11institutetext: Departamento de Astronomía, Universidad de Chile, Camino el
Observatorio 1515, Las Condes, Santiago, Chile
11email<EMAIL_ADDRESS>22institutetext: Instituto de Física,
Pontificia Universidad Católica de Valparaíso, Casilla 4059, Valparaíso, Chile
33institutetext: Franco-Chilean Laboratory for Astronomy, IRL 3386, CNRS and
Universidad de Chile, Santiago, Chile 44institutetext: Institute
d’Astrophysique de Paris, CNRS-SU, UMR 7095, 98bis bd Arago, 75014 Paris,
France 55institutetext: European Southern Observatory, Alonso de Cordova 3107,
Casilla 19001, Santiago, Chile 66institutetext: Dipartimento di Fisica G.
Occhialini, Università degli Studi di Milano Bicocca, Piazza della Scienza 3,
20126 Milano, Italy 77institutetext: Instituto de Estudios Astrofísicos,
Facultad de Ingeniería y Ciencias, Universidad Diego Portales, Av. Ejército
Libertador 441, Santiago, Chile 88institutetext: Max Planck Institut für
Astrophysik, Karl-Schwarzschild-Straße 1, D-85748 Garching bei München,
Germany 99institutetext: Instituto de Física, Pontificia Universidad Católica
de Chile, Av. Vicuña Mackenna 4860, 7820436 Macul, Santiago, Chile.

# Directly constraining the spatial coherence of the $z\sim 1$ circumgalactic
medium

A. Afruni 11 S. Lopez 11 P. Anshul 11 N. Tejos 22 P. Noterdaeme 3344 T. A. M.
Berg 115566 C. Ledoux 55 M. Solimano 77
J. Gonzalez-Lopez 77 M. Gronke 88 F. Barrientos 99 E. J. Johnston 77

One of the biggest puzzles regarding the circumgalactic medium (CGM) is the
structure of its cool ($T\sim 10^{4}$ K) gas phase. While the kinematics of
quasar absorption systems suggests the CGM is composed of a population of
different clouds, constraining the clouds’ extent and spatial distribution has
proven challenging, both from the theoretical and observational points of
view. In this work we study the spatial structure of the $z\sim 1$ CGM with
unprecedented detail via resolved spectroscopy of giant gravitational arcs. We
put together a sample of Mg ii$\lambda\lambda 2796,2803$ detections obtained
with VLT/MUSE in 91 spatially independent and contiguous sight-lines toward 3
arcs, each probing an isolated star-forming galaxy believed to be detected in
absorption. We constrain the coherence scale of this gas ($C_{\rm{length}}$),
which represents the spatial scale over which the Mg ii equivalent width (EW)
remains constant, by comparing EW variations measured across all sight-lines
with empirical models. We find $1.4<C_{\rm{length}}/\rm{kpc}<7.8$ (95%
confidence). This measurement, of unprecedented accuracy, represents the scale
over which the cool gas tends to cluster in separate structures. We argue
that, if $C_{\rm{length}}$ is a universal property of the CGM, it needs to be
reproduced by current and future theoretical models in order to understand the
exact role of this medium in galaxy evolution.

###### Key Words.:

gravitational lensing: strong - galaxies: halos - galaxies: evolution -
(galaxies): intergalactic medium

##  1 Introduction

It is now clear that the evolution of galaxies must be linked to their
interaction with their surrounding gaseous halos, the circumgalactic medium,
or CGM. Of particular importance is the CGM cool phase ($T\sim 10^{4}$ K, see
Tumlinson et al., 2017), which traces the flows of material in and out of
galaxies, known as the baryon cycle, and represents a huge reservoir of fuel
for star-formation. However, granted that this medium holds the key to
understand how galaxies grow and evolve, many of its properties are to date
still unclear.

Due to its low emissivity, observing the cool CGM directly in emission is
challenging and currently most of these observations are focused on the high
redshift Universe ($z\gtrsim 2$), where there seems to be an ubiquity of
extended Ly$\alpha$ nebulae, illuminated by the central quasars or galaxies
(e.g., Cantalupo et al., 2014; Farina et al., 2019). At $z\la 1$, there are
instead only a handful of cases where the cool CGM has been observed in
emission, either around starbursts and galaxy groups (e.g., Chen et al., 2019;
Burchett et al., 2021; Leclercq et al., 2022) or thanks to stacking techniques
(e.g., Zhang et al., 2018). The vast majority of the cool CGM detections come
then from absorption studies, extensively performed for decades both at high
and low redshift, using as tracers the absorption lines of hydrogen and low-
ionization ion species like Mg ii, C ii or Si ii in the spectra of background
quasars (e.g., Bergeron, 1986; Churchill et al., 2000; Zhu & Ménard, 2013;
Keeney et al., 2017; Zahedy et al., 2019; Qu et al., 2023).

A major issue in understanding the elusive properties of this material is the
presence of typically only one single line of sight per galaxy, so that the
properties of the cool CGM are usually studied in a statistical way, combining
together detections from surveys of tens or hundreds of galaxy-QSO pairs
(e.g., Lanzetta et al., 1995; Chen et al., 2010a; Werk et al., 2012; Wilde et
al., 2021). This has proven very useful in unveiling some of the properties of
this medium, like its absorption strength or its covering fraction as a
function of the distance from the central galaxy (e.g., Nielsen et al., 2013;
Huang et al., 2021). However, due to the single-pencil-beam limitation, there
are still very strong uncertainties on the general dynamics and structure of
this gas. High spectral resolution observations have shown that this medium is
likely composed of multiple clumps of gas, or clouds, which produce multiple
kinematic components along the same line of sight and which are bound to the
galaxy halo (e.g., Borthakur et al., 2015), but whether these clouds are
primarily inflowing toward (e.g., Bouché et al., 2013) or outflowing (e.g.,
Schroetter et al., 2019) from the central galaxies is still a matter of
debate.

From a theoretical perspective, the cool CGM properties across the galaxy halo
can be studied using cosmological and zoom-in simulations. However, given the
low resolution in the CGM (kpc-scale), such works are not yet converged in
terms of the cool gas dynamics and structure (e.g., van de Voort et al., 2019)
and the picture is to date uncertain. On the other hand, high-resolution (pc-
scale) hydrodynamical simulations, focused on the motion of a single cool
cloud interacting with the surrounding hotter CGM phase ($T\sim 10^{6}$ K),
have shown that the dynamics and the survival (e.g., Marinacci et al., 2010;
Gronke & Oh, 2020a) of such clouds highly depend on their initial size and
mass, as found also by analytical models (e.g., Nipoti & Binney, 2007; Afruni
et al., 2019; Fielding & Bryan, 2022). Constraining these sizes is therefore
crucial to advance our understanding of the cool CGM dynamics and its
connection with the central galaxy. But estimating these values directly from
observations has proven extremely challenging.

Several studies have resorted to photoionization models like CLOUDY (e.g.,
Ferland et al., 1998) to estimate the thickness of the cool CGM clouds (e.g.,
Werk et al., 2014), but these estimates are subject to large uncertainties (up
to several orders of magnitude) due to the inherent assumptions of the models
(Haehnelt et al., 1996). A more direct way to estimate the spatial extension
of these clouds is to use multiply lensed quasars (e.g., Smette et al., 1995;
Lopez et al., 1999; Ellison et al., 2004; Lopez et al., 2007; Chen et al.,
2014; Zahedy et al., 2016; Rubin et al., 2018b), which allow one to probe the
absorbers through more than just one line of sight. However, the results on
absorber sizes are inconclusive due to the small number of bright lenses and
the small number (2 or 3) of lines of sight per system, preventing a broader
view of the gas structure.

The present work is largely motivated by the promising approach of using
extended background sources (Steidel et al., 2010; Bordoloi et al., 2014;
Rubin et al., 2018a). Using a sample of 27 galaxy-galaxy pairs taken from the
PRIMUS survey (Coil et al., 2011), Rubin et al. (2018a) estimated the scale
over which the absorption strength of the cool CGM, traced by Mg ii
absorption, is expected to vary. Through their analysis, they were able to put
a lower limit on this quantity at about $2$ kpc. This means that they did not
observe variations in the absorption strength on scales smaller than this
threshold and that even if the single clouds have smaller sizes, their
distribution and kinematics result in a constant absorption strength on scales
of at least a few kpc. This coherence scale gives therefore an idea of what is
the clustering scale of the cool medium. In this work we aim at estimating
this quantity with greater accuracy, through the use of gravitational arcs.

In recent years, the ARCTOMO
collaboration111https://sites.google.com/view/arctomo has developed a new
technique to spatially resolve the cool CGM in absorption across individual
galaxy halos. The technique, hereafter ”arc-tomography”, uses resolved
spectroscopy of giant gravitational arcs as background sources, observed with
integral-field spectrographs like VLT/MUSE (Bacon et al., 2010). The
observations enable studies of the cool CGM distribution and kinematics (as
traced by Mg ii absorption) of single galaxies (e.g., Lopez et al., 2018,
2020; Tejos et al., 2021; Fernandez-Figueroa et al., 2022) in an unprecedented
fashion by directly mapping its properties across the arc. In the present
work, we use this unique dataset (hereafter ”ARCTOMO data”), combined with the
predictions from empirical models, to constrain the cool CGM coherence scale,
defined in the same way as in Rubin et al. (2018a).

In Sect. 2, we summarize our data, describing the sample selection, data
reduction, and absorption line search and we define the main observational
constraints that we will utilize in this work; in Sect. 3, we outline the
creation of our fiducial CGM model and the technique used to compare the model
outputs with the data; in Sect. 4, we report our findings regarding the cool
CGM coherence scale, in Sect. 5 we discuss the interpretation of the coherence
scale, the limitations of our analysis, the comparison with previous studies
and the implications of our findings for theoretical models, while in Sect. 6
we summarize the main conclusions of this study.

Throughout the paper, we assume a flat $\Lambda$CDM cosmology with $H_{0}=70$
km s-1 Mpc-1, $\Omega_{\Lambda}=0.7$ and $\Omega_{M}=0.3$.

##  2 Observational data

Table 1: ARCTOMO data

Field | G1 emission propertiesa |  | absorption properties |
---|---|---|---|---
| $z$ | $L_{B}/L^{*}_{B}$ | $\log(M_{*}/M_{\sun})$ | SFR |  | # sp. | # det. | EW | $R$ | scale | refs.
|  |  |  | [$M_{\sun}\leavevmode\nobreak\ {\rm yr}^{-1}$] |  |  |  | [Å] | [kpc] | [kpc] |
| (1) | (2) | (3) | (4) |  | (5) | (6) | (7) | (8) | (9) | (10)
PSZ1 G311.65$-$18.48 | 0.73379 | 0.14$\pm$0.03 | 9.7$\pm$0.3 | 1.1$\pm$0.3 |  | 148 | 29 | 0.2 – 3.7 | 0 – 43 | 2.8 | L20
RCS2 032727$-$132623 | 0.98238 | 0.11$\pm$0.01 | 9.8$\pm$0.2 | 0.4$\pm$0.1 |  | 103 | 26 | 0.3 – 1.9 | 17 – 52 | 2.9 | L18
SGAS J1226+2152 | 0.77138 | 0.27$\pm$0.02 | 10.1$\pm$0.1 | 0.6$\pm$0.2 |  | 90 | 36 | 0.6 – 4.2 | 1 – 21 | 2.6 | T21

Table LABEL:table_data lists summary properties of the 3 systems studied in
this work, that is, the sample of spatially independent Mg ii detections and
their identified absorbing galaxies (hereafter called ‘G1’). These systems
were found in giant gravitational-arc fields observed by us with VLT/MUSE. The
selection for this work was based on absorbing galaxies (1) having a large
number of spatially independent Mg ii detections in their CGM; (2) sharing
similar redshifts, stellar masses, luminosities and star formation rates; and
(3) being considered “isolated”. The “isolation” character holds strictly only
for G1 toward PSZ1 G311.65$-$18.48, where no galaxies at the same redshift
($\Delta v\la 1\,000\ \rm{km}\ \rm{s}^{-1}$) are found in the MUSE field
(Lopez et al., 2020), while for G1 toward SGAS J1226+2152 one galaxy at the
same redshift is found over $225$ kpc away (de-lensed projected distance)
(Tejos et al., 2021) and for G1 toward RCS2 032727$-$132623 two galaxies are
found at the same redshift over 200 kpc away (Lopez et al., 2018). In addition
RCS2 032727$-$132623 G1 itself is resolved into 3 galaxies in the HST images
(Lopez et al., 2018) and hence integrated properties are provided in Table
LABEL:table_data.

Although these systems have been presented individually in the indicated
references, the present work includes improved data reduction and a re-
analysis of the absorption line profiles, about which we provide details in
the following.

###  2.1 MUSE observations and data reduction

VLT/MUSE integral-field observations of these 3 ARCTOMO fields, PSZ1
G311.65$-$18.48, RCS2 032727$-$132623, and SGAS J1226+2152, were carried out
in service mode within ESO programs 297.A-5012(A) (PI Aghanim), 098.A-0459(A)
(PI Lopez), and 0101.A-0364(A) (PI Lopez), respectively. All observations were
carried out in wide-field mode, which provides a field-of-view of $\approx
1\arcmin\times 1\arcmin$, spaxel size of 0.2″, and spectra of resolving power
$R=2\,000$–$4\,000$ in the range $4\,650$–$9\,300$ Å, at a dispersion of
$1.25$ Å. In addition, SGAS J1226+2152 was observed using the Adaptive-Optics
extended wavelength mode.

Observations proceeded during dark time and under good weather conditions. To
minimize instrumental and sky artifacts, short (640–700 s) individual
exposures were taken having small spatial ditherings and cumulative 90∘
rotations. The number of exposures and the total integration times varied
according to arc brightness, resulting in relatively homogeneous continuum
signal-to-noise (S/N) levels near the relevant Mg ii absorption. For more
details on the individual observing conditions we refer the reader to the
references shown in Table LABEL:table_data.

Data of all 3 fields were reduced in an homogeneous fashion using the MUSE
pipeline (Weilbacher et al., 2012) using the EsoRex (ESO Recipe Execution
Tool) tool. The wavelength solution was calibrated to vacuum directly from the
pipeline. The effective PSF FWHM in the combined cubes ranges between 0.7″ and
0.8″. Residual sky contamination was removed using the Zurich Atmosphere Purge
code (ZAP v2.1, Soto et al., 2016), using the default parameters, on the final
combined cubes.

###  2.2 Automated absorption line analysis

To obtain spatially resolved rest-frame equivalent width maps, the absorption
line analysis proceeds automatically, in two basic stages: (i) extraction of
arc spectra and (ii) absorption line fitting. For each data cube the specific
tasks are as follows.

We first rebin the cube spatially. This task serves two purposes: account for
the cross-talk between neighbor MUSE native spaxels due to the seeing, and
increase the S/N. In a sense this is equivalent to extracting spectra of many
resolved sources in a coherent fashion. In this work we use “$3\times 3$
binning”, meaning that each binned cube spaxel is $0.6\arcsec$ on a side and
contains a weighted average spectrum out of 9 native spaxels. This choice
maximizes spatial sampling while still keeping seeing-induced cross-
correlation low between neighbor spaxels (Tejos et al., 2021). We then proceed
to select binned spaxels in a masked region that encompasses the arc based on
S/N. The S/N ratio is calculated in a featureless spectral region close to the
expected Mg ii absorption using the actual flux RMS. We use $\rm{S/N}=2$ as a
threshold. This limit is arbitrary; it serves the purpose of pre-selecting arc
spectra while ruling out spaxels that are contaminated by the sky specially at
the arc borders. The final Mg ii selection used in this work is done at the
absorption-line fitting level.

To obtain equivalent widths, the spectra are normalized and two Gaussians that
account for the Mg ii $\lambda\lambda 2796,2803$ doublet are fitted.
Describing the profiles with Gaussians assumes the absorption signal is
dominated by the instrumental profile. Thus, the 4 free parameters are 2
amplitudes222The final combined equivalent width distribution shows a cutoff
at $\approx 0.3$ Å (see Fig. 10), which indicates that our spectra do not
select individual Mg ii clouds but rather their unresolved line-of-sight
kinematics, which is likely dominated by saturated lines. This justifies the
use of double Gaussians with untied amplitudes., 1 common line width and 1
common redshift. Fits are considered successful if the equivalent width
measurement is larger than 2 times its 1-$\sigma$ error in both doublet lines.
The scheme yields 91 detections among the three fields. The fitted parameters
are reported in Appendix A. The present work is based on the (rest-frame)
equivalent width of the 2796 doublet line, hereafter called EW. EWs are
assigned to the binned spaxels to create the maps shown in Fig. 1 (both in the
image and the absorber planes). Only successful fits are considered. Also,
since we are using an updated selection criterion, these maps are different
from those in Lopez et al. (2018, 2020) and (Fernandez-Figueroa et al., 2022).

###  2.3 Transverse distances in the absorber plane

At the absorber redshifts all RA-DEC coordinates are affected by lensing. We
transform those to ”de-lensed” coordinates using the lens models described in
detail in the references listed in Table LABEL:table_data. Such a
transformation produces a map of de-lensed spaxels in the absorber plane (also
called ”re-constructed” plane, see Fig. 1) at the redshift of the absorber.
The de-lensed coordinates are used to measure projected physical separations
between spaxel centers and also between spaxels and the central galaxy. We
call this latter quantity ”impact parameter”, or $R$.

Lens models are precise to around 5% (Lopez et al., 2020) and this uncertainty
propagates to $R$. We have checked that such uncertainty is however negligible
for the purposes of the present study and does not affect our findings. But
the models are also subject to systematic uncertainties depending on how well
they are constrained observationally, so the solutions might not be unique.
For instance, our model for the SGAS J1226+2152 field considers two perturbers
while the model presented in Mortensen et al. (2021) just one, resulting in
two different reconstructed absorber planes for this field. However, the two
different re-constructed spaxel maps can be reasonably well matched with each
other by a translation of coordinates only. Hence, $R$ and transverse
distances between spaxels are unlikely to be strongly affected by systematic
uncertainties in the lens model (Tejos et al., 2021).

Figure 1: Maps of equivalent widths of the 2976 Mg ii doublet line, for all
the detections found in the three fields analyzed in this work: PSZ1
G311.65$-$18.48 (top), RCS2 032727$-$132623 (center) and SGAS J1226+2152
(bottom). Both the image (observed) plane and the reconstructed absorber plane
(dashed-line rectangle) are shown, with the crosses showing the positions of
the three identified absorbing galaxies, named G1. In SGAS J1226+2152 the
absorption on top of G1 is shown for completeness but not used in the
analysis.

###  2.4 Main observational constraints

Figure 2: Main observational constraints extracted from the ARCTOMO data.
Left: Mg ii$\lambda 2796$ rest-frame equivalent width as a function of
$R^{\prime}$ (Eq. 1). In blue, arctomographic data, in orange, data from Huang
et al. (2021), with 2-$\sigma$ upper limits denoted by downwards triangles.
The black curve shows the best-fit of the Huang et al. (2021) sample (Eq. 2)
and the vertical dashed line marks the region where our data tend to diverge
from this curve (these data are excluded from our analysis). Right:
distribution of the fractional difference of EW in the arctomo data, as a
function of the spaxel separation. The blue points show the values of all the
spaxel pairs, while the blue curve and the gray band show the median values,
with the 1-$\sigma$ uncertainties, of a binning of the same distribution (each
bin contains 150 values). The black curves show the results for distributions
obtained by choosing different orientation angles for the three MUSE fields
(see Sect. 2.4 for more details).

In this work we combine EWs and spaxel positions with respect to the central
galaxy of the three fields described above. The underlying assumption is that
the cool CGM has self-similar properties across our sample of galaxies. This
is a common practice in absorption studies of the CGM, but while it is
normally assumed for samples of tens or even hundreds of galaxies, here we
focus instead on the halos of only three galaxies, with similar properties
(see Table LABEL:table_data). To take into account the slight differences in
the stellar masses of the three galaxies, we define a re-scaled version of
$R$, as:

$R^{\prime}=R/(M_{\ast}/M_{\ast,0})^{a}\ ,$ (1)

where $M_{\ast,0}=10^{10.3}\ M_{\odot}$ and $a=0.26$. This definition was
introduced in Huang et al. (2021), who conducted a survey with MagE of Mg ii
absorbers around almost 400 galaxies at $z<0.4$ using more than 150 quasars as
background sources. In particular, the values of $M_{\ast,0}$ and $a$ reported
above are derived from fitting the Mg ii EW as a function of the impact
parameter and the stellar mass (see below) for their sample of isolated star-
forming galaxies, similar to the objects analyzed in the present study. We
then investigate how the Mg ii equivalent width varies in our data as a
function of $R^{\prime}$ and as a function of the spaxel separation, as
described below. We emphasize that, for simplicity, we use only confirmed Mg
ii detections. We consider that most of non-detections are actually due to
spectra that (despite our selection based on S/N, see Sect. 2.2) are at the
arc borders and therefore contaminated by the sky. We plan to address this
information in future works, as it cannot be trivially implemented in our
models (see Sect. 3).

In the left panel of Fig. 2, we show in blue the Mg ii EWs of the three
ARCTOMO fields with their respective 1-$\sigma$ uncertainties, all as a
function of $R^{\prime}$: we can see that the EW decreases with increasing the
distance from the central galaxy. This distribution represents the first
observational constraint that will be used in this study. As a comparison, we
also report, in orange, the data coming from the survey of Huang et al. (2021)
(considering only the sample of isolated star-forming galaxies), with the
corresponding best-fit curve in black, given by:

$\log EW=a_{0}+a_{1}\log R^{\prime}\ ,$ (2)

where $a_{0}=1.42,a_{1}=-1.07$. The ARCTOMO data lie on top of those obtained
from the quasar survey and are well-described by the quasar-based best-fit
relation. The main difference can be seen in the region where
$R^{\prime}\lesssim 10$ kpc, where however there are no data available from
the Huang et al. (2021) sample and the best-fit relation represents only an
extrapolation. All the data with $R^{\prime}\lesssim 10$ kpc are therefore
excluded from our analysis (see Appendix A).

The second main observable that we use to constrain the coherence length of
the cool CGM is given by the distribution of the EW fractional differences,
called $\Delta EW$, as a function of spaxel separation. We define the
fractional EW difference between two spaxels as (e.g., Ellison et al., 2004;
Rubin et al., 2018b; Okoshi et al., 2021):

$\Delta EW=\frac{|EW_{1}-EW_{2}|}{\max{(EW_{1},EW_{2})}}\ .$ (3)

We calculate this quantity for all the pairs of spaxels with detections and
combining together the three MUSE fields. The error on $\Delta EW$ is
propagated from the uncertainties of the individual EWs (assuming that the
errors of $EW_{1}$ and $EW_{2}$ are uncorrelated). We then create a
distribution of $\Delta EW$ as a function of spaxel separation. To calculate
the distances between spaxels located in different MUSE fields, we assume
random orientation angles between the fields. A Monte Carlo simulation shows
that these relative orientations do not affect the distribution of $\Delta EW$
vs. separations. One of these realizations is shown on the right-hand panel of
Fig. 2 (uncertainties are not shown for clarity). We bin separations by equal
number of pairs (150) and compute the median $\Delta EW$ (blue solid line) and
the 32nd and 68th percentiles (gray band) in each bin. We can appreciate how
$\Delta EW$ is on average already larger than 0 for contiguous spaxels and
tends to increase with spaxel separation. Note that the peculiar shape of this
distribution for separations between 30 and 60 kpc is likely not physical, but
due to the specific spaxel configuration we are using (a similar shape is
indeed visible also in the synthetic observations of our models, Fig. 5). In
this work, we will anyway focus on the small spaxel separations.

Finally, the three additional black curves in the right-hand-side of Fig. 2
show the median values for three other distributions with different choices of
the field orientations: these lines are all within the gray band, showing how,
as mentioned above, the orientation does not affect this distribution. This
suggests that the EW distribution is isotropic, such that it does not show any
increment or decrement of EW in a particular direction with respect to the
host galaxy.

##  3 Models

Figure 3: EW maps obtained using the method described in Sect. 3.2, for a
coherence length of 1 kpc (top), 4 kpc (center) and 10 kpc (bottom). The
spaxel distribution used to create the synthetic observations (Sect. 3.3) is
superimposed in black.

###  3.1 Basic approach

To interpret our observables, we build empirical models of the cool CGM
distribution. We create 2-dimensional maps of Mg ii EW, where a crucial
parameter is given by the gas coherence length, $C_{\rm{length}}$. As in Rubin
et al. (2018a), we define the coherence length (or scale) as the size over
which the equivalent width of Mg ii remains constant. In practice, our maps
are divided in cells, whose size is equal to $C_{\rm{length}}$: the EW cannot
vary within a cell, while different cells can have different EW values.

The two main assumptions when creating our models are i) the ‘intrinsic’
distribution of EWs across the halo follows the results of the quasar survey
from Huang et al. (2021), justified by the fact that our data are consistent
with their findings; ii) the EW distribution is isotropic across the plane,
justified by the distribution of fractional EW differences in our data (see
Sect. 2.4 and Fig. 2). We then perform synthetic observations, tracing MUSE-
like spaxels that resemble the observations outlined in Sect. 2 and assuming
different values for the coherence scale. The outputs of such artificial
observations can be directly compared with the observational constraints
outlined in Sect. 2.4, allowing us to estimate which choice of
$C_{\rm{length}}$ better reproduces the ARCTOMO data.

###  3.2 Creation of EW maps

We populate the re-scaled (Eq. 1) projected plane $(x^{\prime},y^{\prime})$
surrounding the central galaxy (located at $x^{\prime}=0,\ y^{\prime}=0$),
using the EW-vs-$R^{\prime}$ relation described by Eq. (2) and a declining
covering fraction profile based on the constraints found by Huang et al.
(2021)333We use a power-law profile obtained by fitting the values reported in
Table 4 of Huang et al. (2021).. In detail, we first divide the plane into a
grid with a cell-size equal to the assumed coherence length and then into
concentric rings with a width equal to the coherence length itself; we then
populate each ring with a number of ‘systems’ $n_{\rm{ring}}$ equal to the
total number of cells in the ring multiplied for the predicted covering
fraction. This is done by assigning to the selected $n_{\rm{ring}}$ cells a
value of EW taken from a Gaussian distribution centered on the mean value of
EW predicted by Eq. (2) for those distances and with a width equal to the
intrinsic scatter $\sigma$ of the data from Huang et al. (2021). This scatter
is equal to 0.278 for $R^{\prime}<100$ kpc, which is the region analyzed in
this paper. Finally, the ‘empty’ cells have EW $=0\ \AA$444Choosing an
arbitrary low value instead of 0 does not impact the results of this paper..

Figure 4: Outputs of the three models presented in Fig. 3, used to infer the
likelihood term $\mathcal{L}_{\rm{scatter}}$. From top to bottom, the models
have a coherence length of 1, 4 and 10 kpc. Left: model EW distributions as a
function of $R^{\prime}$, with the black curve showing the relation of Eq.
(2). Downward arrows show spaxels whose predicted EWs are below the detection
limit of 0.2 Å and are therefore classified as ‘model non-detections’. Right:
distributions of the differences between the EW values and the corresponding
values predicted by Eq. (2), for data (blue) and models (red).

The outputs of the method described above are shown in Fig. 3, with their
dependence on the gas coherence length. In particular, $C_{\rm{length}}$ is
equal to 1, 4 and 10 kpc going from top to bottom. We can see that, even
though the general trend is similar (by construction), the maps look quite
different between each other, due to the different scale over which the EW is
allowed to vary. Note how all maps show ‘holes’, resembling the assumption
that the cool CGM is made by clouds whose total covering fraction is lower
than one. Both $C_{\rm{length}}$ and covering fractions have an effect on the
ARCTOMO Mg ii detections.

###  3.3 Synthetic observations

To use the observational constraints outlined in Sect. 2.4 one needs to
perform synthetic observations. To this aim, we mimic the MUSE observations by
creating distributions of spaxels that follow the configuration of our three
fields combined, correcting the position of each spaxel vertex for the galaxy
mass (see Eq. 1) and choosing a random orientation angle. An example of these
distributions is shown in black overlapped to the maps of Fig. 3. We select
only those spaxels for which we have detections in the data.

We then extract, from each one of these spaxels, the resulting equivalent
width, given by the average of all the EW values (both ‘systems’ and ‘holes’)
intercepted by the spaxel, each of them weighted by the fraction of the cell
area that overlaps with the spaxel. We exclude all the spaxels within 10 kpc
from the center: at these distances Eq. (2) is indeed only an extrapolation
and we have seen in Fig. 2 that our arctomographic data depart from this
prediction. This region therefore does not satisfy the assumption that our
data are consistent with the results from Huang et al. (2021). We have anyway
verified that including these data does not affect the results reported in
Sect. 4. The final EW distribution represents the model output that we can
compare directly with the observations, in order to constrain $C_{\rm
length}$, as we describe in detail in the following section.

###  3.4 Comparison with data and likelihood definition

Figure 5: Outputs of the three models presented in Fig. 3, used to infer the
likelihood term $\mathcal{L}_{\rm{fractional}}$. From top to bottom, the
models have $C_{\rm{length}}=1,4,10$ kpc. Left: distributions of the
fractional differences of EW as a function of the spaxel separation (same as
right panel of Fig. 2, but for the models). Right: distributions for all the
pairs with spaxel separations smaller than 5 kpc, for data (blue) and models
(red).

The likelihood is composed of two different terms, related to the two main
observational constraints reported in Sect. 2.4, which we call
$\mathcal{L}_{\rm{scatter}}$ and $\mathcal{L}_{\rm{fractional}}$ and that are
explained in detail below. The total likelihood is given by:

$\ln{\mathcal{L}_{\rm{tot}}}=\ln{\mathcal{L}_{\rm{scatter}}}+\ln{\mathcal{L}_{\rm{fractional}}}\
.$ (4)

$\mathcal{L}_{\rm{scatter}}$ is used to compare the variation of EW as a
function of the impact parameter, focusing in particular on the intrinsic
scatter of this distribution with respect to the best fit relation, as this
represents a proxy for the gas coherence length. This can be seen in Fig. 4,
where we show the results of our synthetic observations on the three different
model EW maps shown in Fig. 3. On the left panels, we can see the EW
distributions extracted from our models as a function of $R^{\prime}$, for a
coherence length of 1 (top), 4 (center) and 10 kpc (bottom). The downward
triangles depict the ‘model non-detections’ (excluded from our analysis),
which we define as equivalent widths lower than the observational detection
limit in the arctomographic data, equal to $0.2$ Å (see Table
LABEL:table_data). Note how the predictions of the three models follow the
best-fit relation of Eq. (2), as expected by construction. The outputs are
however very different from each other in terms of the intrinsic scatter of
the distributions, which is much smaller for smaller $C_{\rm{length}}$.
Indeed, if the scale of variation of the equivalent width is smaller than the
extension of the source, different EW values will be averaged along the source
extension, resulting in an overall smaller scatter than if the coherence scale
was larger than the source. This was hypothesized and then demonstrated by
Rubin et al. (2018a), where they performed such an analysis for the first
time, using galaxies as extended background sources; here our extended sources
are the MUSE spaxels.

On the right-hand column of Fig. 4, we show in red the binned differences
($\log EW-\log EW_{\rm{exp}}$) between our synthetic data and the expectations
from Eq. (2) (i.e., the black curve in the left panels), for the same models
shown on the corresponding left panels. The blue histogram shows the
distribution of the same quantity for the observational data. Both in models
and data we exclude the values with $R^{\prime}<10$ kpc, for the reasons
discussed above. We can see how the modeled distributions in the right panels
of Fig. 4 differ from each other, as models with larger coherence lengths
yield wider distributions. By comparing models and data, we can therefore
estimate which coherence length is in a better agreement with the
observations. We finally define the likelihood term
$\mathcal{L}_{\rm{scatter}}$ as the p-value obtained by comparing the $\log
EW-\log EW_{\rm{exp}}$ distributions of models and data with a Kolmogorov-
Smirnov (KS) test555We also performed an Anderson-Darling test (Anderson &
Darling, 1954), which produces very similar results. (Massey, 1951).

$\mathcal{L}_{\rm{fractional}}$ is instead related to the comparison of the
fractional differences. In Fig. 5 we show the outputs of our synthetic
observations for the three maps shown in Fig. 3, for three different coherence
lengths of 1, 4 and 10 kpc. The left panels show the distribution of $\Delta
EW$ as a function of the spaxel separation, calculated in the same way as for
the data, explained in Sect. 2.4 and shown in the right panel of Fig. 2. We
can see how the shape of the distribution varies when changing
$C_{\rm{length}}$. To compare with the data, we focus only on the small spaxel
separations. Indeed, while we have seen that our isotropic model is a good
representation of our observations, any deviation in the data from this
isotropic assumption (see Sect. 5.4) will affect more significantly the shape
of the $\Delta EW$ distribution at large separations, which will be therefore
not consistent with the predictions of our models. By focusing only on small
separations, we are more confident that any difference between model and data
is entirely due to the cool gas coherence scale.

In detail, we select all the pairs with spaxel separations smaller than 5 kpc
and the resulting distributions for the three models are shown in red in the
right panels of Fig. 5, with overlapped in blue the distribution obtained from
the data. We can see how the width of the model distribution increases when
increasing the coherence length from 1 to 4 kpc. Indeed, if $C_{\rm{length}}$
is significantly smaller than the length of the spaxel, different EW values
will be averaged with each other, with the overall effect of reducing the
differences between the EW values of different nearby spaxels, with respect to
models where $C_{\rm{length}}$ is more similar to the spaxel size. However,
models with $C_{\rm{length}}=10$ kpc have a large amount of low $\Delta EW$,
since, if the coherence scale is considerably larger than the length of the
spaxels, adjacent spaxels will be probing the same equivalent width values,
significantly reducing $\Delta EW$. Therefore, this observational constraint
seems to represent a good diagnostic to infer the coherence scale of the cool
CGM. The likelihood term $\mathcal{L}_{\rm{fractional}}$ is given by the
p-value of a KS test performed between the model and data distributions shown
in the right panels of Fig. 5.

###  3.5 Bayesian analysis

Having entirely determined the likelihood of Eq. (4), we can use it to
constrain $C_{\rm length}$. We perform a Bayesian analysis, using the nested
sampling method adopting the python package Dynesty (Skilling, 2004), over our
only free parameter $C_{\rm{length}}$. We utilize the likelihood function
$\mathcal{L}_{\rm{tot}}$ defined in Sect. 3.4 (Eq. 4) and a uniform prior on a
range that varies from $0.5$ to $12$ kpc. In fact, just by simply looking at
the observed EW maps of Fig. 1, one can already see large EW variations on
adjacent spaxels, which implies that $C_{\rm{length}}$ must be of the order of
a spaxel linear size (a few kpc, see Table LABEL:table_data). If
$C_{\rm{length}}\ll 1$ kpc or $C_{\rm{length}}\gg 10$ kpc, we would indeed
expect a much smoother EW variation across the halo. Therefore, with this
prior we are able to explore the full range of possible cool gas coherence
scales that might be in agreement with the data. In each call, once the value
of $C_{\rm{length}}$ is chosen, to estimate the likelihood we extract a random
realization of the model and the data. The latter is obtained by randomly
drawing the values of the observed EWs from a Gaussian distribution centered
on the EW value and with a standard deviation equal to its error (see also
Rubin et al., 2018a) and by assuming orientation angles for the three fields
equal to those used to generate the synthetic observations from the models.

##  4 Results

###  4.1 Coherence scale of the cool CGM

Before reporting the results of the Bayesian analysis, we look at how our
models compare with the two observational constraints as a function of the
assumed coherence length, by analyzing separately the two parts of our
likelihood, $\mathcal{L}_{\rm{scatter}}$ and $\mathcal{L}_{\rm{fractional}}$.
Indeed, while we use the Bayesian analysis to accurately constrain the range
of $C_{\rm{length}}$ that better reproduces the observations, it is important
to know whether or not the predictions of our models are effectively in
agreement with the data.

We explore values of the coherence length varying from 0.5 to 12 kpc and, for
each choice of $C_{\rm{length}}$, we create 1000 model and data realizations.
We finally compare, for each realization, models and data using a KS test, as
in Sect. 3.4. The top panel of Fig. 6 shows the results related to the scatter
in the EW-vs-$R^{\prime}$ relation ($\mathcal{L}_{\rm{scatter}}$) and the
bottom panel those related to the distributions of fractional difference of EW
($\mathcal{L}_{\rm{fractional}}$). The black curves show the median values of
the KS-test p-value, while the gray bands show the extent of the 32nd and 68th
percentiles of the distributions. We consider models with p-values higher than
0.05 (marked by the horizontal dashed line) in agreement with the data.

Figure 6: Comparison between models with different coherence lengths and the
ARCTOMO data: the two plots show the p-values of the KS test between model and
data, as a function of the coherence length adopted in the model. The black
curve shows the median value of the 1000 realizations created for each
$C_{\rm{length}}$ value (see Sect. 4.1), with the gray band showing the 32nd
and 68th percentiles. The horizontal dotted line depicts
$p_{\rm{value}}=0.05$, so that models with a probability higher than this
threshold can be considered in agreement with the data. Top, comparison of the
scatter in the EW-vs-$R^{\prime}$ relation; bottom, comparison of the EW
fractional differences for spaxel separations smaller than 5 kpc. This figure
shows that models with $1.5\lesssim C_{\rm{length}}/\rm{kpc}\lesssim 7$ are
consistent with the data. Figure 7: Posterior distribution of the coherence
length, as found with our Bayesian analysis. The vertical lines show the 2.5,
50 and 97.5 percentiles of the distribution, corresponding to the median value
and its 2$\sigma$ uncertainties, whose values are shown on top. The ARCTOMO
data are best reproduced by coherence lengths varying from 1.4 to 7.8 kpc.

We can see how the consistency with the data depends on the adopted value of
$C_{\rm{length}}$. In particular, the constraint given by the scatter in the
EW-vs-$R^{\prime}$ relation (top panel of Fig. 6) can be used to put a lower
limit on the value of the coherence length, at about 1.3 kpc (where the median
p-value is equal to 0.05), in line with the findings of Rubin et al. (2018a),
who used the same constraint to evaluate the coherence scale of the cool CGM.
The upper limit on $C_{\rm{length}}$, using this constraint, is at about 10
kpc or more. The comparison with the fractional differences of equivalent
width (bottom panel) allows us to put again a lower limit at about 1.5 kpc and
a much tighter constraint on the upper limit of the coherence length, at about
7 kpc, and represents therefore a more stringent constraint for
$C_{\rm{length}}$. Together, the two constraints define a range of models with
$1.5\lesssim C_{\rm{length}}/\rm{kpc}\lesssim 7$ that are in agreement with
the ARCTOMO data.

We move then to the more quantitative results of the Bayesian analysis, where
we adopt the full likelihood described by Eq. (4). The main result of this
work is shown in Fig. 7, where we report the posterior distribution of the
coherence length, the free parameter in our analysis. The vertical dashed
lines show the value of the median of the distribution, with the 2$\sigma$
uncertainties (0.025 and 0.975 quantiles, see also Table 2). We can see that
$C_{\rm length}$ is well defined between 1.4 and 7.8 kpc, with a median value
equal to 4.2 kpc. The 2$\sigma$ limits of the posterior distribution capture
the whole range of models that are in agreement with the ARCTOMO data, as
shown in the two panels of Fig. 6. These are the tightest bounds on $C_{\rm
length}$ so far. We stress that this is possible thanks to the unique nature
of the ARCTOMO data, which allow us to have multiple and extended sources (the
MUSE spaxels) around individual galaxies.

###  4.2 How universal is our result?

Given the complexity of the gaseous halos of galaxies, one could expect the
coherence length of the cool CGM to depend for example on its distance from
the central galaxy or to vary across different galaxy halos. In the following
sections we investigate more in detail these aspects, by analyzing sub-samples
of the original set of data that was used to obtain the results reported
above.

####  4.2.1 Variation of $C_{\rm{length}}$ with impact parameter

Figure 8: Same as Fig. 7, but for the three sub-samples obtained by selecting
the spaxels with $R^{\prime}<30$ kpc (left), $30<R^{\prime}/\rm{kpc}<50$
(center) and $R^{\prime}>50$ kpc (right).

Figure 9: Same as Fig. 7, but selecting separately the spaxels of the three
MUSE fields: PSZ1 G311.65$-$18.48 (left), RCS2 032727$-$132623 (center) and
SGAS J1226+2152 (right).

We have seen that at least some of the properties of the cool CGM, like EW and
covering fraction (e.g., Huang et al., 2021), seem to be related to its
projected distance from the galaxy host (or impact parameter). Even though it
is important to keep in mind that these are all projected quantities, such
trends might be related to the origin and dynamics of the cool CGM and to its
connection with the central galaxy. Using our dataset, we investigate whether
a similar dependency is present also for the gas coherence length.

We divide our original sample in three bins, for spaxels with $R^{\prime}<30$
kpc, $30<R^{\prime}/\rm{kpc}<50$ and $R^{\prime}>50$ kpc. For each sub-sample
we then perform the same Bayesian analysis performed on the whole sample. The
results of this procedure are shown in Fig. 8 and in Table 2. We find a slight
trend of increasing coherence length at larger projected distances from the
central galaxies, with the median values of the posterior distributions going
from 3.9 kpc in the innermost bin to almost 7 kpc in the outermost one.
However, it is hard to say whether this trend is physical or simply due to the
low-number statistics. The three posterior distributions share indeed the same
value of the 2.5 percentile (1.3 kpc) and the main difference is in the
distribution width. The large ‘tails’ at large $C_{\rm{length}}$ in the middle
and right panels are likely due to the fact that most of the spaxels in our
sample are within 30 kpc from the galaxy (47 detections for $R^{\prime}<30$
kpc, 21 detections for $30<R^{\prime}/\rm{kpc}<50$ and 7 detections for
$R^{\prime}>50$ kpc) and therefore the constraints on $C_{\rm{length}}$ at
large $R^{\prime}$ are very loose666Note also that, extending the prior on
$C_{\rm{length}}$, the tails shown in the middle and right panels of Fig. 8
would be even more extended and the median and 97.5 percentile values would
increase. This is most likely not due to a physical property of the CGM, but
to the small number of spaxels present at large $R^{\prime}$..

Given the considerations above, we argue that, if any, the dependence of the
coherence length with the impact parameter is only minor. This could imply
that the cool CGM is originated by similar physical processes across the
galactic halo, which tend to create similar gas structures independently of
the distance from the central galaxies. More data will be needed to study this
in a greater detail.

####  4.2.2 Variation of $C_{\rm{length}}$ between single galaxies

We perform the same analysis described in Sect. 3.5, but selecting the three
halos separately, selecting each time only the spaxels from one of the three
MUSE fields. The results of these tests are shown in Fig. 9 and Table 2. We
can note how the posterior distributions of the gas coherence length are
slightly different in the three cases, indicating that this property can vary
across different galaxy halos. While the distributions of the PSZ1
G311.65$-$18.48 and the SGAS J1226+2152 fields are very similar, with the main
difference being the smaller width of the latter (due to the larger number of
detections in SGAS J1226+2152), the posterior distribution for the RCS2
032727$-$132623 field seems to point to a larger gas coherence length in the
cool CGM of this galaxy with respect to the other two. This difference seems
however to not be significant, given that in all cases the posterior
distributions are relatively extended and the three coherence lengths are
consistent with each other when considering the uncertainties. The slight
difference in the $C_{\rm{length}}$ of RCS2 032727$-$132623 might be due to
the fact that the spaxels in this field are on average at larger impact
parameters with respect to the other two fields, possibly reinforcing the idea
of a connection between $C_{\rm{length}}$ and the impact parameter. A larger
number of spaxels for each halo would be needed to more accurately constrain
the coherence length of the cool CGM of single galaxies.

It is particularly important to note that the three median values of the
distributions are all within the range of coherence lengths that we find when
combining the three fields together (Fig. 7). This suggests that the range of
coherence lengths between about 1.5 and 8 kpc that we found in Sect. 4.1 is
not driven by one particular galaxy, but likely represents a common property
of the cool CGM of these three objects (as we initially assumed). This result
seems to confirm that similar galaxies are surrounded by cool CGM with a
similar structure. A larger sample of galaxies will be needed to confirm this
finding.

Table 2: Results of the Bayesian analysis. TEST | 2.5 perc. | Median | 97.5 perc.
---|---|---|---
| (kpc) | (kpc) | (kpc)
Full sample | 1.4 | 4.2 | 7.8
$R^{\prime}/\rm{kpc}<30$ | 1.3 | 3.9 | 7.7
$30<R^{\prime}/\rm{kpc}<50$ | 1.3 | 5.5 | 11.5
$R^{\prime}/\rm{kpc}>50$ | 1.3 | 6.9 | 11.7
PSZ1 G311.65$-$18.48 | 1.4 | 4.8 | 10.8
RCS2 032727$-$132623 | 2.1 | 6.2 | 11.5
SGAS J1226+2152 | 1.2 | 4.2 | 8.8
Free $\sigma$ | 1.3 | 4.5 | 8
$4\times 4$ binning | 1.8 | 5.7 | 10.6

##  5 Discussion

###  5.1 What is the coherence scale?

We have constrained the typical scale over which the absorption strength of
the cool CGM around three galaxies at $z\lesssim 1$ does not vary. From an
observational point of view, a coherence scale, or length, of a few kpc means
that when observing the cool CGM around a single galaxy with multiple probes,
we do not expect to observe strong variations in Mg ii EW for (projected)
separations smaller than this scale, while we expect little correlation for
larger separations. This can be seen in the right-hand-side of Fig. 2 and has
been found previously by other studies (see Sect. 5.2). The fact that we find
an entire range of coherence lengths that seem to well reproduce our data
($1.4\lesssim C_{\rm{length}}/\rm{kpc}\lesssim 7.8$), means that likely the
exact value of this scale may vary across the halo, although we found that
there is little (if any) dependence of $C_{\rm{length}}$ with the projected
distance from the central galaxy. To better understand the variation of the
coherence length across the halos of galaxies, more data will be needed.

What is the physical nature of this scale? As already discussed extensively in
Rubin et al. (2018a), a variation in EW may be due not only to differences in
the number of clouds intercepted by the line of sight, but also to differences
in the clouds’ kinematics (and partly also in gas metallicity). Higher
spectral resolution observations (e.g., Lopez et al., 1999; Rauch et al.,
2002; Zahedy et al., 2016; Krogager et al., 2018) and/or more sophisticated 3D
modeling (see for example Li et al. 2023 and Sect. 5.2) are needed to
distinguish between different kinematic components along the line of sight and
to better understand what property is driving the change in EW. The coherence
length that we derived using our empirical models of EW is instead a projected
quantity and does not allow us to make conclusions on the physical properties
of the individual CGM clouds. We however speculate that, regardless of the
exact property that drives the change in EW, the coherence scale should
represent the size of the cool gas structures over which such properties
change. Each one of such structures could be composed by a relatively large
cloud with a size comparable to $C_{\rm{length}}$, or by multiple smaller
clouds whose properties are related to each other and that cluster over a
coherence scale. This length is a direct observable that can be tested on
upcoming observational data and on theoretical models (see Sect. 5.3) and it
is independent of any physical assumption on the cool CGM.

###  5.2 Comparison with previous works

The main point of comparison for our study is the work of Rubin et al.
(2018a), on which the idea for the present exploration was based. In a similar
fashion to what we have later developed in this paper, they used a fiducial
model for the cool CGM based on the quasar surveys of Chen et al. (2010b) and
Werk et al. (2013) and they compared the predictions of such model with their
data (a sample of 27 galaxy-galaxy pairs), to estimate the gas coherence
length. The main constrain was given by the comparison of the scatter in the
EW-vs-$R^{\prime}$ distribution of models and data. This coincides with the
first part of the likelihood that we used in this work
($\mathcal{L}_{\rm{scatter}}$).

In our analysis we applied two main refinements to the models of Rubin et al.
(2018a): i) we created 2-dimensional distributions of absorbers across the
halo, also considering the covering fraction as a function of the projected
distance from the center, while Rubin et al. (2018a) explored ‘slices’ of the
CGM at fixed impact parameters, with a unity covering fraction; ii) we used
the actual distribution of MUSE spaxels and their shape to extract the
synthetic observations, while Rubin et al. (2018a) used more idealized
distributions of circular beams to resemble the spatial extension of their
background galaxies.

Regardless of the above differences, and of the use of a model based on a
different survey (Huang et al., 2021), our results are in perfect agreement
with those of Rubin et al. (2018a), who estimated a lower limit for the
coherence length of $\sim 1.9$ kpc. Our estimate
($1.4<C_{\rm{length}}/\rm{kpc}<7.8$) is consistent with this value and
especially the plot in the top panel of Fig. 6 shows that, using the same
observational constraint of Rubin et al. (2018a), we would obtain a very
similar result, with very loose constraints on the coherence length upper
limit. By using the additional observable given by the equivalent width
differences across the spaxel pairs (which was evidently not possible in Rubin
et al. (2018a), given the different observational setup) we were also able to
put a firm upper limit on the coherence length of the gas, at about 8 kpc. The
similarity between our results and those of Rubin et al. (2018a), who used
around 30 absorbing galaxies for their analysis, further points to the
conclusion that the cool CGM of galaxies with similar properties has a similar
’universal’ structure (see Sect. 4.2.2).

Various other observational studies have tried to estimate the typical scale
of the cool circumgalactic medium, either directly through the use of multiply
lensed quasars, or indirectly, using photo-ionization models. Here, we
summarize some of the main results of these works and we refer to Rubin et al.
(2018a) for a more complete and extensive review (see their Section 3.2). The
use of lensed quasars to spectroscopically study the CGM has started more than
40 years ago (e.g., Young et al., 1981; Smette et al., 1995) and has been
extensively adopted in the last decades to investigate the structure of this
cool gas (e.g., Rauch et al., 1999; Lopez et al., 1999; Rauch et al., 2002;
Ellison et al., 2004; Lopez et al., 2005; Chen et al., 2014; Zahedy et al.,
2016; Rubin et al., 2018b; Okoshi et al., 2021). It is not trivial to compare
our findings with those of these works, as different studies use different gas
properties, like column densities of various ions and/or the gas velocity
structure (Chen et al., 2014) to derive a coherence scale of the CGM. Even
when looking at Mg ii absorption, the coherence scale is often based only on
the number of coincidences and anti-coincidences (detection or non-detection
of the absorption) in the QSO lines of sight (e.g., Ellison et al., 2004), a
method that has been pointed out to have limitations (Martin et al., 2010).
Here instead, besides taking full advantage of the arc-tomography spatial
sampling, we also use the variations of the absorption strength across the
halo. The general picture that emerges, putting together the results from
previous studies, is anyway that weak Mg ii absorbers (EW $\lesssim 0.4$ Å)
tend to exhibit a smaller coherence scale (less than 1 kpc), with respect to
stronger systems (like those present in the ARCTOMO data), which show a
coherence length of a few kpc or larger, in agreement with our findings.

Additional constraints on the structure of the cool CGM have been derived by
using photo-ionization models (e.g., CLOUDY Ferland et al., 1998, 2013, 2017),
which allow to estimate the thickness of the absorbers777Note that these
models probe the absorber size along the line of sight, while in this work we
evaluated the spatial coherence length in the transversal direction (even
though these two lengths are not necessarily different from each other)..
Overall, cloud sizes lower than 1 kpc are found at redshifts $z\gtrsim 2$
(e.g., Crighton et al., 2015), while at $z\lesssim 1$ the scales seem in
general to be larger, of the order of several up to even hundreds of kpc
(e.g., Werk et al. 2014; Keeney et al. 2017, but see Stern et al. 2016, who
assuming that the cool gas spans a variety of densities, found that the cool
clouds traced by Mg ii in the COS-Halos sample have sizes of only a few tens
of pc). More recently, Zahedy et al. (2021) applied photo-ionization models to
four Lyman-limit systems at redshift lower than 1, taken from the CUBS survey
and found that the clouds have thicknesses that can vary from 0.01 to 10 kpc,
in agreement with our findings (although their mode is around 0.1 kpc). It is
however important to note that, due to the various assumptions of ionization
models, as for example the strength and shape of the adopted UV background
(see Acharya & Khaire, 2022; Gibson et al., 2022), the cloud sizes inferred
from these works remain quite uncertain. Our estimate of the coherence length
is to date one of the most accurate constraints of the structure of the cool
CGM around galaxies at $z\lesssim 1$.

In the last years, also more complex and physically motivated models have been
used to interpret observations of the cool CGM and to infer, among other
properties, the size of the single clouds. Faerman & Werk (2023) built a model
of the cool gas assuming pressure equilibrium between the $T\sim 10^{4}$ K
medium and a hot corona (see Faerman et al., 2020) and allowing for non-
thermal pressure support. They then compared their models with the COS-Halos
data (e.g., Werk et al., 2013) and found that the observed column densities of
neutral hydrogen and low and intermediate ions (among which Mg ii) are best
reproduced by filling factors of the cool gas of the order of 1%. This,
together with the observed number of kinematic components (or clouds) along
the line of sight, provides an upper limit for the cloud sizes of
$R_{\rm{cl}}\lesssim 0.5$ kpc. This finding is not in disagreement with our
results, since, as discussed in Sect. 5.1, we are not modeling single clouds,
and each absorbing system in our maps could be made of multiple clouds, which
could therefore be smaller than our estimates for the coherence scale. As also
mentioned in Faerman & Werk (2023), multiple clouds could be segregated in
larger complexes and structures, consistently with our work and with
complementary implications from high-resolution data of the high-velocity
clouds in the halo of the Milky Way (e.g., Tripp, 2022). Finally, using
dynamical semi-analytical models of infalling clouds accreted from the
intergalactic medium, Afruni et al. (2022) have reproduced the observed
covering fraction, silicon column density and kinematics of the cool absorbers
detected by the AMIGA project (Lehner et al., 2020) in the halo of the
Andromeda galaxy. In their best-fit models, typical cool clouds have radii of
about 5 kpc at the virial radius and of less than 1 kpc at distances of a few
tens of kpc from the central galaxy. Even though we do not have in our
empirical models any indication of the intrinsic location of the single
absorbing clouds, our coherence length going from about 1.5 to 8 kpc seems to
be in perfect agreement with the results of these models.

###  5.3 Implications for hydrodynamical simulations of the CGM

While the coherence scale that we estimated in this work does not directly
reflect the physical 3D distribution and the size of the single clouds, it
gives an important constraint on the typical scales over which the cool gas is
expected to cluster in separate structures (see Sect. 5.1). Our finding
represents therefore a very useful observable to compare to current (and
future) theoretical models of the circumgalactic medium.

From the point of view of high-resolution hydrodynamical simulations, single
clouds can have sizes smaller than a pc, due to processes like fragmentation,
or shattering, that might lead to small cloudlets with lengths similar to the
gas cooling length (see McCourt et al., 2018). In the hypothesis that these
small cloudlets are long-lived, one possible picture is that the cool CGM is
composed by a fog of these cloudlets that permeate the whole galaxy halo: this
has been proposed especially for the cool CGM at redshifts $z\gtrsim 2$, to
explain a small filling factor and at the same time a covering fraction close
to unity seen in observations (see references in McCourt et al., 2018).
However, to fully understand the structure of the cool gas one has to consider
many other physical effects: small droplets of gas could quickly evaporate
into the hot gas that surrounds them, due to thermal conduction (Armillotta et
al., 2017; Afruni et al., 2023) or they can coagulate to form larger
structures (e.g., Gronke & Oh, 2020b, 2023), which could later even grow due
to the induced condensation of the hot gas (e.g., Marinacci et al., 2010;
Gronke & Oh, 2018; Kooij et al., 2021). The final picture emerging from this
type of simulations is then that the cool CGM is likely composed of a spectrum
of structures, going from a fog of cloudlets to more ‘monolithic’ clouds and
the predominance of one or the other depends on the physical properties of the
gas (Gronke & Oh, 2020b). Our results suggest that, at least in the galaxies
probed in this work, the cool CGM presents substantial differences (in Mg ii
EW) on scales between 1.4 and 7.8 kpc. We speculate that this finding can be
used by hydrodynamical simulations to put upper limits on the larger scales of
the cascade of structures that is believed to form the cool gas around
galaxies.

Given their pc-scale resolution, the studies mentioned above can only be
focused on small parts of the galactic halo. To have a more extended view, one
needs to resort to cosmological and zoom-in simulations, which have been
extensively used in the last decade to study the cool CGM (see Faucher-Giguère
& Oh, 2023, for a recent review). These studies can be very useful to have a
full picture of the distribution of the CGM across the halos of galaxies, but
have necessarily much lower resolutions with respect to the idealized
simulations. Even in the best zoom-in cases (e.g., Peeples et al., 2019) the
resolutions can reach a maximum of about 500 pc, which are not sufficient to
accurately predict the structure of the cool gas. van de Voort et al. (2019)
have indeed shown that increasing the resolution in this type of simulations
drastically changes the distribution and the covering fraction of the cool
gas, indicating no sign of convergence. Therefore, while a considerable effort
has been done to compare the predictions of this type of simulations with
observations of the cool CGM (see for example Appleby et al., 2021; DeFelippis
et al., 2021), the results are still necessarily uncertain. With this caveat
in mind, it is worth mentioning that recently cool clouds with sizes of a few
kpc, possibly in agreement with our findings, have started to be identified
even in cosmological simulations like TNG-50 (Nelson et al., 2020; Ramesh et
al., 2023). A CGM coherence scale could be directly extracted from these
simulations and compared with the findings of this study.

Future simulations and in particular works that aim to link small and large-
scale simulations (e.g., Huang et al., 2022; Fielding & Bryan, 2022;
Weinberger & Hernquist, 2023) will hold the key for a more accurate comparison
with the available observational data. Our result regarding the coherence
length is one of the fundamental observational constraints that such models
should aim to reproduce.

###  5.4 Limitations and assumptions of this study

An important assumption we made in this study is that the gas is distributed
isotropically across the halo, so that the average strength of the absorption
is the same along rings centered on the host galaxy. This is likely a
simplistic assumption. Previous works (e.g., Schroetter et al., 2019) have
found a segregation of absorbers along the galaxy minor and major axis,
hinting at the cool CGM being described by a wind plus disk model. In
particular, Fernandez-Figueroa et al. (2022) found a dependence of the EW
strength with the azimuthal angle for one of the three MUSE fields analyzed in
this work, SGAS J1226+2152, even though the effect dilutes when considering
the three fields together, as we have seen in Sect. 2.4. Moreover, the
presence of satellites in the halo of the main absorbing galaxy might affect
the cool gas distribution: part of the absorbing cool gas might indeed come
from stripping from these satellite galaxies, as extensively observed in the
Local Universe (e.g., Poggianti et al., 2017; Putman et al., 2021), thus
breaking our assumption of isotropy.

Nevertheless, by considering in our analysis only the EW fractional
differences at small spaxel separations (see Sect. 3.4), we are minimizing the
effects due to possible anisotropic distributions, which occur at larger
scales. We have verified that using in our analysis also larger spaxel
separations does not significantly affect our main finding on the coherence
length range, but it reduces, for any given choice of $C_{\rm{length}}$, the
agreement between models and data, indicating a slight level of anisotropy in
our data. We therefore acknowledge the importance of exploring more in detail
also anisotropic models, but we leave it for future studies, as it is outside
the scope of this work. In future works we will also address more realistic
shapes for the gas absorbers, as the squared approximation adopted here is
clearly simplistic. However, a more accurate absorber morphology (which is
also essentially unknown to date) will likely not affect our findings on the
gas coherence scale.

Figure 10: Distribution of Mg ii EW in the data (blue) and models with
$C_{\rm{length}}=4$ kpc (red). Both distributions are built by combining 10
different realizations of models and data (see Sect. 5.4). Both data and
models are incomplete for $\rm{EW}\lesssim 1\ \AA$.

Another main assumption of this work is the creation of our empirical models
based on the quasar survey from Huang et al. (2021), assuming that the cool
CGM around our galaxies has properties similar to the galaxies of their
sample. Our galaxies are at a median redshift $z\sim 0.8$, while those
analyzed in Huang et al. (2021) are at a significantly lower redshift ($z\sim
0.21$). During this cosmic time the properties of the cool circumgalactic
medium might change and while we have seen that our data are consistent with
the EW-vs-$R^{\prime}$ relation found by Huang et al. (2021), the intrinsic
scatter of this relation could be different, affecting our results on the gas
coherence length. However, there is to date no evidence of any dependence of
this scatter across this range of redshifts. Nielsen et al. (2013) explored
the relation between the Mg ii absorption strength and the projected distance
from the galaxy in the MAGiiCAT sample, composed of almost 200 galaxies whose
redshifts vary from 0.072 to 1.120. They found that there is no dependence
with redshift in the scatter of this relation, which implicates that our
results, although based on a sample of galaxies at lower redshift with respect
to our data, are most likely robust. To further test this, we have run an
additional Bayesian test on $C_{\rm{length}}$, leaving also the intrinsic
scatter $\sigma$ used to create our models (see Sect. 3.2) free to vary
between 0.05 and 0.5. We find that the posterior distribution of $\sigma$ has
a median at $0.282^{+0.175}_{-0.124}$ (2$\sigma$ uncertainties), interestingly
very similar to the original value of 0.278 (based on the data from Huang et
al. 2021) and that the findings on $C_{\rm{length}}$ (see Table 2) do not
change with respect to our fiducial model.

A limitation of this study is that our sample is composed of only three
galaxies, whose CGM does not necessarily resemble the average best-fit
relation of Huang et al. (2021), which is based on hundreds of objects. The
fact that our EWs lie on top of the Huang et al. (2021) relation seems to
suggest that this is a good representation of the cool CGM of our galaxies, at
least for $R^{\prime}>10$ kpc; however, including a larger number of halos in
our sample might increase the scatter in our data, potentially affecting our
findings. The tests described in Sect. 4.2.2 indicate that adding more
galaxies to our sample (as long as they have similar properties to those
analyzed here) might not influence our results, but a larger and more
statistically significant sample of galaxies will be needed to confirm this.

We have chosen to work with binned spaxels of $0.6\arcsec$ to maximize the
number of probes, while still reducing the cross-talk between adjacent
spaxels. The (slight) amount of cross-talk has the effect of producing smaller
EW differences between adjacent spaxels and this might affect our findings. To
evaluate the significance of this effect, we ran the same Bayesian analysis
but using a $4\times 4$ binning for the data (corresponding to $0.8\arcsec$,
see also Lopez et al. 2018; Tejos et al. 2021), which further reduces the
seeing-induced cross-correlation. The results, reported in Table 2, are
consistent with our fiducial findings, with the median value of the posterior
distribution well within the fiducial range of coherence lengths. The main
difference is given by the fact that the posterior distribution is much wider
(the 97.5 percentile is equal to almost 11 kpc), meaning that the constraints
on the coherence scale are looser with respect to our fiducial case, due to
the smaller number of spaxels. We therefore conclude that, as long as the
binned spaxels have dimensions comparable to the observational seeing, our
results are rather insensitive to the exact choice of spatial binning.

Finally, an inherent limitation of our dataset is given by its incompleteness
at $\rm{EW}\lesssim 1\ \AA$. This can be seen in Fig. 10, where we show in
blue the EW distribution obtained by combining 10 different bootstrap
realizations of the full data sample, drawing values from Gaussian
distributions centered on the observed EWs and with a standard deviation equal
to the observed uncertainties (as in the Bayesian analysis, see Sect. 3.5). We
can see how, for EWs weaker than about $1\ \AA$, the data do not exhibit the
exponential growth in the distribution expected for Mg ii toward quasars (see
Zhu & Ménard, 2013). On the other hand, the synthetic distribution appears
also to be incomplete below $1$ Å (red distribution of Fig. 10). This is not
surprising as we are excluding all the spaxels with non-detections in the
data, which would yield a large number of low EW values in the model (being
most of them at large impact parameters). The synthetic distribution was
obtained by combining 10 different model realizations with a coherence length
of 4 kpc, consistent with our estimated best-fit value. Even though the shape
of this distribution slightly changes depending on the chosen
$C_{\rm{length}}$, in all cases models and data are consistent with each other
in the region of weak EWs. This feature holds also when looking at the three
fields separately.

Would a higher S/N affect our results? Having spectra with a higher S/N would
allow us to increase the number of detections in our data, especially at
$R^{\prime}\gtrsim 50$ kpc, where most of our current data show only non-
detections. While it is not trivial to predict how a more complete sample
would affect our current findings, such a data-set would be crucial to more
accurately study the possible variation of the coherence length with the
distance from the central galaxy, since we would have a much better sampling
at large impact parameters.

To conclude, even though a higher S/N is desirable to conduct a more detailed
analysis, S/N incompleteness does not affect the results of the current study.

##  6 Summary and conclusions

In this paper, we investigated the coherence length ($C_{\rm{length}}$) of the
cool circumgalactic medium traced by Mg ii absorption. We utilized arc-
tomographic MUSE data of the cool CGM of three different star-forming disk
galaxies at redshift $\lesssim 1$, which allowed us to have a sample of Mg ii
EWs coming from almost 100 spaxels spanning the galaxy halos up to several
tens of kpc from the center. We developed 2D empirical models that describe
the EW distribution and gas covering fraction, based on a recent QSO-survey of
the cool CGM of low-redshift galaxies. We then compared models and data
through a Bayesian analysis, in order to find the best coherence length that
reproduces both the observed scatter in the distribution of EWs as a function
of the impact parameter and the observed distribution of the fractional
difference of EWs. Our main findings are the following:

  1. 1.

our data are best reproduced by models with a cool gas coherence length in the
range between 1.4 and 7.8 kpc (corresponding to the 2$\sigma$ limits of our
posterior distribution), meaning that the Mg ii absorption strength does not
vary significantly below these scales;

  2. 2.

we find that there is no significant trend of the coherence length with the
impact parameter, possibly indicating similar formation mechanisms for the
cool gas at different distances from the central galaxies. There is a slight
tendency of having larger coherence lengths at larger impact parameters, but
the variation is likely due to the low-number statistics and more data are
needed to properly probe this trend;

  3. 3.

we observe a small variation in the coherence lengths of the three individual
galaxy halos, but this variation is well within our fiducial range. This
suggests that similar galaxies are likely to be surrounded by CGM with a
similar structure.

To conclude, with this study we were able to put the most stringent
constraints to date on the coherence scale of the cool CGM, thanks to the use
of unique data that allowed us to spatially resolve this medium around
individual galaxies at $z\sim 1$. While here we made use of only three galaxy
halos, a future larger sample of such data will possibly confirm and refine
the findings of this paper. Moreover, while this analysis is entirely based on
the strength of the Mg ii absorption lines and on simple empirical models,
future studies involving also the gas kinematics, the absorption from other
ions and the use of more sophisticated and physically motivated models, will
help in building a more complete picture of the CGM structure. Such results
will be extremely useful to inform and constrain our current and future
theoretical models of the gaseous halos of galaxies and to eventually
understand how galaxies form and evolve in the Universe.

###### Acknowledgements.

The authors would like to thank the referee for a constructive report. A.A.
acknowledges the financial support of the Joint Committee ESO-Chile grant.
S.L. acknowledges support by FONDECYT grant 1231187. E.J.J. acknowledges
support from FONDECYT Iniciación en investigación 2020 Project 11200263 and
the ANID BASAL project FB210003. L.F.B. acknowledges the support of FONDECYT
project 23050313.

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##  Appendix A Absorption line fits

In Tables LABEL:table_EWs_pks, LABEL:table_EWs_rcs and LABEL:table_EWs_sgas
and in Fig.s 11, 12 and 13 we report the results of the automated Gaussian
fits (Sect. 2.2) that were used to derive the EWs used in this work, for
respectively the PSZ1 G311.65$-$18.48, RCS2 032727$-$132623, and SGAS
J1226+2152 fields. We show only the results concerning the successful fits, or
detections (significance in the EW larger than 2 in both doublet lines), as
the non-detections were excluded from our analysis. In the three figures, the
normalized MUSE spectra are shown in green, with their respective
uncertainties in blue, while the fits of the Mg ii$\lambda\lambda 2796,2803$
doublet are reported in red, with the red ticks showing the positions of the
line velocity centroids. Note that, for the SGAS J1226+2152 field, the Mg ii
lines are partially blended with Si ii$\lambda 1260$ absorption, so this is
modeled by adding an additional Gaussian fit in our analysis for this field
(see Tejos et al. 2021).

For the reasons explained in Sect. 2.4, we do not consider all the spaxels
with $R^{\prime}\lesssim 10$ kpc888Note that in Tables
LABEL:table_EWs_pks–LABEL:table_EWs_sgas we are reporting the impact parameter
$R$, not $R^{\prime}$, which is re-scaled based on the galaxy mass (Eq. 1).,
therefore the spectra from number 21 to 28 in the PSZ1 G311.65$-$18.48 field
and those from 6 to 10 and from 12 to 14 in the SGAS J1226+2152 field are
excluded from our analysis. The results of Sect. 4 are based on all the other
spectra shown in Fig.s 11, 12 and 13. We note that, despite fulfilling the
criteria of our automated fitting analysis, some of the Gaussian fits look
‘dubious’ upon visual inspection. In particular, we consider poor fits those
of the spectra 12 and 26 in the RCS2 032727$-$132623 field and that of
spectrum 36 in the SGAS J1226+2152 field. To assess how much these ‘dubious’
spectra influence our results, we ran an additional Bayesyan analysis (Sect.
3.5), excluding the EWs coming from the three spectra mentioned above. The
resulting posterior distribution has a median $C_{\rm{length}}$ equal to $4.1\
\rm{kpc}$ and the 2$\sigma$ limits are equal to 1.3 and 7.7 kpc, therefore
perfectly consistent with our fiducial findings. We conclude that this (small)
set of uncertain fits does not affect the results of this paper.

Table 3: Absorption line measurements for PSZ1 G311.65$-$18.48. Id. | $\Delta\alpha$ | $\Delta\delta$ | $R$ | EW | $\sigma_{\rm{EW}}$
---|---|---|---|---|---
| (arcsec) | (arcsec) | (kpc) | (Å) | (Å)
1 | -7.29 | -8.94 | 43 | 0.22 | 0.09
2 | -4.89 | -5.94 | 28 | 0.67 | 0.17
3 | -4.29 | -5.34 | 25 | 0.23 | 0.1
4 | -4.89 | -4.74 | 25 | 0.78 | 0.22
5 | -2.49 | -4.14 | 18 | 1.5 | 0.6
6 | -3.09 | -4.14 | 19 | 0.74 | 0.24
7 | -3.69 | -4.14 | 20 | 0.56 | 0.13
8 | -4.29 | -4.14 | 22 | 0.74 | 0.3
9 | -2.49 | -3.54 | 16 | 1.23 | 0.21
10 | -3.09 | -3.54 | 17 | 1.11 | 0.11
11 | -3.69 | -3.54 | 19 | 0.84 | 0.17
12 | -1.89 | -2.94 | 13 | 1.02 | 0.39
13 | -2.49 | -2.94 | 14 | 0.94 | 0.15
14 | -3.09 | -2.94 | 16 | 1.09 | 0.15
15 | -3.69 | -2.94 | 17 | 0.83 | 0.28
16 | -1.89 | -2.34 | 11 | 1.05 | 0.28
17 | -2.49 | -2.34 | 13 | 1.56 | 0.31
18 | -1.29 | -1.74 | 8 | 1.11 | 0.32
19 | -1.89 | -1.74 | 9 | 1.32 | 0.56
20 | 0.51 | -1.14 | 7 | 3.73 | 1.03
21 | -0.09 | -1.14 | 6 | 2.68 | 0.29
22 | -0.69 | -1.14 | 5 | 2.53 | 0.21
23 | -1.29 | -1.14 | 6 | 1.99 | 0.34
24 | 0.51 | -0.54 | 5 | 1.76 | 0.45
25 | -0.09 | -0.54 | 3 | 2.19 | 0.26
26 | -0.69 | -0.54 | 3 | 1.61 | 0.24
27 | -0.09 | 0.06 | 0 | 2.04 | 0.45
28 | -0.69 | 0.06 | 3 | 1.34 | 0.48
29 | 6.51 | 4.26 | 29 | 0.35 | 0.11
Table 4: Same as Table LABEL:table_EWs_pks, for RCS2 032727$-$132623. Id. | $\Delta\alpha$ | $\Delta\delta$ | $R$ | EW | $\sigma_{\rm{EW}}$
---|---|---|---|---|---
| (arcsec) | (arcsec) | (kpc) | (Å) | (Å)
1 | 2.77 | -11.09 | 52 | 1.89 | 0.69
2 | 3.37 | -10.49 | 50 | 0.61 | 0.26
3 | 3.97 | -9.89 | 47 | 0.69 | 0.23
4 | 3.97 | -9.29 | 45 | 0.32 | 0.15
5 | 3.37 | -9.29 | 45 | 0.59 | 0.19
6 | 3.37 | -6.89 | 35 | 0.88 | 0.38
7 | 2.77 | -6.89 | 36 | 0.37 | 0.14
8 | 0.37 | -5.69 | 33 | 1.05 | 0.4
9 | 0.97 | -5.09 | 29 | 0.62 | 0.22
10 | 0.37 | -5.09 | 30 | 0.4 | 0.17
11 | 0.37 | -4.49 | 27 | 0.76 | 0.27
12 | -0.83 | -4.49 | 29 | 1.72 | 0.69
13 | -1.43 | -3.29 | 24 | 0.54 | 0.15
14 | -2.03 | -3.29 | 25 | 0.73 | 0.21
15 | -1.43 | -2.69 | 20 | 0.9 | 0.15
16 | -2.03 | -2.69 | 22 | 0.94 | 0.09
17 | -2.63 | -2.69 | 24 | 0.95 | 0.23
18 | -1.43 | -2.09 | 17 | 1.11 | 0.46
19 | -2.03 | -2.09 | 19 | 1.4 | 0.26
20 | -2.63 | -2.09 | 21 | 1.74 | 0.31
21 | -3.23 | -2.09 | 23 | 1.25 | 0.37
22 | -4.43 | -2.09 | 27 | 1.04 | 0.36
23 | -4.43 | -1.49 | 25 | 1.01 | 0.18
24 | -5.03 | -1.49 | 27 | 0.63 | 0.24
25 | -5.63 | -0.89 | 27 | 0.78 | 0.36
26 | -9.83 | -0.89 | 44 | 1.57 | 0.72
Table 5: Same as Table LABEL:table_EWs_pks, for SGAS J1226+2152. Id. | $\Delta\alpha$ | $\Delta\delta$ | $R$ | EW | $\sigma_{\rm{EW}}$
---|---|---|---|---|---
| (arcsec) | (arcsec) | (kpc) | (Å) | (Å)
1 | 2.15 | -3.08 | 17 | 0.78 | 0.31
2 | 1.55 | -3.08 | 17 | 0.64 | 0.29
3 | 0.95 | -3.08 | 17 | 0.92 | 0.41
4 | -0.85 | -2.48 | 15 | 1.89 | 0.48
5 | -1.45 | -2.48 | 16 | 2.37 | 0.77
6 | 0.35 | -0.08 | 1 | 1.18 | 0.46
7 | -0.85 | -0.08 | 4 | 2.19 | 0.64
8 | 0.35 | 0.52 | 3 | 2.11 | 0.64
9 | -0.25 | 0.52 | 3 | 1.96 | 0.76
10 | -0.85 | 0.52 | 4 | 3.0 | 0.79
11 | 0.95 | 1.72 | 10 | 0.91 | 0.3
12 | 0.35 | 1.72 | 9 | 1.05 | 0.19
13 | -0.25 | 1.72 | 8 | 1.13 | 0.25
14 | -0.85 | 1.72 | 8 | 1.06 | 0.29
15 | 0.95 | 2.32 | 13 | 1.15 | 0.2
16 | 0.35 | 2.32 | 12 | 1.14 | 0.12
17 | -0.25 | 2.32 | 11 | 1.37 | 0.1
18 | -0.85 | 2.32 | 11 | 1.62 | 0.14
19 | -1.45 | 2.32 | 12 | 2.28 | 0.35
20 | -2.05 | 2.32 | 13 | 2.3 | 0.77
21 | 0.95 | 2.92 | 15 | 1.04 | 0.21
22 | 0.35 | 2.92 | 14 | 1.3 | 0.15
23 | -0.25 | 2.92 | 14 | 1.16 | 0.08
24 | -0.85 | 2.92 | 14 | 1.27 | 0.1
25 | -1.45 | 2.92 | 14 | 1.2 | 0.12
26 | -2.05 | 2.92 | 15 | 1.87 | 0.4
27 | 0.95 | 3.52 | 18 | 1.25 | 0.4
28 | -0.25 | 3.52 | 16 | 1.11 | 0.15
29 | -0.85 | 3.52 | 16 | 1.32 | 0.13
30 | -1.45 | 3.52 | 17 | 1.19 | 0.14
31 | -2.05 | 3.52 | 17 | 1.92 | 0.31
32 | -0.25 | 4.12 | 19 | 1.56 | 0.42
33 | -0.85 | 4.12 | 19 | 0.85 | 0.23
34 | -1.45 | 4.12 | 19 | 1.37 | 0.27
35 | -2.05 | 4.12 | 19 | 1.01 | 0.38
36 | -1.45 | 4.72 | 21 | 4.2 | 1.27
Figure 11: Results of the fitting analysis. Each panel shows a different
spaxel (numbered arbitrarily) with Mg ii detections in the PSZ1
G311.65$-$18.48 field. The normalized MUSE spectra are shown in green
(uncertainties in light-blue). The Gaussian fits of the Mg ii doublet are
reported in red. Figure 12: Same as Fig. 11, for the RCS2 032727$-$132623
field. Figure 13: Same as Fig. 11, for the SGAS J1226+2152 field. An
additional Gaussian that accounts for the Si ii $\lambda$1260 absorption is
fitted.