# Shape asymmetries and the relation between lopsidedness and radial alignment
in simulated galaxies

Jinzhi Shen11affiliation: CAS Key Laboratory for Research in Galaxies and
Cosmology, Department of Astronomy, University of Science and Technology of
China, Hefei 230026, China;  22affiliation: School of Astronomy and Space
Science, University of Science and Technology of China, Hefei 230026, China
Xufen Wu11affiliation: CAS Key Laboratory for Research in Galaxies and
Cosmology, Department of Astronomy, University of Science and Technology of
China, Hefei 230026, China;  22affiliation: School of Astronomy and Space
Science, University of Science and Technology of China, Hefei 230026, China
Yirui Zheng33affiliation: Department of Astronomy, Shanghai Jiaotong
University, Shanghai, P.R. China Beibei Guo11affiliation: CAS Key Laboratory
for Research in Galaxies and Cosmology, Department of Astronomy, University of
Science and Technology of China, Hefei 230026, China;  22affiliation: School
of Astronomy and Space Science, University of Science and Technology of China,
Hefei 230026, China

(Accepted for publication in The Astrophysical Journal, 2023 October 28)

###### Abstract

Galaxies are observed to be lopsided, meaning that they are more massive and
more extended along one direction than the opposite. In this work, we provide
a statistical analysis of the lopsided morphology of 1780 isolated satellite
galaxies generated by TNG50-1 simulation, incorporating the effect of tidal
fields from halo centres. The isolated satellites are galaxies without nearby
substructures whose mass is over $1\%$ of the satellites within their virial
radii. We study the radial alignment (RA) between the major axes of satellites
and the radial direction of their halo centres in radial ranges of
$0$-$2\,{\rm R_{\rm h}}$, $2$-$5\,{\rm R_{\rm h}}$ and $5$-$10\,{\rm R_{\rm
h}}$ with $\,{\rm R_{\rm h}}$ being the stellar half mass radius. According to
our results, the RA is virtually undetectable in inner and intermediate
regions, yet it is significantly evident in outer regions. We also calculate
the far-to-near-side semi-axial ratios of the major axes, denoted by
$a_{-}/a_{+}$, which measures the semi-axial ratios of the major axes in the
hemispheres between backwards (far side) and facing (near side) the halo
centres. In all the radial bins of the satellites, the numbers of satellites
with longer semi-axes on the far side are found to be almost equal to those
with longer semi-axes on the near side. Therefore, the tidal fields from halo
centres play a minor role in the generation of lopsided satellites. The long
semi-major-axes radial alignment (LRA), i.e., an alignment between the long
semi-major-axes of satellite galaxies and the radial directions to their halo
centres, is further studied. No clear evidence of LRA is found in our sample
within the framework of $\Lambda$CDM Newtonian dynamics. Finally, we briefly
discuss the possible origins of the asymmetry of galaxies in TNG50-1.

###### Subject headings:

Galaxy dynamics (591) - Galaxy dark matter halos (1880) - Astronomical
simulations (1857) - Galaxy mass distribution (606)

††thanks: Email<EMAIL_ADDRESS>
##  1\. Introduction

The mass distributions of gaseous and stellar components in many galaxies are
observed to be lopsided. Since the 1960s, optical or HI observations have
revealed that some spiral galaxies are asymmetric between their northern and
southern sides (e.g., Dieter, 1962; Arp, 1966; Beale & Davies, 1969; Rogstad,
1971; Bosma, 1978; Sancisi & Allen, 1979). Baldwin et al. (1980) were the
first to find that in a sample of about twenty lopsided galaxies, most of them
are isolated galaxies. Later, systematic studies on the observed disc galaxies
showed that about $1/3$ of field disc galaxies are significantly lopsided with
an azimuthal $m=1$ Fourier amplitude $A_{1}\geq 0.20$ for the stellar light
(Rix & Zaritsky, 1995; Zaritsky & Rix, 1997). In a sample of 54 early-type
disc galaxies (S0 to Sab), $20\%$ of them are lopsided, with $A_{1}\geq 0.19$
for the R-band surface brightness (Rudnick & Rix, 1998). Conselice et al.
(2000) and Bournaud et al. (2005a) found that the lopsidedness of galaxies
strongly correlates with the morphological type, that late-type discs and
irregulars are more asymmetric and ellipticals and lenticular galaxies are
less lopsided. However, Angiras et al. (2006) unveiled that the early-type
galaxies are more lopsided than the late-type galaxies. Moreover, the
lopsidedness of galaxies strongly correlates with stellar surface density and
tends to increase with radius, as revealed by a systematical study on the
light distribution of 25,155 present-day galaxies from the Sloan Digital Sky
Survey Data Release 4 (SDSS DR4, Reichard et al., 2008). It is also confirmed
that the lopsidedness of stellar light distribution corresponds to the
lopsided stellar mass distribution (Reichard et al., 2008). About $50\%$ in
the Westerbork HI sample of Spiral and Irregular Galaxies (WHSIP) survey are
observed to be strongly lopsided within small radii, $R_{\rm 25}$, (i.e., 2.5
times of disc scale lengths, van Eymeren et al., 2011), and there are $20\%$
galaxies in this sample displaying noticeably increasing lopsidedness out to
large radii. In a most recent study on the HI emission line of 29,958 nearby
galaxies by Yu et al. (2022), $20\%$ of the sample of galaxies are
significantly asymmetric considering systematic bias due to S/N. Overall, a
certain fraction of galaxies are morphologically lopsided in observations.

The lopsided morphologies of galaxies are presumably caused by a time-
dependent non-equilibrium dynamical state. Several mechanisms have been
proposed to understand the morphological lopsidedness of galaxies within the
framework of standard Newtonian dynamics. Such mechanisms are divided into two
classes, namely internal and external ones (eg. a review of lopsided galaxies,
Jog & Combes, 2009). The internal mechanisms include the self-gravitational
instability (Zaritsky et al., 2013), the offset centres between the disc and
the dark matter halo in a galaxy (Levine & Sparke, 1998; Noordermeer et al.,
2001), and the lopsided dark matter host halo that provides a lopsided
internal potential (Weinberg, 1995; Jog, 1997).

The external mechanisms include galaxy mergers (Walker et al., 1996; Zaritsky
& Rix, 1997; Bok et al., 2019), asymmetric gas accretion from cosmological
filaments (Bournaud et al., 2005a; Mapelli et al., 2008), close tidal
encounters (Kornreich et al., 2002) and ram pressure stripping (Mapelli et
al., 2008; Scott et al., 2010; Kenney et al., 2015). Bournaud et al. (2005b)
showed that the lopsidedness induced by minor mergers disappears when the
companion is disrupted and most of the lopsided galaxies are not undergoing
mergers. In a flyby interaction between a disc target galaxy and a companion
galaxy (Mapelli et al., 2008), the lopsided feature can be produced in the
target galaxy. Since the disc galaxy is rotating, the perturbation appears on
the opposite side of the intruder galaxy after half the rotating period
(approximately 300 Myr) and this explains well the observed lopsided galaxy
NGC 891. Moreover, Mapelli et al. (2008) indicated that the overall stellar
component is not lopsided in the case of gas accretion, and that ram pressure
only generates moderate tidal tails. The observational results that a larger
fraction of lopsided galaxies are early-type implies a tidal origin for the
lopsidedness (Angiras et al., 2006; van Eymeren et al., 2011). However, the
lopsidedness does not correlate with the tidal parameter, $T_{p}$ (van Eymeren
et al., 2011), which quantifies the effect of tidal force.

Within the framework of $\Lambda$CDM Newtonian dynamics, galaxies are supposed
to be embedded in dark matter haloes. Ciotti & Dutta (1994) made the first
prediction that the tidal force from a host halo causes an alignment between
the major axes of satellites and the radial direction of the central dark
matter haloes. As satellites orbit around the gravitational centre of their
host halos, they are continuously reshaped by tidal fields. The tidal forces
elongate the mass distribution of a satellite along the radial direction and
squeeze the satellite along the directions perpendicular to the radial
direction. This prediction appears to agree with the early observations that
the projected major axes of galaxies preferentially align with the radial
direction of the cluster centres (Hawley & Peebles, 1975; Thompson, 1976). The
radial alignment (hereafter short for RA) between galaxies and cluster centres
has been further confirmed in the SDSS observations (Pereira & Kuhn, 2005;
Faltenbacher et al., 2007; Wang et al., 2008) and in the cosmological
simulations (Faltenbacher et al., 2008; Pereira et al., 2008; Knebe et al.,
2020). However, the transit through the pericenter of the orbit of the
satellite is so rapid that the time could not be long enough for the satellite
galaxy to realign itself towards the host halo centre. Indeed, some other
observations (Hung & Ebeling, 2012; Schneider et al., 2013; Chisari et al.,
2014; Sifón et al., 2015) point towards the absence of RA in large samples of
clusters at different redshifts. More recently, Singh et al. (2015) claimed to
find an RA in a sample of early-type galaxies in the SDSS-III. Wang et al.
(2019) also observed a strong signal for radial alignment in satellites
orbiting host galaxy pairs in the SDSS 13. Since the lopsidedness and the RA
can both be explained by the tidal force from a massive gravitational source,
a natural question arises as to whether the long semi-major axis of a lopsided
galaxy aligns with the radial direction of the cluster centre. Tidal forces
exerted by a halo centre are roughly symmetrical along both near and far sides
of a satellite relative to its halo centre. Nevertheless, when considering the
higher-order terms of the Taylor expansion for the gravitational force
produced by the halo, an asymmetry in these forces emerges. This asymmetry
could potentially result in a satellite galaxy adopting an uneven shape.

In principle, the dark matter halo provides a deep gravitational potential.
The lopsided shapes and the RA of satellite galaxies indicate that the semi-
axial ratio of the major axis $a_{-}/a_{+}<1.0$, which means that the semi-
axis facing to the halo centre is longer than that on the opposite side.
However, the asymmetry in the shape of a satellite is a subtle consequence of
the higher-order terms in gravitational calculations. It remains uncertain
whether this asymmetry can be consistently observed in cosmological
simulations. The accurate definition of $a_{-}$ and $a_{+}$ are to be
presented in §2.3. However, these problems have not yet been systematically
addressed in the framework of $\Lambda$CDM cosmological simulations. The
existing studies on the RA only focus on the alignment angles between the
overall major axes and the radial direction to the cluster centres. We shall
provide a more in-depth analysis below.

In this work, we present a detailed analysis of the asymmetric shapes of
present-day satellite galaxies from the public data release of the
cosmologically simulated galaxies, TNG50-1 (Nelson et al., 2019). We aim to
examine whether LRA exists in these satellites. The manuscript is organised as
follows. In §2, a sample of isolated satellite galaxies is selected. The major
axes of the satellites are determined by calculating the eigenvectors of the
inertia tensor. Then the lopsidedness of satellites is analysed based on the
semi-major axial ratios $a_{-}/a_{+}$ §3. The existence of RA and LRA are
investigated in §4. Furthermore, a table of strongly lopsided satellites in
the TNG50-1 simulation is provided and the mechanisms to generate the
lopsidedness are discussed in §5. Finally, our results are summarised in §6.

##  2\. Simulated satellite galaxies and their axial ratios

The present-day “galaxies” studied here are obtained from the TNG50-1
simulation, which is part of the IllustrisTNG simulation suite (Marinacci et
al., 2018; Naiman et al., 2018; Nelson et al., 2018, 2019; Springel et al.,
2018; Pillepich et al., 2018). The IllustrisTNG project, using a moving-mesh
code AREPO (Springel, 2010), is the next generation of the
magnetohydrodynamical (MHD) cosmological simulation, Illustris (Vogelsberger
et al., 2014a, b). In the TNG suite of cosmological simulations, the TNG50-1
is recognized for having the highest resolution within a box size of
$51.7\,{\rm Mpc}$ (Nelson et al., 2019). There are $2160^{3}$ gas particles
and also $2160^{3}$ dark matter particles in the TNG50-1 simulation, and the
mass resolutions for gas and dark matter are $8.5\times 10^{4}{\rm M}_{\odot}$
and $4.5\times 10^{5}{\rm M}_{\odot}$, respectively. Such a high resolution
allows us to extract satellite galaxies with high enough resolution to analyse
their shapes.

###  2.1. Sample selection

We select the satellite galaxies at z=0 using the following criteria. First of
all, the haloes at the boundary of the simulation box, i.e., if there are
stellar particles in a halo that travel through the periodic boundary and
appear at the other side of the box, are excluded in our halo sample. The
satellite galaxies are then selected from the haloes containing at least one
satellite galaxy. Note that the central galaxies in the haloes are not
included in our sample. The satellites studied in this work are restricted
within a stellar mass range of $[10^{8},~{}10^{11}]{\rm M}_{\odot}/h$. We
carefully checked the number of stellar particles in each galaxy model. Even a
satellite galaxy with a mass of $10^{8}{\rm M}_{\odot}$ contains at least
$2400$ stellar particles, which ensures the shapes of galaxies calculated from
the stellar particles are reliable. Moreover, the radial distances between the
satellites and their halo centres are further than $0.3R_{\rm vir}$. Here
$R_{\rm vir}$ is the virial radius of a halo, within which the mean
overdensity of the halo mass is $200$ times the critical density of the
Universe, i.e., $\rho_{\rm crit}=\frac{3H_{0}^{2}}{8\pi G}$ and
$H_{0}=67.74\,{\rm km\,s}^{-1}{\rm Mpc}^{-1}$ in the TNG50-1 simulation. There
are $2923$ galaxies that satisfy the above selection constraints.

Finally, to overcome the perturbation from recent mergers or collisions, we
require the satellite galaxies to be isolated, which means that there are no
nearby galaxies or substructures with a mass over $1\%$ of a satellite within
the virial radius of the subhalo, $r_{\rm vir,sub}$. The final sample contains
$1780$ isolated galaxies. The isolated satellites do not refer to field
galaxies, but satellites without any nearby galaxies within $r_{\rm vir,sub}$
of themselves. $r_{\rm vir,sub}$ is the truncation radius for the subhalo of a
satellite, within which the mean overdensity of the subhalo mass is $200$
times the critical density of the Universe in the TNG50-1 simulation. Apart
from the isolated galaxies, in this work, we define the rest of $1409$
selected satellites with nearby substructures from the same halo within their
$r_{\rm vir,sub}$ as non-isolated galaxies.

We shall study the semi-axial ratio of the major axes and then analyse the LRA
of the satellite galaxies in the following sections. We shall also provide a
comparison between the isolated and non-isolated samples of satellites.

###  2.2. Intrinsic major axes of the satellites

To study the lopsidedness of satellite galaxies, we need to find out the
intrinsic major axes. The centre of a satellite is defined by the densest
region of the stellar component. The principle axes of a satellite are
determined in a radial range of $0$-$2\,{\rm R_{\rm h}}$ for the inner
regions, $2$-$5\,{\rm R_{\rm h}}$ for the intermediate regions and
$5$-$10\,{\rm R_{\rm h}}$ for the outer regions by the following approach.
Here $\,{\rm R_{\rm h}}$ is the half-mass radius of the stellar component in a
galaxy. Since there are fewer particles in the larger radius of a galaxy, the
particle noise increases as the radius grows. In the intermediate and outer
radial bins, we only calculate the principle axes of a galaxy if there are
more than a thousand particles in the corresponding radial bins. In the
beginning, we assume that the galaxy is a triaxially symmetric system whose
isodensity surface can be described by the ellipsoidal equation,
$\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}+\left(\frac{z}{c}\right)^{2}=r^{2}$.
Initially, $(x,~{}y,~{}z)$ are the coordinates of stellar particles of a
satellite obtained from the TNG50-1 simulation, and $a,~{}b$ and $c$ are the
characteristic scale lengths and take the value of unity. $r$ is the distance
to the satellite centre.

The moments of inertia tensor of stellar particles inside a sphere with a
radius of $r$ are calculated by

$\displaystyle I_{\rm xx}(r)$ $\displaystyle=$
$\displaystyle\frac{\sum_{i}m_{i}(y_{i}^{2}+z_{i}^{2})}{\sum_{i}m_{i}}$ (1)
$\displaystyle I_{\rm xy}(r)$ $\displaystyle=$
$\displaystyle\frac{\sum_{i}-m_{i}x_{i}y_{i}}{\sum_{i}m_{i}}$ (2)

and similar expressions for other components. The diagonalised inertia tensor
is computed, and the eigenvalues of the system, $I_{\rm
x^{\prime}x^{\prime}},~{}I_{\rm y^{\prime}y^{\prime}}$ and $I_{\rm
z^{\prime}z^{\prime}}$, are obtained. The principle axes of the satellite
align with the eigenframe. The characteristic scale lengths are now updated to

$\displaystyle a^{\prime}(r)=\sqrt{[I_{\rm y^{\prime}y^{\prime}}+I_{\rm
z^{\prime}z^{\prime}}-I_{\rm x^{\prime}x^{\prime}}]/2},$ (3) $\displaystyle
b^{\prime}(r)=\sqrt{[I_{\rm x^{\prime}x^{\prime}}+I_{\rm
z^{\prime}z^{\prime}}-I_{\rm y^{\prime}y^{\prime}}]/2},$ (4) $\displaystyle
c^{\prime}(r)=\sqrt{[I_{\rm x^{\prime}x^{\prime}}+I_{\rm
y^{\prime}y^{\prime}}-I_{\rm z^{\prime}z^{\prime}}]/2}.$ (5)

The intrinsic major, intermediate and minor axes are the $x^{\prime}-$,
$y^{\prime}-$ and $z^{\prime}-$axes in the eigenframe of the inertia tensor,
and the values of $a^{\prime}(r),~{}b^{\prime}(r)$ and $c^{\prime}(r)$ are the
corresponding intrinsic characteristic scale lengths, respectively.

###  2.3. Semi-axial ratios of the satellites

For a system that has rotated into its eigenframe of the inertia tensor, the
positive $x^{\prime}$-axis is defined as the major axis pointing to the halo
centre. Thus the far- and near-side hemispheres are the hemispheres backward
and facing the cluster centre, respectively, segmented by a plane
perpendicular to the major axis at the centre of the satellite. The
characteristic scale lengths of the major axis on the far- and near-sides are
described by the root mean square semi-axes, $a_{-}$ and $a_{+}$, which are

$\displaystyle a_{-}(r)=\left(\frac{\sum_{i}m_{i}x_{i}^{\prime
2}}{\sum_{i}m_{i}}\right)^{1/2},~{}~{}~{}~{}x_{i}^{\prime}<0,$ (6)
$\displaystyle a_{+}(r)=\left(\frac{\sum_{i}m_{i}x_{i}^{\prime
2}}{\sum_{i}m_{i}}\right)^{1/2},~{}~{}~{}~{}x_{i}^{\prime}>0,$ (7)

respectively. For a perfectly triaxially symmetric system, $a_{-}$ and $a_{+}$
should be equal. However, as aforementioned, the galaxies are lopsided, i.e.,
$a_{-}$ and $a_{+}$ can be different. Here we use the semi-axial ratio,
$a_{-}/a_{+}$, to quantify the lopsidedness of a satellite galaxy. The semi-
axial ratios are calculated within the radial bins of $0$-$2\,{\rm R_{\rm
h}}$, $2$-$5\,{\rm R_{\rm h}}$ and $5$-$10\,{\rm R_{\rm h}}$ for the sample of
isolated galaxies.

In addition, in the whole sample of satellite galaxies, the non-isolated
galaxies are strongly influenced by the tidal fields from the nearby massive
galaxies, i.e., locating within $10\,{\rm R_{\rm h}}$ of the satellite galaxy.
For non-isolated galaxies, the tidal fields come from both the halo centres
and the nearby galaxies. Thus the definition of positive $x^{\prime}$-axis is
different from that of the isolated galaxies. For a non-isolated satellite,
the near side is defined by the direction of the object, be it a nearby
massive galaxy or the halo centre, along which the tidal field is stronger. To
make a comparison to the isolated sample, the semi-axial ratios of the non-
isolated galaxies are also computed within the radial bins of $0$-$2\,{\rm
R_{\rm h}}$, $2$-$5\,{\rm R_{\rm h}}$ and $5$-$10\,{\rm R_{\rm h}}$.

Table 1 The statistics of semi-axial ratios of major axes, $a_{-}/a_{+}$, of the satellites within the radial bins of $0$-$2\,{\rm R_{\rm h}}$, $2$-$5\,{\rm R_{\rm h}}$ and $5$-$10\,{\rm R_{\rm h}}$ in mass bins of $10^{8}$-$10^{9}{\rm M}_{\odot}/h$, $10^{9}$-$10^{10}{\rm M}_{\odot}/h$ and $10^{10}$-$10^{11}{\rm M}_{\odot}/h$ for the isolated ($3^{\rm rd}$-$7^{\rm th}$ columns) and non-isolated ($8^{\rm th}$-$12^{\rm th}$ columns) samples. Here $N_{-}$ and $N_{+}$ denote the numbers of galaxies with semi-axial ratios $a_{-}/a_{+}<1$ and $a_{-}/a_{+}>1$, respectively. The mean values, $\mu$, and the standard deviation, $\sigma$, for the semi-axial ratios, together with the skewness, are listed in the table. The last column displays the p-value of the Anderson-Darling 2-sample test for the isolated and non-isolated galaxies.  |  | Samples: isolated | Non-isolated | AD 2-sample
---|---|---|---|---
Mass | Radial range | $N_{-}$ | $N_{+}$ | $\mu$ | $\sigma$ | Skewness | $N_{-}$ | $N_{+}$ | $\mu$ | $\sigma$ | Skewness | p-value
$(M_{\odot}/h)$ | ($R_{h}$) |  |  |  |  |  |  |  |  |  |  |
$10^{8}$-$10^{9}$ | 0-2 | 625 | 636 | 1.00 | 0.05 | -0.03 | 475 | 462 | 1.00 | 0.12 | 3.50 | 0.0005
| 2-5 | 384 | 359 | 1.00 | 0.05 | 0.45 | 309 | 252 | 1.00 | 0.12 | 11.07 | 0.0040
| 5-10 | 56 | 54 | 1.00 | 0.05 | -0.01 | 27 | 21 | 0.99 | 0.07 | -0.33 | 0.3295
$10^{9}$-$10^{10}$ | 0-2 | 211 | 197 | 1.00 | 0.06 | 2.37 | 186 | 173 | 1.00 | 0.08 | 0.44 | 0.0445
| 2-5 | 192 | 216 | 1.00 | 0.05 | 0.24 | 184 | 174 | 1.00 | 0.06 | 0.34 | 0.1095
| 5-10 | 121 | 120 | 1.00 | 0.06 | 0.76 | 109 | 77 | 0.98 | 0.10 | -0.67 | 0.0003
$10^{10}$-$10^{11}$ | 0-2 | 51 | 60 | 1.01 | 0.05 | 1.26 | 64 | 49 | 1.00 | 0.07 | 0.27 | 0.0880
| 2-5 | 54 | 57 | 1.00 | 0.06 | 0.35 | 57 | 56 | 1.00 | 0.06 | -0.08 | 0.5655
| 5-10 | 44 | 58 | 1.01 | 0.07 | 0.71 | 63 | 39 | 0.98 | 0.11 | -0.25 | 0.0055
Figure 1.— The fraction of satellite galaxies versus their semi-axial ratio,
$a_{-}/a_{+}$, of the major axes in the radial bins of $0$-$2\,{\rm R_{\rm
h}}$ (red curves), $2$-$5\,{\rm R_{\rm h}}$ (golden curves) and $5$-$10\,{\rm
R_{\rm h}}$ (purple curves) in mass bins of $10^{8}$-$10^{9}{\rm M}_{\odot}/h$
(left panels), $10^{9}$-$10^{10}{\rm M}_{\odot}/h$ (right panels). The upper
and lower panels show the results from samples of isolated and non-isolated
satellites, respectively.

##  3\. Lopsidedness of the satellites

For a self-gravitationally bound system embedded in a static potential of a
dark halo and deviating from the halo centre, the tidal field from the dark
halo may change the shape of the self-bound system. The semi-axis facing the
halo centre is expected to be longer than the semi-axis backward halo centre.
This implies that the system is lopsided and the semi-axial ratio
$a_{-}/a_{+}<1.0$. However, previous studies on tidal effect mainly focused on
galaxies interactions such as minor mergers (Walker et al., 1996; Bok et al.,
2019) or close encounters (Kornreich et al., 2002). Considering the tidal
force decays with distance as a power law of $D^{-3}$, the tidal effect caused
by halo centres is minor. It is unlikely that the tidal field from halo
centres can lead to a detectable lopsidedness. However, a systematic test on
the lopsidedness of satellite galaxies surrounding their host halos is still
lacking.

Moreover, apart from the tidal interactions due to minor mergers or close
encounters, lopsidedness can be produced by several other mechanisms,
including such external effects as asymmetric gas accretion (Bournaud et al.,
2005a; Mapelli et al., 2008) and ram pressure stripping (Mapelli et al., 2008;
Kenney et al., 2015) and such internal effects as self-gravitational
instability (Zaritsky et al., 2013), offset centres between the disc and the
dark matter halo (Levine & Sparke, 1998; Noordermeer et al., 2001), and the
lopsided dark matter host halo (Weinberg, 1995; Jog, 1997). On the other hand,
for a large sample of galaxies, if the tidal fields from the halo centres are
not taken into account, the above effects do not produce a preferred alignment
for the long semi-major axes with respect to the radial directions of halo
centres. The directions of the long semi-major axes are random for a large
sample of galaxies. Therefore, the numbers of galaxies with $a_{-}/a_{+}<1.0$
and $a_{-}/a_{+}>1.0$ are expected to be approximately equal. However, it
remains unclear whether the number of galaxies with long semi-major axes on
the near side of halo centres is larger than that on the far side in case the
minor effect of the tidal field from the halo centres is incorporated.

An analysis of a large sample of 1912 disc galaxies in Illustris-TNG100
simulation (Łokas, 2022) has unveiled that only $8\%$ of discs are moderately
lopsided with the Fourier amplitude $A_{1}\geq 0.10$, and that only $0.37\%$
are strongly lopsided galaxies with $A_{1}\geq 0.2$, which is a substantially
lower fraction compared to the observations (e.g., $30\%$ in Jog & Combes,
2009). In this work, we analyse the lopsidedness of satellite galaxies
generated by the TNG50-1 simulation, of which the resolution is higher. It is
thus possible to include the low-mass satellites in our sample. Moreover, we
study the orientations of the long semi-major axes of the satellites with
respect to the radial directions to halo centres (i.e., the LRA), which have
not yet been systematically explored.

###  3.1. Distribution of the semi-axial ratios

To examine the distribution of $a_{-}/a_{+}$ near the value of $1.0$, we
calculate $a_{-}/a_{+}$ of the stellar components of all galaxies in two
samples of isolated and non-isolated satellites. Each sample is further
divided into three subsamples according to their stellar mass, including low-
mass satellites, intermediate-mass satellites and massive satellites with
stellar masses being in ranges of $[10^{8},10^{9}]{\rm M}_{\odot}/h$,
$[10^{9},10^{10}]{\rm M}_{\odot}/h$ and $[10^{10},10^{11}]{\rm M}_{\odot}/h$,
respectively. The fractions of galaxies as a function of semi-axial ratios,
$a_{-}/a_{+}$, in bins of $0.02$, of the stellar components are shown in Fig.
1, in the radial bins of $0$-$2\,{\rm R_{\rm h}}$ (red curves), $2$-$5\,{\rm
R_{\rm h}}$ (golden) and $5$-$10\,{\rm R_{\rm h}}$ (purple). Moreover, the
numbers of galaxies with $a_{-}/a_{+}<1$ and $a_{-}/a_{+}>1$, denoted as
$N_{-}$ and $N_{+}$, are listed in the fourth and fifth columns of Table 1 for
the samples of isolated and non-isolated satellite galaxies in three mass
bins.

For the isolated satellite galaxies (the upper panels in Fig. 1), the
fractions of galaxies peaked at around $a_{-}/a_{+}\approx 1.0$ are almost
symmetric. Significant fluctuations appear in the fraction of galaxies versus
$a_{-}/a_{+}$ in the most massive subsample (the upper right panel) since the
total number of galaxies within this mass range is the smallest. The almost
symmetric shapes of the galaxy fractions indicate that the tidal field from
the halo centres plays a minor role in determining the asymmetric shapes of
satellite galaxies. For the non-isolated satellites (the lower panels in Fig.
1, the fractions of galaxies also appear to be symmetric around
$a_{-}/a_{+}\approx 1.0$. However, more fluctuations are deviating from
$a_{-}/a_{+}=1.0$. There are more asymmetric satellite galaxies in the non-
isolated subsamples. The tidal field effect from a close encounter or a minor
merger influences the semi-major axial ratio of a non-isolated satellite more
significantly. Below we introduce a Gaussian distribution function to fit the
dependence of fractions of satellites on the varying ratio $a_{-}/a_{+}$.

Table 2The lopsidedness of satellites is defined by two criteria, namely deviation to the near side semi-major axes $\Delta a$ and the Fourier $m=1$ mode values $A_{1}$. The fractions of moderately and strongly lopsided satellites are listed in the $2^{nd}-3^{rd}$ and $4^{th}-5^{th}$ columns, respectively.  Sample | $f_{\Delta a>0.05}$ | $f_{A_{1}>0.1}$ | $f_{\Delta a>0.1}$ | $f_{A_{1}>0.2}$
---|---|---|---|---
Isolated | $25.9\%$ | $19.9\%$ | $5.8\%$ | $2.5\%$
Non-isolated | $33.6\%$ | $34.2\%$ | $12.8\%$ | $11.1\%$

We show how the peak of $a_{-}/a_{+}$ in a Gaussian fitting deviates from 1.0
in Table 1, which quantifies the systematical tidal effect from the halo
centres. For the subsample of low-mass galaxies within the mass range of
$[10^{8},10^{9}]{\rm M}_{\odot}/h$, the numbers $N_{-}$ and $N_{+}$ are almost
equal within all radial ranges, except in the intermediate region, the number
of galaxies with $a_{-}/a_{+}<1.0$ is approximately $10\%$ more than that with
$a_{-}/a_{+}>1.0$. However, the mean values of $a_{-}/a_{+}$ are $\mu=1.00$ in
all ranges of radii, with a standard deviation of $\sigma=0.05$. To judge the
shapes of the fractions of galaxies versus semi-axial ratios, we also
calculate the skewness of the subsamples,

${\textit{S}kewness}=\frac{n}{(n-1)(n-2)}\sum_{\rm i=1}^{\rm
n}\left[\frac{(a_{-}/a_{+})_{i}-\mu}{\sigma}\right]^{3}.$ (8)

We find that the absolute skewness values for this subsample in different
radial ranges are all smaller than 0.5, indicating that the distributions of
fractions of galaxies are perfectly symmetric around $a_{-}/a_{+}=1.00$. Thus
the long side of the major axes of the low-mass satellites does not point
towards the host halo centres. The tidal fields from halo centres do not lead
to clearly visible lopsidedness.

In the subsample of low-mass isolated satellites, the tidal fields from halo
centres do not lead to clearly visible lopsidedness. Such a conclusion still
holds in the subsamples of intermediate-mass and massive isolated satellites.
The shapes of the fraction of galaxies versus the semi-axial ratio within the
three radial ranges appear to be similar and symmetric. The values of skewness
of the fractions of galaxies within $0$-$2\,{\rm R_{\rm h}}$ are 2.37 and 1.26
for the intermediate-mass and massive isolated subsamples, respectively, which
are positively skewed, with long tails on the right sides. In the upper middle
panel of Fig. 1, a few strongly lopsided outliers with semi-axial ratios
beyond $5\sigma$ of the subsample lead to the right skewness of fraction of
galaxies within $0$-$2\,{\rm R_{\rm h}}$. The strongly asymmetric shapes of
these galaxies are not caused by the tidal fields from their halo centres. We
will discuss the possible mechanisms in §5.2.

Figure 2.— The accumulative fraction of satellite galaxies with different
values of m=1 Fourier mode, $A_{1}$. The two vertical lines show the maximal
values of $A_{1}$ for the isolated and non-isolated samples.

In Table 1, the values of $a_{-}/a_{+}$ for subsamples of isolated and non-
isolated galaxies in different mass bins and radius bins are assumed to follow
a Gaussian distribution. From the values of $\mu$ and $\sigma$, it is possible
to determine whether each pair of subsamples, which share the same mass and
radius bins, are from the same distribution. However, it is undetermined if
the values of $a_{-}/a_{+}$ in these subsamples align with a Gaussian
distribution, especially for those subsamples with large values of skewness.
The Anderson-Darling two-sample tests (Anderson & Darling, 1952) provide a way
to examine whether two samples of data are from the same distribution without
a pre-assumption of a specific distribution. Moreover, the Anderson-Darling
two-sample test is tail-weighted and thus is more sensitive than the commonly
used Kolmogorov-Smirnov test (Kolmogorov-Smirnov et al., 1933). The null
hypothesis that two samples are from the same distribution is rejected when
the p-value is larger than 0.05 in an Anderson-Darling 2-sample test.

We perform Anderson-Darling 2-sample tests to investigate whether the
distributions of semi-axial ratios $a_{-}/a_{+}$ are different between the
isolated and non-isolated galaxies within the three radial bins for the same
mass range. In Table 1, the last column shows the p-values of Anderson-Darling
two-sample tests 111We use the twosamples package from the CRAN R project,
which is a statistical computing project. The URL for the twosamples package
is htpps://github.com/cdowd/twosamples . for the distributions of
$a_{-}/a_{+}$ of nine pairs of the isolated and non-isolated galaxies binned
by their masses and radii. For five of the pairs of galaxy subsamples, the
p-values are below 0.05, indicating a deviation from the same distribution of
$a_{-}/a_{+}$. However, p-values are greater than 0.05 for the rest four
pairs, which suggests the isolated and non-isolated subsamples with the same
mass bin and with the same radius bin may be from the same $a_{-}/a_{+}$
distribution. Our findings agree with the previous analysis on values of $\mu$
and $\sigma$. For instance, in the first row of Table 1, the values of
$\sigma$ from a Gaussian fitting are quite different for the isolated and non-
isolated subsamples. This indicates that the two subsamples are not from the
same Gaussian distribution. The p-value obtained from the Anderson-Darling
two-sample test is 0.0005, indicating a null rejected result. Thus the two
subsamples are not from the same distribution without a pre-assumption of a
Gaussian form.

Further, let’s consider the whole sample (including the above three
subsamples) of isolated satellites. We find only $104$ significantly lopsided
galaxies whose semi-axial-scale-length deviation to the near side semi-axis,
$\Delta a\equiv\frac{|a_{-}-a_{+}|}{a_{+}}>0.1$, within $2\,{\rm R_{\rm h}}$.
That means among the isolated sample of satellites, only $5.8\%$ are
significantly lopsided.

Now let’s consider the data of $a_{-}/a_{+}$ ungrouped by masses of galaxes
for the isolated and non-isolated samples. Unbinned methods may potentially
reveal nuanced effects. Here we examine whether the two unbinned samples of
data follow the same distribution in 2D space of $a_{-}/a_{+}$ and galactic
stellar masses. We used the nonparameter two-sample test (Rizzo, 2019) with
bootstrap probabilities to perform the test (Feigelson & Babu, 2012) by using
the cramer package in the CRAN R project. 222The URL for this package is
https://CRAN.R-project.org/package=cramer The p-values are 0.0030, 0.0050 and
0.0020 within the radial ranges of $0$-$2\,{\rm R_{\rm h}}$, $2$-$5\,{\rm
R_{\rm h}}$ and $5$-$10\,{\rm R_{\rm h}}$. Thus the $a_{-}/a_{+}$ and galactic
stellar masses of the isolated and non-isolated sample of galaxies are not
from the same distribution.

In the above analysis, $a_{-}/a_{+}$ is used to define the lopsidedness, which
is different from that adopted in the previous analysis based on disc galaxies
obtained from the TNG100-1 simulation by Łokas (2022). It is helpful to make a
comparison between our results with theirs. To directly compare the
lopsidedness with that in the discs sample (Łokas, 2022) in TNG-100
simulation, we also calculated the Fourier m=1 mode values, $A_{1}$, for the
isolated satellites on the $x^{\prime}-y^{\prime}$ planes in their eigenframes
of inertia tensor. The $A_{1}$ values of satellites are calculated in a radial
bin within $1.0-2.0\,{\rm R_{\rm h}}$, which is the same as the range of
radius used in Łokas (2022). We show the accumulative fraction of satellites
with growing $A_{1}$ values in Fig. 2 and also in Table 2. The fraction of
strongly lopsided satellites with $A_{1}>0.2$ is $2.5\%$, which is almost an
order of magnitude larger than that ($0.37\%$) in Łokas (2022). One of the
possible reasons might be that the sample selections are different. Our sample
includes all isolated satellites in a broader stellar mass range of
$10^{8}$-$10^{11}{\rm M}_{\odot}/h$, whereas the sample of Łokas (2022)
contains only disc galaxies with stellar masses beyond $10^{10}{\rm
M}_{\odot}$. To confirm this, we have further calculated the $A_{1}$ values
for galaxies with the same mass range of Łokas (2022). There are 876 such
galaxies in the Illustris TNG-50 simulation, among which there are $3.5\%$
galaxies moderately lopsided ($A_{1}>0.1$), and only one galaxy ($0.1\%$) is
strongly lopsided with $A_{1}>0.2$. Thus in the same range of masses, the
fraction of strongly lopsided galaxies obtained from our analysis agrees with
the result from Łokas (2022), that the simulated satellite galaxies do not
exhibit significantly asymmetric.

In addition, in order to examine whether the decreased lopsidedness in our
samples, relative to observational samples, is due to our particular choice of
satellite galaxies, we calculate the $A_{1}$ values for a sample of 4001
central galaxies with stellar masses $>10^{8}{\rm M}_{\odot}$ in the TNG 50-1
simulation. The mean values of $A_{1}$ are $0.059$, $0.048$, $0.032$ and
$0.032$ for galaxies within the stellar mass ranges of
$10^{8}$-$10^{9}M_{\odot}/h$, $10^{9}$-$10^{10}M_{\odot}/h$,
$10^{10}$-$10^{11}M_{\odot}/h$, and greater than $10^{11}M_{\odot}/h$,
respectively. Compared to the satellite galaxies, the simulated central
galaxies are even less lopsided, in general.

Although the fraction of strongly lopsided satellites in a broader range of
masses obtained by us for $A_{1}>0.2$ is one order of magnitude larger than
that of Łokas (2022), it is still much smaller than the observed one (van
Eymeren et al., 2011; Yu et al., 2022). It is noteworthy that the stellar mass
ranges of our work and of Yu et al. (2022) are very similar, while the stellar
masses are $\in[10^{8},~{}10^{11.5}]~{}{\rm M}_{\odot}$ with a median of
$10^{9.7}{\rm M}_{\odot}$ in the latter case. Considering the lopsidedness
decreases as stellar mass increases in the above studies, it seems unlikely
the small fraction of galaxies with stellar masses of $10^{8}{\rm
M}_{\odot}$-$10^{8}{\rm M}_{\odot}/h$ or of $10^{11}{\rm
M}_{\odot}/h$-$10^{11.5}{\rm M}_{\odot}$ contribute to the majority of the
proportion of asymmetric galaxies, making it statistically consistent with
observations.

Figure 3.— The lower left panel: The correlation between semi-axial ratios,
$a_{-}/a_{+}$ at $2\,{\rm R_{\rm h}}$, for the 1780 present-day isolated
satellites in $3\sigma$ of Gaussian fitting and the distances, $D$, of
satellites to their host halo centres. The 2-dimensional histograms of data
are binned in $0.02R_{\rm vir}$ intervals in distance and $0.01$ intervals in
$a_{-}/a_{+}$, and are represented using the colour bars. The ratio
$a_{-}/a_{+}=1.0$ is shown with a horizontal line, as well. The fractions of
satellites with $a_{-}/a_{+}<1.0$ at all distances are presented in the upper
left panel and there is a histogram of satellites with different values of
$a_{-}/a_{+}$ in the right panel.  Figure 4.— The lower left panel: The
correlation between semi-axial ratios, $a_{-}/a_{+}$ at $10\,{\rm R_{\rm h}}$,
for the present-day isolated satellites in $3\sigma$ of Gaussian fitting and
the distances, $D$, of satellites to their host halo centres. The binning
method is the same as that in Fig. 3. The fractions of satellites with
$a_{-}/a_{+}<1.0$ at all distances are presented in the upper left panel. A
histogram of satellites with different values of $a_{-}/a_{+}$ is given in the
right panel.

Next, we turn to consider the sample of non-isolated satellites. In this case,
the values of $N_{-}$ are larger than those of $N_{+}$. Moreover, $\Delta a$
is further enhanced. In addition, we find $180$ significantly lopsided
satellites with $\Delta a>0.1$ in the radial range of $0$-$2\,{\rm R_{\rm h}}$
out of $1409$ non-isolated satellites. The fraction of significantly lopsided
non-isolated satellites is about $12.8\%$ under the criteria of $\Delta
a>0.1$, nearly a factor of 2 larger than the fraction in the isolated sample.
If $A_{1}$ is used instead of $\Delta a$, $11.1\%$ of the non-isolated
satellites are strongly lopsided with $A_{1}>0.2$. The larger fraction
measured by $A_{1}$ can be attributed to the usage of a locally radial range
of $[1.0,~{}2.0\,{\rm R_{\rm h}}]$ in calculations. It is the asymmetric
distribution of stars within the range of $[1.0,~{}2.0\,{\rm R_{\rm h}}]$ that
makes $A_{1}$ larger. All stellar particles within $2\,{\rm R_{\rm h}}$ are
included in the calculation of $\Delta a$, so the global asymmetry is weakened
if the central regions are symmetric.

In a word, the non-isolated satellites are made more lopsided by the
perturbations from a close encounter or a minor merger (see §5.2). The
fraction of strongly lopsided satellites in our non-isolated sample is much
higher than that obtained by Łokas (2022), but is also lower than that in
observations van Eymeren et al. (e.g., 2011); Yu et al. (e.g., 2022), as in
the case of isolated satellites.

Figure 5.— The relation between the semi-axial ratios in the inner (upper
panel), intermediate (middle panel) and outer (lower panel) regions of the
overall sample of 3189 satellites and their host halo masses. The satellites
are presented in colour-filled circles. Different colours and sizes of the
circles show the masses and the sizes of the satellites. The axial ratios
$a_{-}/a_{+}=1.0$ are plotted as black dashed lines. The red dashed lines and
pink shadows show the areas of $1\sigma$ and $3\sigma$ of $a_{-}/a_{+}$ in the
Gaussian fitting.

###  3.2. Shape correlations

As aforementioned, the three subsamples of isolated galaxies display weak
lopsidedness in their shapes. The long semi-axes of satellites do not
preferably face the direction of halo centres. This confirms that the lopsided
shapes of satellites are not mainly caused by the tidal fields from the halo
centres. To examine the tidal field effects from the halo centres more
carefully, we further check the inner and outer semi-axial ratios,
$a_{-}/a_{+}$, in different environments of the haloes, and show the results
in Figs. 3-4.

The isolated sample does not contain galaxies near their host central
galaxies, say, no isolated galaxies located within $0.3R_{\rm vir}$ of the
halo. The histograms of $a_{-}/a_{+}$ for the inner (Fig. 3) and outer (Fig.
4) regions of satellites show that the asymmetry of the galaxies is weak. The
values of $a_{-}/a_{+}$ for most of the satellites are between $0.9-1.1$,
which agrees with the conclusion in Table 1. In the dense regions of
satellites, i.e., within $0$-$2\,{\rm R_{\rm h}}$ of the satellites, the
fraction of satellites with $a_{-}/a_{+}<1.0$ (hereafter denoted as
$f_{N_{-}}$), oscillates around $0.5$ for the isolated sample of satellites.
The number of isolated satellite galaxies should be reduced when studying the
outer regions of the satellites since we always require more than a thousand
stellar particles in the radial ranges of a satellite studied. As shown in
Fig. 4, the conclusion that asymmetry is absent in the isolated sample is
still valid in the outer regions ($5$-$10\,{\rm R_{\rm h}}$). In summary, the
tidal fields from halo centres do not contribute to the lopsidedness of the
satellites.

The Jacobi radius of a satellite on a circular orbit is approximately (Binney
& Tremaine, 2008)

$\,{\rm r_{\rm t}}=D\left(\frac{M_{\rm sat}}{\zeta M_{\rm
halo}}\right)^{1/3},~{}~{}~{}\zeta=3$ (9)

and the tidal radii on other kinds of orbits have the same form but with
different values of $\zeta$. The lopsidedness is not directly related to the
tidal force from halo centres for most of the satellites outside $0.2R_{\rm
vir}$. An implication is that there are little correlations between the
physical quantities appearing in Eq. 9 ($M_{\rm sat}$, $M_{\rm halo}$ and $D$)
and the lopsidedness. Note that $M_{\rm sat}$ includes both baryonic and
subhalo dark matter masses of a satellite galaxy. We use the results shown in
Fig. 5 to confirm that in all truncation radii of satellites, the semi-axial
ratios $a_{-}/a_{+}$ do not correlate with the halo masses nor the overall
satellite masses.

Observations have revealed that the HI distribution in galaxies is typically
asymmetric (e.g., Watts et al., 2020a). In simulations, the HI distribution of
satellite galaxies in TNG100-1 is generally asymmetric (Watts et al., 2020b).
Recently, Łokas (2022) pointed out the lack of a direct relationship between
the global asymmetries of the stellar and the gaseous components in the TNG100
simulation. Below, we study the relation between the gas fractions and the
lopsidedness of satellite galaxies. The values of $a_{-}/a_{+}$ for satellites
truncated at $2\,{\rm R_{\rm h}}$ versus the fraction of gas, $f_{\rm gas}$ in
a satellite, are plotted in Fig. 6 for isolated (upper panel) and non-isolated
(lower panel) satellites. The fraction of gas is defined by $f_{\rm
gas}=\frac{m_{\rm sat,gas}}{m_{\rm sat,b}}$, where $m_{\rm sat,gas}$ and
${m_{\rm sat,b}}$ are the masses of gas and baryons of a satellite galaxy. In
both samples, there is a clear signal that more massive satellites tend to be
less gas-rich, and the dispersion of $a_{-}/a_{+}$ is smaller around 1.0. More
massive satellites are more symmetric with smaller values of $\Delta a$. As
satellite mass decreases, $\Delta a$ becomes larger. Satellites with a larger
fraction of gaseous components appear to be more lopsided.

Figure 6.— A relation between $a_{-}/a_{+}$ and the fraction of gaseous
component in the isolated sample (upper panel) and non-isolated sample (lower
sample). The satellites are truncated at $2\,{\rm R_{\rm h}}$ in the
calculations of $a_{-}/a_{+}$. The red dashed lines and pink shadows show the
areas of $1\sigma$ and $3\sigma$ of $a_{-}/a_{+}$ in the Gaussian fitting.

##  4\. Alignments

Figure 7.— The spatial distribution and orientation of the major axes of low
mass satellites in the most massive halo, Halo 0, viewed along the
$z$-direction obtained directly in TNG50-1 simulation. The whole halo is
recentred and the original point is the halo centre. The central pink region
shows the radius of the central galaxy within $10\,{\rm R_{\rm h}}$. The two
dashed circles indicate the radii of $0.5R_{\rm vir}$ and $R_{\rm vir}$ of the
halo. The satellites with different semi-axial ratios are plotted in different
colour circles. The radii of the circles are $10\,{\rm R_{\rm h}}$ of the
satellites. The solid lines going through the satellite circles indicate their
major axes, blue for the inwards and red for the backward directions in
$3-$dimensional space to the halo centres for the long semi-axes, together
with the arrows. Figure 8.— The histograms of the cosines of aligned angles
between the long semi-major-axes of satellites and their halo centres in the
inner regions within $0$-$2\,{\rm R_{\rm h}}$ (upper panel), intermediate
regions within $2$-$5\,{\rm R_{\rm h}}$ (middle panel) and outer regions
within $5$-$10\,{\rm R_{\rm h}}$ (lower panel). The $\cos\theta$ are binned in
$0.05$ for the overall sample of isolated satellites (the flavogreen
histograms) and the triaxial isolated satellites with intermediate-to-long
axial ratios $b:a<0.9$ (the green histograms).

The existence of RA is still controversial. The RA between the major axis of a
satellite and the radial direction of its central halo was found in
observations in the 1970s (Hawley & Peebles, 1975; Thompson, 1976). More
recently, some observations further confirmed the existence of the RA in large
samples of galaxies (Pereira & Kuhn, 2005; Faltenbacher et al., 2007; Wang et
al., 2008, 2019), but other observations do not find evidence of RA at
different redshifts (Hung & Ebeling, 2012; Schneider et al., 2013; Chisari et
al., 2014; Sifón et al., 2015). On the other hand, RA was discovered in the
cosmological simulations by Faltenbacher et al. (2008); Pereira et al. (2008);
Knebe et al. (2020). Especially, it is more pronounced on the small scale of
$1.5R_{\rm vir}$ (Faltenbacher et al., 2008). The mean value of the alignment
angle, $\phi$, i.e., the angle between the major axis of a satellite and the
radial direction of the halo centre, is adopted to determine the existence of
RA in Faltenbacher et al. (2008). The long and short semi-major-axes are not
distinguished in the above studies, thus the values of $\phi$ are taken to lie
in $[0^{\circ},~{}90^{\circ}]$. For a sample of satellites radially aligned
with their halo centres, the mean value of $\phi$ is less than $45^{\circ}$.
In addition, besides the radial alignment of satellites, Wang et al. (2019)
observed a tangential alignment in the satellites around bound primary galaxy
pairs in the SDSS DR13. There is a clear trend that the major axes of
satellites located in the area between two primary galaxies are perpendicular
to the radial direction of their host halo. The tangential alignment is beyond
the scope of this work, and we shall not further discuss this issue.

The RA of isolated satellites with a mass of $10^{8}$-$10^{9}{\rm
M}_{\odot}/h$ in the most massive halo (Halo 0) in TNG50-1 is qualitatively
shown in Fig. 7. These satellites are the ones most substantially perturbed by
the tides from their host halo. The three panels present the RA in all chosen
truncation radii of satellites. The orientations of long semi-major axes of
satellites are plotted with blue (inwards) and red (backward) arrows on the
circles of satellites. We observe no significant RA in the radial bins of
$0$-$2\,{\rm R_{\rm h}}$, $2$-$5\,{\rm R_{\rm h}}$ and $5$-$10\,{\rm R_{\rm
h}}$ at the first glance of the figure. To examine whether RA exists or not,
below we perform a more careful analysis.

###  4.1. RA and LRA

As aforementioned, it remains unclear whether LRA can be produced in the
simulated galaxies within the $\Lambda$CDM framework. Here we provide the
first systematical analysis of the existence of LRA in the sample of satellite
galaxies generated from the TNG50-1 simulation.

We need to consider the RA before analysing LRA. For this purpose, here we
define a 3D alignment angle, $\theta$, as follows,

$\cos\theta={\rm\mathbb{x}^{\prime}_{sat}}\cdot{\rm\mathbb{D}},$ (10)

where ${\rm\mathbb{x}^{\prime}_{sat}}$ is the eigenvector of the long semi-
major axis of a satellite, and ${\rm\mathbb{D}}$ is the radial direction to
the halo centre. The values of $\cos\theta$ are calculated for the sample of
isolated satellites. The definition of the 3D alignment angle in the existing
work of alignments in galaxies is in the range of $[0^{\circ},~{}90^{\circ}]$
(e.g., Libeskind et al., 2009; Velliscig et al., 2015; Tempel et al., 2015;
Zhang et al., 2023) under the assumption of triaxially symmetric satellite
galaxies. Given that the long and short semi-major axes are distinguishable in
the lopsided galaxies in this work, $\theta$ takes all the possible values
within $[0^{\circ},~{}180^{\circ}]$. As a result, $\cos\theta\in[-1,1]$. For a
satellite with $\cos\theta\in[0,1]$, the long semi-major axis is radially
aligned with the direction of the halo centre, and for $\cos\theta\in[-1,0]$,
the short semi-major-axis is radially aligned.

The values of $\theta$ can be used to determine the existence of RA and the
LRA. It is convenient to use ${\langle\cos\theta_{1}\rangle}$ and
${\langle\cos\theta_{2}\rangle}$ to denote the mean values of $\cos\theta$ in
the ranges of $[0,1]$ and $[-1,0]$, respectively. An RA emerges when
${\langle\cos\theta_{1}\rangle}>0.5$ or ${\langle\cos\theta_{2}\rangle}<-0.5$.
Moreover, LRA refers to the special case in which the long semi-major axis of
a lopsided satellite radially aligns with the halo centre, corresponding to
$\cos\theta>0$. The existence of LRA can be determined by the
$\cos\theta$-dependence of the number of satellites.

###  4.2. On the existence of RA

In Fig. 8, the values of $\cos\theta$ are binned in $0.05$ in the inner,
intermediate and outer regions for the overall sample of isolated galaxies
with flavogreen histograms. The alignment signal shows a strong dependence on
the radial ranges. For the radial ranges of $0$-$2\,{\rm R_{\rm h}}$ and
$2$-$5\,{\rm R_{\rm h}}$, the number of galaxies in each bin exhibits a
relatively random $\cos\theta$-dependence, without discernible anisotropy. To
investigate whether the distributions of $\cos\theta$ for the isolated sample
in each radius bin follow the uniform distribution, we perform the Anderson-
Darling 2-sample tests with random numbers following a uniform distribution
within $[-1,~{}1]$. The resulting p-values are 0.224 and 0.089 in radial bins
of $0-2\,{\rm R_{\rm h}}$ and $2$-$5\,{\rm R_{\rm h}}$, suggesting that the
distributions of $\cos\theta$ in these two radial ranges may follow uniform
distributions. In the radius bin of $5$-$10\,{\rm R_{\rm h}}$, the number
distribution of galaxies appears as a U-shape distribution, and the AD test
with a uniform distribution returns a p-value of 0.0005. Apparently, the
distribution of 3D alignment angles is not uniform. Such a U-shape
distribution is compelling evidence for the existence of radial alignment.

We calculate the values of ${\langle\cos\theta_{1}\rangle}$ and
${\langle\cos\theta_{2}\rangle}$ at all truncation radii for the isolated
galaxies binned in radius. We find that ${\langle\cos\theta_{1}\rangle=0.52}$
and ${\langle\cos\theta_{2}\rangle=-0.51}$ for $0$-$2\,{\rm R_{\rm h}}$,
${\langle\cos\theta_{1}\rangle=0.52}$ and
${\langle\cos\theta_{2}\rangle=-0.53}$ for $2$-$5\,{\rm R_{\rm h}}$ and
${\langle\cos\theta_{1}\rangle=0.61}$ and
${\langle\cos\theta_{2}\rangle=-0.58}$ for $5$-$10\,{\rm R_{\rm h}}$. The mean
values of the misalignment angle agree with the Anderson-Darling tests. A
statistically significant signal of the RA emerges in the outer region of
galaxies, which is in agreement with previous results obtained in cosmological
simulations (Kuhlen et al., 2007; Pereira et al., 2008; Knebe et al., 2020).
Contrarily, the presence of RA is not clearly observed in the inner and
intermediate regions. Although the mean values of
${\langle\cos\theta_{1}\rangle}>0.5$ and
${\langle\cos\theta_{2}\rangle}<-0.5$, the differences between these values
and the mean value of a uniform distribution so small that it is impossible to
confirm the existence of RA in the radial bins of $0$-$2\,{\rm R_{\rm h}}$ and
$2$-$5\,{\rm R_{\rm h}}$.

In the samples of Pereira et al. (2008); Knebe et al. (2008, 2020), triaxial
galaxies are selected. Our sample includes both triaxially and axially
symmetric satellites. For an oblate elliptical satellite or a disc satellite,
the major and intermediate axes are indistinguishable. For such an oblate/disc
galaxy, the value of $\cos\theta\approx 0$ might arise due to the intermediate
axis being identified as the major axis, and therefore a well-aligned galaxy
is artificially classified as misaligned. To avoid this problem, we remove the
nearly axially symmetric satellites (with an axial ratio $b:a\in[0.9,~{}1.0]$)
from our sample, following the strategy used by Pereira et al. (2008); Knebe
et al. (2008) and Knebe et al. (2010). After doing so, the rest subsample
contains only triaxial systems. 333The irregular satellites are also
considered to be triaxially symmetric. The $\cos\theta$ values of the triaxial
satellites are computed and presented in Fig. 8 with green histograms.
${\langle\cos\theta_{1}\rangle}=0.53$ and
${\langle\cos\theta_{2}\rangle}=-0.53$ within $0$-$2\,{\rm R_{\rm h}}$,
${\langle\cos\theta_{1}\rangle}=0.46$ and
${\langle\cos\theta_{2}\rangle}=-0.54$ within $2$-$5\,{\rm R_{\rm h}}$ and
${\langle\cos\theta_{1}\rangle}=0.58$ and
${\langle\cos\theta_{2}\rangle}=-0.61$ within $5$-$10\,{\rm R_{\rm h}}$. The
mean values of the misalignment angles suggest a clear signal of RA only
appears in the outer regions of the new subsample of triaxial galaxies.
Therefore the absence of RA in the inner and intermediate regions is not
caused by the axial symmetry of satellite galaxies. However, our findings in
the inner and intermediate regions do not conflict with the earlier studies
(such as Kuhlen et al., 2007; Pereira et al., 2008; Faltenbacher et al., 2008;
Knebe et al., 2020). In some of those studies, despite satellite galaxies, the
RA between subhalos and their host halos are explored (e.g., Kuhlen et al.,
2007; Pereira et al., 2008). The dark matter subhalos can extend to a radius
much further than $5\,{\rm R_{\rm h}}$ of the stellar component. Furthermore,
in the aforementioned research, even when satellite galaxies are recognized as
the stellar components of subhalos, the radial alignment between these
satellites and their host halos is examined by considering all stellar
particles (e.g., Knebe et al., 2020). In fact, the faint signal of RA in the
inner and intermediate regions in this work aligns with the results found by
Faltenbacher et al. (2008). The trend of decreasing
${\langle\cos\theta_{1}\rangle}$ or increasing
${\langle\cos\theta_{2}\rangle}$ as the truncation radius decreases is
consistent with the earlier analysis (e.g., Kuhlen et al., 2007).

In conclusion, our research indicates that RA only appears in the outermost
regions of isolated galaxies in the TNG 50-1 simulation, a region where tidal
interactions from the centres of host halos are prominent. This could
potentially suggest a tidal origin for the occurrence of RA, which is
consistent with the previous analysis based on other cosmological simulations.

###  4.3. On the existence of LRA

For a sample of satellites, LRA exists if there is a significantly larger
number of satellites with $\cos\theta>0$ than that with $\cos\theta<0$. Note
that the existence of LRA is determined by ${\langle\cos\theta\rangle}>0$
rather than ${\langle\cos\theta\rangle}<0.5$ for the whole sample, since a
dominant fraction of the satellites is axial-symmetric (as seen from Fig. 8).
Fig. 8 shows almost mirror-symmetric distributions, centred at $\cos\theta=0$,
of the numbers of isolated satellites (flavogreen histograms) in the radial
ranges of $0$-$2\,{\rm R_{\rm h}}$ and $2$-$5\,{\rm R_{\rm h}}$. The average
values of the $\cos\theta$ in the inner and intermediate regions are $0.003$
and $-0.005$, respectively. Moreover, the skewness of $\cos\theta$ are $0.002$
and $0.042$ in these radial bins. The mean value and skewness indicate that
the long semi-major axes of the satellites in these radii do not prefer to
point towards the direction facing their halo centres, i.e., the near side.
Thus the LRA is not found in the isolated satellites in the inner and
intermediate regions.

In the outer regions of $5$-$10\,{\rm R_{\rm h}}$, the distribution of the
numbers of satellites is different. The histogram exhibits a random
$\theta$-dependence, and ${\langle\cos\theta\rangle}=0.002$ with a skewness of
$0.008$. Thus the LRA is not observed in the outer regions, either. The
absence of the LRA agrees with our previous findings in §3. In summary, the
longer side of the major axis of a lopsided isolated satellite galaxy does not
preferentially point towards the host halo centre. As aforenoted, the tidal
fields induced by the host halos may be the origin for the existence of RA in
the TNG 50-1 satellite galaxies, since the RA emerges in the outskirts of
these galaxies where tidal fields are substantial.

The lack of LRA points out that the far-distance tidal fields triggered by the
host halos do not lead to detectable systematic signals of lopsidedness for
the satellites. However, the lopsidedness of galaxies may still be caused by
short-distance tidal forces generated by nearby massive objects such as a
massive galaxy. For a satellite galaxy moving along an extremely radial orbit,
the mass distribution of the satellite might be tidally reshaped at its
pericentre. For instance, if the satellite is close enough to the massive
galaxy, the outskirts of the galaxy fill the tip and the broad ends of the
Roche lobes. Thus the galaxy is lopsided in morphology. When the galaxy is
moving radially outwards the massive galaxy, the lopsided shape can exist for
a longer time scale than the orbital time. Moreover, since the majority of the
satellites are disc or oblate galaxies, they are rotationally-supported
systems. The rotation of the satellites might change the orientations of the
long semi-major axes (Mapelli et al., 2008) and thus may erase the signals of
LRA.

##  5\. Discussions

###  5.1. Extremely lopsided satellites and absence of LRA

Table 3Extremely lopsided satellites with semi-axial ratios within the radial range of $0$-$2\,{\rm R_{\rm h}}$, $(a_{-}/a_{+})_{2\,{\rm R_{\rm h}}}$, out of 3$\sigma$ of the Gaussian fitting. The information on isolated and non-isolated satellites is shown in the upper and the lower sub-tables, respectively. The IDs of the subhaloes within which the satellites are embedded are listed in the first column. The IDs of the host haloes for the satellites are listed in the second column. The masses of the stellar, gaseous and dark matter of the satellites are provided in the third to the fifth columns. The half-mass radii and tidal radii of satellites are itemised in the sixth and eighth columns, together with the distances between the satellites and their host halo centres in the seventh column. The semi-axial ratios of the major axes in the eigenframe of the satellites calculated within $0$-$2\,{\rm R_{\rm h}}$, and the values of the $m=1$ Fourier mode, $A_{1}$, computed within a local radial range of $1$-$2\,{\rm R_{\rm h}}$, are presented in the ninth and tenth columns, respectively.  subID | haloID | $m_{\rm sat,*}$ | $m_{\rm sat,gas}$ | $m_{\rm sat,dm}$ | $R_{h}$ | $D$ | $\,{\rm r_{\rm t}}$ | $(a_{-}/a_{+})_{2\,{\rm R_{\rm h}}}$ | $(A_{1})_{2\,{\rm R_{\rm h}}}$
---|---|---|---|---|---|---|---|---|---
|  | $(10^{8}M_{\odot}/h)$ | $(10^{8}M_{\odot}/h)$ | $(10^{8}M_{\odot}/h)$ | $({\rm kpc}/h)$ | $(R_{\rm vir})$ | $({\rm kpc}/h)$ |  |
Isolated satellites with extreme lopsidedness
96819 | 2 | 3.0 | 0.0 | 137.0 | 3.4 | 0.73 | 18.88 | 1.19 | 0.13
282875 | 15 | 1.5 | 0.0 | 0.7 | 0.5 | 0.56 | 3.55 | 0.80 | 0.36
429476 | 68 | 10.8 | 12.2 | 169.7 | 2.1 | 0.36 | 9.89 | 1.20 | 0.45
460724 | 95 | 72.3 | 168.3 | 570.4 | 4.3 | 0.78 | 36.18 | 0.78 | 0.12
108 | 0 | 31.9 | 0.0 | 0.0 | 0.2 | 0.37 | 5.84 | 1.37 | 0.39
320 | 0 | 5.9 | 0.0 | 4.2 | 0.3 | 0.56 | 5.96 | 0.69 | 0.51
96782 | 2 | 83.7 | 0.0 | 493.5 | 0.9 | 0.46 | 18.84 | 1.24 | 0.43
96804 | 2 | 31.0 | 0.0 | 82.8 | 0.6 | 0.41 | 9.90 | 0.82 | 0.43
96883 | 2 | 5.6 | 0.0 | 0.0 | 0.2 | 0.33 | 2.91 | 0.82 | 0.30
143916 | 4 | 1.7 | 1.6 | 218.5 | 2.8 | 1.34 | 34.20 | 1.23 | 0.10
184943 | 6 | 150.4 | 0.1 | 154.3 | 0.8 | 0.57 | 18.57 | 1.27 | 0.40
185048 | 6 | 1.0 | 0.0 | 26.3 | 1.1 | 1.07 | 15.57 | 0.84 | 0.20
282802 | 15 | 10.5 | 0.0 | 5.3 | 0.4 | 0.54 | 6.51 | 0.82 | 0.32
300908 | 18 | 15.1 | 77.3 | 535.7 | 3.3 | 1.16 | 48.13 | 1.25 | 0.01
433290 | 71 | 4.1 | 6.5 | 214.7 | 1.2 | 0.84 | 25.91 | 0.83 | 0.31
482891 | 120 | 44.1 | 0.0 | 23.1 | 0.5 | 1.26 | 26.13 | 1.21 | 0.28
507786 | 157 | 5.3 | 16.1 | 204.3 | 1.1 | 0.87 | 23.24 | 0.77 | 0.35
529366 | 198 | 18.7 | 37.7 | 289.5 | 2.0 | 0.87 | 30.79 | 1.50 | 0.48
564827 | 296 | 3.0 | 13.8 | 273.8 | 1.1 | 1.57 | 49.17 | 0.80 | 0.29
567005 | 303 | 8.2 | 40.5 | 468.2 | 1.5 | 2.13 | 79.41 | 1.25 | 0.38
581059 | 356 | 3.4 | 50.0 | 444.3 | 1.4 | 2.61 | 97.97 | 0.79 | 0.41
584877 | 372 | 9.6 | 26.0 | 158.8 | 0.9 | 0.48 | 13.85 | 1.25 | 0.19
585518 | 375 | 4.3 | 33.1 | 247.8 | 2.4 | 1.23 | 38.26 | 1.19 | 0.17
Non-isolated satellites with extreme lopsidedness
365 | 0 | 5.5 | 0.0 | 0.0 | 0.2 | 0.71 | 0.02 | 1.53 | 0.53
1121 | 0 | 1.4 | 0.0 | 0.0 | 0.06 | 0.71 | 0.01 | 1.50 | 0.48
63990 | 1 | 11.0 | 0.0 | 4.8 | 0.3 | 0.03 | 0.42 | 1.32 | 0.39
185045 | 6 | 1.3 | 5.7 | 0.0 | 3.3 | 2.07 | 2.22 | 2.19 | 0.63
275577 | 14 | 1.8 | 13.6 | 0.0 | 1.3 | 0.10 | 1.23 | 1.52 | 0.68
275579 | 14 | 9.2 | 0.0 | 0.0 | 0.2 | 0.16 | 0.02 | 1.40 | 0.37
275622 | 14 | 1.6 | 5.2 | 0.0 | 0.9 | 0.10 | 0.95 | 0.58 | 0.75
300911 | 18 | 41.2 | 3.0 | 0.0 | 0.1 | 0.07 | 0.07 | 0.64 | 0.40
300972 | 18 | 4.4 | 0.0 | 0.0 | 0.03 | 0.07 | 0.01 | 2.32 | 0.76
342487 | 26 | 1.1 | 4.7 | 0.0 | 1.3 | 1.38 | 0.87 | 1.71 | 0.69
371134 | 36 | 1.3 | 11.9 | 0.0 | 1.5 | 0.12 | 1.29 | 1.57 | 0.52
435761 | 73 | 2.2 | 11.3 | 0.0 | 2.0 | 0.18 | 2.08 | 0.22 | 0.80
435762 | 73 | 1.4 | 10.6 | 0.0 | 1.9 | 0.16 | 1.76 | 0.40 | 0.89
474022 | 109 | 1.2 | 7.6 | 0.0 | 1.4 | 0.10 | 1.08 | 2.29 | 0.82
479954 | 116 | 3.4 | 0.0 | 2.4 | 0.4 | 0.05 | 0.42 | 1.71 | 0.66
510286 | 161 | 1.1 | 5.6 | 0.0 | 1.0 | 0.15 | 1.49 | 0.44 | 0.81
535781 | 212 | 1.4 | 1.9 | 0.0 | 0.6 | 0.04 | 0.28 | 0.39 | 0.79
603005 | 463 | 8.4 | 4.4 | 15.0 | 1.1 | 0.09 | 1.35 | 0.54 | 0.58
63944 | 1 | 21.1 | 0.0 | 20.1 | 0.5 | 0.11 | 1.83 | 1.31 | 0.37
253885 | 12 | 32.7 | 0.1 | 0.0 | 0.1 | 0.05 | 0.66 | 1.24 | 0.28
452981 | 88 | 38.8 | 114.4 | 159.4 | 3.2 | 0.34 | 11.57 | 1.45 | 0.41
494013 | 135 | 27.0 | 2.7 | 0.2 | 0.2 | 0.04 | 0.63 | 0.66 | 0.24
122 | 0 | 28.7 | 0.0 | 0.0 | 0.2 | 0.14 | 2.08 | 1.26 | 0.36
125 | 0 | 26.5 | 0.0 | 0.0 | 0.2 | 0.14 | 2.13 | 0.67 | 0.50
161 | 0 | 18.9 | 0.0 | 0.0 | 0.2 | 0.25 | 3.33 | 0.72 | 0.45
96783 | 2 | 125.7 | 0.2 | 70.2 | 0.7 | 0.27 | 7.89 | 1.22 | 0.29
117308 | 3 | 33.6 | 0.0 | 0.0 | 0.2 | 0.23 | 3.16 | 1.23 | 0.38
275567 | 14 | 14.9 | 0.0 | 0.0 | 0.2 | 0.18 | 2.13 | 1.39 | 0.38

The extremely lopsided galaxies with an $a_{-}/a_{+}$ beyond the $3\sigma$ of
the Gaussian fitting are listed in Table 3 for the isolated sample (the upper
sub-table) and the non-isolated sample (the lower sub-table).

Eighteen out of 23 extremely lopsided non-isolated satellites are strongly
influenced by the tidal fields from the halo centres or nearby massive
galaxies. As a result, the tidal radius $\,{\rm r_{\rm t}}$ is smaller than
$2\,{\rm R_{\rm h}}$. There are seven satellites with $a_{-}/a_{+}<1.0$ and
eleven with $a_{-}/a_{+}>1.0$ among the satellites close to halo centres or
nearby massive galaxies. These galaxies are listed in the $1^{st}-18^{th}$
rows of the lower sub-table in Table 3. There are no such strongly tidally
perturbed satellites due to the selection criteria of the isolated sample.
Moreover, isolated and non-isolated satellites within $\,{\rm r_{\rm
t}}<10\,{\rm R_{\rm h}}$ are shown in the $1^{th}-4^{th}$ rows of the upper
sub-table and in the $19^{th}-22^{th}$ rows of the lower sub-table,
respectively. There are only two satellites with $a_{-}/a_{+}<1.0$ in the
upper sub-table and one in the lower sub-table. Interactions from the halo
centres or nearby massive galaxies are responsible for the lopsidedness of the
satellites close to the halo centres or massive galaxies. However, the LRA is
not found even in these strongly tidally disrupted satellites. The reason
could be the rotation of the satellite galaxies, similar to the case of NGC
891 (Mapelli et al., 2008). For a satellite spinning approximately one-half
cycle around its minor axis, the longer side of the major axis appears on the
opposite side of the massive nearby galaxy or the host halo centre.
Considering most of the satellites are oblate systems or discs (see Fig. 8),
the rotation of these satellites might be the reason for the absence of LRA. A
systematic study on the rotation and angular momentum of satellites will be
performed in a follow-up project.

At the bottom of the two sub-tables of Table 3, there are nineteen isolated
and six non-isolated satellites extremely lopsided and with $\,{\rm r_{\rm
t}}>10\,{\rm R_{\rm h}}$. Tidal stripping from halo centres of nearby massive
galaxies is not the driving force for the lopsidedness of these satellites.

###  5.2. Other mechanisms to produce lopsided satellites

The external perturbations and internal dynamics accounting for the
lopsidedness of galaxies have been extensively studied (e.g., Bournaud et al.,
2005a; Mapelli et al., 2008; Kenney et al., 2015; Zaritsky et al., 2013;
Levine & Sparke, 1998; Noordermeer et al., 2001; Weinberg, 1995; Jog, 1997).
We briefly discuss the possible origins of lopsidedness below.

Figure 9.— The projected stellar mass densities of satellite subID 566663 are
exhibited on $x^{\prime}-z^{\prime}$ (upper left), $x^{\prime}-y^{\prime}$
(lower left) and $z^{\prime}-y^{\prime}$ (lower right) planes of inertia
tensor’s eigenframe, with black dotted circles showing the radii of $2\,{\rm
R_{\rm h}}$. This satellite’s radial phase-space density distribution, $r$
versus $v_{r}$, is displayed in the upper right panel.

####  5.2.1 External origins

As aforementioned, the interactions from the halo centres or massive galaxies
make a small part of satellites extremely lopsided, as summarised in Table 3.
These satellites reside in the central regions of their host haloes or the
vicinity of nearby massive galaxies. The tidal radii of these galaxies are
smaller than $2\,{\rm R_{\rm h}}$. Moreover, there are a large number of
satellites in Table 3 that do not have a dark matter subhalo.

Several dark-matter-free (DM-free) galaxies have been recently observed by Oh
et al. (2015); van Dokkum et al. (2018a, b, 2019). The existence of DM-free
galaxies is at odds with the standard stellar-to-halo mass relation, which
challenges the standard $\Lambda$CDM framework (Haslbauer et al., 2019). On
the other hand, some research groups claimed that the DM-free galaxies are the
remnants of a dwarf galaxy undergone tidal disruptions through close
encounters (e.g., Jing et al., 2019; Moreno et al., 2022). Since the dark
matter particles are dynamically hotter than the stellar particles in a dwarf
galaxy, the dwarf galaxy loses its dark matter halo by the tidal shocks as the
consequence of a close encounter with another galaxy (Jing et al., 2019). But
some other researchers found that dark matter-deficient galaxies form through
encounters of disc galaxies (Barnes & Hernquist, 1992). These DM-free
galaxies, formed in the tidal arms of the disc galaxies, are named tidal dwarf
galaxies. We have examined the formation scenario of the DM-free galaxies
appearing in Table 3.

Below, we shall show four of the extremely lopsided DM-free satellites,
including subID 300911, subID 435762 subID 535781 and subID 566663 in Table 3.
Such satellites are passing through or have just passed through the disc
planes of their central host galaxies. The dark matter subhaloes of these
satellites have been tidally disrupted by the central galaxy or a nearby
massive galaxy. Although the dark matter subhalos of these galaxies seem to be
removed by tidal shocking, more careful tests need to be performed by tracing
their mass loss history backwards in time.

We have examined the origins of the four example DM-free satellites. The
galaxy subID 566663 forms through a merger between subID 563017 and subID
565398 at z=0.02 (the $97_{th}$ snapshot). At the redshift of z=0.02, subID
563017 is the more massive galaxy, whose progenitor, subID 554374, can be
traced backwards in time until a redshift of z=0.08 (the $93_{rd}$ snapshot).
Both progenitors of the DM-free galaxy, subID 566663 at the redshift of zero,
are dark-matter deficient galaxies at a redshift of 0.08. Furthermore, we
traced the more massive progenitor, subID 554374 at z=0.08, backwards in time.
We found that $65\%$ stars of this progenitor form in gas-rich environments
during the $90_{th}-92_{nd}$ snapshots. Thus the DM-free satellite galaxy,
subID566663 at the present day, forms in the dark-matter deficient tidal arms.
The other three example galaxies have similar tidal formation histories. In
the satellite subID 300911, $99\%$ of stars form between $z=0.14$ and $z=0.13$
(the $88_{th}$-$89_{th}$ snapshots) through mergers between two satellite
galaxies (subIDs 268865 and 268866) and the central galaxy (subID 268862). The
DM-free spiral galaxies subIDs 535781 and 435762 formed recently at $z=0.01$
and $z=0.02$ (the $98_{th}$ and $97_{th}$ snapshots, respectively). The dark
matter subhalos of these example galaxies are not disrupted by external
interactions from nearby galaxies. Instead, they are tidally formed dwarf
galaxies. These galaxies were born without dark matter subhalos.

The lopsidedness of the DM-free satellite galaxies may result from multiple
origins. For instance, in the central regions of a cluster, satellites are
perturbed by tidal stripping caused by the central galaxy. Meanwhile, the
satellites are interacting with each other because the number density of
galaxies is larger in the cluster centre. Considering their relative
velocities are also higher when they are in a cluster’s central regions, the
gravitational interaction integration time is also shorter. Thus is it hard to
draw a simple conclusion on the satellite-to-satellite-interaction-induced
asymmetric shape. Let us take a DM-free satellite subID 566663 as an example
galaxy. The distance between this satellite and its halo centre is $0.14R_{\rm
vir}$. The halo-centric position of the satellite is
$(x^{\prime},~{}y^{\prime},~{}z^{\prime})=(13.0,~{}11.5,~{}0.6)\,{\rm kpc}/h$
and the relative velocity to the halo centre is $(v_{\rm x^{\prime}},~{}v_{\rm
y^{\prime}},~{}v_{\rm z^{\prime}})=(-107.3,~{}133.4,~{}-8.6)\,{\rm
km\,s}^{-1}$, when the whole halo is rotated to the eigenframe of inertia
tensor of the central galaxy. The orbit of this satellite on the scale of the
cluster of galaxies implies that it is passing through the stellar plane of
its host halo. The ratio $(a_{-}/a_{+})_{2\,{\rm R_{\rm h}}}=1.17~{}>1.0$.
Moreover, we do not find any tidal tails in this satellite galaxy. Therefore,
the long semi-major axis is on the far side of its central galaxy, indicating
that the lopsidedness is not due to tidal stripping. Other physical origins
are required. To find out the true mechanism, we examine the stellar surface
density of this satellite in the left two panels as well as the lower-right
one of Fig. 9. There is a clear structure located outside $2\,{\rm R_{\rm
h}}$. Moreover, an infalling clump at the same radius is found in the phase-
space density distribution of subID 566663 displayed in the upper right panel
of Fig. 9. Since the particles within the clump are identified as part of the
satellite subID 566663, the clump is infalling shows that the satellite is a
non-fully relaxed merger remnant.

Early studies have demonstrated that asymmetric gas accretion with subsequent
star formation is responsible for the lopsidedness of galaxies (e.g., Bournaud
et al., 2005a; Mapelli et al., 2008). The cold gas accreted from the cosmic
filaments could form new stars and thus lead to an asymmetric shape of the
stellar component in a satellite galaxy (Kereš et al., 2005). We have examined
the possible origins of the strongly lopsided galaxies in Table 3 and found
that none of their lopsidedness is caused by star formation from gas
accretion.

On the other hand, the accretion of stars from external structures might
contribute to the asymmetric of galaxies. A typical example satellite galaxy,
subID 455296, is presented in Fig. 10. The phase-space density distribution
(left panel) reveals that there is a substructure falling into the satellite
centre. The gas distribution in this satellite is examined (middle panel) to
exclude other physical processes such as a merger of galaxies. On the scale of
$\approx 1\,{\rm kpc}$, the projected stellar density appears to be quite
symmetric on the $x^{\prime}-z^{\prime}$ plane of its eigenframe of inertia
tensor. The semi-axial of the major axis is $1.02$ within $0$-$2\,{\rm R_{\rm
h}}$. Extended structures are in the upper right corner of the projected
densities of stellar and gaseous components. The value of $A_{1}$ within the
radial range of $1$-$2\,{\rm R_{\rm h}}$ is $0.12$. The gas and stars are
accreted into the satellite centre. The metallicity of stars in this satellite
is studied in the right panel. The metallicity of stars in the regions of
infalling gas appears to be lower, which suggests that the asymmetric comes
from the accretion of stars from external structures.

Figure 10.— The phase-space density distribution of the satellite subID 455296
is displayed in the left panel. In the middle panel, the projected surface
densities of the stellar and gaseous components are presented with colours
provided in the two colour bars. The metallicity of the stellar component of
the satellite is demonstrated in the right panel. Figure 11.— The projected
stellar mass density (rainbow colours) and projected gaseous density (green-
to-yellow colours) of the satellite subID 313700. An example of ram pressure
stripping on a satellite galaxy. The two arrows indicate the directions of the
halo centre (red arrow) and the motion of the satellite (blue arrow). The
distance between the satellite and the halo centre is $0.21R_{\rm vir}$. The
black dashed circles show the position of $2\,{\rm R_{\rm h}}$.

Ram pressure stripping from the diffuse intergalactic medium (IGM) plays only
a minor role in producing the lopsidedness of galaxies (Mapelli et al., 2008).
The gaseous component in a satellite is compressed by the IGM in the leading
side and is stretched in the opposite direction of the orbit (e.g., a recent
review by Boselli et al., 2022). The gaseous component can be stripped outside
the stellar component by ram pressure (Smith et al., 2012). As a consequence,
the stellar and dark matter components would be influenced by the drag force
imposed by the gas, which in turn leads to an offset of a few $\,{\rm kpc}$
between the disc and the dark matter halo (Smith et al., 2012). The satellite
subID 313700 displayed in Fig. 11 is a representative example of a ram
pressure-stripped lopsided galaxy. In the eigenframe of the inertia tensor of
the host central galaxy, the galactocentric distance and relative speed
between subID 313700 and the central galaxy are $74\,{\rm kpc}/h$ and
$765.3\,{\rm km\,s}^{-1}h^{-1}$, respectively. This satellite is in an
incoming orbit to the central galaxy. The gaseous component of the satellite
has been stripped out and left behind the stellar disc. Given the large gas
fraction ($f_{\rm gas}=82\%$) in this satellite, the gravitational force
introduced by the gaseous component is significant. The gravitation from the
stripped gas then drags the stars in the direction opposite to the direction
of motion. We, therefore, conclude that the lopsidedness of the stellar
component is caused by ram pressure in the subID 313700-like satellites.

####  5.2.2 Internal origins

Apart from the external origins for the lopsidedness of galaxies, internal
dynamics can also produce significantly lopsided galaxies. For instance, the
gravitational instability of disc galaxies may result in a significantly
lopsided shape of a galaxy (Zaritsky et al., 2013). The asymmetric shapes of
the strongly lopsided galaxies in Table 3 all have an external origin. Thus
the internal perturbations only result in moderate or mild asymmetry for the
satellite galaxies. For instance, the satellite subID 63869, is a typical
isolated disc galaxy with spiral arms. The stellar disc is thin, with
$c:a=0.34$ within the radial range of $0$-$2\,{\rm R_{\rm h}}$. In addition,
this satellite is far away from the central galaxy and other satellite
galaxies. Moreover, there is neither asymmetric gas accretion nor recent minor
merger events in this satellite. The semi-axial ratio of the major axis is
$a_{-}/a_{+}=1.05$ within $0$-$2\,{\rm R_{\rm h}}$. Thus the gravitational
instability of the thin disc might be the mechanism of the lopsidedness.
Moreover, since this work mainly focuses on the alignments between the
asymmetric shapes and the halo centres, the internal origins of lopsidedness
will not be discussed further.

###  5.3. Asymmetric tidal tails in observations and symmetric tidal tails in
simulations

Now we briefly review some recent progress on tidal tails which are
complementary to our present work. Observations on Palomar 5 streams showed
that the tidal structures for this star cluster are asymmetry (Bernard et al.,
2016). Very recently, there are more asymmetric tidal tails observed in star
clusters, including Hyades, Praesepe, Coma Berenices, COIN-Gaia 13 and NGC 752
(Kroupa et al., 2022). However, the tidal tails of satellites are found to be
extremely symmetric in a smooth and static host halo potential in previous
studies of Newtonian dynamics (Dehnen et al., 2004). The sole tidal field from
the Milky Way cannot reproduce the asymmetric tidal tails of Palomar 5 (Dehnen
et al., 2004), unless other consequences of gravitational interactions are
introduced, such as perturbations induced by giant molecular clouds (Amorisco
et al., 2016), subhaloes (Erkal et al., 2017), spiral arms (Banik & Bovy,
2019) or a prograde Galactic bar (Pearson et al., 2017). Bonaca et al. (2020)
emphasised that none of the currently existing scenarios can explain all the
observational features of the stellar streams of Palomar 5. To explain all the
observational results of Palomar 5 within one scenario is out of the scope of
this work. The symmetric tidal tails of satellite galaxies predicted in
Newtonian dynamics agree with our results on the absence of the LRA.

##  6\. Summaries and conclusions

In this work, we have presented a detailed analysis of the semi-major axial
ratios ($a_{-}/a_{+}$) of the satellite galaxies generated by the TNG50-1
simulation. We have found that the distribution of $a_{-}/a_{+}$ is almost
symmetric around the central value of $a_{-}/a_{+}=1.0$. This result indicates
that the probabilities for the long semi-major axes of satellites to point
towards and outwards the halo centres are nearly equal. We have adopted two
criteria to define the lopsidedness of a satellite. More concretely, a
satellite galaxy is regarded as lopsided if $\Delta>0.1$ or $A_{1}>0.2$. Under
the former criterion, the fractions of strongly lopsided satellites are
$7.5\%$ and $11\%$ for the isolated and non-isolated satellites truncated at
$2\,{\rm R_{\rm h}}$, respectively. Under the latter criterion, the two
numbers become $7.8\%$ and $18.1\%$, respectively. While the fractions of
strongly lopsided satellites in our analysis are at least one order of
magnitude larger than those reported in Łokas (2022), they are still smaller
than observations (e.g., Jog & Combes, 2009; van Eymeren et al., 2011; Yu et
al., 2022).

The correlations between $a_{-}/a_{+}$ and other physical parameters have been
further studied. We have shown that $a_{-}/a_{+}$ does not correlate with the
distance to the halo centres, $D$, in both the inner and outer regions of the
satellites. The fraction of galaxies with long semi-major axes in the near
side to halo centres, $f_{N_{-}}$, is almost equal to $50\%$. These findings
imply that the tidal fields from the halo centres do not lead to the
lopsidedness of satellites. Moreover, we have found that neither the masses of
the host haloes $M_{\rm halo}$ nor the satellite masses $M_{\rm sat}$
correlate with $a_{-}/a_{+}$. However, we have revealed a correlation between
the values of $a_{-}/a_{+}$ and the fractions of gas, $f_{\rm gas}$, in the
satellites. In both isolated and non-isolated samples of satellites, more
massive galaxies tend to be more gas-poor. Moreover, the more gas-rich
galaxies are more lopsided, with a larger $\Delta a$.

In this work, we have explored the existence of RA in an isolated sample of
satellite galaxies. The RA signal within the inner and intermediate regions of
satellite galaxies is so weak that the existence of radial alignment cannot be
asserted. However, RA is statistically significant present in the outer
regions. Our findings are in agreement with the previous cosmological
simulations (Kuhlen et al., 2007; Faltenbacher et al., 2008; Pereira et al.,
2008; Knebe et al., 2020) as well as certain observations (Hawley & Peebles,
1975; Thompson, 1976; Pereira & Kuhn, 2005; Faltenbacher et al., 2007; Wang et
al., 2008, 2019).

We have known from the above analysis that the lopsidedness of satellites is
not mainly induced by the tidal fields from the halo centres. Although this
result provides us with evidence that LRA is unlikely to present in the
simulated satellites within the framework of $\Lambda$CDM. More careful
investigation is required to reach a conclusive answer. This motivated us to
examine the existence of such a new alignment in the TNG50-1 simulation. The
alignment angle, $\cos\theta$, defined in Eq. 10 has been systematically
analysed. No LRA has been observed in all radial ranges.

Finally, we have briefly discussed the possible origins of the lopsidedness of
satellites in the TNG50-1 simulation. The lopsidedness could be induced by the
tidal stripping from the nearby central galaxies or massive satellite
galaxies, the recent mergers, the asymmetric accretion of stellar streams from
external structures and the ram pressure stripping. The internal origins such
as gravitational instability only lead to moderate or mild lopsidedness in the
TNG50-1 satellite galaxies.

##  7\. Acknowledgements

The authors acknowledge the public release of Illustris-TNG data. We thank the
anonymous referee and the statistics editor for the constructive and detailed
comments. We appreciate Dr Hongsheng Zhao (St Andrews)’s suggestions regarding
the alignment angle and the Roche lobes issues. We want to express our
heartfelt gratitude to the contributors of the CRAN R project. Their work has
provided powerful tools that have enabled us to understand better and
interpret our data. We deeply respect and appreciate their hard work.
Additionally, we would like to extend our thanks to the open-source packages,
including twosamples, cramer (Rizzo, 2019; Feigelson & Babu, 2012) and
circular (Fisher et al., 1993; Mardia & Jupp, 2000; Pewsey et al., 2013;
Pewseu, 2018; Ley & Verdebout, 2017; Mohammad et al., 2021). We appreciate the
efforts of all those involved in creating and maintaining these valuable
resources.

XW is financially supported by the Natural Science Foundation of China (Number
NSFC-12073026, NSFC-11421303) and “the Fundamental Research Funds for the
Central Universities”. YZ is financially supported by the “Fund for Fostering
Talents in Basic Science of the National Natural Science Foundation of China
NO.J1310021”.

XW motivated and supervised the project. JS and XW analysed the released
TNG50-1 data with the assistance of BG. YZ contributed to the early analysis
of Illustris-1 data as an undergraduate thesis project. XW and JS wrote the
manuscript with contributions from BG and YZ.

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