# Lopsided Satellite Distributions around Isolated Host Galaxies in a
$\Lambda$CDM Universe

Adam Samuels Boston University, Department of Astronomy & Institute for
Astrophysical Research
725 Commonwealth Ave, Boston, MA 02215, USA  Tereasa G. Brainerd Boston
University, Department of Astronomy & Institute for Astrophysical Research
725 Commonwealth Ave, Boston, MA 02215, USA

###### Abstract

A recent observational study found that the projected spatial distributions of
the satellites of bright, isolated host galaxies tend to be lopsided with
respect to the locations of the hosts. Here, we examine the spatial
distributions of the satellites of a large number of bright, isolated host
galaxies that were obtained from mock redshift surveys of a $\Lambda$CDM
simulation. Host galaxies and their satellites were identified using selection
criteria that are identical to those used in the observational study, allowing
a direct comparison of the results for the simulated and observed systems. To
characterize the spatial distribution of the satellites, we adopt two
statistics: [1] the pairwise clustering of the satellites and [2] the Mean
Resultant Length. In agreement with the observational study, we find a strong
tendency for satellites in the simulation to be located on the same side of
their host, and the signal is most pronounced for the satellites of blue
hosts. These lopsided satellite distributions are not solely attributable to
incompleteness of the observed satellite catalog or the presence of objects
that have been falsely identified as satellites. In addition, satellites that
joined their hosts’ halos in the distant past ($\gtrsim 8$ Gyr) show a
pronounced lopsidedness in their spatial distributions and, therefore, the
lopsidedness is not solely attributable to late-time accretion of satellites.

Companion galaxies (290) — Dark matter distribution (356) — Dwarf galaxies
(416) — Galaxy dark matter halos (1880)

##  1 Introduction

In the Cold Dark Matter (CDM) model, non-linear structure in the universe
grows hierarchically (e.g., Davis et al. 1985) and, at the present day, the
dark matter halos that surround galaxies, groups of galaxies, and clusters of
galaxies are triaxial in shape (e.g., Jing & Suto 2002). In order to test
these robust theoretical predictions in the observed universe, some type of
luminous tracer of infall history and halo shape is needed. In the case of
bright, isolated galaxies that host nearby, faint satellite galaxies, the
spatial distribution of the satellites may provide insight into the shapes and
formation histories of the host galaxies’ dark matter halos. For example, if
the satellites arrived within their hosts’ halos in the far distant past, they
could be expected to be a virialized population that is both symmetrically
distributed about the hosts and directly traces the shapes of the hosts’ dark
matter halos. If, instead, the satellites arrived within their hosts’ halos
relatively recently, they are less likely to be a virialized population and
their spatial distributions may be neither symmetrical about the hosts nor
indicative of the shapes of the hosts’ halos.

Previous studies have shown that satellites around pairs of host galaxies in
the observed universe are not, in fact, symmetrically distributed about each
host. Libeskind et al. (2016) showed that satellites around pairs of host
galaxies in the Sloan Digital Sky Survey (SDSS; Fukugita et al. 1996; Hogg et
al. 2001; Strauss et al. 2002; York et al. 2000) tend to occupy the space
between the hosts rather than being spherically or axis-symmetrically
distributed about each host. Locally, 20 of M31’s 27 satellites are located
between M31 and the Milky Way (Conn et al., 2013). Lopsided satellite
distributions have also been found around pairs of host galaxies in $\Lambda$
Cold Dark Matter ($\Lambda$CDM) simulations (Pawlowski et al., 2017; Gong et
al., 2019), and the lopsidedness has been attributed to satellites on their
first orbit around their host (Gong et al., 2019).

Due to the gravitational potential of a massive neighboring galaxy, lopsided
satellite distributions between pairs of host galaxies are less surprising
than might be expected for the satellites of isolated host galaxies. A recent
observational study by the authors (Brainerd & Samuels 2020, hereafter BS20),
however, found significantly lopsided satellite distributions around isolated
hosts in the NASA-Sloan Atlas (NSA). BS20 computed the pairwise clustering of
the satellites and found a strong tendency for pairs of satellites to be
located on the same side of their host, especially in systems with blue hosts
and systems with two or three satellites.

Lopsided satellite distributions have also been found around isolated hosts in
$\Lambda$CDM simulations. Wang et al. (2021), hereafter W21, performed an
analysis of the projected spatial distribution of the satellites of isolated
hosts in the Illustris-TNG300 simulation (Marinacci et al., 2018; Naiman et
al., 2018; Nelson et al., 2018; Pillepich et al., 2018; Springel et al.,
2018). In general agreement with BS20, W21 found significant lopsidedness,
especially in systems with low-mass blue hosts. However, the W21 satellite
sample was selected in 3D space, with satellites being defined as all other
galaxies that are located within the hosts’ dark matter halo. As a result, the
properties of the W21 host-satellite sample differ considerably from those in
BS20. For example, in the W21 sample, the mean number of satellites per host
is a factor of $\sim 2.2$ larger than it is in BS20. In addition, the W21
sample has many systems for which the number of satellites per host exceeds a
total of 100, whereas the maximum number of satellites per system in BS20 is
only 27. Furthermore, W21 restricted the stellar masses of their hosts to be
$>10^{11}h^{-1}M_{\odot}$, resulting in the majority of their host-satellite
systems being more massive than those in BS20.

In this work, we directly compare the spatial distributions of satellite
galaxies in the observed universe to the spatial distributions of satellite
galaxies in a $\Lambda$CDM universe. To do this, we obtain host and satellite
galaxies from the $\Lambda$CDM universe using redshift space selection
criteria (rather than 3D selection criteria) . To obtain the $\Lambda$CDM
host-satellite sample, we employ two mock, all-sky galaxy redshift surveys
based on the Millennium simulation (Springel et al., 2005). These mock
redshift surveys allow us to select our sample from the $\Lambda$CDM universe
in the same way we selected the BS20 sample from the NSA.

In addition to allowing us to directly compare $\Lambda$CDM predictions to our
previous observational results, selecting the $\Lambda$CDM host-satellite
sample using redshift space criteria allows us to address potential sources of
uncertainty in the observational sample that could result in a spurious
detection of lopsidedness. These include candidate satellites that are within
the on-sky search radius, but which lack spectroscopic redshifts (i.e., $\sim
10\%$ of galaxies in the SDSS with $r<17.77$ lack spectroscopy due to either
fiber positioning constraints or having a surface brightness that was too low
for target selection). In addition, redshift space selection of host-satellite
systems always results in some fraction of the satellites being “interlopers”.
These are galaxies that pass the satellite selection criteria but are not
physically associated with the host galaxies (i.e., interlopers are objects
that are falsely identified as satellites). Further, unknown infall histories
for observed satellite galaxies make it impossible to determine when these
objects entered their hosts’ dark matter halos, and it is possible that the
lopsidedness of the satellite distribution is largely attributable to objects
that joined their host galaxies only recently. Here, we can directly address
these issues using our simulated host-satellite sample since the 3D distances
of all galaxies are known (i.e., we can determine which objects are legitimate
satellites and which are interlopers), and the redshifts at which legitimate
satellites joined their host galaxies’ halos are also known. This allows us to
investigate the degree to which the lopsided spatial distributions of the
satellites in BS20 is reproduced by a $\Lambda$CDM universe, whether the
lopsidedness found by BS20 is spurious (i.e., caused by legitimate satellites
that are “missing” from the sample due to spectroscopic incompleteness, or
caused by false satellites that are located far from the host galaxies in 3D),
and whether more recently accreted satellites show greater lopsidedness in
their spatial distribution than do satellites that entered their hosts’ halos
in the distant past.

The paper is organized as follows. In §2 we discuss the way in which the host-
satellite sample was obtained from the mock redshift surveys and we compare
the properties of the simulated sample to the observational sample in BS20. In
§3 we outline the methods we use to characterize the lopsidedness of the
satellite distribution, we compare results for the simulated sample to results
obtained for the BS20 observational sample, and we investigate the effects of
interlopers and satellite infall redshift on the lopsidedness. In §4 we
present a summary and discussion of the results, and in §5 we state the major
conclusions of our work. Throughout, we adopt values of the fundamental
cosmological parameters obtained by Planck Collaboration et al. (2014):
$H_{0}=67.3$ km s-1 Mpc-1, $\Omega_{m}=0.315$, and $\Omega_{\Lambda}=0.685$.

##  2 Host-satellite Sample

Host galaxies and their satellites were obtained from two mock all-sky galaxy
redshift surveys that were created by Henriques et al. (2015). Luminous
galaxies were produced with the L-Galaxies semi-analytic galaxy formation
model, implemented on the Millennium dark matter simulation, and updated with
Planck Collaboration et al. (2014) cosmology. Henriques et al. (2015)
generated two light cones by stacking the simulation box ($500h^{-1}$ Mpc per
side) with no additional transformations. One light cone places the observer
at the origin and the other places the observer at $x=y=z=250h^{-1}$ Mpc. The
light cone galaxy catalogs include observer-frame magnitudes for 9 bandpasses,
including SDSS $g$ and $r$, both of which we use in this study. The mass
resolution of the Millennium simulation dark matter particles is $1.43\times
10^{9}h^{-1}$M⊙, and the stellar mass resolution of the luminous galaxies is
$\sim 10^{7}$ M⊙. The light cone galaxy catalogs are publicly available via
the Henriques2015a table in the Cosmology Science Domain on the SDSS
SciServer.

In order to facilitate direct comparison with the NSA sample from BS20, we
restrict our analysis to Millennium galaxies with $r<17.77$, consistent with
the SDSS spectroscopic completeness limit. Furthermore, host and satellite
galaxies were selected using the same redshift space criteria as BS20.
Potential host galaxies were required to have radial velocities $v>500$ km s-1
and to be at least 1 mag brighter than all other galaxies within a projected
distance $r_{p}<700$ kpc and radial velocity difference $|dv|<1000$ km s-1.
Additionally, as in BS20, potential host galaxies were required to have
apparent magnitudes in the range $9<r<15$, which ensures that the majority of
hosts (90.2%) have luminosities brighter than $L_{*}$ (i.e., $M_{r,*}=-20.83$;
Blanton et al. 2001). Satellite galaxies were required to be located within a
projected distance $r_{p}<500$ kpc of their hosts and a radial velocity
difference $|dv|<500$ km s-1. Since we are interested in analysing the
pairwise clustering of the satellites, we further restricted the sample to
systems for which two or more satellites were identified. Below, we refer to
the host-satellite sample drawn from the mock all-sky redshift surveys as the
“Millennium sample” and to the host-satellite sample drawn from BS20 as the
“NSA sample”. Various properties of both samples are summarized in Figure 1
and Table 1.

From Table 1, the Millennium sample is considerably larger than the NSA sample
($\sim 7$ times more hosts and $\sim 6$ times more satellites). From Table 1
and Figure 1, the majority of the properties are similar in the two samples
(i.e., redshift distribution, number of satellites per system, extinction-
corrected and $k$-corrected absolute magnitudes of the hosts and satellites,
as well as the dependence of the mean satellite color on host color). Compared
to the NSA sample, the Millennium hosts are somewhat more intrinsically
luminous than their satellites (i.e., the mean host-to-satellite luminosity
ratio is $\sim 40\%$ larger for the Millennium sample; see Table 1). In
addition, the Millennium hosts and satellites have somewhat larger stellar
masses than those in the NSA sample (i.e., the median stellar mass for hosts
and satellites is $\sim 1.4$ times larger in the Millennium sample). Here the
stellar masses for the NSA galaxies correspond to the elpetro_mass parameter
in the NSA database. While there is a systematic offset between the stellar
mass distributions of the NSA and Millennium hosts, the difference between the
stellar masses as a function of host color is consistent between the samples.
That is, as expected, the red hosts are somewhat more massive than the blue
hosts and in both samples the median and mean stellar mass of the red hosts
exceeds that of the blue hosts by a factor of $\sim 3$ (see Table 1). Lastly,
from Figure 1 (panels c and d), the two samples differ somewhat in terms of
the distribution of $(g-r)$ colors for both the hosts and the satellites. The
Millennium hosts show a pronounced color bi-modality that is not shown by the
NSA hosts and the fraction of blue satellites is considerably larger in the
Millennium sample.

Figure 1: Properties of the Millennium (black) and NSA (green) samples. (a) Host redshift distributions. (b) Number of satellites per system, truncated for clarity. (c) Host rest-frame $(g-r)$ color distributions. (d) Satellite rest-frame $(g-r)$ color distributions. (e) Host and satellite $r$-band absolute magnitude distributions. (f) Dependence of mean satellite color on host color. (g) Host and satellite stellar mass distributions. (h) Host stellar mass distributions, computed separately for red and blue hosts. Table 1: Host-Satellite Samples property | Millennium | NSA
---|---|---
number of hosts | 23,652 | 3,575
number of satellites | 80,703 | 13,090
mean number of satellites | 3.4 | 3.7
median number of satellites | 3 | 3
maximum number of satellites | 46 | 26
mean host redshift | 0.039 | 0.046
median host redshift | 0.037 | 0.042
median host $M_{r}$ | -22.0 | -22.8
median satellite $M_{r}$ | -19.6 | -19.9
mean host-to-satellite luminosity ratio | 27.3 | 19.9
mean host stellar mass $[M_{\odot}]$ | 1.3x$10^{11}$ | 9.2x$10^{10}$
median host stellar mass $[M_{\odot}]$ | 9.6x$10^{10}$ | 6.7x$10^{10}$
mean satellite stellar mass $[M_{\odot}]$ | 1.7x$10^{10}$ | 9.2x$10^{9}$
median satellite stellar mass $[M_{\odot}]$ | 7.8x$10^{9}$ | 5.3x$10^{9}$
mean red host stellar mass $[M_{\odot}]$ | 1.9x$10^{11}$ | 1.1x$10^{11}$
median red host stellar mass $[M_{\odot}]$ | 1.4x$10^{11}$ | 8.1x$10^{10}$
mean blue host stellar mass $[M_{\odot}]$ | 6.3x$10^{10}$ | 3.6x$10^{10}$
median blue host stellar mass $[M_{\odot}]$ | 4.2x$10^{10}$ | 2.5x$10^{10}$
mean red satellite stellar mass $[M_{\odot}]$ | 3.3x$10^{10}$ | 1.4x$10^{10}$
median red satellite stellar mass $[M_{\odot}]$ | 2.4x$10^{10}$ | 1.1x$10^{10}$
mean blue satellite stellar mass $[M_{\odot}]$ | 6.4x$10^{9}$ | 3.3x$10^{9}$
median blue satellite stellar mass $[M_{\odot}]$ | 3.2x$10^{9}$ | 1.6x$10^{9}$
number of red hosts | 13,385 | 2,719
number of blue hosts | 10,267 | 856
number of red satellites | 33,125 | 6,873
number of blue satellites | 47,578 | 6,217

In order to subdivide the Millennium sample into “red” and “blue” galaxies, we
fit the distribution of the intrinsic $(g-r)$ colors of all $z=0$ Millennium
galaxies (i.e., not just the galaxies that were selected as hosts and
satellites) with the sum of two Gaussian distributions. From this, we find
that the Gaussians intersect at $(g-r)=0.74$ and we therefore take Millennium
galaxies with $(g-r)<0.74$ to be “blue” and those with $(g-r)\geq 0.74$ to be
“red”. In the case of the NSA sample, the division between red and blue
galaxies was taken to be $(g-r)=0.7$ (e.g., Blanton et al. 2003). Using these
color divisions, we find that, overall, the Millennium sample is bluer than
the NSA sample. From Table 1, the fraction of Millennium hosts that are blue
in color (43.4%) is considerably higher than the fraction of NSA hosts that
are blue in color (23.9%). In addition, the fraction of satellites that are
blue in color is somewhat higher for the Millennium sample (59.0%) than it is
for the NSA sample (50.9%).

##  3 Methods and Results

We use two complementary metrics to quantify the degree of lopsidedness in the
spatial distributions of the satellites. First, we use the same metric as
BS20: the probability distribution of the polar angle differences between
pairs of satellites, $P(\Delta\phi)$, aggregated over all systems. Second, we
use the Mean Resultant Length, which quantifies the directionality of a
distribution of points (i.e., the directionality, relative to the host, of all
satellites in a given system).

###  3.1 Pairwise Polar Angle Differences

As in BS20, we construct probability distributions for the polar angle
differences between each pair of satellites in a system, $P(\Delta\phi)$. Here
$\Delta\phi$ is defined such that pairs of satellites with $\Delta\phi\sim
0^{\circ}$ are located on the same side of their host, and pairs of satellites
with $\Delta\phi\sim 180^{\circ}$ are located on opposite sides of their host
(see Figure 3 of BS20 for an illustration of the definition of $\Delta\phi$).
To make a fair comparison between the Millennium and NSA samples, we also
restrict the maximum number of satellites in a given system to be the lowest
maximum number of satellites between the two samples. That is, the maximum
number of satellites around blue hosts in the Millennium sample is 22, while
in the NSA sample the maximum number of satellites around blue hosts is 17.
So, when comparing results for the satellites of blue hosts, we restrict both
samples to systems that contain at most 17 satellites. Furthermore, to ensure
that the results are not affected by different numbers of systems containing
$n$ satellites, we compare the NSA sample to 1,000 bootstrap resamplings of
the Millennium sample, each containing the same number of systems with $n$
satellites as the NSA sample (i.e., each Millennium sample has the same number
of systems with 2 satellites, 3 satellites, 4 satellites, etc. as the NSA
sample).

Figure 2 shows $P(\Delta\phi)$ for various subsets of the Millennium and NSA
samples. Dashed lines show $P(\Delta\phi)$ for a uniform circular distribution
of satellites. Here, $1\sigma$ error bars were generated from 10,000 bootstrap
resamplings of the ensemble set of $\Delta\phi$ values for each sample, and
error bars are omitted from figures when they are smaller than the sizes of
the data points.

Figure 2: Probability distributions, $P(\Delta\phi)$, for the polar angle
difference between pairs of Millennium satellites (circles) and NSA satellites
(diamonds). Note that the range of ordinate values differs amongst the panels.
(a) All host-satellite systems. (b) Satellites of red hosts. (c) Satellites of
blue hosts. (d) Systems with 2 to 3 satellites. (e) Systems with 4 to 6
satellites. (f) Systems with 7 to 21 satellites. Dashed lines show
$P(\Delta\phi)$ for a uniform circular distribution of satellites.

From Figure 2, pairs of satellites in the Millennium sample show a strong
tendency to be located on the same side of their hosts, resulting in lopsided
distributions. The result is highly significant with $\chi^{2}$ tests
indicating that, in all six panels of Figure 2, $P(\Delta\phi)$ for the
Millennium sample is inconsistent with a uniform circular distribution at
confidence levels $>99.9999\%$. In addition, in panels a), b), e), and f)
(i.e., complete sample, satellites of red hosts, systems with 4 to 6
satellites, and systems with 7 to 21 satellites), $\chi^{2}$ tests indicate
that the distribution of Millennium satellites is inconsistent with the
distribution of NSA satellites at confidence levels $>99.5\%$. The sense of
the disagreement between the two samples is that the Millennium sample shows a
greater degree of lopsidedness than the NSA sample. In the case of systems
with blue hosts and systems with 2 to 3 satellites (which are not fully
independent, since blue hosts tend to have fewer satellites than red hosts),
the lopsidedness of the satellite distribution is the greatest, and $\chi^{2}$
tests indicate that the Millennium and NSA results for $P(\Delta\phi)$ are
consistent with one another.

We further characterize the lopsidedness of the satellite distributions by
computing the ratio of the number of pairs of satellites with
$0^{\circ}\leq\Delta\phi\leq 20^{\circ}$ and $160^{\circ}\leq\Delta\phi\leq
180^{\circ}$ (i.e., the ratio of the number of pairs of satellites on the same
side of their host to the number of pairs on opposite sides of their host). We
refer to this ratio as the lopsidedness fraction, $f_{\rm lop}$, the results
of which are shown in Figure 3. From Figure 3, the lopsidedness fraction for
the Millennium satellites significantly exceeds the lopsideness fraction for
the NSA satellites in the complete sample, in systems with red hosts, in
systems with 4 to 6 satellites, and in systems with 7 to 21 satellites. For
systems with only 2 to 3 satellites, the lopsideness fraction is identical in
both samples. For systems with blue hosts, the lopsideness fraction for the
Millennium satellites is somewhat lower than it is for the NSA satellites.

Figure 3: Lopsidedness fractions, $f_{\rm lop}$, for the host-satellite
samples. Circles: Millennium sample. Diamonds: NSA sample.

###  3.2 Mean Resultant Length (MRL)

The MRL is a circular summary statistic that measures the directionality of a
distribution of points. For a host-satellite system with $n$ satellites, and
satellite polar angles measured in a coordinate system centered on the host,
the MRL is defined as

$\overline{R}=\frac{1}{n}\sqrt{\left(\sum_{i=1}^{n}\cos\phi_{i}\right)^{2}+\left(\sum_{i=1}^{n}\sin\phi_{i}\right)^{2}}$
(1)

where $\phi_{i}$ are the polar angles of the satellites (see, e.g., Feigelson
& Babu 2012). If the satellites are clustered towards one direction relative
to the host, $\overline{R}\rightarrow 1$. If the satellites are distributed in
a balanced way about the host, $\overline{R}\rightarrow 0$. The MRL can be
interpreted as the magnitude of the vector sum of individual vectors, each
with length $1/n$ and each pointing in the direction of one of the satellites,
whose length would be unity if all $n$ satellites had the same polar angle.

A key feature of the MRL is that it takes into account the locations of all
satellites in a given system, not merely pairs of satellites in the system
(i.e., as is the case with the probability distribution of polar angle
differences in the previous subsection). Furthermore, the value of
$\overline{R}$ within a system gives an indication of whether the satellites
in that particular system are distributed in a lopsided or balanced way, as
opposed to $P(\Delta\phi)$, which only measures the aggregate distribution of
polar angle differences between pairs of satellites within the sample (i.e.,
not the degree to which individual systems in the sample show lopsided
satellite distributions).

Figure 4 shows the expected probability distributions, $P(\overline{R})$, for
randomly distributed satellites in systems with various numbers of satellites.
From this, it is clear that the shape of $P(\overline{R})$ depends strongly on
the number of satellites per system. Therefore, throughout our analysis below,
we are careful to compare samples that contain an identical number of systems
with $n$ satellites. To compare results for the NSA sample to results from the
Millennium sample, we again use 1,000 bootstrap resamplings of the Millennium
sample, each of which contains the same number of systems with $n$ satellites
as the NSA sample. To compare results for the Millennium and NSA samples to
the expected distribution for randomly distributed satellites, we also match
the number of systems with $n$ randomly distributed satellites to the number
of systems with $n$ satellites in the NSA sample.

Figure 4: Expected probability distributions, P$(\overline{R})$, for the Mean
Resultant Length, $\overline{R}$, in systems that contain $n$ randomly
distributed satellite galaxies.

Figure 5 shows $P(\overline{R})$ for various subsets of the Millennium and NSA
samples (circles and diamonds, respectively) as well as results for 10,000
iterations of the sample with randomized values of $\phi$ (dashed line). For
clarity of the plot (i.e., to highlight the differences between the
expectations for randomly distributed satellites vs. the Millennium and NSA
satellites), Figure 5 displays the results for $P(\overline{R})$ for the
randomly distributed satellites as spline fit lines through the values at the
centers of each of the bins. Here, $1\sigma$ error bars were generated from
10,000 bootstrap resamplings of the ensemble set of $\overline{R}$ values for
each sample.

It is clear from Figure 5a) that the distributions of $\overline{R}$ for the
full Millennium and NSA samples show similar levels of directionality, as well
as significantly greater directionality than the sample with randomized
satellite locations (i.e., more systems with $\overline{R}$ values close to
1). Similar to our results for the dependence of $P(\Delta\phi)$ on host color
in Figure 2, Figure 5 indicates that the satellites of blue hosts (panel c)
show greater directionality in their locations than do the satellites of red
hosts (panel b). Further, from Figure 5d), the lopsidedness in both the
Millennium sample and the NSA sample is driven largely by systems with 2 to 3
satellites. When we compare the results for the Millennium satellites and the
NSA satellites in Figure 5, $\chi^{2}$ tests indicate that, with the exception
of systems with 7 to 21 satellites, $P(\overline{R})$ for both samples is
consistent with being drawn from the same underlying distribution (i.e., as
quantified by the probability distribution of MRL values, the samples show the
same degree of lopsidedness). In the case of systems with 7 to 21 satellites,
a $\chi^{2}$ test shows that the probability distributions for the two samples
are moderately inconsistent with being drawn from the same underlying
distribution (confidence level of 99.72%, corresponding to a reduced
$\chi^{2}$ value of 2.80).

Figure 5: Probability distributions, P$(\overline{R})$, for the Mean Resultant
Length of each host-satellite system. Note that the range of ordinate values
differs amongst the panels. Circles: Millennium satellites. Diamonds: NSA
satellites. Dashed lines: expected distributions for samples with random
satellite locations. (a) All host-satellite systems. (b) Satellites of red
hosts. (c) Satellites of blue hosts. (d) Systems with 2 to 3 satellites. (e)
Systems with 4 to 6 satellites. (f) Systems with 7 to 21 satellites.

###  3.3 Interlopers

In an observational sample of host galaxies and their satellites, line of
sight distances are rarely known. Because of this, host-satellite systems are
obtained using redshift space selection criteria that are based on proximity
in the plane of the sky (i.e., angular separation) and proximity in line-of-
sight velocity. The majority of the objects that are identified as satellite
galaxies are, in fact, legitimate satellites. However, some fraction of the
objects that are identified as satellites are actually objects that pass the
selection criteria but are not physically associated with the central, host
galaxy. Such objects are known as “interlopers” (or “false satellites”). In an
observational sample it is, of course, not possible to distinguish legitimate
satellites from interlopers. In a sample obtained from a simulation it is,
however, possible to determine which objects are interlopers since, in
addition to the mock redshift surveys, full 3D spatial information is known
for each galaxy.

Here we investigate the effects of interlopers on our detections of
lopsidedness in the Millennium satellite distribution, with an emphasis on
determining whether the lopsidedness is solely attributable to the presence of
interlopers. We define legitimate satellites to be those Millennium objects
that pass the redshift space selection criteria for satellites and are also
located within the 3D virial radius of their host galaxy. From this, we find
that 60.8% of the Millennium satellites are legitimate satellites. The
fraction of objects that qualify as legitimate satellites in a simulation will
vary with both the redshift space selection criteria and the maximum host-
satellite distance in 3D, but the interloper fraction is typically of order
one-third of the objects. The results for our sample are reasonably consistent
with the results from Agustsson & Brainerd (2010), who used an earlier mock
redshift survey of the Millennium simulation to obtain a sample of isolated
host galaxies and their satellites. Agustsson & Brainerd (2010) used somewhat
different selection criteria and defined legitimate satellites to be those
objects that were located within a 3D distance of 500 kpc from their host.
From this, Agustsson & Brainerd (2010) found a legitimate satellite fraction
of 66.8%.

Figures 6 and 7 show the probability distributions, $P(\Delta\phi)$ and
$P(\overline{R})$, for legitimate Millennium satellites (“interlopers
excluded”; circles) and all Millennium satellites (“interlopers included”;
squares). The “interlopers included” sample was constructed from 1,000
bootstrap resamplings of the data, each of which contain the same number of
systems with $n$ satellites as the “interlopers excluded” sample.

From Figures 6 and 7, it is clear that the lopsidedness of the Millennium
satellite distribution persists even when interlopers are excluded from the
sample. Therefore, the lopsidedness of the complete Millennium satellite
distribution (i.e., with interlopers included) is not solely attributable to
the presence of interlopers. In all cases shown in Figures 6 and 7, $\chi^{2}$
tests indicate that both $P(\Delta\phi)$ and $P(\overline{R})$ for the
legitimate Millennium satellites are inconsistent with random satellite
distributions at confidence levels $>99.999\%$.

Figure 8 shows the lopsidedness fractions for legitimate Millennium satellites
and for the the full sample (i.e., with interlopers included). From this
figure, it is clear that the presence of interlopers results in a slight
enhancement of the lopsidedness fraction compared to the samples in which
interlopers have been excluded.

We explore the locations of the interlopers, relative to the hosts, in Figure
9. The top panel of the figure shows $P(\Delta\phi)$ and the bottom panel
shows $P(\overline{R})$. In both panels, the values of $\Delta\phi$ and
$\overline{R}$ were computed solely from the interlopers that were identified
using the full sample of Millennium hosts. From Figure 9, the interlopers show
a significant level of lopsidedness (as measured by both metrics), indicating
that interlopers are not a random background. We find that, in 3D, 63% of the
interlopers are located between a host’s virial radius and a distance of 500
kpc; that is, the majority of interlopers are not distant galaxies that are
completely unrelated to the hosts. Although they adopted a somewhat different
definition of “interloper” (i.e., objects that were $>500$ kpc from the host),
Agustsson & Brainerd (2010) obtained a similar result, finding that the median
host-interloper distance was only 630 kpc. The high level of lopsidedness in
the Millennium interloper population suggests that the majority of interlopers
are associated with the local large-scale structure that surrounds the hosts
and are likely approaching the hosts’ halos for the first time, or are on an
early fly by.

Figure 6: Probability distributions, $P(\Delta\phi)$, for the Millennium
sample. Circles: legitimate satellites only (see text for definition).
Squares: complete satellite samples, including interlopers. (a) All host-
satellite systems. (b) Satellites of red hosts. (c) Satellites of blue hosts.
(d) Systems with 2 to 3 satellites. (e) Systems with 4 to 6 satellites. (f)
Systems with 7 to 21 satellites. Dashed lines show $P(\Delta\phi)$ for a
uniform circular distribution. Figure 7: Probability distributions,
$P(\overline{R})$, for the Millennium sample. Circles: legitimate satellites
only (see text for definition). Squares: complete satellite samples, including
interlopers. Dashed lines: expected distributions for samples with random
satellite locations. (a) All host-satellite systems. (b) Satellites of red
hosts. (c) Satellites of blue hosts. (d) Systems with 2 to 3 satellites. (e)
Systems with 4 to 6 satellites. (f) Systems with 7 to 21 satellites. Figure 8:
Lopsidedness fractions, $f_{\rm lop}$, for the Millennium sample. Squares:
complete satellite samples, including interlopers. Circles: legitimate
satellites only.  Figure 9: Probability distributions, $P(\Delta\phi)$ and
$P(\overline{R})$, computed using only the interlopers that are found in the
complete Millennium sample. Top: $P(\Delta\phi)$, where the dashed line shows
the expectation for a uniform circular distribution. Bottom:
$P(\overline{R})$, where the dashed line shows the expected distribution for a
sample with random satellite locations.

###  3.4 Satellite Infall Redshift

A potential source of lopsidedness in the satellite distributions is the late-
time accretion of small groups of satellites along cosmic filaments. These
small groups of 2 to 3 satellites may still be on their first orbit around the
host and are unlikely to be a virialized population. To investigate this, we
use the infall redshift, $z_{\rm infall}$, at which each legitimate Millennium
satellite first entered its host’s dark matter halo. Here, $z_{\rm infall}$ is
the redshift at which the satellite’s distinct halo became a subhalo of the
host’s halo, as defined by the friend-of-friends algorithm. Figure 10 shows
the probability distribution, $P(z_{\rm infall})$, for the infall redshifts of
legitimate satellites that are classified as being “red” or “blue” at the
present day. The mean and median infall redshifts of the satellites are listed
in Figure 10. As expected, most satellites that are red at the present day
entered their host’s halo earlier than most satellites that are blue at the
present day (i.e., there is a difference of approximately 3.2 Gyr in lookback
time between the median infall redshifts of the red and blue satellites).

Figure 10: Probability distributions, $P(z_{\rm infall})$, for the infall
redshifts of legitimate red and blue Millennium satellites, truncated at high
$z$ for clarity.

Figures 11 and 12 show the probability distributions, $P(\Delta\phi)$ and
$P(\overline{R})$, for legitimate Millennium satellites, separated into
samples with $z_{\rm infall}<1$ and $z_{\rm infall}>1$. In all panels in
Figures 11 and 12, the spatial distributions are clearly lopsided, and
$\chi^{2}$ tests indicate that both $P(\Delta\phi)$ and $P(\overline{R})$ for
satellites with $z_{\rm infall}>1$ and $z_{\rm infall}<1$ are inconsistent
with random satellite distributions at confidence levels $>99.999\%$.

Satellites with $z_{\rm infall}>1$ have been inside their host’s dark matter
halo for $\gtrsim 7.8$ Gyr and might be expected to be a virialized
population. That is, since the dynamical time for a spherical halo is
$\tau_{\rm dyn}\sim{0.1/H_{0}}$ (e.g., Gan et al. 2010), satellites with
$z_{\rm infall}>1$ have been within their host’s halos for a time that is
$\gtrsim 5.5\tau_{\rm dyn}$. However, from Figures 11 and 12, these “oldest”
satellites are not distributed symmetrically about their hosts. Compared to
satellites with $z_{\rm infall}<1$ (lopsidedness fraction, $f_{\rm
lop}=1.26\pm 0.03$), however, satellites with $z_{\rm infall}>1$ exhibit
somewhat less lopsidedness in their spatial distribution ($f_{\rm lop}=1.11\pm
0.03$), which is consistent with the spatial distribution of the oldest
satellites having had more time to be influenced by the gravitational
potentials of their hosts’ halos.

Figure 11: Probability distributions, $P(\Delta\phi)$, for legitimate
Millennium satellites. Open triangles: $z_{\rm infall}<1$. Filled triangles:
$z_{\rm infall}>1$. Dashed lines: $P(\Delta\phi)$ for a uniform circular
distribution. Figure 12: Probability distributions, $P(\overline{R})$.
Triangles: legitimate Millennium satellites. Dashed lines: expected
distributions for samples with random satellite locations. Top: satellites
with $z_{\rm infall}<1$. Bottom: satellites with $z_{\rm infall}>1$.

##  4 Summary and Discussion

We have investigated the spatial distributions of the satellites of bright,
isolated host galaxies in a $\Lambda$CDM universe. The host-satellite sample
was selected from mock redshift space lightcones (built upon the Millennium
simulation, and updated for the Planck 2014 cosmology) using the criteria
adopted by BS20 in our study of the spatial distributions of observed
satellite galaxies in the NASA-Sloan Atlas (NSA). Obtaining the theoretical
host-satellite sample in the same way as the observational host-satellite
sample allows the most direct comparison of the observed universe to the
predictions of $\Lambda$CDM, and also allows potentially spurious sources of
lopsidedness in the observed satellite distribution to be assessed.

From Table 1 and Figure 1, the physical properties of the $\Lambda$CDM and
observational host-satellite samples are quite similar overall. The most
notable differences between the two samples are that Millennium sample is
somewhat bluer than the NSA sample, and the mean Millennium host-satellite
luminosity ratio is somewhat higher than the mean NSA host-satellite
luminosity ratio.

We used two metrics to quantify the spatial distributions of the satellite
galaxies, relative to their hosts: the pairwise polar angle difference,
$P(\Delta\phi)$, and the mean resultant length, $\overline{R}$. Both metrics
indicate that the spatial distributions of the Millennium satellites are
lopsided, with a strong tendency for satellites to be found preferentially on
one side of the host (in the plane of the sky). In agreement with the
observational results of BS20, the lopsidedness is greatest for Millennium
systems with blue hosts and Millennium systems with 2 to 3 satellites.

The pairwise polar angle difference characterizes the clustering of pairs of
satellites, aggregated over all host-satellite systems. In contrast, the mean
resultant length characterizes the “directionality” (relative to the host) of
the satellites within each individual system. These are complementary metrics
and comparison of our results for the Millennium and NSA samples leads to
different conclusions regarding the degree to which the results from the two
samples agree with each other. We find that the degree of clustering of pairs
of satellites around blue host galaxies and pairs of satellites in systems
with only 2 to 3 satellites is essentially the same in both the Millennium and
NSA samples. However, the degree of clustering of pairs of satellites in all
other cases (complete sample, satellites of red hosts, systems with 4 to 6
satellites, systems with 7 to 21 satellites) is much greater in the Millennium
sample than it is in the NSA sample. When we examine the directionality of the
satellites within each system, we find that, with the exception of the largest
systems (7 to 21 satellites), the overall directionality of the satellite
distributions in the Millennium sample is consistent with the overall
directionality of the satellite distributions in the NSA sample. In the case
of the largest host-satellite systems, the directionality of the satellite
distribution is somewhat greater for the Millennium sample than it is for the
NSA sample.

Two obvious sources of potentially spurious lopsidedness in the distribution
of NSA satellites are: [1] satellites that are missing from the NSA sample
because their spectra do not exist in the source databases (i.e.,
incompleteness of the satellite catalog itself) and [2] interloper galaxies
(i.e., galaxies that pass the satellite selection criteria but are not
physically associated with a host galaxy). Since the redshifts of all
Millennium galaxies are known, and since there is a pronounced lopsidedness to
the distribution of Millennium satellites, it is highly unlikely that the
lopsidedness shown by the NSA satellites is solely attributable to
incompleteness of the satellite catalog. Because the 3D distribution of all
Millennium galaxies is known, it is straightforward to determine which objects
in the Millennium satellite sample are outside the virial radii of the hosts.
When we eliminate these objects from our analysis, the distribution of the
remaining (“legitimate”) satellites is still distinctly lopsided. We find
that, while the presence of interlopers increases the lopsidedness of the
Millennium satellite distribution, the lopsidedness of the Millennium
satellite distribution is by no means solely attributable to the interlopers.
Therefore, it is unlikely that the lopsidedness of the NSA satellite
distribution is solely attributable to interlopers. Since satellites that have
been within their hosts’ dark matter halos for several dynamical times can be
expect to be a virialized population, it is also possible that the
lopsidedness of the satellite distribution is solely attributable to
satellites that have only recently arrived within their hosts’ halos. However,
we find that this is not the case for the Millennium satellites. While
satellites that entered their hosts’ halos most recently ($z_{\rm infall}<1$)
do show a greater degree of lopsideness in their spatial distribution,
satellites that have been within their hosts’ halos for $\sim 8$ Gyr or more
also have spatial distributions that are significantly lopsided. In agreement
with W21, who obtained isolated host galaxies and their satellites from the
$\Lambda$CDM TNG300 simulation using 3D selection criteria and whose host
galaxies were typically more massive than ours (i.e., $\sim 70$% of the
Millennium host galaxies have stellar masses that are less than the minimum
host stellar mass adopted by W21), we find that the lopsidedness of the
Millennium satellite distributions depends upon the color of the host galaxy,
with the satellites of blue host galaxies having a greater degree of
lopsidedness in their spatial distribution than the satellites of red host
galaxies. Also, while the host samples are entirely different (i.e., isolated
hosts vs. pairs of hosts), it is notable that even the oldest “legitimate”
Millennium satellites have significantly lopsided spatial distributions, which
contrasts strongly with the results of Gong et al. (2019) who found that the
lopsided satellite distribution for pairs of large galaxies is attributable to
satellites that have not been present long enough to have experienced a “fly-
by”.

##  5 Conclusions

The main conclusions from this work are the following. When samples of
isolated host galaxies and their satellites are obtained from a $\Lambda$CDM
simulation using the same type of criteria that are used to obtain host-
satellite samples from a redshift survey, the spatial distribution of the
satellites shows a pronounced degree of lopsidedness. In agreement with
observations, the lopsidedness is driven by the satellites of blue host
galaxies and satellites in systems with relatively few (2 to 3) satellites.
The lopsidedness found in the observational samples is not solely attributable
to incompleteness in the satellite sample. In addition, while the presence of
interlopers (i.e., false satellites) accounts for some degree of lopsidedness
in the satellite distribution, the lopsidedness is not solely attributable to
interlopers. Furthermore, the lopsidedness is not solely attributable to
satellites that have only recently arrived within their hosts’ halos.

This work was supported by National Science Foundation grant AST-2009397.

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