Tidal evolution of galaxies in the most massive cluster
of IllustrisTNG-100

In this work, we used the publicly available simulation data from IllustrisTNG described by Nelson et al. (2019). For the purpose of this study, we selected the simulation TNG-100 from the IllustrisTNG set, where the number 100 refers to the size of the simulation box in Mpc. The simulation data are available in 100 snapshots from redshift $z=20$ to the present epoch, $z=0$, which is sufficient to trace the evolution of galaxies. The simulation box contains two massive galaxy clusters at $z=0$ from which we select the most massive one (with total mass $M=4\times 10^{14}$ M_⊙), which is more regular and relaxed as indicated by its temperature distribution. This object bears the identification number ID=0 in the catalog of halos and subhalos identified in this simulation at $z=0$.

Using the simulation output 99 corresponding to $z=0,$ we selected subhalos identified with the Subfind algorithm (Springel et al., 2001) parented by this halo with stellar masses $M_{*}>10^{10}$ M_⊙. This threshold translates to the number of stars per galaxy above $10^{4},$ which allows us to reliably measure their global structural properties. After rejecting spuriously identified subhalos with almost no dark matter, we are left with 112 galaxies whose properties we study. We trace the history of these galaxies using main progenitor branches of Sublink merger trees for the subhalos (Rodriguez-Gomez et al., 2015) to the earliest time possible, that is, to the snapshot when the subhalo was first identified. For the selected galaxies, these earliest snapshots of subhalo existence are among the first few snapshots available, that is, the galaxies can be traced down to redshifts $z=10-20$.

One of the most interesting properties of galaxies from the point of view of tidal evolution is their mass. Using the galaxy properties provided in the Sublink catalogs, we constructed the evolutionary histories of the galaxies in terms of their mass evolution in three components: dark matter, stars and gas. Examples of such evolution for six galaxies in the cluster are shown in Fig. 1. The most characteristic feature of these curves is that all the galaxies increase their dark masses up to the point when they start to lose it after entering the cluster, as we discuss in detail below.

Using the subhalo positions provided in the Sublink catalogs, we calculated the orbits of the galaxies in the cluster in terms of their physical distance from the cluster center as a function of time. Examples of the evolution of this distance for six galaxies on tight orbits around the cluster are shown in the second row of panels in Figs. 2 and 3.

Inspection of the orbital history of the whole sample of 112 galaxies allows us to divide them into three distinct subsamples. It turns out that 48 out of 112 galaxies (43%) are on their first infall into the cluster and did not have any pericenter passage on their orbit. These objects are bound to the cluster and approaching it, but in some cases are still well beyond the virial radius of the cluster, which at $z=0$ is equal to 2 Mpc. This sample comprises a variety of galaxies of different types, properties, and histories that may have interacted with other objects or evolved in groups prior to their accretion by the cluster but were not yet significantly affected by the cluster environment. We do not study these galaxies further in detail here, but we use them as a reference for comparison with the properties of galaxies that were affected by the cluster. In the following, we refer to this subsample as our reference or unevolved sample.

The next, more interesting subsample contains 37 out of 112 galaxies (33%) that experienced one pericenter passage on their orbit around the cluster with the pericenter distance within the virial radius of the cluster at that time. We refer to these galaxies as weakly evolved. The most interesting sample from the point of view of the effects of environment is the one containing 27 out of 112 galaxies (24%) that experienced more than one pericenter passage. We consider this subsample as strongly evolved.

In order to quantify the strength of the interaction of each galaxy with the cluster we used the orbital data to calculate the integrated tidal force (ITF) from the cluster experienced by the galaxy over its entire history. Looking at the dependence of different galaxy properties on this quantity rather than just redshift or time allows us to isolate the effect of the cluster from the cosmological evolution of galaxies in general. ITF was obtained by multiplying the tidal force experienced by the galaxy at a given simulation snapshot by the timestep between this and the following snapshot, and summing up such terms for all snapshots. The tidal force was approximated as in Łokas et al. (2011a) by the formula $F_{\rm tidal}\propto rM(<d)/d^{3}$, where $r$ is the characteristic scale length of the galaxy at which the tidal force operates, $d$ is the distance between the center of the galaxy and the center of the cluster, and $M(<d)$ is the mass of the cluster within this distance.

As the characteristic scale of the galaxy, we adopted $r=1$ kpc for all of them, which is always of the order of the radius containing half stellar mass. The distance $d$ was taken from the orbital histories of the galaxies, such as those shown in the second row of panels in Figs. 2 and 3. The mass of the cluster $M(<d)$ was calculated for each snapshot using the virial masses provided by the Sublink tree catalog for the cluster and approximating the mass dependence on radius by the Navarro-Frenk-White profile (Navarro et al., 1997; Łokas & Mamon, 2001). The integrated tidal force thus calculated is in units of M_⊙ Gyr kpc^-2, but since its values are large, in the following we use $\log$ ITF without quoting the units, since only the relative differences between its values for different galaxies are important. The values of $\log$ ITF for our selection of six galaxies in Figs. 2 and 3 are listed in the second column of Table 1, and their present distances from the cluster center in the third column.

The evolution of different mass components in time for the six selected galaxies shown in Fig. 1 is replotted in the upper rows of panels in Figs. 2 and 3 with the mass scale adjusted to the mass of gas and stars, to enable easy comparison with the orbital history and other properties of the galaxies. The stellar and dark masses of these galaxies in the final simulation snapshot, at $z=0$, are listed in the fourth and fifth columns of Table 1.

The dark versus stellar masses of the 112 galaxies from our sample at $z=0$ are shown in the upper panel of Fig. 4 with points of different color indicating their orbital history. Here, and in the following Figures, the galaxies that are infalling and did not yet have any pericenter passage are denoted with blue circles, those with a single pericenter with green dots, and those with multiple pericenters with red symbols. As expected, for the majority of objects dark matter masses are higher than the stellar ones. However, the dark masses are systematically much higher than the stellar ones for the unevolved and weakly evolved sample (blue and green points) than for the strongly evolved one (red symbols). This already suggests that the strongly evolved sample lost much more dark matter than the other samples.

In the lower panel of Fig. 4, we plot the ratio of the dark to stellar mass for each galaxy as a function of the ITF. We see that the samples are now clearly separated by the total tidal force experienced by the galaxies. The division between the unevolved and weakly evolved samples (blue and green points) occurs around log ITF = 5.5 and between the weakly and strongly evolved samples (green and red points) at $\log$ ITF = 6.5. In addition, the weakly and strongly evolved samples (green and red) form a distinct branch with a strong dependence of the mass ratio on ITF. The correlation between the two quantities is almost linear between the log of mass ratio and log ITF. Along this branch, the galaxies from the weakly evolved sample (green) have dark to stellar mass ratios roughly between 5 and 20, while the strongly evolved objects only between 0.4 and 6.

In both panels of Fig. 4, the solid line indicates the equality between the dark and stellar masses. There are seven data points for which the dark mass is smaller than the stellar mass. Six of these seven points belong to the sample of strongly evolved galaxies and three of them correspond to the galaxies with Sublink catalog identification numbers ID = 87, 92, and 101 whose mass and orbital evolution is shown in Fig. 3. In these cases, the galaxies evolved on very tight orbits with many pericenters and lost a large fraction of their dark masses. Since the dark matter distribution is usually more extended than the stars, it is more easily stripped by tidal forces. While stellar mass is also lost, the effect is much milder than in the case of dark matter. We emphasize that these galaxies are genuine cosmologically formed objects that were strongly dark matter dominated prior to their evolution in the cluster. Their present low dark matter content is the result of strong tidal stripping by the tidal forces from the cluster.

The mass history of the galaxies shown in Fig. 1 and the upper rows of Figs. 2 and 3 is typical for all galaxies associated with the cluster. The dark mass (blue lines) reaches a maximum value at some point in the past, after which it declines. In the case of evolved samples, this decline occurs when the galaxy approaches the first pericenter of its orbit around the cluster. The maximum gas mass (green lines) usually coincides in time with the time of the maximum dark mass. As the galaxy approaches the cluster center, the gas is gradually depleted, but it continues to form stars so that the time when the maximum stellar mass is reached (red lines) occurs much later than the maximum of the dark and gas content. At the same time, the gas is stripped by ram pressure from the hot gas halo of the cluster, so the gas is completely lost soon after the first or second pericenter passage. This process is reflected in the final (present) gas content of the galaxies: while only 16 out of 48 galaxies (33%) of the unevolved sample contain no gas at redshift $z=0$, this fraction rises to 25 out of 37 galaxies (67%) for the weakly evolved sample, and to 27 out of 27 objects (100%) for the strongly evolved one.

The mass loss in the galaxies is further illustrated in Fig. 5, where we show the relative mass loss in dark matter (upper panel) and stars (lower panel) as a function of ITF. The values were calculated as the difference between the maximum and the present mass divided by the maximum mass. We can see that, again, the weakly and strongly evolved samples follow a branch of tight dependence on the ITF and are clearly separated within this branch. While the galaxies of the weakly evolved sample lose between 10 and 80% of their dark mass, the objects of the strongly evolved one always lose more than 70%. On the other hand, the mass loss in stars is much weaker, the galaxies with one pericenter passage lose between 0 and 10% of the stellar mass, while those with multiple pericenters lose between 10 and 55% of the stars.

The results of this subsection concerning the mass loss are in good agreement with the studies of Smith et al. (2016), who used zoom-in cosmological simulations of clusters, and Niemiec et al. (2019), who studied subhalo populations in clusters of the original Illustris simulations. Both these studies found that, in general, dark matter is tidally stripped more effectively than stars and that the galaxies continue to form stars after they are accreted by the cluster, which makes the cluster galaxies less dominated by dark matter.

Figure 6 illustrates the timescales of galaxy evolution in the cluster. In the upper panel, we plot the times elapsed between the time when the galaxy reached its maximum dark matter mass and the present time as a function of log ITF, which is the look-back time to the moment of maximum dark matter mass. The time ranges for the evolved samples are again clearly separated and depend on the ITF reflecting the amount of time the galaxies spent in the cluster. While the times for the weakly evolved sample (green points) cover the range between 2 and 11 Gyr, those for the strongly evolved sample (red points) are in the range between 6 and 12 Gyr. This is expected, since galaxies with more pericenters tend to enter the cluster and start losing mass earlier.

The middle panel of Fig. 6 shows the timescales of gas loss as a function of log ITF. Here, only galaxies that lost all their gas until $z=0$ are shown. The timescales are measured as the time elapsed between the time of the maximum gas mass the galaxy possessed during its lifetime and the time it lost all the gas. The dependence on the strength of the tidal interaction is again very pronounced. The weakly evolved galaxies (green points) needed between 3 and 10 Gyr to lose all their gas, while the strongly evolved objects (red points) need only between 2 and 6 Gyr to do so.

In the lower panel of Fig. 6, we compare the time (in terms of the age of Universe) of the first pericenter passage to the time when the galaxy loses all its gas. The comparison is obviously shown only for galaxies that experienced at least one pericenter passage. In the case of strongly evolved galaxies (red points) the causal connection between these two events is stronger: for all of these objects, except two, the gas is lost at or after the first pericenter passage. For the weakly evolved sample (green points), this connection is not as tight, since for a significant fraction of these galaxies the gas is lost even before the first pericenter. We note, however, that in most cases of the evolved sample, the gas is not lost immediately at the first pericenter passage, and the process may take up to 4 Gyr from the first pericenter (it sometimes coincides with the second pericenter passage).

The gas loss discussed in the previous subsection has a great impact on the evolution of the cluster galaxies in terms of star formation, which is suppressed. This environmental quenching leads to the change in galaxy color. IllustrisTNG has been shown to provide a significant improvement over the original Illustris in reproducing galaxy colors in comparison with the data from the Sloan Digital Sky Survey (SDSS) (Nelson et al., 2018). Thus, we may also expect to see some signal in the color evolution for the cluster galaxies.

The Sublink catalogs of IllustrisTNG provide the magnitudes of galaxies in eight bands based on the summed-up luminosities of all the stellar particles of the galaxy, from which we chose SDSS broad-band filters $g$ and $r$. We used $g-r$ as a measure of galaxy color, in which case $g-r=0.6$ may be used as a value separating the blue ($g-r<0.6$) from the red population ($g-r>0.6$). The present ($z=0$) values of the color for all our galaxies are plotted in the upper panel of Fig. 7 as a function of ITF. We can see that while the infalling galaxies (blue dots) show a wide range of colors, the galaxies become redder as we go to the evolved samples. Among the weakly evolved galaxies (green points), there are still four with $g-r<0.6,$ but all the galaxies from the strongly evolved sample (red points) are decidedly red with $g-r>0.7$.

In the lower panel of Fig. 7, we also show the metallicities of the same sample of galaxies estimated as the mass-weighted average metallicity (of all elements above He) of the stellar particles within twice the stellar half-mass radius. Here, the dependence on the evolutionary stage of the galaxies is much weaker, but still the galaxies from the strongly evolved sample tend to be more metal-rich. For each galaxy in the sample, we traced the evolution of the color and metallicity in time. In the lower panels of Figs. 2 and 3, we plot examples of such evolution for our six selected, strongly evolved galaxies. In all cases, the curves look similar: while the metallicities increase rather fast early on and reach a constant level, the colors seem more related to the orbital history. The galaxies remain blue until about the time of their first pericenter passage around the cluster and soon after, when they lose all the gas, their color index rather quickly increases to above $g-r=0.6,$ and they become red. The values of $g-r$ continue to grow slowly and reach $g-r=0.8$ by the present time. The $z=0$ values of these quantities for the six galaxies are listed in the sixth and seventh column of Table 1, respectively.

It is interesting to check when exactly the transition from the blue to red population takes place, that is when the galaxies become red and dead as a result of their evolution in the cluster. To address this issue, in Fig. 8 we show the distribution of the $g-r$ color at different redshifts for our sample of 64 evolved galaxies, meaning for the weakly and strongly evolved samples combined. At redshift $z=1$ (upper panel), most of the galaxies are still blue, with $g-r<0.6$, and at redshift $z=0.5$ (middle panel), the sample is divided equally into blue and red galaxies as 32 objects (exactly 50%) have $g-r<0.6$ and the other half has the color larger than this value. At redshift $z=0$ (lower panel), the population is strongly dominated by red galaxies. Therefore, redshift $z=0.5$ may be considered as the time when the population of galaxies in the cluster becomes predominantly red. This is understandable in the context of the orbital history of the galaxies in the evolved sample. As shown in the lower panel of Fig. 6, at redshift $z=1$ (corresponding to the age of the Universe $t=5.9$ Gyr) only a small fraction of galaxies that lost all their gas by now have already experienced their first pericenter passage. Instead, at $z=0.5$ ($t=8.6$ Gyr) about half of them did.

Although no similar comparison can be made for any real, observed cluster since different clusters are seen at different redshifts, the evolution shown in Fig. 8 clearly explains the origin of the Butcher-Oemler effect, namely that galaxy clusters at higher redshift contain a larger fraction of blue galaxies than the low-redshift ones (Butcher & Oemler, 1978, 1984; Rakos & Schombert, 1995; Margoniner et al., 2001; Goto et al., 2003a). If we treat the simulated cluster as representative of the population of clusters at different redshifts, then given the large scatter in the fraction of blue galaxies among clusters, different ways to measure color (Goto et al., 2003a), and the dependence on limiting distance and cluster richness, the value of the blue fraction of the order of 0.5 at redshift $z=0.5$ found here seems to be in reasonable agreement with observations (Margoniner et al., 2001).

In addition to the tidal stripping that manifests itself in mass loss, the galaxies in the cluster also experience what is sometimes referred to as tidal stirring. This process, proposed as a way to transform originally disky dwarf galaxies into dwarf spheroidals in the context of the Local Group evolution, has been discussed in detail in a number of studies (Kazantzidis et al., 2011; Łokas et al., 2011b; Buck et al., 2019). The repeated action of the tidal forces from a bigger host, mostly during the pericenter passages, is supposed to reduce the ordered motions of the stars in the form of rotation and increase the amount of random motion as well as thicken their disks, thus turning oblate rotating disks into triaxial, prolate, or spherical objects dominated by random motions.

In order to quantify these changes in the case of our galaxies evolving in the cluster, we used the measures of the shape and rotation provided by the Illustris team and calculated as described in Genel et al. (2015). The shape is described using the axis ratios of the stellar component determined within two stellar half-mass radii. They were estimated from the eigenvalues of the mass tensor of the stellar mass inside this radius obtained by aligning each galaxy with its principal axes and calculating three components ($i$=1,2,3): $M_{i}=(\Sigma_{j}m_{j}r^{2}_{j,i}/\Sigma_{j}m_{j})^{1/2}$, where $j$ enumerates over stellar particles, $r_{j,i}$ is the distance of stellar particle $j$ in the $i$ axis from the center of the galaxy, and $m_{j}$ is its mass. The eigenvalues are sorted so that $M_{1}<M_{2}<M_{3},$ therefore the shortest to longest axis ratio is $c/a=M_{1}/M_{3}$, while the intermediate to longest is $b/a=M_{2}/M_{3}$. In order to describe the shape of the galaxies with just one number, we also used the triaxiality parameter $T=[1-(b/a)^{2}]/[1-(c/a)^{2}]$. With this parameter, one can classify the galaxy shapes into oblate ($T<1/3$), triaxial ($1/3<T<2/3$) and prolate ($T>2/3$).

The evolution of the axis ratios $c/a$, $b/a$ and the triaxiality parameter $T$ in time for our six selected galaxies from the strongly evolved sample is shown in the third row of panels in Figs. 2 and 3. The measurements of these parameters are restricted to subhalos with stellar mass $M_{*}>3.4\times 10^{8}$ M_⊙ within twice the stellar half-mass radius, and at least 100 stars so they do not extend so far into the past as the measurements of the mass and orbital history. The final ($z=0$) values of these parameters for these six galaxies are listed in the eighth, ninth, and tenth column of Table 1. The $z=0$ values of triaxiality for all our galaxies are plotted in the upper panel of Fig. 9 as a function of log ITF. We can see that while the weakly evolved sample (green points) is dominated by oblate objects (majority of galaxies have $T<0.5$), prolate galaxies are more numerous (majority of galaxies have $T>0.5$) in the strongly evolved sample (red points).

In order to quantify the amount of rotation support in the galaxies, we used the rotation parameter $f$ defined as the fractional mass of all stars with circularity parameter $\epsilon>0.7,$ where $\epsilon=J_{z}/J(E)$. Here $J_{z}$ is the specific angular momentum of the star along the angular momentum of the galaxy, while $J(E)$ is the maximum angular momentum of the stellar particles at positions between 50 before and 50 after the particle in question in a list where the stellar particles are sorted by their binding energy. The evolution of the rotation parameter $f$ for our selection of six galaxies is shown in the fourth row of the panels in Figs. 2 and 3. The final values of this parameter (at $z=0$) are given in the 11th column of Table 1.

The final snapshot values of $f$ for all galaxies are plotted in the lower panel of Fig. 9 as a function of log ITF. As in the case of shape parameters, there is a systematic trend of $f$ evolution when we go from the least to the most evolved samples, to such an extent that there is less rotation support in the more evolved galaxies. While the galaxies from our weakly evolved sample (green points) take values from a wide range of $f$ between 0 and 0.5, the galaxies from the strongly evolved sample (red points) almost all have $f<0.2$. We note that $f>0.2$ is a reasonable threshold to define late-type or disky galaxies in Illustris (Peschken & Łokas, 2019; Peschken et al., 2020), so $f<0.2$ means that our evolved galaxies are no longer disks.

The results on the shape and kinematics evolution of this subsection combined with the color changes discussed in the previous subsection indicate that indeed as the galaxies orbit the cluster, preferably on tight orbits with many pericenter passages, they change their shapes from disky to triaxial or prolate, and their rotation is suppressed and replaced by random motions while they become increasingly red. In Fig. 10, we again plot the rotation parameters $f$ and colors $g-r$ of the galaxies at the present time, but as a function of their distance from the cluster center. Since the distance is the indicator of cluster density, the Figure illustrates the presence of the morphology-density relation in the simulated cluster. The galaxies with little rotation support and red colors, which would be classified as early type by observers, are predominantly found near the cluster center, while those with a significant amount of rotation and blue colors (late type) are mostly present in the outskirts of the cluster. The trend is thus similar to the morphology-density (or morphology-radius) relation found in real clusters (Dressler, 1980, 1984; Goto et al., 2003b; Postman et al., 2005).

The evolution of the shape and kinematics parameters shown in the third and fourth rows of the panels in Figs. 2 and 3 reveals that all these objects were in the past oblate rotating disks, meaning they had their rotation parameters $f>0.2$ and their triaxiality parameters $T<1/3$. Later on, at some point, the rotation parameters started to decrease and the triaxiality parameters increased. For these six galaxies, the time of this transition clearly coincides with one of the first pericenter passages of the galaxy on its orbit around the cluster. With time, $f$ decreased down to almost zero while $T$ reached values above 2/3. This means that the disks turned into prolate, slowly rotating spheroids, with properties similar to tidally induced bars.

To verify this, we extracted, from the last simulation snapshot, the cutouts containing the stellar particle data for these galaxies and rotated the stellar positions to align the $x$, $y,$ and $z$ coordinates with the principal axes using the same procedure as was used to determine the axis ratio. The coordinate system was chosen so that the $x$ axis is along the longest axis of the stellar component and the $y$ and $z$ axes are along the intermediate and shortest ones, respectively. The projections of the stellar mass density along the shortest $z$ axis for the six galaxies from Figs. 2 and 3 are shown in Fig. 11. They correspond to the face-on views of the stellar component and clearly reveal the presence of elongated shapes along the $x$ axis, which is characteristic of bars.

In order to further confirm that these are indeed tidally induced bars, we calculated the measure of the bar strength as the $m=2$ mode of the Fourier decomposition of the surface density distribution of stellar particles projected along the short axis: $A_{m}(R)=|\Sigma_{j}m_{j}\exp(im\theta_{j})|/\Sigma_{j}m_{j}$. Here, $\theta_{j}$ is the azimuthal angle of the $j$th star, $m_{j}$ is its mass and the sum is up to the number of particles in a given radial bin. The radius $R$ is the standard radius in cylindrical coordinates in the $xy$ plane, $R=(x^{2}+y^{2})^{1/2}$. The results of such measurements within two stellar half-mass radii are shown as a function of time in the fourth row of panels in Figs. 2 and 3 together with the rotation parameter $f$. We can see that before accretion the galaxies were normal disks with a lot of rotation and no bar, as the $A_{2}$ parameter remains close to zero. After one of the pericenter passages $A_{2}$ starts to increase (while the rotation decreases) signifying the formation of a tidally induced bar for all galaxies.

The profiles of the bar mode as a function of radius, $A_{2}(R)$, are plotted in Fig. 12 for the six galaxies. The profiles show a behavior characteristic of galactic bars: they increase to a maximum value and then decrease to almost zero. The maximum values of the profiles, $A_{2,{\rm max}}$, can be used as a measure of the bar strength and are listed in the 12th column of Table 1. All these values are above 0.4, signifying rather strong bars, and they are well within the range of values for bars in observed galaxies (Diaz et al., 2016). The profiles in Fig. 12 can be also used to estimate the length of the bars as the radius $R,$ where $A_{2}(R)$ drops to half the maximum value. These lengths turn out to be of the order of 4-6 kpc, in agreement with the visual impression from Fig. 11 and with the observed values (Aguerri et al., 2005; Diaz et al., 2016; Font et al., 2017).

Since the six galaxies we identified as tidally induced bars are prolate spheroids rather than bars embedded in disks, we also looked at the velocity distribution of their stars. One example, for our strongest tidally induced bar (ID=53), is shown in Fig. 13. In all cases, the velocity field measured in projection along the three principal axes of the stellar component shows features characteristic of bars (Athanassoula & Misiriotis, 2002). Namely, there is a strong rotation signal when the galaxy is seen edge-on and end-on, while there is no systematic rotation in the face-on view, so the galaxies rotate around their shortest axis. This means that these objects are indeed bars and not prolate spheroids with prolate rotation (around the long axis) of the kind studied by Ebrová & Łokas (2017).

We also estimated the pattern speeds $\Omega_{\rm b}$ of these tidally induced bars. Since the simulation outputs are spaced quite widely in time, they cannot be measured directly by the change of the position of the bar major axis between outputs and we had to use the kinematic method proposed by Tremaine & Weinberg (1984). Although the velocity field of the bars seems quite regular, judging by the example in Fig. 13, these measurements are rather noisy due to the low resolution of the simulated galaxies. Still, for all the bars, the method yields values of the order of 10-20 km s^-1 kpc^-1, with high uncertainty, but similar to those obtained by Peschken & Łokas (2019) for the population of tidally induced and all bars in Illustris. These values are, however, on the low end of the range typically measured in real barred galaxies (Font et al., 2017).

The six galaxies whose properties are illustrated in Figs. 2-3 and 11-12 were selected as the most convincing examples of bars induced by the galaxy interaction with the cluster. These galaxies did not experience any significant merger with another subhalo since their accretion onto the cluster and the abrupt change in their morphology and kinematics coincides with one of their first pericenter passages. However, it is difficult to prove beyond reasonable doubt that their transformation is caused by their interaction with the cluster, as some other small accretion events or interactions (not recorded in the merger trees) are still possible, and the nature of bar instability may be to some extent stochastic (Sellwood & Debattista, 2009).

The picture painted by the analysis of these six examples is, however, strongly supported by the more statistical approach we presented in the previous subsection. The differences in the distributions of $T$ and $f$ values for the weakly and strongly evolved samples in Fig. 9 demonstrate that indeed on average as the galaxies orbit the cluster, their rotation tends to decrease, and their prolateness to increase, which supports the interpretation in terms of tidally induced bars.

Among the sample of strongly evolved galaxies in Fig. 9, in addition to the six objects already described in detail, there are seven other galaxies with $T>2/3,$ which suggests a bar-like shape. One of them (ID=11) is prolate but also quite spherical with axis ratios $b/a$ and $c/a$ of the order of 0.8, and so it cannot be considered a bar. The remaining six (ID=34, 45, 58, 66, 96, 105) are genuine bars, but their morphology and kinematics changes occur before or after the pericenter passage, or their stellar distribution was already a bit elongated before the interaction so they could be enhanced by the cluster but not necessarily induced by it. We note that there is no correlation between the strength of the bar, as measured by $A_{2,{\rm max}}$, and the ITF, because although strong tidal interaction can induce a bar, a repeated action of tidal forces can also reduce its strength later on by making the galaxy more spherical.

The remaining 14 galaxies of the strongly evolved sample have intermediate or low values of $T$ and one may wonder why they did not form bars despite their significant interaction with the cluster. These cases have very different histories, but their failure to produce strong bars by $z=0$ can be traced to one or a combination of the following reasons: (1) they had bars earlier, but they were weakened by $z=0$; (2) they did not possess enough rotation initially; (3) their pericenter distances were rather large, so the strength of the interaction was smaller; (4) their orbits in the cluster were retrograde, which is known to suppress bar formation (Łokas et al., 2015; Łokas, 2018; Peschken & Łokas, 2019). In particular, the three galaxies of this sample that remain oblate with $T\sim 0.1$ at $z=0$ with ID = 21, 90, 124 fall into the categories (3), (2)-(3), and (2)-(3)-(4), respectively.

Finally, we note that out of six galaxies with $T>2/3$ in the weakly evolved sample (green points in Fig. 9), four are quite spherical with high axis ratios, and two are more bar-like, but in all cases the transition to prolate shapes is not related to the pericenter passage, which is usually quite recent, but rather to earlier mergers and interactions. On the other hand, the unevolved sample (blue points in Fig. 9), which also contains a significant number of prolate objects, includes bona fide bars corresponding to the points with the highest $T>0.8$ (ID = 44, 55, 91). The formation of these bars is not related to the fact that the galaxies are infalling into the cluster and took place earlier.