Resolved Molecular Gas and Star Formation Properties of the Strongly Lensed z=2.26 Galaxy SDSS J0901+1814

Abstract

We present ~1" resolution (~2 kpc in the source plane) observations of the CO(1-0), CO(3-2), Halpha, and [N II] lines in the strongly-lensed z=2.26 star-forming galaxy SDSS J0901+1814. We use these observations to constrain the lensing potential of a foreground group of galaxies, and our source-plane reconstructions indicate that SDSS J0901+1814 is a nearly face-on (i~30 degrees) massive disk with r_{1/2}>~4 kpc for its molecular gas. Using our new magnification factors (mu_tot~30), we find that SDSS J0901+1814 has a star formation rate (SFR) of 268^{+63}_{-61} M_sun/yr, M_gas=(1.6^{+0.3}_{-0.2})x10^11x(alpha_CO/4.6) M_sun, and M_star=(9.5^{+3.8}_{-2.8})x10^10 M_sun, which places it on the star-forming galaxy "main sequence." We use our matched high-angular resolution gas and SFR tracers (CO and Halpha, respectively) to perform a spatially resolved (pixel-by-pixel) analysis of SDSS J0901+1814 in terms of the Schmidt-Kennicutt relation. After correcting for the large fraction of obscured star formation (SFR_Halpha/SFR_TIR=0.054^{+0.015}_{-0.014}), we find SDSS J0901+1814 is offset from "normal" star-forming galaxies to higher star formation efficiencies independent of assumptions for the CO-to-H_2 conversion factor. Our mean best-fit index for the Schmidt-Kennicutt relation for SDSS J0901+1814, evaluated with different CO lines and smoothing levels, is n=1.54+/-0.13; however, the index may be affected by gravitational lensing, and we find n=1.24+/-0.02 when analyzing the source-plane reconstructions. While the Schmidt-Kennicutt index largely appears unaffected by which of the two CO transitions we use to trace the molecular gas, the source-plane reconstructions and dynamical modeling suggest that the CO(1-0) emission is more spatially extended than the CO(3-2) emission.

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Resolved Molecular Gas and Star Formation Properties of the Strongly Lensed z= 2.26 Galaxy SDSS J0901+1814∗
Chelsea E. Sharon,1, 2 Amitpal S. Tagore,3Andrew J. Baker,4Jesus Rivera,4Charles R. Keeton,4
Dieter Lutz,5Reinhard Genzel,5David J. Wilner,6Erin K. S. Hicks,7Sahar S. Allam,8and
Douglas L. Tucker8
1Yale-NUS College, Singapore, 138527
2Department of Physics & Astronomy, McMaster University, Hamilton, ON, L8S-4M1, Canada
3Jodrell Bank Centre for Astrophysics, The University of Manchester, Manchester, M13 9PL, UK
4Department of Physics and Astronomy, Rutgers, the State University of New Jersey, Piscataway, NJ, 08854-8019, USA
5Max-Planck-Institut f¨ur extraterrestrische Physik (MPE), Giessenbachstr. 1, 85748 Garching, Germany
6Harvard-Smithsonian Center for Astrophysics, Cambridge, MA, 02138, USA
7Department of Physics and Astronomy, University of Alaska, Anchorage, AK, 99508, USA
8Center for Particle Astrophysics, Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL, 60510, USA
(Accepted to ApJ 5/17/2019)
ABSTRACT
We present ∼100 resolution (∼2 kpc in the source plane) observations of the CO(1–0), CO(3–2),
Hα, and [N ii] lines in the strongly-lensed z= 2.26 star-forming galaxy SDSS J0901+1814. We
use these observations to constrain the lensing potential of a foreground group of galaxies, and our
source-plane reconstructions indicate that SDSS J0901+1814 is a nearly face-on (i≈30◦) massive
disk with r1/2&4 kpc for its molecular gas. Using our new magnification factors (µtot ≈30), we
find that SDSS J0901+1814 has a star formation rate (SFR) of 268+63
−61 Myr−1,Mgas = (1.6+0.3
−0.2)×
1011(αCO /4.6) M, and M?= (9.5+3.8
−2.8)×1010 M, which places it on the star-forming galaxy “main
sequence.” We use our matched high-angular resolution gas and SFR tracers (CO and Hα, respec-
tively) to perform a spatially resolved (pixel-by-pixel) analysis of SDSSJ0901+1814 in terms of
the Schmidt-Kennicutt relation. After correcting for the large fraction of obscured star formation
(SFRHα/SFRTIR = 0.054+0.015
−0.014), we find SDSS J0901+1814 is offset from “normal” star-forming galax-
ies to higher star formation efficiencies independent of assumptions for the CO-to-H2conversion factor.
Our mean best-fit index for the Schmidt-Kennicutt relation for SDSS J0901+1814, evaluated with dif-
ferent CO lines and smoothing levels, is ¯n= 1.54 ±0.13; however, the index may be affected by
gravitational lensing, and we find ¯n= 1.24 ±0.02 when analyzing the source-plane reconstructions.
While the Schmidt-Kennicutt index largely appears unaffected by which of the two CO transitions we
use to trace the molecular gas, the source-plane reconstructions and dynamical modeling suggest that
the CO(1–0) emission is more spatially extended than the CO(3–2) emission.
Keywords: galaxies: high-redshift—galaxies: individual (SDSSJ0901+1814)—galaxies: ISM—
galaxies: starburst—galaxies: star formation—ISM: molecules
1. INTRODUCTION
Star forming galaxies at high redshift have been se-
lected using a variety of methods, most notably on
Corresponding author: Chelsea E. Sharon
chelsea.sharon@yale-nus.edu.sg
∗Based on observations carried out with the IRAM Plateau
de Bure Interferometer. IRAM is supported by INSU/CNRS
(France), MPG (Germany) and IGN (Spain).
the basis of rest-ultraviolet (UV) colors (e.g., Lyman
break galaxies, LBGs; Steidel et al. 1996;Giavalisco
2002) and large (observed frame) submillimeter fluxes
(e.g., submillimeter galaxies, SMGs; Blain et al. 2002;
Casey et al. 2014). Historically, galaxies selected using
these two methods have been described as two separate
populations, with UV-bright galaxies characterized as
“normal” galaxies at high redshifts (mostly disks, and
falling along a star-forming “main sequence” (MS) in
star formation rate (SFR) vs. stellar mass; e.g, F¨orster
arXiv:1905.09845v1 [astro-ph.GA] 23 May 2019

2Sharon et al.
Schreiber et al. 2006;Genzel et al. 2006;Bouch´e et al.
2007;Wright et al. 2007;Noeske et al. 2007;Elbaz et al.
2007;Daddi et al. 2007;Genzel et al. 2008;F¨orster
Schreiber et al. 2009;Wisnioski et al. 2015) whose SFRs
are an order of magnitude lower than those of dusty
starbursts that are SMGs (e.g., Rodighiero et al. 2011).
However, fits to spectral energy distributions (SEDs;
e.g., Wuyts et al. 2011), the decomposition of many of
the brightest SMGs into multiples, and stacking (e.g.,
Lindner et al. 2012;Decarli et al. 2014;Walter et al.
2014) suggest there is substantial overlap in the under-
lying physical properties of UV- and IR-bright high-z
galaxies, at least for higher masses.
Despite the increasing evidence of overlap between
these populations, comparing their directly observ-
able properties remains difficult. The substantial dust
masses that give SMGs their large far-infrared (FIR) lu-
minosities obscure their UV-emission (e.g., Smail et al.
2002;Hodge et al. 2012), including common short-
wavelength SFR tracers such as Hα. Similarly, UV-
bright galaxies are comparatively dust and gas poor,
and therefore frequently require substantial investments
of telescope time and/or magnification from gravita-
tional lensing to achieve mere detections of dust and
molecular gas (e.g., Baker et al. 2001,2004;Coppin
et al. 2007;Daddi et al. 2010b;Saintonge et al. 2013;
Dessauges-Zavadsky et al. 2015). Only recently have
larger samples of high-redshift optical/UV color-selected
galaxies been detected in CO (e.g., Tacconi et al. 2013;
Freundlich 2017; see also: Genzel et al. 2015;Tacconi
et al. 2017 and references therein), the canonical tracer
of molecular gas, in numbers comparable to those of
dusty galaxies (see Carilli & Walter 2013 for a review
of gas in high-redshift galaxies). Of the UV or optical
color-selected galaxies with CO detections, few have
spatially resolved or multi-JCO detections (e.g., Gen-
zel et al. 2013). With the wide bandwidths and the
sensitivities of telescopes like the Atacama Large Mil-
limeter/submillimeter Array, it has been suggested that
dust continuum measurements may be a more efficient
way to measure the masses of galaxies’ interstellar media
(ISMs; e.g., Scoville et al. 2014,2016), including their
molecular gas components, even for UV-bright/dust-
poor systems. However, using different observables to
trace the same intrinsic galaxy parameter (e.g., infrared
vs. UV-tracers of the total SFR, or dust vs. CO tracers
of the molecular gas) may generate false differences be-
tween galaxy populations due to systematic factors like
extinction or AGN contamination (Kennicutt & Evans
2012).
The lack of data at complementary wavelengths also
makes resolved multi-wavelength analyses applied to
low-redshift galaxies, such as the Schmidt-Kennicutt re-
lation (the correlation between galaxies’ SFR and gas
mass surface densities; e.g., Schmidt 1959;Kennicutt
1989,1998) significantly less common at high redshift.
High-resolution CO observations are critical for evalu-
ating where high-redshift galaxies fall on the true sur-
face density version of the Schmidt-Kennicutt relation,
where ΣSFR and Σgas can be compared on a pixel-by-
pixel basis within individual galaxies (as done for local
galaxies; e.g., Kennicutt et al. 2007;Bigiel et al. 2008;
Wei et al. 2010;Bigiel et al. 2011;Leroy et al. 2013).
Many high-redshift analyses use star formation and gas
properties averaged over the entire galaxy (e.g., Kenni-
cutt 1989;Buat et al. 1989;Kennicutt 1998;Genzel et al.
2010;Daddi et al. 2010a;Tacconi et al. 2013) or avoid
the additional uncertainties in source size and scaling
factors by using the total luminosities of the star forma-
tion and gas tracers (e.g., Young et al. 1986;Solomon
& Sage 1988;Gao & Solomon 2004). These different
methods for determining SFRs and gas masses make
it difficult to compare studies that focus on different
galaxy populations, leading to significant uncertainties
in the power-law index of the Schmidt-Kennicutt rela-
tion and the relative placement of different galaxy types
in the ΣSFR–Σgas plane. Accurately characterizing the
Schmidt-Kennicutt relation is important, since offsets
imply a difference in star formation efficiency (SFE),
and the power law index probes the underlying phys-
ical processes of star formation (for example, a linear
correlation would imply supply-limited star formation,
whereas super-linear correlations occur if star formation
depends on cloud-cloud collisions or total gas free-fall
collapse times; e.g., Larson 1992;Tan 2000;Krumholz
& McKee 2005;Ostriker & Shetty 2011). Systematic
differences in the Schmidt-Kennicutt relation between
different galaxy populations would imply important dif-
ferences in their star formation processes.
Kennicutt & Evans (2012) present a compilation of
disk-averaged SFR and gas mass surface densities whose
values have been calculated in a uniform manner across
different galaxy types (including normal disk galaxies
and dusty starburst galaxies selected in the IR), and
find a power law index of n∼1.4. However, this result
may be an artifact of combining galaxies of different in-
teraction states. For a sample of z∼1–3 MS galaxies,
Tacconi et al. (2013) find an index consistent with unity
and only a slight offset between their high-redshift sam-
ple and a low-redshift sample with similar masses. How-
ever, SMGs and other ultra-/luminous infrared galaxies
(U/LIRGs) are further offset above the correlation for
star-forming disk galaxies even when similar CO-to-H2
conversion factors are used for all galaxy populations

Resolved Properties of J0901 3
(see also Bigiel et al. 2008;Daddi et al. 2010a;Genzel
et al. 2010,2015;Tacconi et al. 2017). In analyses of
the resolved star formation properties of nearby disks, a
near-unity index for the Schmidt-Kennicutt relation is
also found in regimes where the molecular gas dominates
the total gas mass surface density (Σgas >9Mpc−2;
e.g., Bigiel et al. 2008,2010;Schruba et al. 2011). The
surface density version of the Schmidt-Kennicutt rela-
tion has been evaluated within only eight high-redshift
galaxies: SMM J14011+0252 at z= 2.56 (Sharon et al.
2013), EGS 13011166 at z= 1.53 (Genzel et al. 2013),
HLS0918 at z= 5.24 (Rawle et al. 2014), GN20 at
z= 4.05 (Hodge et al. 2015), PLCK G244.8+54.9 at
z= 3.00 (Ca˜nameras et al. 2017), AzTEC-1 at z= 4.34
(Tadaki et al. 2018), and the two components of HAT-
LAS J084933 at z= 2.41 (G´omez et al. 2018)1. These
studies find a range of Schmidt-Kennicutt relation in-
dices (n= 1–2). It is particularly worth noting that
Genzel et al. (2013) find that their measured index de-
pends strongly on which spatially-resolved extinction
correction they apply to their Hαmeasurements.
Comparisons between the Schmidt-Kennicutt rela-
tions for high- and low-redshift galaxies may be affected
by the different CO lines observed (Narayanan et al.
2011); the molecular gas in local galaxies is probed via
the CO(1–0) and/or CO(2–1) lines, while the molecular
gas at high redshift has typically been probed via mid-J
CO lines (i.e., CO(3–2), CO(4–3), and CO(5–4)). Dif-
ferent transitions have different excitation temperatures
and critical densities and are therefore sensitive to dif-
ferent density regimes in the molecular ISM (Krumholz
& Thompson 2007;Narayanan et al. 2008,2011), mak-
ing the observed index dependent on the physical con-
ditions of the star-forming gas. Using either global lu-
minosities or mean surface densities, substantial differ-
ences in Schmidt-Kennicutt indices have been found us-
ing molecular gas tracers with different critical densities
in local galaxies (all with n < 1.5; e.g., Gao & Solomon
2004;Narayanan et al. 2005;Graci´a-Carpio et al. 2008;
Bussmann et al. 2008;Iono et al. 2009;Juneau et al.
2009;Greve et al. 2014;Kamenetzky et al. 2016), but
no significant difference in index has been found between
CO(1–0) and CO(3–2) studies of z > 1 galaxies (Tacconi
et al. 2013;Sharon et al. 2016). So far there have been
no comparisons between Schmidt-Kennicutt indices for
different molecular gas tracers in spatially resolved stud-
ies of high-redshift galaxies.
1Freundlich et al. (2013) and Sharda et al. (2017) also examine
the Schmidt-Kennicutt relation at z > 1, but they analyze indi-
vidually resolved clumps within high-redshift galaxies rather than
performing full pixel-by-pixel comparisons.
Here we present high-resolution (∼100 observed; ∼
2 kpc in the source plane) observations of the molec-
ular gas and star formation tracers in the UV-bright
galaxy SDSS J0901+1814 (J0901 hereafter). J0901 was
discovered by Diehl et al. (2009) in a systematic search
of the Sloan Digital Sky Survey (York et al. 2000) for
strongly lensed galaxies (identified as blue arcs near
known brightest cluster galaxies or luminous red galax-
ies). Followup observations at the Astrophysics Re-
search Consortium (ARC) 3.5m telescope at Apache
Point Observatory confirmed that J0901 is a z= 2.26
galaxy (Diehl et al. 2009;Hainline et al. 2009) that
is multiply imaged (into a pair of bright arcs to the
north and south that nearly connect to the east, and
a fainter western counter-image) by a z= 0.35 lumi-
nous red galaxy. Single-slit spectroscopy at rest-frame
optical wavelengths using Keck II/NIRSPEC show large
[N ii] (λ= 6583 ˚
A)/Hαratios in the two brightest im-
ages (Hainline et al. 2009), indicating the presence of an
AGN (e.g., Baldwin et al. 1981;Kauffmann et al. 2003)
that includes a prominent broad-line component (Gen-
zel et al. 2014). However, the strong PAH features de-
tected in Spitzer/IRS spectra and weak continuum fea-
tures in the (observed frame) mid-IR suggest that the
AGN contribution to the IR luminosity of J0901 is neg-
ligible (Fadely et al. 2010). Further observations have
revealed that J0901 is one of the brightest high-redshift
UV-selected galaxies in terms of its dust emission (e.g.,
Baker et al. 2001;Coppin et al. 2007); Saintonge et al.
(2013) estimate a total IR luminosity (magnification cor-
rected) of LIR ∼7×1012(µ/8) Lusing Herschel/PACS
and SPIRE photometry. The substantial dust content
implied by the IR luminosity makes J0901 a natural
target for observations of molecular emission lines and
other gas-phase coolants; Rhoads et al. (2014) observe
a double-peaked profile in (spatially unresolved) Her-
schel/HIFI observations of the [Cii] 158 µm line and
infer that J0901 is a rotating disk galaxy. The addi-
tional spatial resolution provided by gravitational lens-
ing allows us to resolve the velocity structure of J0901
and verify its structure in this paper, as well as study
the variation of gas and star formation conditions with
J0901.
We describe our observations of J0901 and basic mea-
surements in Sections 2and 3, respectively. In Section 4
we describe our lens model for J0901 (Section 4.1); the
resulting magnification-corrected gas mass, stellar mass,
SFR, and dynamical mass (Section 4.2); resolved anal-
yses of CO excitation (Section 4.3), metallicity (Sec-
tion 4.4), the Schmidt-Kennicutt relation (Section 4.5),
and the SFR-CO excitation correlation (Section 4.6);
and finally, the potential radio continuum emission from

4Sharon et al.
the central AGN (Section 4.7). Our results are summa-
rized in Section 5. We assume the WMAP7+BAO+H0
mean ΛCDM cosmology throughout this paper, with
ΩΛ= 0.725 and H0= 70.2 km s−1Mpc−1(Komatsu
et al. 2011).
2. OBSERVATIONS & REDUCTION
2.1. IRAM Plateau de Bure Interferometer
We observed CO(3–2) emission from J0901 using the
IRAM Plateau de Bure Interferometer (PdBI; Guil-
loteau et al. 1992) in four separate configurations. Three
tracks in a five-antenna version of the compact D config-
uration were obtained in September and October 2008
(project ID S040; PI Baker), with a single pointing cen-
tered on the southern image that had been strongly de-
tected in 1.2 mm continuum photometry (6.4±0.6 mJy)
with the Max-Planck Millimeter Bolometer (MAMBO)
array (Kreysa et al. 1998). The PdBI data confirmed
that all three images were detected at high significance
in CO(3–2), motivating the acquisition of four further
tracks from 2009 November through 2010 February with
all six PdBI antennas in their more extended C (1),
B (1), and A (2) configurations (project ID T0AB; PI
Baker). All observations targeted a J2000 position of
α(J2000) = 09h01m22.s59 and δ(J2000) = 18◦14024.2000,
and a redshifted CO(3–2) line frequency of 106.082 GHz
in the upper sideband. We employed a narrow-band
correlator mode with 5 MHz channels and a total band-
width of 1 GHz, which recorded both horizontal and
vertical polarizations. The final combination of seven
tracks yielded 52 distinct baselines with lengths ranging
from 24 m to 760 m, and a total on-source integration
time equivalent to 18.0 hr with a six-telescope array.
Phase and amplitude variation were tracked by inter-
leaving observations of J0901 and the bright quasar
PG 0851+202, only 2.4◦away on the sky. Band-
pass calibrators included PG 0851+202, 3C273, and
0932+392; our overall flux scale was tied to observa-
tions of MWC349 and the quasars 3C273 and 0923+392,
which are regularly monitored with IRAM facilities,
and is accurate to ∼10%. Calibration and flagging
for data quality used the CLIC program within the
IRAM GILDAS package (Guilloteau & Lucas 2000).
The resulting uv data set was exported to FITS format
and imaged with AIPS. We created an initial set of
channel maps to explore possible uv weighting schemes,
and after comparing these settled on a robustness of 1,
which delivered slightly higher resolution than natural
weighting without compromising image fidelity or flux
recovery. Our final data cube has a synthesized beam of
1.3300 ×0.9800 at a position angle of 41.1◦, and a mean
rms noise of 0.62 mJy beam−1per 5 MHz ↔14.1 km s−1
channel. Following confirmation that it contained no
continuum emission at the sensitivity/resolution of these
observations (as expected), the resulting data cube was
cleaned with the IMAGR task in AIPS, corrected for
primary beam attenuation, and analyzed further with a
custom set of IDL scripts.
2.2. Karl G. Jansky Very Large Array
We observed J0901 at the Karl G. Jansky Very Large
Array (VLA) in three different configurations (project
IDs AB1347, AS1057, AS1144; PIs Baker, Sharon); the
configurations, maximum baselines, observation dates,
numbers of antennas used, and weather conditions are
summarized in Table 1. The minimum uv-radius of the
full dataset is 3.67 kλ. We observed with the WIDAR
correlator in the “OSRO Dual Polarization” mode using
the lowest spectral resolution (256 channels ×500 kHz
resolution) and a single intermediate frequency pair (IF
pair B/D). The total bandwidth of 128 MHz was cen-
tered at the observed frequency of CO(1–0) for z=
2.2586 (35.363 GHz). Observations were centered at
α(J2000) = 09h01m23.s00, δ(J2000) = +18◦14024.000,
the position of the southernmost and brightest (at opti-
cal wavelengths) of the three lensed images (Diehl et al.
2009). At the beginning of each track, we observed
3C 138 as both passband and flux calibrator, adopt-
ing Sν= 1.1786 Jy using the CASA2(McMullin et al.
2007) package’s default “Perley-Butler 2010” flux stan-
dard. Phase and amplitude fluctuations were tracked
by alternating between the source and a nearby quasar,
J0854+2006, with a cycle time of 6 minutes. A total of
16 hours was spent on source across the various config-
urations in Table 1.
We performed calibration in CASA version 3.3.0, map-
ping in CASA version 4.1.0, and subsequently used
CASA version 4.2.2 for image smoothing and some later
analysis steps. A Hogbom cleaning algorithm was used
to construct the image model; model components were
restricted to an arc-shaped region that encompassed the
northern and southern images, and a circle at the po-
sition of the western image, for all channels. The fi-
nal data cube was created to match the channelization
of the CO(3–2) data (rest frame spectral resolution of
14.129 km s−1). Since the naturally weighted channel
maps synthesized beam (0.0079 ×0.00 68 at a position an-
gle of −70.76◦) already provided higher angular resolu-
tion than our CO(3–2) data, we chose not to pursue still
higher resolution (at the cost of degraded SNR) with al-
ternative weighting schemes. Since the spatial extent of
J0901 is a substantial fraction of the VLA antenna pri-
2http://casa.nrao.edu

Resolved Properties of J0901 5
Table 1. J0901 VLA Observations
Configuration Date NAnt Weather
Max. Baseline
D 2010 April 3 17 Clear
1.031 km 2010 April 15 20 Clear
2010 May 4 19 Clear
2010 May 8 19 Average sky cover 25%; mixed clouds
2010 May 15 20 Sky cover 20%; cumuliform clouds
B 2011 February 14 26 Sky cover <30%; stratiform clouds
10.306 km
C 2012 January 29 26 Clear
3.289 km 2012 January 30 25 Clear
2012 January 31 25 Clear
2012 March 26 25 Sky cover 90%; stratiform clouds
2012 March 30 27 Sky cover 20%; stratiform clouds
mary beam FWHM, we applied a primary beam correc-
tion in order to retrieve the correct flux from the source
(a ∼10% correction for the northern image). The av-
erage noise for each channel is 0.136 mJy beam−1(prior
to correcting for the primary beam).
2.3. Submillimeter Array
We observed J0901 in continuum emission at the Sub-
millimeter Array using the 345 GHz receivers on 2010
May 20 and 2011 March 26 (project ID 2010A-S068,
2010B-S068; PI Baker). The observations were taken
with the array in its compact configuration, using seven
antennas (maximum baseline 66.5m) in 2010 and eight
antennas (maximum baseline 75.25m) in 2011. We
observed with the standard correlator setup that pro-
vided a maximum bandwidth of 4 GHz per sideband (for
a single receiver), with a channel width of 3.25 MHz.
The central frequency of the correlator was tuned to
312 GHz. During the observations, phase and amplitude
variations were tracked with interleaved observations of
the quasars 0854+201 and 0840+132. Mars and Titan
were observed as flux calibrators, and the quasar 3C279
was used for bandpass calibration.
Data calibration and mapping were carried out in
CASA version 4.1.0 after using the sma2casa.py and
smaImportFix.py scripts3to perform the initial system
temperature correction and convert the data format to
CASA measurement sets. The naturally weighted con-
tinuum map has a total bandwidth of 7.96 GHz and total
time on source of 9.15 hours, resulting in an RMS noise
of 0.75 mJy beam−1for a 2.00 09×2.0009 synthesized beam.
3http://www.cfa.harvard.edu/sma/casa/
2.4. SINFONI/VLT
We obtained integral field observations of Hαemission
from J0901 using the Spectrograph for Integral Field
Observations in the Near Infrared (SINFONI) instru-
ment (Eisenhauer et al. 2003;Bonnet et al. 2004) on the
Very Large Telescope (VLT) of the European South-
ern Observatory (ESO; program 087.A-0972, PI Baker).
Observations were obtained in seeing-limited mode with
0.2500 pixels, for which the SINFONI field of view is
800 ×800. Data were taken at three pointings correspond-
ing to the northern (observed 2012 January 7), south-
ern (observed 2012 January 8), and western images (ob-
served 2011 November 21 and December 17), targeted
via blind offsets from a reference star; for each pointing,
8×300 s exposures alternated between source and off-
set sky positions, with small dithers between successive
exposures to facilitate background subtraction. The to-
tal on-source integration time was therefore 1200s per
pointing (2400 s per pointing for the fainter western im-
age, which was deliberately visited twice). All data were
reduced with standard ESO pipeline routines using the
Gasgano interface. Point spread function (PSF) and flux
calibration relied on contemporaneous observations of a
nearby star with published 2MASS photometry.
After the pipeline calibration, we used noise clipping
to identify and mask out cosmic rays and channels af-
fected by sky lines. Since the three images of J0901
were observed on different nights, the PSFs were slightly
different for the three images (∆FWHM <0.001). We
smoothed the observations to the largest PSF among the
three images (the western image; 0.0075×0.00 65 at 11.5◦),
and we also created versions smoothed to the CO beam
size (for multi-line comparisons) if this was larger than

6Sharon et al.
the HαPSF. The three pointings were then combined
into a common cube, with no additional astrometric cor-
rections applied to the blind offset positions. In order
to make preliminary maps of the noise, continuum emis-
sion, line emission, and detector defects, we performed a
linear fit to each pixel (excluding the channels with Hα,
[N ii], or sky lines) and subtracted the fit cube from the
data. This process over-subtracts the background (due
to edges of skylines and cosmic rays that are not ex-
cluded), so we use these preliminary maps to mask out
J0901, foreground galaxies, and chip defects, and then
calculate the median sky level per channel within the
three sub-images. The sky level is then subtracted from
the data cube and then the data are re-fit to produce
our final continuum-subtracted data cube and contin-
uum map. Chip defects not removed by this process are
still somewhat noticeable near the edges of the images
(particularly the regions where dither patterns did not
overlap), but they dominate the continuum image due
to its low noise, so we mask out the outer 1.0025 of the
three sub-images for the continuum map. We calculate
the standard deviation of each pixel (excluding chan-
nels with emission lines) to produce an average noise
map. We then perform an additional astrometric correc-
tion using the integrated Hαand CO(3–2) maps and an
imaging cross-correlation algorithm provided by Adam
Ginsburg4to find and remove a 1.0032 offset between the
near-IR and radio data.
The spectral resolution of SINFONI is λ/∆λ∼4000;
the channel widths are 36.75 km s−1at the frequency of
the Hαline. We apply a 16 km s−1correction to convert
velocities to the same local kinematic standard of rest
used in the radio data. Since the three sub-images were
observed on different dates, we use the average heliocen-
tric corrections for the observations (which range from
12–22 km s−1) when analyzing the aggregate data, but
for the analysis of the spectral line profiles in each sub-
image, we apply their individual velocity corrections.
2.5. Hubble Space Telescope
We also use Hubble Space Telescope (HST) observa-
tions of J0901 to constrain the lens model. J0901 was
observe in Cycle 17 (Program ID 11602, PI S. Allam).
Imaging was performed with HST’s Wide Field Cam-
era 3 (WFC3) using filters F475W, F814W, F606W,
F160W, and F110W. We processed the data using the
standard AstroDrizzle reduction pipeline5. In order to
use these data for lens modeling, we also remove con-
4pixshift:http://casa.colorado.edu/∼ginsbura/corrfit.htm
5Part of DrizzlePac: http://drizzlepac.stsci.edu
taminating light from the foreground lens galaxies using
GALFIT (Peng et al. 2010).
3. RESULTS
We successfully detect the three images of J0901 in
both the CO(1–0) and CO(3–2) maps (Figure 1). In or-
der to make a fair comparison between the two maps,
we also analyze versions of the data cubes (including
the VLT data) that have been smoothed to a common
beam/PSF (the smallest Gaussian resolution FWHM
that all datasets can be smoothed to is 1.00 34 ×1.00 10
at a position angle of 41.10◦, which is close to the na-
tive resolution of the CO(3–2) data); we refer to the
two sets of maps as the “natural” and “matched” maps
below. For the matched CO(1–0) data, in addition to
smoothing to the common beam, we also exclude base-
lines that have uv distances smaller than the minimum
for the CO(3–2) data; the uv-clipping ensures that flux
distributed on large spatial scales that cannot be de-
tected at the PdBI is also excluded from the CO(1–0)
maps. The smoothing most strongly changes the sur-
face brightness distribution in the southern image for the
CO(1–0) data, increasing the peak surface brightness by
∼30% and thus exaggerating the asymmetry between
the two peaks in brightness (see Figure 1). However,
the uv-clipping removes only a small fraction of the to-
tal CO(1–0) flux (<10%).
The measured line fluxes are summarized in Table 2
and are extracted over identical image areas for the three
maps; the uncertainties include an assumed 10% flux
calibration error. For the spectra in Figure 2, we use the
natural maps 6. We find that the spectra of the CO(1–0)
and CO(3–2) lines have a consistent FWZI ≈350 km s−1
centered at the Hα-determined systemic redshift from
Hainline et al. (2009), but that the shapes of the two
CO line profiles differ for the same images. The different
relative line profiles for the two CO lines in all three
images suggest that differential lensing is occurring; i.e.,
the spatial variation of the magnification factor across
J0901 is amplifying the light in regions with different
CO(3–2)/CO(1–0) line ratios (e.g., Blain 1999;Serjeant
2012). In Figure 6we show the overlaid channel maps
of the natural CO(1–0) and CO(3–2) lines, rebinned by
a factor of two; there is a clear velocity gradient across
the three images, suggesting J0901 is either disk-like or
a merging galaxy.
6Due to the velocity structure of J0901 and the small synthe-
sized beam of the natural CO(1–0) map, we extract the spectra
over slightly smaller regions; the larger regions used in the rest
of the analysis include enough signal-free pixels in the individual
channel maps to significantly increase the noise for the integrated
spectra.

Resolved Properties of J0901 7
40″
35″
30″
25″
18° 14′ 20″
Declination (J2000)
Right Ascension (J2000)
21s.822s.022s.222s.422s.622s.823s.009h 01m 23s.2
CO(1–0) 40″
35″
30″
25″
20″
Declination (J2000)
Right Ascension (J2000)
21s.822s.022s.222s.422s.622s.823s.023s.2
CO(3–2)
Figure 1. Integrated CO(1–0) (left) and CO(3–2) (right) intensity maps (with primary beam corrections applied); the CO(1–0)
map is the “natural” map that has not been uv-clipped to match the CO(3–2) map. Contours are multiples of ±2σ1–0 for the
CO(1–0) map and are powers of 2 × ±σ3–2 (i.e. ±2σ,±4σ,±8σ, etc.) for the CO(3–2) map (σ1–0 = 9.1 mJy km s−1beam−1;
σ3–2 = 41 mJy km s−1beam−1). Negative contours are dotted and the synthesized beams are shown at lower left. Blue lines
indicate the lens model critical curves (Section 4.1). Black crosses mark the mean dynamical center determined from the
source-plane reconstructions and dynamical modeling (see Section 4.2).
Velocity Offset (km s–1)
-400 -200 0 200 400
40
30
20
10
0
Sν (mJy)
-10
CO(3–2)
northern
southern
western
Sν (mJy)
Velocity Offset (km s–1)
-400 -200 0 200 400
4
2
0
-2
6CO(1–0)
northern
southern
western
Figure 2. VLA CO(1–0) spectra (left) and PdBI CO(3–2) spectra (right) extracted from the “natural” maps for the northern
(black/solid), southern (red/dashed), and western (blue/dotted) images, plotted relative to the z= 2.2586 Hαsystemic redshift
from Hainline et al. (2009).
We also detect the three images of J0901 in Hαand
[N ii] using the VLT/SINFONI data (Fig. 3). The mea-
sured line fluxes are given in Table 2; the statistical un-
certainties are determined by weighted Gaussian fits to
the line shapes. The spectra for the Hαand [N ii] lines
do not show the double-peaked structure seen in the CO
lines. However, the FWHMs derived from fitting Gaus-
sians to the Hαand [N ii] line profiles (Table 3; after
accounting for instrumental broadening) are consistent
with single Gaussian fits to the CO line profiles.
We successfully detect continuum emission from J0901
at the SMA (295 µm rest frame), the VLA (2.6 mm rest
frame), and the VLT (0.66 µm rest frame; Fig. 5). Our
continuum flux measurements are given in Table 2. We
detect all three images for both the SMA and VLT
continuum maps. For the VLA continuum map, we
definitely detect rest-2.6 mm continuum emission from
the southern image, we marginally detect the north-
ern image, and we do not detect the western image
(Fig. 5). For both the VLA and VLT maps, we also
detect continuum emission from the lensing group galax-
ies (corresponding to rest wavelengths of ∼3.5 mm and
∼0.27 µm at the redshift of the lensing group), although
most group members are masked out in the VLT con-
tinuum image since they are near the edges of the field
of view. For the VLA and SMA data, we compare the

8Sharon et al.
40″
35″
30″
25″
18° 14′ 20″
Declination (J2000)
Right Ascension (J2000)
21s.822s.022s.222s.422s.622s.823s.009h 01m 23s.2
Hα 40″
35″
30″
25″
20″
Declination (J2000)
Right Ascension (J2000)
21s.822s.022s.222s.422s.622s.823s.023s.2
[NII]
Figure 3. Integrated Hα(left) and [N ii] (right) intensity maps of J0901. Due to SINFONI’s small field of view, the three
images of J0901 were observed separately and have been smoothed to the same PSF (0.00 75 ×0.00 65) shown at the bottom left
corners. Contours are multiples of ±2¯σ(where ¯σ= 8.0×10−16 erg s−1cm−2is the average noise for the three sub-images);
negative contours are dashed. Cyan lines indicate the lens model critical curves (Section 4.1). Black crosses mark the mean
dynamical center determined from the source-plane reconstructions and dynamical modeling (see Section 4.2).
Hα
[NII]
northern
southern
western
0.650 0.652 0.654 0.656 0.658 0.660 0.662 0.664
0
1
2
3
4
S
ν
(10
–13
erg s
–1
cm
–2
μm
–1
)
−2000 −1000 0 1000 2000 3000
Δv
Hα
(km s
–1
)
λ
rest
(μm)
Figure 4. VLT spectra showing the Hαand [N ii] lines (as
well as continuum emission) for the northern (black/solid),
southern (red/dashed), and western (blue/dotted) images,
plotted relative to rest wavelength using the z= 2.2586 Hα
systemic redshift from Hainline et al. (2009). Channels with
zero emission correspond to sky-line masks.
distribution of the continuum emission to the CO(3–2)
line emission (smoothed to the continuum maps’ spatial
resolutions; the results are qualitatively similar when
comparing to the smoothed CO(1–0) line maps). The
rest 295 µm continuum emission peaks at the same lo-
cation as the CO emission for the three lensed images.
However, for the northern image, the 295 µm continuum
emission is not as spatially extended as the CO. The
missing extended emission is either below the sensitiv-
ity of our current maps, or the dust distribution does not
perfectly trace the molecular gas within J0901 (regard-
less of any complications caused by lensing). While the
SNR for the VLA continuum map is limited, the rest
2.6 mm emission in the southern image is offset from
the peak in CO emission. As the rest 2.6 mm contin-
uum emission would likely trace either star formation or
a central AGN, the offset is somewhat peculiar.
4. ANALYSIS
4.1. Lens modeling and source-plane reconstruction
4.1.1. Methods
J0901 is lensed by a group of galaxies, which needs to
be accounted for explicitly in order to reconstruct the
galaxy’s source-plane structure. Our lens model there-
fore comprises one component representing the group
halo and others representing the group members. The
former is described by an elliptical power-law density
distribution, whose (spherical) convergence profile is
given by
κ(~x) = b2−α
2|~x|2−α,(1)
where bis the Einstein radius. The group members
within two Einstein radii are represented by singular
isothermal ellipsoids (SIEs) given by equation (1) with
α= 1. In this case, bnot only represents the Einstein
radius, but is also related to the velocity dispersion σv
by b∝σ2
v.7
7This relation does not strictly hold for elliptical mass distribu-
tions, but the corrections are negligible for small ellipticities (e.g.,

Resolved Properties of J0901 9
40″
35″
30″
25″
18° 14′ 20″
Declination (J2000)
Right Ascension (J2000)
21s.822s.022s.222s.422s.622s.823s.009h 01m 23s.2
8.5 mm 40″
35″
30″
25″
20″
Declination (J2000)
Right Ascension (J2000)
21s.822s.022s.222s.422s.622s.823s.023s.2
877 μm 40″
35″
30″
25″
20″
Declination (J2000)
Right Ascension (J2000)
21s.822s.022s.222s.422s.622s.823s.023s.2
2.2 μm
Figure 5. VLA 8.5 mm (left/contours; plotted over the CO(3–2) integrated line map smoothed to the same resolution),
SMA 877 µm (center/contours; plotted over the CO(3–2) integrated line map smoothed to the same resolution), and VLT
2.2µm continuum maps of J0901 (where the wavelengths listed are in the observed frame). For the VLA data, a 100 taper
was applied, resulting in a 2.0022 ×2.00 00 resolution map (beam shown at bottom left). Contours are multiples of ±1.5σ
(σ= 26.6µJy beam−1). The SMA 2.00 09 ×2.00 09 beam FWHM is shown at lower left. Contours are multiples of ±2σ(σ=
0.75 mJy beam−1). Due to SINFONI’s small field of view, the three images of J0901 were observed separately and have been
smoothed to same PSF (0.0075 ×0.00 65) shown at the bottom left. Contours are powers of 2 × ± ¯σ(i.e., ±2 ¯σ,±4 ¯σ,±8¯σ, etc.;
where ¯σ= 5.2×10−18 erg s−1cm−2µm−1is the average noise for the three sub-images). An additional 1.0025 was masked around
the image edges compared to the integrated line maps for clarity (the edges have significant defects which are only apparent in
the high S/N of the continuum map). The other bright continuum sources are members of the lensing cluster. For all maps,
negative contours are dashed and crosses mark the mean dynamical center determined from the source-plane reconstructions
and lens modeling (see Section 4.2). Cyan lines indicate the lens model critical curves (Section 4.1).
While the position and ellipticity of the group halo
are allowed to vary, the group members’ positions and
ellipticities are fixed to the observed values. Addi-
tionally, a log-normal prior about the nominal Faber-
Jackson relation (Faber & Jackson 1976) is placed on
their velocity dispersions. For any two galaxies G1and
G2, equation (1) and the Faber-Jackson relation give
b2/b1∝σ2
v,2/σ2
v,1∝√L2/√L1, where Liis the observed
luminosity of Gi. Using mass as a proxy for luminosity,
we set priors, noting that Gallazzi et al. (2006) find that
the scatter in the logarithmic mass-velocity dispersion
relation is ≈0.07 for early-type galaxies selected from
the Sloan Digital Sky Survey (Abazajian et al. 2004).
We also note that the presence of a galaxy at the loca-
tion of the southern image represents a unique challenge
given its close proximity. Due to its small halo mass, fits
with a SIE model are challenging since deflections due
to that potential never reach zero. Since this is a smaller
galaxy in a dense environment, its mass profile may be
tidally truncated, and we therefore adopt a truncated,
elliptical pseudo-Jaffe profile (Keeton 2001) to represent
Chae 2003;Huterer et al. 2005). The proportionality constant
depends on the ellipticity.
this component. The spherical convergence profile for
this model is given by
κ(~x) = b0
2|~x|2+s2−1
2−|~x|2+a2−1
2,(2)
where sand aare the core and truncation radii, respec-
tively. The truncated pseudo-Jaffe assumption allows us
to explore truncated mass models, but preserves more
extended profile options in the limit that the truncation
radius (a) approaches infinity. The best-fit lens model
parameters for all components are listed in Table 4.
The data used to constrain the model consist of the
HST F606W imaging and the integrated CO(3–2) in-
tensity map (Figure 7). The pair of merging images
comprising the northern arc lie across a critical curve
in the image plane and are more highly magnified than
the southern and western images (Figure 8). A larger
magnification can allow for a more detailed analysis, but
only over the fraction of the source that has crossed the
caustic. There is also a larger uncertainty associated
with the source-plane reconstruction using the northern
arc, as the magnification varies rapidly near the critical
curve (Figure 8). For these reasons, we do not include
the northern arc when constraining the lens model pa-
rameters or performing the source-plane reconstructions
presented throughout.

10 Sharon et al.
Table 2. J0901 emission line and continuum measurements (magnification-corrected where indicated)
Line/Map Parameter Units North South West Total
CO(1–0) S1–0∆vJy km s−11.41 ±0.16 0.94 ±0.12 0.60 ±0.08 2.95 ±0.32
natural L0
CO(1–0) 1010 K km s−1pc238.4±4.5 25.5±3.2 16.3±2.2 3.53+0.57
−0.45
b
r3,10.74 ±0.11 0.84 ±0.14 0.62 ±0.11 0.75 ±0.11
CO(1–0) S1–0∆vJy km s−11.34 ±0.16 0.87 ±0.11 0.59 ±0.08 2.80 ±0.30
matched L0
CO(1–0) 1010 K km s−1pc236.4±2.1 23.8±3.0 16.0±2.1 1.62+0.35
−0.27
b
r3,10.78 ±0.12 0.91 ±0.15 0.63 ±0.12 0.79 ±0.12
CO(3–2) S3–2∆vJy km s−19.35 ±1.00 7.10 ±0.76 3.36 ±0.41 19.8±2.0
L0
CO(3–2) 1010 K km s−1pc228.3±3.0 21.5±2.3 10.2±1.2 1.99+0.32
−0.29
b
Hα SHα∆v10−16 erg s−1cm−25.72 ±0.57 7.69 ±0.74 4.29 ±0.36 18.46 ±1.01
LHα1042 erg s−124.8±2.5 33.3±3.2 18.6±1.6 2.70+0.39
−0.32
b
[N ii]S[N ii]∆v10−16 erg s−1cm−22.80 ±0.57 4.00 ±0.74 2.16 ±0.35 9.60 ±1.01
L[N ii]1042 erg s−112.1±2.5 17.4±3.2 9.4±1.5 1.53+0.36
−0.28
b
8.5 mm S8.5 mm mJy 0.33 ±0.09 0.25 ±0.07 <0.08a0.66 ±0.12
877 µmS877 µmmJy 17.7±5.3 13.0±3.8 4.3±2.9 35.0±8.8
2.2µmS2.2µm10−14 erg s−1cm−2µm−11.40 ±0.08 3.10 ±0.10 0.76 ±0.10 5.26 ±0.14
a3σupper limit assuming a point-like flux distribution.
bThe total line luminosities are magnification corrected assuming the corresponding magnification factors
listed in Table 5(i.e. the “natural” magnification factors calculated using the native resolution data pre-
sented in the bulk of this table, or the “matched” magnification factors for the CO(1–0) data uv-clipped to
create the matching resolution datasets).
Note—The VLT observations include statistical uncertainties only. The integrated line fluxes are from
Gaussian fits to the spectra. Since each image was observed on a different night, the spectra were cor-
rected for their different heliocentric velocities before being combined. Therefore, the total integrated line
fluxes/luminosities differ slightly from the sum from the individual images.
In addition to optimizing the lens model parameters,
we include a registration offset between these data sets
(referenced to the CO(3–2) data). For each set of lens
model parameters and registration offsets, a goodness-
of-fit statistic is computed by multiplicatively combin-
ing the Bayesian evidence from the optical and radio.
We use the framework described in Tagore & Keeton
(2014), Vegetti & Koopmans (2009), and Suyu et al.
(2006) to reconstruct the pixelated source distribution
of J0901 in the source plane, as seen in each band. An
irregular, adaptive source grid is used with priors on
the sources’ surface brightness in the form of curvature
regularization; the Bayesian evidence is maximized at
each step. After optimization, slight discrepancies be-
tween the optical data and the model remain. We add
smoothly varying, non-parametric perturbations to the
potential to compensate for limitations of the macro-
model (Figure 8). These lens potential perturbations are
at the 1–2% level, which correspond to changes in the
deflection angle of 100 mas or less. For the optical HST
data, such changes are significant; however, because the
beam size is ∼100 in the radio bands, the effect on the
CO data is negligible.
Lens modeling of interferometric maps is complicated
by the imaging process, which does not conserve sur-
face brightness, can be strongly affected by choices in
mapping parameters (e.g., visibility weights), and yields
noise that is correlated in the resulting image. All
of these effects can potentially cause the lens model
and source-plane reconstruction to diverge from real-
ity. While a number of routines have been developed
in recent years to constrain lens models using visibility
data directly (e.g., Bussmann et al. 2012,2013;Heza-
veh et al. 2013,2016;Rybak et al. 2015;Spilker et al.
2016;Dye et al. 2018), many rely on parametric source
models, which are overly simplistic compared to the re-
solved observations we have for J0901. Recognizing that
lens models derived from visibility data and from decon-
volved maps are both fundamentally limited by incom-
plete sampling in the uv plane, we prefer to exploit the
well-resolved structure in our maps of J0901 to derive
our lens model. We defer comparisons with source-plane

Resolved Properties of J0901 11
Table 3. Gaussian fits to the spectral lines
Line/Map Parameter North South West Total
CO(1–0) Sν,peaka5.92 ±0.55/5.54 ±0.61 4.53 ±0.36 2.72 ±0.27 10.1±2.0/10.5±1.2
natural FWHM b135 ±22/110 ±20 210 ±22 226 ±29 134 ±27/161 ±35
voffsetb−81 ±9/80 ±9 24 ±9−38 ±12 −81 ±18/64 ±22
CO(1–0) Sν,peaka5.60 ±0.53/5.41 ±0.57 4.24 ±0.35 2.47 ±0.26 9.4±1.4/10.4±0.8
matched FWHMb124 ±20/109 ±18 196 ±21 227 ±30 118 ±22/157 ±28
voffsetb−81 ±8/77 ±8 26 ±8−36 ±12 −83 ±13/61 ±15
CO(3–2) Sν,peaka33.5±3.2/36.6±2.3 29.8±1.3 15.0±1.6/12.3±1.6 69.0±4.0/79.7±3.4
FWHM b105 ±14/151 ±19 237 ±13 125 ±22/124 ±27 122 ±12/138 ±12
voffsetb−92 ±7/59 ±8 19 ±5−95 ±9/71 ±12 −86 ±6/66 ±6
Hα Sν,peakc2.41 ±0.15 3.13 ±0.19 1.63 ±0.09 7.12 ±0.25
FWHMb312 ±24 323 ±24 347 ±23 341 ±14
voffsetb13 ±10 25 ±10 −29 ±9 9 ±6
[Nii]Sν,peak c1.19 ±0.15 1.65 ±0.19 0.87 ±0.09 3.78 ±0.25
FWHM b308 ±49 318 ±45 324 ±41 333 ±27
voffsetb20 ±20 19 ±19 5 ±17 16 ±11
aIn units of mJy.
bIn units of km s−1.
cIn units of 10−13 erg s−1cm−2µm−1.
Note—Multiple values are listed for double Gaussian fits where those fits preferred two spectral peaks. Centroid
velocity offsets are measured relative to the z= 2.2586 Hαsystemic redshift from Hainline et al. (2009).
reconstructions inferred from non-parametric visibility-
based models to future work.
In order to account for the image-plane correlated
noise in our lens modeling, we follow the noise scal-
ing technique of Riechers et al. (2008). For an indi-
vidual data set, we scale the noise (for input into the
lens modeling code) by some factor greater than unity
that could be determined and verified by comparing the
statistical properties of noise residuals in areas where
lensed features are present and absent. However, be-
cause we are comparing source reconstructions across
various data sets with different noise properties, we fix
the noise scaling. A large noise scaling factor allows
the code to under-fit the data in the formal reduced-χ2
sense, since the code assumes there is more noise in the
data, which leads to a higher regularization strength.
Qualitatively, this approach smooths the source over
a larger physical scale, and the resulting source-plane
beam is larger.
Our source-plane reconstructions yield a spatially
varying synthesized beam/PSF. In Figure 9we show
a grid of the beam HWHMs overlaid on a contour
plot of the source-plane reconstruction for the matched
CO(3–2) integrated line map as an example of the vari-
ation in beam/PSF shape that results from de-lensing.
Although the beam shape varies by a factor of a few over
the entire reconstruction, the beam is smaller and more
consistent in the direction of the emission for J0901.
We therefore adopt surface-brightness weighted average
beams/PSFs when analyzing the spatial information for
J0901; these have FWHMs of 0.2–0.300 (corresponding
to physical scales of 1.7–2.6 kpc).
The lens model uncertainties are explored via Markov
chain Monte Carlo modeling for the CO(3–2) data only
to save computational time. As the CO(3–2) moment
map was the primary input used to constrain the lens
model, this method accurately captures the uncertain-
ties in lens model parameters. Magnification factors are
then derived for the individual maps by de-lensing the
emission for the distribution of model parameters. The
magnification factor uncertainties thus take into account
uncertainties in both the surface brightness of the source
and in the lens model parameters.
4.1.2. Resulting magnification factors and image
reconstructions
With the lens model optimized, we perform source
reconstructions of the integrated line maps, the individ-
ual velocity channel maps, and the 2.2µm continuum
map. We present the natural resolution source-plane re-

12 Sharon et al.
RA Offset (arcsec)
DEC Offset (arcsec)
0 -5 -10510
0
-5
-10
5
10 CO(1–0) Velocity
(km s-1)
-159.94
-131.57
-103.21
-74.84
-46.48
-18.11
10.25
38.62
66.99
95.35
123.72
152.08
RA Offset (arcsec)
DEC Offset (arcsec)
0 -5 -10510
0
-5
-10
5
10 CO(3–2) Velocity
(km s-1)
-159.94
-131.57
-103.21
-74.84
-46.48
-18.11
10.25
38.62
66.99
95.35
123.72
152.08
RA Offset (arcsec)
DEC Offset (arcsec)
0 -5 -10510
0
-5
-10
5
10 Hα Velocity
(km s-1)
-170.86
-136.53
-102.19
-67.85
-33.52
0.82
35.15
69.49
103.82
138.16
172.50
RA Offset (arcsec)
DEC Offset (arcsec)
0 -5 -10510
0
-5
-10
5
10 [NII]Velocity
(km s-1)
-154.35
-120.12
-85.89
-51.66
-17.44
16.79
51.02
85.25
119.48
153.70
187.93
Figure 6. Overlaid contours of the natural resolution CO(1–0) (upper left), CO(3–2) (upper right), Hα(lower left), and [Nii]
(lower right) channel maps, colorized by their velocities relative to the z= 2.2586 Hαsystemic redshift from Hainline et al.
(2009). The images are centered at α(J2000) = 09h01m22.s42 and δ(J2000) = +18◦14030.009. For the two CO lines we show only
the ±3σcontours (σ1–0 = 0.21 mJy beam−1,σ3–2 = 0.82 mJy beam−1) where the channels have been rebinned by a factor of two
to 28.37 km s−1. For the VLT/SINFONI data we show only the ±2σcontours (σVLT = 2.2×10−16 erg s−1cm−2µm−1) and have
not done any additional channel binning. Negative contours are dotted. For clarity we do not use the primary beam-corrected
data for the two CO lines and we mask out the outer 1.00 25 (10 pixels from the dither pattern) for the VLT data. Synthesized
beams and PSFs are shown at lower left. Gray lines indicate the lens model critical curves (Section 4.1). Black crosses mark
the mean dynamical center determined from the source-plane reconstructions and lens modeling (see Section 4.2).
constructions of the CO, Hα, and [N ii] lines for J0901
in Figures 10 and 11. In Table 5, we present the 50th
percentile magnification factors and 68% confidence in-
tervals derived using the “natural” resolution data and
the magnification factors derived from the “matched”
resolution data, for each image separately and in aggre-
gate.
While the CO, Hα, and [N ii] lines all show two emis-
sion peaks in the southern arc in the image plane, those
peaks do not correspond to one another across all lines.
In the source-plane reconstructions, the CO peaks re-
main distinct but the Hαand [N ii] peaks do not. The
two peaks seen in the VLT maps are nearly aligned
with the positions of the average dynamical center deter-
mined from the channelized source reconstructions (see
Section 4.2), and are potentially multiple images of the
same region within J0901 caused by a foreground mem-
ber of the lensing group. However, the two peaks may

Resolved Properties of J0901 13
Table 4. Best-fit lens model parameters
Object(s) Model b∆RA ∆DEC e P A s a α
(00) (00 ) (◦) (00) (00 )
Group halo SPLE 2.1157 −0.0157 −0.1954 0.331 −82.7 1.51
Central galaxies SIS 0.7184 0.0585 −0.0147 1.0
SIS 0.9551 −0.5820 −0.7580 1.0
Southern perturber p-Jaffe 1.0833 3.7344 −8.4207 0.244 21.0 0.3295 0.5045
Other galaxies SIS 0.26420 −2.2917 6.8895 1.0
SIS 0.0735 −4.4900 7.5848 1.0
SIS 0.1843 −5.6797 6.3216 1.0
SIS 0.0290 −4.9115 10.7200 1.0
SIS 0.0475 −4.8044 2.3711 1.0
SIS 0.3614 −7.4541 −0.2359 1.0
SIS 0.9159 −9.7170 −6.1960 1.0
SIS 0.1484 −10.8208 −9.2987 1.0
SIS 0.0605 3.5408 7.5583 1.0
SIS 0.1418 9.3071 −5.0192 1.0
SIS 0.0854 3.1367 −4.9471 1.0
SIS 0.0882 0.4415 −7.6127 1.0
SIS 0.0344 6.4117 −8.7968 1.0
SIS 0.0502 −2.4900 −13.3387 1.0
Note—From left to right, the columns are: a description of the model component, the assumed model
for the shape of the lensing potential (either a softened power law ellipsoid (SPLE), single isothermal
sphere (SIS), or pseudo-Jaffe ellipsoid (p-Jaffe)), normalized amplitude (varied), offset in right ascension
(from 09h01m22.s3865; fixed), offset in declination (from 18◦14032.006303; fixed), ellipticity (only relevant
for SPLE and p-Jaffe models; fixed), position angle (only relevant for SPLE and p-Jaffe models; fixed),
the core radius (only relevant for p-Jaffe model), the truncation radius (only relevant for p-Jaffe model),
and index of the power law (only relevant for SPLE model and assumed to be 1.0 for SIS models).
Table 5. Magnification factors
Transition Map North South West Total
CO(1–0) natural 10.2+1.2
−0.97.4+0.6
−0.55.3+0.4
−0.422.7+2.1
−1.5
matched 20.9+4.3
−2.915.1+2.7
−1.911.1+2.0
−1.547.2+8.8
−5.7
CO(3–2) natural 14.2+1.8
−1.610.4+1.4
−1.35.5+0.8
−0.730.1+3.7
−3.2
matched 14.1+1.8
−1.510.7+1.2
−1.15.7+0.7
−0.630.6+3.3
−2.9
Hαnatural 11.8+2.2
−1.911.4+1.7
−1.36.3+0.9
−0.729.6+4.0
−3.1
matched 12.2+2.0
−1.711.9+1.9
−1.75.7+0.9
−0.729.9+4.3
−3.3
[N ii] natural 11.2+2.7
−2.111.5+3.3
−2.04.5+1.1
−0.827.2+5.8
−4.1
matched 9.7+1.2
−1.18.8+1.3
−1.03.5+0.5
−0.421.9+2.5
−2.1
2.2µm natural 11.1+5.4
−3.716.7+9.5
−4.28.6+3.8
−1.937.1+16.1
−8.1
matched 8.9+4.6
−3.316.9+12.1
−5.17.9+5.3
−2.333.7+19.2
−8.9

14 Sharon et al.
35″
30″
25″
18° 14′ 20″
Declination (J2000)
21s.822s.022s.222s.422s.622s.809h 01m 23s.0
CO(3–2)
data 35″
30″
25″
20″
21s.822s.022s.222s.422s.622s.823s.0
CO(3–2)
model 35″
30″
25″
20″
21s.822s.022s.222s.422s.622s.823s.0
CO(3–2)
residual
35″
30″
25″
18° 14′ 20″
Declination (J2000)
Right Ascension (J2000)
21s.822s.022s.222s.422s.622s.809h 01m 23s.0
HST
data 35″
30″
25″
20″
Right Ascension (J2000)
21s.822s.022s.222s.422s.622s.823s.0
HST
model 35″
30″
25″
20″
Right Ascension (J2000)
21s.822s.022s.222s.422s.622s.823s.0
HST
residual
Figure 7. Residual differences (right panels) between the observed image-plane data (left panels) and best-fit lensing model
image-plane reconstructions (center panels) for the two datasets used to constrain the model: the CO(3–2) map (top row) and
HST F606W image (bottom row). Pixels not used in constraining the data are masked out (most notably the northern image;
see text for discussion). Critical curves are shown in blue. Contours are powers of 2 × ±σ(i.e. ±2σ,±4σ,±8σ, etc.); negative
contours are dotted.
Right Ascension (J2000)
21s.822s.022s.222s.422s.622s.823s.0
log(Magnification)
1.81.20.60.0–0.6–1.2–1.8 3.63.02.4
40″
35″
30″
18° 14′ 25″
Declination (J2000)
40″
35″
30″
25″
Declination (J2000)
Right Ascension (J2000)
21s.822s.022s.222s.422s.622s.809h 01m 23s.0
2D Gravitational Potential Perturbations (radians2)
0.100.0–0.10 0.300.20 0.40
Figure 8. The log of the magnification (left) and the 2D projection of the non-parametric perturbations to the lensing potential
(right; normalized by the critical lensing density) for the best-fit lens model. Contours for the integrated line CO(3–2) map are
shown in black (left) or white (right). Contours are powers of 2 × ±σ(i.e. ±2σ,±4σ,±8σ, etc.); negative contours are dotted.
In the right panel we also show the lens model critical curves in grey.

Resolved Properties of J0901 15
09
h
01
m
22
s
.45 22
s
.40 22
s
.35 22
s
.30 22
s
.25
Right Ascension (J2000)
18° 14′ 30.0″
30.5″
31.0″
31.5″
32.0″
32.5″
33.0″
Declination (J2000)
Figure 9. The synthesized beam/PSF for the matched
datasets as a function of position in the source plane. The
black vectors are the beam/PSF HWHM at the pixel for
their common origin; every eighth pixel is shown for clarity.
The red contours show the source-plane reconstruction of the
CO(3–2) integrated line map using the matched resolution
data. Contours are multiples of ±3σobs, but note that due to
spatial variation in the noise, these surface brightness levels
do not correspond to lines of constant significance. Negative
contours are dashed. The blue lines indicate the source-plane
lensing caustics.
also have disappeared on reconstruction due to the de-
gree of regularization (i.e., the smoothness prior may
have “won” over fitting the data due to noise or flaws in
the lens model), and/or because the CO and HST data
used to constrain the model may not have much power
over the relatively small region encompassed by the two
VLT peaks.
We also reconstruct J0901 in the source plane for
the individual channel maps (Figure 12). The promi-
nent velocity gradient observed in the image plane is
also apparent in the reconstructed channel maps. The
well-resolved and smooth velocity gradient seen in all
lines suggests that J0901 is likely a disk galaxy, despite
the two bright peaks seen in the integrated line maps.
We extract the spectra from the reconstructed channel
maps and compare the line profiles to the observed pro-
files from the image plane (Figure 13). Since the per-
channel magnification factors were not computed to in-
clude the northern image, we use the sum of the south-
ern and western observed spectra scaled by the mean
per-channel magnification factor in order to understand
what effects differential lensing might have on the line
profile8. We extracted the source-plane spectra in aper-
8The per-channel magnification factors are, on average, lower
than what was determined for the integrated line maps, and they
tures defined by the SNR >2 regions in the correspond-
ing integrated line maps. We note that this method is
not a perfect match to the procedure used to extract the
image-plane spectra; a more perfect match would require
de-lensing the image-plane aperture for each channel.
Since the area occupied by a channel’s emission varies
with velocity (as expected, particularly when consider-
ing the variation in magnification factor), the aperture
defined by the integrated line map reconstruction may
miss some emission in individual channels. However,
this method is adequate for revealing any dramatic or
velocity-correlated differential lensing effects.
Differential lensing does not appear to strongly affect
the shape of the line profile of J0901 in the southern
and western images. Since it is the bright northern im-
age that only captures a portion of J0901’s source plane
structure (and thus only a portion of the velocity struc-
ture), one might suspect that any distortions of the line
profile are most likely to appear in analyses that include
the northern image. However, it is the northern image’s
CO spectral profile that shows the double-peaked struc-
ture typical of rotating disks (Figure 2), which is perhaps
only hinted at in the combined spectrum of the southern
and western images and their reconstruction (Figure 13).
While the spatial structure of the least-distorted west-
ern image best matches the source-plane reconstruction,
as expected, the de-lensed Hαand [N ii] spectral lines
appear to peak at redder wavelengths than seen in the
observed spectrum of the western image. In addition,
some of the internal structure of J0901 is multiply im-
aged within the Southern arc due to a foreground lensing
group member. We are therefore unable to firmly con-
strain the intrinsic profiles of the Hαand [N ii] spectral
lines for J0901.
4.2. Integrated properties: masses and SFR
4.2.1. Gas mass and dust-to-gas ratio
In order to estimate a gas mass for J0901, we use
the magnification-corrected natural CO(1–0) line lu-
minosity derived from all three images, obtaining
Mgas = (1.6+0.3
−0.2)×1011(αCO /4.6) M(Solomon &
Barrett 1991). We use the Milky Way CO-to-H2con-
version factor due to J0901’s disk-like ordered rota-
tion (Figure 12), but it is also the value favored by
the Narayanan et al. (2012) continuous metallicity and
surface-brightness dependent version of the CO-to-H2
conversion factor. The metallicity-dependent form of
the CO-to-H2conversion factor presented in Genzel
are much noisier. We therefore exclude unphysical magnification
factors outside the range of 0–100 when computing the mean mag-
nification factor for this comparison.

16 Sharon et al.
33.0″
32.0″
31.0″
30.5″
18° 14′ 30.0″
Declination (J2000)
Right Ascension (J2000)
22s.2522s.3022s.3522s.4009h 01m 22s.45
32.5″
31.5″
CO(1–0)
0.120.090.060.030.0–0.03–0.06
Flux (Jy km s–1 beam–1)
33.0″
32.0″
31.0″
30.5″
30.0″
Declination (J2000)
Right Ascension (J2000)
22s.2522s.3022s.3522s.4009h 01m 22s.45
32.5″
31.5″
CO(3–2)
1.20.90.60.30.0–0.3–0.6
Flux (Jy km s–1 beam–1)
Figure 10. Source-plane reconstructions of the integrated CO(1–0) (left) and CO(3–2) (right) intensity maps, derived from the
natural resolution observed images with primary beam corrections (shown in Figure 1). Since the reconstructions have spatially
varying noise, the contours are generated from the SNR maps and show multiples of ±3σ(negative contours are dotted), which
do not strictly follow the surface brightness (color bar; where the minimum and maximum values of the images are shown with
vertical dotted lines). The images also have spatially varying resolution, so we show the intensity-weighted average beams at the
lower left. Blue lines indicate the image-plane lensing caustics. Black crosses mark the mean dynamical center (see Section 4.2).
33.0″
32.0″
31.0″
30.5″
30.0″
Declination (J2000)
Right Ascension (J2000)
22s.2522s.3022s.3522s.4009h 01m 22s.45
32.5″
31.5″
NII
Flux (10–15 erg s–1 cm–2)
7.56.04.53.01.50–1.5
33.0″
32.0″
31.0″
30.5″
18° 14′ 30.0″
Declination (J2000)
Right Ascension (J2000)
22s.2522s.3022s.3522s.4009h 01m 22s.45
32.5″
31.5″
Hα
7.56.04.53.01.50–1.5
Flux (10–15 erg s–1 cm–2)
33.0″
32.0″
31.0″
30.5″
30.0″
Declination (J2000)
Right Ascension (J2000)
22s.2522s.3022s.3522s.4009h 01m 22s.45
32.5″
31.5″
2.2 μm
0.1250.0100.0750.0500.0250–0.025
Flux density (10–15 erg s–1 cm–2 μm–1)
Figure 11. Source-plane reconstructions of the Hαintegrated line (left), [Nii] integrated line (middle), and 2.2µm (observed
frame) continuum (right) intensity maps, derived the natural resolution observed images (shown in Figures 3and 5). SNR
contours, the intensity-weighted average PSF, caustics, and their descriptions are as given in Figure 10. Black crosses mark the
mean dynamical center (see Section 4.2).

Resolved Properties of J0901 17
RA Offset (arcsec)
DEC Offset (arcsec)
0 –11
0
–1
1
CO(1–0) Velocity
(km s–1)
–159.94
–131.57
–103.21
–74.84
–46.48
–18.11
10.25
38.62
66.99
95.35
123.72
152.08
RA Offset (arcsec)
DEC Offset (arcsec)
0 –11
0
–1
1
CO(3–2) Velocity
(km s–1)
–159.94
–131.57
–103.21
–74.84
–46.48
–18.11
10.25
38.62
66.99
95.35
123.72
152.08
DEC Offset (arcsec)
Velocity
(km s–1)
–170.86
–136.53
–102.19
–67.85
–33.52
0.82
35.15
69.49
103.82
138.16
172.50
RA Offset (arcsec)
0 –11
0
–1
1
Hα Velocity
(km s–1)
–154.35
–120.12
–85.89
–51.66
–17.44
16.79
51.02
85.25
119.48
153.70
187.93
RA Offset (arcsec)
DEC Offset (arcsec)
0 –11
0
–1
1
[NII]
Figure 12. Overlaid contours of the source-plane reconstructions for the CO(1–0) (upper left), CO(3–2) (upper right), Hα(lower
left), and [Nii] (lower right) channel maps using the natural resolution observed images, colorized by their velocities relative
to the z= 2.2586 Hαsystemic redshift from Hainline et al. (2009). The images are centered at α(J2000) = 09h01m22.s34 and
δ(J2000) = +18◦14031.005. Contours are for the same surface brightness levels as in Figure 6(±3σobs level for the two CO lines,
and ±2σobs for the VLT/SINFONI data), but these surface brightness contours are not necessarily at the same significance
as for the observed data, since the source-plane reconstructions have spatially varying noise. Negative contours are dashed.
Channels that do not have emission above the required surface brightness are separated by black lines in the legends. The
images also have spatially varying resolution, so we show the intensity-weighted average beams and PSFs at the lower left. Gray
lines indicate the source-plane lensing caustics. Black crosses mark the mean dynamical center (see Section 4.2).

18 Sharon et al.
−200 −100 0 100 200
Velocity Offset (km s
–1
)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
S
ν
(mJy)
CO(1–0)
−200 −100 0 100 200
Velocity Offset (km s
–1
)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
S
ν
(mJy)
CO(3–2)
−200 −100 0 100 200
Velocity Offset (km s
–1
)
0.0
1.0
2.0
3.0
S
ν
(10
–14
erg s
–1
cm
–2
μm
–1
)
Hα
4.0
−200 −100 0 100 200
Velocity Offset (km s
–1
)
0.0
0.5
1.0
1.5
2.5
S
ν
(10
–14
erg s
–1
cm
–2
μm
–1
)
2.0
[NII]
3.0
Figure 13. Spectral line profiles for CO(1–0) (upper left), CO(3–2) (upper right), Hα(lower left) and [N ii] (lower right)
extracted from the channelized image plane lensing reconstructions (black). We also show the observed line profiles (gray)
extracted from the southern and western images (the images used to construct the lens model) divided by the average channelized
magnification factor (since the per channel magnifications are on average lower than what was determined for the integrated line
maps and are noisier/more uncertain). Since the flux is extracted in different ways for the image- and source-plane channel maps
(as the image-plane aperture would be warped into different shapes in different source plane channels), the southern+western
image-plane comparison spectra are not expected to scatter evenly below and above the source-plane spectra, despite being
scaled by the mean per-channel magnification. The vertical bars denote ±1σuncertainties. Channels far from the systemic
redshift were not delensed.
et al. (2015) and Tacconi et al. (2017) yields a slightly
lower value of αCO = 3.8 MK−1km−1s pc−2. How-
ever, the inferred gas mass is consistent with our Milky
Way αCO-derived mass within the uncertainties. We
note that there is also some uncertainty on the metal-
licity of J0901 (see Section 4.4). As a sanity check
on the ∼30% difference between the CO(1–0) and
CO(3–2) lines’ magnification factors, we also calculate
Mgas using the CO(3–2) map and its corresponding
magnification (corrected for excitation using our mea-
sured global r3,1without magnification correction) and
find Mgas = (1.2±0.3) ×1011(αCO /4.6) M; this value
is consistent with the CO(1–0)-derived gas mass and
therefore gives additional credibility to the difference
in the two lines’ magnification factors (at least for the
natural resolution images).
Following Scoville et al. (2016), we also use the 877 µm
(observed frame) SMA continuum detection as an alter-
native probe of the gas mass. This method relies on the
adoption of a dust temperature; Scoville et al. (2014,
2016) recommend against using dust temperatures de-
rived from multi-band SED fits (Tdust = 36 K in the
case of J0901; Saintonge et al. 2013), since they are
luminosity weighted and thus biased towards the hot-
ter components of the ISM that do not make up the
bulk of the mass, and instead recommend the adop-
tion of Tdust = 25 K. Both values result in ∼2–
3×lower ISM masses than the CO-derived gas masses
(Mmol = (4.8±1.3) ×1010 Mfor Tdust = 36 K and
Mmol = (7.1±2.0)×1010 Mfor Tdust = 25 K, when cor-
rected by the CO(3–2) “natural” magnification factor).
These continuum-derived ISM masses suggest a lower
value of αCO ∼1.4–2 would be more appropriate for

Resolved Properties of J0901 19
J0901 (closer to values derived for low-metallicity sys-
tems, or to the canonical value used for local U/LIRGs).
However, since we do not independently derive a mag-
nification factor for the 877 µm continuum data due to
its low angular resolution and S/N, there is some addi-
tional uncertainty in the continuum-derived ISM mass
and implied CO-to-H2conversion factor.
Given the uncertainty in αCO for J0901, we adopt the
magnification-corrected, natural resolution CO(1–0)-
derived value of Mgas = (1.6+0.3
−0.2)×1011(αCO /4.6) M,
carrying the uncertainty in αCO as a free parameter.
Even with conversion factor uncertainties, we note
that the gas mass of J0901 is comparable to those of
other galaxies selected at submillimeter wavelengths,
but larger than those of other UV-selected high-redshift
galaxies (e.g., Riechers et al. 2010).
Adopting the dust mass from Saintonge et al.
(2013), corrected to our CO(3–2) magnification fac-
tor, we obtain a dust-to-gas mass ratio of (4.7+1.4
−1.2)×
10−3(αCO/4.6)−1for J0901. This ratio is within the nor-
mal range for disk galaxies in the local universe (e.g.,
Draine et al. 2007) but is a bit low for those with the
same metallicity (as seen for the high-redshift galaxies
in Saintonge et al. 2013). However, the dust-to-gas mass
ratio strongly depends on the assumed CO-to-H2con-
version factor as well as the properties of dust adopted
by the Draine & Li (2007) dust models. Lower CO-to-H2
conversion factors would increase the dust-to-gas mass
ratio by a factor of ∼5, bringing it more in line with
the dust-to-gas ratios of systems where authors tend
to adopt those lower values (i.e., SMGs and U/LIRGs;
e.g., Santini et al. 2010).
4.2.2. SFR and stellar mass
Using our new magnification factors and Hαmeasure-
ments, we determine improved SFRs for J0901. We use
the SFR scaling factor from Hao et al. (2011)/Murphy
et al. (2011) (as compiled in Kennicutt & Evans 2012)
scaled to a Kroupa (2001) initial mass function. We
find SFRHα= 14.5+2.1
−1.7Myr−1using the total LHα
and native magnification factor without correction for
obscuration. Hainline et al. (2009) measured the Hα
and Hβlines for two regions within J0901, finding ex-
treme obscuration corrections from Hα/Hβthat would
increase the SFR by a factor of &20. However, that
ratio could have been affected by the coincidence of
a skyline with the Hβemission. Using the total in-
frared luminosity (LTIR from 8–1000 µm) derived from
the Draine et al. (2007) fits to J0901’s dust SED in
Saintonge et al. (2013) (LTIR = 1.80+0.42
−0.41 ×1012 L
assuming our new magnification factor for the native-
resolution CO(3–2) data) and our choice in in IMF
yields SFRTIR = 268+63
−61 Myr−1, comparable to the
expected value based on the Hβextinction correction
to Hα.Kennicutt & Evans (2012)/Kennicutt et al.
(2009) also give an alternative method for correcting
Hαto account for obscured star formation using the
observed LTIR, but this method yields a much smaller
value of SFRHα+TIR = 103+21
−20 Myr−1(where we have
corrected the luminosities for the different magnifica-
tion factors for Hαand TIR as above). This hybrid
method for calculating obscured SFRs involves a num-
ber of assumptions that may not apply to galaxies in the
early universe, and was calibrated using galaxies with
infrared luminosities lower than that of J0901 (albeit
with similar LTIR/LHαratios and attenuation levels).
We therefore adopt SFRTIR = 268+63
−61 Myr−1for our
subsequent analysis, since it likely accounts for the bulk
of the star formation in J0901 and is not likely contam-
inated by significant emission from the AGN (Fadely
et al. 2010).
The fraction of the total SFR that can be accounted
for by our Hαmeasurements is consistent with the
SFRUV/SFRIR derived in Saintonge et al. (2013):
SFRHα/SFRTIR = 0.054+0.015
−0.014 vs. SFRUV/SFRIR =
0.040 ±0.007. Since J0901 is known to have an AGN
(Hainline et al. 2009) on the basis of its high [N ii]/Hα
line ratio and large HαFWHM, it is possible that the
Hα-determined SFR is contaminated by emission from
the AGN; IFU observations of the Hαemission from the
nuclear region of J0901 obtained using adaptive optics
show signs of a broad low-level outflow once disk rota-
tion is corrected for (Genzel et al. 2014). However, for
the emission from both the disk and nucleus analyzed
here, the HαFWHM is no wider than one would expect
based on single-Gaussian fits to the double-peaked CO
line profiles (at least for the line profile derived from
the sum of the three images). It seems likely that most
of the Hαemission is due to star formation, and that
some emission from the AGN, near the systemic red-
shift, masks J0901’s double peaked profile (particularly
given the slightly poorer ∼40 km s−1velocity resolution
of the VLT data and ∼150 km s−1CO peak separa-
tions) but contributes only a small amount to the total
Hαluminosity. Higher S/N would be necessary to do
a pixel-by-pixel decomposition of the broad and narrow
line emission components to correct for the Hαemission
from the AGN, as done for the nucleus in Genzel et al.
(2014).
If we re-scale the stellar mass from Saintonge et al.
(2013) to use the same Kroupa IMF that we assume for
our SFR and apply our Hα-determined magnification
factor, we find J0901 has M?= (9.5+3.8
−2.8)×1010 M.
Combined with the TIR-derived SFR, J0901 has a spe-

20 Sharon et al.
cific star formation rate of sSFR = 2.8+1.3
−1.1Gyr−1.
Since we have simply corrected the Saintonge et al.
(2013)-derived values by our new magnification factors
(the CO and Hαmagnification factors are very simi-
lar), choice of IMF, and TIR/SFR conversion factor,
J0901 still falls along the star-forming main sequence
(MS; e.g., Noeske et al. 2007;Speagle et al. 2014),
with an upward offset of just 0.27+0.20
−0.16 dex. We also
compare J0901’s sSFR to the bi-modal MS and star-
burst (SB) populations parameterized in Sargent et al.
(2012)/Rodighiero et al. (2011), who find a MS scatter of
0.188 dex and a second Gaussian peak for starbursts off-
set by log(hsSFRSBi/hsSFRMSi)=0.59 with a 0.243 dex
scatter. In this scheme, J0901 falls between the distribu-
tions for MS and starbursts at 0.22+0.20
−0.16 dex, but with
considerable uncertainty. Based on these parameteri-
zations of the MS, J0901 appears to be a massive but
otherwise “normal” MS galaxy that falls a little to the
high side of the sSFR distribution.
4.2.3. Dynamical mass
Using our de-lensed images, we can measure the phys-
ical size of J0901 and its dynamical mass. Despite
the complications potentially introduced by the spa-
tially varying resolution that results from the de-lensing,
the size of J0901 is quite robust. Gaussian fits to
the de-lensed integrated CO emission maps (without
accounting for beam/resolution effects) are consistent
for the two lines, with major and minor axis FWHMs
of 1.100 ±0.100 and 0.8500 ±0.0500 respectively (posi-
tion angle of 82 ±7◦). The VLT observations have
slightly smaller and more elliptical de-lensed angular
sizes, (1.000 ±0.100)×(0.6000 ±0.0200 ) for Hαand (0.6800 ±
0.0600)×(0.2200 ±0.0200 ) for [N ii], at position angles simi-
lar to those of the CO lines. At these angular scales, the
adopted beam/PSF values do not significantly affect the
source sizes, and both the convolved and de-convolved
(reported) source sizes are consistent within their uncer-
tainties.
In order to estimate the dynamical mass, we ana-
lyze our de-lensed three dimensional data using the
Bayesian Monte Carlo Markov Chain tool GalPaK3D
(Bouch´e et al. 2015), which constrains parametric fits to
galaxy morphologies and dynamics while accounting for
instrumentation-induced correlations in both the spatial
and spectral directions. For the parametric model, we
assume either a Gaussian or exponential intensity dis-
tribution originating from an inclined thick disk with a
rotation profile of v(r) = vcirc tan−1(r/rv) and intrinsic
velocity dispersion σv. In Table 6, we list the best-fit
parameters for both models, and the resulting dynami-
cal mass estimates using Mdyn = 233.5(2r1/2)v2
circ (from
the standard Mdyn =rv2/G with units of the dynam-
ical mass, half-light radius, and circular velocity set to
solar masses, parsecs, and kilometers per second, respec-
tively). For the radius, we use twice the half-light ra-
dius since that is a reasonable approximation for the
radius that encompasses 90% of the emission for both
assumptions of Gaussian and exponential flux profiles.
We fit dynamical models to the CO(1–0), CO(3–2), and
Hαdata for both the Gaussian or exponential flux dis-
tributions in order to estimate systematic uncertainties
caused by model assumptions that may not accurately
describe the underlying emission. Attempts to fit the na-
tive resolution reconstruction of the [N ii] maps did not
converge. We suspect this failure is due to a combina-
tion of factors, including models that poorly describe the
observed emission (which might be expected if the [N ii]
emission is mostly associated with the central AGN), re-
constructed velocity channels that are limited in number
and do not fully trace the broad emission wings, and the
lower S/N of these data. The fit to the reconstructed Hα
map for the assumption of a Gaussian intensity distribu-
tion converges to circular velocities significantly larger
than that of the other emission lines and flux profiles,
likely for the same reasons that the [N ii] does not con-
verge at all. The sub-unity reduced χ2values for both
Hαfits are due to the small number of reconstructed
velocity channels (11) and the large number of model
parameters being fit (10).
Using the five consistent best-fit models for the three
successfully fit lines, we calculate a mean dynamical
center for J0901 of R.A. 09h01m22.s3523 and Dec.
+18◦14031.004813. We then use the lens model to project
the position of the dynamical center to the image plane;
these positions are shown as black crosses in Figures 1,
3,5, and 6. As the coordinates of the mean dynamical
center are outside the (primary) lensing caustic, that
position only appears in the southern and western im-
ages. For the southern image, the foreground member
of the lensing group creates two sub-images of the mean
dynamical center position. As the two peaks of emis-
sion in the VLT Hα, [N ii], and continuum maps are
nearly aligned with the image plane positions of the av-
erage dynamical center, these peaks may correspond to
multiple images of nuclear emission associated with the
central AGN (higher angular resolution observations are
necessary to confirm whether these peaks are multiple
images or unrelated internal structures).
From these fits, J0901 appears to be consistent with
a relatively face-on disk with a half-light radius of ∼
4.25 kpc (consistent with sizes from the Gaussian fits
we previously derived from the de-lensed integrated line
maps). This size is consistent with what has been found

Resolved Properties of J0901 21
Table 6. Kinematic fit parameters
Model Parameter Transition
CO(1–0) CO(3–2) Hα
Exponential disk R.A. 09h01m22.s3518 09h01m22.s3533 09h01m22.s3523
Dec. +18◦14031.004922 +18◦14031.00 4944 +18◦14031.004387
r1/24.83 kpc 4.76 kpc 3.65 kpc
i36◦34◦23◦
P.A. 51◦47◦43◦
rv0.09 kpc 0.67 kpc 0.56 kpc
vcirc 188 km s−1230 km s−1345 km s−1
σv38 km s−139 km s−160 km s−1
χ2
red 1.2 1.6 0.97
Mdyn 0.8×1011 M1.2×1011 M2.0×1011 M
Gaussian R.A. 09h01m22.s3515 09h01m22.s3527 09h01m22.s3519
Dec. +18◦14031.004887 +18◦14031.00 4925 +18◦14031.004506
r1/24.07 kpc 3.95 kpc 3.14 kpc
i30◦27◦17◦
P.A. 51◦45◦42◦
rv0.05 kpc 0.95 kpc 0.92 kpc
vcirc 218 km s−1306 km s−1500 km s−1
σv39 km s−135 km s−157 km s−1
χ2
red 1.1 1.5 0.86
Mdyn 0.9×1011 M1.7×1011 M3.7×1010 M
Note—Since GalPaK3D does not produce meaningful uncertainties and the assumed models
may not accurately reflect the underlying emission and dynamics of J0901, the best-fit
values should be treated as approximate. As discussed in the text, we do not consider the
fit of the Hαkinematics for a Gaussian flux profile to be credible.
for other star-forming galaxies with similar masses and
redshifts (e.g., van der Wel et al. 2014). The circu-
lar velocity is somewhat degenerate with the source
size and inclination angle, so the best-fit models either
find higher circular velocities with lower inclination an-
gles or lower circular velocities with large inclination
angles. On average (neglecting the more questionable
fit to the Hαdata), we find vcirc ≈260 km s−1and
i≈30◦. The Rhoads et al. (2014) measurement of
vcirc = (120 ±7)/sin(i) km s−1is consistent with our
average best-fit circular velocity and inclination angle.
Based on the models’ best fit circular velocities and ve-
locity dispersions, the molecular gas kinematics appear
to be consistent with other high-zdisks (e.g., Tacconi
et al. 2013), with vcirc/σv∼6.
These models yield an average dynamical mass esti-
mate of ∼1.3×1011 M(again, neglecting the likely
unphysical fit to the Hαdata for an assumed Gaussian
intensity distribution). All of the five best-fit models’
dynamical mass estimates are lower than the total bary-
onic mass of 2.6+0.5
−0.3×1011 Mthat we infer from our
adopted gas and stellar masses. However, adopting a
lower value of the CO-to-H2conversion factor signifi-
cantly alleviates this tension, dropping the total bary-
onic mass to 1.2+0.4
−0.3×1011 Mfor αCO = 0.8. Intermedi-
ate values of the CO-to-H2conversion factor (favored by
metallicity-dependent models, for example) could also
be possible if new constraints on the lensing of the stel-
lar mass tracers yield larger magnification factors, or if
the dynamical mass is evaluated out to a larger radius
(than our assumed value of 2r1/2) that captures more of
the CO emission. Better models of the lensing poten-
tial, morphology, and dynamics of J0901 (from data with
higher resolution and/or S/N, and/or models that more
closely match the true flux distribution and kinematics)
may also alleviate some of the tension with the baryonic
mass estimates. Models of low inclination systems are
particularly sensitive to assumptions of azimuthal sym-
metry that may not be valid for J0901 or many rotating
systems in the early universe; lower inclination angles

22 Sharon et al.
(which would imply higher circular velocities) may also
alleviate tensions between the baryonic and dynamical
masses.
4.3. Spatial variation in CO excitation
In order to understand the gas conditions in J0901,
we examine the CO(3–2)/CO(1–0) line ratio in units
of brightness temperature (Table 2). We find that the
global line ratios of the three images do not differ signif-
icantly. Using the matched CO(1–0) image-plane data,
we find that J0901 has a global r3,1= 0.79 ±0.12.
This value is comparable to the r3,1found for SMGs
and LBGs (Riechers et al. 2010;Sharon et al. 2016; al-
though the sample size is small), and larger than the
value implied from excitation analyses of z∼1.5BzK-
selected galaxies (Dannerbauer et al. 2009;Daddi et al.
2015). We note that the r3,1value implied by the natu-
ral maps is only slightly lower but not significantly dif-
ferent from that of the matched maps, with a global
r3,1= 0.75 ±0.11. It is therefore unlikely that the dif-
ferent observations’ uv sampling are leading to a recov-
ery of emission on very different angular scales. How-
ever, if we fold in magnification corrections, r3,1sig-
nificantly decreases for comparisons using the natural
resolution data and their corresponding magnification
factors (r3,1= 0.56+0.13
−0.10), and increases for the matched
resolution data (r3,1= 1.23+0.33
−0.27).
The strong gravitational lensing of J0901 yields ad-
ditional angular resolution, which allows us to exam-
ine spatial variation in the CO excitation. For com-
parisons to the CO(3–2) map, we used the matched
CO(1–0) map. Figure 14 shows the integrated line ra-
tio map for J0901. The average value of r3,1in the
line ratio map is ∼0.8, in line with the r3,1calculated
from the integrated line flux of the uv-clipped CO(1–0)
map. However, if we look at distribution of r3,1values
in the map (Figure 15), we see that the distributions
peak at slightly lower values of r3,1∼0.6–0.7 for all im-
ages and for the source-plane reconstruction. Given this
lower peak r3,1in the source-plane reconstruction, we do
not trust the large magnification factor derived for the
matched-resolution CO(1–0) data that yields the unusu-
ally large global r3,1≈1.2. For the image-plane r3,1dis-
tributions, a strong tail out to higher excitations biases
the average r3,1value, and most of the gas has a lower
CO(3–2)/CO(1–0) line ratio. While the image plane r3,1
distribution appears roughly log-normal, which may hint
at emission from higher density gas phases, we do not
ascribe much significance to this shape, given the un-
derlying noise in the two maps and the 2σsignificance
clipping that is applied. Given the Gaussian noise in the
individual CO maps, the ratio map noise should follow a
Cauchy distribution, which could skew the distribution
of per-pixel r3,1values if it is not properly accounted
for. However, the noise distribution is further compli-
cated by the primary beam corrections required to ac-
curately measure the flux in an extended source such as
J0901. We therefore trust only the peak values of the
r3,1distributions.
For the integrated line ratio map, the lower-excitation
gas (areas in the map with lower values of r3,1) appears
to be more spatially extended than the higher excitation
gas, especially on the basis of the southern image and
reconstructed source plane maps. The line ratio map
for the source-plane reconstruction looks similar to that
of the western image, which we expect since the western
image is the least distorted. For the northern image, it is
difficult to determine whether the large r3,1values near
the image’s edge are caused by noise and weak emission
or by genuine differences between the CO emission in the
two maps (potentially amplified by lensing). Examining
the line ratio maps as a function of channel does not
reveal any significant velocity trend, in either the image
or the source plane, due to the lower SNR of individual
channel maps (which is then amplified when taking their
ratio).
For the source-plane reconstruction maps using the
matched-resolution data, in Figure 16 we show r3,1as a
function of the physical radius from J0901’s dynamical
center. Unlike the mapped values of r3,1in Figure 14,
we include all pixels, regardless of their statistical signif-
icance. In order to calculate each pixel’s distance from
the center, including inclination corrections, we use the
mean dynamical center, position angle, and inclination
angle from the best-fit models in Table 6, omitting the
model for the Hαdata using a Gaussian flux profile since
that model does not converge to sensible values. The
distribution of r3,1values decreases as a function of ra-
dius, which is clearest in the variance-weighted mean
r3,1values calculated in bins of 1kpc. Since the pixels
are correlated, the binned average r3,1values are also
correlated. However, since the intensity-weighted av-
erage PSF’s major axis FWHM (which approximately
gives the resolution and thus correlation length of the
data) is ∼2 kpc when tilted by J0901’s inclination an-
gle, every other bin is approximately uncorrelated. By
using the variance-weighted means in our radial bins, we
can retrieve average values that are not biased by noise-
dominated pixels that scatter to large r3,1or have un-
physical negative r3,1values. The spatial distribution of
line ratios in J0901 is consistent with a picture of multi-
phase gas in which the bulk of the molecular ISM is in
an extended cool/low-density phase, containing smaller
embedded regions of gas in a warm/high-density phase

Resolved Properties of J0901 23
0.50.30.1 0.7 0.9 1.1 1.3
r3,1
0.20.10.0 0.3 0.4 0.5
σr3,1
0.6
RA Offset (arcsec)
DEC Offset (arcsec)
0 –5 –10510
0
–5
–10
5
10 Image plane
RA Offset (arcsec)
DEC Offset (arcsec)
0 –5 –10510
0
–5
–10
5
10
RA Offset (arcsec)
DEC Offset (arcsec)
0 –11
0
–1
1
Source plane
RA Offset (arcsec)
DEC Offset (arcsec)
0 –11
0
–1
1
Figure 14. Map of the CO(3–2)/CO(1–0) line ratio (left) and statistical uncertainty in the line ratio (right) in units of
brightness temperature in the image plane (upper row) and in the de-lensed source-plane reconstruction (lower row). Both
ratio maps use the “matched” datasets with the same spatial resolution and inner uv radius. Negative and <2σsignificance
pixels have been blanked out. For the ratio maps, contours are in steps of ∆r3,1= 0.2, and the color mapping is saturated at
r3,1= 1.3. For the uncertainty maps, contours are in steps of ∆σr3,1= 0.1, and the color mapping is saturated at σr3,1= 0.6.
Blue lines indicate the image-plane lensing critical curves or source-plane caustics (Section 4.1). Black crosses mark the mean
dynamical center determined from the source-plane reconstructions and lens modeling (see Section 4.2).
(e.g., Ivison et al. 2011;Thomson et al. 2012) that is
somewhat more centrally concentrated.
4.4. Spatial variation in metallicity
Using the [N ii] and Hαmaps, we also examine spatial
variations in the metallicity of J0901. We estimate the
metallicity using
12 + log(O/H) = 8.90 + 0.57 log([N ii]/Hα) (3)
from Pettini & Pagel (2004), which is valid for 7.5>
12 + log(O/H) >8.75 (using 8.66 as the solar abun-
dance; Asplund 2004). In our map of the metallicity
(Figure 17) we blank out any pixels with <2σsignifi-
cance in the Hαmap. We find that a substantial fraction
of the source has 12 + log(O/H) values larger than the
range where the [N ii]/Hαaccurately traces the metallic-
ity (although it has been suggested that at high redshift,
the threshold at which the [N ii]/Hαratio becomes af-
fected by the AGN is higher; e.g., Kewley et al. 2013a,b);
the average pixelized value is 12+log(O/H) = 8.73±0.21

24 Sharon et al.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
r3,1
0
100
200
300
400
Npix
Source plane
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
r3,1
0
100
200
300
400
500
600
Npix
Image plane
northern
southern
western
total
Figure 15. Distribution of CO(3–2)/CO(1–0) pixel values (in units of brightness temperature) in both the image plane (left)
and reconstructed source plane (right). The pixels used in these distributions are the same as in Figure 14, which are clipped
at the 2σlevel. For the image plane maps, we show the pixel distribution for the northern (red), southern (blue), and western
(gold) images separately as well as in aggregate (black).
vs. 12 + log(O/H) = 7.3±1.1 calculated from the ratio
of the total luminosities (without magnification correc-
tion). Larger values of [N ii]/Hαcannot be produced
in the photoionization regions of massive stars, indicat-
ing potential heating or shocked excitation by a central
AGN or its winds (e.g., Baldwin et al. 1981;Kauffmann
et al. 2003). The high central [N ii]/Hαratio seen in the
source-plane reconstruction, least-distorted western im-
age, and southern image is in line with previous evidence
of an AGN in J0901 (Hainline et al. 2009;Diehl et al.
2009;Genzel et al. 2014). However, we note that the
average pixelized metallicity is also much closer to the
metallicity predicted by the mass-metallicity relation for
high-zgalaxies, which implies 12 + log(O/H) = 8.5–8.7
for J0901’s new magnification-corrected stellar mass (de-
pending on which relation we use; Genzel et al. 2012;
Wuyts et al. 2014;Sanders et al. 2018).
Caveats on the validity of using [N ii]/Hαto trace
metallicity aside, in Figure 18, we examine the radial
decrease in metallicity in more detail. Like the radial
r3,1plot, we calculate 12+log(O/H) for each pixel in the
matched-resolution source-plane reconstructions regard-
less of SNR. We calculate each pixel’s radial distance
from the average dynamical center, corrected for incli-
nation angle, using the best-fit models in Table 6(again,
omitting the model for the Hαdata using a Gaussian
flux profile). While there is a weak radial gradient in
metallicity out to ∼5 kpc, any trends at larger radii are
lost in the noise. However, the roughly linear radial gra-
dient in [N ii]/Hα(rather than its log ↔the metallicity)
may extend to ∼10 kpc with a slope of ∼ −0.1 kpc−1
(from a linear best-fit to the binned values with no cor-
rection for beam smearing). The radial metallicity gra-
dient of ∼ −0.03 dex kpc−1(from a linear best-fit to the
binned values with r≤5kpc and no correction for beam
smearing) is on the flatter end of (albeit consistent with)
the distribution for disk galaxies in the local universe
(e.g., Rupke et al. 2010). However, high-redshift galax-
ies appear to have a wide range of metallicity gradients
(e.g., Wuyts et al. 2016, and references therein), within
which J0901 falls, making the physical interpretation of
the gradient difficult even without accounting for the
potential influence of the central AGN.
4.5. Spatially resolved Schmidt-Kennicutt relation
4.5.1. Methods
We examine the spatially resolved Schmidt-Kennicutt
relation (Schmidt 1959;Kennicutt 1998) for J0901 us-
ing the Hαand CO maps smoothed to the same spatial
resolution. We use the LHα-SFR conversion factor given
in Kennicutt & Evans (2012), which assumes a Kroupa
(2001) initial mass function (IMF). The Hαbrightness
has not been corrected for extinction. Properly account-
ing for spatially varying extinction can significantly af-
fect the slope of the Schmidt-Kennicutt relation (Genzel
et al. 2013), but our current dust continuum data lack
the spatial resolution for us to effectively perform such
a correction. A global correction for the extinction (as
in Sharon et al. 2013) would simply offset the relation to
higher SFR surface densities (discussed further below).
In order to fit the Schmidt-Kennicutt relation in J0901
to a power law, we follow the methodology of Blanc et al.
(2009) and Leroy et al. (2013) and perform a Bayesian
analysis, since standard orthogonal least squares regres-
sion fits are biased by clipping of the molecular gas
and star formation surface densities at a chosen signif-
icance level. While the full methodology is presented
in Blanc et al. (2009) and Leroy et al. (2013), in short,

Resolved Properties of J0901 25
4 10 12 14
radius (kpc)
0.0
0.5
1.0
r3,1
2 6 80
Figure 16. The distribution of CO(3–2)/CO(1–0) line ra-
tios for pixels in the matched-resolution source-plane recon-
structions as a function of radius relative to the dynamical
center of J0901. For each bin (with width ∆r3,1= 0.05
and ∆r= 0.25 kpc), one of the eight red tones is assigned,
starting at one pixel per bin, and in steps of three pixels
per bin thereafter. We include all pixels regardless of their
statistical significance. Radial positions account for the in-
clination of the source. We use the mean dynamical center,
position angle, and inclination angle from the best-fit mod-
els in Table 6, omitting the model for the Hαdata using a
Gaussian flux profile since that model does not converge to
sensible values. The black squares are the variance-weighted
mean r3,1values for pixels in bins of 1kpc. Associated un-
certainties are calculated from a bootstrap analysis (with
replacement) in which we calculate the dispersion from the
variance-weighted mean for 104iterations of the underlying
CO(1–0) and CO(3–2) pixels, after randomly perturbing the
pixels’ fluxes in each iteration by their uncertainties as de-
termined from the lens reconstructions. Since the pixels are
correlated, adjascent binned average r3,1values are also cor-
related; however, the intensity-weighted average PSF’s major
axis FWHM is ∼2 kpc (when tilted by J0901’s inclination
angle), so every other bin is approximately uncorrelated. The
dashed line corresponds to the approximate peak value in the
r3,1histogram for the reconstructed source as shown in Fig-
ure 15 (r3,1= 0.7). The dotted line corresponds to r3,1= 0
for reference.
we iteratively calculate the SFR surface density for a
random sample (with replacement) of the observed pix-
elized molecular gas surface densities in J0901 for a grid
of potential normalization factors (A), indices (n), and
intrinsic scatter values (σ) in the equation
ΣSFR
1Myr−1kpc−2=AΣgas
1000 Mpc−2n
×10N(0,σ),
(4)
where N(0, σ) is a normal distribution with mean zero
and standard deviation σ. For each possible combina-
tion of Schmidt-Kennicutt relation parameters, we grid
the resulting model values of ΣSFR and Σgas and com-
pare them to a grid of the measured values to calculate
aχ2value. As in Leroy et al. (2013), we apply a 2σ
cut in gas mass surface density before comparing the
grids of the observed and model data points in order
to define a clear y-axis; since this cut is applied after
the data are simulated, it does not bias the selection of
the best-fit model in the same way as more conventional
linear fitting algorithms. For each of the three Schmidt-
Kennicutt parameters, we fit polynomials to the shape of
their χ2values (taking the minimum χ2along the com-
plementary parameters’ axes, collapsing the model grid
to a distribution of χ2values for each parameter sepa-
rately), and use the polynomials’ minima as the best-fit
values of the parameters. We then perform this com-
parison multiple times, each time removing a pixel at
random, perturbing the grid on which we compare the
source and model, and perturbing the emission for both
tracers by both the additive statistical uncertainty (on
a per-pixel basis) and the multiplicative flux calibration
uncertainty (applied to all pixels). Our best-fit values
for the Schmidt-Kennicutt relation and their uncertain-
ties (both statistical and systematic) are given by the
mean and standard deviation of the resulting distribu-
tion of fitted parameters.
In addition to the different denominator of Eqn. 4
(which must be accounted for in comparisons to other
Schmidt-Kennicutt studies and is chosen to reduce the
fitting covariance), our implementation of the algorithm
differs from that of Blanc et al. (2009) and Leroy et al.
(2013) in the following ways: (1) We randomly draw
104values of Σgas for calculating model values of ΣSFR
and allow repeats, but we only perform the iterative fit-
ting routine for 100 perturbations of the model/source
(due to computational/time limits). (2) Since our Σgas
and ΣSFR uncertainties are both dominated by mea-
surement uncertainties, we additively perturb both the
model gas and SFR surface densities. (3) We sample the
Schmidt-Kennicutt parameters in ∆ log(A)=0.03 from
−1.25 ≥log(A)≥ −0.5, ∆n= 0.05 from 0.8≥n≥2.3,
and ∆σ= 0.015 from 0 ≥σ≥0.3. (4) We assume
a 10% flux calibration uncertainty for the Hαand CO
data. (5) Our minimum χ2curves are fit by a third or-
der polynomial rather than a second order polynomial
as in Leroy et al. (2013) in order to better fit our skewed
χ2curves.
4.5.2. Results for J0901
In Figures 19 and 20 we show the Schmidt-Kennicutt
relation using the native resolution integrated CO(1–0)
and CO(3–2) maps. In order to determine whether dif-
ferential lensing affects the observed Schmidt-Kennicutt
relation in J0901, we analyze each image of J0901 sep-
arately and all three images combined. We also com-

26 Sharon et al.
7.5
8.0
8.5
9.0
12 + log(O/H)
0.6 0.4 0.2 0.0 –0.2 –0.4 –0.6
RA Offset (arcsec)
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
DEC Offset (arcsec)
Source plane
−10−510
RA Offset (arcsec)
−10
−5
0
5
10
DEC Offset (arcsec)
5 0
Image plane
Figure 17. Map of the metallicity as estimated from the [N ii]/Hαline ratio (Pettini & Pagel 2004) in both the image
plane (left) and source-plane reconstruction (right). The black contour at 12 + log(O/H) = 8.75 represents the upper limit
on the range for which [N ii]/Hαaccurately estimates the metallicity. Pixels with <2σsignificance in the Hαline have
been blanked out (excluding <2σsignificance pixels in the image plane [N ii] map would remove nearly all pixels below
12 + log(O/H) = 8.75). PSFs are shown at lower left. Gray lines indicate the image-plane lensing critical curves or source-plane
caustics (Section 4.1). Black crosses mark the mean dynamical center determined from the source-plane reconstructions and
lens modeling (see Section 4.2).
410 12 14
radius (kpc)
–1.0
–0.5
0.0
0.5
1.0
1.5
N
II
/Hα
6 80 2
410
radius (kpc)
8.2
8.4
8.6
8.8
9.0
9.2
12+log(O/H)
–1.0
–0.5
0.0
0.5
log(N
II
/Hα)
28
06
9.4
Figure 18. The metallicity (or log([N ii]/Hα); left) and [N ii]/Hαratio (right) for individual pixels in the matched-resolution
source-plane reconstructions as a function of radius relative to the dynamical center of J0901. For each bin (with width
∆r= 0.25 kpc and either ∆Z= 0.025 or ∆([N ii]/Hα)= 0.05), one of the six (left) or five (right) red tones is assigned,
starting at one pixel per bin, and in steps of three pixels per bin thereafter. We include all pixels regardless of their statistical
significance. Radial positions account for the inclination of the source. We use the mean dynamical center, position angle, and
inclination angle from the best-fit models in Table 6, omitting the model for the Hαdata using a Gaussian flux profile since
that model does not converge to sensible values. The black squares are the variance-weighted mean values for pixels in bins of
1 kpc. Associated uncertainties are calculated from a bootstrap analysis (with replacement) in which we calculate the dispersion
from the variance-weighted mean for 104iterations of the underlying Hαand [N ii] pixels; the pixels’ fluxes in each iteration
are randomly perturbed by their uncertainties as determined from the lens reconstructions. Since the pixels are correlated,
adjascent binned average values are also correlated; the intensity-weighted average PSF’s major axis FWHM is ∼2 kpc (when
tilted by J0901’s inclination angle), so every other bin is approximately uncorrelated. The dashed lines correspond to the value
above which the [Nii]/Hαratio is no longer believed to be an accurate tracer of the metallicity (at least in the local universe).
For the right panel, we also show [N ii]/Hα= 0 for clarity (dotted line; negative values are caused by noise), and the best-fit
linear relation for r≤10 kpc (solid line).

Resolved Properties of J0901 27
pare these results to a Schmidt-Kennicutt analysis us-
ing the matched maps (not shown) and the source-plane
reconstructions of the matched maps for both CO lines
(Figure 21). Table 7lists the best-fit parameters of the
Schmidt-Kennicutt relation in Equation 4. The power
law fits are roughly consistent with super-linear indices
of n≈1.5 for both CO transitions, with a mean value
of ¯n= 1.54±0.13, although individual fits for the image
plane analyses range from n= 1.38–1.73.
We note that in Figures 19–22, we show much smaller
ΣSFR and Σgas bins in our Schmidt-Kennicutt plots
(0.025 dex) than we use in the fitting analysis (0.05–
0.2 dex, randomly assigned for each iteration of the
Schmidt-Kennicutt fitting routine). These smaller bins
show some structure in the pixel densities due to differ-
ences in the brightness distributions of our Hαand CO
maps that are amplified by the beams/PSFs (causing
neighboring pixels to be correlated and blending real
source structure together, in some cases causing mis-
aligned peaks of emission). The pixel density structures
are particularly apparent in the analysis of the source
plane reconstructed images since correlations in the ob-
served images can be warped and exaggerated by the
de-lensing process. The larger bins we use in the fit-
ting analysis (plus the Monte Carlo perturbations in
the model realizations) do not show these structures,
and therefore these structures do not bias our power-law
fits. The numbers of correlated pixels and correspond-
ing numbers of independent beams/PSFs used in the fits
are listed in Table 7as Npix and Nind , respectively.
The smoothing and inner-radius uv clipping used
to create the “matched” dataset lowers the best-fit
Schmidt-Kennicutt index for the CO(1–0) line only (ex-
cept for the southern image where the index is un-
changed). We suspect that the best-fit index for the
CO(3–2) data is unchanged between the “native” and
“matched” resolution maps because considerably less
smoothing (and no uv clipping) is required to create its
matched map. Therefore, while the CO(1–0) line might
better represent the true distribution of the total molec-
ular gas mass, one must compare the matched maps in
order to make fair comparisons between how the choice
of molecular gas tracer affects the Schmidt-Kennicutt
relation.
We find significant differences between the Schmidt-
Kennicutt indices determined from the image-plane data
and from the source-plane reconstructions; the best-fit
indices for the source-plane reconstructions are much
lower with ¯n= 1.24 ±0.02. It is difficult to explain this
difference since lensing conserves surface brightness. We
do expect the distributions of pixels in the parameters
space of SFR vs. gas mass to change due both to the
existence of multiple images of the same region within
the source and to the larger number of samples per re-
gion within the source given the uniform image-plane
pixel size. These effects should be particularly strong
for the northern and southern images since they cross
critical curves where the magnification is highest. The
western image is the least distorted of the three images,
and although it suffers from the lowest S/N, it shows a
similar distribution of pixels in the parameters space of
SFR vs. gas mass to the source-plane reconstruction: the
highest concentration of pixels is at the highest SFR and
gas mass surface densities. The lack of difference in the
Schmidt-Kennicutt index between the three images de-
spite these lensing effects is reassuring since it indicates
that Schmidt-Kennicutt relation does not change much
between different regions within J0901 (at least at the
spatial resolutions probed here). The similar pixel dis-
tribution for the western image and source plane recon-
struction is similarly reassuring since it indicates that
our lens modeling is working correctly. However, neither
of these points explains why the best-fit index differs be-
tween the image plane and source plane analyses.
We note that there is one significant methodolog-
ical difference between the image-plane and source-
plane Schmidt-Kennicutt analyses that does influence
the best-fit index. The source-plane reconstructions for
the Hαdata are more compact than for the CO data,
and thus the gas and SFR surface densities are largely
uncorrelated beyond the edges of the Hαemission. The
uncorrelated pixels, if left in the Schmidt-Kennicutt
analysis, result in indices of n∼2, which is even steeper
than found in the image plane. Clipping the data at a
fixed value of the gas mass or SFR surface density to
remove uncorrelated data would both bias the fits and
leave too few pixels to support a robust fit. Instead,
we only include pixels that have SNR≥2 for both the
gas and SFR maps. Due to the spatially varying noise
in the source-plane reconstructions, this significance cut
does not result in a constant surface brightness cut. We
re-derived the fits with SNR cuts between 2 and 4σand
found no difference in the best-fit values of the Schmidt-
Kennicutt parameters, so we believe this method is reli-
able for the source-plane reconstructions. Therefore we
do not think this methodological difference causes the
difference in the Schmidt-Kennicutt index between the
image and source planes.
One potential way to resolve the difference in the
Schmidt-Kennicutt index between the source and im-
age planes would be to observe J0901 at higher angular
resolution. While the fit to the western image is deter-
mined by a large number of pixels (∼800–1000), those
pixels only correspond to a small number of independent

28 Sharon et al.
2.4 2.6 2.8 3.0 3.2 3.4
–1.6
–1.4
–1.2
–1.0
–0.8
–0.6
log(Σ
SFR
/M yr
–1
kpc
–2
)
log(Σ
gas
/M pc
–2
)
northern
2σ
2σ
2.4 2.6 2.8 3.0 3.2 3.4
–1.6
–1.4
–1.2
–1.0
–0.8
–0.6
log(Σ
SFR
/M yr
–1
kpc
–2
)
log(Σ
gas
/M pc
–2
)
southern
2σ
2σ
2.4 2.6 2.8 3.0 3.2 3.4
–1.6
–1.4
–1.2
–1.0
–0.8
–0.6
log(Σ
SFR
/M yr
–1
kpc
–2
)
log(Σ
gas
/M pc
–2
)
western
2σ
2σ
2.4 2.6 2.8 3.0 3.2 3.4
–1.6
–1.4
–1.2
–1.0
–0.8
–0.6
log(Σ
SFR
/M yr
–1
kpc
–2
)
log(Σ
gas
/M pc
–2
)
total
2σ
2σ
Figure 19. Star formation rate surface density as measured by Hαsurface brightness (uncorrected for extinction) vs. CO(1–0)-
determined molecular gas mass surface density of J0901 (using the natural resolution/weighted data). From left to right, the
four panels plot the density of pixels in 0.025 dex bins in gas mass and SFR surface density for the northern image (red),
southern image (blue), western image (gold), and all images combined (gray). The color tones start at one pixel per bin and
are in steps of two pixels per bin thereafter; there are six, three, three, and seven color steps in the northern, southern, western,
and total panels, respectively. The two dashed lines mark 2σsurface brightness cuts, but only the gas surface density cut was
applied during the linear fit (as described in the text). In the rightmost plot we include the linear fits for the individual images
for easier comparisons; note that the blue line for the southern image falls nearly directly under the gold line for the western
image.
2.2 2.4 2.6 2.8 3.0 3.2
–1.6
–1.4
–1.2
–1.0
–0.8
–0.6
log(Σ
SFR
/M yr
–1
kpc
–2
)
log(Σ
gas
/M pc
–2
)
western
2σ
2σ
2.2 2.4 2.6 2.8 3.0 3.2
–1.6
–1.4
–1.2
–1.0
–0.8
–0.6
log(Σ
SFR
/M yr
–1
kpc
–2
)
log(Σ
gas
/M pc
–2
)
total
2σ
2σ
2.2 2.4 2.6 2.8 3.0 3.2
–1.6
–1.4
–1.2
–1.0
–0.8
–0.6
log(Σ
SFR
/M yr
–1
kpc
–2
)
log(Σ
gas
/M pc
–2
)
northern
2σ
2σ
2.2 2.4 2.6 2.8 3.0 3.2
–1.6
–1.4
–1.2
–1.0
–0.8
–0.6
log(Σ
SFR
/M yr
–1
kpc
–2
)
log(Σ
gas
/M pc
–2
)
southern
2σ
2σ
Figure 20. Star formation rate surface density as measured by Hαsurface brightness (uncorrected for extinction) vs. CO(3–2)-
determined molecular gas mass surface density of J0901 (using the natural resolution data). All lines and colors are as described
in Figure 19, but there are seven, six, three, and eight steps in the northern, southern, western, and total panels, respectively,
and the axis ranges differ as well.
resolution elements (.10). If the western image is ex-
pected to better represent the image-plane structure and
have a different distribution of pixels in the parameter
space of SFR vs. gas mass relative to the other images,
then more high-SNR independent data points would be
helpful. While our calculated uncertainties for the best-
fit correlations fold in the effects of excluding individual
pixels, they do not fold in the effects of excluding entire
resolution elements. Therefore, the uncertainties quoted
in Table 7are likely underestimates of the true uncer-
tainties, particularly for the western image.
We also see a significant difference in the observed
scatter of the Schmidt-Kennicutt relation, σ, between
the source-plane and image-plane data. While the un-
certainty in the scatter is likely an underestimate as de-
scribed above, we suspect that the difference in the scat-
ter is an artifact of the lens modeling noise scaling and
regularization. As discussed in Section 4.1, we scale up
the assumed image noise during the lens modeling in or-
der to allow the code to under-fit the flux distributions
and thus minimize the effects of the correlated noise in
the interferometric data. Since this process increases
the regularization strength, it effectively smooths the
source-plane emission, and thus increases the source-
plane SNR while decreasing the scatter of the SFR and
gas mass surface densities.
Lastly, we find no significant systematic differences in
the best-fit indices between the matched datasets for the
two CO lines using either the image-plane data or the
source-plane reconstructions, except in the case of the
southern image. However, given that the native reso-
lution CO(1–0) and CO(3–2) data produce significantly

Resolved Properties of J0901 29
Table 7. J0901 Schmidt-Kennicutt fit parameters
Image line Npix Nind A n σ
North CO(1–0) 1891 23.5−0.96 ±0.08 1.67 ±0.06 0.21 ±0.02
CO(1–0)m1785 17.7−0.97 ±0.06 1.40 ±0.08 0.24 ±0.01
CO(3–2) 2278 24.1−0.97 ±0.06 1.38 ±0.04 0.23 ±0.01
CO(3–2)m2366 22.1−0.96 ±0.07 1.41 ±0.04 0.23 ±0.01
South CO(1–0) 1478 18.4−0.89 ±0.07 1.73 ±0.08 0.23 ±0.01
CO(1–0)m1570 14.7−0.86 ±0.07 1.70 ±0.09 0.24 ±0.01
CO(3–2) 1757 18.6−0.84 ±0.05 1.43 ±0.04 0.22 ±0.01
CO(3–2)m1851 17.3−0.83 ±0.07 1.42 ±0.04 0.21 ±0.01
West CO(1–0) 858 10.7−0.88 ±0.08 1.71 ±0.10 0.24 ±0.02
CO(1–0)m793 7.4−0.82 ±0.07 1.56 ±0.10 0.21 ±0.02
CO(3–2) 954 10.1−0.82 ±0.07 1.54 ±0.07 0.26 ±0.01
CO(3–2)m982 9.2−0.80 ±0.08 1.51 ±0.07 0.25 ±0.01
Total CO(1–0) 4227 52.6−0.92 ±0.07 1.72 ±0.06 0.23 ±0.01
CO(1–0)m4148 38.8−0.91 ±0.05 1.55 ±0.08 0.24 ±0.01
CO(3–2) 4989 52.8−0.90 ±0.04 1.43 ±0.03 0.239 ±0.004
CO(3–2)m5199 48.6−0.89 ±0.06 1.44 ±0.03 0.237 ±0.004
De-lensed CO(1–0)m1688 15.8−0.86 ±0.06 1.22 ±0.03 0.18 ±0.01
CO(3–2)m1673 15.7−0.80 ±0.06 1.25 ±0.03 0.13 ±0.01
Note—The lines with subscript muse the matched maps with the same smoothed
resolution and the same inner uv radius. Npix lists the total number of pixels used
in the fitting procedure, which are not all independent from one another due to
the beam/PSF size, and Nind lists the number of independent resolution elements
(beams/PSFs) to which those pixels correspond.
different indices (n∼1.7 using the CO(1–0) data and
n∼1.4 for the CO(3–2) data) and that the source-plane
reconstruction produce yet a different index (n∼1.2),
the potential differences between indices for the two CO
lines are inconclusive. Since we apply a global excitation
correction in determining Σgas based on our measured
r3,1values, we find consistent offsets (A) in the linear
fits between the two CO transitions, regardless of map
(native, matched, or reconstructed) or lens-plane image
(north, south, west, or total).
4.5.3. Comparisons to other spatially resolved galaxies and
important caveats
In Figure 22 we compare the results for J0901 to other
high-redshift galaxies in which resolved pixel-by-pixel
Schmidt-Kennicutt analyses have been performed. Only
eight high-redshift galaxies besides J0901 have been ana-
lyzed on a pixel-by-pixel basis on the Schmidt-Kennicutt
relationship: seven SMGs and one normal disk galaxy
(Sharon et al. 2013;Genzel et al. 2013;Rawle et al. 2014;
Hodge et al. 2015;Ca˜nameras et al. 2017;Tadaki et al.
2018;G´omez et al. 2018; but see also Freundlich et al.
2013;Sharda et al. 2017). We also compare J0901 to the
sample of local disk galaxies from Leroy et al. (2013).
For these comparisons, we convert all measurements to
the same (Kroupa 2001) initial mass function. The po-
sition of J0901 (or any galaxy) on the star formation
relation strongly depends on the assumed value of the
CO-to-H2conversion factor. While αCO is still uncer-
tain for high-redshift galaxies, different authors’ choices
in CO-to-H2conversion factors are often justified based
on metallicity or dynamical arguments. Therefore we
do not correct gas measurements to the same αCO, but
instead show a horizontal bar to indicate how gas mass
surface density measurements would scale for the range
of possible conversion values (0.7≤αCO ≤4.6). For
GN20, EGS13011166, PLCK G244.8+54.9, and HAT-
LAS J084933, where the CO measurements were not
made for the J= 1–0 transition, we include additional
factors in this rough systematic molecular gas uncer-
tainty to account for the unknown gas excitation; we
allow the ratios to the ground state to be as low as
those found for the Milky Way (r2,1= 0.50, r3,1= 0.26,
r4,1= 0.15, or r7,1= 0.015; Fixsen et al. 1999) and as
high as thermalized (r2,1=r3,1=r4,1=r7,1= 1.0).

30 Sharon et al.
2.2 2.4 2.6 2.8 3.0 3.2 3.4
–1.6
–1.4
–1.2
–1.0
–0.8
–0.6
–0.4
log(Σ
SFR
/M yr
–1
kpc
–2
)
log(Σ
gas
/M pc
–2
)
2σ
2σ
CO(1–0)
Source plane
–1.8 3.6
2.0 2.2 2.4 2.6 2.8 3.0 3.2
–1.6
–1.4
–1.2
–1.0
–0.8
–0.6
–0.4
log(Σ
SFR
/M yr
–1
kpc
–2
)
log(Σ
gas
/M pc
–2
)
2σ
2σ
CO(3–2)
Source plane
–1.8 3.4
Figure 21. Star formation rate surface density as mea-
sured by Hαsurface brightness (uncorrected for extinction)
vs. CO(1–0)-determined molecular gas mass surface density
(top) and CO(3–2)-determined molecular gas mass surface
density (bottom) using the de-lensed CO images of J0901
derived from maps with matched beams/PSFs and inner uv
radii. For each 0.025 dex bin in gas mass and SFR surface
density, one of six or ten red tones is assigned (for the up-
per or lower panels, respectively), starting at one pixel per
bin, and in steps of two pixels per bin thereafter. The two
dashed lines mark 2σsurface brightness cuts applied to the
image-plane data, which are not applied here since pixels
with at least 2σsignificance correspond to different surface
brightnesses in the de-lensed data.
One significant source of uncertainty in the SFE for
J0901 is that we do not have comparable spatial resolu-
tion tracers of obscured star formation. Our dust map
from the SMA is not sufficiently resolved to map local
variations of the dust column, and we do not have Hβ
maps or enough resolved multi-band optical/UV data to
perform a spatially resolved SED analysis as in Genzel
et al. (2013). In Figure 22, the contours are for star for-
mation traced by Hαbut scaled to account for the total
SFR as inferred from LTIR; this rescaling is analogous
to applying a uniform global extinction correction (the
same technique used to correct the SFR surface density
for J14011 in Sharon et al. 2013). The obscured star
formation as probed by the total infrared luminosity
is significantly higher than the Hα-traced star forma-
tion, and moves J0901 to higher star formation efficien-
cies, making J0901 significantly offset from the Schmidt-
Kennicutt relation found for local normal disk galaxies.
For comparison, we also show how the locus of points
would change for different extinction corrections to the
Hα-derived SFR surface density in Fig. 22, including no
extinction corrections, the total infrared-corrected Hα
emission to measure the SFR as in Kennicutt & Evans
(2012), and the global extinction value calculated from
the Hα/Hβratio in Hainline et al. (2009) (using the
standard assumption of Case B recombination). Using
the total infrared-corrected Hαemission to measure the
SFR moves J0901 to higher star formation efficiencies,
but not as high as using LTIR alone. The global extinc-
tion value calculated from the Hα/Hβratio is highly
uncertain since the Hβline was coincident with a sky
line; however, using this line ratio to correct for extinc-
tion (and thus obscured star formation) produces a large
SFR, comparable to that calculated using the infrared.
Global extinction corrections preserve the index of the
Schmidt-Kennicutt relationship, but different extinction
laws and patchy/localized extinction could significantly
change the correlation’s slope. Genzel et al. (2013) find
that the index of the Schmidt-Kennicutt relation for
EGS13011166 varied between 0.8≤n≤1.7 for the
different extinction corrections they explore. Such ex-
tinction corrections are particularly challenging for star-
burst SMGs where nearly all of the star formation is ex-
pected to be obscured. For GN20, HLS0918, AzTEC-1,
and J084933, their SFR surface densities are inferred
from maps of the continuum emission near the peak
of the dust SED (∼170 µm rest frame) scaled to their
LTIR-determined SFRs (Rawle et al. 2014;Hodge et al.
2015); a similar LTIR-determined SFR scaling is used to
infer ΣSFR for G244, but they scale continuum emission
from further down the Rayleigh-Jeans tail of the dust
SED (∼750 µm rest frame; Ca˜nameras et al. 2017),
which may better trace dust mass than the SFR. Ac-
counting for additional obscured star formation in J0901
(or unobscured star formation in the case of SMGs) may
change the index of the Schmidt-Kennicutt relation, but
J0901 would remain at elevated SFE relative to the local
relation (regardless of the choice of CO-to-H2conver-

Resolved Properties of J0901 31
−2
0
2
4
log(Σ
SFR
/M yr
–1
kpc
–2
)
−1
3
3
−3
0123456
log(Σ
gas
/M pc
–2
)
τ
=
1
0
8
y
r
H
I
saturation
Normal
Starburst
τ
=
1
0
9
y
r
τ
=
1
0
10
yr
J0901 (z=2.26) CO(1–0); this work
Local Galaxies CO(2–1); Leroy et al. 2013
EGS13011166 (z=1.53) CO(3–2); Genzel et al. 2013
HATLAS J084933 W (z=2.41) CO(7–6); Gomez et al. 2018
HATLAS J084933 T (z=2.41) CO(7–6); Gomez et al. 2018
J14011 (z=2.56) CO(1–0); Sharon et al. 2013
G244 (z=3.00) CO(4–3); Cañameras et al. 2017b
GN20 (z=4.05) CO(2–1); Hodge et al. 2015
AzTEC-1 (z=4.34) CO(4–3); Tadaki et al. 2018
HLS0918 (z=5.24) CO(1–0); Rawle et al. 2014
Figure 22. Comparison between the star formation rate surface density and gas mass surface density for high-redshift galaxies
with resolved pixel-by-pixel analyses: the modestly lensed SMG J14011 (stars; Sharon et al. 2013), the z= 1.5 normal disk galaxy
EGS13011166 (wide diamonds with error bars; Genzel et al. 2013), the strongly lensed SMG HLS0918 (narrow diamonds; Rawle
et al. 2014), the SMG GN20 (squares with error bars; Hodge et al. 2015), the strongly lensed SMG G244 (circles; Ca˜nameras
et al. 2017), the SMG AzTEC-1 (pentagons with error bars; Tadaki et al. 2018), and the two components of the HyLIRG
HATLAS J084933 (upward and downward pointing triangles with error bars; G´omez et al. 2018). The red region shows the
density of pixels for all three images of J0901 using the natural resolution CO(1–0) data, but the SFR has been scaled to match
the TIR-derived SFR. The gray shaded region shows the Leroy et al. (2013) sample of local galaxies. For all high-redshift
galaxies, the star formation rates have been converted to the Kroupa initial mass function (the same IMF as used for the local
sample). However, we respect the different authors’ choices of αCO, and instead show how the locus of points (centered at the
mean surface density; black/red symbols) would translate for 0.7≤αCO ≤4.6 using the horizontal lines. Since the molecular gas
for GN20, EGS13011166, G244, AzTEC-1, and HATLAS J084933 was not observed in the CO(1–0) line, we include an additional
excitation uncertainty for a range of possible line ratios, using that of the Milky Way as a lower limit (r2,1= 0.50, r3,1= 0.26,
r4,1= 0.15, r7,1= 0.015; Fixsen et al. 1999), and thermalized excitation as an upper limit (r2,1=r3,1=r4,1=r7,1= 1.0). For
J0901, the vertical bar denotes how the SFR surface density would scale for different global extinction corrections. From bottom
to top, we mark the average SFR surface density if the SFR was determined from the Hαdata without extinction correction
(as in Fig. 19), from the Hαluminosity corrected to include obscured star formation traced by the TIR luminosity following
Kennicutt & Evans (2012), from the TIR luminosity only (current location; red circle), and from the Hαluminosity corrected for
extinction using the Hα/Hβvalue from Hainline et al. (2009). Dashed lines are as in Bigiel et al. (2008), and include diagonal
lines of constant SFE (or the inverse of the gas consumption timescale), the threshold at which atomic gas converts entirely to
molecular gas (left vertical line), and a proposed threshold for the transition between “normal” and “starburst” modes of star
formation (right vertical line; Bigiel et al. 2008).

32 Sharon et al.
sion factor). Resolved dust maps or extinction correc-
tions are necessary to more firmly pin down the index of
the Schmidt-Kennicutt relationship for UV-bright high-
redshift galaxies and for J0901 specifically.
A second source of uncertainty in the position of J0901
relative to the star formation law is that we do not cor-
rect for contaminating AGN emission. While Fadely
et al. (2010) determine that the AGN in J0901 is not a
significant contributor to its FIR luminosity, the AGN
may contribute to the Hαluminosity used in our pix-
elized analysis. If the AGN is producing Hαemission
in excess of that expected from star formation (Genzel
et al. (2014) suggest the AGN is responsible for ∼10% of
the Hαemission), then some regions of J0901 would have
an unexpectedly large SFE, which could either glob-
ally bias the ΣSFR-Σgas relation to higher SFEs (e.g.,
if the AGN emission were uncorrelated with the molec-
ular gas) or bias the fit to steeper slopes (if the AGN
were fueled by molecular gas, the excess Hαemission
could correspond to high molecular gas surface bright-
ness). In order to determine if there are regions in J0901
that might be affected by the AGN, we examined SFE
as a function of r3,1and the [N ii]/Hαratio, since gas
fueling the AGN may be at higher density, in a higher
excitation state, and/or shocked. While the distribu-
tion of SFE has a tail towards higher values, we found
no significant correlation between SFE and r3,1or SFE
and metallicity. Absent some additional tracer of AGN-
affected Hαemission in J0901, we err on the side of using
all pixels in the Schmidt-Kennicutt analysis. An alter-
native method for identifying and excluding Hαemission
from the AGN would be to perform a velocity decom-
position for each SINFONI pixel; the broader Hαline
profile could be associated with the AGN rather than
star formation, although a narrow-line AGN component
might still masquerade as star formation. However, our
current data lack the S/N (and likely the spatial resolu-
tion) to do such a decomposition.
Adopting αCO = 4.6M(K km s−1pc2)−1and scal-
ing to the IR-determined SFR, we find J0901 appears
slightly offset to higher SFEs relative to the “sequence
of disks” (e.g., Daddi et al. 2010a;Genzel et al. 2010),
while lower values of αCO move J0901 to the “sequence
of starbursts,” in line with the scenario that the distinc-
tion between such sequences is at least partly a product
of the assumed αCO factors. If we assume the CO-to-H2
conversion factors are correct for all of the resolved high-
redshift sources, J0901 appears to fall along or slightly
below a track of high-redshift starbursts with a net in-
dex that is potentially super-unity (Figure 22). How-
ever, given the variety of assumptions involved (extinc-
tion corrections, excitation corrections, and αCO), J0901
and the other eight individual galaxies studied to date
may lie within the normal scatter of SFEs.
Individual Schmidt-Kennicutt indices range from
1.0.n.2.0 for these high-redshift sources. Given
that many of the sources explored here use low-JCO
lines (CO(1–0) or CO(2–1)), and that we find there is
no clear excitation difference for J0901 up to CO(3–2),
we do not think the range of indices reflects the critical
densities of different observed gas tracers. For the local
disk galaxies in Leroy et al. (2013), there is also a wide
range of Schmidt-Kennicutt indices (all mapped in the
CO(2–1) line), and it is only the distribution of indices
that peaks at n∼1. Wei et al. (2010) also find a range of
Schmidt-Kennicutt indices (n∼1.6–1.9, mapped in the
CO(1–0) line) for nearby low-mass E/S0 galaxies with
a median index of n∼1.2. While some of the apparent
variation in local galaxies is due to other factors (like
spatially varying CO-to-H2conversion factors), some of
the scatter is real, and we suspect this is also the case
for high-redshift galaxies. Therefore, larger samples of
high-redshift galaxies need to be analyzed in a uniform
way if we are to say conclusively whether they have a
different Schmidt-Kennicutt index or higher SFE, or if
there are differences between galaxy populations (like
starbursts vs. normal galaxies).
While different choices of molecular gas tracers can
complicate comparisons between studies of the Schmidt-
Kennicutt relation, these tracers’ density sensitivities
are valuable tools for probing the underlying volumetric
“Schmidt law” (SFR ∝ρn
gas;Schmidt 1959). In a series
of hydrodynamic galaxy simulations with 3-D non-LTE
radiative transfer modeling, Narayanan et al. (2008) and
Narayanan et al. (2011) demonstrate that the change in
the Schmidt-Kennicutt index with CO rotational line
(for the surface density or integrated versions of the re-
lation) differs depending on the index of the underlying
volumetric star formation law. They argue that the cold
gas less directly involved in star formation will be under-
luminous in higher-excitation emission lines; therefore,
while the intrinsic star-formation relation using a cold
gas tracer might have an index of n= 1.5, higher excita-
tion emission lines would trace less mass per unit of star
formation, resulting in observed indices closer to n= 1.
Variation in the power law index with gas tracer
has been seen in the integrated form of the Schmidt-
Kennicutt relation (e.g., Sanders et al. 1991;Yao et al.
2003;Gao & Solomon 2004;Narayanan et al. 2005;Buss-
mann et al. 2008;Graci´a-Carpio et al. 2008;Bayet et al.
2009;Iono et al. 2009;Juneau et al. 2009; cf. Tac-
coni et al. 2013;Sharon et al. 2016), but these stud-
ies do not observe all tracers for the same galaxies, nor
do they examine spatially resolved star formation prop-

Resolved Properties of J0901 33
erties. Greve et al. (2014) analyze the integrated CO
and FIR properties for a comprehensive sample of local
U/LIRGs and high-redshift SMGs (although not every
CO line is detected in every galaxy) and find strong
trends in the integrated form of the Schmidt-Kennicutt
index with critical density. They find n∼1 for CO
rotational transitions .Jupper = 6 and decreasing in-
dices for higher-excitation lines, a pattern that does not
match the predictions of Narayanan et al. (2008). How-
ever, this discrepancy is not entirely robust: Kamenet-
zky et al. (2016) perform a similar analysis using largely
the same sample, and find near-unity indices for all
CO lines up to CO(13–12), unless AGN host galaxies
were included in the analysis. For high-redshift galaxies
with surface density measurements, neither J0901 nor
the (current) high-redshift sources exhibit the change
in index with CO rotational transition predicted by
Narayanan et al. (2011) for any of the underlying po-
tential Schmidt laws. Given the variation in indices seen
for local disk galaxies (Leroy et al. 2013), it seems likely
that intrinsic variations between galaxies will mask the
population average when only small numbers of obser-
vations are available at high redshift. Larger samples of
galaxies will therefore need to be mapped in multiple gas
tracers (in addition to efforts addressing the extinction
corrections mentioned previously) in order to properly
test the Narayanan et al. (2011) surface density pre-
dictions and determine the underlying volumetric star
formation law.
4.6. Correlation between CO excitation and SFR
We also compare our observations of J0901 to the cor-
relation between the shape of the CO spectral line en-
ergy distribution (SLED) and ΣSFR predicted from a
suite of galaxy simulation by Narayanan & Krumholz
(2014). Since the correlation is for the total SFR sur-
face density, we scale the individual pixels of the Hα
map such that their sum is equal to the TIR-determined
SFR (effectively a global extinction correction as de-
scribed above). In Figures 23 and 24 we plot the r3,1
value of each pixel (determined from the uv-matched
CO maps) vs. its Hα-determined SFR surface density,
scaled to the TIR-determined SFR. We also show the
r3,1values predicted for our range of SFR surface den-
sities for both the “resolved” and “unresolved” parame-
terizations of Narayanan & Krumholz (2014); the reso-
lution of our observations, while good for z∼2, is still
significantly worse than the ∼70pc resolution of their
simulations. While the average ΣSFR-predicted value of
r3,1= 0.77±0.02 is consistent with our measured global
average r3,1= 0.79 ±0.12, the pixelized values of r3,1
are generally offset by ∆r3,1≈0.2.
Since density plots of the southern image, combi-
nation of all three images, and the de-lensed source
plane reconstruction are all suggestive of correlation be-
tween r3,1and the SFR surface density, we attempt a
more detailed comparison to the results of Narayanan
& Krumholz (2014) in order to test if their models can
be extended to predict the range of CO excitation vari-
ations within galaxies. Since any correlation between
these two parameters is weak, and the Narayanan &
Krumholz (2014) relation for r3,1is effectively linear over
the relatively narrow range of ΣSFR probed by J0901, we
attempt a linear fit between r3,1and log(ΣSFR). Since,
as in fitting the Schmidt-Kennicutt relation, any signif-
icance cut on the included pixels could bias the fit (al-
though, for the reasons presented above, we do exclude
pixels with fluxes <2σfor the source-plane reconstruc-
tions), we follow a similar procedure as in our Schmidt-
Kennicutt fitting: we randomly sample ΣSFR 105times,
use those values to predict r3,1for a specific choice of
linear model parameters, perturb r3,1and ΣSFR by the
noise, compare the binned grid of the model results to
that of the observations to calculate a χ2value, and
repeat for a range of model parameters, ultimately find-
ing the model parameters that produce the lowest χ2
value. We repeat this procedure 100 times, each time
choosing a random bin size (between 0.025–0.1 dex for
both ΣSFR and ∆r3,1), perturbing the gridding (by up
to half a bin width in any direction), perturbing ΣSFR
and r3,1values by their flux calibration uncertainties
(∼10%), and perturbing individual values of ΣSFR and
r3,1by their statistical uncertainties. We use the vari-
ation in the best-fit values for these 100 iterations to
estimate uncertainties. We note that in this case, the
choice of noise for r3,1is not trivial, since it is a ratio
of two values, and thus the uncertainty depends on the
measured values of both the CO(1–0) and CO(3–2) lines
(see Figure 14), which are not separately predicted by
the model. We therefore produce 105iterations of the
r3,1map in which both image-plane integrated line maps
have been perturbed by their own Gaussian noise, and
then produce corresponding r3,1uncertainty maps for
each iteration.9We then collect all values of σr3,1(from
the 105uncertainty maps) that correspond to values of
r3,1in ∆r3,1= 0.05 bins. When perturbing our ΣSFR-
predicted values of r3,1for each model, we randomly
choose a σr3,1from the appropriate ∆r3,1bin.
As for the Schmidt-Kennicutt relation, we include a
Gaussian component in case there is intrinsic scatter
9We did not scale the noise by the primary beam corrections,
which were already applied to the integrated line maps, so the r3,1
iterations’ scatter is only approximately correct.

34 Sharon et al.
0.80.4 0.6−0.2 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
r3,1
0.2
log(ΣSFR/M yr–1 kpc–2)
total
−0.2 0.0 0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
1.0
1.2
1.4
r3,1
log(ΣSFR/M yr–1 kpc–2)
northern
0.2 0.4 0.6 0.8
log(ΣSFR/M yr–1 kpc–2)
−0.2 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
r3,1
southern
0.80.0 0.2 0.4 0.6
log(ΣSFR/M yr–1 kpc–2)
−0.2
0.2
0.4
0.6
0.8
1.0
1.2
1.4
r3,1
western
Figure 23. r3,1vs. star formation rate surface density as measured by Hαsurface brightness (corrected globally for extinction)
for J0901 (using the uv- and resolution-matched maps). From left to right, the four panels plot the density of pixels in 0.05
dex bins for the northern image (red), southern image (blue), western image (gold), and all images combined (gray). The color
tones start at one pixel per bin and are in steps of three pixels per bin thereafter; there are eight, six, five, and twelve color
steps in the northern, southern, western, and total panels, respectively. The best fit linear relations are shown as solid lines.
The black dashed and dotted lines show the correlations between r3,1and ΣSFR from Narayanan & Krumholz (2014) for the
unresolved and resolved cases, respectively.
–0.2 0.0 0.2 0.4 0.6 0.8
log(Σ
SFR
/M yr
–1
kpc
–2
)
0.2
0.4
0.6
0.8
1.0
1.2
1.4
r
3,1
Source plane
Figure 24. r3,1vs. star formation rate surface density as
measured by Hαsurface brightness (corrected globally for
extinction) for J0901 using the de-lensed uv- and resolution-
matched maps. The color tones plot the density of pixels in
0.05 dex bins, starting at one pixel per bin and in steps of
five pixels per bin thereafter; there are nine color steps. The
best fit linear relation is shown as the solid red line. The
black dashed and dotted lines show the correlations between
r3,1and ΣSFR from Narayanan & Krumholz (2014) for the
unresolved and resolved cases, respectively.
in addition to the noise. The resulting relationship we
attempt to fit is thus
r3,1=A+B×log ΣSFR
2 Myr−1kpc−2+N(0, σ).(5)
For the Narayanan & Krumholz (2014) model of r3,1, a
first order Taylor expansion about a typical log(ΣSFR) =
Table 8. r3,1-ΣSFR fit parameters
Image A B σ
North 0.69 ±0.04 0.08 ±0.08 0.33 ±0.04
South 0.61 ±0.03 0.18 ±0.05 0.18 ±0.05
West 0.65 ±0.02 −0.09 ±0.07 0.28 ±0.03
Total 0.69 ±0.04 0.09 ±0.04 0.32 ±0.03
De-lensed 0.67 ±0.15 0.14 ±0.30 0.03 ±0.03
0.4 yields A= 0.75 and B= 0.08 for unresolved sources
and A= 0.85 and B= 0.10 for resolved sources. Our
best-fit values for this relationship are given in Table 8
and are shown in Figures 23 and 24. Based on the un-
certainties on the best-fit slopes, we do not find evi-
dence for a trend in r3,1with SFR surface density in
the northern image, western image, or the source-plane
reconstruction. However, our analysis does suggest that
the observed r3,1values are correlated with ΣSFR for the
southern image and tentatively correlated for all images
combined. While the slopes of the best-fit models for
southern and total correlations are similar to the results
of Narayanan & Krumholz (2014) (as is the slope for the
source-plane reconstruction, if we neglect its significant
uncertainty), the overall normalization is lower. This
offset indicates that consistency between the observed
integrated r3,1and its predicted value is largely an ef-
fect of the asymmetric distribution of r3,1pixels, which
has a tail to large values (Figure 15).
All of the best-fit correlations derived for the image
plane that are consistent with Narayanan & Krumholz
(2014) require additional scatter beyond the statistical
uncertainty of the individual image-plane maps, sug-

Resolved Properties of J0901 35
gesting that other physical processes besides the SFR
density affect the molecular excitation (if the predicted
correlation is real). However, the source-plane recon-
struction does not require additional scatter; the spa-
tially varying uncertainty associated with the source-
plane CO maps appears to be sufficient, and therefore
suggests that the scatter contains no astrophysical in-
formation. The larger uncertainty associated with the
source plane and relatively limited range in SFR sur-
face densities allowed by the 2σsignificance requirement
(not shown in Fig. 24, but it removes most pixels with
log(ΣSFR/(Myr−1kpc−2)) <0.1 and leaves the spur
of high r3.1values at log(ΣSFR/(Myr−1kpc−2)) ∼0.4)
likely contributes to the uncertainty in the correlation.
We emphasize that although these coefficients are the
best-fit values for an assumed linear relationship, that
does not mean there is actually a statistically significant
correlation between r3,1and ΣSFR for J0901. The mod-
els of Narayanan & Krumholz (2014) were derived to
reproduce a much wider range in ΣSFR than probed by
any one galaxy, and were intended to predict the global
CO excitation for a galaxy-wide average SFR surface
density. Therefore, it is perhaps unsurprising that their
correlation does not reproduce our observed distribu-
tion of r3,1values for a single galaxy. Our results do not
clearly indicate that the SFR surface density depends on
the gas excitation for the physical scales probed in our
images of J0901. This result is consistent with the un-
changing Schmidt-Kennicutt index for the different CO
lines. Similar, albeit unresolved, comparisons by Yao
et al. (2003) for a sample of infrared bright galaxies in
the local universe and by Sharon et al. (2016) for a sam-
ple of z∼2 SMGs also do not find a correlation between
the luminosity of a total SFR tracer (LFIR) and r3,1.
However, Kamenetzky et al. (2016) do find some corre-
lation of r3,1with LFIR for all nearby galaxies observed
with the Herschel SPIRE Fourier Transform Spectrom-
eter (mostly U/LIRGs and IR-bright AGN, but includes
a substantial number of galaxies with LFIR ∼1010 L).
4.7. Possible AGN origin for excess 35 GHz continuum
emission
Synchrotron emission at radio wavelengths can be an
alternative probe of galaxies’ SFRs. However, J0901
is known to contain an AGN, which may corrupt long-
wavelength estimates of its SFR. In addition, our 35 GHz
(observed frame) continuum detection is in the region of
the SED where the Rayleigh-Jeans tail of the dust emis-
sion, synchrotron emission, and free-free emission can
all contribute to the observed flux density. We therefore
estimate the contribution from each of these emission
components to the observed 35 GHz flux density to de-
termine whether the observed emission is driven by star
formation and/or the AGN.
J0901 falls off the radio-FIR correlation presented
in Magnelli et al. (2015). Using the (observed) TIR
luminosity from Saintonge et al. (2013) rescaled for
our magnification, the standard spectral index for syn-
chrotron emission (Sν∝ν−0.8), and the (weakly)
redshift-dependent form the of radio-IR correlation from
Magnelli et al. (2015), we would expect the 35 GHz (ob-
served frame; 115 GHz rest frame) continuum emission
in J0901 to be only ∼1µJy, which is significantly less
than our measured flux density of 0.66 ±0.12 mJy. Al-
ternatively, we can use J0901’s SFR (based on LTIR
from Saintonge et al. (2013)) and invert the relation-
ship for determining SFR from 1.4 GHz continuum lu-
minosity (Kennicutt & Evans 2012) to determine the
expected contribution to the 35 GHz flux density from
synchrotron emission (again, assuming Sν∝ν−0.8and
our CO(3–2)-determined magnification factor). Using
this method, we expect ∼9µJy of synchrotron emission
at 35 GHz, which is more than we would expect based
on the radio-FIR correlation, but still nearly two orders
of magnitude smaller than the observed emission.
Thermalized free-free emission from the H ii regions of
massive (>5M) stars can also be used as a tracer of
(high mass) star formation. Following Condon (1992),
the TIR-determined SFR yields an expected 35 GHz flux
density of ∼60 µJy (for an assumed magnification factor
of 31.3; we lack adequate S/N to independently deter-
mine the magnification for the VLA continuum map).
Since this estimate does not account for the formation
of lower mass stars, we estimate that .20% of the ob-
served 35 GHz flux density comes from free-free emis-
sion.
The Rayleigh-Jeans tail of the dust continuum peak
is unable to account for the difference between the ob-
served 35 GHz continuum emission and the expected
contributions from synchrotron and free-free emission
associated with star formation. Using our observed
858 µm SMA detection and β= 1.5 (Saintonge et al.
2013), we predict a 11.5±2.9µJy contribution to the
35 GHz flux density. Similar calculations using the
Rayleigh-Jeans tail continuum measurements of J0901
in Saintonge et al. (2013) produce consistently low ex-
trapolated 35 GHz flux densities.
Regardless of the discrepancy between the two meth-
ods for determining the synchrotron emission from star
formation, which dust continuum estimate we use, and
reasonable perturbations for assumed spectral indices,
the expected combined contribution from star formation
and dust emission is .20% of our observed 35 GHz con-
tinuum detection. Therefore, either the free-free emis-

36 Sharon et al.
sion has a significantly different magnification factor
from the CO(1–0) emission, or the bulk of the VLA con-
tinuum emission is due to an AGN. The magnification
factor required to bring our observed 35 GHz emission
into alignment with our total SFR is µ&140 (assum-
ing it is dominated by free-free emission), several times
larger than our other magnification factors, which seems
unlikely if they are all tracing the same star forming ma-
terial within J0901 (i.e., there should not be much differ-
ential lensing). Given that J0901 is known to contain an
AGN, we conclude that its large 35GHz flux density is
most likely due to synchrotron emission from the AGN.
The synchrotron emission from the AGN may also be
affected by differential lensing, and to the extent that
we trust the morphology of the low-S/N 35GHz emis-
sion, it does not appear to originate from the brightest
emission regions at longer wavelengths, but may corre-
spond to the bright emission seen in Hαand [N ii]. High-
resolution observations at lower frequencies are neces-
sary to test this hypothesis.
As a final possibility, some fraction of the radio contin-
uum emission may not be associated with J0901 at all,
and may instead be due to members of the foreground
group of galaxies. This scenario may explain the incon-
sistencies between star formation rate predictors and the
offset of the peak emission in the northern image. As the
central galaxies in the lensing cluster definitely do pro-
duce 35 GHz continuum emission, the southern emission
peak may also be due to emission from the foreground
interloper that is nearly aligned with the southern arc.
Higher SNR or better spatial resolution observations are
necessary to determine what fractions of the radio con-
tinuum emission are associated with J0901 and the lens-
ing galaxies.
5. SUMMARY
We present ∼100 resolution (∼2 kpc in the source
plane) observations of the CO(1–0), CO(3–2), Hα, and
[N ii] lines in a strongly-lensed star-forming galaxy at
z= 2.26, SDSS J0901+1814 (J0901). We use our high-
est S/N line detection (the CO(3–2) line) and existing
HST data to constrain the lensing potential of the fore-
ground group of galaxies, and find a typical magnifi-
cation factor µ≈30 (depending on wavelength). Dy-
namical modeling of the source-plane reconstruction us-
ing both the CO and Hαdata indicates that J0901 is a
nearly face-on (i≈30◦) massive disk with r1/2&4 kpc.
Our Hαobservations of J0901 trace only a small fraction
of the total star formation rate implied by the galaxy’s
LTIR. Applying our new magnification factors to LTIR
from Saintonge et al. (2013), we find the SFR for J0901 is
268+63
−61 Myr−1. J0901’s magnification-corrected SFR
and stellar mass place it only ∼0.25 dex above the star-
forming galaxy main sequence, consistent with its be-
ing a “normal” galaxy considering the significant uncer-
tainty in its sSFR.
Our CO observations yield a total molecular gas mass
of Mgas = (1.6+0.3
−0.2)×1011(αCO /4.6) M. The molec-
ular gas is nearly equal to the magnification-corrected
stellar mass, which yields a total baryonic mass of
(2.6+0.5
−0.3)×1011 Mthat is significantly larger than our
dynamical mass estimate of ∼1.3×1011 M. Reduc-
ing the assumed CO-to-H2conversion factor to the typ-
ical “starburst” values of αCO ∼0.8 would bring the
baryonic and dynamical masses into alignment (assum-
ing moreover J0901 is baryon dominated). For our two
integrated CO lines, we find an average line ratio of
r3,1= 0.79 ±0.12, which is skewed somewhat higher
than the peak of the the pixelized r3,1distribution. Af-
ter correcting for the inclination angle, we find evidence
for a significant decrease in r3,1as a function of radius
out to ∼10 kpc. However, there is no significant cor-
relation between r3,1and the [N ii]/Hαratio (used as a
metallicity tracer), nor does there appear to be a signif-
icant trend in r3,1with velocity channel.
Using our CO and Hαmaps, we analyze where J0901
falls relative to the Schmidt-Kennicutt relation for other
galaxies. The relative positions of galaxies strongly de-
pend on the extinction correction used to determine the
spatially resolved SFR and assumed CO-to-H2conver-
sion factor. Since we do not have a spatially resolved
tracer of the obscured star formation, we plot J0901 us-
ing the Hα-derived SFR surface density and show how it
would shift assuming the obscured star formation traces
the unobscured star formation. With the correction to
account for the obscured star formation, J0901 appears
to be slightly offset to higher SFEs than “normal” disk
galaxies found in the local universe (e.g., Leroy et al.
2013). Given J0901’s dynamics and high metallicity, we
assume a typical Galactic conversion factor. Contrary
to results claiming that galaxies are offset relative to
the local Schmidt-Kennicutt relation solely due to the
assumed conversion factor, J0901 would be offset even
further for lower values of αCO. We find the average
slope for the Schmidt-Kennicutt relation for J0901 to
be ¯n= 1.54 ±0.13 in the image plane. We do not find
significantly different slopes when using the CO(1–0)
and CO(3–2) lines to trace the molecular gas for the
matched resolution/inner uv-radius data (in either the
image or source plane), but we do find some difference
for the native resolution CO data. The observed slope of
the Schmidt-Kennicutt relation does differ between the
CO(1–0) maps using the natural resolution data and
the CO(1–0) maps that have been smoothed and uv-

Resolved Properties of J0901 37
clipped (to match the CO(3–2) data). We also find a
slightly flatter slope of ¯n= 1.24 ±0.02 when using the
source-plane reconstructions of J0901. While the true
index for J0901 is somewhat uncertain, all of these in-
deces are higher than the average for normal disk galax-
ies in the local universe (e.g., Leroy et al. 2013) but
within their observed scatter. Few measurements of the
resolved Schmidt-Kennicutt relation exist at high red-
shift, but J0901 is within the measured range of indices
of n= 1–2. However, our analysis assumes a global ex-
tinction correction to the Hαdata used to trace the star
formation, and Genzel et al. (2013) find significant vari-
ation in the index depending on the assumed extinction
correction.
We also use our resolved observations to assess
whether the correlation between SFR surface density
and CO excitation identified in the simulations of
Narayanan & Krumholz (2014) holds within individ-
ual galaxies. For the limited range in ΣSFR in J0901,
we do not reproduce the Narayanan & Krumholz (2014)
correlation, although the galaxy-wide average r3,1is
comparable to the value predicted from the measured
average SFR surface density. This distinction is likely
tied to the limited range of ΣSFR and skewed distri-
butions of r3,1values in our data. However, as any
correlations are weak, we do not ascribe much mean-
ing to the difference between the observed fits and the
correlation predicted in Narayanan & Krumholz (2014).
We find a significant excess of 35 GHz (observed
frame) continuum emission relative to the expected
contributions from the Rayleigh-Jeans tail of the dust
emission, synchrotron emission, and free-free emission
predicted for the measured SFR. Given that the mag-
nification factor for the free-free emission would need
to be about five times larger than what we have found
at other wavelengths to account for the excess flux, we
conclude that the 35 GHz continuum emission is either
synchrotron emission from the AGN and/or contamina-
tion from the foreground group of lensing galaxies.
Although J0901’s SFE and sSFR are slightly elevated,
and it contains an AGN, J0901 appears to be a relatively
normal massive disk galaxy at z= 2.26. The nearly
face-on orientation and additional physical resolution
enabled by gravitational lensing (for the same observed
angular resolution) makes J0901 a valuable laboratory
for probing galaxy evolution in galaxies at z∼2 (see
also Johnson et al. 2017), such as feedback from star for-
mation and AGN. However, larger samples of resolved
systems with comparable-quality multi-wavelength ob-
servations are necessary to test whether these results
(e.g., pertaining to the Schmidt-Kennicutt index as a
function of CO line and spatial variations in CO line ra-
tios) generalize to other (populations of) high-zgalaxies.
We thank the anonymous referee for helpful com-
ments. This work has been supported by the Na-
tional Science Foundation through grant AST-0955810.
The National Radio Astronomy Observatory is a facil-
ity of the National Science Foundation operated un-
der cooperative agreement by Associated Universities,
Inc. The Submillimeter Array is a joint project be-
tween the Smithsonian Astrophysical Observatory and
the Academia Sinica Institute of Astronomy and Astro-
physics and is funded by the Smithsonian Institution
and the Academia Sinica. This work is based in part on
observations made with the ESO Very Large Telescope
at the La Silla Paranal Observatory under program ID
087.A-0972.
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