11institutetext: Institut de Recherche en Astrophysique et Planétologie
(IRAP), Université de Toulouse, CNRS, UPS, CNES, 31400 Toulouse, France
22institutetext: Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France
33institutetext: Canada-France-Hawaii Telescope, CNRS, 96743 Kamuela, Hawaii,
USA 44institutetext: Gemini Observatory, NSF’s NOIRLab, 670 N. A’ohoku Place,
Hilo, Hawaii, 96720, USA 55institutetext: Space Telescope Science Institute
3700 San Martin Drive Baltimore, MD 21218, USA 66institutetext: Leibniz-
Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, D-14482 Potsdam,
Germany 77institutetext: Max Planck Institute for Astronomy, Königstuhl 17,
D-69117 Heidelberg, Germany 88institutetext: UCO/Lick Observatory, Department
of Astronomy & Astrophysics, UC Santa Cruz, 1156 High Street, Santa Cruz, CA
95064, USA

# Stellar angular momentum of disk galaxies at $z\approx 0.7$ in the MAGIC
survey††thanks: Table 4 is only available in electronic form at the CDS via
anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/,††thanks: Based on observations
made with ESO telescopes at the Paranal Observatory under programs 094.A-0247,
095.A-0118, 096.A-0596, 097.A-0254, 099.A-0246, 100.A-0607, 101.A-0282,
102.A-0327, and 103.A-0563.

I. Impact of the environment

W. Mercier, Stellar angular momentum of disk galaxies at $z\approx 0.7$ in the
MAGIC survey††thanks: Table 4 is only available in electronic form at the CDS
via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/,††thanks: Based on observations
made with ESO telescopes at the Paranal Observatory under programs 094.A-0247,
095.A-0118, 096.A-0596, 097.A-0254, 099.A-0246, 100.A-0607, 101.A-0282,
102.A-0327, and 103.A-0563.Stellar angular momentum of disk galaxies at
$z\approx 0.7$ in the MAGIC survey††thanks: Table 4 is only available in
electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/,††thanks:
Based on observations made with ESO telescopes at the Paranal Observatory
under programs 094.A-0247, 095.A-0118, 096.A-0596, 097.A-0254, 099.A-0246,
100.A-0607, 101.A-0282, 102.A-0327, and 103.A-0563<EMAIL_ADDRESS>B.
Epinat Stellar angular momentum of disk galaxies at $z\approx 0.7$ in the
MAGIC survey††thanks: Table 4 is only available in electronic form at the CDS
via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/,††thanks: Based on observations
made with ESO telescopes at the Paranal Observatory under programs 094.A-0247,
095.A-0118, 096.A-0596, 097.A-0254, 099.A-0246, 100.A-0607, 101.A-0282,
102.A-0327, and 103.A-0563.Stellar angular momentum of disk galaxies at
$z\approx 0.7$ in the MAGIC survey††thanks: Table 4 is only available in
electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/,††thanks:
Based on observations made with ESO telescopes at the Paranal Observatory
under programs 094.A-0247, 095.A-0118, 096.A-0596, 097.A-0254, 099.A-0246,
100.A-0607, 101.A-0282, 102.A-0327, and 103.A-0563.Stellar angular momentum of
disk galaxies at $z\approx 0.7$ in the MAGIC survey††thanks: Table 4 is only
available in electronic form at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-
bin/qcat?J/A+A/,††thanks: Based on observations made with ESO telescopes at
the Paranal Observatory under programs 094.A-0247, 095.A-0118, 096.A-0596,
097.A-0254, 099.A-0246, 100.A-0607, 101.A-0282, 102.A-0327, and
103.A-0563.Stellar angular momentum of disk galaxies at $z\approx 0.7$ in the
MAGIC survey††thanks: Table 4 is only available in electronic form at the CDS
via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/,††thanks: Based on observations
made with ESO telescopes at the Paranal Observatory under programs 094.A-0247,
095.A-0118, 096.A-0596, 097.A-0254, 099.A-0246, 100.A-0607, 101.A-0282,
102.A-0327, and 103.A-0563. T. Contini Stellar angular momentum of disk
galaxies at $z\approx 0.7$ in the MAGIC survey††thanks: Table 4 is only
available in electronic form at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-
bin/qcat?J/A+A/,††thanks: Based on observations made with ESO telescopes at
the Paranal Observatory under programs 094.A-0247, 095.A-0118, 096.A-0596,
097.A-0254, 099.A-0246, 100.A-0607, 101.A-0282, 102.A-0327, and
103.A-0563.Stellar angular momentum of disk galaxies at $z\approx 0.7$ in the
MAGIC survey††thanks: Table 4 is only available in electronic form at the CDS
via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/,††thanks: Based on observations
made with ESO telescopes at the Paranal Observatory under programs 094.A-0247,
095.A-0118, 096.A-0596, 097.A-0254, 099.A-0246, 100.A-0607, 101.A-0282,
102.A-0327, and 103.A-0563. D. Krajnović Stellar angular momentum of disk
galaxies at $z\approx 0.7$ in the MAGIC survey††thanks: Table 4 is only
available in electronic form at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-
bin/qcat?J/A+A/,††thanks: Based on observations made with ESO telescopes at
the Paranal Observatory under programs 094.A-0247, 095.A-0118, 096.A-0596,
097.A-0254, 099.A-0246, 100.A-0607, 101.A-0282, 102.A-0327, and
103.A-0563.Stellar angular momentum of disk galaxies at $z\approx 0.7$ in the
MAGIC survey††thanks: Table 4 is only available in electronic form at the CDS
via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/,††thanks: Based on observations
made with ESO telescopes at the Paranal Observatory under programs 094.A-0247,
095.A-0118, 096.A-0596, 097.A-0254, 099.A-0246, 100.A-0607, 101.A-0282,
102.A-0327, and 103.A-0563. L. Ciesla Stellar angular momentum of disk
galaxies at $z\approx 0.7$ in the MAGIC survey††thanks: Table 4 is only
available in electronic form at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-
bin/qcat?J/A+A/,††thanks: Based on observations made with ESO telescopes at
the Paranal Observatory under programs 094.A-0247, 095.A-0118, 096.A-0596,
097.A-0254, 099.A-0246, 100.A-0607, 101.A-0282, 102.A-0327, and
103.A-0563.Stellar angular momentum of disk galaxies at $z\approx 0.7$ in the
MAGIC survey††thanks: Table 4 is only available in electronic form at the CDS
via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/,††thanks: Based on observations
made with ESO telescopes at the Paranal Observatory under programs 094.A-0247,
095.A-0118, 096.A-0596, 097.A-0254, 099.A-0246, 100.A-0607, 101.A-0282,
102.A-0327, and 103.A-0563. B. C. Lemaux V. Abril-Melgarejo Stellar angular
momentum of disk galaxies at $z\approx 0.7$ in the MAGIC survey††thanks: Table
4 is only available in electronic form at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-
bin/qcat?J/A+A/,††thanks: Based on observations made with ESO telescopes at
the Paranal Observatory under programs 094.A-0247, 095.A-0118, 096.A-0596,
097.A-0254, 099.A-0246, 100.A-0607, 101.A-0282, 102.A-0327, and
103.A-0563.Stellar angular momentum of disk galaxies at $z\approx 0.7$ in the
MAGIC survey††thanks: Table 4 is only available in electronic form at the CDS
via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/,††thanks: Based on observations
made with ESO telescopes at the Paranal Observatory under programs 094.A-0247,
095.A-0118, 096.A-0596, 097.A-0254, 099.A-0246, 100.A-0607, 101.A-0282,
102.A-0327, and 103.A-0563.Stellar angular momentum of disk galaxies at
$z\approx 0.7$ in the MAGIC survey††thanks: Table 4 is only available in
electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/,††thanks:
Based on observations made with ESO telescopes at the Paranal Observatory
under programs 094.A-0247, 095.A-0118, 096.A-0596, 097.A-0254, 099.A-0246,
100.A-0607, 101.A-0282, 102.A-0327, and 103.A-0563.Stellar angular momentum of
disk galaxies at $z\approx 0.7$ in the MAGIC survey††thanks: Table 4 is only
available in electronic form at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-
bin/qcat?J/A+A/,††thanks: Based on observations made with ESO telescopes at
the Paranal Observatory under programs 094.A-0247, 095.A-0118, 096.A-0596,
097.A-0254, 099.A-0246, 100.A-0607, 101.A-0282, 102.A-0327, and 103.A-0563. L.
Boogaard Stellar angular momentum of disk galaxies at $z\approx 0.7$ in the
MAGIC survey††thanks: Table 4 is only available in electronic form at the CDS
via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/,††thanks: Based on observations
made with ESO telescopes at the Paranal Observatory under programs 094.A-0247,
095.A-0118, 096.A-0596, 097.A-0254, 099.A-0246, 100.A-0607, 101.A-0282,
102.A-0327, and 103.A-0563.Stellar angular momentum of disk galaxies at
$z\approx 0.7$ in the MAGIC survey††thanks: Table 4 is only available in
electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/,††thanks:
Based on observations made with ESO telescopes at the Paranal Observatory
under programs 094.A-0247, 095.A-0118, 096.A-0596, 097.A-0254, 099.A-0246,
100.A-0607, 101.A-0282, 102.A-0327, and 103.A-0563. D. Pelliccia Stellar
angular momentum of disk galaxies at $z\approx 0.7$ in the MAGIC
survey††thanks: Table 4 is only available in electronic form at the CDS via
anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/,††thanks: Based on observations
made with ESO telescopes at the Paranal Observatory under programs 094.A-0247,
095.A-0118, 096.A-0596, 097.A-0254, 099.A-0246, 100.A-0607, 101.A-0282,
102.A-0327, and 103.A-0563.Stellar angular momentum of disk galaxies at
$z\approx 0.7$ in the MAGIC survey††thanks: Table 4 is only available in
electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/,††thanks:
Based on observations made with ESO telescopes at the Paranal Observatory
under programs 094.A-0247, 095.A-0118, 096.A-0596, 097.A-0254, 099.A-0246,
100.A-0607, 101.A-0282, 102.A-0327, and 103.A-0563.

(Received ; accepted )

###### Abstract

Aims. At intermediate redshift, galaxy groups/clusters are thought to impact
galaxy properties such as their angular momentum. We investigate whether the
environment has an impact on the galaxies’ stellar angular momentum and
identify underlying driving physical mechanisms.

Methods. We derive robust estimates of the stellar angular momentum using
Hubble Space Telescope (HST) images combined with spatially resolved ionised
gas kinematics from the Multi-Unit Spectroscopic Explorer (MUSE) for a sample
of $\sim 200$ galaxies in groups and in the field at $z\sim 0.7$ drawn from
the MAGIC survey. Using various environmental tracers, we study the position
of the galaxies in the the angular momentum-stellar mass (Fall) relation as a
function of environment.

Results. We measure a $0.12\text{\,}\mathrm{d}\mathrm{e}\mathrm{x}$ ($2\sigma$
significant) depletion of stellar angular momentum for low-mass galaxies
($M_{\star}<${10}^{10}\text{\,}\mathrm{M_{\sun}}$$) located in groups with
respect to the field. Massive galaxies located in dense environments have less
angular momentum than expected from the low-mass Fall relation but, without a
comparable field sample, we cannot infer whether this effect is mass- or
environmentally-driven. Furthermore, these massive galaxies are found in the
central parts of the structures and have low systemic velocities.

The observed depletion of angular momentum at low stellar mass does not appear
linked with the strength of the over-density around the galaxies but it is
strongly correlated with (i) the systemic velocity of the galaxies normalised
by the dispersion of their host group and (ii) their ionised gas velocity
dispersion.

Conclusions. Galaxies in groups appear depleted in angular momentum,
especially at low stellar mass. Our results suggest that this depletion might
be induced by physical mechanisms that scale with the systemic velocity of the
galaxies (e.g. stripping or merging) and that such mechanism might be
responsible for enhancing the velocity dispersion of the gas as galaxies lose
angular momentum.

###### Key Words.:

Galaxies: evolution – Galaxies: kinematics and dynamics – Galaxies: clusters:
general – Galaxies: groups – Galaxies: high-redshift – Galaxies: fundamental
parameters

##  1 Introduction

In the current paradigm of galaxy evolution, galaxies are expected to form
through the condensation of baryons in the centres of dark matter (DM) haloes
where, because of external tidal torques, the gas in the proto-galaxy acquires
angular momentum before condensing into a disk and forming stars (e.g. Peebles
1969; Fall & Efstathiou 1980). The same is true for DM haloes that also
acquire angular momentum during their linear phase of structure growth (e.g.
Doroshkevich 1970; White 1984). Initially, it was thought that the angular
momentum of the baryons (e.g. stars or gas) traces that of the DM component.
However, recent simulations have shown that this picture is not entirely
correct as processes exist that can either add or remove angular momentum from
either component independently of the halo and proto-galaxy early formation
phase (e.g. Genel et al. 2015). For instance, there is strong evidence
supporting galaxies having smoothly accreted large amounts of cold gas from
their circum-galactic medium to sustain high values of star formation rate
(SFR) across cosmic time (e.g. Bouché et al. 2013, 2016; Zabl et al. 2019).
This accretion of fresh gas is thought to take place predominantly in the disk
plane of late-type galaxies and, thus, does not only drive their star forming
events throughout their formation history but also increases the angular
momentum associated with their baryonic component (e.g. Danovich et al. 2015;
Cadiou et al. 2022). Similarly, feedback processes such as galactic winds have
also been found as potential mechanisms to increase the angular momentum of
the baryons in a galaxy (e.g. DeFelippis et al. 2017). In addition, galaxy
mergers can also substantially redistribute angular momentum by increasing
that of the DM halo (e.g. Hetznecker & Burkert 2006) and by redistributing the
mass, thus, either decrease or increase the angular momentum of the baryons
depending on the spins and mass ratio of the two galaxies (e.g. Bois et al.
2011; Genel et al. 2015; Lagos et al. 2018)

Numerous studies have tried to constrain the angular momentum of the stellar
and gas components in galaxies at various redshifts since the seminal work of
Fall (1983) by studying the angular momentum-stellar mass relation, also known
as the Fall relation. In the local Universe, some studies have focussed on the
shape of the relation as a function of galaxy type (e.g. Fall 1983; Romanowsky
& Fall 2012; Cortese et al. 2016; Rizzo et al. 2018), surface brightness (e.g.
Salinas & Galaz 2021), stellar mass (e.g. Posti et al. 2018; Di Teodoro et al.
2021), cold gas fraction (e.g. Mancera Piña et al. 2021b; Kurapati et al.
2021), or bulge fraction (e.g. Obreschkow & Glazebrook 2014; Fall & Romanowsky
2018). The general conclusion that can be drawn is that, in the local
Universe, galaxies exhibit a linear relation (in log-log space) between
stellar mass and angular momentum otherwise known as the Fall relation (i.e.
no deviation at low or high stellar masses). The scatter in the relation seems
to be mainly correlated with the galaxies’ morphology, with elliptical and
disk galaxies with prominent bulges located below the relation found for
bulge-less disks. The Fall relation has also been studied at higher redshift
($z\sim 1-2$, e.g. Burkert et al. 2016; Swinbank et al. 2017; Harrison et al.
2017; Bouché et al. 2021) where it is found to hold but with a lower zero-
point that is consistent with galaxies gaining angular momentum at fixed
stellar mass with cosmic time (i.e. with decreasing redshift). Interestingly,
the recent analysis by Bouché et al. (2021) hinted at a correlation between
the zero-point of the Fall relation and the dynamical state of galaxies in the
sense that dispersion dominated systems have lower SFR-weighted angular
momentum than rotationally supported galaxies at fixed stellar mass. This
correlation was already observed in a few previous studies (e.g. Contini et
al. 2016; Burkert et al. 2016) and it helps to explain the separation in the
Fall relation between spirals and ellipticals seen at $z=0$ (e.g. Romanowsky &
Fall 2012).

From a methodological perspective, measuring the angular momentum in observed
galaxies is not straightforward as it would require to know the full mass
distribution (including stars, gas, and DM) as well as the amplitude and
orientation of the velocity vector of the component of interest (e.g. stars
for the stellar angular momentum) at each position in the galaxy. Hence, given
the impossibility to measure the intrinsic gas or stellar angular momentum of
the galaxies, proxies have been used in the literature that rely on various
assumptions. A common proxy is to assume that the gas and the stars are
located in a disk that is dynamically stable against its own gravity through
its rotation. With a few additional assumptions on the stellar or gas
kinematics and on the mass distribution of the stellar and gas disks, it is
possible to derive the Romanowsky & Fall (2012, hereafter RF12) approximation
that is widely used in the literature (e.g. Romanowsky & Fall 2012; Contini et
al. 2016; Burkert et al. 2016; Swinbank et al. 2017; Rizzo et al. 2018).
Though already mentioned in Romanowsky & Fall (2012) and in subsequent
studies, a recent discussion was given in Bouché et al. (2021) where it was
shown that, depending on the radius at which the angular momentum is measured,
the RF12 approximation can overestimate the galaxies’ stellar or gas angular
momentum by nearly 20%. Furthermore, thanks to the increasing number of robust
kinematic measurements for the last ten years, some authors have started using
other estimates that rely on fewer assumptions. For instance, some studies
(e.g. Posti et al. 2018; Bouché et al. 2021; Mancera Piña et al. 2021a, b)
assumed an axisymmetric disk model (typically exponential) and either
numerically integrated the analytical expression of the angular momentum for a
given rotation curve model or substituted the integral with a sum on the
pixels of a galaxy’s image (e.g. Rizzo et al. 2018). Similarly, Cortese et al.
(2016) proposed another method that numerically integrates the angular
momentum by summing the contribution of each spatial pixel (spaxel) in a data
cube. The main advantage is that it alleviates the assumption of axial
symmetry since the observed flux distribution is directly taken into account.
However, its main drawback is that it suffers from the usually poor spatial
resolution of 3D spectroscopic observations compared to high spatial
resolution images obtained, for instance, by the Hubble Space Telescope (HST).
Other authors have also implemented more complex methods that combine semi-
analytical models with N-body simulations to derive the angular momentum of
both the baryonic and DM components of local galaxies (Ansar et al. 2023).

Finally, a few studies have tried to probe the impact of the environment on
the galaxies’ angular momentum (e.g. Pelliccia et al. 2019; Pérez-Martínez et
al. 2021). The current picture that emerges from these studies is that
galaxies found in galaxy groups and galaxy clusters at $z\sim 1$ seem to have
a deficit of angular momentum with respect to field galaxies located at the
same redshift. Because physical mechanisms can affect the angular momentum of
the baryons in different ways (i.e. increase or decrease it), an
interpretation given in Pelliccia et al. (2019) is that this reduction of
angular momentum at fixed stellar mass could be due to galaxy mergers, in line
with their prevalence in dense environments (e.g. Tomczak et al. 2019) and
with recent simulations (e.g. Lagos et al. 2018). However, these results
required to compare galaxies from different surveys and observed with various
instruments (e.g. Pérez-Martínez et al. 2021) to reach these conclusions. As
in Abril-Melgarejo et al. (2021) and Mercier et al. (2022), we argue that
there might be systematic effects when doing so, mainly driven by different
selection functions between different datasets.

This paper is the first of a series. In this analysis, we propose to study the
impact of the environment on the stellar angular momentum of galaxies in the
MUSE-gAlaxy Groups In Cosmos (MAGIC) survey (Epinat et al., in prep.). As
shown in Mercier et al. (2022), this survey is ideal to probe the impact of
the environment on galaxy dynamics at $z\sim 1$ because it allows for the
simultaneous observation of galaxies located in structures of various masses
(mostly galaxy groups) and foreground/background galaxies in more rarefied
environments, as well as to analyse them in a consistent manner. In the second
paper of the series, we will investigate how different physical mechanisms can
affect galaxies across the Fall relation as a function of their stellar mass,
morphology, and dynamical properties.

The structure of the paper is as follows. In Sect. 2, we give a brief
description of the MAGIC sample, the HST and MUSE observations, how we
performed the morphological and kinematics modellings, and how we
characterised the galaxies’ environment. More importantly, we also describe
the sample selection used to study the stellar angular momentum. In Sect. 3,
we describe the method used to derive the angular momentum with HST images,
and in Sect. 4 we assess the reliability of the method. Afterwards, we perform
the analysis of the Fall relation in Sect. 5 and we conclude in Sect. 6.
Throughout this paper, we assume a $\Lambda$ cold dark matter cosmology with
${\rm{H}}_{0}=$70\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{M\text{pc}}}^{-1}$$,
$\Omega_{\rm{M}}=0.3$, and $\Omega_{\Lambda}=0.7$.

##  2 Sample selection and main properties

The sample used in this analysis is part of the MAGIC survey (Epinat et al.,
in prep.), a deep Multi-Unit Spectroscopic Explorer (MUSE) Guaranteed Time
Observation (GTO) survey targetting 14 groups in the Cosmic Evolution Survey
(Cosmos) area (Scoville et al. 2007) with 17 different MUSE pointings. The
main goal of this survey is to study the impact of the environment on the
properties of galaxies at intermediate redshift by combining multi-band
photometry, in particular Hubble Space Telescope Advanced Camera for Surveys
(HST-ACS) observations (Koekemoer et al. 2007; Massey et al. 2010), with
spatially resolved spectroscopic properties from MUSE. The analysis performed
in this paper is the continuation of two previous ones. In the first paper
(Abril-Melgarejo et al. 2021), we studied the impact of the galaxies’
environment on the Tully-Fisher Relation (TFR) for a subsample of galaxies
located in dense groups. In the second paper (Mercier et al. 2022), we
focussed our analysis on the impact of the galaxies’ environment on three
major galaxy scaling relations (size-mass, Main Sequence - MS, and TFR) for
the full MAGIC sample. This latter work was carried out by comparing galaxies
located in the field with galaxies found in groups with a large dynamical
range of density, all observed in MAGIC. A summary of the survey is presented
below and for a complete description, see Epinat et al. (in prep.).

###  2.1 Observations and physical parameters

Observations are split in 17 different MUSE pointings, resulting in data and
variance cubes that range from $4750\text{\,}\mathrm{\SIUnitSymbolAngstrom}$
to $9350\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ with a spatial sampling of
$0.2\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ and a spectral sampling of
$1.25\text{\,}\mathrm{\SIUnitSymbolAngstrom}$. The observing strategy and data
reduction can be found in Epinat et al. (in prep., but see also Mercier et al.
2022 and Abril-Melgarejo et al. 2021). The MUSE Line Spread Function (LSF) was
modelled with a second order polynomial function as in Bacon et al. (2017) and
Guérou et al. (2017) and the MUSE Point Spread Function (PSF) was modelled by
extracting $100\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ wide narrow-band
images of stars in each MUSE field and by fitting them with a Moffat profile
with free $\beta$ index. The wavelength dependence of the PSF Full Width at
Half Maximum (FWHM) was then derived field-by-field by fitting a linear
relation to the median curve (FWHM vs. wavelength) and the $\beta$ parameter
as the mean value weighted by the uncertainties. The median value of the MUSE
PSF FWHM for the 17 fields is $0.67\text{\,}\mathrm{\SIUnitSymbolArcsecond}$
at $4000\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ and
$0.53\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ at
$8000\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ which correspond respectively to
$4.8\text{\,}\mathrm{k\text{pc}}$ and $3.8\text{\,}\mathrm{k\text{pc}}$ at
$z=0.7$. In addition, we also used high-resolution
$$4\text{\,}\mathrm{\SIUnitSymbolArcsecond}$\times$4\text{\,}\mathrm{\SIUnitSymbolArcsecond}$$
HST stamps in the F814W filter111This is the best HST spatial resolution
available in the Cosmos field (PSF FWHM below
$0.1\text{\,}\mathrm{\SIUnitSymbolArcsecond}$, pixel scale of
$0.03\text{\,}\mathrm{\SIUnitSymbolArcsecond}$). to model the galaxies’
morphology. The HST PSF profile was measured in Abril-Melgarejo et al. (2021)
by fitting a Moffat profile onto 27 non saturated stars found in the MUSE
fields and adopting the median values for the PSF parameters. The PSF
parameters used for the morphological modelling are
FWHM${}_{\rm{HST}}=$0.0852\text{\,}\mathrm{\SIUnitSymbolArcsecond}$$ and
$\beta=1.9$.

Additionally, we also estimated the galaxies’ stellar mass and SFR values with
Spectral Energy Distribution (SED) fitting using the Code Investigating GALaxy
Emission (Cigale, see Boquien et al. 2019) by fixing the redshift of the
galaxies to their MUSE spectroscopic redshift and using Cosmos2020 catalogue
of Weaver et al. (2022). We used Bruzual & Charlot (2003) single stellar
populations with a Salpeter (1955) initial mass function (IMF) and a single
metallicity value of $0.02\text{\,}\mathrm{\text{dex}}$, a truncated delayed
exponential star formation history (SFH, described in Ciesla et al. 2018,
2021), and a Charlot & Fall (2000) attenuation law with a total-to-selective
extinction ratio $R_{V}=3.1$. More details about the grid of parameters and
the choice of models can be found in Epinat et al. (in prep.). As an
indication, we note that different IMF (Chabrier 2003) and SFH (exponentially
declining) models were used in Abril-Melgarejo et al. (2021) and Mercier et
al. (2022). In particular, this affects the stellar mass of the galaxies that
will be used when investigating the impact of the environment on the galaxies’
angular momentum. A comparison between these different models shows overall
consistent values within $0.5\text{\,}\mathrm{\text{dex}}$ for galaxies more
massive than ${10}^{8}\text{\,}\mathrm{M_{\sun}}$. Furthermore, the latest
Cigale-based stellar masses are slightly offset by roughly
$0.05\text{\,}\mathrm{\text{dex}}$ from the previous values which can be
accounted for by the use of a Salpeter (1955) IMF instead of a Chabrier (2003)
one, as previously done.

###  2.2 Morpho-dynamical modelling

In Mercier et al. (2022), we carried out a dynamical modelling of the entire
MAGIC survey, focussing on galaxies at $0.2<z<1.5$, a redshift range where the
[O ii] doublet can theoretically be detected given the wavelength coverage of
our MUSE observations. Galaxies for which the [O ii] doublet is indeed
detected form what we refer to as the kinematics sample (since we can use
their [O ii] doublet to extract their ionised gas kinematics). For a complete
description of the morphological and kinematics modellings, see Sect. 4.1 and
5.1 of Mercier et al. (2022). In what follows, a quick summary is presented.

First, we modelled the morphology of 1142 galaxies in the redshift range
$0.2<z<1.5$ detected in MAGIC. For each galaxy, we performed a bulge-disk
decompositions with Galfit (Peng et al. 2002) using an exponential disk model
for the stellar disk and a circular de Vaucouleurs profile for the bulge. From
this initial sample, 890 galaxies could be reliably modelled (see Sect. 4 of
Mercier et al. 2022 for more details). For each galaxy, various morphological
parameters were derived, including:  (i) their global effective radius
$R_{\textit{eff}}$ (i.e. taking into account the mass distribution of both the
disk and bulge components, see Eq. 6 of Mercier et al. 2022),  (ii) the bulge-
to-total flux ratio (B/T) evaluated at one global effective radius,  (iii) the
disk major axis position angle (PA), and  (iv) the apparent disk axis ratio
$q=b/a$, with $a$ and $b$ defined as the major and minor axes, respectively.
Furthermore, we introduced two expressions to correct the central surface
brightness and the observed axis ratio $q$ of the stellar disk for its non-
zero thickness, assuming an intrinsic double exponential 3D mass distribution
(see Sect. 4.4 and Appendix D.4 of Mercier et al. 2022 for more details). In
addition, because the rotation velocity that was used for the TFR in Abril-
Melgarejo et al. (2021) and Mercier et al. (2022) was measured at the radius
$R_{22}=2.2R_{\rm{d}}$ in the plane of the disk, with $R_{\rm{d}}$ defined as
the disk scale length, we also introduced a corrected stellar mass
($M_{\star,\rm{corr}}$) so that it is also evaluated at $R_{22}$ (see Sect.
4.3 of Mercier et al. 2022). The same morphological models as in Mercier et
al. (2022) have been used to derive the stellar angular momentum of the
galaxies in what follows, except for 17 galaxies for which the morphology was
updated (mainly to take into account the impact of bars and spiral arms on the
disk parameters, see Appendix C for a full description of how the models
changed for these galaxies). Then, we extracted the kinematics of the ionised
gas component in the galaxies using the [O ii] doublet as kinematics tracer.
We fitted in each spaxel the [O ii] doublet using
Camel222https://gitlab.lam.fr/bepinat/CAMEL and then we cleaned the maps to
remove isolated spaxels and those with large velocity discontinuities with
respect to their neighbours (see Sect. 3.5 of Abril-Melgarejo et al. 2021 or
Sect. 5.1 of Mercier et al. 2022 for a complete description). This led to the
removal of 271 galaxies that had no remaining spaxels in their kinematics maps
with ${\rm{S/N}}>5$ in the [O ii] doublet. We note that we did not extract the
stellar kinematics in MAGIC that would only be available for a small sub-
sample of relatively bright/massive galaxies. Therefore, in what follows, we
only use the ionised gas kinematics as a proxy of the stellar kinematics to
estimate the stellar angular momentum.

Finally, we fitted the velocity field of the ionised gas component of the
galaxies using MocKinG333https://gitlab.lam.fr/bepinat/MocKinG. We implemented
for the rotation curve a mass modelling approach comprising three velocity
components:  (i) a double exponential stellar disk (constrained from the
morphology),  (ii) a Hernquist stellar bulge (also constrained from the
morphology), and  (iii) an unconstrained Navarro-Frenk-White (NFW Navarro et
al. 1996) DM halo (see Sect. 5.1 of Mercier et al. 2022 for the details of the
models).  Note that we do not have any constraints on the cold gas components
(i.e. atomic and molecular) of the galaxies in MAGIC. Thus, we did not
introduce any additional velocity components for the cold gas, which means
that in practice the best-fit NFW profile implicitly includes this component
along with the DM halo. We used MocKinG to model the ionised gas velocity
field of the galaxies using the method of line moments (Epinat et al. 2010)
assuming the gas is located in a razor-thin disk. During the fitting process
we fixed the parameters of the stellar disk and bulge components, as well as
the centre position and the disk’s inclination in order to remove
degeneracies. Thus, the only free parameters are:  (i) the DM halo parameters
(scale length $r_{s}$ and maximum velocity $V_{\rm{h,max}}$),  (ii) the
kinematics position angle, and  (iii) the systemic redshift $z_{s}$ of the
galaxies.  Once a best-fit velocity field model is found, MocKinG also derives
a beam-smearing and LSF-corrected ionised gas velocity dispersion map. The gas
kinematics of the galaxies in MAGIC were re-modelled with respect to Mercier
et al. (2022) using updated Moffat MUSE PSF profiles in each MUSE field (see
Epinat et al., in prep.). We also removed four additional galaxies because
they were on the edge of the MUSE field-of-view or had signs of merger in
their gas kinematics maps, leading to a kinematics sample of 571 galaxies.

###  2.3 Environment characterisation

The galaxies’ environment can be characterised in various ways. For instance,
the friends-of-friends (FoF) algorithm used in Knobel et al. (2012) and Iovino
et al. (2016) is easy to implement but it only uses spectroscopic redshifts
while the Voronoi tessellation Monte-Carlo mapping (VMC) technique discussed
in Lemaux et al. (2017, 2022) and Hung et al. (2020, 2021) combines both
photometric and spectroscopic redshifts. Still, every technique has its
limitations as it always remains sensitive to the completeness of redshift
measurements and the size of the field-of-view. The details of the environment
characterisation, including the determination of the groups and the density
estimation can be found in the MAGIC survey paper (Epinat et al., in prep.).
In what follows, we provide a quick description. To begin with, we defined the
groups the galaxies belong to using an iterative 3D FoF algorithm (as in
Abril-Melgarejo et al. 2021 and Mercier et al. 2022). For groups with more
than seven galaxy members, the maximum sky projected separation used was
$375\text{\,}\mathrm{k\text{pc}}$ and the maximum line-of-sight velocity
separation was $500\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$. The
location of the centre of the groups, their systemic redshift, and their
dispersion $\sigma_{V}$ are determined from the distribution of their galaxy
members, and their radius is determined from $\sigma_{V}$. In particular, the
velocity dispersion of the groups is evaluated using the gapper method (see
Beers et al. 1990, Cucciati et al. 2010, or Epinat et al., in prep. for more
details). In this analysis we consider three environmental tracers.

First, we use the richness (i.e. number of galaxy members as given by the FoF
algorithm) of the groups. Because it does not take into account the
distribution of the galaxies in the groups, we use a second estimate defined
as (e.g. Noble et al. 2013; Pelliccia et al. 2019)

$\eta=\frac{R_{\rm{proj}}}{R_{200}}\times\frac{|\Delta v|}{\sigma_{V}},$ (1)

where $R_{\rm{proj}}$ is the projected distance of a galaxy with respect to
the centre of the group, $R_{200}$ is the radius where the density of the
group is equal to 200 times the critical density of the Universe, and $\Delta
v$ is the systemic velocity of a galaxy along the line-of-sight with respect
to its host group’s redshift. Thus, this tracer does not take into account the
distribution of galaxies but only their position in phase-space, providing us
with an idea of how dynamically bound the galaxy is with respect to its host
group.

The last estimate directly probes the over-density in the vicinity of the
galaxies. It relies on the VMC technique (see Lemaux et al. 2017, 2022 and
Hung et al. 2020, 2021) where both photometric and spectroscopic redshifts in
Cosmos are used to estimate the density $\Sigma$ in cells of
$$75\text{\,}\mathrm{k\text{pc}}$\times$75\text{\,}\mathrm{k\text{pc}}$$ and
$\pm$3.75\text{\,}\mathrm{M\text{pc}}$$ wide (proper distances, see also
Epinat et al., in prep.). The over-density $\delta$ is then estimated as
$1+\delta=\Sigma/\Sigma_{\rm{med}}$, where $\Sigma_{\rm{med}}$ is the median
density over the entire map in the same redshift slice as is used to compute
$\Sigma$. For the analysis carried out in Sect. 5, we use the over-density
$\delta$ in order to not be biased by sampling variations from one redshift
slice to another.

###  2.4 Sample selection

Figure 1: Criteria used for the selection of the kinematics sample. The black
points represent galaxies selected according to the surface, S/N, and
inclination (removing face-on galaxies only) criteria. Removed galaxies are
represented as follows and in this specific order: (i) those removed by the
inclination criterion (pink squares), (ii) those removed by the S/N criterion
among the remaining galaxies (orange downward pointing triangles), (iii) those
removed by the surface criterion among the remaining galaxies but that would
have been kept by our previous size criterion used in Mercier et al. (2022,
blue upward pointing triangles), and (iv) those removed by the full selection
and by our previous size criterion as well (grey dots). We also show galaxies
flagged with peculiar kinematics with red contours. The vertical black line
shows the surface selection criterion given in Eq. 2 and the other black lines
represent the S/N selection for different disk axis ratios (see Eq.3).

Because precisely measuring the stellar angular momentum requires robust
estimates of both the morphology and the kinematics of the galaxies, we have
updated our sample selection compared to Mercier et al. (2022). Since the
morphological models were re-inspected for this analysis, instead of applying
a B/T criterion to remove galaxies with loosely constrained disks, we manually
picked those galaxies and removed them from the sample. In particular, this
includes galaxies whose stellar disk component appears noise-dominated in the
HST image (as evaluated from the S/N after removing the best-fit bulge
component from the image; see Sect. 5 of Mercier et al. 2022 for more
details). The five galaxies that were manually removed are 24_CGr32, 29_CGr23,
240_CGr30, 240_CGr84, and 76_CGr172. Because we do not use a B/T criterion,
contrary to Mercier et al. (2022), we will be able in this analysis to study
both disk- and bulge-dominated galaxies. Furthermore, we also decided to apply
a selection criterion on the extent of the galaxies on the plane of the sky to
make sure that there are enough resolution elements in their MUSE kinematics
maps. In this analysis, galaxies are selected if their surface on the plane of
the sky within an ellipse with major axis equal to one disk effective radius
($R_{\textit{eff},\rm{d}}$) and apparent axis ratio ($q$) is larger than the
surface of the MUSE PSF within its FWHM. In other words, the size selection
writes

$2\sqrt{q}\times
R_{\textit{eff},\rm{d}}/{\rm{FWHM}_{[{O\textsc{ii}}]{}}}(z)>1,$ (2)

where FWHM${}_{[{O\textsc{ii}}]{}}(z)$ is the MUSE PSF FWHM evaluated at the
redshift $z$ of the galaxy and where $R_{\textit{eff},\rm{d}}$ and
FWHM${}_{[{O\textsc{ii}}]{}}$ must have the same angular or length unit 444We
remind that the [O ii] PSF FWHM in Eq. 2 was measured in the MUSE cubes
whereas the disk effective radius and apparent axis ratio were both measured
in high resolution HST images.. This criterion naturally excludes galaxies
with small sizes with respect to the MUSE PSF and edge-on galaxies because
their axis ratio is too small. We also removed face-on galaxies by selecting
those with a stellar disk inclination
$i>$25\text{\,}\mathrm{\SIUnitSymbolDegree}$$. This value seemed a good
compromise between removing galaxies that are too face-on to be well
constrained (especially their kinematics) and keeping a sufficiently large
sample to perform the analysis. As an indication a more conservative criterion
of $i>$30\text{\,}\mathrm{\SIUnitSymbolDegree}$$ would only remove eight
additional galaxies. Similarly, we also applied a S/N criterion on the
detected [O ii] doublet in the kinematics maps. We selected galaxies if they
have an average S/N per spaxel of at least eight across the observed disk’s
surface on the plane of the sky within the disk’s effective radius. Because
the PSF smears the stellar disk distribution on the plane of the sky, we
assume that the observed disk’s extent can be written as the quadratic sum of
the intrinsic extent (i.e. $R_{\textit{eff},\rm{d}}$ and
$qR_{\textit{eff},\rm{d}}$ along the major and minor axes, respectively) and
half the PSF FWHM555In other terms, the observed surface of the stellar disk
at an intrinsic distance $R_{\textit{eff},\rm{d}}$ writes
$\pi\sqrt{\left(R_{\textit{eff},\rm{d}}^{2}+({\rm{FWHM}}/2)^{2}\right)\left(q^{2}R_{\textit{eff},\rm{d}}^{2}+({\rm{FWHM}}/2)^{2}\right)}$..
Putting everything together, the S/N selection criterion writes666The
derivation assumes constant flux and S/N maps, as well as Poisson’s statistics
for the noise.

$\frac{({\rm{S/N}})_{\rm{tot}}}{{\rm{FWHM}}_{[{O\textsc{ii}}]{}}(z)}\geq
20\sqrt{\pi}\left[\left(x^{2}+1\right)\left(qx^{2}+1\right)\right]^{1/4},$ (3)

where $x=2R_{\textit{eff},\rm{d}}/{\rm{FWHM}}_{[{O\textsc{ii}}]{}}(z)$, with
${\rm{FWHM}}_{[{O\textsc{ii}}]{}}(z)$ and $R_{\textit{eff},\rm{d}}$ in
arcsecond. Compared to the previous S/N criterion used in Mercier et al.
(2022) that did not take into account the ellipticity, this one adds 30 new
galaxies. However, when including the inclination and surface (size for the
old selection) criteria, it removes 25 galaxies. The new sample selection is
shown in Fig. 1. Selected galaxies are shown in black and those removed by the
selection with other symbols (one symbol per criterion). Because a galaxy can
be removed by different selection criteria, we show them in the following
order:  (i) galaxies removed by the inclination criterion (pink squares),
(ii) among the remaining galaxies, those removed by the S/N criterion (i.e.
Eq. 3; orange downward pointing triangles), and  (iii) among the remaining
galaxies, those removed by the surface criterion (i.e. Eq. 2). As an
indication, we split the latter between galaxies that would have been selected
by the size selection criterion used in Mercier et al. (2022) (blue upward
pointing triangles) and galaxies that that would have been removed by this
size selection criterion (grey dots).  The black vertical line shows the limit
of the surface selection criterion (see Eq. 2) and the other black lines show
various S/N criteria with different disk axis ratios $q$ (see Eq. 3). The
combination of the surface, S/N and inclination criteria yields a sample of
182 galaxies. As an indication, the previous selection used in Mercier et al.
(2022) would have yielded 207 galaxies instead. Among them, those that would
have been added by the size criterion are mostly located at the limit of the
selection (thus quite small) and are significantly inclined
($i\gtrsim$60\text{\,}\mathrm{\SIUnitSymbolDegree}$$).

Finally, we decided to visually inspect the kinematics maps and the rotation
curves of the remaining galaxies and to flag the 30 following galaxies as
having ’peculiar kinematics’777See Table F.1 of Mercier et al. 2022 or Epinat
et al., in prep. for the galaxies’ IDs.:  (i) 38_CGr172, 54_CGr51, 70_CGr79,
74_CGr172, 90_CGr23, 93_CGr114, 101_CGr32, 104_CGr28, 104_CGr172, 105_CGr114,
113_CGr23, 148_CGr30, 185_CGr30, 226_CGr84, 257_CGr84, 267_CGr84, 313_CGr84,
442_CGr32, and 454_CGr32 because there is no visible velocity gradient in
their velocity fields so that their best-fit rotation curve can hardly
describe the intrinsic rotation of the gas,  (ii) 28_CGr26, 96_CGr28, and
172_CGr32 because there is no central peak observed in the HST image as would
be expected for an exponential disk even without a bulge so that the
contribution of their disk component is overestimated in the inner parts,
(iii) 23_CGr84because it has a very massive disk whose contribution is
overestimated in the inner parts so that its contribution to the total
rotation curve is too high and therefore produces large velocity residuals,
(iv) 87_CGr35and 345_CGr32 because their velocity field and [O ii] emission
are quite off-centred from their morphological centre so that their velocity
gradient is not correctly fitted in the inner parts,  (v) 37_CGr84, 85_CGr23,
and 278_CGr84 because, even though they do show signs of rotation, their
velocity fields are too perturbed for a rotating razor-thin gas disk model to
properly fit their kinematics,  (vi) 106_CGr84because there are two
kinematically distinct components in its velocity field whose centres match
the locations of two morphologically distinct component in its HST image
(perhaps following a merging event), therefore rendering the kinematics model
uncertain, and  (vii) 85_CGr35because it is a double-peaked galaxy in its HST
image with little rotation and whose velocity field residuals show that a
rotating razor-thin gas disk model is not sufficient.  Rather than removing
these galaxies, we will keep them in what follows and we will clearly identify
them when analysing their stellar angular momentum.

The choice of whether a galaxy has enough rotation to be flagged or not is
ultimately a question of perspective. Thus, we have tried to remain as
conservative as possible. To illustrate the difference between a galaxy that
does not have clear signs of rotation and one that does, as well as galaxies
at the limit of this selection, we show examples of dynamical models for such
cases in Fig. 14. For each sub-figure the morphological model is shown in the
leftmost column, the velocity field model in the middle one, and the velocity
dispersion in the rightmost one. We show on the top left galaxy 18_CGr114
whose morphology and kinematics are sufficiently well fitted for the analysis
of the angular momentum and on the top right 70_CGr79 that does not have any
velocity gradient in its velocity field (i.e. flagged). On the bottom row, we
show on the left-hand side galaxy 17_CGr34 that has a small and slightly
disturbed velocity field (not flagged) and on the right-hand side 278_CGr84, a
massive galaxy with a large but quite disturbed velocity field as can be seen
in its velocity field residual map. Because the rotating razor-thin gas disk
model produces large residuals, it is unlikely that we correctly constrain the
intrinsic gas kinematics and so we decided to flag it.

##  3 Deriving the stellar angular momentum

We are interested in deriving as precisely as possible the stellar angular
momentum of the galaxies. We therefore implement a method, discussed in Sects.
3.1 and 3.2, that makes full use of the morphological information contained in
the high-resolution HST images and of the best-fit gas kinematics (velocity
field and dispersion map) derived from the MUSE data cubes in the previous
section. We assume that the ionised gas kinematics traces sufficiently well
the stellar kinematics to derive the stellar angular momentum. This assumption
is based on comparisons performed between the kinematics of the stellar and
ionised gas components of galaxies at intermediate redshift (e.g. Guérou et
al. 2017) and is discussed in more details in Sect. 3.3. The formalism
presented below only applies for (potentially thick) stellar disks. Because we
assume that bulges are spherically symmetric, we therefore cannot use this
formalism to estimate their angular momentum. In addition, we expect bulges to
be dispersion dominated and therefore to have a negligible contribution to the
total stellar angular momentum compared to that of the disk component (see
Sect. 3.3 for more details). Thus, in what follows, we focus our analysis on
the angular momentum of the stellar disk component only.

###  3.1 General derivation

Deriving an accurate estimate of the stellar angular momentum
$\@vec{J}_{\star}$ in galaxies is not straightforward given that it involves
knowing a priori the 3D stellar mass density $\rho_{\rm{M}}$ as well as the 3D
velocity vector $\@vec{V}$ of the stars. In general terms, the stellar angular
momentum integrated in a volume $\mathcal{V}$ writes

$\@vec{J}_{\star}(\@vec{r})=\int_{\mathcal{V}}d^{3}\@vec{r}\,\rho_{M}(\@vec{r})\,\@vec{r}\times\@vec{V}(\@vec{r}),$
(4)

where $\times$ represents the cross product operation. However, Eq. 4 is
hardly directly usable unless one is working with simulations (e.g. Brook et
al. 2012; Stewart et al. 2013; Zavala et al. 2016; Cadiou et al. 2022) or in
the vicinity of the Milky Way where 6D phase-space positions are available
from Gaia (e.g. del Pino et al. 2021). Therefore, assumptions must be made on
both the stellar mass density and the velocity of the stars in order to
constrain their angular momentum from morphological and kinematics data.
Perhaps the most widely used expression in the literature is that of RF12
which assumes that the stellar mass distribution can be described by a razor-
thin exponential disk with a constant rotation curve. Recently, Posti et al.
(2018) and Bouché et al. (2021) showed that this approximation may not
correctly compute the stellar angular momentum of intermediate redshift
galaxies, especially in the case of low-mass galaxies (typically
$\log_{10}M_{\star}/{\rm{M_{\odot}}}\lesssim 9-9.5$) since they tend to have a
shallower inner slope in their rotation curve than higher-mass counterparts. A
more general expression than RF12 can be derived from Eq. 4 assuming a razor-
thin disk under rotation only, that is neglecting radial and vertical motions
(see Appendix D for a derivation), without making any assumptions on the shape
of the rotation curve. In this case, the stellar angular momentum becomes
orthogonal to the plane of the stellar disk. Furthermore, it is common
practice to normalise it by the stellar mass of the galaxy, in which case it
is referred to as the specific angular momentum defined as

$j_{\star}(\mathcal{S})=\int_{\mathcal{S}}dSR\leavevmode\nobreak\
\Sigma_{\rm{M}}(R,\theta)\,V_{\theta}(R,\theta)/M_{\star},$ (5)

where $\mathcal{S}$ is the surface in the plane of the stellar disk over which
the angular momentum is integrated, $dS=Rd\theta dR$ is the integrand, $R$ is
the distance to the centre of the galaxy, $\theta$ is the azimuthal angle in
the plane of the disk, $\Sigma_{\rm{M}}$ is the surface mass density of the
stellar disk component (i.e. the integral of the 3D mass distribution along
the vertical direction with respect to the plane of the disk), and
$V_{\theta}(R,\theta)$ is the rotation velocity of the stars at position
$(R,\theta)$. Even though there is no restriction on the surface $\mathcal{S}$
onto which the stellar angular momentum is integrated, it is nevertheless a
common practice to integrate it in a circular aperture of radius $R$. If we
further assume that both $\Sigma_{\rm{M}}$ and $V$ are independent of $\theta$
and that the mass-to-light ratio $\Upsilon_{\star}$ is constant throughout the
galaxy’s stellar disk, Eq. 5 simplifies to

$j_{\star}\,(R)=2\pi\Upsilon_{\star}\int_{0}^{R}dR^{\prime}\,R^{\prime
2}\leavevmode\nobreak\ \Sigma(R^{\prime})\,V_{\theta}(R^{\prime})/M_{\star},$
(6)

where $\Sigma$ is the intrinsic surface brightness distribution (i.e. not sky
projected). Equation 6 corresponds to the expression used in Bouché et al.
(2021) and Mancera Piña et al. (2021a). On the other hand, Eq. 5 is slightly
more general because it can account for both axisymmetric and non-axisymmetric
stellar disks. Both equations are also valid for thick stellar
disks888Equations 5 and 6 are not valid in the case of spherical bulges since
it is not possible to write the mass distribution as the product of a surface
density and a vertical profile. Therefore, in what follows, we restrict
ourselves to estimating the stellar angular momentum of the disk component
only. as long as the disks are  (i) symmetric with respect to the plane of the
disk and that  (ii) it is possible to separate the vertical profile from the
surface brightness distribution in the 3D stellar mass density (see Appendix
D.2).  The choice of normalisation is technically free. In this analysis, we
always normalise the stellar angular momentum using the mass of the stellar
disk component integrated in the same radius within which the angular momentum
is estimated (i.e. within $R_{22}=2.2R_{\rm{d}}$, where $R_{\rm{d}}$ is the
stellar disk’s scale length, see also Sect. 4).

###  3.2 Estimating angular momentum from HST maps

Instead of directly using Eq. 6, we decided to use the more general expression
given by Eq. 5 in combination with HST images to take into account the non-
axisymmetric mass distribution of the galaxies’ stellar disk component. To do
so, we approximate Eq. 5 by discretising it along the HST images’ spatial
dimensions $x^{\prime}$ and $y^{\prime}$. We consider $R$, $V_{\theta}$, and
$\Sigma_{M}$ constant within a pixel’s surface and we assume a constant mass-
to-light ratio throughout the galaxy’s stellar disk so that
$\Sigma_{\rm{M}}=\Upsilon_{\star}\Sigma$, with $\Sigma$ the intrinsic surface
brightness distribution of the disk, and $M_{\star}=\Upsilon F_{\rm{tot}}$,
with $F_{\rm{tot}}$ the total flux integrated in the same aperture as
$M_{\star}$. Then, each pixel at position $(x^{\prime},y^{\prime})$
contributes to the specific stellar angular momentum as

$j_{\star,\rm{pix}}(x^{\prime},y^{\prime})=R\,F(x^{\prime},y^{\prime})V_{\theta}(x^{\prime},y^{\prime})/F_{\rm{tot}},$
(7)

where $R=\sqrt{x^{2}+y^{2}}$ is the radial distance at position $(x,y)$ in the
plane of the disk and $F$ is the flux in the pixel. The conversion from the
apparent position $(x^{\prime},y^{\prime})$ on the the sky to its location
$(x,y)$ in the plane of the disk is done by taking into account the position
of the galaxy’s centre, the position angle (PA), and the inclination of the
disk, all derived in Mercier et al. (2022) from the bulge-disk decomposition
performed with Galfit, assuming a razor-thin disk geometry (i.e. elliptical
isophotes). The specific stellar angular momentum within a circular aperture
of radius $r$ in the plane of the stellar disk is then given as the sum of the
contribution of each pixel in the aperture, that is

$j_{\star}(<r)=\displaystyle\sum_{\\{x^{\prime},y^{\prime}\,|\,R<r\\}}j_{\star,\rm{pix}}(x^{\prime},y^{\prime}).$
(8)

We cannot directly use the ionised gas velocity fields extracted from the MUSE
cubes to compute the stellar angular momentum for a few reasons. First, the
MUSE observations are much less spatially resolved than the HST data
($0.2\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ per spaxel for a PSF FWHM of
$0.5\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ on average for MUSE versus
$0.03\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ per pixel for a PSF FWHM of
roughly $0.1\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ for HST, see Sect. 2.1).
Second, the velocity fields extracted from the cubes are too severely affected
by beam smearing, especially in the inner parts where the velocity gradient
and the ionised gas flux are large. Third, the velocity fields are projected
onto the sky and there is no trivial way to invert the projection (especially
along the minor axis). Thus, we use instead the rotation curves obtained from
the forward mass models performed on the velocity fields in Mercier et al.
(2022). These have the advantage of being intrinsic (i.e. before the impact of
beam-smearing and projection effects) and they can be interpolated to any
radius $r$.

###  3.3 Application of the HST formalism to MAGIC

Our goal is to estimate the stellar angular momentum of intermediate redshift
galaxies in MAGIC. To do so, we need both the distribution and the velocity of
the stars in the plane of the galaxies’ stellar disk. The former can be
estimated from the HST F814W images, assuming this band traces sufficiently
well the stellar mass of the galaxies at $z\approx 1$, but the latter would
only be available for a limited number of galaxies. Indeed, our sample is
comprised mostly of star-forming galaxies with strong emission lines (e.g. [O
ii]) for which we expect only a small fraction (roughly less than 25%) to have
strong enough absorption lines to derive their stellar kinematics. Hence, we
have decided to estimate the stellar angular momentum by using the ionised gas
rotation curves as proxy for the stellar kinematics. This approximation
assumes  (i) that there is co-rotation between the gas and the stars and  (ii)
that the stars are located in dynamically cold disks (i.e. low stellar
velocity dispersion).  If a galaxy is in equilibrium, then the dynamics of
both the stars and the gas should be governed by the galaxy’s total
gravitational potential and its components should have comparable circular
velocities. However, one caveat is that each component might have different
velocity dispersions which will contribute to the dynamical support (i.e.
asymmetric drift) and will therefore lower the rotation velocities by
different amounts. For instance, this is illustrated in Guérou et al. (2017)
who showed that even though there are variations between the rotation of the
gas and the stars, both components have similar kinematics when taking into
account the effect of the velocity dispersion. However, because we do not have
any estimates of the stellar velocity dispersion, we cannot correct a
posteriori for this effect.

We note that similar methods to the one we have implemented have already been
used in a few previous studies (e.g. Cortese et al. 2016; Di Teodoro et al.
2021). However, they usually take into account the contribution of all the
components (in general stellar disk and bulge) to the observed surface
brightness distribution. Doing so may not be entirely appropriate because  (i)
the equations that are used are only valid in the case of a disk, and  (ii)
bulges might be dispersion dominated systems in which case their angular
momentum should be nearly null.  Thus, to avoid being biased by the bulge
component that can significantly contribute to the flux in the inner parts, we
removed the best-fit Galfit bulge model from the HST images before computing
the stellar angular momentum. In the rare cases where the flux in the bulge-
removed images becomes negative near the centre (e.g. because its contribution
was slightly overestimated by Galfit), we replace the pixels with negative
values by the flux of the disk model at the same location. Because we use in
Eq. 7 the flux of the pixels, we know that the estimate of the stellar angular
momentum is also going to be impacted by the noise in the HST images. Given
that we had already removed the background signal from the images, either
beforehand or by using an additional sky background component during the
morphological modelling performed with Galfit, we expect the noise to have a
null mean. Hence, if there is a sufficiently large number of pixels in the sum
in Eq. 8, then the contribution of the noise to the stellar angular momentum
should be close to null. This argument does not hold near the centre where
there are not enough pixels to properly sample the noise distribution.
However, since the rotation velocity also quickly drops to zero there, the
impact of noise should also be reduced in the inner parts.

Finally, we note that the we could have also used the best-fit Galfit disk
model instead of relying on bulge-removed HST images to derive the stellar
angular momentum of the galaxies. This would alleviate the difficulties of
dealing with projection effects and the impact of noise but the main drawback
in doing so is that we would not be able to take into account asymmetries in
the disks of the galaxies since our model assumes axial symmetry. In Sect.
4.2, we perform a comparison between the angular momentum derived from HST
images and that derived from the best-fit Galfit disk model but, in the
following parts of Sect. 4, we solely rely on bulge-removed HST images to
estimate the angular momentum.

##  4 Analysis

We present in this section the framework used to analyse the impact of the
environment on the stellar angular momentum of the galaxies. First, we discuss
in Sect. 4.1 the methodology used to fit the Fall relation and we discuss the
effect of the selection on the shape of the Fall relation for the entire
kinematics sample. Then, we asses in Sect. 4.2 the reliability of the HST
formalism used in this work by comparing it with other angular momentum
estimates.

###  4.1 Methodology

We use the 182 galaxies from the kinematics sample, 30 of which have been
flagged in Sect. 2.4 as having peculiar kinematics. In what follows, we
consider the formalism based on HST images discussed in Sect. 3.2. In this
analysis, the angular momentum (including its normalisation taken as the mass
of the disk component only) and the stellar mass appearing in the Fall
relation are always measured within $R_{22}$999We note that photometric and
kinematics measurements beyond this radius become less robust because of the
significant drop in S/N.. Similarly, we always remove the contribution of the
bulge component to the HST images before estimating the galaxies’ stellar
angular momentum.

The Fall relation for the entire kinematics sample is shown in Fig. 2, where
we use symbols similar to those in Fig. 1. In particular, selected galaxies
are represented with black circles and those flagged as having peculiar
kinematics are shown with red contours. We recover the usual shape of the Fall
relation with the angular momentum that seems to scale roughly linearly with
stellar mass. Our range of angular momentum values and the scatter in the
relation are also consistent with recent results obtained with MUSE using a
SFR-weighted angular momentum tracer (Bouché et al. 2021). There is no
significant difference in the shape of the relation between galaxies that are
kept by the selection (i.e. black circles) and those that are removed (i.e.
other symbols). Low-mass objects
($M_{\star}\lesssim${10}^{9}\text{\,}\mathrm{M_{\sun}}$$) mostly correspond to
small galaxies, as would be expected from the shape of the size-mass relation
(see Fig. 10 of Mercier et al. 2022). Incidently, this means that very few
galaxies are kept below this limit (18 in total versus 152 for the galaxies
removed by the selection) and that this analysis will therefore focus on
galaxies in the stellar mass range $M_{\star}\sim
10^{9}-${10}^{11}\text{\,}\mathrm{M_{\sun}}$$.

Another important point is that flagged galaxies mostly populate the bottom
part of the relation. This can be understood by noting that the majority of
these galaxies do not exhibit visible velocity gradients, meaning that their
rotation velocity is poorly constrained. But, because the contribution to the
rotation curve of the stellar disk and bulge components is entirely fixed by
the morphology, applying a mass modelling approach for these galaxies can
result in a velocity field model that rotates faster than the observed one. In
such cases, no DM halo can be fitted because the rotation velocity is already
too high with just the bulge and disk components. The net effect is that it
will produce a spurious correlation between the stellar mass and angular
momentum of the galaxies (described by Eq. 27 in the case of a bulge-less
galaxy; see also the end of Sect. 6 of Mercier et al. 2022 for a discussion of
the limits of mass models for such cases). One way to alleviate these spurious
correlations is to use for comparison a non-physically motivated rotation
curve model (i.e. a model that does not use the morphology as a prior or
constraint for the kinematics). Thus, in what follows, we will always analyse
our results using both the mass model rotation curve, as well as a flat
rotation curve model (composed of a linearly rising part followed by a
plateau, see also Eq. 7 of Abril-Melgarejo et al. 2021).

To study the impact of the galaxies’ environment on the shape of the Fall
relation, we use the same methodology as applied to the TFR in Mercier et al.
(2022), that is we fit

$\log_{10}j_{\star}\,[\unit{\kilo\kilo\per}]=\beta+\alpha(\log_{10}M_{\star}\,[\unit{}]-p),$
(9)

where $\beta$ is the the zero-point, $\alpha$ is the slope, and
$p=$9.8\text{\,}\mathrm{\text{dex}}$$ is the pivot point taken as the decimal
logarithm of the median stellar mass (in solar masses) of the kinematics
sample101010We include a pivot point in order to reduce the correlation
between the slope and the zero point during the fit.. We always consider
$\log_{10}j_{\star}$ as the dependent variable and $\log_{10}M_{\star}$ as the
independent variable. Following Mercier et al. (2022), we will mostly
investigate variations in the zero-point of the Fall relation as a function of
galaxy environment for a fixed slope. Nevertheless, we will also quickly
discuss a potential variation of the slope of the relation, especially at the
high-mass end, though a more thorough discussion will be given in the second
paper of the series. We use a slightly updated version of
LtsFit111111https://pypi.org/project/ltsfit/ (Cappellari et al. 2013) to fit
the relation which implements the Least Trimmed Squares (LTS) technique (see
Rousseeuw & Driessen 2006) to find and remove outliers from the fit. The only
feature that this new version adds is the possibility to fit with a fixed
slope. When doing so, we always use the following procedure:  (i) we fit the
entire sample with a free slope, and  (ii) we fix the slope to this value
before fitting each subsample separately.  When fitting the Fall relation, we
take into account the uncertainties on both dependent and independent
variables. For the stellar mass, we use the uncertainty derived from SED
fitting and we add a systematic uncertainty of
$0.2\text{\,}\mathrm{\text{dex}}$, as in Mercier et al. (2022). For each
galaxy, we estimate the uncertainty on $j_{\star}$ by generating 100 Monte-
Carlo realisations of the flux distribution and kinematics parameters of the
rotation curve. For each realisation, the stellar angular momentum is
estimated and the uncertainty on $j_{\star}$ is taken as the standard
deviation of the 100 realisations. Furthermore, in order not to treat
asymmetrically the dependent and independent variables (i.e. having one
variable with underestimated uncertainties) during the fitting process, we
also add a systematic uncertainty of $0.1\text{\,}\mathrm{\text{dex}}$ to the
angular momentum. Finally, we estimate for each fit the unbiased average,
standard-deviation, and 95% confidence interval (CI) of the best-fit zero-
point using jackknife resampling (Quenouille 1956; Tukey 1958) on the
considered sub-sample with the slope and outliers fixed.

Figure 2: Fall relation for the entire kinematics sample, using bulge-removed
HST images. Galaxies selected from the kinematics sample are represented with
black points and other symbols represent galaxies removed by different
selection criteria (see Fig. 1). Red contours correspond to galaxies flagged
with peculiar kinematics. The typical uncertainty (including systematic
uncertainties added when fitting the relation) is shown as an error bar on the
bottom right corner. †Galaxies removed by the surface criterion defined in
Sect. 2.4 but that would have been kept by the size selection criterion used
in Mercier et al. (2022).

###  4.2 Assessing the reliability of the method

Figure 3: Fall relation for the kinematics sample using various formalisms. On
the top row is shown the angular momentum derived using HST maps and on the
bottom row that derived using an exponential disk model. The leftmost column
represents the Fall relation when using the rotation curve derived from the
mass modelling and the rightmost column the Fall relation when using a flat
model rotation curve (the top-left plot is similar to Fig. 2). Galaxies
selected for the analysis are shown as black points and those flagged with
peculiar kinematics with red contours. Other symbols correspond to galaxies
removed by the selection (see Fig. 1 for more details). The typical
uncertainty is shown on the bottom of each plot as a black error-bar. The two
black lines show the limit of the Fall relation for an exponential disk
without DM halo and with two different disk scale lengths of
$5\text{\,}\mathrm{k\text{pc}}$ (dashed line) and
$15\text{\,}\mathrm{k\text{pc}}$ (plain line, see Eq. 27).

In this section, we estimate the reliability of the method we used to derive
the stellar angular momentum. To do so, we can first check the impact of using
HST images instead of assuming a stellar mass distribution that follows an
axisymmetric disk. Then, we can see how changing the shape of the rotation
curve affects the value of the stellar angular momentum. For that, we can
compare its value when using the mass model rotation curve (i.e. including the
contributions of the stellar bulge, stellar disk, and DM halo) with the value
obtained when using a simpler flat model rotation curve (see Abril-Melgarejo
et al. 2021 for a definition). Even though the latter is not physically
motivated, it is nevertheless a robust model that was fitted in Mercier et al.
(2022) to the ionised gas kinematics to asses the reliability of the recovered
gas intrinsic velocities. The Fall relations for four different estimates of
the specific stellar angular momentum are shown in Fig. 3. On the top left,
the specific stellar angular momentum uses the bulge-removed HST images in
combination with the mass model rotation curve (i.e. similarly to Fig. 2). On
the top right, it uses the bulge-removed HST images combined with a flat
rotation curve model. On the bottom-left, it uses the best-fit exponential
disk mass distribution (derived with Galfit) in combination with the mass
model rotation curve (the angular momentum integrated numerically, see Eq. 6).
And, on the bottom-right, is uses an exponential disk mass distribution
combined with a flat model rotation curve (the angular momentum is evaluated
analytically, see Appendix D). The symbols are the same as in Fig. 1.

We start by considering the effect of the rotation curve. Comparing the plots
on the first row of Fig. 3, we see large variations of the specific stellar
angular momentum. Overall, the difference between the estimate that uses the
mass model and the one that uses the flat model rotation curve is roughly
$0.35\pm$0.35\text{\,}\mathrm{\text{dex}}$$ for the entire kinematics sample.
This becomes $0.1\pm$0.15\text{\,}\mathrm{\text{dex}}$$ when considering the
selected sample only and it is amplified to
$0.45\pm$0.4\text{\,}\mathrm{\text{dex}}$$ when considering the galaxies
removed by the selection. Galaxies flagged as having peculiar kinematics are
also greatly impacted with a difference of around
$0.3\pm$0.25\text{\,}\mathrm{\text{dex}}$$, in between the samples of galaxies
kept and removed by the selection. In particular, we see that flagged galaxies
tend to be aligned along a lower limiting line (as indicated by the black
lines in Fig. 3) when using the mass model rotation curve, whereas they are
spread throughout and below the relation when using a flat model rotation
curve. As discussed in the previous section, the reason for that effect is
that most of these galaxies lack a visible velocity gradient (e.g. 70-CGr79 in
Fig. 14). Thus, it is unlikely that their kinematics is properly fitted,
meaning that the contributions of their stellar disk and bulge components are
overestimated and that no DM halo could be fitted, hence producing a spurious
correlation between the galaxies’ stellar mass and specific angular
momentum121212The reason for this correlation is that the stellar mass is
linked to the galaxies’ stellar mass distribution. In the case of a DM halo-
free galaxy, the gravitational potential is dominated by that of the stars, so
that, under the assumption of equilibrium, the kinematics is also directly
governed by the stellar mass distribution.. Because of varying bulge
contributions to the rotation curve and non-axisymmetric features in the HST
images (e.g. clumps), this effect is not systematic for every flagged galaxy
(e.g. 17% of them have a stellar angular momentum difference below 15%). Then,
we investigate the effect of using HST images instead of an exponential disk
model. This is shown in Fig. 3 by vertically comparing the plots in a same
column (i.e. left-hand column for the mass model rotation curve and right-hand
column for the flat rotation curve). We find that the use of HST images only
has a small impact on the position of the galaxies in the Fall relation.
Selected galaxies are barely affected with a difference (angular momentum from
HST images minus that from the exponential disk model) of
$0.0\pm$0.1\text{\,}\mathrm{\text{dex}}$$, whereas unselected galaxies tend
have slightly larger values with an average difference of roughly
$-0.15\pm$0.25\text{\,}\mathrm{\text{dex}}$$. This can be explained by the
fact that most of the unselected galaxies are removed by the surface selection
criterion (blue upward pointing triangles and grey dots) which tends to mostly
remove small galaxies (grey dots only). Since their best-fit disk’s model is
not as precisely constrained as larger galaxies, it is likely that it
overestimates the stellar mass in the disk, therefore producing higher
specific angular momentum values.

Hence, these comparisons suggest that the choice of the rotation curve is what
mainly drives the values of the specific stellar angular momentum. In
particular, one must be careful when using the mass model rotation curve not
to be biased by galaxies with large stellar mass fraction uncertainties (i.e.
galaxies that seemingly appear baryon-dominated, though they are not
intrinsically so) given that these objects produce spurious correlations at
the bottom of the Fall relation (i.e. for low $j_{\star}$ values at fixed
stellar mass). Because the same models and selection function were applied to
the entire kinematics sample independently of the galaxies’ environment, we do
not expect this effect to be directly correlated to the density estimates.
Thus, this should not affect our conclusions.

##  5 Results

We present in this section the results of the analysis of the impact of the
environment. To begin with, we probe in Sect. 5.1 the environment using the
groups’ richness. In Sect. 5.2, we consider two different environment tracers
that probe the position of the galaxies in phase-space and the over-density
around them. In Sect. 5.3, we discuss the link between the angular momentum of
the galaxies and their normalised systemic velocity with respect to their host
group. And, in Sect. 5.4, we discuss the link between the angular momentum of
the galaxies and the velocity dispersion of their ionised gas component.

###  5.1 Environment traced by groups’ richness

####  5.1.1 Comparison between galaxies in the field and in rich groups

Figure 4: Fall relation for galaxies in the field (black squares) and those in
rich groups with more than ten members (red circles). A stellar mass cut
$M_{\star}<${10}^{10}\text{\,}\mathrm{M_{\sun}}$$ and a redshift cut
$0.5<z<0.9$ were applied to each sub-sample. Outliers detected during the
fitting process are identified with yellow stars and the typical uncertainty
is shown with a black error-bar. The best-fit lines are shown for the field
(black) and groups’ (red) sub-samples, along with their 95% confidence
intervals (coloured areas) determined with jackknife resampling. Galaxies
removed by the mass and redshift cuts are identified with semi-transparent
symbols.

First, we consider the impact of the environment using the simplest density
tracer that we computed: the richness of the groups. This parameter was
already used in Mercier et al. (2022) to study the impact of the environment
on the size-mass, MS, and Tully-Fisher relations. We consider the two
following sub-samples:  (i) field galaxies, that is galaxies that were not
associated to any group by the FoF algorithm or that were associated to a
group with less than three members, and  (ii) galaxies located in rich groups
that have at least ten galaxy members ($N>10$).  Their Fall relations are
illustrated in Fig. 4. Field galaxies are shown with black points, those in
groups with red circles, and outliers identified during the fitting process
with yellow stars. The black line represents the best-fit linear relation for
the field galaxies sample and the red line that for the galaxies in groups
(both were obtained using the same slope derived on the entire sample). We
also show the 95% confidence intervals for both fits, estimated using
jackknife resampling. In this figure, we apply a stellar mass cut
$M_{\star}<${10}^{10}\text{\,}\mathrm{M_{\sun}}$$ and a redshift cut
$0.5<z<0.9$ in order not to be biased by massive galaxies that are almost
entirely found in the richest groups (i.e. we do not have a statistically
significant sample of field galaxies at
$M_{\star}\gtrsim${10}^{10}\text{\,}\mathrm{M_{\sun}}$$, for more details see
the discussion in Mercier et al. 2022), as well as by a potential redshift
evolution of the Fall relation131313Galaxies in the groups are nearly all
found at $z\approx 0.7$, whereas field galaxies are spread throughout the
entire redshift range. The interested reader can find the resdhift and stellar
mass distributions in Figs. 1 and 2 of Mercier et al. (2022).. As an
indication, we also show the galaxies removed by the two cuts with semi-
transparent symbols. To gain deeper insights into the impact of the
environment on the Fall relation, we also fitted it without any cut, with the
stellar mass cut only, and with the redshift cut only. We also did the same
but using the flat model rotation curve instead. The results of all these fits
in terms of slope, best-fit zero-point, and confidence intervals are
summarised in Table 1.

Table 1: Best-fit values for the Fall relation using the groups’ richness as
density estimate.

| Cut | Sub-sample | Nb. galaxies | $\alpha$ | $\beta$ | 95% CI
---|---|---|---|---|---|---
| (1) | (2) | (3) | (4) | (5) | (6)
Mass model | None | Field | 85 | 0.50 | 2.80 | 2.73, 2.86
Large | 75 | 2.68 | 2.62, 2.74
Mass | Field | 67 | 0.83 | 2.96 | 2.89, 3.04
Large | 36 | 2.84 | 2.73, 2.95
Redshift | Field | 33 | 0.52 | 2.76 | 2.67, 2.86
Large | 66 | 2.68 | 2.61, 2.75
| Both | Field | 27 | 0.80 | 2.92 | 2.81, 3.02
| Large | 30 | 2.81 | 2.71, 2.92
Flat model | None | Field | 85 | 0.55 | 2.73 | 2.65, 2.80
Large | 75 | 2.55 | 2.48, 2.62
Mass | Field | 67 | 0.94 | 2.92 | 2.83, 2.99
Large | 36 | 2.74 | 2.62, 2.87
Redshift | Field | 33 | 0.57 | 2.66 | 2.54, 2.78
Large | 66 | 2.52 | 2.44, 2.60
| Both | Field | 27 | 0.91 | 2.86 | 2.74, 2.99
| Large | 30 | 2.74 | 2.60, 2.88

Notes: (1) Selection cut applied, (2) sub-sample (i.e. galaxies in the field
or in rich groups), (3) number of galaxies in the sub-sample, (4) best-fit
slope, (5) best-fit zero-point, and (6) jackknife resampling-derived 95%
confidence intervals for the best-fit zero-point. The same slope was used for
both sub-samples. It was derived by fitting first with a free slope the entire
sample with the considered selection cuts applied. The stellar mass cut
selects galaxies according to
$M_{\star}<${10}^{10}\text{\,}\mathrm{M_{\sun}}$$ and the redshift cut
according to $0.5<z<0.9$.

Without any cut, we obtain a $3.5\sigma$-significant difference (with $\sigma$
the jackknife-derived standard-deviation) in zero-points of
$0.12\text{\,}\mathrm{\text{dex}}$ between galaxies in the field and those in
rich groups, consistent with a depletion of specific stellar angular momentum
at fixed stellar mass for the galaxies in the groups with respect to the
field. When applying both the stellar mass and the redshift cuts, the zero-
points of the two sub-samples increase each by roughly
$0.13\text{\,}\mathrm{\text{dex}}$ (i.e. the difference remains the same), but
their uncertainties increase as well (changing from
$0.03\text{\,}\mathrm{\text{dex}}$ to $0.05\text{\,}\mathrm{\text{dex}}$), so
that the difference in zero-points reduces to $2\sigma$-significant only.
Thus, it is not straightforward to conclude whether the observed difference is
statistically induced or environmentally-driven. Especially when applying both
the mass and redshift cuts, we cannot completely rule out the null hypothesis
that both sub-samples share the same zero-point. Besides, independently of
whether we apply the cuts or not, removing galaxies flagged as having peculiar
kinematics (identified with red contours in Figs. 2 and 3) has a negligible
impact on the results (up to $0.02\text{\,}\mathrm{\text{dex}}$). We also find
the same trend, that is a depletion of specific stellar angular momentum for
the galaxies in the groups with respect to the field, when using the flat
model rotation curve (see Fig. 8). Compared to the mass models, the zero-point
changes by $-0.06\text{\,}\mathrm{\text{dex}}$ for either sub-sample. Without
any cuts, we get a difference in zero-points of
$0.22\text{\,}\mathrm{\text{dex}}$ ($4.5\sigma$-significant), which reduces to
$0.12\text{\,}\mathrm{\text{dex}}$ (around $2\sigma$-significant) when
applying both stellar mass and redshift cuts.

We note that this observed depletion of angular momentum for galaxies located
in rich groups with respect to the field is compatible with previous results
from both Pelliccia et al. (2019) and Pérez-Martínez et al. (2021). In
particular, looking at Fig. 8 and Table 5 of Pérez-Martínez et al. (2021), we
determine that they found a similar depletion of angular momentum of around
$0.1-$0.15\text{\,}\mathrm{\text{dex}}$$ between their sample of galaxies
located in clusters and the various field samples they used for comparison.
Similarly, Pelliccia et al. (2019) also found a
$0.1\text{\,}\mathrm{d}\mathrm{e}\mathrm{x}$ depletion in groups/clusters
compared to the field (see their Fig. 8 and Sect. 4.4).

####  5.1.2 Effect of stellar mass and redshift

It might be interesting to comment on the effect of stellar mass and redshift
cuts. First, the mass cut increases the slope from $\alpha=0.5$ when using the
mass models ($\alpha=0.55$ for the flat model) to $\alpha=0.83$ after the cut
is applied ($\alpha=0.94$). It also increases the zero-points of the field
sub-sample by $0.2\text{\,}\mathrm{\text{dex}}$
($0.25\text{\,}\mathrm{\text{dex}}$ for the flat model) and of the rich groups
sub-sample by $0.14\text{\,}\mathrm{\text{dex}}$
($0.21\text{\,}\mathrm{\text{dex}}$). The main impact of this change of slope
is to bring the most massive galaxies in the sample
($M_{\star}\gtrsim${10}^{10}\text{\,}\mathrm{M_{\sun}}$$) below the best-fit
lines obtained when applying either the redshift cut only or no cut at all.
Thus, massive galaxies appear depleted in angular momentum with respect to
what would be expected by extrapolating the Fall relation found at lower mass,
as is visible in Fig. 4. In particular, the more massive the galaxies, the
lower is their average angular momentum (up to
$-0.5\text{\,}\mathrm{d}\mathrm{e}\mathrm{x}$ with respect to the best-fit
line, see Table. 3 for a complete list of values). However, the lack of a
statistically significant sample of field galaxies in this mass range prevents
us from drawing conclusions as to whether this effect is mass- or
environmentally-driven (or both). Furthermore, this effect might also be
statistically induced by the fact that, when applying the mass cut, we are
giving more weight to low-mass galaxies with larger uncertainties and with a
coarser sampling.

On the contrary, the redshift cut has nearly no effect on the slope (changing
it by $\pm$0.03\text{\,}\mathrm{\text{dex}}$$ at most) and on the zero-point
of the sub-sample of galaxies in rich groups. However, it does reduce that of
the field galaxies sub-sample ($-0.04\text{\,}\mathrm{\text{dex}}$ with the
redshift cut and $-0.07\text{\,}\mathrm{\text{dex}}$ with both mass and
redshift cuts). Galaxies located in rich groups are not affected by the
redshift cut mostly because more than 90% are located in the range
$0.5<z<0.9$. The visible effect of the redshift cut on the field sub-sample
might be an indication of a redshift evolution of the Fall relation’s zero-
point. This is strengthened by the fact that around 65% of galaxies at $z>0.9$
are found above the best-fit Fall relation whereas galaxies at $z<0.5$ are
located evenly throughout and along the relation. However, given the low
statistics at $z<0.5$ and $z>0.9$ (about 20 galaxies each), it is difficult to
assess whether this effect is statistically-induced or due to an evolution of
the zero-point of the Fall relation with redshift. In particular, massive
galaxies that are depleted in angular momentum (i.e. located below the best-
fit line) are mostly found at $z<0.9$, meaning that they will mainly lower the
zero-point for samples that are at $z<0.9$ and give the impression that
galaxies at $z>0.9$ have higher angular momentum values.

###  5.2 Link with global and local density estimates

Figure 5: Fall relation colour coded by $\eta$ (see Sect. 2.3). Galaxies in
rich groups with more than ten members are shown as circles and field galaxies
as black squares (both have the redshift cut $0.5<z<0.9$ applied). The mass
model rotation curve is used (see Fig. 9 for the flat model rotation curve
instead). As an indication, the best-fit linear relations (with redshift cut
applied, see also Table 1) for the field and rich groups sub-samples are shown
as grey and black plain lines, respectively. The typical uncertainty is shown
as the black error-bar on the bottom right corner.

We refine the previous analysis by using more precise density estimates.
Indeed, the groups’ richness is a simple proxy to probe the effect of the
environment and, as such, it potentially mixes galaxies located in regions of
various densities that might have been affected by their environment on
different time-scales. To do so, we use the global density estimate $\eta$
that encompasses in a single parameter the position and dynamics of the
galaxies in their host group (see Sect. 2.3). Following Noble et al. (2013),
we separate galaxies associated to groups between those that have been
accreted early ($\eta<0.1$), ”backsplash” galaxies that have already passed
the pericentre of their orbit once ($0.1<\eta<0.4$, e.g. Balogh et al. 2000;
Gill et al. 2005), and recently accreted galaxies infalling into the groups
($0.4<\eta<2$)141414We do not have galaxies with $\eta>2$ because all these
objects have already been associated to the field sub-sample by the FoF
algorithm.. In our case, we lack statistics across the entire stellar mass
range in MAGIC to properly sample  (i) early accreted galaxies when applying
both mass and redshift cuts (eight galaxies remaining) and  (ii) infalling
galaxies ($\eta>0.4$) when applying either the mass cut only (13 galaxies
remaining) or both mass and redshift cuts (seven galaxies remaining).  Second,
as can be seen in Fig. 5, different $\eta$ classes populate different mass
ranges, meaning that we would be biased by the different mass distributions
when fitting each class separately. Thus, instead of fitting these classes, we
compare their distributions in the $j_{\star}-M_{\star}$ plane with the field
sample and with respect to the best-fit lines (with redshift cut applied, see
Table 1) obtained in Sect. 5.1. The Fall relation for these three $\eta$
classes is shown in Fig. 5 with early accreted galaxies represented as red
circles, backsplash galaxies as blue circles, infalling galaxies as orange
circles, and field galaxies as black points. We note that $\eta$ can also be
computed for smaller groups, but including the galaxies from these groups does
not affect the conclusions. Thus, to remain consistent with previous sections
we only focus on rich groups in what follows.

More than 90% of the backsplash galaxies ($0.1<\eta<0.4$) are in the stellar
mass range $10^{9}<M_{\star}/\unit{}<10^{10.5}$ and around 75% of the early
accreted galaxies ($\eta<0.1$) have
$M_{\star}>${10}^{10}\text{\,}\mathrm{M_{\sun}}$$ (more than 50% above
$10^{10.5}\,\unit{}$). Thus, there is a strong dichotomy between these two
classes which seems to be mostly driven by stellar mass, with a transition at
around ${10}^{10}\text{\,}\mathrm{M_{\sun}}$. Overall, 65% of the early
accreted galaxies are located below the best-fit line (with redshift cut
applied, see Table 1), whereas infalling galaxies are spread evenly
throughout. This number rises to 75% when compared to the best-fit line
obtained when applying both mass and redshift cuts.

Hence, early accreted galaxies are located on average below the best-fit Fall
relation. If we separate the sub-sample in various mass bins, we see that low-
mass galaxies ($M_{\star}<10^{10.5}\,\unit{}$) are mostly associated with
backsplash galaxies ($\eta\approx 0.2$), whereas massive galaxies are nearly
all identified as early accreted with $\eta<0.1$ (see Table 2 for the complete
list of values). In other words, low $\eta$ values for galaxies located at the
bottom of the Fall relation are mostly driven by massive galaxies. These
results also hold when using the flat model rotation curve (see Fig. 9).

Table 2: Global and local density in various stellar mass bins for the sub-
sample of galaxies located in rich groups.

Nb. | Stellar mass bin | $\eta$ | Local over-
---|---|---|---
galaxies | [$\log_{10}\unit{}$] |  | density $\delta$
(1) | (2) | (3) | (4)
13 | 8.5, 9.5 | $0.19^{0.71}_{0.09}$ | $0.81^{1.00}_{0.40}$
22 | 9, 10 | $0.20^{0.34}_{0.10}$ | $0.82^{1.10}_{0.48}$
37 | 9.5, 10.5 | $0.17_{0.09}^{0.31}$ | $0.86^{1.11}_{0.54}$
31 | 10, 11 | $0.10_{0.04}^{0.24}$ | $0.98^{1.34}_{0.64}$
18 | 10.5, 11.5 | $0.04_{0.04}^{0.06}$ | $1.19^{1.48}_{0.83}$

Notes: (1) Number of galaxies located in rich groups in the given stellar mass
range, (2) minimum and maximum bounds of the stellar mass bin, (3) median,
16th, and 84th percentiles of $\eta$, and (4) median, 16th (subscript), and
84th (superscript) percentiles of the VMC-based over-density $\delta$. For
definitions of $\eta$ and $\delta$, see Sect. 2.3.

We also investigate whether this dichotomy is observed with a direct tracer of
the over-density around the galaxies. To do so, we measured the median value
of the VMC-based over-density $\delta$ (see Sect. 2.3) for low- and high-mass
galaxies (see Table. 2 for the complete list of values and Figs. 10 and 11 for
the Fall relation colour-coded with $\delta$). We do find an increase from
$\delta\approx 5$ for $M_{\star}<${10}^{10}\text{\,}\mathrm{M_{\sun}}$$ to
$\delta\approx 9$ above this mass threshold. However, when splitting the
sample between galaxies located below and above the best-fit Fall relation
(with redshift cut applied, see Table 1), we do not find any significant
difference in the distributions of $\delta$. Moreover, this result is
independent of the choice of rotation curve.

Hence, there is no evidence that the depletion of angular momentum for
galaxies in the groups with respect to the field is associated with the over-
density around the galaxies. Furthermore, even though galaxies located below
the Fall relation tend to have a larger fraction of early accreted galaxies,
the effect seems to be dominated by their stellar mass rather than the
environment.

###  5.3 Motion-induced loss of angular momentum ?

A potential caveat of using $\eta$ in Sect. 5.2 is that it combines in a
single parameter the position with the velocity of the galaxies in their host
group but, depending on the underlying physical mechanism, each can contain
different information (e.g. ram-pressure stripping primarily scales with
velocity). Thus, in this section, we split $\eta$ into a velocity component
$|\Delta v|/\sigma_{V}$, where $\Delta v$ is the systemic velocity of a galaxy
with respect to the dynamical centre of its host group and $\sigma_{V}$ is the
velocity dispersion of the group (see Sect. 2.3), and a distance component
$R/R_{200}$, where $R_{200}$ is the virial radius of the group.

To begin with, we do not find any obvious correlation between the position of
the galaxies along, above, or below the Fall relation and their normalised
distance $R/R_{200}$. However, we do find that the most massive galaxies
($M_{\star}\gtrsim 10^{10.5}\,\unit{}$) in the groups are primarily found in
the inner parts of the groups (on average $R/R_{200}\approx 0.15$ above this
mass threshold and $R/R_{200}\approx 0.3$ below). We note that this is in line
with previous results found for galaxy groups in the local Universe (e.g.
Roberts et al. 2015).

On the other hand, we do find a significant correlation between  (i) $|\Delta
v|/\sigma_{V}$,  (ii) the stellar mass of the galaxies, and  (iii) their
position orthogonal to the Fall relation (i.e. above or below it).  The Fall
relation colour-coded by $|\Delta v|/\sigma_{V}$ is shown in Fig. 6 for the
mass model rotation curve and in Fig. 12 for the flat model rotation curve
(both for galaxies in rich groups only). First, we find that massive galaxies
($M_{\star}\gtrsim 10^{10.5}\,\unit{}$) tend to have the lowest $|\Delta
v|/\sigma_{V}$ values with, on average, $|\Delta v|/\sigma_{V}\approx 0.17$
and 0.8 for galaxies above and below this mass threshold, respectively. This
dichotomy in $|\Delta v|/\sigma_{V}$ strongly correlates with the dichotomy
seen for $\eta$. Thus, this means that the massive early accreted galaxies
located at the high-mass end are so mostly because of their low systemic
velocity with respect to the velocity dispersion (or alternatively dynamical
mass) of their host group.

Interestingly and as clearly visible in Fig. 6, we also find a strong
dichotomy in $|\Delta v|/\sigma_{V}$ at lower stellar masses
($M_{\star}\lesssim 10^{10.5}\,\unit{}$), but this time orthogonal to the Fall
relation. Overall, galaxies above the best-fit line (when applying the
redshift cut only) have a median $|\Delta v|/\sigma_{V}$ equal to 0.9, whereas
it is only 0.7 below. Furthermore, we also find that around 70% of the
galaxies with $|\Delta v|/\sigma_{V}<0.7$ are located below this line. When
further splitting $|\Delta v|/\sigma_{V}$ into the systemic velocity of the
galaxies $|\Delta v|$ and their host group’s dispersion $\sigma_{V}$, we find
that this dichotomy is mostly driven by the systemic velocity of the galaxies.
In other words, galaxies located below the Fall relation appear to have lower
systemic velocities than those located above it. This includes the majority of
the low-mass early accreted galaxies ($\eta<0.1$) for which we expect, by
definition, to have relatively low $|\Delta v|/\sigma_{V}$ values, as well as
some of the backsplash galaxies ($0.1<\eta<0.4$). This result suggests that
the mechanism responsible for the loss of specific stellar angular momentum
for the galaxies located in the groups must be tightly linked to the velocity
at which galaxies move through the intra-group medium.

Figure 6: Fall relation for galaxies in rich groups (with more than ten
members) colour coded by the galaxies’ line-of-sight systemic velocity
$|\Delta v|$ normalised by the velocity dispersion of their host group
$\sigma_{V}$. Galaxies with peculiar kinematics identified in Sect. 2.4 are
shown with a black cross. The mass model rotation curve is used (see Fig. 12
for the flat model rotation curve instead). As an indication, the best-fit
linear relations (with redshift cut applied, see also Table 1) for the field
and rich groups sub-samples are shown as grey and black plain lines,
respectively. The typical uncertainty is shown as the black error-bar on the
bottom right corner.

###  5.4 An anti-correlation between gas dispersion and angular momentum

We complete this analysis by investigating whether there is a link between the
depletion of specific stellar angular momentum for galaxies in rich groups
with respect to the field (see Sect. 5.1) and the velocity dispersion of the
gas. Finding such a link would be interesting because it could pinpoint at
mechanisms that are able to either increase or decrease the dispersion of the
gas as the galaxies move through the groups (e.g. mergers or thermal
evaporation). The Fall relation colour-coded according to the median value of
the galaxies’ ionised gas velocity dispersion151515Measured from the beam-
smearing- and LSF-corrected velocity dispersion map extracted from the MUSE
cubes. is shown in Fig. 7 for the mass model rotation curve and in Fig. 13 for
the flat model rotation curve.

Similarly to $|\Delta v|/\sigma_{V}$, we also find a strong dichotomy in terms
of the ionised gas velocity dispersion orthogonal to the Fall relation.
Galaxies located below the best-fit line (when applying the redshift cut only)
have an average gas velocity dispersion of around
$40\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$161616We note that the
typical LSF dispersion is around
20-$30\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$, whereas those located
above it rather have values around
$10\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$. Furthermore, 75% of the
galaxies above the best-fit line have an ionised gas velocity dispersion below
$25\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ (hence the low median
value), whereas 75% of the galaxies below the best-fit line have values above
$25\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$, with a peak at around
$50\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$. It is also interesting to
note that, contrary to what was found in Sect. 5.3, this dichotomy is not
correlated with the stellar mass of the galaxies. Hence, massive early
accreted galaxies discussed in the previous sections do not appear to have
either higher or lower gas velocity dispersion values than lower mass
galaxies.

Thus, galaxies located above the Fall relation appear to be strongly
rotationally-supported with very low ionised gas velocity dispersion, whereas
galaxies located below the Fall relation have significantly higher values.
This result is not entirely new and a similar correlation was already observed
in previous analyses that used the dynamical state of the galaxies $V/\sigma$
rather than the velocity dispersion of the gas alone (e.g. Contini et al.
2016, but in particular Bouché et al. 2021 where this correlation was
highlighted). However, these studies focussed mostly on field galaxies,
therefore pointing to an underlying mechanism that is not environmentally-
induced. This is further supported by the fact that a similar dichotomy can be
found for field galaxies in our sample. More than 70% of the field galaxies
located above the Fall relation have ionised gas velocity dispersion values
below $25\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$, whereas roughly 70%
of those below have a gas velocity dispersion above
$25\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$. Still, we do see an
enhancement of the ionised gas velocity dispersion for galaxies in the groups.
In particular, we find that more than 30% of the galaxies located in rich
groups have a ionised gas velocity dispersion above
$50\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$, whereas only 5% of the
field galaxies show a dispersion this high.

Figure 7: Fall relation colour coded according to the median velocity
dispersion of the ionised gas component in the galaxies. Galaxies in rich
groups with more than ten members are shown as circles and field galaxies as
smaller squared (both have the redshift cut $0.5<z<0.9$ applied). Galaxies
with peculiar kinematics identified in Sect. 2.4 are shown with a black cross.
The mass model rotation curve is used (see Fig. 12 for the flat model rotation
curve instead). As an indication, the best-fit linear relations (with redshift
cut applied, see also Table 1) for the field and rich groups sub-samples are
shown as grey and black plain lines, respectively. The typical uncertainty is
shown as the black error-bar on the bottom right corner.

###  5.5 Summary and interpretation of the results

Interpreting these results in terms of underlying driving physical mechanisms
that are either linked to the mass of the galaxies or to the environment
around them is difficult without relying on hydrodynamical simulations, or at
least analytical models of galaxy evolution. Our aim with the second paper of
this series is actually to investigate how different physical processes can
affect galaxies and move them along or across the Fall relation. Still, given
the visible impact of the environment found in the previous sections and the
interesting new correlations with the galaxies systemic velocity and gas
velocity dispersion, we can propose a few scenarios that might produce such
effects.

In the previous sections, we have found that galaxies located in rich groups
tend to be depleted in specific stellar angular momentum with respect to the
field, especially at low mass where we have a comparable field sample. This
depletion does not appear correlated neither with the position of the galaxies
in their host group, nor with the over-density around them. However, low-mass
galaxies located below the Fall relation show lower values of $|\Delta
v|/\sigma_{V}$ than galaxies located at the top of the relation (i.e. they
move the slowest in their host group) and they also have significantly higher
ionised gas velocity dispersion values. Even though field galaxies show a
comparable dichotomy in terms of gas velocity dispersion, there is
nevertheless evidence that the groups might enhance it.

Physical mechanisms compatible with these results must therefore be able to
reduce the angular momentum either by changing the morphology of the galaxies
(i.e. transferring mass from high to low circular velocity regions) or by
decreasing the amplitude of the circular velocity (or alternatively change the
shape of the rotation curve). We did show in Mercier et al. (2022) that MAGIC
galaxies located in rich groups are on average more compact than galaxies in
the field at the same redshift (by roughly
$0.03\text{\,}\mathrm{\text{dex}}$). However, early calculations about the
impact of such a difference on the Fall relation have convinced us that the
offset is certainly too small to explain the large difference observed in
Sect. 5.1. Furthermore, we also showed in Mercier et al. (2022) that low-mass
($M_{\star}<${10}^{10}\text{\,}\mathrm{M_{\sun}}$$) galaxies in groups and in
the field have similar circular velocities, meaning that the impact of the
environment is certainly due to variations in the shapes of the rotation
curves. This effect will be investigated in much more details in the second
paper of the series.

Since our stellar angular momentum estimate uses the ionised gas velocity as a
proxy for the stellar kinematics, it should be sensitive to hydrodynamical
mechanisms. The clear correlation with $|\Delta v|/\sigma_{V}$ presented in
Sect. 5.3, the enhanced velocity dispersion of the ionised gas seen for
galaxies in the groups, and the lack of correlation with the strength of the
over-density around the galaxies could be an indication that we are seeing the
effect of such physical processes (e.g. stripping or evaporation). This is
strengthened by the fact that hydrodynamical mechanisms scale with the density
of the intra-group medium rather than the density of galaxies and that some
mechanisms, such as ram-pressure stripping, also scale with the velocity of
the galaxies. Assuming there is enough pressure in the intra-group medium,
these physical processes could affect the kinematics of the gas in the
galaxies and, therefore, their angular momentum. On the other hand,
gravitational interactions, in particular mergers, might also be an acceptable
explanation. Indeed, they would be more likely to happen for galaxies that
move slowly in the groups, they would affect both the morphology (contraction
of baryons) and the kinematics of the gas and the stars, in particular
enhancing their velocity dispersion. Furthermore, mergers could also be
compatible with the lack of correlation with the strength of the over-density
around the galaxies discussed in Sect. 5.2 as there is some observational
evidence (e.g. Shen et al. 2021) that such physical processes can locally
reduce the density after the merging event.

##  6 Conclusions

We have studied the dependence of the specific stellar angular momentum with
environment using the Fall relation for nearly 200 galaxies at $z\sim 0.7$
from the MAGIC survey. These include galaxies in the field and found in groups
of various densities. Their stellar angular momentum was estimated with a new
method that combines robust rotation curves of the ionised gas obtained with
MUSE, and used as a proxy of the stellar kinematics, with high-resolution HST
images. In particular, this approach allows us to alleviate the assumption of
axial symmetry for the stellar disk component.

We have shown that there is a visible impact of the environment on the angular
momentum of galaxies located in rich groups with respect to the field. This
effect is compatible with a depletion of angular momentum in the groups with
respect to the field of around $-0.12\text{\,}\mathrm{\text{dex}}$, which is
$2\sigma$ significant after accounting for biases induced by different mass
and redshift distributions between sub-samples. This first result is
compatible with previous studies performed on galaxy clusters rather than
groups, such as Pelliccia et al. (2019) or Pérez-Martínez et al. (2021).
Furthermore, we find that massive galaxies have much lower specific stellar
angular momentum values than what would be expected from extrapolating at the
high-mass end the Fall relation found at low mass. However, we cannot rule out
the possibility that this effect is statically induced rather than mass- or
environmentally-driven. In particular, we will need to combine MAGIC with
other MUSE-GTO surveys, such as MUSCATEL171717ESO ID 1104.A-0026, MUSE-Wide
(Herenz et al. 2017; Urrutia et al. 2019), MUSE-HDFS (Bacon et al. 2015), or
MUSE-HUDF (Bacon et al. 2017, 2023), to increase the statistics at the high-
mass end for the field sample.

When probing in more details the environment around the galaxies, we find
little evidence for a correlation between their stellar angular momentum and
the over-density around them, meaning that the observed depletion of angular
momentum does not seem primarily driven by the local density in the groups.
However, we do find that galaxies located below the Fall relation tend to be
slightly more dynamically bound (as traced by $\eta$, see Sect. 2.3) to their
host group than those located above. Nevertheless, the effect appears to be
mostly mass-driven with nearly all massive galaxies ($M_{\star}\gtrsim
10^{10.5}\,\unit{}$) identified as early accreted galaxies ($\eta<0.1$).

We also looked at whether the depletion of angular momentum in the groups with
respect to the field is associated to the position of the galaxies in their
host group or to their systemic velocity (normalised by their host group
velocity dispersion) with respect to the group’s redshift. Even though we do
not find any evidence that the position of the galaxies is linked to their
angular momentum, we find a strong dichotomy in terms of systemic velocity
between galaxies located below and above the Fall relation. Those below tend
to have a significantly lower systemic velocity than those located above, with
$|\Delta v|/\sigma_{V}\approx 0.7$ and 0.9 for those below and above,
respectively, and with 70% of the galaxies with $|\Delta v|/\sigma_{V}<0.7$
found on the bottom of the Fall relation. This result hints at physical
mechanisms that scale with the velocity at which galaxies move through the
intra-group medium.

Finally, we also find evidence for a strong link between the position of the
galaxies below and above the Fall relation and their ionised gas velocity
dispersion. Galaxies located above the relation appear to be strongly
rotationally supported, with a median velocity dispersion of around
$10\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$, whereas galaxies located
below the relation have a higher median velocity dispersion value of about
$40\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$. Nevertheless, we find a
similar correlation using field galaxies, thus highlighting the fact that this
effect might not be solely environmentally-driven. Still, the lower values
found in the field sub-sample point to a scenario where the groups enhance the
velocity dispersion of the gas as the galaxies lose angular momentum.

From these results, we infer that the most likely pathway through which
galaxies in groups lose partly their stellar angular momentum is by a change
of the shape of their rotation curve rather than by reducing their maximum
circular velocity or by redistributing their stellar mass. Both hydrodynamical
processes, in particular stripping, and gravitational interactions, in
particular mergers, can be viable scenarios that could produce both the
depletion of angular momentum in the groups with respect to the field and the
aforementioned correlations with $|\Delta v|/\sigma_{V}$ and with the ionised
gas velocity dispersion.

###### Acknowledgements.

We thank the referee for providing useful and constructive comments on the
submitted version of this paper. We dedicate this article in memory of Hayley
Finley. We acknowledge David Carton for his investment in the build-up of the
project. This work was supported by the Programme National Cosmology et
Galaxies (PNCG) of CNRS/INSU with INP and IN2P3, co-funded by CEA and CNES.
This work has been carried out through the support of the ANR FOGHAR
(ANR-13-BS05-0010-02), the OCEVU Labex (ANR-11-LABX-0060), and the A*MIDEX
project (ANR-11-IDEX-0001-02), which are funded by the “Investissements
d’avenir” French government program managed by the ANR. This work has been
carried out thanks to the support of the Ministry of Science, Technology and
Innovation of Colombia (MINCIENCIAS) PhD fellowship program No. 756-2016. This
research has made use of matplotlib (Hunter 2007), scipy (Virtanen et al.
2020), numpy (Harris et al. 2020) and astropy (Astropy Collaboration et al.
2013, 2018).

## References

  * Abril-Melgarejo et al. (2021) Abril-Melgarejo, V., Epinat, B., Mercier, W., et al. 2021, A&A, 647, A152
  * Ansar et al. (2023) Ansar, S., Kataria, S. K., & Das, M. 2023, MNRAS, 522, 2967
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33
  * Bacon et al. (2023) Bacon, R., Brinchmann, J., Conseil, S., et al. 2023, A&A, 670, A4
  * Bacon et al. (2015) Bacon, R., Brinchmann, J., Richard, J., et al. 2015, Astronomy & Astrophysics, 575, A75
  * Bacon et al. (2017) Bacon, R., Conseil, S., Mary, D., et al. 2017, A&A, 608, A1
  * Balogh et al. (2000) Balogh, M. L., Navarro, J. F., & Morris, S. L. 2000, ApJ, 540, 113
  * Beers et al. (1990) Beers, T. C., Flynn, K., & Gebhardt, K. 1990, AJ, 100, 32
  * Bois et al. (2011) Bois, M., Emsellem, E., Bournaud, F., et al. 2011, MNRAS, 416, 1654
  * Boquien et al. (2019) Boquien, M., Burgarella, D., Roehlly, Y., et al. 2019, Astronomy & Astrophysics, 622, A103
  * Bouché et al. (2016) Bouché, N., Finley, H., Schroetter, I., et al. 2016, The Astrophysical Journal, 820, 121
  * Bouché et al. (2013) Bouché, N., Murphy, M. T., Kacprzak, G. G., et al. 2013, Science, 341, 50
  * Bouché et al. (2021) Bouché, N. F., Genel, S., Pellissier, A., et al. 2021, Astronomy & Astrophysics, 654, A49
  * Brook et al. (2012) Brook, C. B., Stinson, G., Gibson, B. K., et al. 2012, Monthly Notices of the Royal Astronomical Society, 419, 771
  * Bruzual & Charlot (2003) Bruzual, G. & Charlot, S. 2003, Monthly Notices of the Royal Astronomical Society, 344, 1000
  * Burkert et al. (2016) Burkert, A., Förster Schreiber, N. M., Genzel, R., et al. 2016, The Astrophysical Journal, 826, 214
  * Cadiou et al. (2022) Cadiou, C., Dubois, Y., & Pichon, C. 2022, Monthly Notices of the Royal Astronomical Society, 514, 5429
  * Cadiou et al. (2022) Cadiou, C., Pontzen, A., & Peiris, H. V. 2022, Stellar angular momentum can be controlled from cosmological initial conditions, arXiv:2206.11913 [astro-ph]
  * Cappellari et al. (2013) Cappellari, M., Scott, N., Alatalo, K., et al. 2013, MNRAS, 432, 1709
  * Chabrier (2003) Chabrier, G. 2003, PASP, 115, 763
  * Charlot & Fall (2000) Charlot, S. & Fall, S. M. 2000, The Astrophysical Journal, 539, 718
  * Ciesla et al. (2021) Ciesla, L., Buat, V., Boquien, M., et al. 2021, Astronomy & Astrophysics, 653, A6
  * Ciesla et al. (2018) Ciesla, L., Elbaz, D., Schreiber, C., Daddi, E., & Wang, T. 2018, Astronomy & Astrophysics, 615, A61
  * Contini et al. (2016) Contini, T., Epinat, B., Bouché, N., et al. 2016, A&A, 591, A49
  * Cortese et al. (2016) Cortese, L., Fogarty, L. M. R., Bekki, K., et al. 2016, MNRAS, 463, 170
  * Cortese et al. (2016) Cortese, L., Fogarty, L. M. R., Bekki, K., et al. 2016, Monthly Notices of the Royal Astronomical Society, 463, 170
  * Cucciati et al. (2010) Cucciati, O., Marinoni, C., Iovino, A., et al. 2010, A&A, 520, A42
  * Danovich et al. (2015) Danovich, M., Dekel, A., Hahn, O., Ceverino, D., & Primack, J. 2015, Monthly Notices of the Royal Astronomical Society, 449, 2087
  * DeFelippis et al. (2017) DeFelippis, D., Genel, S., Bryan, G. L., & Fall, S. M. 2017, The Astrophysical Journal, 841, 16
  * del Pino et al. (2021) del Pino, A., Fardal, M. A., van der Marel, R. P., et al. 2021, The Astrophysical Journal, 908, 244
  * Di Teodoro et al. (2021) Di Teodoro, E. M., Posti, L., Ogle, P. M., Fall, S. M., & Jarrett, T. 2021, Monthly Notices of the Royal Astronomical Society, 507, 5820
  * Doroshkevich (1970) Doroshkevich, A. G. 1970, Astrofizika, 6, 581
  * Epinat et al. (2010) Epinat, B., Amram, P., Balkowski, C., & Marcelin, M. 2010, MNRAS, 401, 2113
  * Fall (1983) Fall, S. M. 1983, in Internal Kinematics and Dynamics of Galaxies, ed. E. Athanassoula, Vol. 100, 391–398
  * Fall & Efstathiou (1980) Fall, S. M. & Efstathiou, G. 1980, Monthly Notices of the Royal Astronomical Society, 193, 189
  * Fall & Romanowsky (2018) Fall, S. M. & Romanowsky, A. J. 2018, The Astrophysical Journal, 868, 133
  * Genel et al. (2015) Genel, S., Fall, S. M., Hernquist, L., et al. 2015, The Astrophysical Journal Letters, 804, L40
  * Gill et al. (2005) Gill, S. P. D., Knebe, A., & Gibson, B. K. 2005, Monthly Notices of the Royal Astronomical Society, 356, 1327
  * Graham et al. (2005) Graham, A. W., Driver, S. P., Petrosian, V., et al. 2005, AJ, 130, 1535
  * Guérou et al. (2017) Guérou, A., Krajnović, D., Epinat, B., et al. 2017, A&A, 608, A5
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357
  * Harrison et al. (2017) Harrison, C. M., Johnson, H. L., Swinbank, A. M., et al. 2017, MNRAS, 467, 1965
  * Herenz et al. (2017) Herenz, E. C., Urrutia, T., Wisotzki, L., et al. 2017, Astronomy & Astrophysics, 606, A12
  * Hetznecker & Burkert (2006) Hetznecker, H. & Burkert, A. 2006, Monthly Notices of the Royal Astronomical Society, 370, 1905
  * Hung et al. (2020) Hung, D., Lemaux, B. C., Gal, R. R., et al. 2020, Monthly Notices of the Royal Astronomical Society, 491, 5524
  * Hung et al. (2021) Hung, D., Lemaux, B. C., Gal, R. R., et al. 2021, Monthly Notices of the Royal Astronomical Society, 502, 3942
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90
  * Iovino et al. (2016) Iovino, A., Petropoulou, V., Scodeggio, M., et al. 2016, Astronomy & Astrophysics, 592, A78
  * Knobel et al. (2012) Knobel, C., Lilly, S. J., Iovino, A., et al. 2012, The Astrophysical Journal, 753, 121
  * Koekemoer et al. (2007) Koekemoer, A. M., Aussel, H., Calzetti, D., et al. 2007, ApJS, 172, 196
  * Kurapati et al. (2021) Kurapati, S., Chengalur, J. N., & Verheijen, M. A. W. 2021, MNRAS, 507, 565
  * Lagos et al. (2018) Lagos, C. d. P., Stevens, A. R. H., Bower, R. G., et al. 2018, MNRAS, 473, 4956
  * Lemaux et al. (2022) Lemaux, B. C., Cucciati, O., Le Fèvre, O., et al. 2022, Astronomy & Astrophysics, 662, A33
  * Lemaux et al. (2017) Lemaux, B. C., Tomczak, A. R., Lubin, L. M., et al. 2017, Monthly Notices of the Royal Astronomical Society, 472, 419
  * Mancera Piña et al. (2021a) Mancera Piña, P. E., Posti, L., Fraternali, F., Adams, E. A. K., & Oosterloo, T. 2021a, Astronomy & Astrophysics, 647, A76
  * Mancera Piña et al. (2021b) Mancera Piña, P. E., Posti, L., Pezzulli, G., et al. 2021b, Astronomy & Astrophysics, 651, L15
  * Massey et al. (2010) Massey, R., Stoughton, C., Leauthaud, A., et al. 2010, MNRAS, 401, 371
  * Mercier et al. (2022) Mercier, W., Epinat, B., Contini, T., et al. 2022, VizieR Online Data Catalog, J/A+A/665/A54
  * Mercier et al. (2022) Mercier, W., Epinat, B., Contini, T., et al. 2022, arXiv:2204.08724 [astro-ph], arXiv: 2204.08724
  * Navarro et al. (1996) Navarro, J. F., Frenk, C. S., & White, S. D. M. 1996, ApJ, 462, 563
  * Noble et al. (2013) Noble, A. G., Webb, T. M. A., Muzzin, A., et al. 2013, The Astrophysical Journal, 768, 118
  * Obreschkow & Glazebrook (2014) Obreschkow, D. & Glazebrook, K. 2014, The Astrophysical Journal, 784, 26
  * Peebles (1969) Peebles, P. J. E. 1969, Astrophysical Journal, 155, 393
  * Pelliccia et al. (2019) Pelliccia, D., Lemaux, B. C., Tomczak, A. R., et al. 2019, Monthly Notices of the Royal Astronomical Society, 482, 3514
  * Peng et al. (2002) Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H.-W. 2002, The Astronomical Journal, 124, 266
  * Pérez-Martínez et al. (2021) Pérez-Martínez, J. M., Ziegler, B., Dannerbauer, H., et al. 2021, A&A, 646, A53
  * Posti et al. (2018) Posti, L., Fraternali, F., Di Teodoro, E. M., & Pezzulli, G. 2018, Astronomy & Astrophysics, 612, L6
  * Quenouille (1956) Quenouille, M. H. 1956, Biometrika, 43, 353
  * Rizzo et al. (2018) Rizzo, F., Fraternali, F., & Iorio, G. 2018, MNRAS, 476, 2137
  * Roberts et al. (2015) Roberts, I. D., Parker, L. C., Joshi, G. D., & Evans, F. A. 2015, MNRAS, 448, L1
  * Romanowsky & Fall (2012) Romanowsky, A. J. & Fall, S. M. 2012, The Astrophysical Journal Supplement, 203, 17
  * Romanowsky & Fall (2012) Romanowsky, A. J. & Fall, S. M. 2012, The Astrophysical Journal Supplement Series, 203, 17
  * Rousseeuw & Driessen (2006) Rousseeuw, P. & Driessen, K. 2006, Data Min. Knowl. Discov., 12, 29
  * Salinas & Galaz (2021) Salinas, V. H. & Galaz, G. 2021, ApJ, 915, 125
  * Salpeter (1955) Salpeter, E. E. 1955, The Astrophysical Journal, 121, 161
  * Scoville et al. (2007) Scoville, N., Aussel, H., Brusa, M., et al. 2007, ApJS, 172, 1
  * Sérsic (1963) Sérsic, J. L. 1963, Boletin de la Asociacion Argentina de Astronomia La Plata Argentina, 6, 41
  * Shen et al. (2021) Shen, L., Lemaux, B. C., Lubin, L. M., et al. 2021, ApJ, 912, 60
  * Stewart et al. (2013) Stewart, K. R., Brooks, A. M., Bullock, J. S., et al. 2013, The Astrophysical Journal, 769, 74
  * Swinbank et al. (2017) Swinbank, A. M., Harrison, C. M., Trayford, J., et al. 2017, MNRAS, 467, 3140
  * Tomczak et al. (2019) Tomczak, A. R., Lemaux, B. C., Lubin, L. M., et al. 2019, MNRAS, 484, 4695
  * Tukey (1958) Tukey, J., W. 1958, The Annals of Mathematical Statistics, 29, 614
  * Urrutia et al. (2019) Urrutia, T., Wisotzki, L., Kerutt, J., et al. 2019, Astronomy & Astrophysics, 624, A141
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261
  * Weaver et al. (2022) Weaver, J. R., Kauffmann, O. B., Ilbert, O., et al. 2022, ApJS, 258, 11
  * White (1984) White, S. D. M. 1984, Astrophysical Journal, 286, 38
  * Zabl et al. (2019) Zabl, J., Bouché, N. F., Schroetter, I., et al. 2019, Monthly Notices of the Royal Astronomical Society, 485, 1961
  * Zavala et al. (2016) Zavala, J., Frenk, C. S., Bower, R., et al. 2016, Monthly Notices of the Royal Astronomical Society, 460, 4466

##  Appendix A Additional figures and tables

Figure 8: Fall relation between galaxies in the field (black squares) and
those in rich groups with more than ten members (red circles) using the flat
model rotation curve. See Fig. 4 for details regarding the legend. Figure 9:
Fall relation colour coded according to $\eta$ (see Sect. 2.3). This figure is
similar to Fig. 5 but uses the flat model rotation curve instead. Figure 10:
Fall relation colour coded according to the VMC method over-density estimate
$\delta$. Only galaxies located in rich groups with more than ten members are
shown as circles (black squares correspond to field galaxies). The mass model
rotation curve is used with the redshift cut ($0.5<z<0.9$) applied. As an
indication, the best-fit linear relations (with only the redshift cut applied,
see also Table 1) for the field and rich groups sub-samples with the stellar
mass cut ($M_{\star}<${10}^{10}\text{\,}\mathrm{M_{\sun}}$$) applied are shown
as grey and black plain lines, respectively, and without the mass cut applied
as grey and black dotted lines, respectively. The typical uncertainty is shown
as the black error-bar. Figure 11: Fall relation colour coded according to the
VMC method over-density estimate $\delta$. This figure is similar to Fig. 10
but uses a flat model rotation curve instead. Figure 12: Fall relation colour
coded according to the galaxies’ line-of-sight systemic velocity $|\Delta v|$
normalised by the velocity dispersion of their host group $\sigma_{V}$. This
figure is similar to Fig. 6 but uses a flat rotation curve model instead
Figure 13: Fall relation colour coded according to the median velocity
dispersion of the ionised gas component in the galaxies. This figure is
similar to Fig. 7 but uses a flat rotation curve model instead Table 3:
Median, 16th, and 84th percentiles estimates of the depletion of specific
stellar angular momentum†† at the high-mass end when using the groups’
richness as density estimate.

| Sub-sample | Stellar mass bin | $\Delta\beta$ | 16th, 84th
---|---|---|---|---
|  | [$\log_{10}\unit{}$] | [] | percentiles []
| (1) | (2) | (3) | (4)
Mass model | Field† |  $9.5$, $10.5$ | $-0.07$ |  $-0.49,$ | $0.15$
$10$, $11$ | $-0.74$ |  $-0.83,$ | $-0.57$
$10.5$, $11.5$ | $-0.78$ |  $-0.85,$ | $-0.73$
Large |  $9.5$, $10.5$ | $-0.17$ |  $-0.47,$ | $0.13$
$10$, $11$ | $-0.32$ |  $-0.59,$ | $0.02$
|  $10.5$, $11.5$ | $-0.53$ |  $-0.77,$ | $-0.23$
Flat model | Field† |  $9.5$, $10.5$ | $-0.11$ |  $-0.62,$ | $0.14$
$10$, $11$ | $-1.04$ |  $-1.22,$ | $-0.71$
$10.5$, $11.5$ | $-0.91$ |  $-1.01,$ | $-0.87$
Large |  $9.5$, $10.5$ | $-0.22$ |  $-0.65,$ | $0.07$
$10$, $11$ | $-0.35$ |  $-0.76,$ | $-0.08$
|  $10.5$, $11.5$ | $-0.63$ |  $-1.01,$ | $-0.29$

Notes: (1) Sub-sample (i.e. galaxies in the field or in rich groups), (2)
minimum and maximum bounds of the stellar mass bin, (3) median depletion of
specific stellar angular momentum in the given mass bin with respect to the
best-fit line obtained when fitting low-mass
($M_{\star}<${10}^{10}\text{\,}\mathrm{M_{\sun}}$$) galaxies in the same sub-
sample, and (4) 16th and 84th percentiles for the considered mass bin.
† Values for the field sub-sample are given as an indication but there are too
few galaxies in this sub-sample for all these stellar mass ranges to derive
robust constraints.
†† The depletion is evaluated as the best-fit model (with mass and redshift
cuts applied) minus the galaxies’ specific stellar angular momentum. The same
slope, corresponding to mass and redshift cuts applied (see Table 1), was used
for both sub-samples.

##  Appendix B Examples of various types of morpho-dynamical models

Figure 14: Examples of morpho-dynamical models obtained in Mercier et al.
(2022) and used in this analysis. For each sub-figure and from top to bottom
we have the HST image, the Galfit model, and the residuals in the leftmost
column, the velocity field map extracted with Camel, the best-fit MocKinG
velocity field, and the residuals in the middle column, and the raw velocity
dispersion map extracted with Camel, the beam-smearing model from MocKinG, and
the beam-smearing corrected velocity dispersion map from MocKinG in the
rightmost column. The top left sub-figure shows a galaxy with a seemingly
correct dynamical model, the top right one shows a low S/N galaxy without any
velocity gradient in its velocity field map that was flagged, the bottom left
one shows a low S/N galaxy with a weak but non-negligible velocity gradient
that was not flagged, and the bottom right one shows a massive galaxy with a
kinematically disturbed velocity field that was flagged as well.

##  Appendix C Updates on the morphological models

For this analysis and after inspecting the HST images, we decided to remodel
the morphology for the 17 following galaxies: 85_CGr23, 82_CGr30, 97_CGr32,
35_CGr35, 74_CGr35, 127_CGr35, 94_CGr51, 36_CGr61, 39_CGr61, 52_CGr61,
121_CGr61, 41_CGr79, 23_CGr84, 122_CGr84, 79_CGr114, 31_CGr172, and 68_CGr172
(see Table F.1 of Mercier et al. 2022 or Epinat et al., in prep. for the
galaxies’ IDs). Among these galaxies, we included a logarithmic spiral pattern
to the disk component181818Available since version three of Galfit for the
four following galaxies: 85_CGr23, 121_CGr61, 36_CGr61, and 79_CGr114. We did
so because these galaxies showed strong spiral and bar features so that their
previous disk morphological model used to fit the bar rather than the extended
disk. Even if the modelling of the spiral arms and of the bar is not trivial
and far from perfect, we did find more robust disk parameters once these
features were included, especially for the disk axis ratio and PA. Overall,
the remaining 13 galaxies had a centre that was slightly misaligned with the
bulge component which had the effect to mildly underestimate its contribution.
With the updated models, we now have better constraints on the galaxy’s centre
and on the parameters of the bulge component. Furthermore, after trying to
redo their morphology without more success, we decided to remove the 22
following galaxies:  (i) 12_CGr61, 24_CGr32, 26_CGr61, 29_CGr23, 54_CGr79,
76_CGr30, 76_CGr84 76_CGr172, 89_CGr79, 240_CGr30, and 240_CGr84 because they
are strongly bulge-dominated (probably elliptical) whose disks are too faint
to be efficiently constrained,  (ii) 55_CGr61, 66_CGr30, 67_CGr87, 73_CGr61,
75_CGr84, 76_CGr172, 79_CGr61, 82_CGr87, and 97_CGr34 because their disks are
detected but with too low S/N to be properly constrained, and  (iii)
79_CGr30and 84_CGr35 because they both show complex edge-on morphologies that
do not match their velocity fields. Indeed, 79_CGr30 looks like two edge-on
galaxies whose major axes are aligned but with a velocity field that spans the
two objects and that shows undisturbed rotation with a PA rotated by
$90\text{\,}\mathrm{\SIUnitSymbolDegree}$ with respect to the stellar
components. Similarly, galaxy 84_CGr35 seems to show multiple diffuse
components that might be the result of a past merger or a stripping event even
though its velocity field also shows an extended rotation which is not aligned
with the brightest component visible in the HST image. Therefore, the