# Stellar angular momentum of disk galaxies at z ≈ 0.7 in the MAGIC survey

W. Mercier, B. Epinat, T. Contini, D. Krajnović, L. Ciesla, B C Lemaux, V.

Abril-Melgarejo, L. Boogaard, D. Pelliccia

► To cite this version:

W. Mercier, B. Epinat, T. Contini, D. Krajnović, L. Ciesla, et al. Stellar angular momentum of disk galaxies at z ≈ 0.7 in the MAGIC survey. Astronomy & Astrophysics - A&A, 2023, 677, pp.A143. 10.1051/0004-6361/202346700. hal-04289519

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Submitted on 16 Nov 2023

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# Stellar angular momentum of disk galaxies at $z \approx 0.7$
in the MAGIC survey

## I. Impact of the environment*,**

W. Mercier<fnref n="1" />, B. Epinat<fnref n="2" />,<fnref n="3" />, T. Contini<fnref n="1" />, D. Krajnović<fnref n="4" />, L. Ciesla<fnref n="2" />, B. C. Lemaux<fnref n="5" />, V. Abril-Melgarejo<fnref n="6" />, L. Boogaard<fnref n="7" />, and D. Pelliccia<fnref n="8" />

Received 19 April 2023 / Accepted 22 June 2023

## ABSTRACT

**Aims.** At intermediate redshift, galaxy groups and 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 derived 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 ~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 angular momentum-stellar mass (Fall) relation as a function of environment.

**Results.** We measured a 0.12 dex (2σ significant) depletion of stellar angular momentum for low-mass galaxies ($M_* < 10^{10} M_\odot$) 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 a 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: general – 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 that galaxies have smoothly accreted large amounts of cold gas from their circum-galactic medium to sustain high values for the 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

<footnote n="1">¹ Institut de Recherche en Astrophysique et Planétologie (IRAP), Université de Toulouse, CNRS, UPS, CNES, 31400 Toulouse, France
e-mail: wilfried.mercier@lam.fr</footnote>

<footnote n="2">² Aix-Marseille Univ., CNRS, CNES, LAM, Marseille, France</footnote>

<footnote n="3">³ Canada-France-Hawaii Telescope, CNRS, 96743 Kamuela, HI, USA</footnote>

<footnote n="4">⁴ Gemini Observatory, NSF's NOIRLab, 670 N. A'ohoku Place, Hilo, HI 96720, USA</footnote>

<footnote n="5">⁵ Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA</footnote>

<footnote n="6">⁶ Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany</footnote>

<footnote n="7">⁷ Max Planck Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany</footnote>

<footnote n="8">⁸ UCO/Lick Observatory, Department of Astronomy & Astrophysics, UC Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA</footnote>

<footnote>* Table E.1 is only available at the CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/677/A143</footnote>

<footnote>** 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.</footnote>
take place predominantly in the disk plane of late-type galaxies and, thus, it not only drives 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. 2022a). 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, and 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 momen-
tum 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), stel-
lar 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 rela-
tion (in log-log space) between stellar mass and angular momen-
tum 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 rela-
tion found for bulge-less disks. The Fall relation has also been
studied at a 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 con-
sistent with galaxies gaining angular momentum at a fixed stel-
lar mass with cosmic time (i.e., with decreasing redshift). Inter-
estingly, 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 a lower SFR-weighted angular momentum than
rotationally supported galaxies at a fixed stellar mass. This cor-
relation 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 angu-
lar momentum in observed galaxies is not straightforward as it
would require one 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 of measuring 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 kine-
matics 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 has appeared in Bouché et al. (2021)
where it is 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 num-
ber 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 angu-
lar 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) have proposed
another method that numerically integrates the angular momen-
tum by summing the contribution of each spatial pixel (spaxel) in
a data cube. The main advantage is that it alleviates the assump-
tion 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 spectro-
scopic 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 angu-
lar 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 a fixed stellar
mass could be due to galaxy mergers, which is 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,
the results from Pelliccia et al. (2019) and Pérez-Martínez et al.
(2021) 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 func-
tions between different datasets.

This paper is the first of a series. In this analysis, we pro-
pose to study the impact of the environment on the stellar angu-
lar 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 and
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, and we explain how we performed the morphological and kinematic modellings, as well as 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 Λ cold dark matter cosmology with $H_0 = 70 \text{ km s}^{-1} \text{ Mpc}^{-1}$, $\Omega_M = 0.3$, 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 targeting 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 sub-sample 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 Å to 9350 Å with a spatial sampling of 0.2" and a spectral sampling of 1.25 Å. The observing strategy and data reduction can be found in Epinat et al. (in prep., but see also Abril-Melgarejo et al. 2021 and Mercier et al. 2022). 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 Å 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" at 4000 Å and 0.53" at 8000 Å which correspond respectively to 4.8 kpc and 3.8 kpc at $z = 0.7$. In addition, we also used high-resolution 4" × 4" HST stamps in the F814W filter<fnref n="1" /> 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 $FWHM_{HST} = 0.0852''$ and $\beta = 1.9$.

Additionally, we also estimated the galaxies' stellar mass and SFR values with a 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 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 dex for galaxies more massive than $10^8 M_\odot$. Furthermore, the latest CIGALE-based stellar masses are slightly offset by roughly 0.05 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, readers can refer to Sects. 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 MUSE-HUDF (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 $R_{eff}$ (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).

<footnote n="1">¹ This is the best HST spatial resolution available in the COSMOS field (PSF FWHM below 0.1", pixel scale of 0.03").</footnote>
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 $R_{22} = 2.2R_d$ in the plane of the disk, with $R_d$ defined as the disk scale length, we also introduced a corrected stellar mass ($M_{\star,corr}$) 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 CAMEL<fnref n="2" /> 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 $S/N > 5$ 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 and 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 MocKinG<fnref n="3" />. 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). We 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 $V_{h,max}$), (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; Mercier et al. 2022). For groups with more than seven galaxy members, the maximum sky projected separation used was 375 kpc and the maximum line-of-sight velocity separation was 500 km 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)

$$ \eta = \frac{R_{\text{proj}}}{R_{200}} \times \frac{|\Delta v|}{\sigma_v}, \qquad (1) $$

where $R_{\text{proj}}$ 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; Hung et al. 2020, 2021) where both photometric and spectroscopic redshifts in COSMOS are used to estimate the density $\Sigma$ in cells of 75 kpc × 75 kpc and ±3.75 Mpc wide (proper distances, see also Epinat et al., in prep.). The over-density $\delta$ is then estimated as $1 + \delta = \Sigma/\Sigma_{\text{med}}$, where $\Sigma_{\text{med}}$ 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

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

<footnote n="2">² https://gitlab.lam.fr/bepinat/CAMEL</footnote>

<footnote n="3">³ https://gitlab.lam.fr/bepinat/MocKinG</footnote>
<figure><img src="image_2.png" /><figcaption>Fig. 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)).</figcaption></figure>

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 are 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 ($R_{eff,d}$) and apparent axis ratio ($q$) is larger than the surface of the MUSE PSF within its FWHM. In other words, the size selection writes

$$2 \sqrt{q} \times R_{eff,d}/\mathrm{FWHM}_{[OII]}(z) > 1, \quad (2)$$

where FWHM$_{[OII]}(z)$ is the MUSE PSF FWHM evaluated at the redshift $z$ of the galaxy and where $R_{eff,d}$ and FWHM$_{[OII]}$ must have the same angular or length unit<fnref n="4" />. 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 $i > 25^{\circ}$. 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 $i > 30^{\circ}$ 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., $R_{eff,d}$ and $qR_{eff,d}$ along the major and minor axes, respectively) and half the PSF FWHM<fnref n="5" />. Putting everything together, the S/N selection criterion writes<fnref n="6" />

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

where $x = 2R_{eff,d}/\mathrm{FWHM}_{[OII]}(z)$, with FWHM$_{[OII]}(z)$ and $R_{eff,d}$ 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 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 ($i \ge 60^{\circ}$).

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’<fnref n="7" />: (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

<footnote n="4">⁴ We 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.</footnote>

<footnote n="5">⁵ In other terms, the observed surface of the stellar disk at an intrinsic distance $R_{eff,d}$ writes $\pi \sqrt{(R_{eff,d}^2 + (\mathrm{FWHM}/2)^2)(q^2 R_{eff,d}^2 + (\mathrm{FWHM}/2)^2)}$.</footnote>

<footnote n="6">⁶ The derivation assumes constant flux and S/N maps, as well as Poisson's statistics for the noise.</footnote>

<footnote n="7">⁷ See Table F.1 of Mercier et al. (2022) or Epinat et al. (in prep.) for the galaxies' IDs.</footnote>
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_CGR84 because 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_CGR35 and 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_CGR84 because 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_CGR35 because 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 keep them in what follows and we 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. B.1. 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 $J_\star$ in galaxies is not straightforward given that it involves knowing a priori the 3D stellar mass density $\rho_M$ as well as the 3D velocity vector $V$ of the stars. In general terms, the stellar angular momentum integrated in a volume $V$ writes

$$J_\star(\mathbf{r}) = \int_{V} d^3\mathbf{r} \rho_M(\mathbf{r}) \mathbf{r} \times V(\mathbf{r}), \quad (4)$$

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. 2022b) 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 $\log_{10} M_\star/M_\odot \le 9-9.5$) 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

$$j_\star(S) = \int_S dS R \Sigma_M(R, \theta) V_\theta(R, \theta)/M_\star, \quad (5)$$

where $S$ is the surface in the plane of the stellar disk over which the angular momentum is integrated, $dS = R d\theta dR$ is the integrand, $R$ is the distance to the centre of the galaxy, $\theta$ is the azimuthal angle in the plane of the disk, $\Sigma_M$ 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 $S$ 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 $\Sigma_M$ 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

$$j_\star(R) = 2\pi\Upsilon_\star \int_0^R dR' R'^2 \Sigma(R') V_\theta(R')/M_\star, \quad (6)$$

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 disks<fnref n="8" /> 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 $R_{22} = 2.2R_d$, where $R_d$ 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 $x'$ and $y'$. 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 $\Sigma_M = \Upsilon_*\Sigma$, with $\Sigma$ the intrinsic surface brightness distribution of the disk, and $M_* = \Upsilon_*F_{tot}$, with $F_{tot}$ the total flux integrated in the same aperture as $M_*$. Then, each pixel at position $(x', y')$ contributes to the specific stellar angular momentum as

$$j_{*,\text{pix}}(x', y') = R F(x', y') V_{\theta}(x', y') / F_{\text{tot}}, \quad (7)$$

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 $(x', y')$ on 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 MUSE-HUDF, 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

$$j_{*}(<r) = \sum_{\{x', y' \mid R<r\}} j_{*,\text{pix}}(x', y'). \quad (8)$$

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" per spaxel for a PSF FWHM of 0.5" on average for MUSE versus 0.03" per pixel for a PSF FWHM of roughly 0.1" 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 MUSE-HUDF 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 MUSE-HUDF), 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 MUSE-HUDF, 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

<footnote n="8">⁸ Equations (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.</footnote>
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 MUSE-HUDF 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 MUSE-HUDF 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 assess 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 $R_{22}^9$. 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 ($M_{\star} \le 10^9 M_{\odot}$) 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). Incidentally, this means that very few galaxies are kept below this limit (18 in total vs. 152 for the galaxies removed by the selection) and that this analysis will therefore focus on galaxies in the stellar mass range $M_{\star} \sim 10^9-10^{11} M_{\odot}$.

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 rota-

tion 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. (D.18) 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 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

$$ \log_{10} j_{\star} [\text{kpc km s}^{-1}] = \beta + \alpha(\log_{10} M_{\star} [M_{\odot}] - p), \quad (9) $$

where $\beta$ is the zero point, $\alpha$ is the slope, and $p = 9.8$ dex is the pivot point taken as the decimal logarithm of the median stellar mass (in solar masses) of the kinematics sample$^{10}$. 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 mostly investigate variations in the zero point of the Fall relation as a function of galaxy environment for a fixed slope. Nevertheless, we 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 LTSFit$^{11}$ (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 sub-sample 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 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 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.

<footnote>$^9$ We note that photometric and kinematics measurements beyond this radius become less robust because of the significant drop in S/N.</footnote>

<footnote>$^{10}$ We include a pivot point in order to reduce the correlation between the slope and the zero point during the fit.</footnote>

<footnote>$^{11}$ https://pypi.org/project/ltsfit/</footnote>
<figure><img src="image_3.png" /><figcaption>**Fig. 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).</figcaption></figure>

## 4.2. Assessing the reliability of the method

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 assess 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 MUSE-HUDF) 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 ± 0.35 dex for the entire kinematics sample. This becomes 0.1 ± 0.15 dex when considering the selected sample only and it is amplified to 0.45 ± 0.4 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 ± 0.25 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. B.1). 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 momentum<fnref n="12" />. 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 ± 0.1 dex, whereas unselected galaxies tend have slightly larger values with an average difference of roughly -0.15 ± 0.25 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 $j_*$ 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.

<footnote n="12">¹² The 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.</footnote>
<figure><img src="image_4.png" /><figcaption>**Fig. 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 kpc (dashed line) and 15 kpc (plain line, see Eq. (D.18)).</figcaption></figure>

**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**

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 with any group by the FoF algorithm or that were associated with 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 $M_{\star} < 10^{10} M_{\odot}$ 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
<figure><img src="image_5.png" /><figcaption>**Fig. 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 $M_{\star} < 10^{10} M_{\odot}$ 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.</figcaption></figure>

galaxies at $M_{\star} \gtrsim 10^{10} M_{\odot}$, for more details see the discussion in Mercier et al. (2022), as well as by a potential redshift evolution of the Fall relation<fnref n="13" />. 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.

Without any cut, we obtain a 3.5σ-significant difference (with σ the jackknife-derived standard-deviation) in zero points of 0.12 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 dex (i.e., the difference remains the same), but their uncertainties increase as well (changing from 0.03 dex to 0.05 dex), so that the difference in zero points reduces to 2σ-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

<table><caption>**Table 1.** Best-fit values for the Fall relation using the groups' richness as density estimate.</caption><thead><tr><th></th><th>Cut<br>(1)</th><th>Sub-sample<br>(2)</th><th>Nb. galaxies<br>(3)</th><th>α<br>(4)</th><th>β<br>(5)</th><th>95% CI<br>(6)</th></tr></thead><tbody><tr><td rowspan="6">Mass model</td><td rowspan="2">None</td><td>Field</td><td>85</td><td rowspan="2">0.50</td><td rowspan="2">2.80</td><td rowspan="2">2.73, 2.86</td></tr><tr><td>Large</td><td>75</td></tr><tr><td rowspan="2">Mass</td><td>Field</td><td>67</td><td rowspan="2">0.83</td><td rowspan="2">2.96</td><td rowspan="2">2.89, 3.04</td></tr><tr><td>Large</td><td>36</td></tr><tr><td rowspan="2">Redshift</td><td>Field</td><td>33</td><td rowspan="2">0.52</td><td rowspan="2">2.76</td><td rowspan="2">2.67, 2.86</td></tr><tr><td>Large</td><td>66</td></tr><tr><td rowspan="2"><b>Both</b></td><td><b>Field</b></td><td><b>27</b></td><td rowspan="2"><b>0.80</b></td><td rowspan="2"><b>2.92</b></td><td rowspan="2"><b>2.81, 3.02</b></td></tr><tr><td><b>Large</b></td><td><b>30</b></td></tr><tr><td rowspan="6">Flat model</td><td rowspan="2">None</td><td>Field</td><td>85</td><td rowspan="2">0.55</td><td rowspan="2">2.73</td><td rowspan="2">2.65, 2.80</td></tr><tr><td>Large</td><td>75</td></tr><tr><td rowspan="2">Mass</td><td>Field</td><td>67</td><td rowspan="2">0.94</td><td rowspan="2">2.92</td><td rowspan="2">2.83, 2.99</td></tr><tr><td>Large</td><td>36</td></tr><tr><td rowspan="2">Redshift</td><td>Field</td><td>33</td><td rowspan="2">0.57</td><td rowspan="2">2.66</td><td rowspan="2">2.54, 2.78</td></tr><tr><td>Large</td><td>66</td></tr><tr><td rowspan="2"><b>Both</b></td><td><b>Field</b></td><td><b>27</b></td><td rowspan="2"><b>0.91</b></td><td rowspan="2"><b>2.86</b></td><td rowspan="2"><b>2.74, 2.99</b></td></tr><tr><td><b>Large</b></td><td><b>30</b></td></tr></tbody></table>

**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 $M_{\star} < 10^{10} M_{\odot}$ and the redshift cut according to $0.5 < z < 0.9$. Boldface values represent the main scientific results of the analysis. Other values are provided to separate the effects of stellar mass, redshift, and environment.

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 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. A.1). Compared to the mass models, the zero point changes by −0.06 dex for either sub-sample. Without any cuts, we get a difference in zero points of 0.22 dex (4.5σ-significant), which reduces to 0.12 dex (around 2σ-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 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 dex depletion in groups and 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 dex (0.25 dex for the flat model) and of the rich groups sub-sample by 0.14 dex (0.21 dex). The main impact of this change of slope is to bring the most massive galaxies in the sample ($M_{\star} \gtrsim 10^{10} M_{\odot}$) below the best-fit lines obtained when applying either the redshift cut

<footnote n="13">¹³ Galaxies 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 redshift and stellar mass distributions in Figs. 1 and 2 of Mercier et al. (2022).</footnote>
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 dex with respect to the best-fit line, see Table A.1 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 ±0.03 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 dex with the redshift cut and −0.07 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

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 with 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 ($0.4 < \eta < 2$)$^{14}$. 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

<figure><img src="image_6.png" /><figcaption>**Fig. 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. A.2 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.</figcaption></figure>

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 $10^9 < M_\star/M_\odot < 10^{10.5}$ and around 75% of the early accreted galaxies ($\eta < 0.1$) have $M_\star > 10^{10} M_\odot$ (more than 50% above $10^{10.5} M_\odot$). Thus, there is a strong dichotomy between these two classes which seems to be mostly driven by stellar mass, with a transition at around $10^{10} M_\odot$. 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 ($M_\star < 10^{10.5} M_\odot$) 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. A.2).

<footnote n="14">14 We do not have galaxies with $\eta > 2$ because all these objects have already been associated with the field sub-sample by the FoF algorithm.</footnote>
<table><caption>**Table 2.** Global and local density in various stellar mass bins for the sub-sample of galaxies located in rich groups.</caption><thead><tr><th>Nb. galaxies (1)</th><th>Stellar mass bin [log<sub>10</sub> M<sub>☉</sub>] (2)</th><th>η (3)</th><th>Local overdensity δ (4)</th></tr></thead><tbody><tr><td>13</td><td>8.5, 9.5</td><td>0.19<sup>0.71</sup><sub>0.09</sub></td><td>0.81<sup>1.00</sup><sub>0.40</sub></td></tr><tr><td>22</td><td>9, 10</td><td>0.20<sup>0.34</sup><sub>0.10</sub></td><td>0.82<sup>1.10</sup><sub>0.48</sub></td></tr><tr><td>37</td><td>9.5, 10.5</td><td>0.17<sup>0.31</sup><sub>0.09</sub></td><td>0.86<sup>1.11</sup><sub>0.54</sub></td></tr><tr><td>31</td><td>10, 11</td><td>0.10<sup>0.24</sup><sub>0.04</sub></td><td>0.98<sup>1.34</sup><sub>0.64</sub></td></tr><tr><td>18</td><td>10.5, 11.5</td><td>0.04<sup>0.06</sup><sub>0.04</sub></td><td>1.19<sup>1.48</sup><sub>0.83</sub></td></tr></tbody></table>

**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 overdensity $\delta$. 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. A.3 and A.4 for the Fall relation colour-coded with $\delta$). We do find an increase from $\delta \approx 5$ for $M_{\star} < 10^{10} M_{\odot}$ 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. A potential 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 1.14 times 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 ($M_{\star} \ge 10^{10.5} M_{\odot}$) 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. A.5 for the flat model rotation curve (both for galaxies in rich groups only). First, we find that massive galaxies ($M_{\star} \ge 10^{10.5} M_{\odot}$) 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 ($M_{\star} \le 10^{10.5} M_{\odot}$), 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.

### 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 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 dispersion<fnref n="15" /> is shown in Fig. 7 for the mass model rotation curve and in Fig. A.6 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 $40 \text{ km s}^{-1}$<fnref n="16" />, whereas those located above it rather have values around $10 \text{ km s}^{-1}$. Furthermore, 75% of the galaxies above the best-fit line have an ionised gas velocity dispersion below $25 \text{ km s}^{-1}$ (hence the low median value), whereas 75% of the galaxies below the best-fit line have values above $25 \text{ km s}^{-1}$, with a peak at around $50 \text{ km s}^{-1}$. 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.

<footnote n="15">¹⁵ Measured from the beam-smearing- and LSF-corrected velocity dispersion map extracted from the MUSE cubes.</footnote>

<footnote n="16">¹⁶ We note that the typical LSF dispersion is around $20-30 \text{ km s}^{-1}$.</footnote>
<figure><img src="image_7.png" /></figure>

**Fig. 6.** Fall relation for galaxies in rich groups (with more than ten members) colour coded by the galaxies' line-of-sight systemic velocity $|\Delta v|/\sigma_v$ 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. A.5 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.

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 $25 \text{ km s}^{-1}$, whereas roughly 70% of those below have a gas velocity dispersion above $25 \text{ km s}^{-1}$. 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 $50 \text{ km s}^{-1}$, whereas only 5% of the field galaxies show a dispersion this high.

## 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

<figure><img src="image_8.png" /></figure>

**Fig. 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. A.5 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.

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 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 ($M_{\star} < 10^{10} M_{\odot}$) 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 overdensity 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 overdensity 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 has allowed 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 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 a 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 MUSCATEL$^{17}$, 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 the environment around the galaxies in more detail, we find little evidence for a correlation between their stellar angular momentum and the overdensity 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 ($M_{\star} \ge 10^{10.5} M_{\odot}$) 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 with 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 the systemic velocity between galaxies located below and above the Fall relation. Those below it tend to have a significantly lower systemic velocity than those located above it, 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 in 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 $10 \text{ km s}^{-1}$, whereas galaxies located below the relation have a higher median velocity dispersion value of about $40 \text{ km s}^{-1}$. Nevertheless, we find a similar correlation using field galaxies, which highlights 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 their stellar angular momentum in part is by a change in 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, most notably 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

<footnote n="17">17 ESO ID 1104.A-0026.</footnote>
managed by the ANR. This work has been carried out thanks to the support of the Ministry of Science, Technology and Innovation of Colombia (MINCIEN-CIAS) 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 2013, 2018).

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**Appendix A: Additional figures and tables**

<figure><img src="image_9.png" /><figcaption>**Fig. A.1.** 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.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. A.2.** 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.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. A.3.** 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 ($M_{\star} < 10^{10} M_{\odot}$) 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.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. A.4.** Fall relation colour coded according to the VMC method over-density estimate $\delta$. This figure is similar to Fig. A.3 but uses a flat model rotation curve instead.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Fig. A.5.** 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</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. A.6.** 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</figcaption></figure>

<table><caption>**Table A.1.** 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.</caption><thead><tr><th></th><th>Sub-sample<br>(1)</th><th>Stellar mass bin<br>[log<sub>10</sub> M<sub>☉</sub>]<br>(2)</th><th>Δβ<br>[dex]<br>(3)</th><th>16th, 84th<br>percentiles [dex]<br>(4)</th></tr></thead><tbody><tr><td rowspan="3">Mass model</td><td>Field<sup>†</sup></td><td>9.5, 10.5<br>10, 11<br>10.5, 11.5</td><td>-0.07<br>-0.74<br>-0.78</td><td>-0.49, 0.15<br>-0.83, -0.85<br>-0.73</td></tr><tr><td><strong>Large</strong></td><td><strong>9.5, 10.5</strong><br><strong>10, 11</strong><br><strong>10.5, 11.5</strong></td><td><strong>-0.17</strong><br><strong>-0.32</strong><br><strong>-0.53</strong></td><td><strong>-0.47, 0.02</strong><br><strong>-0.59, -0.77</strong><br><strong>-0.23</strong></td></tr><tr><td>Field<sup>†</sup></td><td>9.5, 10.5<br>10, 11<br>10.5, 11.5</td><td>-0.11<br>-1.04<br>-0.91</td><td>-0.62, 0.14<br>-1.22, -1.01<br>-0.87</td></tr><tr><td><strong>Large</strong></td><td></td><td><strong>9.5, 10.5</strong><br><strong>10, 11</strong><br><strong>10.5, 11.5</strong></td><td><strong>-0.22</strong><br><strong>-0.35</strong><br><strong>-0.63</strong></td><td><strong>-0.65, -0.08</strong><br><strong>-0.76, -0.29</strong><br><strong>-1.01</strong></td></tr></tbody></table>

**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 ($M_{\star} < 10^{10}$ M$_{☉}$) 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><img src="image_15.png" /><figcaption>**Fig. B.1.** 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 MUSE-HUDF 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.</figcaption></figure>
# 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 component<fnref n="18" /> 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_cgr30 and 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° 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 morphological and mass models for these two objects would be too uncertain to compute their angular momentum.

# Appendix D: Deriving the angular momentum

Our goal is to derive the angular momentum of the stars in the MAGIC survey. We can split distribution of stars into two main components: the stellar disk and the stellar bulge. Among the two, the disk is the simplest to derive its angular momentum because of the assumptions that are usually made on its geometry and on its kinematics. Besides, bulges are usually thought of as stellar systems dynamically supported by random motions. In our case, because we model the bulge component as a spherical system, this means that its angular momentum should be nearly null. Thus, we make the assumption that the bulge component does not possess any angular momentum and we focus on deriving that of the disk component only.

## D.1. General derivation

The angular momentum of a system is defined from its 3D mass distribution $\rho(r)$ and 3D velocity field $V(r)$ as

$$J = \int_V d\tau \rho(r) r \times V(r), \qquad (D.1)$$

where $V$ represents the volume over which the integral is computed. In cylindrical coordinates, the cross product writes

$$r \times V(r) = RV_{\theta}\hat{z} + (zV_{R} - RV_{z})\hat{\theta} - zV_{\theta}\hat{R}, \qquad (D.2)$$

where $V(r) = V_R \hat{R} + V_{\theta}\hat{\theta} + V_z \hat{z}$<fnref n="19" />. Assuming we want to derive the angular momentum of a disk-like distribution where only the tangential component of the velocity field is non-zero and does not depend on $z$, that is $V_R = V_z = 0$ and $V_{\theta} = V_{\theta}(R, \theta)$, then the angular momentum simplifies to

$$J = \int_V d\theta \, dR \, dz \, \rho(r) \left[ R^2 V_{\theta}(R, \theta) \hat{z} - Rz V_{\theta}(R, \theta) \hat{R} \right]. \qquad (D.3)$$

## D.2. Angular momentum of thick disks

The angular momentum of a disk galaxy with a non-zero thickness profile $f(z)$ is similar to that of a razor-thin disk galaxy with the same surface mass density $\Sigma_M(R)$ if the three following criteria are met:

(i) $\rho(r) = \Sigma_M(R, \theta)f(z)$,

(ii) $\int_R dz \, f(z) = 1$,

(iii) $f(z) = f(-z)$.

Criterion (i) means that it is possible to separate the surface mass density profile from the thickness profile. For instance, this is the case for a double exponential profile where $f(z) = \exp\{-|z|/h_z|}/h_z$. Among the three conditions, criterion (ii) is always met since it is required in order to recover the surface mass density when integrating the 3D mass distribution along the vertical axis with respect to the disk plane, that is $\Sigma_M(R, \theta) = \int_R dz \, \rho(r)$. Finally, criterion (iii) is usually assumed to simplify the shape of the vertical profile. If these three conditions are met, one can see that the rightmost integral in Eq. D.3 vanishes when integrating along the vertical axis, whereas only the $R$ and $\theta$ integrations remain in the leftmost one. Thus, we get an angular momentum integral similar to that of a razor-thin disk<fnref n="20" />

$$J = \int_S d\theta \, dR \, R^2 \, \Sigma_M(R, \theta) V_{\theta}(R, \theta) \hat{z}, \qquad (D.4)$$

where $S$ represents the surface over which the $R$ and $\theta$ integrations are carried out. It is common practice to normalise the angular momentum by the total stellar mass to define a specific angular momentum $j = |J|/M_*$ (e.g., Fall 1983; Romanowsky & Fall 2012; Bouché et al. 2021) but it is also possible to normalise it by the mass of the disk only. For instance, if we normalise the angular momentum measured over some surface $S$ by the total disk mass, we get

$$j = \frac{\int_S d\theta \, dR \, R^2 \, \Sigma_M(R, \theta) V_{\theta}(R, \theta)}{\int_{[0,2\pi] \times \mathbb{R}_+} d\theta \, dR \, R \, \Sigma_M(R, \theta)}, \qquad (D.5)$$

<footnote n="18">¹⁸ Available since version three of MUSE-HUDF</footnote>

<footnote n="19">¹⁹ In the following, when the velocity field is fully described by $V_{\theta}$, we will refer to it as the galaxy rotation curve.</footnote>

<footnote n="20">²⁰ For a razor-thin disk we have $f(z) = \delta(z)$, with $\delta$ a Dirac distribution, so criteria (i) to (iii) are also met.</footnote>
or in the case where there is no dependence on θ for both the surface mass density and the velocity:

$$j = \frac{\int_{R} dR R^{2} \Sigma_{M}(R) V_{\theta}(R)}{\int_{R_{+}} dR R \Sigma_{M}(R)}. \qquad (D.6)$$

### D.3. Angular momentum for special rotation curves

In most general terms, if one wants to compute the specific angular momentum for any surface mass density and velocity profile, one needs to numerically integrate Eq. D.5 or D.6 depending on whether there is a θ dependence or not. However, there exist for a few specific velocity profiles simplified formulae which can be computed either numerically or analytically in certain cases. We present in the following simplified expressions for some rotation curves, assuming no θ dependence on both the surface mass density and the rotation curve.

#### D.3.1. Constant rotation curve

If one assumes a constant rotation curve $V_{\theta}(R) = V_{\theta}$, then the angular momentum up to radius r is given by

$$j(r) = V_{\theta} \times \frac{M_2\{\Sigma_M\}_0^r}{M_1\{\Sigma_M\}_0^{\infty}}, \qquad (D.7)$$

where $M_k\{\Sigma_M\}_{\alpha}^{\beta}$ is the kth radial moment of the surface brightness distribution $\Sigma_M$ integrated between $\alpha$ and $\beta$, that is

$$M_k\{\Sigma_M\}_{\alpha}^{\beta} = \int_{\alpha}^{\beta} dR R^k \Sigma_M(R). \qquad (D.8)$$

#### D.3.2. Flat model

If one assumes the rotation curve to be a flat model as in Abril-Melgarejo et al. (2021), that is described by the following formula:

$$V_{\theta}(R) = V_t \times \begin{cases} R/R_t, & \text{if } R < R_t \\ 1, & \text{otherwise} \end{cases} \qquad (D.9)$$

then the angular momentum is given by

$$j(r) = \frac{V_t}{R_t M_1 \{\Sigma_M\}_0^{\infty}} \times \begin{cases} M_3 \{\Sigma_M\}_0^R, & \text{if } R < R_t \\ M_3 \{\Sigma_M\}_0^R + R_t M_2 \{\Sigma_M\}_0^r, & \text{otherwise} \end{cases} \qquad (D.10)$$

### D.4. Angular momentum for a Sérsic profile

If the surface mass density can be further described by a Sérsic profile (Sérsic 1963) with Sérsic index n, effective radius $R_{eff}$ and central surface mass density $\Sigma_{M,0}$, then we have

$$J = \Sigma_{M,0} \int_{R} dR R^{2} e^{-b_{n}(R/R_{eff})^{1/n}} \int_{\theta} d\theta V_{\theta}(R, \theta) \hat{z}, \qquad (D.11)$$

where $b_n$ is the solution of the equation $\Gamma(2n) = 2\gamma(2n, b_n)$ (Graham et al. 2005), with $\gamma$ and $\Gamma$ the lower incomplete and complete gamma functions, respectively. Without any $\theta$ dependence, the specific angular momentum writes

$$j = \frac{2 b_n^{2n}}{(2n)!} \int_{R} dR (R/R_{eff})^2 V_{\theta}(R) e^{-b_n(R/R_{eff})^{1/n}}, \qquad (D.12)$$

where the normalisation factor comes from the analytical expression for the total mass of a Sérsic profile (e.g., Mercier et al. 2022). In the case of a flat rotation curve as in Eq. D.9, it is possible to compute analytically Eq. D.8 and therefore Eq. D.10. Indeed, we have

$$M_k\{\Sigma_M\}_{\alpha}^{\beta} = \frac{n \Sigma_{M,0} R_{eff}^{k+1}}{b_n^{n(k+1)}} \left[ \gamma \left( n(k+1), b_n \left( \frac{\beta}{R_{eff}} \right)^{1/n} \right) - \gamma \left( n(k+1), b_n \left( \frac{\alpha}{R_{eff}} \right)^{1/n} \right) \right]. \qquad (D.13)$$

### D.5. Recovering the RF12 approximation

The Romanowsky & Fall (2012) approximation, hereafter referred as RF12, assumes an exponential disk with a constant rotation curve and computes the angular momentum at infinity. Thus, we can use Eq. D.10 and D.13 to check whether we recover it. First, we can see that for a flat model and in the limit where the rotation curve becomes constant, that is $R_t \to 0$, Eq. D.10 reduces to D.7. Furthermore, evaluated at infinity and for an exponential disk model (i.e. $n = 1$) Eq. D.13 simplifies to

$$M_k\{\Sigma_M\}_0^{\infty} = \Sigma_{M,0} R_d^{k+1} \Gamma(k+1), \qquad (D.14)$$

where $\Gamma$ is the complete gamma function and $R_d = R_{eff,d}/b_1$ is the disk scale length, with $b_1 \approx 1.6783$. Thus, given Eq. D.7, we recover the Romanowsky & Fall (2012) approximation

$$j = 2V_{\theta}R_{d}. \qquad (D.15)$$

### D.6. Angular momentum for an exponential disk in equilibrium

It can be interesting to derive the angular momentum for an exponential disk model that is stable against its own gravity, that is without any DM halo or bulge component. We consider the case where the normalisation is taken at the same radius as the angular momentum. Given the expression of the integrated mass and circular velocity for such a model (see Eqs. C.1, D.8, and D.9 of Mercier et al. (2022), its integral writes

$$j(R) = \frac{\sqrt{\pi G R_d \Sigma_{M,d}(0)}}{2\pi R_d^3 \gamma(2, R/R_d)} \int_0^R dr \, r^3 e^{-r/R_d} f\left(\frac{r}{2R_d}\right), \qquad (D.16)$$

where G is the gravitational constant, $R_d$ is the disk scale length, $\Sigma_{M,d}(0)$ is the central surface mass density of the disk, $\gamma$ is the lower incomplete gamma function, and f is a function of Bessel functions defined in Appendix D of Mercier et al. (2022). If we define $y = x/R_d$, then it simplifies to

$$j(R) = \frac{\sqrt{GM_d(R)R_d}}{2\sqrt{2\pi}\gamma^2(2, R/R_d)} \times \int_0^{R/R_d} dy \, y^3 e^{-y} f(y/2), \qquad (D.17)$$

where $M_d(R)$ is the mass of the disk within the radius R. At $R_{22} = 2.2R_d$, the radius of maximum velocity for the disk, both $\gamma$ and the integral on the right-hand side become constants and Eq. D.17 further reduces to

$$j(R_{22}) [\text{kpc km s}^{-1}] \approx 3.373 \times 10^{-4} \times \sqrt{M_d(R_{22})R_d}, \qquad (D.18)$$

where $M_d(R_{22})$ is in $M_{\odot}$ and $R_d$ in kiloparsecs.
## Appendix E: Catalogue of the main sample

<table><caption>**Table E.1.** Column description of the catalogue used in this analysis.</caption><thead><tr><th>No.</th><th>Title</th><th>Description</th></tr></thead><tbody><tr><td>1</td><td>ID</td><td>MUSE galaxy ID in the form X-CGrY, where X refers to the galaxy identification number within the field targeting COSMOS group CGrY</td></tr><tr><td>2</td><td>z</td><td>Spectroscopic systemic redshift derived from the kinematics modelling</td></tr><tr><td>3</td><td>RA</td><td>J2000 Right Ascension of morphological centre in decimal degrees</td></tr><tr><td>4</td><td>Dec</td><td>J2000 Declination of morphological centre in decimal degrees</td></tr><tr><td>5</td><td>N</td><td>Number of galaxies in the host structure. Field galaxies have a value of -99.</td></tr><tr><td>6</td><td>R/R200</td><td>Ratio of the projected distance of a galaxy to its host group's centre with the radius of the group R<sub>200</sub> (see Sect. 2.3). Field galaxies have a value of -99.</td></tr><tr><td>7</td><td>dV/sigma</td><td>Ratio of the systemic velocity of a galaxy along the line-of-sight with its host group's velocity dispersion (see Sect. 2.3). Field galaxies have a value of -99.</td></tr><tr><td>8</td><td>Eta</td><td>Environmental tracer defined in Sect. 2.3, combining parameters 6 and 7. Field galaxies have a value of -99.</td></tr><tr><td>9</td><td>log over-density</td><td>Logarithm of the over-density around the galaxies as measured by the VMC technique (see the end of Sect. 2.3)</td></tr><tr><td>10</td><td>log j_HST_MM</td><td>Logarithm of the specific stellar angular momentum (kpc km s<sup>-1</sup>) derived from HST images using the mass model rotation curve</td></tr><tr><td>11</td><td>log j_HST_flat</td><td>Logarithm of the specific stellar angular momentum (kpc km s<sup>-1</sup>) derived from HST images using the flat model rotation curve</td></tr><tr><td>12</td><td>log j_exp_MM</td><td>Logarithm of the specific stellar angular momentum (kpc km s<sup>-1</sup>) derived by numerically integrating Eq. 6 assuming an exponential disk mass distribution and a mass model rotation curve</td></tr><tr><td>13</td><td>log j_exp_flat</td><td>Logarithm of the specific stellar angular momentum (kpc km s<sup>-1</sup>) analytically derived assuming an exponential disk mass distribution and a flat model rotation curve (see Appendix D)</td></tr><tr><td>14</td><td>Gas dispersion</td><td>Average velocity dispersion of the ionised gas (kpc km s<sup>-1</sup>) measured in the velocity dispersion map of the [O II] doublet after correcting for the LSF and beam-smearing</td></tr><tr><td>15</td><td>log M*</td><td>Logarithm of the stellar mass (M<sub>☉</sub>)</td></tr></tbody></table>

**Notes:** The sample used in this catalogue corresponds to the 182 galaxies selected in terms of their surface, S/N, and stellar disk inclination (see Sect. 2.4). Complementary physical, morphological, and dynamical information can be found in the morpho-kinematics catalogue published in Mercier et al. (2022, catalogues can be cross-matched using the ID field). All logarithmic values are given as decimal logarithms.