properties of WISE J090924.01+000211.1
(WISEJ0909+0002), an extremely luminous infrared (IR) galaxy (ELIRG) at
$z_{\rm spec}$= 1.871 in the eROSITA final equatorial depth survey (eFEDS).
WISEJ0909+0002 is a WISE 22 $\mu$m source, located in the GAMA-09 field,
detected by eROSITA during the performance and verification phase. The
corresponding optical spectrum indicates that this object is a type-1 active
galactic nucleus (AGN). Observations from eROSITA combined with Chandra and
XMM-Newton archival data indicate a very luminous ($L$ (2–10 keV) = ($2.1\pm
0.2)\times 10^{45}$ erg s-1) unobscured AGN with a power-law photon index of
$\Gamma$ = 1.73${}_{-0.15}^{+0.16}$, and an absorption hydrogen column density
of $\log\,(N_{\rm H}/{\rm cm}^{-2})<21.0$. The IR luminosity was estimated to
be $L_{\rm IR}$ = (1.79 $\pm$ 0.09) $\times 10^{14}\,L_{\sun}$ from spectral
energy distribution modeling based on 22 photometric data (X-ray to far-IR)
with X-CIGALE, which confirmed that WISEJ0909+0002 is an ELIRG. A remarkably
high $L_{\rm IR}$ despite very low $N_{\rm H}$ would indicate that we are
witnessing a short-lived phase in which hydrogen gas along the line of sight
is blown outwards, whereas warm and hot dust heated by AGNs still exist. As a
consequence of eROSITA all-sky survey, $6.8_{-5.6}^{+16}\times 10^{2}$ such
X-ray bright ELIRGs are expected to be discovered in the entire extragalactic
sky ($|b|>10\degr$). This can potentially be the key population to constrain
the bright-end of IR luminosity functions.

###### Key Words.:

Galaxies: active – X-rays: galaxies – Infrared: galaxies – quasars:
individual: WISE J090924.01+000211.1

##  1 Introduction

Galaxies with infrared (IR) luminosity ($L_{\rm IR}$111Empirically, $L_{\rm
IR}$ is defined as the luminosity integrated over a wavelength range of 8–1000
$\mu$m (e.g., Sanders & Mirabel, 1996; Chary & Elbaz, 2001). However, since
this definition includes contributions from stellar emissions, we do not adopt
any boundary for the integration range on the wavelength. Instead, we employed
a physically oriented approach to estimate $L_{\rm IR}$ in this work (see
Section 3.1).) greater than 1013$L_{\sun}$ and 1014$L_{\sun}$ have been termed
as hyper-luminous IR galaxies (HyLIRGs; Rowan-Robinson, 2000) and extremely-
luminous IR galaxies (ELIRGs: Tsai et al., 2015), respectively. The IR
luminosity can arise from active galactic nuclei (AGN) and star formation (SF)
activity. During galactic merger events, these luminous IR galaxy populations
may correspond to a phase in which the AGN and SF activity reach a peak,
shrouded by dense gas and dust clouds (e.g., Hopkins et al., 2008; Narayanan
et al., 2010; Toba et al., 2015; Blecha et al., 2018). Systematic
investigations on Hy/ELIRGs are required to cement the understanding on the
co-evolution of such galaxies and their supermassive black holes (SMBHs)
during the peak of AGN and SF activity.

Since the advent of IR satellites such as the Wide-field Infrared Survey
Explorer (WISE; Wright et al., 2010) and the Herschel Space Observatory
(Pilbratt et al., 2010), a large number of Hy/ELIRGs have been discovered
(e.g., Casey et al., 2012; Weedman et al., 2012; Leipski et al., 2014; Toba &
Nagao, 2016; Duras et al., 2017). However, accurate characterization of the
properties of ELIRGs based on multi-wavelength spectral energy distribution
(SED) is limited (see e.g., Toba et al., 2018, 2020b). In particular, X-ray
properties of ELIRGs such as hydrogen column density ($N_{\rm H}$) and
absorption-corrected X-ray luminosity are poorly understood.

The extended ROentgen Survey with an Imaging Telescope Array (eROSITA; Merloni
et al., 2012; Predehl et al., 2021) has recently probed the X-ray properties
of ELIRGs. eROSITA is the primary instrument on the Spectrum-Roentgen-Gamma
(SRG) mission, which was successfully launched on July 13, 2019. Since ELIRGs
are a spatially rare population and may be X-ray faint due to the obscuration,
the eROSITA all-sky survey with high X-ray sensitivity is an ideal platform to
investigate the X-ray properties of ELIRGs. In this letter, we report the
eROSITA view of ELIRG at $z_{\rm spec}$ = 1.871, WISE J090924.01+000211.1
(hereafter WISEJ0909+0002), in the GAMA-09 field observed by eROSITA. We
employed the performance and verification (PV) phase program called eROSITA
Final Equatorial Depth Survey (eFEDS: Brunner et al. in prep.) in our
observation. The eFEDS catalog contains approximately 28,000 X-ray point
sources detected over an area of 140 deg2 in a single broad band, with a
5$\sigma$ sensitivity of $f_{\rm 0.3-2.3\leavevmode\nobreak\ keV}\sim 9\times
10^{-15}$ erg s-1 cm-2. We discuss the AGN and its host properties of
WISEJ0909+0002 based on SED fitting and X-ray spectral analysis. Throughout
this letter, the adopted cosmology is a flat universe with $H_{0}$ = 70 km s-1
Mpc-1, $\Omega_{M}$ = 0.3, and $\Omega_{\Lambda}$ = 0.7, and the initial mass
function of Chabrier (2003) is assumed.

##  2 Data and analysis

###  2.1 The candidate ELIRG WISEJ0909+0002

WISEJ0909+0002 is an ELIRG candidate detected by eROSITA with a positional
uncertainty of 2.0″. It is the only ELIRG candidate detected by eROSITA in a
sample of $\sim$ 300 WISE 22 $\mu$m-selected sources222The corresponding
signal-to-noise ratio of the flux density at 22 $\mu$m is greater than 5.0.
Possible stars and artifacts were removed from the sample. in the eFEDS area
with multi-wavelength data ($\sim$50 deg2) (Toba et al. in prep.). This object
is included in the eFEDS X-ray point source catalog with optical-to-mid-IR
(MIR) counterparts (Salvato et al. in prep.) called eFEDSJ090924.0+000209.9.
The optical-MIR counterparts were identified using data from the DESI Legacy
Imaging Surveys Data Release 8, (LS8: Dey et al., 2019) which includes $g$,
$r$, and $z$ data from the Dark Energy Camera Legacy Survey (DECaLS), and 3.4,
4.6, 12, and 22 $\mu$m data from unWISE (Lang, 2014; Lang et al., 2016). The
optical-MIR counterparts are determined using a Bayesian statistics based
algorithm (NWAY Salvato et al. 2018) and the maximum likelihood method (see
Salvato et al. for details), and the multi-wavelength data were compiled (see
Toba et al. for details). First, we used absorption-corrected fluxes in the
0.5–2.0 and 2.0–10 keV bands obtained via X-ray spectroscopy (see Sect. 2.3).
The UV data were adopted from the Galaxy Evolution Explorer (GALEX: Martin et
al., 2005; Bianchi, 2011). Other optical ($u$, $i$, and $Y$) and near-IR (NIR)
data ($J$, $H$, and $K_{\rm s}$) were adopted from the kilo-degree survey
(KiDS) DR4 (Kuijken et al., 2019). The far-IR (FIR) data (100, 160, 250, 350,
and 500 $\mu$m) were adopted from the H-ATLAS DR1 (Valiante et al., 2016). The
photometry of WISEJ0909+0002 is presented in Table 2 in Appendix A. We could
confirm the classification of WISEJ0909+0002 as an ELIRG from the SED fitting
(see Sects. 2.2 and 3.1), and the redshift was determined spectroscopically
using the Sloan Digital Sky Survey (SDSS; York et al., 2000) to be $z_{\rm
spec}$ = 1.871. The observed properties of WISEJ0909+0002 are summarized in
Table 1.

Table 1: Observed Properties of WISEJ0909+0002. WISE J090924.01+000211.1 |
---|---
R.A. (SDSS) [hh:mm:ss, J2000.0] | 09:09:24.01
Decl. (SDSS) [dd:mm:ss, J2000.0] | +00:02:11.06
Redshift (SDSS) | 1.871 $\pm$ 0.0001
SED fitting with X-CIGALE (Sect. 3.1) |
$E(B-V)_{*}$ | 0.13
$M_{*}$ [$M_{\sun}$] | (4.94 $\pm$ 1.39) $\times 10^{11}$
SFR [$M_{\sun}$ yr-1] | (3.85 $\pm$ 0.06) $\times 10^{3}$
$L_{\rm IR}$ [$L_{\sun}$] | (1.79 $\pm$ 0.09) $\times 10^{14}$
$\nu L_{\nu}$ (6 $\mu$m) [erg s-1] | (3.53 $\pm$ 0.80) $\times 10^{46}$
X-ray spectral analysis (Sect. 3.2) |
$\log\,(N_{\rm H}/{\rm cm}^{-2})$ | $<21.0$
power-law slope ($\Gamma$) | 1.73${}_{-0.15}^{+0.16}$
$L_{\rm 2-10\ keV}$ [erg s-1] | $(2.1\pm 0.2)\times 10^{45}$
BH properties (Sect. 3.2) |
$L_{\rm bol}$ [erg s-1] | ($4.3\pm 0.6)\times 10^{47}$
$M_{\rm BH}$ [$M_{\sun}$] | ($7.4\pm 0.3)\times 10^{9}$
$\lambda_{\rm Edd}$ | $0.4\pm 0.1$
$\kappa_{\rm X}$ (=$L_{\rm bol}/L_{\rm 2-10\ keV})$ | $(2.0\pm 0.3)\times 10^{2}$

###  2.2 SED fitting with X-CIGALE

To estimate the IR luminosity with high precision, we conducted SED fitting by
considering the energy balance between the UV/optical and IR ranges. We
employed the new version of the Code Investigating GALaxy Emission (CIGALE;
Burgarella et al., 2005; Noll et al., 2009; Boquien et al., 2019) called
X-CIGALE333https://gitlab.lam.fr/gyang/cigale/tree/xray (Yang et al., 2020),
which enables SED fitting from the X-ray to radio range. In X-CIGALE, we are
able to handle many parameters, such as the star formation history (SFH),
single stellar population (SSP), attenuation law, AGN emission, dust emission,
radio synchrotron emission, and AGN X-ray emission (see e.g., Boquien et al.,
2014, 2016; Buat et al., 2015; Lo Faro et al., 2017; Toba et al., 2019b,
2020c). A brief description of each parameter is provided in Appendix B and
the parameter ranges used in the SED fitting are presented in Table 3 (see
also Boquien et al., 2019; Yang et al., 2020, and references therein).

Figure 1: The best-fit SED of WISEJ0909+0002. The black points are photometric
data, and the gray solid line represents the resultant best-fit SED. The inset
figure shows the SED at 0.1–500 $\mu$m, where the contributions from the
stellar, nebular, AGN, and SF components to the total SED are shown as blue,
green, yellow, and red lines, respectively. The relative residual (defined as
the best-fit value minus data with respect to data) are shown at the bottom,
where the black line represents the case when the residual is zero.

###  2.3 X-ray spectral analysis

WISEJ0909+0002 was observed by eROSITA with a net exposure time of 2.3 ks in
2019. The eROSITA X-ray spectrum of WISEJ0909+0002 is extracted using eSASS
srctool v1.60444This software derives source level products from calibrated
eROSITA event files and their ancillary meta-data. See
https://erosita.mpe.mpg.de/eROdoc/tasks/srctool_doc.html for details. within a
circle of 50″in diameter, and the background spectrum is extracted inside an
annulus between 114″and 623″, excluding all nearby sources (see Liu et al. in
prep., in details). This source was observed by Chandra in 2004 (Observation
ID = 5703), and XMM-Newton in 2013 (Observation ID = 0725310143). We extracted
the Chandra and XMM spectra to perform a joint fitting of the eROSITA spectrum
in the 0.2–8 keV band (90 net counts), Chandra spectrum in the 0.5–7 keV (17
net counts), and the XMM EPIC-pn, EPIC-MOS1, and EPIC-MOS2 spectra in the
0.5–8 keV band (with net counts of 33, 11, and 14, respectively).

The Chandra data are processed with CIAO 4.12 and CALDB 4.9.0. The background
is stable during observation with a net exposure time of 1.3 ks. The source
spectrum is extracted from a circular region centered at the WISEJ0909+0002
with a radius of 4$\aas@@fstack{\prime\prime}$9\. The background spectrum is
obtained from an annular region around the source. The XMM-Newton data are
processed with the science analysis system version 18.0.0, combined with the
current calibration files as of June 2019. The source spectra are extracted
from a circular region with a radius of 17$\aas@@fstack{\prime\prime}$5\. The
background spectra are adopted from a nearby source free region. The
background was stable during the observation, and the net exposure times for
the PN, MOS1, and MOS2 detectors were 2.5, 2.7, and 2.7 ks, respectively.

We model the source with a power-law model with galactic absorption ($N_{\rm
H}=2.58\times 10^{20}$ cm-2) and intrinsic absorption
(constant*TBabs*zTBabs*powerlaw in xspec). Three constant factors (0.65 for
Chandra and 1.36 for XMM) were used to account for the potential variability
and calibration difference between eROSITA, Chandra, and XMM (but see Appendix
D). We also fitted the spectra with the same model, setting all the constant
factors to unity, in order to measure the averaged luminosity. Fixing these
factors has no impact on the spectral shape parameters, because the XMM and
Chandra spectra have much lower counts than that of eROSITA. To derive
constraints on the parameters, we employed the Bayesian method BXA (Buchner et
al., 2014; Buchner, 2019), adopting wide uniform priors for the power-law
slope and the constant factors, and log-uniform priors for the column density
and normalization. In Table 1, we list the posterior median and 68% confidence
interval of the parameters, as well as the absorption-corrected rest-frame
2–10 keV luminosity (see also Sect.3.2).

##  3 Results and discussion

###  3.1 IR luminosity and host properties

Fig. 1 shows the best-fit SED of WISEJ0909+0002 (see also Fig. 3 for multi-
wavelength images). We find that the observed data points (except for the
observed-frame optical data) are well explained by the combination of stellar,
nebular, AGN, and SF components. The best-fit SED, particularly of the AGN
accretion disk, underestimates the flux densities in the rest-frame 0.1–1
$\mu$m by 0.2–0.3 dex, which induces a relatively large reduced $\chi^{2}$ (=
3.7). This is partially due to the fact that X-CIGALE fixes a slope of the
power-law disk component (i.e., the optical spectral index, $\alpha_{\rm
opt}$) to be 0.5 in $\lambda$–$f_{\nu}$ space where $\alpha_{\rm opt}$ is
defined in the wavelength rage 0.125–1 $\mu$m (see Equation 9 in Yang et al.
2020). However, $\alpha_{\rm opt}$ for WISEJ0909+0002 inferred from the SDSS
spectrum and NIR data is $\sim 0.2$555This value is within a dispersion of
$\alpha_{\rm opt}$ distribution for SDSS quasars (e.g., Vanden Berk et al.,
2001)., which is flatter than that employed in X-CIGALE. The relative residual
defined as (data – best-fit)/data is also plotted in the bottom panel of
Fig.1. In the optical–NIR region, the relative residual is slightly larger at
shorter wavelengths, which can support the above interpretation.

The resultant IR luminosity is $L_{\rm IR}$ = (1.79 $\pm$ 0.09) $\times
10^{14}$ $L_{\sun}$, which establishes the classification of WISEJ0909+0002 as
an ELIRG. The AGN fraction defined as $L_{\rm IR}$(AGN)/$L_{\rm IR}$ is $\sim
0.4$, suggesting that $L_{\rm IR}$ for WISEJ0909+0002 may have contributions
from AGN and SF. We note that recent SED fitting codes such as CIGALE and
Multiwavelength Analysis of Galaxy Physical Properties (MAGPHYS; da Cunha et
al., 2008) employ physically-motivated $L_{\rm IR}$ without any boundary for
the integration range for the wavelength. Hence, X-CIGALE purely considers the
energy re-emitted by dust that absorbs UVoptical photons from AGNSF to
estimate $L_{\rm IR}$ (see also Toba et al., 2020b).

The derived color excess of the stellar emission ($E(B-V)_{*}$), stellar mass
($M_{*}$), and SFR output from X-CIGALE are $E(B-V)_{*}=0.13$, $M_{*}=$ (4.94
$\pm$ 1.39) $\times 10^{11}\,M_{\sun}$, and SFR = (3.85 $\pm$ 0.06) $\times
10^{3}$ $M_{\sun}$ yr-1, respectively, where the SFR was estimated based only
on the resultant parameters of the SFH output by X-CIGALE (see Boquien et al.,
2019, for more details). The $M_{*}$–SFR relation of WISEJ0909+0002 shows a
significant positive offset ($\sim$1 dex) with respect to the main-sequence
galaxies at 1.5 $<z<2$ for the same stellar mass (Tomczak et al., 2016;
Pearson et al., 2018). This indicates that WISEJ0909+0002 can be considered a
starburst galaxy. We note that it is often difficult to decompose the
UV/optical SED of quasars into stellar and AGN emission with limited
photometric data (see e.g., Merloni et al., 2010; Bongiorno et al., 2012; Toba
et al., 2018). Hence, the derived $M_{*}$ may have a large uncertainty due to
possible contamination from emission from the AGN accretion disk. Recent works
have also reported that AGNs with higher luminosities exhibit higher AGN
contribution to FIR emission than less luminous AGNs (e.g. Symeonidis et al.,
2016; Symeonidis, 2017). It has been shown that contribution from the AGN
torus to FIR emission is small, and FIR emission can be predominantly linked
to dust heated by the AGN at kpc-scales. Since the SED modeling in our study
does not take into account such dust, the derived $f_{\rm AGN}$ and SFR may be
underestimated and overestimated, respectively. Nevertheless, given the fact
that possible underestimated/overestimated values for $M_{*}$ and SFR for type
1 AGNs are expected to be 0.3 dex (e.g., Symeonidis et al. 2016; Toba et al.
in prep.), our conclusion (i.e., WISEJ0909+0002 can be considered a starburst
galaxy) is reasonable.

###  3.2 AGN properties

To the best of our knowledge, this is the first work on eROSITA data to
determine the X-ray properties of an ELIRG directly from the X-ray
spectra666Krawczyk et al. (2013) reported $L_{\rm 2-10\ keV}$ for
WISEJ0909+0002 to be $\log\,(L_{\rm 2-10\ keV}/{\rm erg\leavevmode\nobreak\
s}^{-1})=45.45\pm 0.18$, which is in good agreement with our work. However,
their derivation is based on an empirical relation between 2500 Å and the 2
keV luminosity for a given photon index ($\Gamma=2$).. Fig. 2 shows the X-ray
spectra of WISEJ0909+0002, in which the data are well fitted using the model
described in Sect. 2.3. The resulting values of the photon index ($\Gamma$),
$N_{\rm H}$, and absorption-corrected hard X-ray luminosity in the rest-frame
2–10 keV ($L_{\rm 2-10\ keV}$) are $\Gamma$ = 1.73${}_{-0.15}^{+0.16}$,
$\log\,(N_{\rm H}/{\rm cm}^{-2})<21.0$, and $L_{\rm 2-10\ keV}=(2.1\pm
0.2)\times 10^{45}$ erg s-1. These values are typical for luminous unobscured
type-1 AGNs.

Figure 2: X-ray spectra of eROSITA (blue), EPIC-pn (green), and Chandra (red)
folded with the energy responses and fitted with the same model (absorbed
power-law). The data are binned to a minimum significance of 2$\sigma$ for
convenience. The best-fit model and the corresponding 99% confidence interval
are displayed in blue, green, and red for eROSITA, EPIC-pn, and Chandra,
respectively. The lower panel displays the (data-model)/$\sqrt{{\rm model}}$.

WISEJ0909+0002 exhibits broad emission lines of C iv and Mg ii with full width
at half maximum (FWHM) values of 5580$\pm$102 and 6173$\pm$115 km s-1,
respectively (Rakshit et al., 2020) in the SDSS spectrum. Rakshit et al.
(2020) measured the monochromatic luminosity at 3000 Å777Rakshit et al. (2020)
decomposed the host galaxy from spectra only for quasars at $z<0.8$, and thus
the contribution from the host galaxy for WISEJ0909+0002 was not taken into
account to estimate $L_{\rm 3000}$. But since the host contribution to $L_{\rm
3000}$ for luminous quasars is expected to be negligible, we did not apply for
host subtraction (see also Calderone et al., 2017)., ($L_{\rm 3000}$ =
$7.2\times 10^{46}$ erg s-1) and FWHM of Mg ii, which contribute to the
bolometric luminosity ($L_{\rm bol}$) and BH mass ($M_{\rm BH}$), by using
multi-component spectral fitting. $L_{\rm bol}$ is then calculated using
$L_{\rm 3000}\times{\rm BC_{\rm 3000}}$, where BC3000 is bolometric correction
(BC3000 = 5.9 $\pm$ 0.8; Nemmen & Brotherton, 2010; Runnoe et al., 2012).
$M_{\rm BH}$ is estimated using a single epoch method reported by Vestergaard
& Osmer (2009). The uncertainty in $L_{\rm bol}$ and $M_{\rm BH}$ is
calculated through error propagation in the same manner as in Toba et al.
(2021).

The resultant $L_{\rm bol}$ and $M_{\rm BH}$ are $L_{\rm bol}=(4.3\pm
0.6)\times 10^{47}$ erg s-1 and $M_{\rm BH}=(7.4\pm 0.3)\times 10^{9}$
$M_{\sun}$, respectively. The Eddington ratio ($\lambda_{\rm Edd}=L_{\rm
bol}/L_{\rm Edd}$) is determined to be $\lambda_{\rm Edd}=0.4\pm 0.1$.
Following Toba et al. (2017), we estimated $L_{\rm bol}$ by integrating the
best-fit SED template of the AGN component output by X-CIGALE over wavelengths
longward of Ly$\alpha$. We obtained $\log\,(L_{\rm bol}^{\rm SED}/{\rm
erg\,s^{-1}})\sim 47.5$, which is consistent with $L_{\rm bol}$ derived from
the SDSS spectrum within the error.

It has been reported that there is a strong luminosity dependence of the hard
X-ray bolometric correction $\kappa_{\rm X}$ (=$L_{\rm bol}/L_{\rm 2-10\
keV})$ at high luminosities (e.g., Lusso et al., 2010; Duras et al., 2020).
The resulting $\kappa_{\rm X}$ of WISEJ0909+0002 is $(2.0\pm 0.3)\times
10^{2}$, which is consistent with the value obtained via an empirical
relationship between $\kappa_{\rm X}$ and $L_{\rm bol}$ for type-1 AGNs
($\kappa_{\rm X}\sim 207$888If we employ relationships of $\kappa_{\rm
X}$–$\lambda_{\rm Edd}$ and $\kappa_{\rm X}$–$M_{\rm BH}$ presented in Duras
et al. (2020), the resultant $\kappa_{\rm X}$ values are $\sim$36 and
$\sim$116, respectively. Although $\lambda_{\rm Edd}$-based $k_{\rm X}$ is
significantly smaller than what is reported ($\sim$200), it would be hard to
discuss here the origin of this discrepancy, given a huge scatter (0.3–0.4
dex) in those empirical relationships. We note that the $\kappa_{\rm
X}$–$\lambda_{\rm Edd}$ and $\kappa_{\rm X}$–$M_{\rm BH}$ relations in Duras
et al. (2020) are based not only on type 1 but also on type 2 AGNs although
$k_{\rm X}$–$L_{\rm bol}$ relation is derived based only from type 1 AGNs,
which might induce the discrepancy.) with a dispersion of 0.26 dex, reported
by Duras et al. (2020). $\kappa_{\rm X}$ is also correlated with $M_{\rm BH}$
and $\lambda_{\rm Edd}$ (e.g., Vasudevan & Fabian, 2007; Lusso et al., 2012;
Martocchia et al., 2017). The measured value of $\kappa_{\rm X}$ is larger
than the values estimated from empirical relations for type-1 AGNs of (i)
$\kappa_{\rm X}$–$M_{\rm BH}$ ($\sim$113) and (ii) $\kappa_{\rm
X}$–$\lambda_{\rm Edd}$ ($\sim$153) reported in Martocchia et al. (2017) and
Lusso et al. (2012), respectively, but still consistent with them within the
scatters of the correlations. We conclude that the observed AGN properties
($M_{\rm BH}$, $L_{\rm bol}$, $L_{\rm X}$, and $\lambda_{\rm Edd}$) of
WISEJ0909+0002 follow empirical relations for type-1 AGNs reported previously.

###  3.3 Expected surface number density of X-ray bright ELIRGs

We then discuss the number of X-ray bright ELIRGs expected to be discovered as
a consequence of the eROSITA all-sky survey (eRASS) that will continue until
the end of 2023 (Predehl et al., 2021). Following Toba & Nagao (2016), we
estimate the surface number density by taking into account the survey area
addressed in this work and the detection completeness of the WISE and eROSITA
all-sky surveys. The footprint of ELIRG survey in the eFEDS (GAMA-09 field) is
currently determined by the overlapping region of KiDS DR4 and H-ATLAS DR1,
which is approximately 50 deg2 (e.g., Fleuren et al., 2012). WISEJ0909+0002 is
drawn from the 22 $\mu$m-selected sample, and its flux density at 22 $\mu$m
($\sim$30 mJy) is well above the fluxes at the 95% completeness
limit999https://wise2.ipac.caltech.edu/docs/release/allsky/expsup/sec6_5.html.
($\sim$7 mJy). The X-ray flux in the 0.5–2 keV band of WISEJ0909+0002 is $\sim
1.6\times 10^{-13}$ erg s-1 cm-2, which is roughly one order of magnitude
brighter than that at 95% completeness ($\sim 2\times 10^{-14}$ erg s-1 cm-2)
(Merloni et al., 2012). Therefore, the expected surface number density of
X-ray bright ELIRGs is $2.0_{-1.7}^{+4.6}\times 10^{-2}$ deg-2, where the
1$\sigma$ confidence limits are estimated based on the Poisson statistics
reported by Gehrels (1986). Therefore, $2.0_{-1.7}^{+4.6}\times 10^{-2}$
$\times$ 34,100 deg2 (corresponding to the sky area at $|b|>10\degr$)
$=6.8_{-5.6}^{+16}\times 10^{2}$ objects are expected to be discovered in the
entire extragalactic sky, which suggests that X-ray bright ELIRGs are a rare
population, and this phase appears in a significantly short time scale.

###  3.4 The evolutionary stage of WISEJ0909+0002

A remarkable property of WISEJ0909+0002 is the low $N_{\rm H}$ ($<10^{21.0}$
cm-2), despite the fact that its $L_{\rm IR}$ value is extraordinary large
(i.e., this ELIRG is presumably embedded within a large amount of gas and
dust). Furthermore, WISEJ0909+0002 is known as a broad absorption line (BAL)
quasar (Trump et al., 2006; Ganguly et al., 2007; Moravec et al., 2017). To
check the consistency of $N_{\rm H}$ derived from the X-ray spectral analysis,
we estimate $N_{\rm H}$ from an expected ionization parameter ($U$) inferred
from the C iv $\lambda 1549$ and N v $\lambda 1240$ lines as follows. Moravec
et al. (2017) reported that the lower limits of the column density ($N_{\rm
ion}$) of C iv and the corresponding optical depth of the line center
($\tau_{0}$) are $2.89\times 10^{15}$ cm-2 and 0.55, respectively, while those
of $N_{\rm ion}$ and $\tau_{0}$ for N v are $5.61\times 10^{15}$ cm-2 and 0.7,
respectively. This suggests that (i) relative abundance of N v (N4+) is larger
than C iv (C3+) and (ii) those ions are optically-thin. Under the above
conditions101010We are not able to rule out the possibility that the X-ray
absorber and BAL gas are not exactly the same, although these absorbers are
often reported to have similar properties (e.g., velocity and column density)
and are likely to be the same for some quasars (Hamann et al., 2018). A
detailed analysis of the X-ray absorption line is required to address this
issue, which may be the scope of future work., we estimate the lower-limits to
$U$ and $N_{\rm H}$ based on photoionization calculations reported by Hamann
et al. (2011) and Wang et al. (2015). As a result, we obtain $\log\,U>-1.2$
and $\log\,N_{\rm H}>21$ cm-2, which is consistent with what is derived by Xu
et al. (2018) for other quasars with a similar BAL feature as WISEJ0909+0002.

The derived $N_{\rm H}$ is larger than the value obtained from the X-ray
spectral analysis. This discrepancy may be attributed to the variability of
$N_{\rm H}$. The BAL feature of WISEJ0909+0002 was reported based on the
averaged UV/optical spectra measured in 2001–2010 (the averaged epoch is
2006). The Chandra X-ray spectrum was measured in 2004. If we derive $N_{\rm
H}$ only from Chandra, we obtain $\log\,(N_{\rm H}/{\rm
cm}^{-2})=21.5_{-1.63}^{+1.24}$ (see Appendix D), which is roughly consistent
with that estimated from the BAL feature. Since XMM and eROSITA observed this
object in 2013 and 2019, respectively, it may be possible that $N_{\rm H}$ of
WISEJ0909+0002 decreased in 10-15 years. Given the large uncertainty of
Chandra-based $N_{\rm H}$, it may be hard to conclude that this variability is
significant. Nevertheless, if this drastic decline of $N_{\rm H}$ is true, the
BAL feature of the WISEJ0909+0002 spectra is expected to be weak, or even
missing in the current UV/optical spectra. This is an issue to be addressed in
a future work.

Recent galactic merger simulations show that an evolutionary stage with high
$L_{\rm IR}$ and low $N_{\rm H}$ appears for $\sim$ 5–10 Myr during the major
merger event in which hydrogen gas along the line of sight is blown outwards,
whereas warm and hot dust heated by AGNs still exist (Yutani et al. in
prep.111111Yutani et al. conducted high-resolution N-body/SPH simulations with
ASURA (Saitoh et al., 2008, 2009), and investigated the time evolution of SED
of mergers using the the radiative transfer simulation code RADMC-3D
(Dullemond et al., 2012).). We checked images from the Hubble Space Telescope
(HST) to see if there are any clues of potential merger event, but no such
observation is apparent in the image, as shown in Figure 3. The FWHM and
radial profile of WISEJ0909+0002 are comparable to those of possible stars
around our object in the same FoV of HST, suggesting that WISEJ0909+0002 can
be considered as a point source. This may be because the merger sign is too
weak to be detected currently, given the fact that an object with low $N_{\rm
H}$ and high $L_{\rm IR}$ would appear in the late stage of a galactic merger
(Yutani et al. in prep.). In addition, strong nuclear emissions often erase
the trace of the merger sign. However, the optical spectrum exhibits blue wing
for some emission lines such as C iii] (Rakshit et al., 2020), which may
support the idea that this ELIRG is in the blow-out phase. Thus, we are
witnessing a short-lived ELIRG phase in the course of galactic evolution for
WISEJ0909+0002. Owing to the rarity and extreme luminosity, such X-ray bright
ELIRGs will be ideal environments to investigate the bright-end of X-ray and
IR luminosity functions.

###### Acknowledgements.

We gratefully acknowledge the anonymous referee for a careful reading of the
manuscript and very helpful comments. We thank Takashi Horiuch, Keiichi Wada,
and Naomichi Yutani for useful discussion and comments. This work is based on
data from eROSITA, the primary instrument aboard SRG, a joint Russian-German
science mission supported by the Russian Space Agency (Roskosmos), in the
interests of the Russian Academy of Sciences represented by its Space Research
Institute (IKI), and the Deutsches Zentrum für Luft- und Raumfahrt (DLR). The
SRG spacecraft was built by Lavochkin Association (NPOL) and its
subcontractors, and is operated by NPOL with support from the Max-Planck
Institute for Extraterrestrial Physics (MPE). The development and construction
of the eROSITA X-ray instrument was led by MPE, with contributions from the
Dr. Karl Remeis Observatory Bamberg & ECAP (FAU Erlangen-Nuernberg), the
University of Hamburg Observatory, the Leibniz Institute for Astrophysics
Potsdam (AIP), and the Institute for Astronomy and Astrophysics of the
University of Tübingen, with the support of DLR and the Max Planck Society.
The Argelander Institute for Astronomy of the University of Bonn and the
Ludwig Maximilians Universität Munich also participated in the science
preparation for eROSITA. The eROSITA data shown here were processed using the
eSASS/NRTA software system developed by the German eROSITA consortium.
This work is supported by JSPS KAKENHI grant Nos. 18J01050 and 19K14759
(Y.Toba), 16K05296 (Y.Terashima), 20H01949 (T.Nagao), and 20H01946 (Y.Ueda).

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##  Appendix A Photometry of WISEJ0909+0002

The multi-band photometry of WISEJ0909+0002 is presented in Table 2. The X-ray
flux densities are corrected for galactic and intrinsic absorption, where
$\log,(N_{\rm H}/{\rm cm}^{-2})=20.1$ is used for intrinsic absorption
correction (see Fig. 4a). Flux densities in the optical to MIR are corrected
for galactic extinction according to Fitzpatrick (1999) and Schlafly &
Finkbeiner (2011). The SPIRE FIR flux densities are corrected for flux-
boosting (see e.g., Valiante et al., 2016; Toba et al., 2019b).

Table 2: Photometry of WISEJ0909+0002.

Band | Flux density [mJy]
---|---
eROSITA $f_{\rm 2-10\,keV}$ | $(6.18\pm 1.30)\times 10^{-6}$
eROSITA $f_{\rm 0.5-2\,keV}$ | $(18.1\pm 1.47)\times 10^{-6}$
GALEX FUV | $(8.89\pm 0.54)\times 10^{-3}$
GALEX NUV | $(57.8\pm 1.34)\times 10^{-3}$
KIDS $u$-band | $(72.6\pm 0.03)\times 10^{-2}$
DECaLS $g$-band | $(83.2\pm 0.05)\times 10^{-2}$
DECaLS $r$-band | $(93.6\pm 0.05)\times 10^{-2}$
KiDS $i$-band | $(96.0\pm 0.07)\times 10^{-2}$
DECaLS $z$-band | $(10.9\pm 0.01)\times 10^{-1}$
KiDS/VIKING $Y$-band | $(11.1\pm 0.01)\times 10^{-1}$
KiDS/VIKING $J$-band | $(10.5\pm 0.01)\times 10^{-1}$
KiDS/VIKING $H$-band | $(9.46\pm 0.02)\times 10^{-1}$
KiDS/VIKING $K_{\rm S}$-band | $(10.3\pm 0.02)\times 10^{-1}$
unWISE 3.4 $\mu$m | $(12.8\pm 0.02)\times 10^{-1}$
unWISE 4.6 $\mu$m | $(23.1\pm 0.05)\times 10^{-1}$
unWISE 12 $\mu$m | 10.2 $\pm$ 0.13
unWISE 22 $\mu$m | 29.1 $\pm$ 1.88
PACS 100 $\mu$m |  $<124$aa$a$3$\sigma$ upper limit.
PACS 160 $\mu$m |  $<144$aa$a$3$\sigma$ upper limit.
SPIRE 250 $\mu$m | 72.5 $\pm$ 7.20
SPIRE 350 $\mu$m | 71.6 $\pm$ 8.09
SPIRE 500 $\mu$m | 39.2 $\pm$ 8.49
121212

##  Appendix B Parameter ranges used in the SED fitting with X-CIGALE

Table 3 summarizes the parameter ranges used in the SED fitting with X-CIGALE
(see Sect. 2.2). Best-fit values (termed as BEST_$parametername$ in the output
file from X-CIGALE) are denoted by boldface. Here, we briefly explain the each
parameter for the SED fitting.

We assume the delayed SFH with recent starburst (Ciesla et al., 2017) with
parameterizing e-folding time of the main stellar population model ($\tau_{\rm
main}$), age of the main stellar population in the galaxy, age of burst, and
ratio of the SFR after and before the burst (R_sfr). A starburst attenuation
curve (Calzetti et al., 2000; Leitherer et al., 2002) is used for the dust
attenuation in which we parameterize color excess of the nebular emission
lines ($E(B-V)_{\rm lines}$). We chose the SSP model (Bruzual & Charlot,
2003), assuming the IMF of Chabrier (2003), and the standard nebular emission
model included in X-CIGALE (see Inoue, 2011). AGN emission is modeled by using
the SKIRTOR (Stalevski et al., 2016). This torus model consists of 7
parameters; torus optical depth at 9.7 $\mu$m ($\tau_{\rm 9.7}$), torus
density radial parameter ($p$), torus density angular parameter ($q$), angle
between the equatorial plane and edge of the torus ($\Delta$), ratio of the
maximum to minimum radii of the torus ($R_{\rm max}/R_{\rm min}$), the viewing
angle ($\theta$), and the AGN fraction in total IR luminosity ($f_{\rm AGN}$).
Dust grain emission is modeled by Draine et al. (2014) in which we
parameterize the mass fraction of PAHs ($q_{\rm PAH}$), the minimum radiation
field ($U_{\rm min}$), the power-low slope of the radiation field distribution
($\alpha$), and the fraction illuminated with a variable radiation field
ranging from $U_{\rm min}$ to $U_{\rm max}$ ($\gamma$). X-ray emission is
modeled with fixed power-low photon indices of AGN, low-mass X-ray binaries
(LMXB), and high-mass X-ray, binaries (HMXB).

Table 3: Parameter Ranges Used in the SED Fitting with X-CIGALE Parameter | Value
---|---
Delayed SFH with recent starburst (Ciesla et al., 2017)
$\tau_{\rm main}$ [Gyr] | 1.0, 4.0, 8.0, 12
age [Gyr] |  0.5, 1.0, 1.5, 2.0
age of burst [Myr] | 10, 50, 100
R_sfr | 1, 5, 10
SSP (Bruzual & Charlot, 2003)
IMF | Chabrier (2003)
Metallicity | 0.02
Nebular emission (Inoue, 2011)
$\log\,U$ |  $-$3.0, $-$2.0, $-$1.0
line width [km s-1] | 300
Dust attenuation (Calzetti et al., 2000; Leitherer et al., 2002)
$E(B-V)_{\rm lines}$ |  0.3, 0.5, 1.0
AGN Emission (Stalevski et al., 2012, 2016)
$\tau_{\rm 9.7}$ | 3, 7, 11
$p$ |  0.5, 1.5
$q$ |  0.5, 1.5
$\Delta$ [$\degr$] | 80
$R_{\rm max}/R_{\rm min}$ | 30
$\theta$ [$\degr$] |  0, 10, 20
$f_{\rm AGN}$ |  0.4, 0.5, 0.6, 0.7, 0.8, 0.9
Dust Emission (Draine et al., 2014)
$q_{\rm PAH}$ |  2.50, 5.26, 6.63, 7.32
$U_{\rm min}$ | 10.00, 50.00
$\alpha$ | 1.0, 1.5, 2.0
$\gamma$ |  0.01, 0.1, 1.0
X-ray Emission (Yang et al., 2020)
AGN photon index | 1.7
$|\Delta\,\alpha_{\rm OX}|_{\rm max}$ | 0.2
LMXB photon index | 1.56
HMXB photon index | 2.0

##  Appendix C Possibility of gravitationally lensed source

We briefly consider the possibility that $L_{\rm IR}$ of WISEJ0909+0002 is
boosted by gravitational lensing131313For example, Glikman et al. (2018)
reported that an ELIRG (WISE J104222.11+164115.3) is a quadruply lensed system
with a magnification factor of 53–122 (see also Matsuoka et al., 2018)..
WISEJ0909+0002 was observed by an optical instrument called the the Space
Telescope Imaging Spectrograph (STIS) with 50 CCD modes (with a pixel scale of
$\sim$0.05″), with a central wavelength of 5903 Å on board HST (proposal ID =
8202)141414Although the aim of this program was to identify lenses with
component image separations $<1\arcsec$, WISEJ0909+0002 has not been selected
as a lensed system candidate by other lens searches such as H-ATLAS sources
(e.g., Negrello et al., 2010, 2017; González-Nuevo et al., 2012, 2019; Bakx et
al., 2020, and references therein) as well as SDSS quasars (e.g., Oguri et
al., 2006; Inada et al., 2012, and reference therein).. However, we could not
find any features of strong lensing (see Fig. 3), indicating that $L_{\rm IR}$
for WISEJ0909+0002 is intrinsic, and not boosted by lensing.

Figure 3: Multi-wavelength images (FUV, NUV, 50CCD/HST, $u$, $g$, $r$, $i$,
$z$, $Y$, $J$, $H$, $K_{\rm s}$, and 3.4, 4.6, 12, 22, 250, 350, and 500
$\mu$m, and 0.2–2.3 keV, from top left to bottom right) for WISEJ0909+0002.
North is upwards, and east is to the left. R.A. and decl. are relative
coordinates with respect to the object in the LS8. Yellow circles in the
images also correspond to the coordinates of the LS8.

We note that WISEJ0909+0002 is known as a pair/binary quasar with a companion
quasar (SDSS J090923.13+000204.0) at $z_{\rm spec}=1.887\pm 0.001$ located
$\sim$15″ from it (Hennawi et al., 2006; Foreman et al., 2009; Tytler et al.,
2009) (see the point source in the southwest direction in the UV–NIR images of
Fig. 3). The proper comoving transverse separation of this system is
approximately 130 kpc. Since the companion is located at a higher redshift
than that of WISEJ0909+0002, it cannot act as a lens.

Furthermore, given the relatively large beam size ($\sim 30\arcsec$) of
SPIRE/Herschel, its FIR photometry would be affected by FIR emissions from the
companion quasar151515The companion quasar was not detected by ALLWISE (Cutri
et al., 2014) 12 and 22 $\mu$m, but it was detected by unWISE 3.4 and 4.6
$\mu$m with flux densities of $51\pm 3$ and $61\pm 5$ $\mu$Jy, respectively..
We also find that the other two sources also fall within SPIRE’s beam size,
given the limiting magnitude of DECaLS and KiDS (see optical sources in the
northwest direction seen in Fig.3). However, the MIR–FIR emissions of
WISEJ0909+0002 do not exhibit any offsets and elongations expect from 500
$\mu$m. This indicates that the influence of FIR emissions (at least in 250
and 350 $\mu$m) from the neighborhood including the companion quasar is
expected to be small.

We also consider the possibility that the X-ray flux is overestimated due to
the contribution from the companion quasar, because the extraction radii for
the X-ray spectra contains the position of the companion quasar (see Sect.
2.3). We confirm that there is no companion quasar in the XMM and eROSITA
images. The Chandra image with higher angular resolution also shows that the
companion is not clearly visible, where the counts from WISEJ0909+0002 and the
companion quasar are 17 and 2, respectively, including the background. This
suggests that the X-ray flux of WISEJ0909+0002 is more than 8.5 times brighter
than that of the companion quasar. Thus, contamination from the companion
quasar in $L_{\rm X}$ of WISEJ0909+0002 is expected to be small.

##  Appendix D Possible variability of X-ray properties from Chandra to
XMM/eROSITA

As displayed in Fig. 2, Chandra flux is fainter at $<3$ keV, consistent with
absorption, which would be supported by the BAL feature in its UV/optical
spectra obtained in 2001-2009 (see Sect. 3.4). Separated by 15 years, it is
possible that the spectral shape changed between the Chandra and the eROSITA
observations. We fit the Chandra data only using the same method and model as
described in Sect.2.3. We note that we run the Bayesian method on the unbinned
Chandra spectrum. Since there are only $17$ counts in total, any binning of
the spectrum will cause a loss of information.

Figure 4: The posterior distributions of the intrinsic column density and
power-law slope for (a) the joint eROSITA-XMM-Chandra fit and (b) the Chandra-
only fit, respectively, plotted using the “corner” package (Foreman-Mackey,
2016). The contour levels correspond to 30%, 60%, and 90% of the distribution.
The dashed vertical lines mark the 1$\sigma$ percentile interval, which is
also printed on top of the figure.

The resultant $N_{\rm H}$ and $\Gamma$ derived from (i) fitting all the data
simultaneously with the same model and (ii) fitting only the Chandra spectrum
are shown in Fig.4a and b, respectively. We calculated the 68% percentile
around the median from the posterior distribution. The lower and upper limits
respectively correspond to 16% and 84% percentile values of all the randomly
sampled points that constitute the posterior distribution. As displayed in
Fig.4a, the $N_{\rm H}$ distribution is concentrated at the lower boundary,
and the lower limit of $N_{\rm H}$ is unconstrained (with a truncated
contour). Thus, we obtain the upper limit for $N_{\rm H}$. The lower limit of
$N_{\rm H}$ is still unconstrained, even if we fit only the Chandra data, as
shown in Fig.4b. However, the $N_{\rm H}$ distribution has a peak at $\sim
10^{23}$ cm-2. Meanwhile, the unconstrained lower tail of $N_{\rm H}$
corresponds to a very-flat $\Gamma$ of 0.99. These results indicate that the
Chandra spectral shape is harder, possibly due to high $N_{\rm H}$
obscuration. Considering the low number of counts and potential cross-
calibration issue between different facilities, such a spectral variability is
not conclusive (see also Sect. 3.3).

##  Appendix E X-ray deficit given 6 $\mu$m AGN luminosity

It is known from observations that MIR luminous AGNs tend to deviate from the
linear relationship between hard X-ray and MIR luminosities, and are less
X-ray luminous at a given e.g., 6 $\mu$m luminosity ($L_{6}$). This is the so-
called “X-ray deficit (e.g., Stern, 2015; Chen et al., 2017)”. The X-ray
deficit has also been reported for optically luminous quasars (e.g., Vignali
et al., 2003; Just et al., 2007). We checked if WISEJ0909+0002 exhibits X-ray
deficit at a given $L_{6}$, where $L_{6}$ = (3.53 $\pm$ 0.80) $\times 10^{46}$
erg s-1 is purely from the AGN, estimated by the SED decomposition following
Toba et al. (2019a). The expected $L_{\rm 2-10\ keV}$ based on the non-linear
relations reported by Stern (2015) and Chen et al. (2017) are $L_{\rm 2-10\
keV}$ $\sim$ 1.6 and 2.0 $\times 10^{45}$ erg s-1, respectively, which are
consistent with the values obtained in this work. This means that
WISEJ0909+0002 exhibits X-ray deficit, as reported in Stern (2015) and Chen et
al. (2017).

Toba et al. (2019a) suggested that this X-ray deficit can be explained by a
difference in $\lambda_{\rm Edd}$. The ratio of the 2–10 keV and 6 $\mu$m
luminosities ($L_{\rm X}/L_{6}$) decreases with increasing $\lambda_{\rm
Edd}$, which is interpreted as a change in the structure of the accretion flow
(see also e.g., Chen et al. 2017 and Ishikawa et al. 2021, where other
possibilities indicating X-ray deficit have been reported). Toba et al.
(2019a) reported a linear relationship between $L_{\rm X}/L_{6}$ and
$\lambda_{\rm Edd}$ for AGNs (see Equation 3 in Toba et al. 2019a), and this
relationship would be appreciable to a Compton-thick AGN (Toba et al., 2020a).
Fig. 5 shows $\lambda_{\rm Edd}$ as a function of $L_{\rm X}/L_{6}$, where
type 1 AGNs selected from the SDSS and WISE (WISSH quasar: Martocchia et al.,
2017), XMM-Newton (Mateos et al., 2015), and ROSAT (Toba et al., 2019a) are
plotted. A hot, dust-obscured galaxy (hot DOG; Ricci et al., 2017) is also
plotted. The best-fit relation based on the above populations reported in Toba
et al. (2019a) is overlaid in Fig.5. We find that WISEJ0909+0002 follows the
correlation for the observed $L_{\rm X}/L_{6}$ ratio, and the expected
$\lambda_{\rm Edd}$ in Toba et al. (2019a) is $\lambda_{\rm Edd}\sim$ 0.4,
suggesting that this empirical relation could also be applicable to ELIRGs.

Figure 5: Eddington ratio ($\lambda_{\rm Edd}$) as a function of $L_{\rm
X}/L_{\rm 6}$ of ROSAT type-1 AGNs (gray cross: Toba et al., 2019a), XMM
type-1 AGNs (cyan circle: Mateos et al., 2015), a hot DOG (orange square:
Ricci et al., 2017), and WISSH quasars (purple asterisk: Martocchia et al.,
2017). The red star represents WISE 0909+0002. The green solid line with the
shaded region is a linear relationship between $\lambda_{\rm Edd}$ and $L_{\rm
X}/L_{\rm 6}$, as reported by Toba et al. (2019a).