Ram Pressure Stripping and ISM disc Truncation
: Prediction vs. Observation

We observationally define $R_{\rm t}$ at the radius where the H i surface density drops to $1~{}M_{\odot}~{}{\rm pc}^{-2}$. For the sample undergoing active RPS, more than one $R_{\rm t}$ are required to consider the asymmetric H i morphology. Therefore we measure $R_{\rm t}$ toward eight directions as described in Section 3.1.2. The limit of $1~{}M_{\odot}~{}{\rm pc}^{-2}$, which is a commonly adopted level to define H i extents, is also used in this study as it is found to well represent the level where sharp truncations are observed. Consequently, $R_{\rm t}$ to the direction with low surface density tails in a stripped gas trailing region can be underestimated. However, those tail features are not the main topic of interest in this study.

The important characteristics of active RPS commonly found in H i observations are 1) highly disturbed H i morphology and 2) (regional) truncation within the extent of the stellar disc. The galaxies with these characteristics among the sample are located within a clusto-centric distance of $R_{200}$ (1.55 Mpc for Virgo) at which the ICM surface density estimated by the $\beta$-model is more or less consistent with that observed using X-ray surface brightness. Thus for active RPS cases, i.e., Class II galaxies of Yoon et al. (2017), ram pressure can be better constrained than the other classes in the early or later stages of RPS, making them ideal targets to verify the GG’s formula.

In the first and third columns of Fig. 6, the total H i intensity contours of active RPS galaxies are shown by white contours overlaid on the SDSS $\it{i-}$band images. The thick purple line indicates the H i surface density of $\rm 1~{}M_{\odot}~{}{\rm pc}^{-2}$. As mentioned, the observational $R_{\rm t}$ defined at the H i surface density of $\rm 1~{}M_{\odot}~{}{\rm pc}^{-2}$ is generally a good representative of the H i extent in the direction where sharp H i truncation is seen. For each galaxy, $R_{\rm t}$ is measured eight individual segments of an opening angle of 45^∘, i.e., four grey and four yellow sectors. Half of the disc inferred to be facing directly the ICM wind, i.e., the leading side, is shaded in yellow, whereas the tail side is shaded in grey. For reference, the arrow in the upper-right corner of each panel points in the opposite direction of M87. This is a likely ICM wind direction if the galaxy is on the way to the Virgo centre, radially falling in the plane of the sky. It is also the direction in which stripped gas features are expected to roughly point. However, the motion along the observer’s line-of-sight and the galaxy’s recent core-crossing, non radial orbits, or locally non-static ICM can result in some discrepancies in the direction between the arrow and the stripped gas tail. Therefore instead of the location of M87, we infer the effective direction of the ICM wind (i.e., ram pressure) based on the H i compression and tails.

For each galaxy, $R_{\rm t}$ is measured for eight directions using the H i radial profiles of eight sectors. For the highly inclined systems ($i>75^{\circ}$), the H i extent along the stellar minor axis is comparable with the synthesized beam size $\sim$1 kpc. In such cases, $R_{\rm t}$ along the minor axis is measured to be too large to represent the H i truncation, and thus we do not use four $R_{\rm t}$’s along the minor axis for highly inclined galaxies in further analysis. The second and fourth columns of Fig. 6 show the comparisons of $R_{\rm t}$ from the predictions based on (1) (Section 3.1) and the measurements from the observations (Section 3.2). The diagonal grey line is an equal line of the two measurements. The $R_{\rm t}$ of the leading and tail sides are shown in yellow and grey, respectively. The $R_{\rm t}$ from the azimuthally averaged radial profile is shown in black for comparison. The measurement uncertainties in $R_{\rm t}$ are presented as vertical/horizontal bars. The centre of each predicted $R_{\rm t}$ represents the mean of the lower and upper limits for the case with an encounter angle of 45^∘. Meanwhile, the size of each bar shows the possible $R_{\rm t}$ for the entire angle range explored in this study.

For most active RPS cases, the observed $R_{\rm t}$ agrees with the predicted $R_{\rm t}$ within the measurement uncertainties at least for one region of the leading side (yellow colour). The true gas truncation radius is likely to be the shortest $R_{\rm t}$ from one of the leading sectors. This approach is similar to how Gullieuszik et al. (2020) defined their $R_{\rm t}$ as the shortest extension of the H$\alpha$ emission along the galaxy major axis. NGC 4298, NGC 4302, and NGC 4522 generally show a great agreement between $R_{\rm t,obs}$ and $R_{\rm t,pred}$ for all directions. The mean of $\Delta R_{\rm t}/R_{\rm t,obs}$ is $\lesssim$ 0.06 for these three galaxies. On the other hand, half of the active RPS cases show discrepancies on the tail side (i.e., the opposite side of the interface with the ICM wind). They are NGC 4330, NGC 4388, NGC 4402, NGC 4501, and NGC 4654. For this group, the mean $\Delta R_{\rm t}/R_{\rm t,obs}\sim$ 0.07 on the leading side, whereas it is 0.13 on the tail side. This implies that the GG’s equation predicts a truncation radius reasonably well for a diffuse gas like H i on the side which is directly affected by the ICM wind. That is, broadly speaking, RPS can be understood as a process of the ICM momentum pushing the ISM from the galactic disc. Furthermore, this supports the idea that the shortest $R_{\rm t}$ on the leading side is most probable to represent the true gas truncation radius for further similar works.

On the one hand, 2 out of 10 active RPS cases – NGC 4424 and NGC 4694 – show relatively large deviations on a one-to-one line. NGC 4424 tends to show overall larger offsets between the predicted $R_{\rm t}$ and the observed $R_{\rm t}$ in all directions compared to the others in the same group. This galaxy has a hint of minor merging (Kenney et al., 1996; Cortés et al., 2006) which may have disturbed the potential, and thus the estimation of its anchoring pressure may not be straightforward. In addition, this galaxy is located close to M49, the centre of a sizable sub-cluster which is in the process of merging to the main Virgo cluster from the south (Gavazzi et al., 1999; Mei et al., 2007). In fact, the systematic velocity makes this target more likely to be part of the M49 group (Sancisi et al., 1987; Chung et al., 2009). Indeed, the possibility that this galaxy had interacted with the M49’s hot halo has been suggested by Biller et al. (2004). If so, our M87-based estimation must not be suitable for this case. Thus the merging history and membership may result in some discrepancies in the comparison of $R_{\rm t}$ in this case. NGC 4694 also shows somewhat different patterns from the others in the sense that most of the measurements are offset from the grey line. NGC 4694 has a low surface brightness neighbour which is connected through an H i bridge (VCC 2062; van Driel & van Woerden, 1989; Chung et al., 2009). Thus, our simple comparison is unlikely to work properly for this case as well.

In Fig. 7, we compare the samples undergoing various RPS stages. For the pre-peak and active RPS cases (Class I and II of Yoon et al. (2017)), the shortest $R_{\rm t}$ in the leading side (i.e., the opposite side of extended tails) of each galaxy is plotted. Again, the measurements along the minor axis are not considered when $i$>75^∘. For the post-active RPS cases with relatively symmetric H i morphology, a single $R_{\rm t}$ from the azimuthally averaged radial profile is plotted for each galaxy. Among this group, NGC 4293 is excluded as the H i surface density does not exceed $\rm 1~{}M_{\odot}~{}pc^{-2}$ throughout the disc, and its $R_{\rm t}$ is not defined.

First of all, it is quite noticeable that $R_{\rm t}$ decreases from pre-peak to active, and then post-peak RPS cases in both predictions and observations. However, the pre- and post-peak RPS groups show somewhat larger deviations from the equal-$R_{\rm t}$ relation compared to the active RPS cases. On average, the predicted $R_{\rm t}$ of the pre-peak RPS group is smaller than its observationally measured $R_{\rm t}$ by 0.27 $R_{\rm 25}$, whereas the predicted $R_{\rm t}$ is larger than the observed $R_{\rm t}$ by 0.16 $R_{\rm 25}$ for the post-peak RPS group. That is, the other classes deviates from the one-to-one correlation by a few times on average. The mean $R_{\rm t}$’s of individual groups are marked with star symbols in the figure.

The pre-peak RPS cases are found with overall larger offsets compared to the other two groups including four cases (57%) with a noticeably large offset as presented in Table 1 and Fig. 7. In particular, their observed $R_{\rm t}$ is systematically larger than the predicted $R_{\rm t}$, indicating that their H i is not stripped as much as expected based on their properties and locations. It means our estimation of the ram pressure for this group is generally overestimated. In fact, most of the sample in this group is located at the clusto-centric distance where no detectable X-ray emission is present (Fig. 2), yet we estimate the surrounding ICM density by extrapolating the $\beta$-model. The slope of the modelled ICM density could be less steep than that in reality, which might have resulted in the overestimation of the ICM density for this group, so as the strength of ram pressure. Alternatively, at the stage where the gas starts being pushed strongly by ram pressure yet not stripped, the H i density across the disk can be overall decreased, proceeding to the next stage (i.e., Class II). This effect may lead to small predicted $R_{\rm t}$ but still a large observed $R_{\rm t}$ for some pre-stripping cases. Indeed, the three extreme outliers in Fig. 7 are NGC 4294/9 and NGC 4396 which have been identified with a long one-sided tail, i.e., the sign of quite strong ram pressure among the same group (Chung et al., 2007).

On the one hand, almost 80% of the post-peak RPS cases show offsets to the opposite, i.e., $R_{\rm t,pred}$ is generally larger than what is actually measured (see Table 1). The mean $\Delta R_{\rm t}/R_{\rm t,obs}$ for these outliers is 0.57, which is about seven times the value for the other two post-peak RPS cases, NGC 4569 and NGC 4607. Among them, four cases even deviate by more than 1$\sigma$ from one-to-one line in Fig. 7. This is consistent with the inferred history for this group that they have gone through strong RPS in the past in the higher density region as indicated by the severe H i truncation, and they have moved out to lower-density regions.

Our results suggest that the understanding of a ram pressure stripped galaxy from the perspective of GG’s formula is quite secure when the target is undergoing active RPS in the environment where the surrounding pressure due to the ICM can be well defined. If the galaxy is not close to the peak-pressure or RPS is not a dominant mechanism affecting the target, more caution would be needed to assess the impact of ram pressure on the target.

There have been quite a number of challenges to analytically understand GG’s relation by comparing ram pressure and anchoring pressure (e.g. Abadi et al., 1999; Vollmer et al., 2001; Roediger & Hensler, 2005; Jáchym et al., 2007; Steinhauser et al., 2016). Among many previous studies, our work must be most comparable with Köppen et al. (2018) considering that both studies use the same H i data and estimate pressures in similar ways. However, there are further details for which the two studies approach differently, and the comparison of our results with Köppen’s is one way to confirm how broadly GG’s formula can be adopted to interpret the observations in spite of many uncertainties.

In Köppen et al. (2018), they estimated $P_{\rm loc}$, the expected local ram pressure at a given location of individual galaxies, and $P_{\rm cfg}$, the maximum ram pressure that each galaxy could have experienced (see their Fig. 14 of Köppen et al. (2018)). Their $P_{\rm loc}$ is comparable with our $P_{\rm ram}$ in a sense that both have been calculated assuming a smooth and spherically symmetric ICM distribution based on the best fit $\beta$-model of the Virgo cluster (Schindler et al., 1999; Vollmer, 2009; Vollmer et al., 2001) at a given location. However, the two studies use different clusto-centric distance measurements. As in Jaffé et al. (2015), we first scale the projected distance from M87 by $\pi$/2 to estimate the 3D angle, taking the average of the possible projection angle (i.e., $r=cos\phi~{}r_{\rm 3D}$ where $0<\phi<\pi\mathbin{/}2$, and $r_{\rm 3D}\sim(\pi\mathbin{/}2)~{}r$ on average). As stated, we adopted a Virgo distance of 16.5 Mpc to convert $r_{\rm 3D}$ in the sky to a physical distance. Meanwhile, Köppen et al. (2018) used the projected distance from M87 converted with a uniform distance of 17 Mpc for most of the sample, and adopted the stellar Tully-Fisher distance for 11 galaxies from Cortés et al. (2008). Regarding the galaxy’s velocity, whereas we use a velocity range for each system from the Keplerian velocity to the escape velocity, Köppen et al. (2018) used only the escape velocity and even this value may differ from ours depending on the assumed Virgo potential.

The x-axis of Fig. 8 shows the difference between our $P_{\rm ram}$ and $P_{\rm loc}$ of Köppen et al. (2018) which is normalized by our measurement uncertainty of $P_{\rm ram}$ (i.e., $|P_{\rm ram}-P_{\rm loc}|/\Delta P_{\rm ram}$). $\Delta P_{\rm ram}$ has been evaluated by considering the potential encounter angle from 0^∘ to 60^∘, and the relative velocity ranges from Keplerian velocity to escape velocity. In spite of the differences in distance and velocity, $|P_{\rm ram}-P_{\rm loc}|$ is comparable with $\Delta P_{\rm ram}$ for most of the cases. Particularly for the active stripping cases which are the main targets of this study, the mean $|P_{\rm ram}-P_{\rm loc}|/\Delta P_{\rm ram}$ is $\sim$ 0.84, and the ram pressure from the two studies is generally in good agreement.

Meanwhile, $P_{\rm cfg}$ of Köppen et al. (2018) can be compared with our $P_{\rm anch}$ although there are some important differences. Köppen et al. (2018) estimated the maximum ram pressure that the galaxy could have experienced to be observed with the current H i radius under its centrifugal force. For the pre-stripping H i content, they assumed that the gas surface density follows a Miyamoto-Nagai profile (Miyamoto & Nagai, 1975) with constants satisfying the initial H i mass and initial outer H i radius. On the other hand, our anchoring pressure is derived as a function of radii for each galaxy using the radial stellar and gas surface densities based on the SDSS and VIVA observations. In addition, Köppen et al. (2018) took the azimuthally averaged H i radius from Chung et al. (2009) as a stripping radius whereas we make considerations of the H i asymmetry using the isophotal H i radii measured along 8 directions which is particularly important for the active stripping cases (i.e., the main targets of this study), and some early stripping cases.

The y-axis of Fig. 8 shows the difference between our $P_{\rm anch}$ and $P_{\rm cfg}$ of Köppen et al. (2018) which is normalized by our measurement uncertainty of $P_{\rm anch}$ (i.e., $|P_{\rm anch}-P_{\rm cfg}|/\Delta P_{\rm anch}$). $\Delta P_{\rm anch}$ has been evaluated considering the spatial resolution of the VIVA data, i.e, assuming the minimum/maximum $R_{\rm t}~{}\pm\approx 15\arcsec$. Compared to $P_{\rm ram}$, $|P_{\rm anch}-P_{\rm cfg}|/\Delta P_{\rm anch}$ ranges quite broadly from very small values for which the two studies are extremely well consistent to quite large values for which the difference between the two studies is much bigger than our measurement uncertainties. All four outliers are the cases showing significant H i asymmetry. Among these, three galaxies (NGC 4254, 4294, and 4535) are the ones that do not show H i truncation within the extent of the stellar disc yet the H i is quite asymmetric (i.e., early stripping cases). The fourth outlier is NGC 4522 that is undergoing active stripping and shows a very disturbed H i distribution. This comparison thus shows that the consideration of gas asymmetry can make quite a huge difference in the estimation of anchoring pressure, especially for the cases with a H i disc that is more extended than the stellar disc along all or some directions. However, for most of the active stripping cases which are severely stripping within the extent of the stellar disc, the anchoring pressure from the two studies is in quite good agreement, showing discrepancies that is comparable with our measurement uncertainty or less.

To summarize, in spite of subtle differences in estimating both ram pressure and anchoring pressure, GG’s relation seems to hold for the cases which show clear evidence of H i truncation within the extent of the stellar disc. For those with highly asymmetric gas distribution yet not stripped severely, extra caution may be needed in the interpretation as the anchoring pressure across the disc may significantly vary.

To estimate the expected $R_{\rm t}$, we adopted the $\Sigma_{\rm g}$ from the observed H i surface radial profiles regardless of the galaxy’s RPS status. However, the gas column density before stripping is likely higher, and the mean gas density and size must be continuously decreasing during the RPS process as seen in Fig. 9 (see also Cayatte et al., 1994). That is, the actual density/extent right before the galaxy got stripped and reaches the current status must be slightly higher/larger than what we assumed. In this section, we therefore test how the gas conditions before stripping would change our results. Since it is highly uncertain by how far the gas was denser right before a galaxy is observed to be as now, we do the test adopting an exponential profile as Jaffé et al. (2018) do for gas-rich disky galaxies.

The gas has an exponential profile described as follows:

where $R_{\rm s,gas}$ is the gas scale length, $\Sigma_{\rm 0,gas}(=M_{\rm gas}/2\pi\cdot{R_{\rm s,gas}}^{2}$) is the central gas surface density, and $M_{\rm gas}$ is the total gas mass. The gas scale length, $R_{\rm s,gas}$, is assumed to be 1.7 times larger than the stellar disc based on the non-H i-deficient Virgo spirals (Cayatte et al., 1994). The cool gas (H i and H₂) mass fraction to the stellar mass is adopted from the known relation for disc-dominated galaxies with B/T$<$0.4 as follows (Popping et al., 2014; Gullieuszik et al., 2020):

where $M_{\rm gas}$ and $M_{\rm\star}$ are the masses of cool gas and stars, respectively. In Fig. 9, the predicted radial $\Sigma_{\rm g}$ from the model is compared with the observed H i surface density by RPS stages.

As expected from the concept of GG, Fig. 9 clearly shows that the stripping proceeds from outside to inside on a gas disc. In addition, it is also intriguing to see a dramatic decrease of the gas surface density in the inner part ($\lesssim 0.5R/R_{\rm 25}$). This evidently shows that the inner gas disc can be severely affected by the ram pressure in spite of a large potential and higher density as reported in both simulations (e.g. Vollmer et al., 2001; Steinhauser et al., 2016; Lee et al., 2020) and observations (e.g. Wang et al., 2014; Mok et al., 2017; Lee et al., 2017). It should be noted that molecular gas is not included in the observed gas surface density in this comparison. If the molecular phase is added, the gas surface density can be higher by a factor of a few tens in the central region (e.g. Bigiel et al., 2008). However, the molecular gas density rapidly drops with increasing radii, and around $\sim$0.5 $R_{\rm 25}$, two phases become comparable and the atomic phase takes over at larger radii. Therefore, at $R_{\rm t}$ of active and post-peak RPS cases ($\sim$0.54 $R_{\rm 25}$ and $\sim$0.37 $R_{\rm 25}$), the total surface density can be several times higher than that of atomic gas alone, which will lead to a few times higher anchoring pressure than what we estimated. However, a few times higher gas surface density results in the increase of $R_{\rm t}$ only by a kilo-parsec or so as seen in Fig. 5. This is smaller than the usual uncertainties in $R_{\rm t}$, and thus the presence of molecular gas is thought to hardly affect our results such as Fig. 7 and Fig. 10.

In Fig. 10, the comparisons of the re-estimated expected $R_{\rm t}$ and the H i observed $R_{\rm t}$ are presented. By using the pre-stripping $\Sigma_{\rm g}$, the expected $R_{\rm t}$ is overall shifted up when compared with Fig. 7. As discussed in Section 4.2, $\Sigma_{\rm g}$ of some pre-stripping cases is expected to be lowered. Thus this group agrees better in this relation in Fig. 10 compared with in Fig. 7. However, this assumption (i.e., $\Sigma_{\rm g}$ follows that of the model) is another extreme condition since $\Sigma_{\rm g}$ at the time of stripping is more likely to be somewhere between the model and the currently observed $\Sigma_{\rm H\scaleto{\rm I}{4pt}}$. Indeed, the expected $R_{\rm t}$ based on the model does not seem to be too realistic. The ranges of expected $R_{\rm t}$ are comparable regardless of the RPS stage unlike in Fig. 7 where $R_{\rm t}$ is found to decrease from early, active to past RPS stages as the clusto-centric distance of each group also decreases overall.

These indicate that using $\Sigma_{\rm g}$ from H i imaging data for the relevant estimation should be reasonable. Although the observed H i surface density may result in the underestimation of the predicted $R_{\rm t}$, it must still be closer to the actual $\Sigma_{\rm g}$ at the time of stripping than the model of a gas disc that is intact at all.

In the estimation of the ram pressure experienced by each galaxy, $\rho_{\scaleto{\rm ICM}{4pt}}$ and $v_{\rm rel}$ have been inferred based on the projected clusto-centric distance of the target. However, the galaxies that have gone through strong RPS a while ago, i.e., Class III of Yoon et al. (2017), may have lost the gas somewhere closer to the cluster centre than where they are currently located. Therefore, our estimation of ram pressure for those galaxies may not be sufficient to explain how deficient and truncated their gas discs are. As seen in Fig. 7, the expected $R_{\rm t}$ of most post-peak RPS cases are indeed found to be larger than the observed $R_{\rm t}$.

In order to probe how deep the post-peak RPS cases have been in the cluster, we estimate the closest approach to the cluster centre (periapsis) which potentially better explains their H i properties. Assuming that the H i disc of each galaxy has reached its observed size around its periapsis, we first find the ram pressure which balances with the anchoring pressure at the observed H i radius. We then find the clusto-centric distance which matches with this ram pressure.

Fig. 11 shows the estimates for the periapsis of the post-peak RPS cases. The upper panel compares the possible periapsis ($d_{\rm peri}$) with the projected clusto-centric distance ($d_{\rm M87}$) of the current location. From left to right, galaxies are shown in the order of increasing $d_{\rm M87}$. Except for one galaxy (NGC 4607), all the post-peak RPS cases are found to have been closer than $\sim$0.5$R_{\rm 200}$ from M87, suggesting that they experienced stronger ram pressure in the past. In addition, those suggested as back-splashed galaxies by Yoon et al. (2017) indeed show the largest discrepancies between $d_{\rm peri}$ and $d_{\rm M87}$ (shaded in red). In the case of NGC 4607, it is highly inclined ($i>75^{\circ}$) and the vertical scale height of its H i disc is comparable to the H i resolution, and the derived radial profile, hence $R_{\rm t}$ is highly uncertain.

The lower panel of Fig. 11 shows the comparison of the possible $v_{\rm rel}$ to the cluster centre at the periapsis ($v_{\rm peri}$) and the H i observed line-of-sight velocity to the cluster mean. We first calculated the potential velocity that these galaxies might have had in the past at the periapsis. For this, we adopt $P_{\rm ram}$ which balances with $P_{\rm anch}$ (i.e., $v=\sqrt{\frac{P_{\rm ram}}{\rho_{\scaleto{\rm ICM}{4pt}}(d_{\rm peri})}}=\sqrt{\frac{P_{\rm anch}(R_{\rm t,obs})}{\rho_{\scaleto{\rm ICM}{4pt}}(d_{\rm peri})}}$). In addition, we assume that the actual velocity can range between this Keplerian velocity and the escape velocity at the periapsis of each galaxy, which is shown in Fig. 11. All the galaxies in this group are inferred to have had higher velocities than the currently observed line-of-sight velocities. In particular, the back-splashed cases are thought to have been affected by ram pressure that is much stronger than what they are experiencing in their current location by a factor of up to $\sim$100 as inferred from $d_{\rm peri}$ and $v_{\rm peri}$.