Star–Gas Misalignment in Galaxies. II. Origins Found from the Horizon-AGN Simulation

Horizon-AGN (Dubois et al. 2014), one of the major recent collaborative efforts of cosmological large-volume hydrodynamic simulations, was performed using the adaptive mesh refinement code ramses (Teyssier 2002), based on the cosmological context from the seven-year Wilkinson Microwave Anisotropy Probe results (Komatsu et al. 2011). The simulation includes gas cooling, star formation, and feedback from stars and supermassive black holes. The volume of the simulation box is (100 Mpc h⁻¹)³, and the maximum (smallest) force resolution is about 1 kpc. The mass resolution is 8 × 10⁷ M_☉ for dark matter and 2 × 10⁶ M_☉ for stellar particles. A total of 787 snapshots are in Horizon-AGN, each with a time interval of about 17 Myr. However, only 61 snapshots with a time interval of about 250 Myr contain information on gas, dark matter, and sink particles. As our research requires both stars and gas, from now on, the word snapshot refers only to these 61 "gas snapshots." Meanwhile, all 787 snapshots were used to analyze the merger trees, merger processes (Section 2.5), the path of galaxy interaction (Section 4), and perturbation index (Section 4.2). For more details on the simulation, readers are referred to Dubois et al. (2014).

Horizon-AGN galaxies were identified by using HaloMaker through the AdaptaHOP algorithm (Aubert et al. 2004), with the most massive sub-node mode (Tweed et al. 2009) applied for stellar particles. A group of stars had to contain at least 50 star particles to be classified as galaxies, which corresponds to 1.7 × 10⁸ M_☉. Using this method, 124,744 galaxies were identified at z = 0.018, the last gas snapshot of Horizon-AGN.

As mentioned in Paper I, galaxies with a small number of stellar particles are not suitable for studying the kinematics of galaxies (e.g., Snyder et al. 2015; Dubois et al. 2016; Penoyre et al. 2017; Rodriguez-Gomez et al. 2017). Therefore, we used only massive galaxies with a stellar mass of more than 10¹⁰ M_☉, corresponding to about 3000 stellar particles, the same criterion used in the preceding study. The number of galaxies above the criterion is 27,903 out of 124,744 at z = 0.018.

The stellar V/σ represents the kinematic property of a galaxy, where V and σ are the rotational velocity and the velocity dispersion of the galaxy, respectively. The galaxies with a higher V/σ tend to have disk-shaped structures whereas the galaxies with a lower V/σ tend to have a spheroidal structure. We measured V/σ as done by Dubois et al. (2016). After that, we classified the galaxies into early-type galaxies (ETGs) and late-type galaxies (LTGs), based on the V/σ cut of 1, as we performed in Paper I (see also, Dubois et al. 2016). Note that we have already analyzed the misaligned galaxies with respect to the V/σ ratio in Paper I.

Galaxy groups were defined as follows, using the same method as in Paper I (see, Khim et al. 2020). We first identified dark matter halos with M_vir ≥ 10¹² M_☉ and defined a galaxy located within its 1.5 virial radii (R₂₀₀) as a member. If a dark matter halo had more than three member galaxies (M_* > 10¹⁰ M_☉), it was defined as a galaxy group. Based on the above definition, we identified a total of 606 galaxy groups and a total of 5984 member galaxies in the group environment at z = 0.018. The most massive group had 208 group members and a halo mass of 7.4 × 10¹⁴ M_☉ (14.87 in logarithm solar mass). This definition is the same as that of Paper I, but we did not separate "clusters" from groups this time.

We use a linear cut in the logarithmic density–temperature plane using Equation (1) from Torrey et al. (2012) to separate the gases in the simulation into galactic cold gas and non-cold surroundings:

$$\begin{eqnarray}&&\mathrm{log}(T/[{\rm{K}}])=6+0.25\mathrm{log}(\rho /{10}^{10}[{M}_{\odot }{h}^{2}\,{\mathrm{kpc}}^{-3}]).\end{eqnarray} \tag{ 1 }$$

"Galactic cold gas," the gas below the density–temperature criterion, has a temperature of about 10,000–30,000 K, depending on the density. The cold gas is used for star formation in the simulation and represents the kinematic property of the interstellar medium. Also, they have similar properties to the gas component observed by Hα emission lines (≃10,000 K) in IFS, such as SAMI. However, it should be noted that Horizon-AGN cannot resolve extremely cold gas, such as molecular gas. Contrary to the galactic cold gas, (non-cold) "surroundings," the high-temperature side, corresponds to the intergalactic gas or intragroup medium. This relatively hot gas is used to measure ram pressure (Gunn & Gott 1972; Quilis et al. 2000) in Section 5.1.

Galaxies that do not have sufficient gas need to be excluded from the sample because their gas rotation cannot be measured or will show a large error if measured. In our previous paper (Khim et al. 2020) comparing SAMI and Horizon-AGN, it was assumed that the gas flux detection limit would be linked with the cold gas fraction, the mass ratio of cold gas to stars (M_{cold gas}/M_*). We measured the gas fraction within one effective radius (R_eff), encompassing half of the total stellar mass (half projected stellar mass) of the galaxy, which is similar to the range observed by SAMI. We compared the misalignment fractions in SAMI and Horizon-AGN, and suggested that galaxies with the gas fraction higher than 3% were suitable for use as samples. With this criterion, 26,222 out of 27,903 galaxies at z = 0.018 have enough gas fraction and are considered observable.

In order to calculate the rotational axes of a galaxy, we measured the angular momentum of stellar particles and cold gas cells with respect to the galactic center. Then, the direction of the net angular momentum of each component was regarded as the rotational axis. The position angle (PA) offset, which is the difference between the rotation axes of stars and gases (misaligned angle), has a range from 0 (corotation or aligned) to 180° (counterrotation). We calculated the stellar and gas rotational axes inside 1R_eff of the galaxy, similarly to that of many IFS observations. The galaxies with PA offset larger than 30° were classified as misaligned galaxies to be consistent with SAMI and other observational studies. Among the galaxies with sufficient gas contents, 2662 galaxies (10.2%) were misaligned and used as the main targets in this study. Although the snapshots used in this study are slightly different from those used in Paper I (z = 0.055), the misalignment features in terms of number fraction, properties, and PA offset distribution of misaligned galaxies are almost identical.

We have built merger trees to link all identified galaxies reaching far below our mass cut in all 787 snapshots. After merger trees are made, we investigated the stellar particle exchange between interacting galaxies as follows. The "initial masses" of galaxies are measured at the time the secondary (less-massive) galaxy passes through the virial radius of the primary (more-massive) galaxy. The merger was defined when more than 50% of the initial mass of the secondary galaxy is consumed by the primary galaxy. We confirm that a different choice of the cut would not cause a notable difference as long as it is in a reasonable range (50%–90%).

We used a narrow definition for the duration of mergers to distinguish mergers and interactions as much as possible. The beginning of a merger is defined as the moment when the exchange of star particles begins. Similarly, the end point of the merger is defined as the moment when the exchange stops, or when the secondary galaxy is no longer detected (coalescence).

We define three classes of mergers based on the mass ratio: (1) major merger when M_*,s/M_*,p ≥ 1/4, (2) minor merger when 1/4 > M_*,s/M_*,p ≥ 1/50, and (3) tiny merger when M_*,s/M_*,p < 1/50, where M_*,p and M_*,s denote the stellar masses of the primary and secondary galaxies, respectively. As described earlier in Section 2.2, our galaxy-finding scheme detects all of the galaxies above 1.7 × 10⁸ M_☉, while we use the galaxies above 10¹⁰ M_☉ as main targets. Therefore, the galaxies near the mass limit of 10¹⁰ M_☉ would have only major and minor mergers, but few tiny mergers. As a result, a total of 21,122 major mergers and 50,848 minor mergers have taken place in Horizon-AGN since z = 3.

The stellar rotational axis is influenced when there is a significant accretion of stellar mass (Regions Mw, Md), as shown in Figure 1(a). In contrast, the change in gas mass regardless of inflow or outflow (Regions Ga, Gl) does not influence the stellar rotational axis significantly. Some of our target galaxies show a decrease in the stellar mass (Region Sl) mostly due to galaxy interactions, but it hardly affects the stellar rotational axis, either.

The gas rotational axis changes are shown in Figure 1(b). The gas rotational axis is influenced by an inflow of gas from outside (Regions Ga, Mw), which is easily expected. We found that it is also influenced by the accretion of stars (Region Md), which is mostly due to interactions. Interestingly, the greatest magnitude of the gas rotational axis change is found in the galaxies that experience a significant loss of gas while maintaining their stellar mass (Region Gl). Such a major change in the rotational axis is not visible in any other region in panels (a) or (b). We have already demonstrated in Paper I that misalignment can occur when a galaxy loses its gas while falling into the central region of clusters, which has been confirmed by the study based on the Illustris simulation (Starkenburg et al. 2019). We found a considerable number of galaxies within Region Gl in field environments as well. Most of these galaxies appear to be experiencing a gas loss due to the active galactic nuclei (AGNs) or stellar feedback. These massive gas loss events seemed to be linked with mergers.

Figure 1(c) shows the difference between the rotational axis changes of stars and gas: that is, panel (c) is panel (a) minus panel (b), pixel by pixel. Blue pixels mark the galaxies that show larger axis changes in stars than in gas, whereas red pixels show larger changes in gas than in stars. The difference away from zero (red or blue) means that the PA offset, the angle between the two axes, has changed. Thus, these colors are closely linked with the star–gas misalignment. Almost all regions in panel (c) are red, indicating that the rotational axis is more easily changed for gases than for stars. This means that the main drivers of misalignment are the processes that influence gases rather than stars. They are, for example, gas inflow (Regions Ga, Mw) and outflow (Region Gl).

Regions Mw, Md, and Sl show the galaxies that experienced stellar mass changes more dramatically than other regions. These regions also show some misalignment (red or blue pixels); yet, the degree of misalignment is lower than Regions Ga and Gl, where stellar masses hardly changed. This means that galaxy interactions that involve stellar mass changes may not be the primary driver of misalignment, unlike our expectation at the beginning of this section.

As seen in Figure 1, galaxies undergoing mergers have a large accretion of stars and gases; thus, the merging galaxy can easily change its rotational axes. In this section, we will examine the process of merger-driven misalignment and quantify the level of contribution to misalignment formation.

4.1.1. Mergers and Misalignment

Figure 2 shows an example of merger-driven misaligned galaxies. Panel (a) shows the stellar mass (red), gas mass (green), and PA offset (blue) of the primary (target) galaxy as a function of time. The criterion for the misalignment, 30°, is expressed with a gray horizontal dashed line. Some snapshots are numbered (Stages 1–4) to illustrate the merger-driven misalignment formation process. In panel (b), the stellar mass of the secondary galaxy (red) and the distance between the two galaxies (violet) are presented. We mark the first (the red star mark) and the second (the blue star mark) pericenter passes in the panel. In the case where the merging galaxies are very close to each other, HaloMaker cannot accurately distinguish the two galaxies. Thus, the stellar mass curve of the secondary galaxy in particular can be distorted or even partially disconnected near the pericenter. In panel (c), projected stellar and cold gas Doppler maps corresponding to each step are shown. The effective radius and 3 R_eff are expressed in solid and dotted circles, respectively. Black arrows show the rotational axes within 1 R_eff. When the secondary galaxy enters within the field of view, the center of the secondary galaxy is expressed with a purple dot (Stages 1–2). Coalescence happened near Stage 2.

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The secondary galaxy infalls into the primary galaxy with the aligned orbit with respect to the preexisting stellar rotational axis. Also, the stellar mass ratio between the two galaxies is about 1:4. Thus, during Stages 1–4, the merger does not affect the stellar rotational axis significantly.

The gas axis, on the other hand, shows more dramatic features. Before the merger, the primary galaxy has a well-developed aligned gas disk (Stage 1). However, as the secondary galaxy passes through the pericenter for the first time (red star), the cold gas of the secondary galaxy gradually begins to flow to the primary galaxy, affecting the PA offset of the primary galaxy (Stage 2). At the second pericenter pass, a large amount of cold gas, nearly as much as the preexisting gas, flows into the 1 R_eff region, generating a strong star–gas misalignment. At this stage, the gas rotation of the primary galaxy is unstable due to the massive gas outflow (Stage 3). Finally, the gas disk stabilizes with a misaligned rotational axis (Stage 4).

We describe the trajectory of this case in the format of Figure 1. The main galaxy is a typical galaxy with the aligned axes between stars and gas near the origin in the panel (i.e., coordinates 0,0). At Stage 1, the galaxy gets a significant amount of cold gas and moves to Region Ga in Figure 1. After that, the galaxy moves to the right during the merger process (Stage 2) according to its merging mass ratio. The galaxy gives off its cold gas and moves downward (Stage 3). After the merger, the galaxy eventually returns to the origin in the panel and remains misaligned.

This and many other galaxies show a clear correspondence between the formation of misalignment and the gas mass change, which motivated us to focus on the gas and stellar mass changes during the merging and interacting events. However, we also note the possibility of PA change through tidal effects between interacting galaxies even if there is little exchange of stars or gas.

4.1.2. Merger-driven Misalignment Fraction

The epoch of a merger event and the emergence of misalignment do not necessarily coincide. Some galaxies show misalignment even before the merging process becomes violent, for example, through gas accretion. Other galaxies show misalignment as a result of a merger, for example, through stellar accretion. These changes are thought to occur within a merging timescale frame.

The merging timescale is considered to be a couple of times the crossing time (e.g., White 1978; Gerhard 1981; Barnes 1988). It is however difficult to imagine a galaxy that is just touching the virial radius of another (target) galaxy affecting its rotational axes of gas and stars. It is more probable that a shorter distance is needed. An easy and practical way of estimating the crossing time is to measure the time difference between the first and the second pericenter passes, which is shown in Figure 3. We here show all the mergers since z = 1 for which we are able to identify both the first and second pericenter passes. While the data appear to be wildly scattered in the diagram, the median as a function of the galaxy mass ratio is steady and well defined. With a slow decay with increasing mass ratio, the typical value is roughly between 1.0 and 1.5 Gyr. Therefore, the causality connection between merger and misalignment can be searched roughly within this time frame.

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We now examine the correlation between the two events: mergers and misalignment formation. We use the merger trees to identify when the galaxies undergo mergers and compare those with the epochs when the galaxies develop the misalignment. In the case where the merger takes place across multiple snapshots, we select the nearest snapshot to the emergence of the misalignment.

Figure 4 shows the histograms of the time interval between the emergence of misalignment and the merger event for all misaligned galaxies that experience at least one merger. We use either the beginning or the end point of the merger period, whichever is closer to the misalignment. In these histograms, a positive time interval on the x-axis means that the emergence of the misalignment preceded the nearest merger event, which is rare. We have found a peak at SS_merger − SS_MA = 0, where the two events occur at the same time, as we expected. Also, the immediately neighboring bins have significantly more galaxies than the distant bins. Note that the size of the bins in the diagram is 0.25 Gyr, which is the time interval between the gas snapshots in the Horizon-AGN simulation.

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We classify the misaligned galaxies around the peak as "merger driven," showing a sign of circumstantial connection between the mergers and misalignment. The misaligned galaxies with snapshot differences equal to or smaller than three (a total of seven bins) are classified as merger driven (orange area) because we do not see a signal of difference in the histogram beyond this range. The duration of time over the seven snapshots corresponds to 1.75 Gyr which is comparable to the merging timescale shown in Figure 3. Through this classification method, 928 of the total 2662 (35%) misaligned galaxies have been identified as merger driven. A different choice of the interval cut, for example, 9 or 11 instead of 7 bins, would not lead to any notable difference in the result.

It should also be noted that the right wing from the peak in Figure 4 is shorter than the left wing. This makes sense in the sense that misalignment likely trails rather than leads a merging event in most cases (see Figure 2, for example). Another possible explanation for the uneven wing is the epoch of the sampling. Our galaxies are sampled at z = 0.018, and so it is impossible to follow their misalignment beyond z = 0. The presence of some galaxies beyond the peak (the right wing) despite that is because some of the misaligned galaxies formed their misalignment somewhat earlier than z = 0.018. In this sense, the right wing may not fully represent the true shape of the distribution in this diagram. We may attempt to recover some of the true distribution by considering only the left wing (the three orange bars on the left of the peak) and doubling it. This leads to an upper limit of 39%, which is somewhat higher than our previous estimate (35%).

The galaxies with extremely slow rotation (e.g., V/σ < 0.2) must be treated with care. These galaxies are dispersion-dominated systems, and their stellar rotational axes are not well defined. When we exclude the misaligned galaxies with V/σ < 0.2, 711 (30%) out of the total 2333 misaligned galaxies are classified as merger driven.

4.1.3. Merging Mass Ratio and Misalignment

When a merger occurs, the merging mass ratio naturally has a significant impact on the rotation of the primary galaxy (e.g., Choi et al. 2018; Lagos et al. 2018). As the misalignment is also caused by the rotational axis changes, the mass ratio is thought to have a significant impact on the number of merger-driven misaligned galaxies.

Figure 5 shows the stellar (x-axis) and the cold gas (y-axis) mass ratio of mergers in Horizon-AGN. All mergers since z = 0.2 are marked with gray dots, and the merger-driven misaligned galaxies seen above are marked with red dots. Among the 928 merger-driven misaligned galaxies, 426 are due to the major merger (46%), 392 are due to the minor merger (42%), and 110 are due to the tiny merger (12%).

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The contributions from the major and minor mergers are comparable. Therefore, considering that minor mergers are more frequent than major mergers in Horizon-AGN by a ratio of 5:2 (see Section 2.5), major mergers can be said to be more effective than minor mergers in generating misalignments by a factor of 5/2, which is understandable.

Minor mergers with a low mass ratio (i.e., tiny mergers) are unlikely to cause notable misalignments. They may either experience several small mergers at the same time or work together with other (non-merger) mechanisms to create misalignments. Meanwhile, half of the tiny-merger-driven misaligned galaxies (49) have V/σ < 0.2 at the time of misalignment formation, whereas only 15% of the major-merger-driven misaligned galaxies show such low values of V/σ. In addition, their median cold gas fraction is only one-third of that of all Horizon-AGN galaxies at z = 0. Both of these make it very difficult to measure the misalignment of such insignificant mergers reliably.

In Section 4.1, we found that about 35% of the misaligned galaxies originated from mergers. This means that 65% of the misalignment has been created by "non-merger" origins, which is the topic of this section. We investigate non-merger formation channels and classified the non-merger-driven misaligned galaxies into three categories: interaction-driven, group-driven, and secularly driven misaligned galaxies.

4.2.1. Interaction-driven Misalignment

Some galaxies develop misalignment through interactions with nearby galaxies. The misalignment forms when the gas mass exchange occurs between galaxies mainly through gas accretion, or when the gas velocity field is altered as the gas is affected by nearby galaxies.

Figure 6 shows an example of the "interaction-driven" misaligned galaxies in a similar format to Figure 2. There is the primary (target) galaxy in the center having an aligned gas disk, and the secondary galaxy with a mass of about 1/20 of the primary galaxy is approaching (Stage 1). The secondary galaxy gets close, within 50 kpc, and donates its gas to the primary galaxy. The gas from the secondary galaxy, equivalent to the amount of preexisting gas, flows inside 1 R_eff of the primary galaxy. Due to the gas inflow, the gas rotational axis is altered and a misalignment is developed (Stage 2). The misalignment of the primary galaxy continues to grow. However, unlike the merger-driven case, the misalignment is not strong. In particular, the gas in the outer part of the galaxy (≃3 R_eff) more or less maintains its original rotation (Stage 3). As a result, the PA offset of the target galaxy inside 1 R_eff starts to settle down, i.e., realign (Stage 4). Note that there is little stellar accretion onto the target galaxy. Thus, similarly to the minor-merger case, the stellar mass and stellar rotational axis are not affected significantly. As discussed in Section 4.1.1, tidal effects besides mass exchanges may also contribute to the formation of the interaction-driven misalignment.

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4.2.2. Group-driven Misalignment

The gas of group member galaxies can be influenced by interactions with the medium of the dense environment (e.g., Gunn & Gott 1972; Quilis et al. 2000). The member galaxies are also affected by the brightest group galaxy (BGG), which is the most influential object in a group. Therefore, a large number of galaxies develop misalignment when they pass near the BGG, as we noted in Paper I.

Figure 7 shows an example of "group driven," which is the same galaxy shown in Paper I (Khim et al. 2020). As a galaxy infalls to the central region of a group, it loses cold gas by ram pressure stripping and develops the misalignment (Stages 1–4). Most of the group-driven misaligned galaxies develop the misalignment when they pass the pericenter (Stage 4 in the case of the example galaxy). However, some galaxies have a time difference between the misalignment formation and either/both of the epoch of pericenter passes or/and the epoch of the nearest PI maximum. In addition, the PA offsets of group-driven cases are characterized by developing over a longer period of time compared to other cases (e.g., interaction and merger driven). This is understandable because the group-driven process works in the timescale of the orbital (crossing) time, which is much longer inside a group (or cluster) than between two galaxies.

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One of the important features of the group-driven misaligned galaxies is that their gas rotational axis changes significantly, while their stellar rotational axis is virtually fixed. Thus, the group-driven misalignment is mainly due to the changes in the gas rotational axis. The ram pressure stripping can be a possible origin, as mentioned in Paper I (Khim et al. 2020).

We have found that the gas mass of some galaxies does not decrease within 1 R_eff when they develop the misalignment. Some galaxies even show a temporary increase in their gas mass (a solid green curve in panel (a)) accompanied by a significant increase in PA offset (Stages 1–3 and 7). This may be puzzling because we can easily expect that the strongest candidate for group processes, ram pressure likely reduces the gas fraction, which is not visible in this diagram clearly. However, during the same period, their total amount of gas (e.g., within 2 R_eff, shown as a dotted green curve) decreases steadily. This suggests that the gas pushed into 1 R_eff by ram pressure or interaction with the BGG may have created the misalignment.

4.2.3. Secularly Driven Misalignment

4.2.4. Non-merger-driven Misalignment Classification

In this section, we classify non-merger-driven misaligned galaxies into the three categories identified above. We explore the short-range effects due to neighboring galaxies and the long-range effects due to the galaxy group itself. After that, we classify the galaxies as to which formation process plays a leading role in developing the misalignment.

We introduce the perturbation index (PI) to investigate the interactions with the nearby galaxies (Byrd & Valtonen 1990):

$$\begin{eqnarray}&&\mathrm{PI}=\mathrm{log}\left[{\int }_{{t}_{0}}^{t}\displaystyle \sum _{i=0}^{n}\left(\displaystyle \frac{{M}_{{\rm{p}},i}}{{M}_{\mathrm{gal}}}\right)\times {\left(\displaystyle \frac{{R}_{\mathrm{gal}}}{{d}_{{\rm{p}},i}}\right)}^{3}\,{dt}/\mathrm{Gyr}\right].\end{eqnarray} \tag{ 2 }$$

In this equation, M_gal and R_gal represent the mass and the size of the target galaxy, respectively, and M_p,i and R_p,i are the mass and distance of the ith perturber, respectively. As we are interested in the rotational axis changes within 1 R_eff, we have decided to use R_eff for the size of a galaxy in the PI measurement. To calculate PI, all of the identified galaxies within 2 Mpc around the target galaxy are considered perturbers. Due to the steep (cubic) dependence on distance, the use of a limit greater than 2 Mpc would not have a large impact on the PI measurement. In order to obtain more accurate PI values for 61 coarse gas snapshots, we have measured the PIs for the whole of 787 stellar snapshots and derived the mean values for the 61 gas snapshots, which were used in our analysis.

To investigate the effect of nearby galaxies on the misalignment formation, we perform a similar task to that of Section 4.1.2. First, we identify the PI local maxima of each galaxy. To select statistically significant local maxima only, we consider only peak values higher than −5.3, the median value of the PI of the Horizon-AGN galaxies in the local universe. A different choice of this cut makes only a small difference in the result. For example, the use of the 25th quantile instead of the median would decrease the fraction of secularly driven misaligned galaxies by 7.7% (572 to 528.) Next, we measure the snapshot number for the misalignment formation (SS_MA), where the PA offset becomes larger than 30° for the first time. We also measure that of the epoch of the maximum value of PI (SS_PI). Finally, we compute the difference between the two, which is, X = SS_PI − SS_MA.

We investigate the effect of galaxy groups (and the BGG) in a similar way. The BGG is the most influential object in a group, and its surrounding area would also provide a high degree of ram pressure (Jung et al. 2018). Hence, we use the distance to the BGG as a proxy of the magnitude of the group environmental effect. We measure the difference between the snapshot for the misalignment formation epoch (SS_MA) and the snapshot for the epoch of the closest pericenter pass to the BGG in terms of time (SS_peri), that is, Y = SS_peri − SS_MA.

We then draw a two-dimensional histogram by placing X on the abscissa and Y on the ordinate (Figure 8). The range of histograms is set to ±4 snapshots from SS_MA. If the time interval deviates from the given range (−4 to 4), the galaxy is categorized as "unlinked" (UL), that is, unlinked to the environmental effects. When either X or Y is out of range, the galaxy is added to the grids in the bottom row or leftmost column. On the other hand, if both X and Y are out of range, the galaxy is put in the bottom-left grid (coordinate UL–UL). If a galaxy is located in the "UL" grids in the leftmost column, it means there is little or no causal connection between its misalignment and the environmental effects from nearby galaxies (PI). Similarly, galaxies in the grids in the bottom row show misalignment from something other than the environmental effects caused by the BGG. Note that the galaxies in the field environment, whether merging or not, do not have a value for SS_peri and hence would be put into the UL grids on the bottom.

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The darkness of each pixel increases with the number of galaxies in each pixel. A couple of features are outstanding, and these are useful for us to explore the origins of misalignment, as discussed below.

Now, we classify the non-merger-driven misaligned galaxies, based on Figure 8. The background colors present a classification scheme for non-merger-driven misaligned galaxies. We first classify secularly driven misaligned galaxies (the blue area) when ∣X∣ > 3, which developed the misalignment without a significant contribution from external (environmental) perturbers. This is similar to what we have described for the classification of merger-driven cases in Section 4.1. As a result, a total of 572 galaxies are classified as secularly driven (21% of the total misaligned galaxies).

The galaxies whose misalignment seems linked with the nearest PI maximum, i.e., ∣X∣ ≤ 3, can be divided into two categories depending on whether the galaxy was affected by the BGG. One of the notable features in the figure is the diagonal distribution of galaxies with similar values between X and Y, which implies that the BGG is the main driver of both X and Y.

The galaxies along this line are close to their BGG and so have a high value of PI, and their misalignment is likely generated by the interaction with the BGG and the central group environment; hence, we classify them as group driven (the orange area). It should be noted that we widen the diagonal band by one snapshot in both directions in the figure to consider the poor time resolution of the gas snapshots. We have identified 578 group-driven misaligned galaxies (23%) using our classification method.

On the other hand, the galaxies away from the diagonal line have their misalignment origin from something other than the BGG or the central group environment. These galaxies instead develop their misalignment from the strong interaction with nearby galaxies and are hence classified as interaction driven (the green area). Of the total misaligned galaxies, about 21%, or 584 galaxies, are classified as interaction driven based on the classification scheme of Figure 8.

Some of the interaction-driven misaligned galaxies may have been sampled just before a merging process. If we followed them through beyond z = 0, they would be classified as merger driven instead, as was discussed in Section 4.1.2. In this sense, the current estimate for the fraction of interaction-driven misaligned galaxies measured at z = 0.018 is an upper limit. The correction to the estimate for the merger-driven misaligned galaxies discussed in Section 4.1.2 likely affects the estimate for the interaction-driven misaligned galaxies with similar magnitudes but in the opposite direction. In other words, the correction from 35% to 39% for merger-driven misaligned galaxies would require a correction to the interaction driven by a similar magnitude (i.e., from 22% to 18%).

As we performed in Section 4.1.2, we measure the level of contribution from non-merger-driven misalignment formation channels only for galaxies with V/σ ≥ 0.2. However, excluding the non-merger-driven misaligned galaxies with V/σ < 0.2 does not affect our result significantly.

Table 1 shows a summary of our classification scheme, including merger-driven misaligned galaxies. However, we would like to note that a galaxy may have more than one channel to develop misalignment—for example, interacting galaxies inside a large group or cluster. Nevertheless, we will attempt to estimate the degrees of contribution from various channels/categories in misalignment formation. The estimates of fractions are subject to changes depending on the classification details and are uncertain by roughly 10% in each category.

Table 1. The Formation Channels of the Misalignment and Their Level of Contributions at z = 0.018

Origin	Raw a		Symmetry Assumed b		V/σ ≥ 0.2 c
	Number	Fraction d	Number	Fraction	Number	Fraction
Total	2662	100%	2662	100%	2333	100%
Merger	928	35%	1030	39%	711	30%
- Major	426	16%	459	17%	362	16%
- Minor	392	15%	425	16%	288	12%
- Tiny	110	4%	146	5%	61	3%
Non-Merger	1734	65%	1632	61%	1622	70%
- Group	610	23%	610	23%	590	25%
- Interaction	552	21%	450	17%	507	22%
- Secular	572	21%	572	21%	525	23%

Notes.

^a The raw classification of misalignment formation channels in Horizon-AGN. ^b The result from doubling the left wing of Figure 4 (see Section 4.1.2). ^c The result from excluding galaxies with V/σ < 0.2. ^d This shows the fraction of the galaxies in the bin over the total number of the misaligned galaxies. While the binomial standard deviation of each sample is less than 1%, the fractions are subject to changes depending on the classification details and are uncertain by roughly 10% in each category.

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Figure 9 shows the environmental properties of the misaligned galaxies. The misaligned galaxies are classified based on the scheme in Section 4 and expressed in different colors corresponding to their origins. We also present their 0.5 and 1σ distributions with color contours. Meanwhile, all of the Horizon-AGN galaxies at z = 0.018 are shown with gray dots for comparison, regardless of whether they are misaligned. In the side panels, we present the histogram of each parameter. The diamond symbols show the median values.

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Figure 9(a) shows the Horizon-AGN aligned and misaligned galaxies in the plane of D1 and D5, where D1 and D5 are the distance to the closest and the fifth closest galaxies. We use D1 and D5 as a measure of the short-range and long-range environmental density. For D5 measurements, we take all galaxies with a mass ratio 1:10 into account while the target galaxies have M_* > 10¹⁰ M_☉. On the other hand, we use all galaxies without the mass limit for calculating D1 because a very close neighboring galaxy can have a significant impact even if it has a low mass. These parameters show a substantial variation from snapshot to snapshot, especially in the case of D1 during merging events. To catch the moment of the maximum environmental effect, we measure D1 and D5 in the target snapshot and the four preceding ones, and take the minimum value.

The merger-driven misaligned galaxies tend to have a lower value of D1 and a higher value of D5 than the others. A low value of D1 means that there must be another galaxy around for mergers to take place. Similarly, a high value of D5 indicates that it is difficult for mergers to take place in a very dense environment such as in clusters. Meanwhile, secularly driven misaligned galaxies are rarely affected by either neighboring galaxies or dense environments, so they tend to have high values of D1 and D5.

The group-driven and interaction-driven misaligned galaxies have similar values of D1, lying between the merger-driven and secularly driven misaligned galaxies. However, their D5 values are somewhat different from each other. The group-driven misaligned galaxies are naturally distributed in dense areas with a smaller value of D5, whereas the interaction-driven misaligned galaxies have more diverse values. Although the interaction-driven misaligned galaxies with a small value of D5 are affected by the group environment, they have been classified as interaction driven because they developed the misalignment through interactions with non-BGG group member galaxies.

Figure 9(b) shows the misaligned galaxies in the plane of ram pressure and perturbation index (Section 4.2), which are thought to have a significant impact on the misalignment formation. We measure the degree of ram pressure following the description of Gunn & Gott (1972):

$$\begin{eqnarray}&&{P}_{\mathrm{ram}}={\rho }_{\mathrm{ICM}}{v}_{\mathrm{rel}}^{2},\end{eqnarray} \tag{ 3 }$$

where ρ_ICM is the mass-weighted mean density of ambient gas cells, and v_rel is their relative velocity with respect to the galaxy center. We define the ambient gas cells by the surrounding gas cells within 200 kpc from the galaxy. However, even if we change the aperture criterion, there is no significant impact on the results.

Except for the secularly driven ones, misaligned galaxies tend to have both higher values of PI and ram pressure than aligned galaxies do. Among them, the merger-driven misaligned galaxies have a particularly high value of PI, and the group-driven misaligned galaxies have a particularly high value of ram pressure. The interaction-driven ones have lower values of PI and ram pressure compared to the two previous groups. Finally, the secularly driven misaligned galaxies tend to have lower values of both PI and ram pressure than other misaligned galaxies do. Their values of PI and ram pressure are closer to those of aligned galaxies than to the other types of misaligned galaxies.

The apparent separation between different groups of galaxies based on their misalignment origins in this diagram is encouraging and supporting our classification schemes presented in Section 4.

In this subsection, we investigate the physical properties of misaligned galaxies depending on their misalignment formation channel. Some galaxies experience more than one misalignment event. For example, merger-driven misaligned galaxies may have developed another (former) misalignment by interacting with their satellites before the merger in question. Besides, galaxies may have a temporarily aligned stage during one misalignment event. For these misaligned galaxies, it is difficult to investigate how their physical properties are linked with their (most recent) misalignment formation channel. Thus, we use in this section only the galaxies that have been aligned for at least 2.5 Gyr (10 gas snapshots) before the misalignment formation in question.

Figure 10 shows the physical properties of galaxies instead of environmental properties, in a similar format to Figure 9. In the main panel, as all the galaxies with non-merger-driven origins share very similar physical properties, we display their distribution contours together (purple). In the side panels, the histograms for each parameter are separately presented. The diamond symbols show the median values for each group. However, as the physical properties can be changed significantly in the process of developing misalignment, we also measure the properties of their progenitor (three gas snapshots or 0.75 Gyr ago) and mark their median values with triangles. Thus, we can track the change of the properties of misaligned galaxies by comparing the diamonds and triangles in the histograms.

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Figure 10(a) shows the stellar mass and kinematic morphology (V/σ) of the galaxies in Horizon-AGN. Note that we have already presented some results from our analysis of the mass and V/σ ratio of the misaligned galaxies in Paper I. The first to note in this panel is that the merger-driven misaligned galaxies are substantially more massive than the whole sample (gray dots and histograms) and non-merger-driven misaligned galaxies. It is easy to understand because more massive galaxies are more likely to attract neighboring galaxies and experience misalignment.

The mass histogram attached to Figure 10(a) also shows that the non-merger-driven misaligned galaxies are all less massive than the whole sample. This is also understandable. The interaction-driven misaligned galaxies are those that are affected by passers-by, and small galaxies are more likely to feel the impact of passers-by more significantly. The group-driven misaligned galaxies are mostly satellites in groups and hence less massive than the whole sample. The isolated galaxies are also known to be less massive than those in dense environments (e.g., Khim et al. 2015).

We now focus on the impact of the misalignment formation on non-merger-driven misaligned galaxies (purple contours). Comparing the diamonds and triangles in the top side histograms, the change in mass in the last three snapshots, that is, during the misalignment process, was negligible. This result is consistent with the previous examples in Section 4.2. The misalignment process caused V/σ to decrease slightly due to the newborn "misaligned" stars from the misaligned gas disk and the blurring of preexisting stellar rotation motion (V/σ histogram).

We then inspect the merger-driven misaligned galaxies. They undergo a merger event, which resulted in an increase in mass and a decrease in V/σ. The change in mass is small because the majority of mergers were minor rather than major. The decrease in V/σ is more pronounced, indicating that the morphology of the galaxy becomes slightly but notably earlier.

Figure 10(b) shows the cold gas fraction and stellar mass for all the galaxies. There is no significant change in cold gas fraction (between triangles and diamonds) during the process of non-merger misalignment. This may sound strange when it comes to group-driven misaligned galaxies. If ram pressure stripping is the main process that causes misalignment for them, we might naturally expect a decrease in the gas fraction during the misalignment process. As was visible in Figure 7(a), however, the removal of gas due to ram pressure stripping has a time lag from the onset of the misalignment. As we here use three snapshots befor e the snapshot of misalignment for a reference point, the change in the gas fraction appeared negligible in the case of group-driven processes.

On the other hand, the merger-driven misaligned galaxies show a notable difference in the cold gas fraction in terms of the median value and the shape of the distribution. This trend suggests that for the merger-driven misaligned galaxies, a large amount of the newly accreted gas overwhelms the preexisting gas and changes their gas rotational axis, as was demonstrated in Figure 2.

Overall, non-merger-driven misaligned galaxies do not dramatically change their physical properties during the misalignment formation. However, merger-driven misaligned galaxies increase their stellar and gas masses during the merger event.

Once a galaxy develops the misalignment, the stellar and gas disks attract each other and eventually realign. Some physical properties of the galaxy may influence the speed of the "settling down" process, e.g., ellipticity (Bryant et al. 2019).

We first examine the lifetime of the misalignment depending on their kinematic morphology. Figure 11 is a decay histogram of star–gas misalignment. The abscissa shows the lifetime of the misalignment, and each bin shows the number fraction of the galaxies maintaining their misalignment as a function of time. We find that a considerable number of galaxies maintain their misalignment for more than a few billion years. In particular, a total of 138 galaxies maintain their misalignment through the last gas snapshot (z = 0.018), and so their lifetime cannot be fully measured.

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We divide the galaxies into ETGs and LTGs (see Section 2.2) and investigate their lifetimes. The histograms of both ETGs and LTGs show that the number of misaligned galaxies decays gradually and steadily with time. However, there is a clear difference between ETGs and LTGs: ETGs tend to have a longer lifetime than LTGs. For example, during the first gas snapshot interval (≃0.25 Gyr), about 90% of the misaligned ETGs still maintain their misalignment, whereas only about 60% of misaligned LTGs do.

To quantify the lifetime of the misalignment, we fit the histogram using an exponential decay function:

$$\begin{eqnarray}&&N(t)={N}_{0}\exp (-t/\tau ),\end{eqnarray} \tag{ 4 }$$

where N₀ is the initial number of misaligned galaxies, t is time, and τ is the exponential decay timescale. The fit is performed for the range of 0–5 Gyr (the vertical line) of the lifetime only, because 138 galaxies maintain their misalignment through the end of the simulation as mentioned above. The decay timescale from the best fit is 2.34 ± 0.05 Gyr and 0.75 ± 0.04 Gyr for ETGs (orange curve) and LTGs (green curve), respectively. This means that the misalignment lifetime of ETGs is 3.4 times longer than that of LTGs.

Let us extend the analysis further. Paper I demonstrated that misalignment phenomena (in terms of the number fraction and PA offset) are affected by both the kinematic morphology and the cold gas fraction of galaxies. Thus, we investigate whether these parameters have an impact on the lifetime, too. We draw a two-dimensional histogram to analyze the effects of each parameter separately, as shown in Figure 12. In the main panel, each pixel is color-coded by the mean lifetime of the misalignment. Each pixel contains at least five galaxies to ensure statistical significance. In the side panels, we present the lifetime as a function of each parameter.

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We find in the top panel that the lifetime varies depending on V/σ even within the same classification of morphology. Overall, the galaxies of earlier type (i.e., lower values of V/σ) tend to have a longer lifetime of misalignment. This trend is consistent with the result of Bryant et al. (2019), considering that there is a clear positive trend between kinematic morphology and ellipticity.

Next, we examine the impact of the cold gas fraction. We find that the relatively gas-poor galaxies tend to have a longer misalignment lifetime than gas-rich galaxies. While the gas fraction and V/σ (morphology) are closely linked with each other, we find that they independently affect the misalignment fraction. The smaller the two parameters are, the longer the lifetime is. This result is also consistent with the trend shown in the gas-morphology histogram mentioned in Paper I. We present a simple dynamical model that explains the effect of gas fraction in the Appendix.

We can explain the impact of the amount of gas in the same way as mentioned in Paper I. In a galaxy with a higher cold gas fraction, the momentum dissipation due to the interaction between stars and gas is expected to be more efficient. This may impact the settling down process and decrease the PA offset more quickly. Also, star formation in the gas disk may reduce the misalignment lifetime. The new stars born with the kinematic characteristics of the misaligned gas disk are added to the existing stars. As a result, the stellar rotational axis (the direction of the mean angular momentum) would be gradually changed to that of the gas disk.

As noted in Paper I, simulation galaxies show an enhanced misalignment fraction in dense environments, such as massive groups or clusters. Also, we have found that dense environments are closely linked with the misalignment formation (Section 4.2.2). Thus, the galaxies in different environments may have different misalignment lifetimes.

In order to investigate the possible impact of the environment on the lifetime, we divide the galaxies into group members and field (non-group) galaxies (see Section 2.2) and measure their lifetime, as shown in Figure 13. Panel (a) shows the decay histogram in the field environment. The decay timescale is 2.28 ± 0.08 Gyr in ETGs and 0.49 ± 0.02 Gyr in LTGs: 4.7 times different. In contrast, in the dense environment (panel (b)), the decay timescale is 2.52 ± 0.03 Gyr in ETGs and 1.55 ± 0.04 Gyr in LTGs, leading to the 1.62 times difference. Overall, the decay timescale of misalignment is longer in denser environments. Also, the lifetime of LTGs seems to be affected by the environmental effect more significantly.

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Figure 13 can be explained by Figure 12. The group/cluster member galaxies are affected by their environments; member galaxies in groups more easily become early types (morphology–density relation; Dressler 1980), passive in terms of star formation (Gómez et al. 2003), and gas poor due to the gas-stripping process (e.g., Gunn & Gott 1972; Quilis et al. 2000). The infalling ETGs in dense environments steadily lose their cold gas due to the environmental effect. The mean gas fraction when the group ETGs generate the misalignment is 0.27, and it becomes 0.12 after 2.5 Gyr. The group ETGs move to the bottom of Figure 12 and have a longer lifetime than field ETGs.

The infalling LTGs also lose their cold gas (0.52 to 0.25), as ETGs do. Moreover, they become earlier in kinematic morphology (the mean value of V/σ of 1.55 to 1.11 after 2.5 Gyr). They move to the left and bottom in Figure 12 and have a much longer lifetime than field LTGs. In addition, the misaligned galaxies in group environments often regenerate the misalignment when the galaxy passes the pericenter multiple times (e.g., Stage 7 in Figure 7). This helps the misalignment of galaxies survive longer than in the field.

In this section, we explore the misalignment lifetime depending on their origins. We classify the Horizon-AGN galaxies at z = 0.52 with their origins in the same way as in Section 4. Among the 2177 galaxies, there are 1313 (60%) merger-driven, 195 (9.0%) group-driven, 247 (11%) interaction-driven, and 422 (19%) secularly driven misaligned galaxies.

There is significant discrepancy between the results here at z = 0.52 and at z = 0.018 from Section 4 in terms of the number fraction of the merger driven. A small part of the discrepancy can be explained by the tendency of higher merger frequencies in the earlier universe. However, the main reason for the difference is the bias from the sample selection. As we mentioned, we have used only the galaxies that survived through z = 0.018. A larger fraction of the lower mass galaxies disappeared through galaxy mergers between z = 0.5 and z = 0.018. As the merger-driven misaligned galaxies are heavily biased toward massive galaxies, this leads to an increase of the fraction of merger-driven misaligned galaxies, compared to that of z = 0.018. However, it should be noted that the difference in the number fractions cannot affect the subsequent analysis because we will measure the lifetime for each origin.

Figure 14(a) shows the decay histogram of the misalignment in a similar format to Figure 11. Overall, the group-driven misaligned galaxies tend to maintain their misalignment longer than the others, followed by the merger-driven and interaction-driven ones. The misalignment decays most rapidly in the secularly driven misaligned galaxies.

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We have found that the non-merger-driven misaligned galaxies share similar physical properties in terms of stellar mass, morphology, and gas fraction when they develop the misalignment (Figure 10). Thus, the lifetime difference between the three subgroups of the non-merger-driven misalignment is difficult to explain using the information in Figure 12 solely.

Figure 14(b) shows the median PA offset as a function of time. The criterion for misalignment (30°) is marked with a horizontal dashed line. Except for the initial value of PA offset (x = 0), the group driven tend to have a higher PA offset than other types. Their PA offset decreases gradually due to the environmental effect (Section 6.2). In the case of the merger-driven misaligned galaxies, on the other hand, their initial PA offsets are similar to those of the group-driven ones but show a rapid decline. The interaction-driven misaligned galaxies, like the merger-driven ones, show a decline in their PA offset. However, because their initial PA offsets are typically less than those of the merger driven, the time for their PA offsets to reach below our PA offset cut for misalignment (30°) is shorter. Finally, the secularly driven misaligned galaxies have the lowest initial PA offsets on average, and so their misalignment dissipates most quickly.

The results of the misalignment lifetime analysis using Horizon-AGN galaxies at z = 0.52 can be summarized as follows. First, we have investigated the lifetime depending on the physical properties of galaxies in Section 6.1. Morphology (V/σ) and gas fraction are independently anticorrelated with the decay timescale of misalignment (2.28 Gyr for ETGs and 0.49 Gyr for LTGs). This result is consistent with the misalignment scheme and the discussion of the impact of the gas fraction on the realignment process mentioned in Paper I. Next, we have found that the galaxies in dense environments tend to have a longer lifetime than field galaxies (Section 6.2). This tendency seems to reflect the result of Section 6.1. Because infalling galaxies in dense environments lose their cold gas and become earlier in morphology, they tend to have a longer lifetime than field galaxies. Lastly, we have found that the lifetimes of the misalignment are longer in order of group-driven, merger-driven, interaction-driven, and secularly driven misalignment in Section 6.3. While the different properties of misaligned galaxies depending on their origin can partly explain this, the lifetime difference among the three subgroups of the non-merger-driven misalignment can be explained by the environmental effect and their initial PA offsets.

In this section, we examine and discuss how the misalignment develops. We show the stellar and gas rotational axis changes of the misaligned galaxies at z = 0.018 by measuring the rotational axes at the snapshot of misalignment formation and one snapshot before. Figure 15 shows this result in a similar format to Figure 1. We divide the galaxies into merger-driven (the upper panels) and non-merger-driven (the lower panels) misaligned galaxies. Each pixel contains at least three galaxies to ensure statistical significance.

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The merger-driven misaligned galaxies tend to show a significant increase in their stellar mass due to a major or minor merger. The distribution of red pixels in panel (a) shows that the greater change in stellar mass is accompanied by the greater change in the stellar axis. The merger-driven misaligned galaxies also show changes in the cold gas mass due to the merger events. These gas mass changes are also linked with the changes in the gas rotational axis, as shown in panel (b). Some of the merger-driven misaligned galaxies accrete cold gas from the merging galaxies, resulting in changes in their gas rotational axis. On the other hand, we have also found that some galaxies lose their cold gas due to the AGN or stellar feedback after the merger. Although small in number, some galaxies do not show a change in their gas or stellar mass during one gas snapshot interval (i.e., those near the origin in this panel); yet, their gas rotational axis is still altered by more than 20°. Overall, the misalignment of the merger-driven misaligned galaxies is associated with mass changes in both the stars and gas. However, panel (c) shows that most of the pixels are red, which indicates that the rotational axis changes are usually greater in the gas rather than in stars. Thus, the star–gas misalignment of merger-driven misaligned galaxies is mainly due to the change in the gas rotational axis during mergers.

Non-merger-driven misaligned galaxies show little accretion of stellar mass, unlike the merger-driven misaligned galaxies. Some of them even lose their stellar mass due to tidal stripping. Moreover, they show only minute changes in their stellar rotational axes when they develop the misalignment, as shown in panel (d). This means that the examples we have investigated in Section 4.2 (e.g., Figures 6 and 7) are not exceptional cases. On the other hand, the gas mass of the non-merger-driven misaligned galaxies can be changed due to the interactions with nearby galaxies or environments. The changes in their gas mass lead to the changes in the gas rotational axis, as shown in panel (e). Like the merger-driven case, the gas rotation axes are changed by at least 20° even if the galaxies do not show changes in their gas mass. Lastly, panel (f) indicates that the rotational axis changes of the gas outweigh those of stars when the non-merger-driven misaligned galaxies develop the misalignment.

We conclude that the star–gas misalignment of the galaxies is mainly due to the change in the gas rotational axis regardless of its origin.

In Section 4.1.2, we have found that about 35% of the total misaligned galaxies at z = 0.018 are merger driven. It means that mergers play a crucial role in the star–gas misalignment formation (e.g., Balcells & Quinn 1990; Hernquist & Barnes 1991; Barnes & Hernquist 1996; Bekki 1998; Puerari & Pfenniger 2001; Crocker et al. 2009). Recently, Lu et al. (2021) investigated hot and counterrotating disk galaxies in IllustrisTNG and reported that galactic mergers might have played a leading role in building up the counterrotating orbital fraction.

The misalignment study of Starkenburg et al. (2019) based on the Illustris simulation reported that mergers with merging mass ratios larger than 1:10 did not significantly contribute to the misalignment formation, which may appear contradictory with our result.

The difference comes from the use of different mass ranges for the galaxy sample selection. Starkenburg et al. (2019) used galaxies in the range of M_* = 2 × 10⁹ M_☉ to M_* = 5 × 10¹⁰ M_☉ in order to investigate the galaxies that are expected to have a low impact from mergers. On the other hand, we did not set an upper limit on stellar mass, thus our sample included a significant number of massive galaxies. Figure 10 illustrates well that the median value of the stellar mass of the merger-driven misaligned galaxies is roughly 8 × 10¹⁰ M_☉, which exceeds the mass range of Starkenburg et al. (2019).

We plot Figure 16 to visualize the discrepancy more clearly. In this two-dimensional histogram, each pixel is color-coded by the number fraction of merger-driven misaligned galaxies. The vertical line shows the upper limit of the mass range in Starkenburg et al. (2019), and the shaded region shows where the two samples overlap. The majority of the massive misaligned galaxies above the mass range of the sample of Starkenburg et al. (2019) are merger driven (magenta). On the other hand, Horizon-AGN also shows that the low-mass shaded region is dominated by non-merger-driven misaligned galaxies (cyan), such as interaction-driven, group-driven, and secularly driven misaligned galaxies. Therefore, the two different cosmological simulations are not in conflict with each other.

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One of the key issues about the misalignment formation is the contribution of central black hole activities. Starkenburg et al. (2019) suggested that supermassive black hole feedback is one of the main channels of misalignment. Duckworth et al. (2020a) also reported that there is a clear trend between black hole activities and misalignment. However, they mention that the causality between the two is not clear. Similarly, Khoperskov et al. (2021) suggested that the central black hole activity could be linked with the misalignment formations but does not seem to the main cause.

We have found that in some galaxies in Horizon-AGN, misalignment and central black hole activity coincide. However, most of them are also accompanied by galaxy mergers and thus classified as merger-driven misaligned galaxies in our classification scheme. Unlike Starkenburg et al. (2019), we find a small number of misalignments that seem to be associated only with black hole activities (see Section 4.2.3). In this sense, it seems that feedback from the central black holes itself is not a main driver of misalignment.

The time resolution of Horizon-AGN is insufficient to resolve central black hole activities. We anticipate that our next-generation simulations (e.g., NewHorizon; Dubois et al. 2020) with improved resolution and gas tracer particles will provide more information on this issue.

The toy model describes a satellite galaxy falling into a larger group/cluster as a set of two gravitationally coupled rings subject to external tides. Each set represents the gas and the stellar disk, respectively. Each ring represents a set of orbits with a given set of actions (i.e., orbital parameters), which again, for simplicity, are characterized here by a mean radius and some (irrelevant) minor spread in eccentricity and vertical oscillation around circular coplanar orbits. So, in the end, the ring is completely specified by its radius and relative orientation with respect to the reference center-of-mass midplane of the whole gas plus star system. Similarly, the gaseous disk is formally decomposed into such a set of coupled rings characterized by its radius R_i and its orientation (θ_i , ϕ_i ) with respect to the midplane. The set of concentric gravitationally self-interacting rings is qualitatively depicted in Figure A1.

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We will assume here that the radii do not change with cosmic time (ignoring accretion), and that the two disks are embedded in a spherical halo (or that the halo's static asphericity is accounted for in setting up the midplane, and its time-dependent variation is integrated in the "external tides" that the set of disks will be subject to). Conversely, we will assume that the orientations of both sets of rings are time dependent, as they respond both to the tidal force imposed by the position of the satellite galaxy within the cluster, and the time-dependent distortion of both (satellite and cluster) dark (sub)halos. The gas disk is also specifically subject to two extra sets of torques induced by ram pressure and turbulent viscosity.

Let us first give a qualitative account of the expected behavior of such a toy model before spelling out the mathematics. Because we are concerned by the departure from a set of settled coplanar disks, we will assume without loss of generality that the equations of motion describing the different rings are linearized with respect to an unperturbed coplanar configuration. We can therefore Taylor-expand the Hamiltonian of the system with respect to the amplitude of small oscillations of each ring above and below the (time-dependent) midplane. In doing so, each ring will formally become a coupled oscillator with effectively one degree of freedom. It is (tidally) coupled to all other (gas+star) rings and subject to external forcing. After linearization, the set of 2N coupled oscillators will obey a matrix equation. These equations of motion can be decoupled by moving to the eigenframe diagonalizing the matrix of either gas or stellar disk. Each eigenmode will represent a displacement wave of the set of N rings of the chosen component and will obey in isolation (ignoring briefly the other disk) a linearly forced oscillator equation, where the forcing in that frame is simply the projection of the forcing in configuration space projected onto the eigenvectors of the matrix. Accounting for the other disk implies that in such a frame, the evolution of the gas and star eigenmodes follows two sets of coupled forced oscillator equations as we want to explicitly account for the fact that the gaseous component can dissipate oscillations, so we will add a damping term for the gas eigenequation.

It follows quite straightforwardly that if the gas eigenmode alone is subject to a differential torque (e.g., ram pressure), it will react to it and eventually damp it out, but it will also drag the typically more massive coupled stellar disk out of phase so that both modes will oscillate with respect to each other for a while. The relative amplitude of these oscillations will essentially reflect the relative mass in each disk. Because the physical disk response is the sum of their eigenmodes, the misalignment between the disks will reflect that of their eigenmodes.

While this toy model is not meant to be taken too seriously, it has the merit of explaining qualitatively what the simulation is doing. The key relevant ingredients are (i) the (relative) masses of the two disks, i.e., the strength of their gravitational coupling; (ii) dissipation only within the gas component; and (iii) relative forcing on the gas component.

In order to be more quantitative, e.g., about the relative role of gas fraction, let us spell out the toy model presented in the previous section. This is best described in the so-called Laplace–Lagrange theory.

A.2.1. Stellar Disk Setup

Let us assume that stellar orbits with guiding center R in the disk are nearly coplanar (θ ≪ 1) and nearly circular (e ≪ 1), where θ and ϕ are the polar and azimuthal angles specifying the orientation of this orbital plane. For simplicity, let us assume that we are considering the outer part of the disk, so that the potential can be described as nearly Keplerian. Defining the canonical variables, ⁹ p , q as

$$\begin{eqnarray}&&{q}_{i}={\gamma }_{i}{\theta }_{i}\sin ({\phi }_{i}),\quad {p}_{i}=-{\gamma }_{i}{\theta }_{i}\cos ({\phi }_{i}),\end{eqnarray} \tag{ A1 }$$

with ${\gamma }_{i}=\sqrt{{m}_{i}}{\left({{GMR}}_{i}\right)}^{1/4}$, the Hamiltonian describing the coupling between the ring at radius R_i in that limit reads (Kocsis & Tremaine 2011)

$$\begin{eqnarray}&&H({\boldsymbol{p}},{\boldsymbol{q}})={{\boldsymbol{p}}}^{{\rm{T}}}\cdot {\boldsymbol{A}}\cdot {\boldsymbol{p}}+{{\boldsymbol{q}}}^{{\rm{T}}}\cdot {\boldsymbol{A}}\cdot {\boldsymbol{q}},\end{eqnarray} \tag{ A2 }$$

where

$$\begin{eqnarray}&&{A}_{{ij}}=-\displaystyle \frac{1}{8}\displaystyle \frac{{{Gm}}_{i}{m}_{j}{\alpha }_{{ij}}}{\max ({R}_{i},{R}_{j}){\gamma }_{i}{\gamma }_{j}}{b}_{3/2}({\alpha }_{{ij}}),\quad \mathrm{if}\quad i\ne j\end{eqnarray} \tag{ A3a }$$

$$\begin{eqnarray}&&=\,\displaystyle \sum _{k\ne i}\displaystyle \frac{1}{8}\displaystyle \frac{{{Gm}}_{i}{m}_{k}{\alpha }_{{ik}}}{\max ({R}_{i},{R}_{k}){\gamma }_{i}^{2}}{b}_{3/2}({\alpha }_{{ik}}),\quad \mathrm{if}\quad i=j,\end{eqnarray} \tag{ A3b }$$

given ${\alpha }_{{ij}}=\min ({R}_{i},{R}_{j})/\max ({R}_{i},{R}_{j})$ and

$$\begin{eqnarray}&&{b}_{3/2}(\alpha )=\displaystyle \frac{2}{\pi }{\int }_{0}^{\pi }\displaystyle \frac{\cos {xdx}}{{\left(1-2\alpha \cos x+{\alpha }^{2}\right)}^{3/2}},\end{eqnarray} \tag{ A4a }$$

$$\begin{eqnarray}&&=\,\displaystyle \frac{(1+{\alpha }^{2})E(\alpha )-(1-{\alpha }^{2})K(\alpha )}{\pi \alpha {\left(1-{\alpha }^{2}\right)}^{2}},\end{eqnarray} \tag{ A4b }$$

with K and E the elliptic functions of the first and second types, respectively.

The equations of motion following from Equation (A2) are best solved if we move to a frame that diagonalizes the positive semidefinite symmetric matrix A , so that in that frame, Hamilton's equation yields for each eigenmode

$$\begin{eqnarray}&&{\ddot{\hat{q}}}_{i}+{\omega }_{i}^{2}{\hat{q}}_{i}={\hat{\xi }}_{i},\end{eqnarray} \tag{ A5 }$$

where ω_i is the ith eigenvalue and ${\hat{\xi }}_{i}$ is the external specific force applied on the ring projected on the corresponding eigenvector. Here, the hat quantities are (briefly) used for the new frame of coordinates.

A.2.2. Stellar-gas Disk Coupling

Now let us note that so long as the surface density of the gas and the stars are proportional, the matrix M describing gas rings will be formally identical to that for the stars (up to a multiplicative factor reflecting the gas to star mass ratio), so that the eigenspaces of both disks are the same.

In that frame, the gas ring eigenmode formally obeys a similar equation to Equation (A5) with one extra caveat, which is that the gas can shock, so that each gas ring is subject to an extra drag force. For expediency, we consider that the drag term operates on the eigenmode.

Finally, when simultaneously considering the evolution of both gas and star eigenmodes, we need to account for their relative gravitational interaction, which can be accounted for by a supplementary coupling term in both equations.

This then leads us to consider the dynamics of the set of coupled gas plus star eigenmodes for the stars and for the gas components, whose amplitudes are written as q_⋆, and q_g , respectively. For expediency, let us consider only one such mode (which effectively assumes that both disks can be diagonalized in the same frame, which will be the case when assuming that the stellar and gas disks are the same up to a multiplicative constant) so we drop the index i following each mode and the hats.

We will consider that this eigenmode has its own natural frequency, ω_⋆ and ω_g , respectively, a coupling term, ω_⋆g , and a driving, ξ, and damping, η, term specific to the gas component. The amplitude of each mode then obeys the set of coupled equations:

$$\begin{eqnarray}&&{\ddot{q}}_{\star }+{\omega }_{\star }^{2}{q}_{\star }+{\omega }_{\star g}^{2}{q}_{g}=0,\end{eqnarray} \tag{ A6a }$$

$$\begin{eqnarray}&&{\ddot{q}}_{g}+{\omega }_{g}^{2}{q}_{g}+{\omega }_{\star g}^{2}{q}_{\star }+\eta {\dot{q}}_{g}=\xi ,\end{eqnarray} \tag{ A6b }$$

which is the main equation of this appendix. The coupling term follows from writing the Hamiltonian for the two sets of rings, while the damping and driving terms are phenomenological add-ons. Note that we put all the (relative) torquing, ξ, on the gas component, because any torquing that applies to both components will not induce relative misalignment. Note also that typically ω_⋆ ≫ ω_g , given the relative mass ratio of the two disks (see Equation (A3)). Solving for Equation A6(b), each stellar eigenmode will obey

$$\begin{eqnarray}&&{q}_{\star }(t)=-\displaystyle \sum _{\omega \in {S}_{4}}\displaystyle \frac{{\omega }_{g\star }^{2}{\int }_{-\infty }^{t}\exp \left((t-\tau )\omega \right)\xi \left(\tau \right)d\tau }{\eta \left(3{\omega }^{2}+{\omega }_{\star }^{2}\right)+2\omega \left(2{\omega }^{2}+{\omega }_{g}^{2}+{\omega }_{\star }^{2}\right)},\end{eqnarray} \tag{ A7 }$$

where the frequencies, ω, are one of four complex conjugate solutions of the implicit equation ¹⁰

$$\begin{eqnarray}&&\left.{S}_{4}=\{\omega \right|\left({\omega }^{2}+{\omega }_{\star }^{2}\right)\left(\omega \left(\eta +\omega \right)+{\omega }_{g}^{2}\right)={\omega }_{g\star }^{4}\},\end{eqnarray} \tag{ A8 }$$

which will have both a damped (real) component and an oscillatory (complex) one.

Illustratively, Figure A2 shows the (unforced) damping of the two stellar and gas modes (of a given ad hoc initial amplitude) when one increases the drag on the gas component (top panel), while the bottom panel shows the frequencies, which are roots of S₄ in Equation (A8). As expected, as one increases η, the complex roots acquire a larger and larger negative real part (corresponding to damping), and the gas disk will generally more efficiently drag the stellar disk toward itself as it settles.

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A.2.3. Stellar-gas Disk Realignment

Equipped with Equation (A7), we can now investigate the relative orientations of sets of rings corresponding to the stellar and gaseous disk, respectively, to understand within the framework of the linearized Laplace–Lagrange theory how the two disks reorient with respect to each other. We aim here to account for the fact that only the gas disk is subject to forcing by ram pressure on the one hand and dissipation through shocks between rings on the other hand. The gas disk is also typically lighter, hence more responsive to torques.

Let us first consider an idealized experiment when only the gas disk is subject to an impulse that propagates to the other disk before eventually both modes damp and the coupled system settles. Figure A3 shows qualitatively the result of such an experiment. As expected, the gas response is significantly stronger because the gas disk is assumed to be here less massive, in agreement with the findings of the main text. It reacts in phase to the impulse, while the stars are out of phase. As shocks and turbulence in the gas component dissipate energy, both oscillations eventually dephase and damp away.

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Focusing now specifically on the quantities of interest in our paper, Figure A4 illustrates the damping of two modes when one increases both the drag on the gas disk and its mass. As expected, the lighter the gas disk, the longer the settling phase. This is also in qualitative agreement with the findings of the main text, corresponding to the situation where a given galaxy enters a group or a cluster and the gas component feels ram pressure from the hot corona. As expected from the toy model and observed in the simulation, the more gas rich the disk is, the stronger the turbulence, the faster the damping, the shorter the lifetime of misalignment.

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As previously stated, the simplistic ring model is only illustrative and not without flaws. Within the context of this paper, we do not aim to quantify the statistical properties of the forcing, which one could extract from the sets of measured misalignment events. It would also be of interest, beyond the scope of this appendix, to quantify the damping process.

Note that to be more quantitative, the misalignment should be measured in configuration space, which can be extracted from the eigenspace information. Note that nonlinear mode coupling could and will also dampen oscillations, as will angular momentum exchanges at resonances within the stellar disk.

At some very coarse level, one can consider that fly-bys correspond to an extreme case of satellite–cluster interaction, so the arguments presented for the latter would apply to the former. Similarly, cosmic infall could be approximated by a secularly added misaligned ring which will inject some energy into the set of coupled oscillators.