Type I Shell Galaxies as a Test of Gravity Models

The host galaxy, a typical elliptical, is modeled as a luminous homogenous sphere of $5\,\mathrm{kpc}$ radius, and the dark matter contribution to the potential is modeled with a spherical virialized halo surrounding the stellar mass of the primary. We choose the $\mathrm{NFW}$ density profile that Navarro et al. (1997) proposed for describing the cold dark matter (CDM) halos of galactic structures (see also Kroupa et al. 2010). Given the critical density of the Universe, ${\rho }_{0}$, the halo density at distance r from the center is defined by

$$\begin{eqnarray}&&{\rho }_{\mathrm{NFW}}(r)=\displaystyle \frac{{\delta }_{{\rm{c}}}{\rho }_{0}}{r/{r}_{{\rm{s}}}{(1+r/{r}_{{\rm{s}}})}^{2}},\end{eqnarray} \tag{ 1 }$$

where the characteristic radius ${r}_{{\rm{s}}}$ and the parameter ${\delta }_{{\rm{c}}}$ are defined by the total mass of the halo. The mass within a certain radius is obtained by integrating the density over the volume,

$$\begin{eqnarray}&&M(r)=4\pi {\rho }_{0}{\delta }_{c}{r}_{{\rm{s}}}^{3}\ \left[\displaystyle \frac{1}{1+\ r/{r}_{{\rm{s}}}}+\mathrm{ln}(1+\ r/{r}_{{\rm{s}}})-1\right].\end{eqnarray} \tag{ 2 }$$

The virial mass of the halo, the total mass within the virial radius, ${r}_{\mathrm{vir}}$, is defined as

$$\begin{eqnarray}&&{M}_{\mathrm{vir}}=\displaystyle \frac{4\pi }{3}{{\rm{\Delta }}}_{\mathrm{vir}}{\rho }_{0}{r}_{\mathrm{vir}}^{3},\end{eqnarray} \tag{ 3 }$$

where ${{\rm{\Delta }}}_{\mathrm{vir}}{\rho }_{0}$ is equal to the critical density at which matter overcomes the cosmic expansion and collapses into a virialized halo. The value of ${{\rm{\Delta }}}_{\mathrm{vir}}$ for the local universe (redshift equal to zero) is approximately equal to 104.2 for ${\rm{\Lambda }}\mathrm{DCM}$ cosmology according to Dutton & Macciò (2014). Introducing the concentration parameter ${c}_{\mathrm{vir}}={r}_{\mathrm{vir}}/{r}_{{\rm{s}}}$ and setting $r={r}_{\mathrm{vir}}$ in Equation (2), the parameter ${\delta }_{{\rm{c}}}$ is obtained as

$$\begin{eqnarray}&&{\delta }_{{\rm{c}}}=\displaystyle \frac{{{\rm{\Delta }}}_{\mathrm{vir}}}{3}\displaystyle \frac{{c}_{\mathrm{vir}}^{3}}{\mathrm{ln}(1+{c}_{\mathrm{vir}})-{c}_{\mathrm{vir}}/(1+{c}_{\mathrm{vir}})}.\end{eqnarray} \tag{ 4 }$$

The relation between ${c}_{\mathrm{vir}}$ and ${M}_{\mathrm{vir}}$ in the local universe is obtained from simulations (Dutton & Macciò 2014) as

$$\begin{eqnarray}&&{\mathrm{log}}_{10}({c}_{\mathrm{vir}})=1.025-0.097{\mathrm{log}}_{10}\left(\displaystyle \frac{{M}_{\mathrm{vir}}}{{10}^{12}{h}^{-1}{M}_{\odot }}\right).\end{eqnarray} \tag{ 5 }$$

To achieve the goal of this study, which is to compare the theories under the same conditions, we completely focus on the theoretical expectations of each model, instead of using observational data. To model the dark matter halo of an elliptical galaxy, we use the Illustris cosmological simulation, which is based on ${\rm{\Lambda }}\mathrm{CDM}$ Cosmology. The contribution of the total baryonic mass in this simulation is derived from Figure 1 of Haider et al. (2016). Using the curve labeled as "Illustris, total baryons," for a structure with a given total halo mass, one can define the baryonic fraction of the structure. We choose two different masses for the dark matter halo to model the host galaxy. The characteristics of the models are described in Table 1.

Table 1.  Dark and Luminous Matter Properties of the Model Elliptical Galaxy

Models	${M}_{{\rm{b}}}$	${r}_{{\rm{b}}}$	${M}_{\mathrm{vir}}$	${r}_{\mathrm{vir}}$
	$({M}_{\odot })$	$(\mathrm{kpc})$	$({M}_{\odot })$	$(\mathrm{kpc})$
Halo model A	$5\times {10}^{11}$	5	1013	568
Halo model B	1011	5	1012	264
MOG	1011	5	⋯	⋯
MOND	1011	5	⋯	⋯

Note. ${M}_{{\rm{b}}}$ and ${r}_{{\rm{b}}}$ are the total baryonic mass and radius, respectively. ${M}_{\mathrm{vir}}$ is the total halo mass within ${r}_{\mathrm{vir}}$, the virial radius of the dark halo.

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MOG has been proposed by Moffat (2006) to explain galactic dynamics using the existing baryonic matter. Being a covariant extension of GR, this model is derived from the action principle that introduces two scalar fields and a vector field in addition to GR fields. The effective potential in the weak-field approximation of MOG is a combination of a strong Newtonian-like attraction and a Yukawa-like repulsive term with two additional parameters α and μ (Moffat & Rahvar 2013).

For a nonrelativistic test-mass particle in the distribution of matter $\rho ({\boldsymbol{r}})$, the gravitational acceleration ${\boldsymbol{a}}$ (${\boldsymbol{r}}$) is equal to

$$\begin{eqnarray}{\boldsymbol{a}}({\boldsymbol{r}}) & = & -G{\displaystyle \int }_{0}^{R}\rho ({\boldsymbol{r}}^{\prime} )\displaystyle \frac{({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )}{| {\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} {| }^{3}}\\ & & \times \,\{1+\alpha -\alpha {e}^{-\mu | ({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )| }(1+\mu | ({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )| )\}{d}^{3}{\boldsymbol{r}}^{\prime} .\end{eqnarray} \tag{ 6 }$$

For large scales compared to ${\mu }^{-1}$, the gravity is $(1+\alpha )$ times stronger than the Newtonian case, while for small scales, Newtonian dynamics is fully established.

This model has been successful in describing galactic rotation curves. Moffat & Rahvar (2013) fit theoretical rotation curves of galaxies to observational data and found the best-fit values for the free parameters to be $\alpha =8.89\pm 0.34$ and $\mu =0.042\pm 0.004\,{\mathrm{kpc}}^{-1}$. We use the same values in this paper as the universal model parameters. In addition, MOG has been shown to be consistent with ionized gas and temperature profiles of nearby clusters obtained by the Chandra X-ray telescope (Moffat & Rahvar 2014).

MOND is another alternative to the non-baryonic dark matter proposed by Milgrom (1983). Based on MOND, gravity or inertia does not follow the prediction of Newtonian dynamics for accelerations smaller than a specified threshold of ${a}_{0}\approx 1.2\times {10}^{-10}\,{\rm{m}}\,{{\rm{s}}}^{-2}$. For the case that the acceleration of a test-mass particle in the gravitational field is slower than the characteristic acceleration of the theory, ${a}_{0}$, the acceleration is related to the Newtonian acceleration ${a}_{{\rm{N}}}$ by $a=\sqrt{{a}_{0}{a}_{{\rm{N}}}}$. This regime is called "deep-MOND". In the intermediate range, the interpolating function $\mu (x)$ describes the transition from the Newtonian gravity to the deep-MONDian regime:

$$\begin{eqnarray}&&{a}_{{\rm{N}}}=a\mu (a/{a}_{0}),\end{eqnarray} \tag{ 7 }$$

where the standard form is

$$\begin{eqnarray}&&\mu (x)=\displaystyle \frac{x}{\sqrt{1+{x}^{2}}}.\end{eqnarray} \tag{ 8 }$$

The model recovers the Newtonian gravity for large accelerations in the strong-field regime.

Since its proposal in 1983, MOND succeeded in interpreting many observations without invoking the existence of dark matter. The most prominent observational pieces of evidence of MOND are galactic rotation curves, e.g., Begeman et al. (1991), the Milgromain dynamics tests for the spacetime scale-invariance (Milgrom 2009), the baryonic Tully–Fisher relation, and the mass discrepancy-acceleration relation (McGaugh 2004; Wu & Kroupa 2015), or alternatively, the radial acceleration anomaly (McGaugh et al. 2016; Lelli et al. 2017). For a major review see Famaey & McGaugh (2012).

The faster any star is when it leaves its host dwarf galaxy (the socondary), the farther it moves away, and the larger the shell it contributes to form. Thus, the surface brightness of a shell, determined by the number density of the stars in that shell, depends on the initial velocity distribution of its stars. When we assume the initial velocity of the released stars to have a Maxwellian distribution around the velocity of the secondary at the time of passage through the center of the host (v), the stellar speeds mainly lie in the range $v\pm {\rm{\Delta }}v$, considering the velocity dispersion they had in the dwarf before leaving it (${\rm{\Delta }}v$). This will limit the visibility of the large shells because of the small number density of the stars beyond this range.

Our goal is to find the positions where shell formation is possible. Thus, without loosing generality, one can assume that the energies of the released stars cover a continuous interval so that the motion of the stars poses all possible velocities between zero and infinity and the shells live forever once they are generated after a passage. This assumption is compatible with the statement by Bílek et al. (2013) and their Figure 3. As in the case of the limited velocity distribution for the stars, in this case, one has to keep in mind that the surface brightness of the shells should vary in time because the initial velocity distribution of the stars is not uniform. According to simulations, the surface brightness profile of the shells depends on the original size and velocity dispersion of the secondary in addition to the potential of the primary (Hernquist & Quinn 1988). Figure 2 schematically shows the effect of limitating the initial stellar velocities in the visibility of shells. The large shells will fade after a long time and escape from observational accessibility, although it is theoretically possible for them to occur. Moreover, very small shells would not be visible owing to the small number density of the stars. Studying the surface brightness profile of the shells in a quantified manner is beyond the scope of this paper.

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According to Equation (10), the relevant factor that determines the position of the shells is the gravitational field of the host galaxy, or equivalently, the acceleration of the particles in that field. Prior to the analysis of the shell formation mechanism in different models, it is useful to have a qualitative comparison between the accelerations that the secondary, or its liberated stars, undergo on their trajectory around the primary. This comparison between the halo model A, MOG, and MOND, is shown for a $5\,\mathrm{kpc}$ luminous sphere with uniform mass distribution in the left panel of Figure 3. One interesting point is that for the smaller scales, comparable to the non-dark matter scales, the MOND and MOG accelerations are almost equal and behave like the Newtonian case. On larger scales in the range between 10 and $100\,\mathrm{kpc}$, the MOG gravitational field is stronger than the others, so that the test particle will be more accelerated.

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Here we have a brief description for the asymptotic behavior of the accelerations in MOG and MOND. In the gravitational potential of MOG on scales much larger than the length scale of the theory, ${\mu }^{-1}\approx 24\,\mathrm{kpc}$, Equation (6) reduces to the form of a Newtonian gravity enhanced by a factor of $(1+\alpha )$. Thus, in this limit, which we call deep-MOG, the test particle acceleration behaves as $1/{r}^{2}$. Moreover, according to Equation (7), the acceleration of a test particle in the deep-MOND regime changes as $1/r$. Defining the characteristic scale of deep-MOND as

$$\begin{eqnarray}&&{r}_{0}=\sqrt{{GM}/{a}_{0}},\end{eqnarray} \tag{ 12 }$$

the ratio of acceleration in MOG over MOND is given by ${a}_{\mathrm{MOG}}/{a}_{\mathrm{MOND}}=(1+\alpha ){r}_{0}/r$. For $r\lt (1+\alpha ){r}_{0}$, we would therefore expect ${a}_{\mathrm{MOG}}\gt {a}_{\mathrm{MOND}}$ and for $r\gt (1+\alpha ){r}_{0}$, ${a}_{\mathrm{MOG}}\lt {a}_{\mathrm{MOND}}$. The ratio of the accelerations between MOG and MOND is shown in the right panel of Figure 3 for the two uniform spheres with different luminous masses. We have this $1/r$ property for the ratio of accelerations for $r\gt 30\mbox{--}40\,\mathrm{kpc}$, depending on the mass of luminous matter. On shorter scales ($r\ll (1+\alpha ){r}_{0}$), we have to take into account the full acceleration formula for MOG and MOND.

To find the shell radii at any time, one first needs to trace the motion of the incoming galaxy, which is released to move radially toward the host center. This is done by solving the equation of motion of the dwarf and the individual stars under certain conditions. The incoming galaxy moves as a damped oscillator around the center of the primary and releases a part of its mass each time it passes through the center of the host galaxy and forms a new generation of shells. The oscillations continue until the dwarf loses all its mass or completely merges with the host and the centers of density coincide. Multi-generation shell structures, caused by multiple crossings, commonly form in self-consistent simulations (Seguin & Dupraz 1996; Bartošková et al. 2011; Cooper et al. 2011). It has been shown in test-particle simulations (Dupraz & Combes 1986; Ebrova et al. 2012; Bílek et al. 2013) that for a minor merger model, the observed wide range of shell radii requires the shell system to be formed in several generations. In this study, consistently with the simulations, we choose the dwarf to lose a part of its mass in every passage and all of the remaining mass in the fourth transit, such that after creating four shell generations, dissolution occurs.

Releasing the secondary at rest from different initial distances, Figures 4 and 5 show the trajectory of the secondary and its core debris after the first passage and compare MOG and MOND with the dark matter halo models A and B, respectively. We let the secondary start its motion from three different points: the edge, the inside, and the outside of the dark halo. Figure 4 is related to halo model A, where the dark mass is 10 times more massive than in halo model B. In this case, where the virial radius of the halo is equal to $568\,\mathrm{kpc}$, the starting points have been chosen to be $300,568$, and $1000\,\mathrm{kpc}$, respectively. For Figure 5 these distances take the values of $100,264$, and $500\,\mathrm{kpc}$, and the virial radius of the dark halo is $264\,\mathrm{kpc}$.

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The mass ratio is defined as $q={M}_{{\rm{p}}}^{* }/{M}_{{\rm{s}}}$, where ${M}_{{\rm{p}}}^{* }$ is the baryonic mass of the primary and ${M}_{{\rm{s}}}$ is the total initial mass of the secondary. In the first two columns of the Figures 4 and 5, we compared two different cases for all models such that q takes the values of 10 and 100. We also show the results for $q\to \infty$, representing the condition with relatively no dynamical friction in the last columns. As a consistency check, it has been shown in the last columns of Figures 4 and 5 that for a dwarf with sufficiently low mass, the MOG and MOND potentials act the same for large infall distances, as do the halo models. This identical behavior of the models on large scales was expected from comparing the accelerations in different models, which is demonstrated in Figure 3 (see Section 4.1). Owing to dynamical friction in different matter halo models, having a more massive secondary means greater orbital energy loss within the primary and more slowly released stars, which leads to a brighter system of small shells on a shorter timescale.

When we only consider the red solid curves that represent the dark matter halo models in Figures 4 and 5 for different initial conditions, we can investigate the effect of the existence of dark mattar in the dwarf galaxy by comparing the first two columns. As the dwarf galaxy is assimilated to a point mass in our study, the only important parameter is the total mass of the dwarf apart from its distribution. In this case, we assume the luminous mass of the dwarf galaxy to be the same in both columns, that is, one hundredth of the elliptical galaxy luminous mass. Therefore, we can translate q = 100 to represent a dwarf without dark matter and q = 10 to a dwarf with a dark matter halo whose dark mass is 10 times its luminous mass. Comparison between these two cases shows that the presence of dark matter in the dwarf causes faster merging, ending in tighter shell generations. In the case q = 100 when the second-generation shells appear, the first-generation shells are almost old and wide and have faded because of the long time between the core passage through the center.

Shell patterns at any time after an encounter are derived when we follow the oscillation of the released stars and find the positions where they spend more time, as shown in Figure 1. The distance between the shell generations is controlled by dynamical friction, which depends on the initial mass ratio between the dwarf and the elliptical and the mass-loss ratio of the secondary in each passage. It is clear from Figures 4 and 5 that in the MOND potential (because the dwarf galaxy moves through the stellar body of the host), dynamical friction has a tiny effect on the movement of the secondary core. However, it is larger compared to MOG because of smaller relative velocity, and it strongly influences the motion in the dark matter halo models. It slows down the core so that a massive dwarf quickly sinks into the host and all of the stars generate a bright shell system in only one or two close passages. In the modified gravity models, on the other hand, it takes significantly longer for complete merging to happen, and the shells will be less bright than in the halo models. Therefore, in the same encounter conditions of the two galaxies, shells appear on a shorter timescale in the presence of dark matter. In this case, the resulting bright shells are mostly composed of low-velocity stars and so appear over a smaller range of distances. Higher velocity stars generate large faint shells. Therefore, the observation of small bright shells should be accompanied by large faint ones because dynamical friction dominates in the halo models. In the case of MOG, the velocity of the core remnant after the first passage does not decrease much, and the next shell generations would have a similar velocity distribution as the first. The first-generation shells repeat with approximately the same total luminosity and therefore decreasing surface brightness since later shell orders have larger radii. Successive shell generations, however, produce dimmer shells because fewer stars are liberated from the secondary at each passage.

Figure 6 shows the shell structures in the first-generation shells formed in different models of the gravitational potential. We assumed that the stellar velocities take all possible values from zero to infinity (see Section 3.1), therefore the initial conditions of the impact do not play a role in one-generation shell patterns. A comparison between halo models A and B in Figure 6 shows that the size of the shells in the dark matter scenario obviously depends on the amount of dark matter of the primary galaxy, so that this can be used to estimate the total mass of the dark halo (but not the dark matter distribution) in the host galaxy. The appearance times of the the next shell generations in the presence of dark matter depend on the initial mass of the dwarf. From the observational point of view, measuring the integrated surface brightness of the shells would give the approximate initial dwarf mass before the encounter and make the dynamical friction calculation more accurate in case studies. This figure also shows that there is a degeneracy with time and position in the first shell generation patterns in halo model B and MOG and MOND. The exact shell positions therefore cannot be used to verify the models without studying the next shell generations. As we discussed in Section 4.2, the distance of the next-generation shells depends on the existence of dark matter. Thus, one has to label a given shell to a specific generation in the observational data. This is not possible unless the shell ages are defined. Even in this case, without knowledge of the initial distance of the dwarf, MOG and MOND cannot be distinguished from each other because of the negligible dynamical friction.

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To find a criterion to more easily distinguish between the models, one needs to employ the dynamics of the shells as derived from the equation of motion of the stars in different potentials. The scale-dependent variation of the accelerations in the different models leads to different dynamics of the shells (see Section 4.1). We compare the evolution of the relative distances of the shells in our models defined as

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}r}{\bar{r}}=2\left(\displaystyle \frac{r-{r}^{{\prime} }}{r+{r}^{{\prime} }}\right),\end{eqnarray} \tag{ 13 }$$

where r and ${r}^{{\prime} }$ are the shell radii from the center of the primary galaxy. According to Figure 1, the outermost shell (${l}_{+1}$) always contains the stars that have completed three-fourths of their period. The stars in the second shell in the same direction (${l}_{+2}$) have passed seven-fourths of their oscillation period. The shell ${l}_{-1}$ is the first on the opposite side from the originally incoming dwarf. We show the time evolution of ${\rm{\Delta }}r/\bar{r}$ for the shells in different models in Figure 7 between ${l}_{+1}$ and ${l}_{+2}$ in the left panel and between ${l}_{+1}$ and ${l}_{-1}$ in the right panel. As we can see in Figure 7, the relative distances are significantly different between the different gravitational theories for times longer than $2\,\mathrm{Gyr}$ after the first encounter. Thus, measuring the relative distances of the outer shells on the two sides of the galaxy long enough after the encounter is a potentially powerful tool to decide which gravity theory has governed the merger process. In addition, among the modified gravity theories, MOG and MOND can be distinguished from each other via this probe. Being dimensionless and independent of the distance of the galaxy is the advantage of this probe, in comparison to the position of the shells.

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According to Equation (11), the expansion (phase) velocity of the shells can also be used as a probe to verify the existence of dark matter or to decide whether it is necessary to modify the theory of gravity. Figure 8 depicts this quantity for ${l}_{+1}$ and ${l}_{-1}$ for the first-generation shells. It shows that an almost constant shell expansion velocity is the property of MOND, while in MOG and the dark matter halo models, the expansion velocity decreases with time. Moreover, the more massive the dark matter halo surrounding the galaxy, the faster the expansion of the shells.

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As a practical example, we take the case of NGC 3923, a unique Type I shell galaxy, which is surrounded by many stellar shells (Bílek et al. 2016). There are 18 redshift-independent distances in the NASA/IPAC extragalactic database for this galaxy, with a median of $20.8\,\mathrm{Mpc}$.⁴ If we take the typical expansion velocity of the outermost shell for this galaxy to be $\approx 100\,\mathrm{kpc}\,{\mathrm{Gyr}}^{-1}$, the metrical displacement rate of the shell would be ${\rm{\Delta }}{\rm{\Theta }}/{\rm{\Delta }}t\approx 1\,\mathrm{mas}\,{\mathrm{yr}}^{-1}$. The typical evolution of velocity for this shell would be ${{\rm{\Delta }}}^{2}{\rm{\Theta }}/{\rm{\Delta }}{t}^{2}\approx {10}^{-11}\,\mu \mathrm{as}\,{\mathrm{yr}}^{-2}$. Ebrova et al. (2012) developed the usage of the shell spectral line profile to measure the expansion velocity of the shell. They used an analytical approach and test-particle simulations to predict the line-of-sight velocity profile across the shell structure and showed that spectral peaks are split into two, giving a quadruple-peaked line profile that enables us to directly measure the shell expansion velocity.

To compare the model predictions of the relative distances of the shells (Equation (13)) with observational data, we use the data of Bílek et al. (2016). They observed 42 shells around NGC 3923 with Megacam, which is the highest number of shells among shell galaxies. The summary of their data for the spatially largest shells is given in Table 2 and Figure 9. The luminous mass of halo model A is $5\times {10}^{11}{M}_{\odot }$ (Table 1), which is consistent with the value reported for NGC 3923 by Bílek et al. (2013). Halo model A should therefore be valid for the mass profile when we have dark matter.

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According to the merger theory for shell formation, the shells should be interleaved in radius. Thus, a quick look at the data reveals that there must be some missing shells between S₂ and S₃ and also S₃ and S₄ in the northeastern side of the galaxy. Any further analysis for the largest shells would be possible after observing the missing shells. However, S₂ and S₄ are possibly uncertain shells according to their observational prominence, as are shells S₅ and S₆. Therefore we consider only the most prominent data, i.e., S₁, S₃, and S₇, as certain shells, as if there were no shells in the other reported positions. That is, we discard shells with prominence degrees 3 and 4.

Figure 10 shows the model comparison with the data. The model relative distances are calculated for the first-generation shells. In the right panel the relative distance of the largest opposite shells (S₁ and S₃) are plotted. In this case, MOG is more consistent with the data. The left panel shows the relative distance of the two outermost successive shells. The radial distance of S₁ and S₇ is very large such that it is possible for S₇ to belong to another shell generation. Thus, inconsistency with the models is possible. A possibility is that there should be an unobserved shell at a distance larger than S₇ in the northeastern side of the galaxy. As a possible candidate for this shell we considered S₅ and checked the consistency in the models and the data. MOND is more consistent for this case. However, for a precise comparison, one should have detailed N-body simulations for each model, deliberating the characteristics of this special galaxy.

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