Study of Satellite Plane Structure Characteristics Based on TNG50 Simulations: A Comparative Analysis from Plane to Non-Plane Structures

In this work, the simulation we utilized is the TNG50 simulation of IllustrisTNG suite ¹¹1https://www.tng-project.org. Detailed descriptions of this database can be found in Pillepich et al. (2018a); Springel et al. (2018); Nelson et al. (2018); Naiman et al. (2018); Marinacci et al. (2018). The IllustrisTNG simulations are an improvement of the previous Illustris simulations (Vogelsberger et al., 2014), incorporating a revised active galactic nucleus (AGN) feedback model to ragulate star formation efficiency in massive galaxies (Weinberger et al., 2017), and a galactic wind model to inhibit efficient star formation in low- and intermediate-mass galaxies (Pillepich et al., 2018b). The TNG50-1 simulation (Nelson et al., 2019a, b; Pillepich et al., 2019) is the highest resolution version of the IllustrisTNG suite, with $2160^{3}$ dark matter and $2160^{3}$ gas particles in a cubic box of $(51.7\ \rm Mpc)^{3}$. The mass resolution is $8.5\times 10^{4}\ \rm M_{\odot}$ for baryon particles and $4.5\times 10^{5}\ \rm M_{\odot}$ for dark matter, while the spatial resolution is set as $72\ \rm pc$. The cosmological parameters are derived from Planck (Planck Collaboration et al., 2016): $\Omega_{m}=0.3089$, $\Omega_{\Lambda}=0.6911$, $\Omega_{b}=0.0486$, $n_{s}=0.9667$, $\sigma_{8}=0.8159$, and $h=0.6774$. Dark matter halos are identified using the Friends-of-Friends (FoF) algorithm (Davis et al., 1985), while subhalos are identified as overdense, gravitationally bound substructures using the SUBFIND algorithm (Springel et al., 2001; Dolag et al., 2009).

FoF groups are selected with $\rm M_{200}>10^{9}\>{\rm M_{\odot}}$, where $\rm M_{200}$ represents the mass within a radius where the average density is 200 times the critical density of the Universe. Subhalos are included if they have a total stellar mass $\>{\rm M_{\star}}>10^{5}\>{\rm M_{\odot}}$. In many theoretical studies, subhalos are typically selected to include more than 100 stellar particles, which corresponds to $\rm\>{\rm M_{\star}}>10^{7}\>{\rm M_{\odot}}$ in the TNG50-1 simulation. However, most observed satellite galaxies in plane structures, such as those in the Milky Way and M31, have stellar masses below $10^{7}\ \>{\rm M_{\odot}}$. To enable a meaningful comparison with observations, we adopt a lower stellar mass threshold of $\>{\rm M_{\star}}>10^{5}\>{\rm M_{\odot}}$ in the procedure of satellite plane finding, consistent with the selection criteria used in Xu et al. (2023), which balances simulation resolution limitations with observational completeness. We acknowledge that subhalos with $\rm 10^{5}\>{\rm M_{\odot}}<\>{\rm M_{\star}}<10^{7}\>{\rm M_{\odot}}$ are represented by fewer than 100 stellar particles in the simulation, leading to weaker statistical significance for results derived from this mass range. So we remove the satellite galaxies with $\>{\rm M_{\star}}<10^{7}\>{\rm M_{\odot}}$ in Section 3.2 and  3.3 to reduce the effects of the simulation mass resolution as much as possible. Overall, The selection criterion of $\>{\rm M_{\star}}>10^{5}\>{\rm M_{\odot}}$ allows us to include a larger number of potential samples, enhancing the statistical robustness of the analysis. We emphasize that conclusions drawn for subhalos within the range $\rm 10^{5}\>{\rm M_{\odot}}<\>{\rm M_{\star}}<10^{7}\>{\rm M_{\odot}}$ should be interpreted with caution due to resolution limitations. Only groups containing more than 6 subhalos are considered, excluding the centrals, which are the most massive subhalos within the groups. The N > 6 criterion is based on observations, as the smallest number of satellites in observed satellite plane structures is approximately 5 (e.g., Karachentsev et al., 2003; Martínez-Delgado et al., 2021). The addition of one more satellite ensures statistical reliability across our definitions. The snapshot used for analysis is taken at redshift $z=0$ (snap number = 99). The final sample comprises 699 FoF groups.

We compare previous methods for identifying satellite planes (e.g., Gillet et al., 2015; Pawlowski et al., 2024) and find that most of the random projection ellipsoid fitting approaches do not account for the mass of satellite galaxies, focusing instead solely on fitting the spatial positions of the member galaxies and considering motion coherence. This can lead to significant issues, as low-mass galaxies located outside the satellite plane and at large distances can substantially affect the satellite plane’s orientation. This results in a deviation between the primary satellite plane and the true satellite plane. To mitigate the impact of mass distribution on satellite plane identification, it is crucial to incorporate mass weighting into the analysis.

Nearly half of the satellite galaxies in the Milky Way and M31 have been observed to align within plane structures. However, if the member galaxies not residing on the satellite plane have substantial mass, are numerous, or are located far from the satellite plane structures, this uneven distribution can significantly affect the calculation of shape parameters (e.g., Libeskind et al., 2016; Pawlowski et al., 2017; Gong et al., 2019; Wang et al., 2021), such as the eigenvalues and eigenvectors of the inertia tensor. As with previous random projection ellipsoid fitting methods, this uneven distribution can increase the angular error between the fitted plane and the true plane.

To accurately determine geometric and dynamical properties of the plane structures of satellite galaxies, we develop a method based on the Random Sample Consensus (RANSAC) algorithm. The RANSAC algorithm (Raguram et al., 2013) enables robust fitting by randomly selecting inliers and iteratively managing noise and outliers in the data. By selecting the model with the highest number of inliers as the final result, this approach allows us to obtain relatively accurate fitting even in the presence of noise and outliers. This process involves excluding satellite galaxies farther from the principal plane than a distance $D_{0}$​, enabling a more precise determination of the principal axis direction of the fitted ellipsoid. Here, $D_{0}$ represents the vertical distance from the plane using in the RANSAC algorithm.

Given the influence of gravitational potential energy and the conservation of momentum in kinematics, satellite members within a plane structure must maintain consistent motion within the same plane. As a result, their rotational radii are relatively small, causing these co-rotating satellites to be tightly aligned with the central galaxy. Observational comparisons reveal that the physical sizes of plane structures typically range from 10 to 100 ${\rm\ {kpc}}$. Beyond this distance, the satellite distributions become more random, and no plane structures are observed.

With careful consideration, we utilize a radius of $100{\rm\ {kpc}}$ to filter satellites. Satellites located beyond this distance from the central galaxies are then removed. This step helps avoid the influence of distant satellites on the identification and analysis of satellite planes, as previously discussed. The remaining satellites are then subjected to inertia tensor calculations and corrections. To ensure the accuracy of our calculations, we select galaxy clusters that contain more than 6 satellite galaxies within $100{\rm\ {kpc}}$ of the central galaxies. This criterion results in a sample of 347 out of the 699 groups in the principal sample, with the remaining groups classified as non-plane structures. The procedure we use to define plane structures of these 347 groups is introduced below.

1. 1.

   Inertia Tensor Calculation: We begin by computing the inertia tensor $T$ of the galaxy group using the satellite members within $100{\rm\ {kpc}}$ centred at central galaxy, following the method of Xu et al. (2023). The inertia tensor is calculated as follows:

   $$T=\sum_{i}M_{i}\bf({r}_{i}-r_{0})(r_{i}-r_{0})^{T}$$

   (1)

   where $M_{i},r_{i},r_{0}$ represent the total mass of the satellite galaxies, the position of the satellite galaxies, and the position of the central galaxies, respectively. The three eigenvectors of the inertia tensor correspond to the directions of the fitted ellipsoid’s major, semi-major, and minor axes. The original plane of the satellite structure is defined by the major and semi-major axes.

2. 2.

   RANSAC algorithm: Then, we determine the principal membership of the plane structures by applying the RANSAC algorithm to the original plane defined in the first step. Following careful analysis, we use a threshold distance $D_{0}=50{\rm\ {kpc}}$ to filter the satellites. Satellites located beyond $50{\rm\ {kpc}}$ are excluded, refining the identification of the satellite membership. To accurately determine the membership of the satellite planes, we perform multiple iterations, reducing the threshold distance with $D_{i+1}=0.9\times D_{i}$, where $i$ represents the iteration number. This filtering process is repeated up to 20 times or until the number of satellites falls below 6 or less than half of the original count. Note that the plane used in RANSAC algorithm may increasingly approach the principal plane as the process is repeated. The precise shape of the satellite planes can be computed using the inertia tensor of the principal membership of the satellite planes determined by the RANSAC algorithm.

3. 3.

   Kinematic Analysis: We then project the velocities and coordinates of the satellite galaxies onto the system’s principal plane and calculate the spin $S_{i}$ of each satellite galaxies selected in first step:

   $$S_{i}=\bf({r}_{i}-r_{0})\times(v_{i}-v_{0})$$

   (2)

   where $\bf v_{i}$ and $\bf v_{0}$ are the velocities of the satellite galaxies and central galaxy, respectively. The sign of $S_{i}$ indicates the direction of satellite rotation. By counting the positive and negative direction ($N_{+}$ and $N_{-}$), we calculate $K$, the maximum of these two numbers, and divide it by the total number of satellites in the plane to obtain the scaled system spin $S$. A larger $K$ and $S$ indicate more satellites co-rotating and stronger kinematic coherence among satellite galaxies. $S=1$ means that all the satellite galaxies on the plane structures rotate with a same direction, i.e., $100\%$ co-rotation.

4. 4.

   Structure Height Calculation: We next calculate the root-mean-square (RMS) height $H$ to define the plane thickness, similar to the approach used in Samuel et al. (2021). A smaller $H$ suggests a thinner plane structure.

5. 5.

   Repetition and Classification: Finally, we repeat the procedures 50 times to obtain the mean values $S_{mean}$, $H_{mean}$, and $K_{mean}$ for $S$, $H$, and $K$. Based on these parameterized geometric and dynamical properties, we classify the satellite structures into the following three types:

   1. (a)

      Plane Structures: $H_{mean}<10{\rm\ {kpc}}$, $S_{mean}>0.75$, and $K_{mean}>4.5$, or $10<H_{mean}<15{\rm\ {kpc}}$, $S_{mean}>0.75$, and $K_{mean}>6$;

   2. (b)

      Pseudo-Plane Structures: $10{\rm\ {kpc}}<H_{mean}<15{\rm\ {kpc}}$, $S_{mean}>0.75$, and $4.5<K_{mean}<6$;

   3. (c)

      Non-Plane Structures: $H_{mean}>15{\rm\ {kpc}}$, or $S_{mean}<0.75$, or $K_{mean}<4.5$.

   Additionally, there are 352 non-plane structures defined by having less than 6 satellite galaxies within $100{\rm\ {kpc}}$ of the central galaxies.

$S_{mean}>0.75$ indicates that more than 75% of the projected velocities of satellite galaxies in the planes point in the same rotation direction. $H_{mean}$ quantifies the thickness of the structures, with $10$ and $15{\rm\ {kpc}}$ selected as the best thresholds to differentiate between plane, pseudo-plane, and non-plane structures based on careful evaluation. $K_{mean}>4.5$ signifies that, in multiple grogram runs, the majority of $K$ values are greater than 5, with some less than 5; this threshold of 5 is based on current observations (e.g., Martinez-Delgado et al., 2024). Defining pseudo-plane structures using the criteria of $10{\rm\ {kpc}}<H_{mean}<15{\rm\ {kpc}}$, $S_{mean}>0.75$, and $K_{mean}>4.5$ includes structures similar to those identified as marked planes with $K_{mean}$ values exceeding 6. Consequently, we introduce an additional criterion of $10{\rm\ {kpc}}<H_{mean}<15{\rm\ {kpc}}$, $S_{mean}>0.75$, and $K_{mean}>6$ to distinguish thick plane structures. For instance, the plane structure in $Halo395$ has $H_{mean}=12.89{\rm\ {kpc}}$, $S_{mean}=0.875$, and $K_{mean}=13.6$. The significantly smaller thickness of the plane structure in $Halo395$ compared to the $27{\rm\ {kpc}}$ reported by Xu et al. (2023) is due to our $H_{mean}$ being calculated based on satellites with the planes centred on the central galaxies, which differs sightly from Samuel et al. (2021) and Xu et al. (2023), where the calculations are centred on the the geometric center of the satellites. Structures with $H_{mean}<10{\rm\ {kpc}}$ are classified as thin. Table 2 lists the numbers and selection criteria for the four types of structures in our samples. Note that plane structures (including thin and thick) and pseudo-plane structures must satisfy all four selection criteria simultaneously, while non-plane structures only need to meet any one of these criteria. Fig. 1 illustrates the distribution of $H_{mean}$ and $S_{mean}$. By incorporating $K_{mean}$, we can classify our sample into the four types of structures as shown in Fig. 1. For the analysis, we will treat both thin and thick plane structures as unified plane structures.

Fig. 2 illustrates the three types of structures, presented from top to bottom panels, with face-on (first row) and edge-on (second row) views of 2D coordinates and velocities. Red and gray dots represent satellite galaxies within the plane and outside the plane structures, respectively. The face-on views (first row in the top panel) clearly show that the satellite galaxies within plane structures exhibit distinct co-rotation around the central galaxies. The edge-on views (second row in the top panel) reveal that most of these structures are very thin, with a mean thickness of $5.24{\rm\ {kpc}}$. These structures are primarily found in galaxy groups with intermediate halo viral masses with a narrow range from $\rm 10^{11.5}$ to $10^{12.5}\>{\rm M_{\odot}}$. Thick plane structures (not shown in the figure) also have numerous sub-members rotating around the centers but appear significantly thicker in the edge-on view, with a mean thickness of $13.18{\rm\ {kpc}}$. Pseudo-plane structures resemble planes in the edge-on view (first row in the middle panel). However, the velocities of their sub-members are uncorrelated, displaying a dispersion distribution (second row in the middle panel). In contrast, the satellites in the non-plane structures neither form a plane nor co-rotate around the central galaxies in any directions (bottom panel).

Note that mass weighting is not directly considered in the RANSAC algorithm, as it has already been incorporated in the calculation of the inertia tensor. The RANSAC algorithm effectively removes distant satellite galaxies that deviate significantly from the plane, thereby reducing their impact on the accurate determination of the satellite plane during the inertia tensor calculation. Compared to previous methods, the RANSAC algorithm provides robust outlier rejection and iterative refinement, enhancing the accuracy of satellite plane definitions. For instance, unlike Xu et al. (2023), which calculates the inertia tensor once, our approach involves multiple iterations to refine the plane. Additionally, the combination of RANSAC and inertia tensor calculations improves accuracy compared to the random projection method (e.g., Pawlowski et al., 2024; Gillet et al., 2015) by reducing the influence of distant galaxies.

As shown in Fig. 3, we present the cross-relations and number distributions for five different properties. Error bars represent the standard deviation. The number distributions in this figure clearly indicate that central galaxies with plane structures differ significantly from those without plane structures. In the top panel of Fig. 3, we find that central galaxies with satellite planes exhibit a more concentrated distribution compared to those without satellite planes. Approximately 80% of central galaxies with satellite planes have stellar masses in the range of $10^{10}$ to $10^{11}\>{\rm M_{\odot}}$, whereas only about 50% of central galaxies without satellite planes fall within this intermediate range. Moreover, central galaxies with satellite planes tend to occupy specific parameter ranges, including stellar metallicities of $\log_{10}(\rm Z/Z_{\odot})$ between 0 and 0.4, star formation rates ($\rm SFR$) from $\rm 10^{-2}$ to $\rm 10^{2}\ M_{\odot}/yr$, B-band magnitudes $\rm M_{B}<-20\ mag$, $\rm B-V$ colors range from $0.3$ to $\rm 0.6\ mag$, and formation time scales ($\rm a_{form}$) between 0.3 and 0.6. The stellar metallicity distribution of central galaxies with satellite planes also slightly higher than that of those without satellite planes. These findings suggest that satellite plane formation is more likely to occur around central galaxies that are star-forming within and fall within intermediate ranges of stellar mass, metallicity, age and luminosity. However, it is noted that there is still significant overlap in the properties of central galaxies with and without satellite planes, which makes directly distinguishing between the two populations challenging.

For a given stellar mass, central galaxies with plane structures are typically bluer, metal-poorer, and have higher star formation rates compared to their counterparts without satellite planes. At lower star formation rates, these plane structures still exhibit a blue color. Overall, central galaxies with satellite planes tend to be brighter, bluer, and have slightly higher metallicity compared to those in non-plane structures. The properties of central galaxies with pseudo-plane structures are more dispersed due to the small number of such structures, but their distributions are similar to those of plane structures, falling between plane and non-plane structures. This suggests that pseudo-plane structures may represent an intermediate state between plane and non-plane structures.

Due to the gravitational potential interactions between the satellite and central galaxies – whether through mergers or accretion – it is essential to consider the properties of each satellite galaxy within their respective dark matter halos. In this section, similar to the analysis in Section.3.1, we aggregate data for the satellite galaxies of each structural type and present the results as shown in Fig. 4. Note that we remove the satellite galaxies with $\>{\rm M_{\star}}<10^{7}\>{\rm M_{\odot}}$ to reduce the effects of simulation mass resolution in this section.

From the number distributions in Fig. 4, it is evident that the satellite galaxies associated with plane structures exhibit more pronounced differences compared to those located in the non-plane structures than the central galaxies do. The satellite galaxies in plane structures are primarily concentrated around $\>{\rm M_{\star}}\sim 10^{8}\>{\rm M_{\odot}}$, while those in non-plane structures exhibit a more uniform distribution across stellar mass $\>{\rm M_{\star}}>10^{7}\>{\rm M_{\odot}}$. Additionally, the satellite galaxies in plane structures are much brighter and bluer than those in non-plane structures, with almost all below $\rm-15\ mag$ in $\rm M_{B}$ and 0.25 $\rm mag$ in $\rm B-V$ color index. However, number distributions of the metallicity and star formation rate seem to be similar for both satellite with plane and non-plane structures. These two populations generally exhibit a number distribution peaked at $\rm\log(Z/Z_{\odot})\sim 0$ and $\log\rm SFR\sim-0.5$. For a given stellar mass, it is obvious that satellite galaxies in plane structures are bluer, brighter with higher metallicities and star formation rates than those in non-plane structures. Similar to the analysis in the Section 3.1, pseudo-planes display distributions between plane and non-plane structures but are closer to those planes in both the number distributions or the property cross-relations.

As shown in the cross-relations images in Fig. 4​, there are similarities between satellite galaxies in plane and non-plane structures for masses $\>{\rm M_{\odot}}>10^{10}\>{\rm M_{\odot}}$. In these extreme high-mass ranges, the properties of satellite galaxies in plane structures closely resemble those in non-plane structures. We speculate that these satellites may not genuinely belong to the plane structures but instead appear within the planes by coincidence. Additionally, the mass resolution of simulated satellites could also affect the results at the low-mass end.

In Section 3.2, we focus on satellites within the plane structures, which reveal significant differences from non-plane structures. However, when considering all satellites in plane structures, the differences with non-plane structures become less pronounced, particularly at the low or high mass end. This is likely due to the influence of out-of-plane satellites in plane structures. Satellite galaxies within the plane structures exhibit strong kinematic consistency, with nearly all rotating around the central galaxy, experiencing strong gravitational potential from the central galaxy, and displaying great kinematic correlation. These factors theoretically make them more suitable for studying the properties of satellite planes. Therefore, further analysis of plane structures is necessary to explore the internal properties of systems containing satellite plane structures.

In this section, we distinguish between the in-plane and out-of-plane satellite galaxies within the plane structures, and compare their properties. The satellite galaxies are with $\>{\rm M_{\star}}>10^{7}\>{\rm M_{\odot}}$, similar to that in Section 3.2. We calculate the average properties for all in-plane and out-of-plane satellite galaxies in each structure, as indicated by the plus and triangle symbols in Fig. 5. In general, the properties of in-plane and out-of-plane galaxies differ significantly. It is found that satellite galaxies within the plane structures tend to be relatively more massive and brighter, while those outside the planes exhibit lower stellar masses and luminosities. The difference of metallicity and star formation rate between in-plane and out-of-plane satellites seem weak. With a careful comparison for the two populations located in the same groups, it is found that most of the in-plane satellites exhibit higher metallicities and star formation rates compared to the out-of-plane satellites. However, the color distributions of both in-plane and out-of-plane satellites span a wide range, indicating that there are no clear difference in color index between the two populations. These findings suggest that in-plane satellites may have slightly longer formation times, and exhibit more active interstellar matter cycles, which could influence the star formation activities, compared to out-of-plane satellites.

In previous research, Xu et al. (2023) identified Milky Way analogs in the TNG50-1 simulation by restricting the stellar mass of halos to $1\times 10^{10}\>{\rm M_{\odot}}$​ to $10\times 10^{10}\>{\rm M_{\odot}}$, setting the stellar mass of satellite galaxies to $\>{\rm M_{\star}}>10^{5}\>{\rm M_{\odot}}$​, and requiring a distance from the central galaxy between 15 and 300 ${\rm\ {kpc}}$. They required the groups to contain at least 14 satellite galaxies based on the VPOS and identified 231 MW analogs. However, only one halo (Halo ID: 395, $Halo395$) exhibited similarities to the MW in terms of the angle between the satellite plane structure and the central plane, as well as the radial distribution of satellites. Therefore, we narrow the search range and apply the following selection criteria: the halo mass $\rm M_{200}=0.5\times 10^{12}\>{\rm M_{\odot}}$ to​ $2.5\times 10^{12}\>{\rm M_{\odot}}$, satellite galaxy stellar mass $\>{\rm M_{\star}}>10^{5}\>{\rm M_{\odot}}$​, and a minimum of 14 satellites within the groups, aiming to distinguish between the three types of satellite structures in MW analogs. Based on these criteria, we identify 154 MW analogs. The radial distribution of satellites is shown in the panel (a) of Fig. 6. The observational data of MW is taken from McConnachie (2012); Torrealba et al. (2016a, 2019).

To explore the characteristics of M31 analogs, Buck et al. (2015) utilized the $N$-body simulation PKDGRAV2 and set the selection criteria as $\rm M_{200}$ from $0.74\times 10^{12}\>{\rm M_{\odot}}$ to $2.2\times 10^{12}\>{\rm M_{\odot}}$, satellite masse from $4.4\times 10^{7}\>{\rm M_{\odot}}$ to $1.5\times 10^{10}\>{\rm M_{\odot}}$​​, and distance to the central galaxy from $30{\rm\ {kpc}}$ to $250{\rm\ {kpc}}$. In our research, considering the characteristics of TNG50-1 data, we simplify the selection criteria by setting the mass range from $1\times 10^{12}\>{\rm M_{\odot}}$ to $3\times 10^{12}\>{\rm M_{\odot}}$​. For the number of satellite galaxies, we use the classical count of 27 satellites within $500{\rm\ {kpc}}$ around M31. This results in 57 M31 analogs, and we plot their radial satellite distribution in panel (b) of Fig. 6. The observational data of M31 is taken from Conn et al. (2012).

Pawlowski et al. (2024) searched for NGC 4490 analogs in the IllustrisTNG simulations by selecting systems with masses between $0.2\times 10^{12}\>{\rm M_{\odot}}$ and $2\times 10^{12}\>{\rm M_{\odot}}$, distance within $450{\rm\ {kpc}}$, and mass less then $2\times 10^{11}\>{\rm M_{\odot}}$ within $1{\rm\ {Mpc}}$. They finally identified 141 NGC 4490 analogs. Using the random projection fitting method to calculate the system flattening ratio ($D/L$), they found that the peak of sample distribution ($D/L=0.6$) was significantly larger than the $D/L$ ratio of NGC 4490 ($D/L=0.38$). Therefore, we set the mass range from $0.1\times 10^{12}\>{\rm M_{\odot}}$ to $0.2\times 10^{12}\>{\rm M_{\odot}}$, and utilize the NGC 4490 satellite galaxy data taken from Karachentsev & Kroupa (2024), which provided information on 14 satellite galaxies in NGC 4490 within $450{\rm\ {kpc}}$. This results 134 NGC 4490 analogs. We select the 14 satellites with the largest masses of each analogs to plot their radial distribution, as shown in the panel (c) of Fig. 6.

Using the 27 satellites around Cen A from Kanehisa et al. (2023), we attempt to find Cen A analogs in TNG50-1 using the previous method, but find few samples that include systems with a satellite plane. This is because TNG50-1 contains few dark matter halos as massive as the Cen A halos. Therefore, we chose to search Cen A analogs in the TNG100-1 simulation. We apply the selection criteria from Müller et al. (2018, 2019), i.e., viral mass $\rm M_{200}=4\times 10^{12}\>{\rm M_{\odot}}$ to $12\times 10^{12}\>{\rm M_{\odot}}$, and add the criterion of satellite mass with $\>{\rm M_{\star}}>10^{8}\>{\rm M_{\odot}}$ and including at least 18 satellites within $500{\rm\ {kpc}}$ centred on Cen A. Finally, We identify 36 Cen A analogs, and plot their radial satellite distributions, as shown in the panel (d) of Fig. 6.

From Fig. 6, it is evident that for local systems like MW and M31, with intermediate stellar masses, the spatial distributions of their satellite galaxies resemble the radial distribution of satellite galaxies in groups with satellite plane structures in TNG50-1. These systems exhibit a dense arrangement within $200{\rm\ {kpc}}$ and a more sparse distribution at greater distances. In contrast, groups without satellite plane structures show a more uniform satellite distribution, generally appearing more dispersed. For the satellites around NGC 4490, their radial distribution appears sparse and uniform. Cen A’s satellites display a diffuse distribution, primarily concentrated at larger distances.

Additionally, we compare the properties of MW analogs in TNG50-1 and observational data from McConnachie (2012) on the “11 classical satellites” of the Milky Way and CVn I, including the mass, metallicity, $\rm B-V$ color, and V-band absolute magnitude ($\rm M_{v}$), as well as the mass and metallicity of Antlia II, discovered by Torrealba et al. (2019). We construct the mass-metallicity relation for the Milky Way halo (shown in the left-top panel of Fig. 7) and the $\rm B-V$ versus absolute magnitude diagram (shown in the left-bottom panel of Fig. 7). Simultaneously, we use the satellite mass data from Santos-Santos et al. (2020) and the metallicity of M31 satellites from Jennings et al. (2023) to compare the mass-metallicity relation of 25 M31 satellites with the TNG50-1 simulation (shown in the right-top panel of Fig. 7). We also use the $\rm B-V$ colors of M31 satellites from Conn et al. (2012) and the V-band absolute magnitudes from Savino et al. (2022) to construct the corresponding $\rm B-V$ versus absolute magnitude diagram (shown in the right-bottom panel of Fig. 7).

From Fig. 7, we observe that the masses of Milky Way and Andromeda satellites are generally smaller than the overall masses of plane structures in the simulation. The metallicities of satellite galaxies in the MW plane structure are comparable to some of the massive satellites in MW-like plane structures in TNG50-1. However, the metallicities of all the satellite galaxies in the M31 plane structure appear to be higher than those in M31-like plane structures in TNG50-1. In the bottom panels, it is evident that the satellites in plane structures are distinctly separate from those out-of-plane structures in the magnitude - color digram. Moreover, the satellites in the MW and M31 plane structures are located in the B-V color range of 0.0 to 0.2, while the majority of satellites in the plane structures in TNG50-1 exhibit a B-V color index between 0.0 and 0.5. This indicates that the satellites in the MW and M31 plane structures are broadly consistent in color index with those in the TNG50-1 plane structures. However, the satellites in the MW and M31 plane structures are significantly fainter than their simulated counterparts in TNG50-1. As the stellar masses of all the satellites in M31 plane structure are less than $10^{7}\>{\rm M_{\odot}}$, the comparison with the results in TNG50-1 should be carefully treated.

The discrepancy between the observational data and the TNG50-1 simulation data may arise from several factors. First, galaxy masses in simulations might be overestimated due to the inclusion of intracluster light (ICL; e.g., Tang et al., 2021; Pillepich et al., 2018a). Observationally, galaxies are often defined by a surface brightness cutoff, with light fainter than $\rm\sim 26.5\ mag\ arcsec^{-2}$ typically classified as ICL and excluded. In contrast, simulations define galaxy structures based on gravitationally bound systems identified by algorithms like SUBFIND, which include ICL. This difference can lead to higher mass estimates in simulations compared to observations. Second, simulations might over-predict galaxy growth due to more frequent mergers and accretion events, which could bias mass and metallicity estimates. Finally, the lack of interstellar extinction in simulations may result in satellites appearing brighter than they do in real observations, further contributing to the discrepancy between simulated and observed data.