# COOL-LAMPS. VII. Quantifying Strong-lens Scaling Relations with 177 Cluster-
scale Gravitational Lenses in DECaLS

Simon D. Mork Department of Astronomy and Astrophysics, University of Chicago,
5640 S. Ellis Ave, Chicago, IL 60637, USA  Michael D. Gladders Department of
Astronomy and Astrophysics, University of Chicago, 5640 S. Ellis Ave, Chicago,
IL 60637, USA  Kavli Institute for Cosmological Physics, University of
Chicago, 5640 S. Ellis Ave, Chicago, IL 60637, USA  Gourav Khullar Department
of Physics and Astronomy, University of Pittsburgh, 3941 O’Hara St,
Pittsburgh, PA 15260, USA  Pittsburgh Particle Physics Astrophysics and
Cosmology Center, University of Pittsburgh, 3941 O’Hara St, Pittsburgh, PA
15260, USA  Keren Sharon Department of Astronomy, University of Michigan, 1085
S. University Ave, Ann Arbor, MI 48109, USA  Nathalie Chicoine Department of
Astronomy and Astrophysics, University of Chicago, 5640 S. Ellis Ave, Chicago,
IL 60637, USA  Aidan P. Cloonan Department of Astronomy and Astrophysics,
University of Chicago, 5640 S. Ellis Ave, Chicago, IL 60637, USA  Department
of Astronomy, University of Massachusetts Amherst, 710 N. Pleasant St,
Amherst, MA 01003, USA  Håkon Dahle Institute of Theoretical Astrophysics,
University of Oslo, P.O. Box 1029, Blindern, NO-0315 Oslo, Norway  Diego Garza
Department of Astronomy and Astrophysics, University of Chicago, 5640 S. Ellis
Ave, Chicago, IL 60637, USA  Department of Astronomy and Astrophysics,
University of California, Santa Cruz, CA 95064, USA  Rowen Glusman Department
of Astronomy and Astrophysics, University of Chicago, 5640 S. Ellis Ave,
Chicago, IL 60637, USA  Gravitation & Astroparticle Physics, University of
Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands  Katya Gozman
Department of Astronomy and Astrophysics, University of Chicago, 5640 S. Ellis
Ave, Chicago, IL 60637, USA  Department of Astronomy, University of Michigan,
1085 S. University Ave, Ann Arbor, MI, 48109, USA  Gabriela Horwath Department
of Astronomy and Astrophysics, University of Chicago, 5640 S. Ellis Ave,
Chicago, IL 60637, USA  Benjamin C. Levine Department of Astronomy and
Astrophysics, University of Chicago, 5640 S. Ellis Ave, Chicago, IL 60637, USA
Department of Physics and Astronomy, Stony Brook University, 100 Nicolls Rd,
Stony Brook, NY 11794, USA  Olina Liang  Department of Astronomy and
Astrophysics, University of Chicago, 5640 S. Ellis Ave, Chicago, IL 60637, USA
Daniel Mahronic  Department of Astronomy and Astrophysics, University of
Chicago, 5640 S. Ellis Ave, Chicago, IL 60637, USA  Viraj Manwadkar Department
of Astronomy and Astrophysics, University of Chicago, 5640 S. Ellis Ave,
Chicago, IL 60637, USA  Department of Physics, Stanford University, 382 Via
Pueblo, Stanford, CA 94305, USA  Kavli Institute for Particle Astrophysics and
Cosmology, Stanford University, 382 Via Pueblo, Stanford, CA 94305, USA
Michael N. Martinez Department of Astronomy and Astrophysics, University of
Chicago, 5640 S. Ellis Ave, Chicago, IL 60637, USA  Department of Physics,
University of Wisconsin, Madison, 1150 University Ave, Madison, WI 53706, USA
Alexandra Masegian Department of Astronomy and Astrophysics, University of
Chicago, 5640 S. Ellis Ave, Chicago, IL 60637, USA  Department of Astronomy,
Columbia University, 538 W. 120th St, New York, NY 10027, USA  Owen S.
Matthews Acuña Department of Astronomy and Astrophysics, University of
Chicago, 5640 S. Ellis Ave, Chicago, IL 60637, USA  Department of Astronomy,
University of Wisconsin—Madison, 475 N. Charter St, Madison, WI 53706, USA
Kaiya Merz Department of Astronomy and Astrophysics, University of Chicago,
5640 S. Ellis Ave, Chicago, IL 60637, USA  Yue Pan Department of Astronomy and
Astrophysics, University of Chicago, 5640 S. Ellis Ave, Chicago, IL 60637, USA
Department of Astrophysical Sciences, Princeton University, 4 Ivy Ln,
Princeton, NJ 08544, USA  Jorge A. Sanchez Department of Astronomy and
Astrophysics, University of Chicago, 5640 S. Ellis Ave, Chicago, IL 60637, USA
Isaac Sierra Department of Astronomy and Astrophysics, University of Chicago,
5640 S. Ellis Ave, Chicago, IL 60637, USA  Daniel J. Kavin Stein Department of
Astronomy and Astrophysics, University of Chicago, 5640 S. Ellis Ave, Chicago,
IL 60637, USA  Ezra Sukay Department of Astronomy and Astrophysics, University
of Chicago, 5640 S. Ellis Ave, Chicago, IL 60637, USA  Department of Physics
and Astronomy, Johns Hopkins University, 3400 N. Charles St, Baltimore, MD
21218, USA  Marcos Tamargo-Arizmendi Department of Astronomy and Astrophysics,
University of Chicago, 5640 S. Ellis Ave, Chicago, IL 60637, USA  Kiyan
Tavangar Department of Astronomy and Astrophysics, University of Chicago, 5640
S. Ellis Ave, Chicago, IL 60637, USA  Department of Astronomy, Columbia
University, 538 W. 120th St, New York, NY 10027, USA  Ruoyang Tu Department of
Astronomy and Astrophysics, University of Chicago, 5640 S. Ellis Ave, Chicago,
IL 60637, USA  Department of Anthropology, Yale University, 10 Sachem St, New
Haven, CT 06520, USA  Grace Wagner Department of Astronomy and Astrophysics,
University of Chicago, 5640 S. Ellis Ave, Chicago, IL 60637, USA  Erik A.
Zaborowski Department of Astronomy and Astrophysics, University of Chicago,
5640 S. Ellis Ave, Chicago, IL 60637, USA  Department of Physics, The Ohio
State University, 191 W. Woodruff Ave, Columbus, OH 43210, USA  Center for
Cosmology and Astro-Particle Physics, The Ohio State University, 191 W.
Woodruff Ave, Columbus, OH 43210, USA  Yunchong Zhang Department of Astronomy
and Astrophysics, University of Chicago, 5640 S. Ellis Ave, Chicago, IL 60637,
USA  Department of Physics and Astronomy, University of Pittsburgh, 3941
O’Hara St, Pittsburgh, PA 15260, USA

###### Abstract

We compute parametric measurements of the Einstein-radius-enclosed total mass
for 177 cluster-scale strong gravitational lenses identified by the ChicagO
Optically-selected Lenses Located At the Margins of Public Surveys (COOL-
LAMPS) collaboration with lens redshifts ranging from $0.2\lessapprox
z\lessapprox 1.0$ using only two measured parameters in each lensing system:
the Einstein radius, and the brightest-cluster-galaxy (BCG) redshift. We then
constrain the Einstein-radius-enclosed luminosity and stellar mass by fitting
parametric spectral energy distributions (SEDs) with aperture photometry from
the Dark Energy Camera Legacy Survey (DECaLS) in the $g$, $r$, and $z$-band
Dark Energy Camera (DECam) filters. We find that the BCG redshift, enclosed
total mass, and enclosed luminosity are strongly correlated and well described
by a planar relationship in 3D space. We also find that the enclosed total
mass and stellar mass are correlated with a logarithmic slope of $0.443\pm
0.035$, and the enclosed total mass and stellar-to-total mass fraction are
correlated with a logarithmic slope of $-0.563\pm 0.035$. The correlations
described here can be used to validate strong-lensing candidates in upcoming
imaging surveys—such as Rubin/Legacy Survey of Space and Time (LSST)—in which
an algorithmic treatment of lensing systems will be needed due to the sheer
volume of data these surveys will produce.

Galaxy clusters (584) — High-redshift galaxy clusters (2007) — Scaling
relations (2031) — Spectral energy distribution (2129) — Strong gravitational
lensing (1643)

††journal: ApJ††facilities: APO/2.5m SDSS Telescope, CTIO/4m Blanco
Telescope.††software:  astropy (Astropy Collaboration et al., 2013, 2018,
2022), astro-prospector (Johnson et al., 2021), emcee (Foreman-Mackey et al.,
2013), FSPS (Conroy et al., 2009; Conroy & Gunn, 2010) Jupyter Notebook
(Kluyver et al., 2016), LENSTOOL (Jullo et al., 2007), matplotlib (Hunter,
2007), numpy (Harris et al., 2020), pandas (Wes McKinney, 2010), python-FSPS
(Johnson et al., 2023), SAOImageDS9 (Joye & Mandel, 2003), scipy (Virtanen et
al., 2020).

##  1 Introduction

Strong gravitational lensing is a rare phenomenon in the universe in which an
intervening object with sufficient surface mass density bends light from a
background source relative to the observer such that multiple images of the
source are formed. For cases where the background source is a galaxy, the
lensing effect can create highly distorted and magnified images of the source
galaxy in an arclike shape (e.g., Lynds & Petrosian 1986; Soucail et al. 1987;
Kneib & Natarajan 2011; Meneghetti et al. 2013; Rivera-Thorsen et al. 2017;
Shajib et al. 2022). The discovery of new gravitational lenses of this type is
accelerating rapidly, particularly with the help of machine learning applied
to finding galaxy-galaxy strong lenses in large optical-imaging surveys (e.g.,
Huang et al. 2021; Rojas et al. 2022; Zaborowski et al. 2023). Coupled with
access to a flood of new data from large surveys such as Rubin/LSST in the
coming years, the population of candidate lenses both in the bulk and at the
margins will only grow. Contemporary methodologies often rely on visual
inspection in order to winnow candidate lists to an acceptable number worthy
of detailed follow-up (e.g., Diehl et al. 2017); such inspection has been
successful in locating new systems thus far (e.g., Rojas et al. 2023).
However, with thousands to millions of lines of sight to choose from in an
ever-growing plethora of imaging data, statistical measures and correlations
in existing lensing systems may help hone candidate-lens samples by
quantifying instances of strong lensing in tandem with traditional
morphological identification. This would produce a purer sample and help
streamline the ability to identify targets. In this paper, we analyze a sample
of 177 strong gravitational lenses identified via visual inspection by the
COOL-LAMPS collaboration in DECaLS Legacy Survey Data Release 8 (LS DR8, Dey
et al. 2019) images, and we investigate the scaling relations between the BCG
redshift, Einstein-radius-enclosed (enclosed) total mass, enclosed luminosity,
enclosed stellar mass, and enclosed stellar-to-total mass fraction therein.
The sample we use here consists of a refined subsample of all strong-lensing
candidates identified by COOL-LAMPS, and we consider them unambiguous and
robust as informed by their visual morphology—despite a lack of definite
spectroscopic confirmation.

Throughout this paper, we adopt a $\mathrm{\Lambda CDM}$ WMAP-9 cosmology
(Hinshaw et al., 2013). All photometric calculations were done in the AB
magnitude system.

##  2 Methodology

Galaxies and galaxy clusters host their mass in a variety of forms. Theory and
observations both place strong constraints on observed stellar luminosity as a
function of mass (e.g., Kuiper 1938; Presotto et al. 2014; Wang & Zhong 2018),
hot intracluster gas contributes significantly to the total mass in galaxy
clusters (e.g., Forman et al. 1972; Kellogg et al. 1972; Markevitch &
Vikhlinin 2007), and both galaxies and galaxy clusters are composed of a
significant fraction of dark matter (e.g., Zwicky 1933; Rubin 1986; Navarro et
al. 1996; Bryan & Norman 1998; Hradecky et al. 2000). However, robustly
measuring these mass components on a per-system basis is time and resource
intensive—which is unsustainable for large samples. Conveniently, the
existence of lensing prescribes a specific amount of mass that must exist to
create the observed lensing a priori.

Remolina González et al. (2020) have shown that knowing only the angle between
the BCG and lensed source arc (a proxy for Einstein radius; $\theta_{E}$) and
the line-of-sight geometry (redshift of the lens and the source; $z_{L}$,
$z_{S}$), the core mass interior to the Einstein radius of a strong-lensing
galaxy cluster may be measured with minimal scatter on order of $\approx
10\%$. With these three parameters, the mass can be immediately calculated and
accounts for all of the projected mass density interior to the Einstein
radius. Measuring mass in this way also offers a well-defined aperture with
which to constrain photometric properties of the lens as well. By directly
integrating over pixels inside this aperture in flux-calibrated images, the
lens-galaxy photometry can be automatically utilized by an SED modeller to
obtain measurements of enclosed luminosity and stellar mass.

The COOL-LAMPS collaboration has constructed a large dataset of strong-lensing
candidates using a simple human-led ranking system (e.g., Khullar et al. 2021;
Sukay et al. 2022; Martinez et al. 2023; Zhang et al. 2023; Napier et al.
2023b; Klein et al. 2024). From this, we have collated a sample of 177 galaxy
clusters and groups that exhibit robust visual evidence of strong
gravitational lensing in which the primary lensed source arc is roughly
circular with its projected center located at or very near to the BCG. Any
systems with complex source-galaxy morphology are less likely to be well
described by the simple symmetric dark-matter profiles that inform the
parametric mass estimates described in Remolina González et al. (2020)—such
complex systems were intentionally excluded from this work.

Figure 1: A subset of nine strong gravitational lenses analyzed in this work.
The lenses are arranged by increasing redshift from the top to the bottom of
the figure, and they are arranged by increasing Einstein radius from the left
to the right of the figure. In each subplot, red circles denote positional
constraints used to derive the Einstein radius, white circles represent the
Einstein aperture, and a black ‘x’ marks the exact coordinates of the BCG.
Respective lens redshifts and Einstein radii are also shown in each subplot.
RGB images have equal astrometric scaling and are sourced from DECaLS LS DR9
(Dey et al., 2019) using $z$, $r$, and $g$-band imaging data with custom color
scaling respectively. Figure 2: Observed-frame spectral energy distributions
(SEDs) in maggies for the nine lenses shown in Figure 1 after SED fitting with
Prospector. The top plot of each subplot contains the best-fit SED (red line),
1024 SEDs drawn from their respective posterior distributions (grey lines),
observed lens-galaxy photometry (blue points), and photometric filter
transmission curves for the Dark Energy Camera (DECam, Flaugher et al. 2015)
$g$, $r$, and $z$-band filters from left to right respectively (in arbitrary
units). The bottom plot of each subplot shows the residual between the
observed photometry and the best-fit SED—normalized by the standard deviation
of the observed photometry.

###  2.1 Einstein Radius

We defined the Einstein radius for each system as the radius of a circle
centered on the BCG (Remolina González et al., 2020) which minimizes the total
angular separation between each of the three most readily identifiable bright
“clumps” in the tangentially lensed arcs and the perimeter of said circle.
Henceforth, we refer to this circular aperture centered on the BCG with radius
equal to the Einstein radius as the “Einstein aperture”. Four points (the BCG
center and three points along each arc) physically constrain the Einstein
aperture, and we obtained them by tagging the point of peak surface brightness
for the BCG and similar peaks in surface brightness in the clumps of each
tangentially lensed arc respectively. If three distinct peaks were unable to
be found along the tangential arcs, the main arc was simply traced. We
bootstrapped the three tangential-arc positional constraints 1000 times, and
we adopted the mean and standard deviation from the resulting distribution of
Einstein radii as the Einstein radius and its error respectively for each
system. A descriptive visualization sampling the range of Einstein radii and
lens redshifts considered in this work is shown in Figure 1.

###  2.2 Lens Redshift

BCGs are sui generis red-sequence cluster galaxies which can also be used as
proxies to infer properties of the cluster-scale dark-matter halo (e.g.,
Oegerle & Hoessel 1991; Lauer et al. 2014). Since we assume that the lensing
systems in this work can be well described by a symmetric dark-matter profile,
we take the redshift of the BCG as interchangeable with the systematic
redshift of the cluster-scale dark-matter halo. The lens redshift converts the
measured Einstein radius to a proper transverse distance via the angular
diameter distance, and we queried the Sloan Digital Sky Survey Data Release 15
(SDSS DR15, Aguado et al. 2019; Kollmeier et al. 2019) in tandem with the
Legacy Survey Data Release 9 (LS DR9, Dey et al. 2019; Duncan 2022) for each
system in this work to obtain lens (BCG) redshifts. If the centroid
coordinates for each BCG corresponded to a warning-free spectroscopic redshift
in SDSS DR15, we adopted that redshift as the lens redshift for that system
along with its associated error. If a given BCG lacked any corresponding value
in SDSS DR15, as is the case for the higher redshift clusters in this work
and/or clusters located in SDSS-non-imaged areas, we adopted the LS DR9
photometric redshift (Zhou et al., 2023) as the lens redshift and its
associated error instead. We obtained spectroscopic redshifts for 41 systems
and photometric redshifts for the remaining 136 systems.

Table 1: Parameters used in SED Fitting with Prospector Free | Parameter | Description | Priors
---|---|---|---
Y | $\mathrm{log}(M_{tot}/M_{\odot})$ | Total stellar mass formed in dex solar masses. | Top Hat: [8.0, 14.0].
Y | $\mathrm{log}(Z/Z_{\odot})$ | Stellar metallicity in dex solar metallicity. | Top Hat: [-1.0, 0.2].
Y | $\lambda_{2}$ | Diffuse dust optical depth. | Top Hat: [0.0, 2.0].
Y | $t_{\mathrm{age}}$ | Age of the cluster in Gyr. | Top Hat: [$t_{z=20}$, $t_{z_{BCG}}$].
Y | $\tau$ | SFH e-folding time in Gyr. | Top Hat: [0.1, 10].
N | imf_type | Initial mass function type. | Chabrier (Chabrier, 2003).
N | dust_type | Dust attenuation curve. | Calzetti (Calzetti et al., 2000).
N | sfh | Star formation history model. | Delayed tau (Carnall et al., 2019).

###  2.3 Source Redshift

In strong lensing, the source redshift is typically constrained either with
spectroscopic follow-up (e.g., Sharon et al. 2020) or by inferring a
photometric redshift (e.g., Cerny et al. 2018). However, obtaining these
redshifts for all identified source galaxies in a given system requires
prolonged additional inquiry in tension with the appeal of an efficient mass
estimator. Remolina González et al. (2020) have shown that substituting a
single known source-galaxy redshift with a distribution of source-galaxy
redshifts introduces a statistically insignificant uncertainty into the final
distribution of mass measurements when compared to the magnitude of other
systematic uncertainties. This distribution of lensed source redshifts is
measurable (e.g., Bayliss et al. 2011a, b; Tran et al. 2022), and we simply
adopt a well-described Gaussian with $\mu=2$ and $\sigma=0.2$ from Bayliss et
al. (2011a) in keeping with the methodology of Remolina González et al. (2020)
as the distribution of source redshifts across the entire sample in this work.

###  2.4 Photometry

We obtained aperture photometry for SED fitting in each system by linearly
summing the dereddened flux from pixels within the Einstein aperture in $g$,
$r$, and $z$-band imaging data from DECaLS LS DR9. Since it is critical that
we do not include light from the lensed source arcs as well as any intervening
stars or foreground galaxies in the measured photometry, we created two masks
to exclude non-lens-galaxy regions. The first mask removed flux from the
lensed source arcs, and the second mask removed flux from interlopers as
identified by a color and magnitude screening by eye. Accounting for masking,
this linear integration was done for each of the 1000 bootstrapped Einstein
apertures derived in Section 2.1, and the resulting mean and standard
deviation were adopted as the $g$, $r$, and $z$-band photometry and their
errors respectively for each system.

##  3 Analysis

###  3.1 Total Mass

Following Remolina González et al. (2020), the enclosed total mass within the
Einstein aperture was calculated using Equations (1) and (2)—where $D(z)$
refers to the angular diameter distance at redshift $z$—via a Monte Carlo
approach. The enclosed total mass was computed 1000 times for each system, in
which each calculation used a different Einstein radius, source redshift, and
lens redshift randomly drawn from a Gaussian distribution with mean equal to
the parameter value and standard deviation equal to the parameter error. Also
described in Remolina González et al. (2020) is the need to apply an empirical
correction described in Equation (3) based on the “completeness” of the lensed
source arcs in each system—where $f(\theta_{E})$ is a cubic polynomial
function specified in Table 1 of Remolina González et al. (2020). This is
because the systematic bias and scatter of $M_{\Sigma}(<\theta_{E})$ has a
dependence on the degree of symmetry for each system, in which larger-
Einstein-radius systems have a stronger offset. The correction factor aims to
remove this dependence as a function of Einstein radius in non-perfectly
symmetric systems.

$\Sigma_{\mathrm{cr}}(z_{L},z_{S})=\frac{c^{2}}{4\pi
G}\frac{D_{S}(z_{S})}{D_{L}(z_{L})D_{LS}(z_{L},z_{S})}$ (1)
$M(<\theta_{E})=\pi(D_{L}(z_{L})\theta_{E})^{2}\,\Sigma_{\mathrm{cr}}(z_{L},z_{S})$
(2) $\mathrm{\ }M_{\Sigma}(<\theta_{E})=\frac{M(<\theta_{E})}{f(\theta_{E})}$
(3)

For the purposes of applying this correction in each of the 177 lensing
systems analyzed in this work, we measured the azimuthal coverage
($\phi$)—defined as the percentage of the Einstein aperture that was traced
out by the lensed source arcs in a given system. For example, a lens with a
tangential arc stretching from an azimuthal angle of 10 to 100 degrees would
have $\phi=0.25$. As was done by Remolina González et al. (2020), $\phi$ was
used as an observable proxy for the degree of symmetry in each system to
determine whether or not the correction factor was needed. Larger-Einstein-
radius systems with a predominantly low $\phi$ tend to deviate from a
symmetric dark-matter halo and thus require the empirical correction. For
spherically symmetric systems with a large $\phi$, the measured
$M_{\Sigma}(<\theta_{E})$ is taken to be fairly unbiased, and thus the
correction factor is not needed. We obtained $\phi$ by determining the
fraction of the perimeter of the Einstein aperture that was subtended by the
the regions masking the lensed source arcs described in Section 2.4 in each
system. Adopting the convention used by Remolina González et al. (2020) in
their analysis, if $\phi<0.5$, Equation (3) was applied. If $\phi\geq 0.5$, it
was not applied. Accounting for the empirical correction, the mean and
standard deviation of the 1000 measurements of $M_{\Sigma}(<\theta_{E})$ were
adopted as the enclosed total mass and its error respectively for each system.

###  3.2 Stellar Luminosity and Mass

We conducted parametric SED fitting using the photometry obtained in Section
2.4 with Prospector (Conroy et al., 2009; Conroy & Gunn, 2010; Johnson et al.,
2021, 2023); five free parameters and three assumed parameters parameterized
each SED. A summary of all eight parameters used in fitting can be found in
Table 1. Utilizing emcee (Foreman-Mackey et al., 2013) as implemented in
Prospector with 84 walkers for a total of 6720 iterations, only the last 840
iterations for each of the 84 walkers (70,560 total parameter vectors) were
taken as representing the posterior distribution for each of the five free
parameters in each system in order to eliminate the significant burn-in
sequence of the fitting process. After fitting, we adopted the 50th percentile
of the posterior distribution as the value for each free parameter, and the
greater difference between the 84th-50th percentile and 50th-16th percentile
values as the error. Observed-frame SEDs for the systems shown in Figure 1
after SED fitting are shown in Figure 2.

To derive luminosity, we generated new SEDs with a random sample of 1024 free-
parameter vectors from the posterior distributions of the best-fit SED in each
system. We integrated over the rest-frame wavelength interval of 3000Å to
7000Å, since this corresponds to the longest wavelength interval sampled by
part of at least two bands of photometry across all BCG redshifts in this work
(the lower bound of 3000Å gets redshifted out of the $\mathit{g}$ and into the
$\mathit{r}$ band at high redshifts). This range also allows us to sample the
rest-frame 4000Å break while recovering flux in longer wavelengths which are
significantly more luminous in cluster galaxies. After integrating the SEDs,
we converted flux to luminosity in solar luminosities using the appropriate
luminosity distance and adopted the mean and standard deviation from the 1024
random parameter vectors as the luminosity and its error respectively for each
system.

##  4 Results

As Figure 3 shows, the BCG redshift, enclosed total mass, and enclosed
luminosity are correlated. Indeed, we note that the most massive clusters in
this work also possess a large stellar mass and thus have a large luminosity.
Using emcee, we constrained a plane-of-best-fit using a likelihood
maximization estimation taking into account the error in data on all three
axes to describe this correlation—where X = $z_{{}_{BCG}}$, Y =
$\mathrm{log}_{10}[M_{\Sigma}(<\theta_{E})/M_{\odot}]$, and Z =
$\mathrm{log}_{10}[L(<\theta_{E})/L_{\odot}]$—yielding:

$Z=0.137_{-0.051}^{+0.051}X+0.522_{-0.023}^{+0.022}Y+4.207_{-0.290}^{+0.300}$
(4)

This simple and analyic relationship derived from a sample of known strong
lenses, considering only quantities derived from observations, may have
utility in differentiating bona fide strong lenses from non-lenses in large
samples of candidate lenses where the evidence of lensing is not nearly as
robust as the sample of clusters analyzed here. Moreover, this correlation
offers a future opportunity to test simulations against
observations—particularly those that aim to generate realistic galaxy
populations in dense cluster environments.

Figure 3: A 3D visualization of the BCG redshift, enclosed total mass, and
enclosed luminosity for the systems in this work plotted as black circles. A
plane-of-best-fit is also shown in black as well. Errors in data on each axis
are represented as red, blue, and green lines respectively.

A full accounting of all measured quantities in this work can be found in
Table 2.

##  5 Discussion

###  5.1 Scaling Relations

Besides the enclosed total mass and luminosity, we can also infer the observed
enclosed stellar mass at the redshift of observation for each system using the
stellar-mass posterior distribution from each SED constrained with Prospector.
We obtained this by correcting the derived enclosed stellar mass formed in
each system with the surviving mass fraction returned by Prospector. This is
necessary because the amount of stellar mass that we see at the redshift of
observation is less than the total stellar mass formed over the lifetime of
the galaxy (e.g., Li et al. 2017)—the latter being what Prospector
parametrically fits. While we note that enclosed stellar mass is more complex
to infer and subject to more significant systematics when compared to the more
direct measures of enclosed total mass from strong-lensing geometry and
luminosity, it nevertheless offers another point of comparison with those
measurables.

####  5.1.1 Total Mass — Stellar Mass

Figure 4: The total mass enclosed within the Einstein aperture for each system
on the x-axis plotted against both the enclosed stellar mass and the enclosed
stellar-to-total mass fraction on the y-axes. Points in Panels I and III are
color coded according to the luminosity of the system in units of solar
luminosities from stellar light enclosed within the Einstein aperture as
measured within the rest-frame wavelength interval of 3000Å to 7000Å. Points
in Panels II and IV only represent systems in which multiple cluster members
fall within the Einstein aperture. Linear regression lines in all four Panels
are drawn from the posterior distributions of their respective MCMC.

In Panel I of Figure 4, we see a clearly positive correlation between the
enclosed total mass and the enclosed stellar mass for all systems. By
performing a linear regression with emcee using a likelihood maximization
estimation that accounts for errors in data on both axes, we find that the
slope of this correlation is $0.443\pm 0.035$ from the posterior distribution
of the MCMC fit. We colored the points according to their luminosity to
highlight how the intrinsic differences between individual systems contribute
to the overall intrinsic scatter. In Table 2 of Kravtsov et al. (2018), they
found $M_{500}$ (White, 2001) and the stellar mass of solely the BCG to be
correlated with a logarithmic slope of $0.33\pm 0.11$. In addition, Kravtsov
et al. (2018) found $M_{500}$ and the stellar mass of the entire cluster to be
correlated with a logarithmic slope of $0.59\pm 0.08$. The fact that the slope
for this work falls between the two slopes quoted by Kravtsov et al. (2018) is
consistent with how our methodology samples a physical regime between
exclusively the BCG and that of the entire cluster.

Panel II of Figure 4 further explores this correlation, with only those
clusters where multiple cluster members were contained within the Einstein
aperture are plotted. A linear regression for the multiple-galaxy systems
gives a slope of $0.453\pm 0.049$, which is consistent with the slope using
all systems.

####  5.1.2 Total Mass — Stellar-to-Total Mass Fraction

In Panel III of Figure 4, we plot the enclosed stellar-to-total mass fraction
against the enclosed total mass and see a clearly negative correlation (again
with the same colormapping as Panel I to emphasize intrinsic scatter). Also
using emcee, we find that the slope of this correlation is $-0.563\pm 0.035$.
Categorizing in Panel IV identically to Panel II, we find that the multiple-
galaxy systems are correlated with a slope of $-0.561\pm 0.050$. We again find
that the slope for the multiple-galaxy systems are statistically
indistinguishable from the slope for the entire sample shown in Panel III.
Andreon (2010) measured 52 clusters over a slightly larger mass range and
found that the stellar mass fraction depended on halo mass with a slope of
$-0.55\pm 0.08$ when measuring out to a physical extent of $R_{200}$. Both of
the slopes in Panels III and IV are statistically similar to Andreon (2010),
and while the linear fit derived by Andreon (2010) has a slightly larger
normalization, this is to be expected because the cluster masses are
constrained out to $R_{200}$.

###  5.2 Photometric Bias and Error

Deriving light in Section 2.4 via directly integrating over the Einstein
aperture presents some challenges. Since we identified red-sequence cluster
members by using observed color as a proxy for cluster membership, it is
possible that we may have included intrinsically bluer galaxies at redshift
$z>z_{{}_{BCG}}$ that appear to be the same color as the cluster members.
Conversely, we may have excluded intrinsically bluer bona fide cluster
members. However, star-forming cluster members have less stellar mass and are
intrinsically rarer in cluster cores (e.g., Dressler et al. 1997, 2004), and
we do not expect that these missed cluster galaxies introduce a significant
loss to the measured cluster light. In addition, bluer galaxies at redshift
$z>z_{{}_{BCG}}$ appear relatively fainter, which are also unlikely to bias
our methodology of measuring cluster light. Interlopers such as foreground
stars that appear in front of cluster galaxies are forced to be masked, but
any underlying lens-galaxy flux is typically minimal due to the limited
angular size of such stars.

In general, we expect that all of the unmasked flux from cluster cores at the
angular scales considered in this work originates from genuine cluster
members. Any modest background correction that could be made will also be
effectively captured in the redshift dependence of the fits (e.g., Figure 3).
While we must also consider associated galaxy populations in the immediate
large-scale structure when considering galaxy clusters projected onto the
sky—as such structures may also be projected onto the cluster core—those
galaxies will also contribute to the lensing effect and will have their mass
and light captured by the methodology presented here.

We also note that our sample contains extremely few examples of systems which
have a measured Einstein radius of $\theta_{E}\lessapprox 2^{\prime\prime}$.
To some extent, this is a consequence of our chosen sample of cluster-scale
lenses, which we would generally expect to have larger Einstein radii.
However, we caution that there exists a large scatter between the total halo
mass and Einstein radius (Fox et al., 2022). Systems whose arcs are separated
from the central galaxy at small radii may also be located directly within the
discernable BCG light. This would not only make them hard to visually notice,
but it would also make them unsuitable for this analysis because the source
light would be significantly blended together with the lensing cluster light.
While we thus expect this work to undersample the small-Einstein-radius
regime, it is not apparent that this significantly alters our findings.

In addition, our use of relatively shallow ground-based imaging limits the
$5\sigma$ point-source magnitude depth of objects in the $g$, $r$, and
$z$-band filters to $\approx$ 24.7, 23.9, and 23.0 respectively in the average
case of two exposures per filter (Dey et al., 2019). Systems with a lens
redshift $z\gtrapprox 1$ are barely visible even in $\mathit{z}$-band LS DR9
imaging data, and they are also simply a rarer type of lensing system in the
universe (Li et al., 2019). Both the data depth and intrinsic redshift
distribution of lenses shape the redshift distribution in this work.

###  5.3 Alternative Mass Measurement

In constraining mass, we also considered a second method for measuring
cluster-centric mass elucidated in Remolina González et al. (2021a) using the
parametric lens-modelling software LENSTOOL (Jullo et al., 2007) to measure
the enclosed total mass—as opposed to the simple evaluation of Equations (1),
(2), and (3). For the first 35 clusters that were analyzed in this work, we
generated single-halo lens models by taking the three coordinates in each
lensed arc used to derive the Einstein radius in Section 2.1 as a multiple-
image family with a single pseudo-isothermal elliptical mass distribution
(PIEMD, Kassiola & Kovner 1993) locked to the center of the BCG. Mass
estimates were then obtained from the resulting best-fit lens models according
to the methodology described in Remolina González et al. (2021a).

Figure 5: A comparison of the single-halo LENSTOOL-generated mass measurements
($M_{SHM}$) and the parametrically generated mass measurements ($M_{\Sigma}$)
for the first 35 clusters analyzed in this work. The black line represents an
ideal 1-1 correspondence, and the blue lines represent the uncertainty on the
measured correspondence.

As expected from Remolina González et al. (2021b), these differing
methodologies show similar results. The two mass measurements for the 35
clusters studied here are correlated with a slope of $0.960\pm 0.036$ and
intercept of $0.492\pm 0.476$ dex as shown in Figure 5. In other words, we
find no reason to suggest a preference between one method over the other, and
we abandoned the LENSTOOL approach in favor of the much faster parametric
approach throughout the rest of the work.

In theory, lens modelling has the potential to be the most accurate mass-
constraining methodology. However, this requires high-spatial-resolution
imaging data and ample time. The process of iteratively refining just one lens
model may take upwards of an hour for even a single relatively simple
system—which is untenable when the number of candidate systems reaches into
the thousands. Moreover, the accuracy of measuring mass with lens modelling is
sensitive to the choice of defining multiple-image families along the lensed
source arcs in a way that simple circle fitting is not. These essential
positional constraints are often unresolved even with the highest-quality
ground-based imaging data. Conversely, the positional constraints that inform
the Einstein radius ultimately used by Equations (1), (2), and (3) need not be
extremely accurate with respect to exactly constraining image family
positions. As long as the positional constraints are reasonably placed within
the arcs, something which is resolved by ground-based imaging data, it is
accurate to within the systematic uncertainty.

For more information on lens-modelling software, related techniques, and
applications of strong lensing, we refer the reader to the following non-
exhaustive list of papers for further review: Elíasdóttir et al. (2007), Oguri
(2010), Lefor et al. (2013), Meneghetti et al. (2017), Birrer & Amara (2018),
Schäfer et al. (2020), Sharon et al. (2022), and Napier et al. (2023a).

##  6 Summary and Conclusions

This paper presents measurements of mass and light in 177 strong-lensing
systems at cluster and group mass scales. By directly obtaining the Einstein
radius and BCG redshift for each system, we compute the Einstein-radius-
enclosed total mass parametrically. We use the enclosed $g$, $r$, and $z$-band
photometry with Prospector to parameterize the enclosed luminosity and stellar
mass, and we obtain a planar relationship between the BCG redshift, enclosed
total mass, and enclosed luminosity of the form
$Z=0.137_{-0.051}^{+0.051}X+0.522_{-0.023}^{+0.022}Y+4.207_{-0.290}^{+0.300}$
where X = $z_{{}_{BCG}}$, Y =
$\mathrm{log}_{10}[M_{\Sigma}(<\theta_{E})/M_{\odot}]$, and Z =
$\mathrm{log}_{10}[L(<\theta_{E})/L_{\odot}]$. We find that the enclosed total
mass and stellar mass for the entire sample of strong-lensing systems
considered in this work are correlated with a logarithmic slope,
normalization, and intrinsic stellar-mass scatter of $0.443\pm 0.035$,
$5.958\pm 0.454$ dex, and $0.173\pm 0.013$ dex respectively. We also find that
the enclosed total mass and stellar-to-total mass fraction are correlated with
a logarithmic slope, normalization, and intrinsic stellar-to-total-mass-
fraction scatter of $-0.563\pm 0.035$, $6.034\pm 0.457$ dex, and $0.161\pm
0.013$ dex respectively. Enormous volumes of imaging data from new space-based
and ground-based surveys at deeper and sharper photometric limits are on the
horizon. Probing these upcoming datasets with both visual and machine-learning
driven searches for new strong lenses at a variety of scales is inevitable.
The correlations shown in this work should have utility in filtering rankings
of possible strong-lensing systems as well as offering another measure of
cluster cores that can be compared to cosmological simulations.

## Acknowledgments

This work is supported by The College undergraduate program and the College
Innovation Fund at the University of Chicago and the Department of Astronomy
and Astrophysics at the University of Chicago.

The Legacy Surveys consist of three individual and complementary projects: the
Dark Energy Camera Legacy Survey (DECaLS; Proposal ID #2014B-0404; PIs: David
Schlegel and Arjun Dey), the Beijing-Arizona Sky Survey (BASS; NOAO Prop. ID
#2015A-0801; PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band Legacy
Survey (MzLS; Prop. ID #2016A-0453; PI: Arjun Dey). DECaLS, BASS and MzLS
together include data obtained, respectively, at the Blanco telescope, Cerro
Tololo Inter-American Observatory, NSF’s NOIRLab; the Bok telescope, Steward
Observatory, University of Arizona; and the Mayall telescope, Kitt Peak
National Observatory, NOIRLab. Pipeline processing and analyses of the data
were supported by NOIRLab and the Lawrence Berkeley National Laboratory
(LBNL). The Legacy Surveys project is honored to be permitted to conduct
astronomical research on Iolkam Du’ag (Kitt Peak), a mountain with particular
significance to the Tohono O’odham Nation.

NOIRLab is operated by the Association of Universities for Research in
Astronomy (AURA) under a cooperative agreement with the National Science
Foundation. LBNL is managed by the Regents of the University of California
under contract to the U.S. Department of Energy.

This project used data obtained with the Dark Energy Camera (DECam), which was
constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES
Projects has been provided by the U.S. Department of Energy, the U.S. National
Science Foundation, the Ministry of Science and Education of Spain, the
Science and Technology Facilities Council of the United Kingdom, the Higher
Education Funding Council for England, the National Center for Supercomputing
Applications at the University of Illinois at Urbana-Champaign, the Kavli
Institute of Cosmological Physics at the University of Chicago, Center for
Cosmology and Astro-Particle Physics at the Ohio State University, the
Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M
University, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho
de Amparo, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de
Amparo a Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de
Desenvolvimento Cientifico e Tecnologico and the Ministerio da Ciencia,
Tecnologia e Inovacao, the Deutsche Forschungsgemeinschaft and the
Collaborating Institutions in the Dark Energy Survey. The Collaborating
Institutions are Argonne National Laboratory, the University of California at
Santa Cruz, the University of Cambridge, Centro de Investigaciones
Energeticas, Medioambientales y Tecnologicas-Madrid, the University of
Chicago, University College London, the DES-Brazil Consortium, the University
of Edinburgh, the Eidgenossische Technische Hochschule (ETH) Zurich, Fermi
National Accelerator Laboratory, the University of Illinois at Urbana-
Champaign, the Institut de Ciencies de l’Espai (IEEC/CSIC), the Institut de
Fisica d’Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig
Maximilians Universitat Munchen and the associated Excellence Cluster
Universe, the University of Michigan, NSF’s NOIRLab, the University of
Nottingham, the Ohio State University, the University of Pennsylvania, the
University of Portsmouth, SLAC National Accelerator Laboratory, Stanford
University, the University of Sussex, and Texas A&M University.

BASS is a key project of the Telescope Access Program (TAP), which has been
funded by the National Astronomical Observatories of China, the Chinese
Academy of Sciences (the Strategic Priority Research Program “The Emergence of
Cosmological Structures” Grant # XDB09000000), and the Special Fund for
Astronomy from the Ministry of Finance. The BASS is also supported by the
External Cooperation Program of Chinese Academy of Sciences (Grant #
114A11KYSB20160057), and Chinese National Natural Science Foundation (Grant #
12120101003, # 11433005).

The Legacy Survey team makes use of data products from the Near-Earth Object
Wide-field Infrared Survey Explorer (NEOWISE), which is a project of the Jet
Propulsion Laboratory/California Institute of Technology. NEOWISE is funded by
the National Aeronautics and Space Administration.

The Legacy Surveys imaging of the DESI footprint is supported by the Director,
Office of Science, Office of High Energy Physics of the U.S. Department of
Energy under Contract No. DE-AC02-05CH1123, by the National Energy Research
Scientific Computing Center, a DOE Office of Science User Facility under the
same contract; and by the U.S. National Science Foundation, Division of
Astronomical Sciences under Contract No. AST-0950945 to NOAO.

The Photometric Redshifts for the Legacy Surveys (PRLS) catalog used in this
paper was produced thanks to funding from the U.S. Department of Energy Office
of Science, Office of High Energy Physics via grant DE-SC0007914.

Funding for the Sloan Digital Sky Survey V has been provided by the Alfred P.
Sloan Foundation, the Heising-Simons Foundation, the National Science
Foundation, and the Participating Institutions. SDSS acknowledges support and
resources from the Center for High-Performance Computing at the University of
Utah. The SDSS web site is www.sdss.org.

SDSS is managed by the Astrophysical Research Consortium for the Participating
Institutions of the SDSS Collaboration, including the Carnegie Institution for
Science, Chilean National Time Allocation Committee (CNTAC) ratified
researchers, the Gotham Participation Group, Harvard University, Heidelberg
University, The Johns Hopkins University, L’Ecole polytechnique fédérale de
Lausanne (EPFL), Leibniz-Institut für Astrophysik Potsdam (AIP), Max-Planck-
Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für
Extraterrestrische Physik (MPE), Nanjing University, National Astronomical
Observatories of China (NAOC), New Mexico State University, The Ohio State
University, Pennsylvania State University, Smithsonian Astrophysical
Observatory, Space Telescope Science Institute (STScI), the Stellar
Astrophysics Participation Group, Universidad Nacional Autónoma de México,
University of Arizona, University of Colorado Boulder, University of Illinois
at Urbana-Champaign, University of Toronto, University of Utah, University of
Virginia, Yale University, and Yunnan University.

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Table 2: Measured Quantities for all 177 Systems

ID | RA [∘] | Dec [∘] | $z_{{}_{BCG}}$ |  $\theta_{E}$ [′′] |  $M_{\odot,\Sigma}$ [dex] |  $M_{\odot,\star}$ [dex] |  $L_{\odot}$ [dex]
---|---|---|---|---|---|---|---
J0008$+$1822 m | 2.029829 | 18.372282 | 0.5566 $\pm$ 0.0483 p | 3.2431 $\pm$ 0.1446 | 12.3479 $\pm$ 0.0594 | 11.2363 $\pm$ 0.0944 | 10.5378 $\pm$ 0.0835
J0008$-$0624 o | 2.169667 | -6.405227 | 0.9801 $\pm$ 0.0584 p | 3.7217 $\pm$ 0.1170 | 12.7744 $\pm$ 0.0674 | 11.2099 $\pm$ 0.1202 | 10.7562 $\pm$ 0.0632
J0008$+$2150 m | 2.181223 | 21.836401 | 0.5974 $\pm$ 0.0190 p | 7.3602 $\pm$ 0.4852 | 13.0947 $\pm$ 0.0627 | 11.8165 $\pm$ 0.1114 | 10.9954 $\pm$ 0.0348
J0019$+$0438 m | 4.954375 | 4.649355 | 0.8685 $\pm$ 0.0339 p | 5.8885 $\pm$ 0.9581 | 13.1101 $\pm$ 0.1467 | 11.7218 $\pm$ 0.3601 | 11.0978 $\pm$ 0.0804
J0021$+$0333 m | 5.402546 | 3.556039 | 0.7164 $\pm$ 0.0222 p | 4.5494 $\pm$ 0.2190 | 12.7638 $\pm$ 0.0520 | 11.7067 $\pm$ 0.1184 | 10.9886 $\pm$ 0.0356
J0022$+$1431 ${}^{\mathrm{o}}_{*}$ | 5.670506 | 14.519566 | 0.3805 $\pm$ 0.0001 s | 3.8254 $\pm$ 0.0599 | 12.3208 $\pm$ 0.0169 | 11.1673 $\pm$ 0.0893 | 10.7093 $\pm$ 0.0020
J0023$-$0252 ${}^{\mathrm{o}}_{*}$ | 5.835225 | -2.874259 | 0.7973 $\pm$ 0.0698 p | 2.0058 $\pm$ 0.2289 | 12.2348 $\pm$ 0.1202 | 10.8995 $\pm$ 0.0010 | 10.2946 $\pm$ 0.1102
J0027$-$0413 m | 6.750411 | -4.223259 | 0.4950 $\pm$ 0.0001 s | 5.6052 $\pm$ 0.2712 | 12.7693 $\pm$ 0.0434 | 11.8215 $\pm$ 0.0886 | 11.0519 $\pm$ 0.0053
J0030$-$0101 o | 7.558161 | -1.029405 | 0.7234 $\pm$ 0.0954 p | 3.6596 $\pm$ 0.4608 | 12.7085 $\pm$ 0.1409 | 11.1761 $\pm$ 0.2916 | 10.5556 $\pm$ 0.1754
J0032$+$0740 o | 8.013587 | 7.667835 | 0.6437 $\pm$ 0.0003 s | 3.0485 $\pm$ 0.2216 | 12.4813 $\pm$ 0.0674 | 11.5989 $\pm$ 0.1874 | 10.7664 $\pm$ 0.0158
J0034$+$0225 ${}^{\mathrm{m}}_{*}$ | 8.617335 | 2.422958 | 0.3865 $\pm$ 0.0077 p | 13.3278 $\pm$ 0.7876 | 13.4127 $\pm$ 0.0525 | 11.8401 $\pm$ 0.1610 | 11.2355 $\pm$ 0.0213
J0036$-$0506 o | 9.081628 | -5.108834 | 0.8121 $\pm$ 0.0223 p | 3.8970 $\pm$ 0.0943 | 12.6984 $\pm$ 0.0426 | 11.4701 $\pm$ 0.1397 | 10.9183 $\pm$ 0.0286
J0038$+$0719 m | 9.730731 | 7.323072 | 0.2643 $\pm$ 0.0178 p | 7.3590 $\pm$ 0.0588 | 12.7345 $\pm$ 0.0292 | 11.4604 $\pm$ 0.0443 | 10.7314 $\pm$ 0.0669
J0046$-$0156 m | 11.508417 | -1.941216 | 0.5501 $\pm$ 0.0002 s | 5.3104 $\pm$ 0.3456 | 12.7713 $\pm$ 0.0588 | 11.6781 $\pm$ 0.1068 | 10.9179 $\pm$ 0.0072
J0047$+$0508 m | 11.961840 | 5.138756 | 0.4291 $\pm$ 0.0001 s | 8.9680 $\pm$ 0.1704 | 13.1134 $\pm$ 0.0209 | 11.7715 $\pm$ 0.1798 | 11.1678 $\pm$ 0.0063
J0051$-$0923 o | 12.900786 | -9.390354 | 0.7134 $\pm$ 0.0243 p | 3.6019 $\pm$ 0.2007 | 12.5599 $\pm$ 0.0582 | 11.6749 $\pm$ 0.1024 | 11.0457 $\pm$ 0.0363
J0056$-$1402 m | 14.247809 | -14.042115 | 0.4251 $\pm$ 0.0208 p | 6.1175 $\pm$ 0.3036 | 12.7791 $\pm$ 0.0501 | 11.5599 $\pm$ 0.1234 | 10.8910 $\pm$ 0.0535
J0057$+$2138 o | 14.312163 | 21.642771 | 0.8471 $\pm$ 0.0402 p | 4.0474 $\pm$ 0.0557 | 12.7556 $\pm$ 0.0470 | 11.5082 $\pm$ 0.0201 | 10.9451 $\pm$ 0.0550
J0057$+$0230 o | 14.421865 | 2.504200 | 0.9262 $\pm$ 0.0005 s | 3.8478 $\pm$ 0.1220 | 12.7632 $\pm$ 0.0473 | 11.6204 $\pm$ 0.1193 | 11.0455 $\pm$ 0.0074
J0058$-$0721 ${}^{\mathrm{m}}_{*}$ | 14.703994 | -7.365765 | 0.6379 $\pm$ 0.0001 s | 15.5906 $\pm$ 0.5102 | 13.7754 $\pm$ 0.0354 | 12.4146 $\pm$ 0.0888 | 11.6556 $\pm$ 0.0036
J0100$+$1818 ${}^{\mathrm{o}}_{*}$ | 15.204935 | 18.307730 | 0.5819 $\pm$ 0.0002 s | 6.3332 $\pm$ 0.5963 | 13.0701 $\pm$ 0.0821 | 11.8056 $\pm$ 0.1124 | 10.9924 $\pm$ 0.0077
J0101$+$2055 o | 15.437768 | 20.928614 | 0.8708 $\pm$ 0.0003 s | 4.5847 $\pm$ 0.1863 | 12.8807 $\pm$ 0.0494 | 11.5800 $\pm$ 0.1037 | 10.8803 $\pm$ 0.0171
J0101$-$2126 ${}^{\mathrm{o}}_{*}$ | 15.498254 | -21.448722 | 0.4001 $\pm$ 0.0375 p | 4.4222 $\pm$ 0.2826 | 12.4691 $\pm$ 0.0694 | 11.6006 $\pm$ 0.1268 | 10.8508 $\pm$ 0.0965
J0102$-$1400 o | 15.676298 | -14.015030 | 0.6737 $\pm$ 0.0321 p | 3.5777 $\pm$ 0.0469 | 12.6385 $\pm$ 0.0338 | 11.3892 $\pm$ 0.0428 | 10.8223 $\pm$ 0.0492
J0102$-$2356 o | 15.678569 | -23.943939 | 0.7084 $\pm$ 0.0286 p | 3.1461 $\pm$ 0.0379 | 12.4376 $\pm$ 0.0332 | 11.2482 $\pm$ 0.1321 | 10.6643 $\pm$ 0.0430
J0103$+$2706 m | 15.937230 | 27.102542 | 0.3825 $\pm$ 0.0001 s | 4.3067 $\pm$ 0.5513 | 12.5506 $\pm$ 0.1105 | 11.5542 $\pm$ 0.1512 | 10.7612 $\pm$ 0.0282
J0104$-$0340 o | 16.214582 | -3.671084 | 0.9163 $\pm$ 0.0350 p | 6.8895 $\pm$ 0.0476 | 13.2638 $\pm$ 0.0492 | 11.3674 $\pm$ 0.1768 | 11.1311 $\pm$ 0.0442
J0104$-$0757 ${}^{\mathrm{o}}_{*}$ | 16.220190 | -7.952046 | 0.8988 $\pm$ 0.0401 p | 2.6289 $\pm$ 0.1924 | 12.4196 $\pm$ 0.0831 | 11.3152 $\pm$ 0.1311 | 10.8239 $\pm$ 0.0505
J0105$-$0501 o | 16.462922 | -5.032347 | 0.7015 $\pm$ 0.0002 s | 3.7470 $\pm$ 0.0744 | 12.5842 $\pm$ 0.0294 | 11.6409 $\pm$ 0.0585 | 11.0437 $\pm$ 0.0028
J0109$-$0455 m | 17.294557 | -4.919535 | 0.7623 $\pm$ 0.0352 p | 5.1460 $\pm$ 0.1236 | 12.9069 $\pm$ 0.0439 | 11.7990 $\pm$ 0.1102 | 11.1524 $\pm$ 0.0473
J0109$-$3335 m | 17.469499 | -33.592604 | 0.4730 $\pm$ 0.0114 p | 9.0865 $\pm$ 0.1633 | 13.1676 $\pm$ 0.0239 | 11.9410 $\pm$ 0.0867 | 11.2692 $\pm$ 0.0259
J0111$+$1346 m | 17.779877 | 13.778901 | 0.7203 $\pm$ 0.0002 s | 5.2451 $\pm$ 0.9923 | 12.9065 $\pm$ 0.1627 | 11.7728 $\pm$ 0.3236 | 11.1351 $\pm$ 0.0841
J0111$+$0855 m | 17.881205 | 8.928360 | 0.4855 $\pm$ 0.0001 s | 11.2303 $\pm$ 0.3733 | 13.3637 $\pm$ 0.0327 | 12.1471 $\pm$ 0.1127 | 11.4398 $\pm$ 0.0110
J0112$-$2053 ${}^{\mathrm{m}}_{*}$ | 18.014632 | -20.891104 | 0.6441 $\pm$ 0.0355 p | 5.4284 $\pm$ 0.3018 | 12.8648 $\pm$ 0.0595 | 11.9186 $\pm$ 0.1656 | 11.2702 $\pm$ 0.0612
J0117$-$0527 m | 19.494867 | -5.454945 | 0.5797 $\pm$ 0.0002 s | 2.8465 $\pm$ 0.2686 | 12.3730 $\pm$ 0.0814 | 11.3386 $\pm$ 0.1937 | 10.6179 $\pm$ 0.0259
J0118$-$0526 m | 19.663276 | -5.443448 | 0.5803 $\pm$ 0.0002 s | 12.8405 $\pm$ 0.1036 | 13.5613 $\pm$ 0.0193 | 12.1258 $\pm$ 0.0706 | 11.4273 $\pm$ 0.0033
J0122$-$0831 o | 20.604365 | -8.520537 | 0.4906 $\pm$ 0.0495 p | 2.0523 $\pm$ 0.2209 | 12.0154 $\pm$ 0.1054 | 11.0911 $\pm$ 0.1386 | 10.4452 $\pm$ 0.1081
J0124$-$2401 o | 21.086027 | -24.029980 | 0.6187 $\pm$ 0.0092 p | 2.9426 $\pm$ 0.1948 | 12.3140 $\pm$ 0.0608 | 11.6559 $\pm$ 0.1861 | 10.9180 $\pm$ 0.0192
J0127$+$2713 m | 21.939205 | 27.229009 | 0.7989 $\pm$ 0.0234 p | 7.1353 $\pm$ 0.3294 | 13.2140 $\pm$ 0.0522 | 11.9420 $\pm$ 0.0511 | 11.3248 $\pm$ 0.0325
J0130$-$0305 m | 22.517741 | -3.095758 | 0.6754 $\pm$ 0.1767 p | 17.8274 $\pm$ 0.2720 | 13.9264 $\pm$ 0.1399 | 12.5665 $\pm$ 0.0855 | 11.6481 $\pm$ 0.3212
J0130$+$3216 o | 22.730273 | 32.275463 | 0.5093 $\pm$ 0.0001 s | 9.5797 $\pm$ 0.8274 | 13.3667 $\pm$ 0.0753 | 11.5566 $\pm$ 0.0884 | 10.9050 $\pm$ 0.0070
J0133$-$1650 m | 23.423964 | -16.839019 | 0.6490 $\pm$ 0.0191 p | 6.1055 $\pm$ 0.5181 | 12.9727 $\pm$ 0.0784 | 11.5655 $\pm$ 0.1746 | 11.0693 $\pm$ 0.0423
J0134$+$0433 o | 23.676569 | 4.563916 | 0.5509 $\pm$ 0.0001 s | 5.2927 $\pm$ 0.1836 | 12.8849 $\pm$ 0.0357 | 11.7604 $\pm$ 0.0564 | 11.0052 $\pm$ 0.0039
J0135$-$2033 ${}^{\mathrm{m}}_{*}$ | 23.928317 | -20.559902 | 0.6111 $\pm$ 0.0169 p | 4.7364 $\pm$ 0.3760 | 12.8403 $\pm$ 0.0737 | 11.8398 $\pm$ 0.1447 | 11.1292 $\pm$ 0.0305
J0136$-$2200 o | 24.211948 | -22.007600 | 0.6741 $\pm$ 0.0247 p | 2.6192 $\pm$ 0.1161 | 12.3702 $\pm$ 0.0491 | 11.4187 $\pm$ 0.1177 | 10.6767 $\pm$ 0.0446
J0137$+$2109 o | 24.385656 | 21.156756 | 0.8805 $\pm$ 0.0610 p | 4.4567 $\pm$ 0.5514 | 12.9894 $\pm$ 0.1220 | 11.2638 $\pm$ 0.2091 | 10.9060 $\pm$ 0.0742
J0138$-$2155 ${}^{\mathrm{m}}_{*}$ | 24.515695 | -21.925493 | 0.2468 $\pm$ 0.0451 p | 18.2693 $\pm$ 0.4811 | 13.4977 $\pm$ 0.0793 | 12.0640 $\pm$ 0.0339 | 11.1297 $\pm$ 0.1897
J0138$-$2844 m | 24.597416 | -28.735794 | 0.4005 $\pm$ 0.0141 p | 8.7725 $\pm$ 0.1847 | 13.0635 $\pm$ 0.0260 | 11.6350 $\pm$ 0.1838 | 10.9807 $\pm$ 0.0406
J0139$+$2207 o | 24.778355 | 22.123106 | 0.4770 $\pm$ 0.0001 s | 5.8602 $\pm$ 0.4416 | 12.7931 $\pm$ 0.0670 | 11.5957 $\pm$ 0.0918 | 10.8865 $\pm$ 0.0083
J0140$-$2006 m | 25.052071 | -20.105520 | 0.3617 $\pm$ 0.0070 p | 17.3092 $\pm$ 0.8612 | 13.6105 $\pm$ 0.0447 | 11.9248 $\pm$ 0.1966 | 11.2967 $\pm$ 0.0212
J0143$+$1427 o | 25.953318 | 14.461970 | 0.3305 $\pm$ 0.0001 s | 6.8998 $\pm$ 0.3140 | 12.7743 $\pm$ 0.0408 | 11.3398 $\pm$ 0.2162 | 10.6899 $\pm$ 0.0034
J0144$-$2213 m | 26.168407 | -22.229454 | 0.2773 $\pm$ 0.0068 p | 12.5538 $\pm$ 0.0634 | 13.2187 $\pm$ 0.0130 | 11.8627 $\pm$ 0.0364 | 11.0659 $\pm$ 0.0255
J0144$+$1008 o | 26.172042 | 10.141264 | 0.4649 $\pm$ 0.0001 s | 4.1770 $\pm$ 0.1607 | 12.4857 $\pm$ 0.0358 | 11.3946 $\pm$ 0.1086 | 10.7550 $\pm$ 0.0056
J0145$-$3541 o | 26.444968 | -35.691008 | 0.4986 $\pm$ 0.0182 p | 3.7835 $\pm$ 0.0985 | 12.4300 $\pm$ 0.0328 | 11.4321 $\pm$ 0.0631 | 10.7235 $\pm$ 0.0413
J0145$+$0402 o | 26.484783 | 4.041294 | 0.7800 $\pm$ 0.0410 p | 2.2780 $\pm$ 0.0877 | 12.3276 $\pm$ 0.0560 | 11.3770 $\pm$ 0.1367 | 10.7692 $\pm$ 0.0568
J0146$-$0929 ${}^{\mathrm{m}}_{*}$ | 26.733372 | -9.497905 | 0.4469 $\pm$ 0.0001 s | 12.1801 $\pm$ 0.5248 | 13.5148 $\pm$ 0.0396 | 12.0644 $\pm$ 0.0874 | 11.3973 $\pm$ 0.0034
J0149$-$2834 o | 27.483033 | -28.575331 | 0.5563 $\pm$ 0.0076 p | 5.8990 $\pm$ 0.1144 | 12.8664 $\pm$ 0.0254 | 11.6969 $\pm$ 0.0422 | 10.9577 $\pm$ 0.0154
J0150$+$2725 m | 27.503618 | 27.426752 | 0.3062 $\pm$ 0.0001 s | 5.5224 $\pm$ 0.3406 | 12.5492 $\pm$ 0.0548 | 11.5340 $\pm$ 0.0749 | 10.7487 $\pm$ 0.0069
J0151$-$0407 m | 27.908847 | -4.120915 | 0.6700 $\pm$ 0.0002 s | 6.7660 $\pm$ 0.1959 | 13.0749 $\pm$ 0.0342 | 11.9359 $\pm$ 0.1096 | 11.1376 $\pm$ 0.0060
J0153$-$3235 o | 28.474812 | -32.599633 | 0.6541 $\pm$ 0.0204 p | 4.1855 $\pm$ 0.0550 | 12.6452 $\pm$ 0.0295 | 11.5416 $\pm$ 0.1301 | 10.9411 $\pm$ 0.0345
J0159$-$3413 m | 29.766637 | -34.217907 | 0.4143 $\pm$ 0.0252 p | 10.4481 $\pm$ 0.1989 | 13.2293 $\pm$ 0.0323 | 11.8191 $\pm$ 0.1728 | 11.2005 $\pm$ 0.0679
J0202$-$1109 ${}^{\mathrm{m}}_{*}$ | 30.543766 | -11.153215 | 0.4444 $\pm$ 0.0237 p | 5.6519 $\pm$ 0.7326 | 12.8526 $\pm$ 0.1131 | 11.6920 $\pm$ 0.1024 | 10.9192 $\pm$ 0.0545
J0203$-$2017 m | 30.793373 | -20.289939 | 0.4389 $\pm$ 0.0227 p | 13.3912 $\pm$ 0.4259 | 13.4719 $\pm$ 0.0375 | 11.9077 $\pm$ 0.1197 | 11.2031 $\pm$ 0.0590
J0204$-$2918 m | 31.034313 | -29.302323 | 0.9894 $\pm$ 0.0775 p | 3.9745 $\pm$ 0.0540 | 12.8406 $\pm$ 0.0737 | 11.4984 $\pm$ 0.2316 | 11.0811 $\pm$ 0.0955
J0205$-$3539 m | 31.358759 | -35.663177 | 0.3856 $\pm$ 0.0101 p | 3.4782 $\pm$ 0.0561 | 12.2441 $\pm$ 0.0211 | 11.3942 $\pm$ 0.0211 | 10.7298 $\pm$ 0.0274
J0209$-$3547 m | 32.476474 | -35.799023 | 0.3105 $\pm$ 0.0097 p | 5.8527 $\pm$ 0.4780 | 12.6069 $\pm$ 0.0724 | 11.5478 $\pm$ 0.0768 | 10.7458 $\pm$ 0.0324
J0210$+$2600 o | 32.594801 | 26.011166 | 0.3960 $\pm$ 0.0001 s | 3.2755 $\pm$ 0.0630 | 12.2030 $\pm$ 0.0195 | 11.2954 $\pm$ 0.0693 | 10.7267 $\pm$ 0.0022
J0210$+$3044 o | 32.744179 | 30.739885 | 0.4687 $\pm$ 0.1181 p | 5.5256 $\pm$ 0.9274 | 12.7561 $\pm$ 0.1985 | 11.4055 $\pm$ 0.1522 | 10.7429 $\pm$ 0.3074
J0214$-$0206 ${}^{\mathrm{o}}_{*}$ | 33.533392 | -2.107933 | 0.6477 $\pm$ 0.0322 p | 2.9225 $\pm$ 0.2936 | 12.4520 $\pm$ 0.0920 | 11.5662 $\pm$ 0.1695 | 10.8001 $\pm$ 0.0587
J0224$+$0849 ${}^{\mathrm{o}}_{*}$ | 36.233942 | 8.829935 | 0.3271 $\pm$ 0.0225 p | 7.6845 $\pm$ 0.3879 | 12.8637 $\pm$ 0.0533 | 11.8480 $\pm$ 0.1421 | 11.1563 $\pm$ 0.0718
J0225$-$0737 m | 36.442200 | -7.627366 | 0.5137 $\pm$ 0.0001 s | 6.2327 $\pm$ 0.4870 | 12.8795 $\pm$ 0.0701 | 11.6916 $\pm$ 0.1246 | 10.9776 $\pm$ 0.0172
J0227$+$2934 ${}^{\mathrm{m}}_{*}$ | 36.806665 | 29.579592 | 0.4856 $\pm$ 0.0351 p | 3.9666 $\pm$ 0.5623 | 12.5866 $\pm$ 0.1266 | 11.3633 $\pm$ 0.1055 | 10.6706 $\pm$ 0.0800
J0228$-$2923 o | 37.067992 | -29.396016 | 0.2951 $\pm$ 0.0106 p | 3.8286 $\pm$ 0.2142 | 12.2159 $\pm$ 0.0513 | 11.2861 $\pm$ 0.0695 | 10.4456 $\pm$ 0.0382
J0230$-$2702 m | 37.702673 | -27.041580 | 0.3874 $\pm$ 0.0330 p | 13.3375 $\pm$ 0.4507 | 13.4144 $\pm$ 0.0487 | 11.5716 $\pm$ 0.1364 | 10.9715 $\pm$ 0.1014
J0233$+$0559 m | 38.348967 | 5.999309 | 0.2688 $\pm$ 0.0122 p | 13.0853 $\pm$ 0.1110 | 13.2416 $\pm$ 0.0214 | 12.0689 $\pm$ 0.0758 | 11.2812 $\pm$ 0.0479
J0233$+$0642 o | 38.488527 | 6.711545 | 0.5972 $\pm$ 0.0179 p | 2.9543 $\pm$ 0.1346 | 12.3010 $\pm$ 0.0465 | 11.4323 $\pm$ 0.0624 | 10.7955 $\pm$ 0.0341
J0237$-$3017 m | 39.348438 | -30.292085 | 0.5451 $\pm$ 0.0182 p | 5.3818 $\pm$ 0.2927 | 12.7783 $\pm$ 0.0528 | 11.7743 $\pm$ 0.0951 | 11.0858 $\pm$ 0.0375
J0239$-$2047 o | 39.776954 | -20.788370 | 0.6268 $\pm$ 0.0098 p | 3.6113 $\pm$ 0.0629 | 12.4961 $\pm$ 0.0266 | 11.7599 $\pm$ 0.0719 | 10.9720 $\pm$ 0.0175
J0239$-$0134 m | 39.971351 | -1.582259 | 0.3731 $\pm$ 0.0001 s | 11.2100 $\pm$ 0.6935 | 13.2480 $\pm$ 0.0549 | 11.8979 $\pm$ 0.1383 | 11.2710 $\pm$ 0.0079
J0248$-$0216 m | 42.034802 | -2.276981 | 0.2338 $\pm$ 0.0000 s | 13.3201 $\pm$ 0.2057 | 13.1984 $\pm$ 0.0144 | 11.8312 $\pm$ 0.0675 | 11.0657 $\pm$ 0.0023
J0251$-$1220 m | 42.897099 | -12.333713 | 0.4324 $\pm$ 0.0123 p | 5.1718 $\pm$ 0.2473 | 12.6387 $\pm$ 0.0454 | 11.5625 $\pm$ 0.1004 | 10.8641 $\pm$ 0.0331
J0251$-$2358 o | 42.990479 | -23.978480 | 0.3488 $\pm$ 0.0075 p | 11.5719 $\pm$ 0.0731 | 13.2453 $\pm$ 0.0140 | 11.8767 $\pm$ 0.0078 | 11.2202 $\pm$ 0.0213
J0255$-$1640 m | 43.821469 | -16.681517 | 0.6949 $\pm$ 0.0251 p | 3.5382 $\pm$ 0.0561 | 12.5302 $\pm$ 0.0343 | 11.3902 $\pm$ 0.0480 | 10.8106 $\pm$ 0.0405
J0257$-$2008 m | 44.433800 | -20.147361 | 0.7033 $\pm$ 0.0214 p | 5.5778 $\pm$ 0.0402 | 12.9304 $\pm$ 0.0303 | 11.9830 $\pm$ 0.0824 | 11.2537 $\pm$ 0.0350
J0302$-$1004 m | 45.597783 | -10.076458 | 0.5506 $\pm$ 0.0644 p | 17.7556 $\pm$ 0.4175 | 13.8200 $\pm$ 0.0621 | 12.1914 $\pm$ 0.1028 | 11.5761 $\pm$ 0.1323
J0303$-$2119 m | 45.791775 | -21.325857 | 0.4059 $\pm$ 0.0176 p | 11.3597 $\pm$ 0.0991 | 13.2944 $\pm$ 0.0234 | 11.6924 $\pm$ 0.0288 | 11.1592 $\pm$ 0.0442
J0310$-$1746 ${}^{\mathrm{o}}_{*}$ | 47.708730 | -17.774793 | 0.6691 $\pm$ 0.0524 p | 2.7874 $\pm$ 0.1736 | 12.3081 $\pm$ 0.0716 | 11.4130 $\pm$ 0.1711 | 10.6386 $\pm$ 0.0873
J0318$-$2915 m | 49.619953 | -29.259311 | 0.5432 $\pm$ 0.0246 p | 4.7809 $\pm$ 0.1150 | 12.7895 $\pm$ 0.0351 | 11.6384 $\pm$ 0.0769 | 10.8987 $\pm$ 0.0514
J0327$-$1326 ${}^{\mathrm{m}}_{*}$ | 51.863245 | -13.439585 | 0.5759 $\pm$ 0.0230 p | 21.3502 $\pm$ 2.8938 | 14.0070 $\pm$ 0.1226 | 12.4358 $\pm$ 0.2426 | 11.7787 $\pm$ 0.0585
J0328$-$2140 m | 52.056629 | -21.672094 | 0.5617 $\pm$ 0.0125 p | 19.9614 $\pm$ 1.3808 | 13.9311 $\pm$ 0.0611 | 12.5000 $\pm$ 0.1279 | 11.8501 $\pm$ 0.0283
J0334$-$1311 ${}^{\mathrm{o}}_{*}$ | 53.625432 | -13.186562 | 0.3622 $\pm$ 0.0121 p | 6.7750 $\pm$ 1.1623 | 12.9257 $\pm$ 0.1453 | 11.7358 $\pm$ 0.1275 | 10.9468 $\pm$ 0.0392
J0340$-$2533 m | 55.089050 | -25.558420 | 0.6326 $\pm$ 0.0189 p | 3.6340 $\pm$ 0.3239 | 12.6255 $\pm$ 0.0803 | 11.4958 $\pm$ 0.1199 | 10.7713 $\pm$ 0.0336
J0347$-$2454 o | 56.935631 | -24.908777 | 0.6390 $\pm$ 0.0350 p | 2.6813 $\pm$ 0.1082 | 12.3657 $\pm$ 0.0499 | 11.4301 $\pm$ 0.0796 | 10.7615 $\pm$ 0.0627
J0348$-$2145 m | 57.009695 | -21.750889 | 0.3565 $\pm$ 0.0173 p | 9.3239 $\pm$ 0.9232 | 13.0714 $\pm$ 0.0862 | 11.9305 $\pm$ 0.1862 | 11.1878 $\pm$ 0.0612
J0349$-$1500 o | 57.450133 | -15.002336 | 0.3212 $\pm$ 0.0086 p | 6.8526 $\pm$ 0.3801 | 12.7568 $\pm$ 0.0503 | 11.6309 $\pm$ 0.0772 | 10.8348 $\pm$ 0.0284
J0353$-$1706 o | 58.442671 | -17.110893 | 0.5256 $\pm$ 0.0904 p | 2.0505 $\pm$ 0.0567 | 12.0426 $\pm$ 0.0862 | 11.3743 $\pm$ 0.0267 | 10.4374 $\pm$ 0.1943
J0354$-$1609 ${}^{\mathrm{o}}_{*}$ | 58.576148 | -16.164544 | 0.5887 $\pm$ 0.0082 p | 2.5302 $\pm$ 0.2045 | 12.2775 $\pm$ 0.0718 | 11.5293 $\pm$ 0.0750 | 10.8251 $\pm$ 0.0165
J0400$-$1357 m | 60.241996 | -13.956629 | 0.6457 $\pm$ 0.0335 p | 11.0343 $\pm$ 0.6570 | 13.4828 $\pm$ 0.0620 | 11.9998 $\pm$ 0.1808 | 11.2753 $\pm$ 0.0591
J0401$-$0951 o | 60.314572 | -9.860522 | 0.5816 $\pm$ 0.1755 p | 3.6236 $\pm$ 0.0457 | 12.5847 $\pm$ 0.1502 | 11.7039 $\pm$ 0.0034 | 10.6573 $\pm$ 0.4109
J0411$-$2256 m | 62.783524 | -22.944009 | 0.4954 $\pm$ 0.0267 p | 4.7790 $\pm$ 0.2868 | 12.6303 $\pm$ 0.0589 | 11.6568 $\pm$ 0.1454 | 10.9039 $\pm$ 0.0643
J0420$-$0421 o | 65.242499 | -4.357444 | 0.7245 $\pm$ 0.0421 p | 3.0239 $\pm$ 0.1457 | 12.5342 $\pm$ 0.0589 | 11.5643 $\pm$ 0.1256 | 10.8291 $\pm$ 0.0683
J0423$-$0121 o | 65.796719 | -1.351486 | 0.3851 $\pm$ 0.0245 p | 3.8175 $\pm$ 0.1122 | 12.3238 $\pm$ 0.0398 | 11.3825 $\pm$ 0.0938 | 10.7479 $\pm$ 0.0666
J0424$-$1624 o | 66.011723 | -16.411331 | 0.6681 $\pm$ 0.0379 p | 2.8343 $\pm$ 0.1562 | 12.4363 $\pm$ 0.0595 | 11.4788 $\pm$ 0.1160 | 10.7444 $\pm$ 0.0678
J0424$-$3317 ${}^{\mathrm{o}}_{*}$ | 66.161195 | -33.294925 | 0.5542 $\pm$ 0.0594 p | 4.9734 $\pm$ 0.4561 | 12.8353 $\pm$ 0.0941 | 11.9122 $\pm$ 0.1256 | 11.1048 $\pm$ 0.1203
J0429$-$2957 m | 67.472905 | -29.959746 | 0.6919 $\pm$ 0.0195 p | 11.7104 $\pm$ 0.3467 | 13.5683 $\pm$ 0.0385 | 11.9433 $\pm$ 0.0361 | 11.3918 $\pm$ 0.0306
J0432$-$2000 o | 68.102241 | -20.003820 | 0.5642 $\pm$ 0.0164 p | 4.3046 $\pm$ 0.3815 | 12.6016 $\pm$ 0.0798 | 11.6363 $\pm$ 0.0893 | 10.9457 $\pm$ 0.0326
J0440$-$2658 m | 70.192758 | -26.975392 | 0.5018 $\pm$ 0.0089 p | 9.9881 $\pm$ 0.4193 | 13.2767 $\pm$ 0.0407 | 11.8938 $\pm$ 0.0935 | 11.2158 $\pm$ 0.0194
J0447$-$0251 m | 71.865678 | -2.861248 | 0.4270 $\pm$ 0.0104 p | 10.4768 $\pm$ 2.7785 | 13.2751 $\pm$ 0.2313 | 11.9119 $\pm$ 0.4235 | 11.1838 $\pm$ 0.1023
J0450$-$3302 m | 72.501915 | -33.048279 | 0.4990 $\pm$ 0.0158 p | 6.7920 $\pm$ 0.1347 | 12.9391 $\pm$ 0.0282 | 11.7887 $\pm$ 0.0739 | 11.0779 $\pm$ 0.0346
J0451$+$0006 ${}^{\mathrm{m}}_{*}$ | 72.977704 | 0.105035 | 0.4221 $\pm$ 0.0171 p | 35.8005 $\pm$ 2.0949 | 14.3096 $\pm$ 0.0543 | 12.4608 $\pm$ 0.1270 | 11.7802 $\pm$ 0.0443
J0452$-$3540 o | 73.123877 | -35.674171 | 0.5958 $\pm$ 0.0123 p | 2.9425 $\pm$ 0.0674 | 12.2948 $\pm$ 0.0297 | 11.4572 $\pm$ 0.0617 | 10.7358 $\pm$ 0.0234
J0455$-$2530 ${}^{\mathrm{m}}_{*}$ | 73.903064 | -25.512814 | 0.3524 $\pm$ 0.0374 p | 15.9619 $\pm$ 0.1962 | 13.5310 $\pm$ 0.0466 | 11.5990 $\pm$ 0.0950 | 11.0584 $\pm$ 0.1182
J0458$-$2637 m | 74.673306 | -26.626873 | 0.2677 $\pm$ 0.0075 p | 11.3630 $\pm$ 0.0540 | 13.1170 $\pm$ 0.0141 | 11.6874 $\pm$ 0.0440 | 10.9491 $\pm$ 0.0298
J0459$-$3043 m | 74.964252 | -30.723626 | 0.4329 $\pm$ 0.0106 p | 3.5759 $\pm$ 0.0773 | 12.3176 $\pm$ 0.0256 | 11.3325 $\pm$ 0.1536 | 10.6898 $\pm$ 0.0261
J0513$-$3050 m | 78.356121 | -30.843339 | 0.3814 $\pm$ 0.0152 p | 5.4548 $\pm$ 0.4222 | 12.6330 $\pm$ 0.0698 | 11.5392 $\pm$ 0.1286 | 10.8729 $\pm$ 0.0435
J0524$-$2721 m | 81.098605 | -27.353147 | 0.3081 $\pm$ 0.0086 p | 6.8557 $\pm$ 0.2129 | 12.7382 $\pm$ 0.0305 | 11.6347 $\pm$ 0.0617 | 10.8221 $\pm$ 0.0283
J0527$-$1858 ${}^{\mathrm{m}}_{*}$ | 81.754744 | -18.967375 | 0.4420 $\pm$ 0.0116 p | 7.3056 $\pm$ 0.7421 | 13.0698 $\pm$ 0.0909 | 11.7325 $\pm$ 0.1197 | 11.0333 $\pm$ 0.0279
J0532$-$3545 m | 83.077550 | -35.755547 | 0.4924 $\pm$ 0.0313 p | 5.3516 $\pm$ 0.0409 | 12.7244 $\pm$ 0.0330 | 11.5158 $\pm$ 0.0511 | 10.7890 $\pm$ 0.0736
J0545$-$2635 ${}^{\mathrm{m}}_{*}$ | 86.306565 | -26.588383 | 0.3177 $\pm$ 0.0220 p | 13.3576 $\pm$ 0.0791 | 13.3303 $\pm$ 0.0300 | 11.8456 $\pm$ 0.0523 | 11.3718 $\pm$ 0.0717
J0545$-$3542 o | 86.352690 | -35.714575 | 0.4065 $\pm$ 0.0086 p | 10.8589 $\pm$ 0.2087 | 13.2555 $\pm$ 0.0218 | 11.4552 $\pm$ 0.0915 | 10.9562 $\pm$ 0.0207
J0549$-$2355 m | 87.402875 | -23.925304 | 0.5165 $\pm$ 0.0114 p | 5.7656 $\pm$ 0.6057 | 12.8166 $\pm$ 0.0933 | 11.8464 $\pm$ 0.2017 | 11.1332 $\pm$ 0.0394
J0553$-$2853 o | 88.296789 | -28.893375 | 0.7899 $\pm$ 0.0617 p | 2.3562 $\pm$ 0.1271 | 12.3668 $\pm$ 0.0713 | 11.0630 $\pm$ 0.0692 | 10.6426 $\pm$ 0.0863
J0559$-$3540 ${}^{\mathrm{o}}_{*}$ | 89.956356 | -35.670137 | 0.8034 $\pm$ 0.0381 p | 2.4366 $\pm$ 0.2014 | 12.4059 $\pm$ 0.0836 | 11.4305 $\pm$ 0.1689 | 10.9083 $\pm$ 0.0549
J1132$+$0212 ${}^{\mathrm{o}}_{*}$ | 173.111909 | 2.215971 | 0.9367 $\pm$ 0.0505 p | 10.4815 $\pm$ 1.2337 | 13.6499 $\pm$ 0.1132 | 11.5785 $\pm$ 0.2362 | 11.2598 $\pm$ 0.0688
J2043$-$0609 ${}^{\mathrm{o}}_{*}$ | 310.801950 | -6.164955 | 0.8125 $\pm$ 0.0385 p | 4.9276 $\pm$ 0.3242 | 12.9029 $\pm$ 0.0696 | 11.7136 $\pm$ 0.1030 | 11.1461 $\pm$ 0.0547
J2105$+$0537 m | 316.420521 | 5.620318 | 0.3095 $\pm$ 0.0179 p | 5.5740 $\pm$ 0.2132 | 12.5583 $\pm$ 0.0421 | 11.8041 $\pm$ 0.1454 | 11.0545 $\pm$ 0.0608
J2106$-$0027 m | 316.613480 | -0.458322 | 0.7903 $\pm$ 0.0313 p | 7.2086 $\pm$ 0.3762 | 13.2184 $\pm$ 0.0578 | 11.8610 $\pm$ 0.1772 | 11.3299 $\pm$ 0.0474
J2106$-$0547 m | 316.721396 | -5.783672 | 0.6485 $\pm$ 0.0501 p | 3.8671 $\pm$ 0.9094 | 12.7110 $\pm$ 0.2080 | 11.5791 $\pm$ 0.2754 | 10.8871 $\pm$ 0.1047
J2109$-$0135 o | 317.329931 | -1.593868 | 0.8934 $\pm$ 0.0533 p | 1.5025 $\pm$ 0.2300 | 11.9386 $\pm$ 0.1480 | 11.2400 $\pm$ 0.2550 | 10.6351 $\pm$ 0.0905
J2111$-$0114 ${}^{\mathrm{m}}_{*}$ | 317.830618 | -1.239845 | 0.6386 $\pm$ 0.0001 s | 12.6252 $\pm$ 1.5331 | 13.6005 $\pm$ 0.1091 | 12.1776 $\pm$ 0.2164 | 11.4900 $\pm$ 0.0365
J2114$+$0658 m | 318.724767 | 6.968209 | 0.2081 $\pm$ 0.0052 p | 8.1851 $\pm$ 0.4026 | 12.7274 $\pm$ 0.0449 | 11.5598 $\pm$ 0.0657 | 10.7503 $\pm$ 0.0249
J2126$+$0949 o | 321.560372 | 9.833233 | 0.8025 $\pm$ 0.0291 p | 4.1700 $\pm$ 0.4403 | 12.8715 $\pm$ 0.1005 | 11.5284 $\pm$ 0.1658 | 10.9920 $\pm$ 0.0414
J2129$-$0126 m | 322.287133 | -1.442114 | 0.9581 $\pm$ 0.0949 p | 17.5086 $\pm$ 0.0426 | 14.1089 $\pm$ 0.0852 | 11.4555 $\pm$ 0.2358 | 11.3166 $\pm$ 0.1092
J2143$+$1431 m | 325.794687 | 14.522257 | 0.4102 $\pm$ 0.0087 p | 9.4940 $\pm$ 0.1068 | 13.1423 $\pm$ 0.0184 | 11.6014 $\pm$ 0.0376 | 11.0261 $\pm$ 0.0239
J2146$-$0317 m | 326.623059 | -3.284784 | 0.5759 $\pm$ 0.0213 p | 12.3640 $\pm$ 0.0691 | 13.5257 $\pm$ 0.0263 | 11.4957 $\pm$ 0.0066 | 11.2704 $\pm$ 0.0395
J2151$-$0138 m | 327.858149 | -1.647066 | 0.3131 $\pm$ 0.0001 s | 15.9036 $\pm$ 0.0759 | 13.4753 $\pm$ 0.0092 | 11.8579 $\pm$ 0.0111 | 11.3326 $\pm$ 0.0007
J2206$+$1104 o | 331.567919 | 11.068702 | 0.7772 $\pm$ 0.0002 s | 2.8576 $\pm$ 0.0721 | 12.4034 $\pm$ 0.0368 | 11.8932 $\pm$ 0.0826 | 11.0847 $\pm$ 0.0063
J2207$-$0411 ${}^{\mathrm{o}}_{*}$ | 331.762532 | -4.196620 | 0.7576 $\pm$ 0.0416 p | 3.5709 $\pm$ 0.3867 | 12.7045 $\pm$ 0.1044 | 11.3932 $\pm$ 0.1781 | 10.9579 $\pm$ 0.0641
J2210$+$2604 m | 332.512025 | 26.068904 | 0.3467 $\pm$ 0.0001 s | 7.4681 $\pm$ 0.2191 | 12.8627 $\pm$ 0.0274 | 11.7072 $\pm$ 0.0743 | 10.8734 $\pm$ 0.0027
J2222$+$2745 m | 335.536310 | 27.759241 | 0.4717 $\pm$ 0.0404 p | 7.4138 $\pm$ 0.6700 | 12.9932 $\pm$ 0.0919 | 12.0060 $\pm$ 0.1049 | 11.2401 $\pm$ 0.0919
J2226$+$0041 m | 336.538789 | 0.694997 | 0.6471 $\pm$ 0.0001 s | 3.1267 $\pm$ 0.0892 | 12.3873 $\pm$ 0.0336 | 11.8505 $\pm$ 0.0510 | 10.9511 $\pm$ 0.0024
J2234$-$0630 ${}^{\mathrm{o}}_{*}$ | 338.724911 | -6.515025 | 0.8146 $\pm$ 0.0705 p | 2.5836 $\pm$ 0.2115 | 12.4661 $\pm$ 0.0982 | 11.4292 $\pm$ 0.2681 | 10.7640 $\pm$ 0.1104
J2238$+$1319 ${}^{\mathrm{m}}_{*}$ | 339.630481 | 13.332188 | 0.4129 $\pm$ 0.0001 s | 9.3281 $\pm$ 0.2308 | 13.2479 $\pm$ 0.0237 | 11.8539 $\pm$ 0.1052 | 11.3099 $\pm$ 0.0024
J2240$-$0139 o | 340.054725 | -1.659645 | 0.8515 $\pm$ 0.0393 p | 8.3870 $\pm$ 0.5461 | 13.3927 $\pm$ 0.0687 | 11.4458 $\pm$ 0.1560 | 10.9993 $\pm$ 0.0549
J2241$-$0527 m | 340.298713 | -5.460154 | 0.9632 $\pm$ 0.0418 p | 8.2405 $\pm$ 0.1584 | 13.4557 $\pm$ 0.0596 | 12.2401 $\pm$ 0.2411 | 11.4152 $\pm$ 0.0496
J2244$+$2759 o | 341.020635 | 27.987671 | 0.3429 $\pm$ 0.0001 s | 6.1671 $\pm$ 0.2461 | 12.6916 $\pm$ 0.0357 | 11.7689 $\pm$ 0.0444 | 10.9261 $\pm$ 0.0045
J2246$+$0415 o | 341.687109 | 4.263947 | 1.0241 $\pm$ 0.0503 p | 9.6721 $\pm$ 0.3324 | 13.6355 $\pm$ 0.0783 | 11.5978 $\pm$ 0.1415 | 11.3445 $\pm$ 0.0549
J2247$-$0205 m | 341.801229 | -2.093837 | 0.3383 $\pm$ 0.0234 p | 8.7594 $\pm$ 0.1647 | 12.9912 $\pm$ 0.0346 | 11.7359 $\pm$ 0.1764 | 11.1032 $\pm$ 0.0697
J2248$-$0123 m | 342.151533 | -1.392814 | 0.3983 $\pm$ 0.0001 s | 4.1601 $\pm$ 0.1096 | 12.4137 $\pm$ 0.0256 | 11.2435 $\pm$ 0.0853 | 10.6479 $\pm$ 0.0021
J2248$+$2015 o | 342.151611 | 20.252945 | 0.8046 $\pm$ 0.0320 p | 4.0530 $\pm$ 0.0779 | 12.7274 $\pm$ 0.0433 | 11.5123 $\pm$ 0.1199 | 11.0254 $\pm$ 0.0442
J2249$-$0110 o | 342.410083 | -1.179021 | 0.3399 $\pm$ 0.0001 s | 4.3647 $\pm$ 0.1033 | 12.3878 $\pm$ 0.0227 | 11.3617 $\pm$ 0.0476 | 10.5159 $\pm$ 0.0026
J2252$-$0413 m | 343.040271 | -4.218693 | 0.4273 $\pm$ 0.0139 p | 7.5002 $\pm$ 0.1272 | 12.9566 $\pm$ 0.0231 | 11.7065 $\pm$ 0.0879 | 11.0875 $\pm$ 0.0333
J2256$+$1005 m | 344.205409 | 10.086293 | 0.8824 $\pm$ 0.0395 p | 7.3106 $\pm$ 0.4774 | 13.2962 $\pm$ 0.0748 | 11.6659 $\pm$ 0.1212 | 11.2499 $\pm$ 0.0519
J2257$-$0610 ${}^{\mathrm{o}}_{*}$ | 344.434506 | -6.173864 | 0.7776 $\pm$ 0.0335 p | 5.1725 $\pm$ 0.2591 | 12.9198 $\pm$ 0.0577 | 11.5310 $\pm$ 0.1716 | 10.9212 $\pm$ 0.0471
J2258$+$1709 m | 344.556726 | 17.150617 | 0.7990 $\pm$ 0.0413 p | 8.5561 $\pm$ 0.1622 | 13.3714 $\pm$ 0.0483 | 11.9105 $\pm$ 0.1594 | 11.4468 $\pm$ 0.0633
J2259$+$1212 m | 344.797839 | 12.209790 | 0.8885 $\pm$ 0.1309 p | 4.8219 $\pm$ 0.3680 | 13.0569 $\pm$ 0.1227 | 11.8003 $\pm$ 0.1218 | 10.9898 $\pm$ 0.1751
J2300$+$2213 m | 345.071926 | 22.224908 | 0.4237 $\pm$ 0.0116 p | 17.8867 $\pm$ 0.1814 | 13.7078 $\pm$ 0.0193 | 12.0469 $\pm$ 0.1640 | 11.4679 $\pm$ 0.0296
J2303$+$2328 m | 345.860959 | 23.475984 | 0.2767 $\pm$ 0.0000 s | 9.0970 $\pm$ 0.3456 | 12.9388 $\pm$ 0.0339 | 11.6889 $\pm$ 0.0700 | 10.9067 $\pm$ 0.0057
J2304$+$3327 ${}^{\mathrm{o}}_{*}$ | 346.022626 | 33.459947 | 0.4837 $\pm$ 0.0233 p | 7.5387 $\pm$ 0.9608 | 13.0212 $\pm$ 0.1116 | 11.7765 $\pm$ 0.1300 | 11.0498 $\pm$ 0.0539
J2304$-$0052 m | 346.101457 | -0.877417 | 0.8850 $\pm$ 0.0305 p | 4.9030 $\pm$ 0.2141 | 12.9480 $\pm$ 0.0572 | 11.7937 $\pm$ 0.0456 | 11.1501 $\pm$ 0.0375
J2307$-$1322 ${}^{\mathrm{m}}_{*}$ | 346.766785 | -13.374414 | 0.5087 $\pm$ 0.0223 p | 7.6014 $\pm$ 0.1203 | 13.0455 $\pm$ 0.0288 | 11.8918 $\pm$ 0.0323 | 11.1448 $\pm$ 0.0535
J2308$-$0211 m | 347.092613 | -2.192172 | 0.2949 $\pm$ 0.0097 p | 37.4958 $\pm$ 1.4818 | 14.1956 $\pm$ 0.0379 | 12.3708 $\pm$ 0.1143 | 11.6313 $\pm$ 0.0342
J2312$+$0451 m | 348.245242 | 4.861479 | 0.3102 $\pm$ 0.0001 s | 9.5915 $\pm$ 0.1157 | 13.0323 $\pm$ 0.0132 | 11.7486 $\pm$ 0.0834 | 10.9586 $\pm$ 0.0038
J2313$-$0104 m | 348.477049 | -1.080103 | 0.5312 $\pm$ 0.0002 s | 7.8648 $\pm$ 0.2229 | 13.0941 $\pm$ 0.0286 | 11.8258 $\pm$ 0.0711 | 11.0930 $\pm$ 0.0036
J2318$-$1106 o | 349.549151 | -11.101265 | 0.7203 $\pm$ 0.0264 p | 5.4290 $\pm$ 0.0264 | 12.9220 $\pm$ 0.0344 | 11.7873 $\pm$ 0.0336 | 11.1669 $\pm$ 0.0426
J2319$+$0038 m | 349.972629 | 0.637053 | 0.8295 $\pm$ 0.1561 p | 8.8106 $\pm$ 1.5057 | 13.4456 $\pm$ 0.1916 | 12.0224 $\pm$ 0.4035 | 11.4651 $\pm$ 0.2497
J2320$-$1202 m | 350.027080 | -12.034294 | 0.3967 $\pm$ 0.0130 p | 5.6273 $\pm$ 0.2854 | 12.6744 $\pm$ 0.0472 | 11.6153 $\pm$ 0.1509 | 10.8922 $\pm$ 0.0433
J2326$+$2026 m | 351.748351 | 20.449553 | 0.7895 $\pm$ 0.0289 p | 5.7747 $\pm$ 0.4179 | 13.0236 $\pm$ 0.0739 | 11.8390 $\pm$ 0.1464 | 11.1600 $\pm$ 0.0452
J2331$+$2749 m | 352.825967 | 27.817418 | 0.8688 $\pm$ 0.0618 p | 9.5331 $\pm$ 0.1430 | 13.5153 $\pm$ 0.0609 | 11.1560 $\pm$ 0.0429 | 11.1953 $\pm$ 0.0841
J2334$-$0746 o | 353.528695 | -7.771200 | 0.4018 $\pm$ 0.0130 p | 5.6332 $\pm$ 0.2861 | 12.6809 $\pm$ 0.0476 | 11.4592 $\pm$ 0.1648 | 10.7858 $\pm$ 0.0340
J2335$+$0922 o | 353.981202 | 9.382482 | 0.7536 $\pm$ 0.0302 p | 6.6253 $\pm$ 0.0521 | 13.1183 $\pm$ 0.0368 | 11.3639 $\pm$ 0.0546 | 10.9481 $\pm$ 0.0437
J2347$-$0047 m | 356.811434 | -0.797979 | 0.7889 $\pm$ 0.0188 p | 8.1719 $\pm$ 0.1422 | 13.3231 $\pm$ 0.0377 | 11.9225 $\pm$ 0.1335 | 11.3496 $\pm$ 0.0262
J2347$-$0439 o | 356.954863 | -4.650060 | 0.9785 $\pm$ 0.0842 p | 2.7133 $\pm$ 0.0469 | 12.5039 $\pm$ 0.0822 | 11.1439 $\pm$ 0.2992 | 10.7868 $\pm$ 0.1018
J2348$+$1407 o | 357.034778 | 14.129527 | 0.6546 $\pm$ 0.0002 s | 2.4746 $\pm$ 0.3247 | 12.3132 $\pm$ 0.1142 | 11.5998 $\pm$ 0.0877 | 10.7861 $\pm$ 0.0129
J2356$+$0241 o | 359.142118 | 2.692205 | 0.8204 $\pm$ 0.0304 p | 5.0113 $\pm$ 1.1522 | 13.0628 $\pm$ 0.2042 | 11.3861 $\pm$ 0.3399 | 10.8535 $\pm$ 0.0555
J2359$-$1214 m | 359.802678 | -12.236703 | 0.9131 $\pm$ 0.0672 p | 7.4654 $\pm$ 0.1480 | 13.3350 $\pm$ 0.0684 | 11.5590 $\pm$ 0.2083 | 11.2256 $\pm$ 0.0836
J2359$+$0208 ${}^{\mathrm{m}}_{*}$ | 359.889762 | 2.139947 | 0.4294 $\pm$ 0.0001 s | 9.8445 $\pm$ 0.7516 | 13.1967 $\pm$ 0.0669 | 12.1698 $\pm$ 0.1299 | 11.4403 $\pm$ 0.0098

Note. — All values quoted in dex are constrained exclusively within the
Einstein aperture for each system. Luminosity was measured within the rest-
frame wavelength interval of 3000Å to 7000Å. Superscript labels in the ID
column stand for the following: ‘o’ and ‘m’ stand for whether one or multiple
cluster members fall within the Einstein aperture respectively. Subscript
asterisks in the ID column denote systems which were analyzed with LENSTOOL in
Section 5.3. Superscript labels in the $z_{{}_{BCG}}$ column stand for the
following: ‘p’ stands for LS DR9 photometric redshift; ‘s’ stands for SDSS
DR15 spectroscopic redshift. A csv file containing the information in this
table can be provided upon request to the corresponding author.