# Simulating the arrival of the southern substructure in the galaxy cluster
Abell 1758

Rubens E. G. Machado Departamento Acadêmico de Física, Universidade
Tecnológica Federal do Paraná, Rua Sete de Setembro 3165, Curitiba, Brazil
<EMAIL_ADDRESS>Ricardo C. Volert Departamento Acadêmico de Física,
Universidade Tecnológica Federal do Paraná, Rua Sete de Setembro 3165,
Curitiba, Brazil  Richards P. Albuquerque Departamento Acadêmico de Física,
Universidade Tecnológica Federal do Paraná, Rua Sete de Setembro 3165,
Curitiba, Brazil  Rogério Monteiro-Oliveira Institute of Astronomy and
Astrophysics, Academia Sinica, Taipei 10617, Taiwan  Gastão B. Lima Neto
Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Universidade de
São Paulo, Rua do Matão 1226, São Paulo, SP, Brazil

(Accepted 30 May 2024)

###### Abstract

Abell 1758 ($z\sim 0.278$) is a galaxy cluster composed of two structures:
A1758N and A1758S, separated by $\sim$2.2 Mpc. The northern cluster is itself
a dissociative merging cluster that has already been modelled by dedicated
simulations. Recent radio observations revealed the existence of a previously
undetected bridge connecting A1758N and A1758S. New simulations are now needed
to take into account the presence of A1758S. We wish to evaluate which orbital
configuration would be compatible with a bridge between the clusters. Using
$N$-body hydrodynamical simulations that build upon the previous model, we
explore different scenarios that could have led to the current observed
configuration. Five types of orbital approaches were tested: radial,
tangential, vertical, post-apocentric, and outgoing. We found that the
incoming simulated scenarios are generally consistent with mild enhancements
of gas density between the approaching clusters. The mock X-ray images exhibit
a detectable bridge in all cases. Compared to measurements of Chandra data,
the amplitude of the X-ray excess is overestimated by a factor of $\sim$2–3 in
the best simulations. The scenario of tangential approach proved to be the one
that best matches the properties of the profiles of X-ray surface brightness.
The scenarios of radial approach of vertical approach are also marginally
compatible.

Galaxy clusters (584), Intracluster medium (858), Hydrodynamical simulations
(767)

††journal: ApJ

##  1 Introduction

In merging galaxy clusters, compression of the gas between the two approaching
clusters should boost the X-ray emission in that region. Examples of bridges
or filaments detected in X-rays between clusters include, among others: A3391
and A3395 (Tittley & Henriksen, 2001; Biffi et al., 2022), A222 and A223
(Werner et al., 2008), A3556 and A3558 (Ursino et al., 2015), and A399 and
A401 (Akamatsu et al., 2017). From the simulation point of view, such
enhancements of X-ray surface brightness bridges linking merging clusters were
studied by e.g., Roettiger et al. (1997); Ricker & Sarazin (2001); Ritchie &
Thomas (2002); Donnert et al. (2017); Chadayammuri et al. (2022). These are,
in principle, thermal emissions associated with the hot gas, and the
increments in X-rays surface brightness are due to increments in gas density.

In radio, it is common for merging clusters to exhibit extended diffuse
emission in the form of giant radio relics (e.g., van Weeren et al., 2019;
Monteiro-Oliveira, 2022), which are typically giant arcs in the periphery of
merging clusters, although more complicated radio morphologies are also seen.
On the other hand, bridges of radio emission (diffuse synchrotron radiation)
connecting pre-merging clusters are a recently discovered entity. At the
moment, such radio bridges are known to have been detected in only two pairs
or pre-merging clusters: Abell 399/401 (Govoni et al., 2019) and Abell
1758N/1758S (Botteon et al., 2020). In the case of Abell 399/401, the radio
bridge has a projected length of 3 Mpc. Theoretically, the mechanisms of
magnetic field amplification and particle acceleration are understood to be
caused by the probable compression and turbulence generated in the low-density
regions between the clusters (Brunetti & Vazza, 2020).

Abell 1758 (hereafter A1758), at a redshidft of $z\sim 0.278$, consists of two
major structures: the northern (A1758N) and the southern (A1758S) components,
separated by approximately $\sim$2.2 Mpc. This bimodality has been known since
ROSAT observations (Rizza et al., 1998). The northern component is itself
bimodal, as indicated by gravitational weak lensing analyses (Dahle et al.,
2002; Okabe & Umetsu, 2008; Ragozzine et al., 2012; Monteiro-Oliveira et al.,
2017). X-ray observations (David & Kempner, 2004) combined with optical and
multi-wavelength analyses (Boschin et al., 2012; Durret et al., 2011) further
indicated that A1758N must be a post-merger cluster, i.e. the result of a
collision between two clusters. Following the nomenclature of Monteiro-
Oliveira et al. (2017), the two components within A1758N are the NW and NE
substructures.

Hydrodynamical $N$-body simulations are often used to model individual merging
clusters. These can be used to study different phenomena such as dissociation
between dark matter (DM) and gas (e.g. Springel & Farrar, 2007; Moura et al.,
2021; Lourenço et al., 2020; Albuquerque et al., 2024), sloshing (e.g.
Roediger et al., 2011; Machado & Lima Neto, 2015; Walker et al., 2017;
Doubrawa et al., 2020), and radio relics (e.g. van Weeren et al., 2011; Lee et
al., 2020) among many others. In particular, Ha et al. (2018) studied
simulations of equatorial shocks, which propagate perpendicularly to the
merger axis. Molnar et al. (2013) simulated pre-merger shocks in Abell 1750.

In Machado et al. (2015), we proposed a tailored simulation model of A1758N,
which reproduced several of its observed features, by means of an off-axis
collision between two equal-mass clusters. The main constraint was the
projected separation of approximately 750 kpc between the mass peaks of the NW
and NE substructures, derived from weak lensing (Okabe & Umetsu, 2008;
Ragozzine et al., 2012). Starting from a total A1758N virial mass of
$10^{15}\,{\rm M}_{\odot}$, a mass ratio of $1:1$ was assumed. Furthermore,
the peculiar feature of A1758N is that it is a dissociative cluster, meaning
that in one of its substructures that gas has been separated from the DM.
Namely, the NW substructure coincides with the location of a peak in X-ray
emission, while the NE substructure does not. In other words, the NW cluster
has retained its gas, while the NE cluster has been mostly stripped of its gas
due to the collision. The simulation in Machado et al. (2015) was able to
recover, within an acceptable approximation, the overall morphology of the gas
at the instant when the separation between NW and NE reached the target value
of 750 kpc.

The simulation model of Machado et al. (2015) was later updated using the more
recent weak lensing mass estimates in Monteiro-Oliveira et al. (2017).
Specifically, the new virial masses were $(7.9\pm 1.6)\times 10^{14}\,{\rm
M}_{\odot}$ for NW; and $(5.5\pm 1.6)\times 10^{14}\,{\rm M}_{\odot}$ for NE.
The updated simulation proved to be qualitatively similar to the earlier
model. By requiring it to meet the same criteria as before, the best-matching
instant was found to be approximately 0.27 Gyr after the first pericentric
passage. That new simulation ensured that the more recent mass estimates
remained compatible with the scenario of a DM/gas dissociation induced by the
merger. In the present paper, we will take the definitive simulation from
Monteiro-Oliveira et al. (2017) to be the fiducial model of A1758N.

The presence of A1758S was disregarded in Machado et al. (2015) and Monteiro-
Oliveira et al. (2017), because at the time there was no evidence of
interaction between the northern and southern structures. Moreover, even in
the absence of A1758S, the resulting simulations proved to be sufficient to
explain the internal dynamics of A1758N itself.

A1758S is less studied than its more massive northern companion. Nevertheless,
A1758S is suspected to be also a binary merger in its own turn. From its X-ray
morphology, David & Kempner (2004) proposed that the two substructures within
A1758S might be undergoing a nearly head-on collision. Based on only 27
spectroscopic members, Monteiro-Oliveira et al. (2017) proposed a scenario in
which the two substructures within A1758S would be colliding along a direction
which is closer to the line of sight than to the plane of the sky. This
collision would probably be in an early stage of the merger. Additionally,
there is an X-ray group found by Haines et al. (2018) that was
spectroscopically confirmed to be associated with A1758N, and adds a further
substructure to this already complex system.

In fact, a line-of-sight velocity offset of $-1200$ km s-1 between A1758S and
A1758N was measured in Monteiro-Oliveira et al. (2017), with A1758S at
$z=0.274$ and A1758N at $z=0.279$. More recently, Bianconi et al. (2020)
combined redshifts obtained by Monteiro-Oliveira et al. (2017) with their own
redshifts from the Arizona Cluster Redshift Survey (ACReS), and additional
redshifts from Haines et al. (2013), to conduct a dynamical analysis of A1758.
They identified members of A1758S and of A1758N (within $r_{200}$), with
central redshifts of 0.27386 for A1758S and 0.27879 for A1758N. Consequently,
the calculated line-of-sight velocity offset for A1758S relative to A1758N
would be $-1156$ km s-1. Bianconi et al. (2020), showing the redshift as a
function of declination for spectroscopic members of A1758, concludes that the
member galaxies of A1758S can be clearly distinguished from A1758N.

Regardless of the internal structure of A1758S, the possible interaction
between A1758N and A1758S remained unclear at first. From X-ray observations
with Chandra and XMM-Newton, David & Kempner (2004) had put forth the
possibility of the N and S clusters being gravitationally bound, although at
that time there were yet no direct indications of interaction between them.

Botteon et al. (2018) combined LOFAR (LOw Frequency ARray) observations with
archival data from GMRT (Giant Metrewave Radio Telescope) and VLA (Very Large
Array). They suggested a possible low surface brightness radio bridge
connecting A1758N and A1758S. Furthermore, analysis of X-ray Chandra data
points to the unrelaxed state of the outskirts of the clusters and, possibly,
to some compression and heating of the region between the pair. From this
putative equatorial shock, Botteon et al. (2018) estimate a Mach number of
$\mathcal{M}_{kT}=3.0^{+1.4}_{-1.0}$ from the temperature jump.

Subsequently, Schellenberger et al. (2019) analysed shocks in the edges of
A1758N and of A1758S. Interestingly, they found a temperature structure around
the NW cluster similar to the missing bow shock predicted by the simulation in
Machado et al. (2015). However, no conclusive evidence was found for shocked
gas between A1758N and A1758S.

More recently, Botteon et al. (2020) confirmed the detection of the giant
radio bridge that had been tentatively suggested earlier (Botteon et al.,
2018). The bridge is clearly seen in the LOFAR image, and the radio and X-ray
emissions are correlated.

Once the existence of a subtle but detectable bridge has been established, the
presence of A1758S can no longer be overlooked in a consistent model of A1758.
Specifically, the question arises about what dynamical configuration could
have brought A1758S to its current position. Thus, dedicated simulations are
needed in order to include the arrival of A1758S. In this paper, we explore a
suite of different scenarios which could have led to the observed
configuration of the A1758 system. In particular, we wish to evaluate whether
those possible scenarios would be consistent with a tenuous bridge between the
clusters. In other words: is it conceivable, from simulations, that A1758S
could have arrived at its present location without inducing very substantial
X-ray excess due to compression?

The diffuse radio emission detected between the clusters (Botteon et al.,
2018, 2020) is understood to be due to synchrotron radiation from relativistic
electrons in the presence of amplified magnetic fields. However, in this
paper, we will not simulate radio emission. Furthermore, in the context of
such a radio bridge, it is expected that some additional X-ray emission might
also arise from inverse Compton scattering (Brunetti & Vazza, 2020). This
mechanism will not be the focus of the current paper either. Instead, we will
focus on the thermal emission only. We will explore possible enhancements in
X-ray emission, but only due to gas compression, not due to additional non-
thermal mechanisms.

This paper is structured as follows. In Section 2, we describe the initial
conditions of the simulations and also the proposed scenarios. In Section 3,
we present the results for the radial approach and for the other scenarios, as
well as the analysis of the shocks and the X-ray mocks. Summary and
conclusions are given in Section 4. At a redshift of $z=0.278$, 1 arcsec
corresponds to 4.25 kpc for a cosmology with $\Omega_{\rm m}=0.27$,
$\Omega_{\Lambda}=0.73$ and $H_{0}=70\,{\rm km}\,{\rm s}^{-1}\,{\rm
Mpc}^{-1}$.

##  2 Simulation setup

The simulations presented in this paper build upon the previous results in
Machado et al. (2015, hereafter M15) and in Monteiro-Oliveira et al. (2017,
hereafter MO17). Our simulation of A1758N is thoroughly explained in M15. With
the weak lensing masses of MO17, we had performed an updated simulation – this
is the version taken to be the fiducial one, which is the basis for the
composite model developed in the present paper. However, the two versions
produced qualitatively similar outcomes. In this paper, we will focus on the
arrival of A1758S. The MO17 model of A1758N will be taken as is. Thus, the
same A1758N initial conditions will be applied, and A1758S will be added in
the ways described in Section 2.2

###  2.1 Initial conditions

As a first approximation, we will ignore the possible merging taking place
within A1758S: it will be modelled as a single structure, because our focus
for the moment is on its probable orbit, not on its internal structure. The
total mass of A1758S determined in MO17 is $4.96^{+1.08}_{-1.19}\times
10^{14}\,{\rm M}_{\odot}$.

To create the initial conditions for an idealised cluster representing A1758S,
we followed the same procedures outlined in M15. Here we give a brief summary
of the main properties. The cluster is represented by an initially spherical
distribution of DM and gas. The virial mass is set to $5.2\times 10^{14}\,{\rm
M}_{\odot}$, and the adopted gas fraction is 15 per cent. The DM halo follows
a Hernquist (1990) profile with a scale length of 400 kpc. The gas follows a
Dehnen (1993) profile with the parameter $\gamma=0$, which implies a core that
resembles a $\beta$ model. The choice of a gas scale length of 210 kpc,
together with the halo potential, leads to a central temperature of $\sim$6.4
keV, due to the imposition of hydrostatic equilibrium. Further details of the
numerical realisation can be found in Machado & Lima Neto (2013). The initial
conditions for A1758S are generated with a resolution of $10^{6}$ particles
for the DM and $10^{6}$ particles for the gas, using the
clustep111https://github.com/ruggiero/clustep code (Ruggiero & Lima Neto,
2017). In keeping with the procedures in M15, the simulations are carried out
with the Gadget-2 code (Springel, 2005) with a gravitational softening length
of 5 kpc.

This initial condition created to represent A1758S is then joined with the
initial condition of the A1758N merger to produce the complete $t=0$ snapshot
of the actual simulation. The separations and relative velocities will be
specified in Section 2.2.

For completeness, it should be mentioned that the A1758N initial conditions
are composed of 2 clusters (NW and NE), each created with analogous procedures
and with parameters given in MO17. Likewise, both NW and NE are each composed
of $10^{6}$ particles for the DM and $10^{6}$ particles for the gas.

Figure 1: Schematic representation of the initial conditions in different
scenarios: (A) radial approach, (B) tangential approach, (C) vertical
approach, (D) post-apocentric passage, and (E) outgoing.

###  2.2 Scenarios

Within the A1758N collision, the NW and NE clusters pass by one another at a
pericentric separation of only 80 kpc. Approximately 0.3 Gyr later, the so-
called “best instant” of the simulation is reached. This is the moment when
the gas morphology as well as other quantitative constraints are satisfied.
The challenge of setting up a three-body interaction is to have A1758S arrive
at the desired position precisely at that moment. In all the proposed
scenarios, we assume that A1758S is undergoing its first passage.

Table 1: Parameters of the initial conditions. Position and velocity coordinates of A1758S are given with respect to the centre of mass of A1758N. Model | $x$ | $y$ | $z$ | $|\mathbf{r}|$ | $v_{x}$ | $v_{y}$ | $v_{z}$ | $|\mathbf{v}|$
---|---|---|---|---|---|---|---|---
| (kpc) | (kpc) | (kpc) | (kpc) | (km s-1) | (km s-1) | (km s-1) | (km s-1)
Radial approach (A) | 959 |  $-$2966 | 0 | 3117 | 0 | 0 | 0 | 0
Tangential approach (B) |  $-$2600 |  $-$3100 | 0 | 4043 | 2000 | 0 | 0 | 2000
Vertical approach (C) | 859 |  $-$2657 | 3350 | 4361 | 0 | 0 |  $-$2000 | 2000
Post-apocentric (D) | 3000 |  $-$1964 | 0 | 3586 |  $-$1200 |  $-$500 | 0 | 1300
Outgoing (E) | 0 | 0 | 2000 | 2000 | 640 |  $-$2000 |  $-$450 | 2148

In order to clarify the spatial coordinates of the initial conditions, we need
to recapitulate the A1758N setup. In that collision, the NW and NE clusters
were set initially 3 Mpc apart along the $x$-axis, but also shifted by an
impact parameter of $b=350$ kpc along the $y$-axis. The initial relative
velocity was $-1500$ km s-1 parallel to the $x$-axis direction. However, it
should be noted that (unless otherwise stated) the figures in this paper
display the result of the simulation rotated by an angle around the $z$-axis.
This is done merely to have the simulation result match the position angle of
the observations. It is not an inclination with respect to the line of sight,
and thus the $xy$ simulation plane remains parallel to the plane of the sky.

Having set the stage of the A1758N collision, we may now bring A1758S into the
picture. The initial position of A1758S will be given with respect to the
centre of mass of A1758N. Likewise, the initial velocity of A1758S will be
given as a velocity relative to A1758N.

In order to explore a wide variety of possible orbits, we have set up five
distinct scenarios describing how A1758S could have arrived at its present
position. These configurations are meant to be representative of plausible
regimes for the initial conditions.

In the observed system, A1758S lies at $\sim$2.2 Mpc roughly to the south of
A1758N (actually about $15^{\circ}$ to the west of the north-south direction).
Thus, in each of the five simulations, the A1758S cluster must be required to
arrive at that desired location specifically at $t=1.47$ Gyr. Here, it is not
an issue of redefining the time coordinate arbitrarily, because that specific
moment (by any label one would call it) is the intrinsic snapshot when the
appropriate A1758N morphology is attained. Earlier or later snapshots would
not be suitable. The final three-body choreographies are achieved by trial and
error. Since the clusters are considerably far apart, one may obtain an
initial estimate of a plausible orbit via a two-body calculation by replacing
both and A1758N and A1758S by point masses. This provides a rough educated
guess, which then needs to be fine-tuned, first with low-resolution DM-only
simulations, which are inexpensive, and finally with the full high-resolution
clusters. This was done separately for each of the five scenarios. The
coordinates of initial positions and initial velocities are given in Table 1.
A sketch of each scenario is drawn in Fig. 1. The initial conditions of each
scenario are individually explained in the following:

#### (A) Radial approach

This is the simplest assumption. A1758S falls radially towards A1758N. In the
initial condition, A1758S is placed at rest, at an initial separation of 3117
kpc. Even though in the default model A the initial velocity is effectively
zero, the arrow sketched in Fig. 1 is meant to represent what happens as soon
as the simulation starts. In Section 3.1, variants of model A with higher
(non-zero) initial velocity are also explored.

#### (B) Tangential approach

In this scenario, the initial velocity of A1758S is parallel to the merger
axis of A1758N, rather than being perpendicular to it. In the initial
condition, A1758S is placed at an initial separation of 4043 kpc, with an
initial velocity of 2000 km s-1 entirely in the tangential direction.

#### (C) Vertical approach

In this case, A1758S starts with a separation of 2792 kpc on the $xy$ plane,
but it is also raised 3350 kpc above the $xy$ plane. The initial velocity is
2000 km s-1 pointing entirely down in the vertical direction. In Fig. 1, the
cross represents the vertical velocity going into the page. The sketched arrow
should be understood as a velocity component that will develop towards A1758N
once the simulation starts.

#### (D) Post-apocentric

In this scenario, A1758S starts at a separation of 3586 kpc, moving initially
away from A1758N in a general direction intermediate between radial and
tangential. The small velocity of 1300 km s-1 is such that it soon reaches its
apocenter and then turns into an incoming trajectory. With some graphic
liberty, the curved arrow in Fig. 1 should be seen not as the $t=0$ velocity
vector, but rather as the qualitative shape of the trajectory around the point
of return.

#### (E) Outgoing

In this scenario, A1758S starts moving outwards, radially away from A1758N.
A1758S could neither be placed between NW and NE, nor be made to cross that
region, because this would disrupt too dramatically the already settled NW–NE
interaction. The solution was to raise A1758S 2000 kpc above the centre of
mass of A1758N, and set it with an outwards velocity of 2148 km s-1,
represented by the arrow sketched in Fig. 1. The into-the-page cross depicts
the downward vertical velocity component that will develop as soon as the
simulation starts.

##  3 Results

###  3.1 Radial approach

In this section, we present the results of the radial approach, which is the
simplest assumption. In the default version of scenario A, the A1758S is
released with zero velocity and then falls toward A1758N.

Fig. 2 shows the time evolution of scenario A around the best-matching
instant, which is the middle frame ($t=1.47$ Gyr). The frames are 3.5 Mpc wide
and display the projected gas density and the emission-weighted temperature in
the plane of the orbit, with the position angle having been rotated to match
the orientation of the observations.

Regarding the internal morphology of A1758N, the best moment had been selected
based on the criteria presented in M15, mainly the separation between the DM
peaks of the NW and NE structures, and the DM/gas dissociation. Now, the best
moment corresponds, additionally, to the time at which A1758S arrives at a
position 2.2 kpc away from A1758N. The initial conditions were retroactively
constructed in order to ensure this synchronicity.

In spite of the gradual approach of A1758S, the time evolution in Fig. 2
reveals that the morphology of A1758N remains sufficiently undisturbed with
respect to the original model (see Fig. 3). Mild increase in gas densities can
be seen in the region between A1758S and A1758N.

For comparison, we display the A1758N-only simulation model in Fig. 3. This
corresponds to the best model of A1758N in MO17, i.e. in the absence of
A1758S. This model serves as the reference for further comparisons to evaluate
the impact of including A1758S.

The impact of the arrival of A1758S seems more noticeable in the temperature
maps of Fig. 2. In the intermediate region between the clusters, gas is heated
to around 10 keV. This temperature structure in the intermediate region was
not seen in the A1758N-only simulation (Fig. 3). The very hot outwards-
propagating shock fronts due to the internal collision within A1758N were
already present in the earlier model and remain quite similar.

Figure 2: Time evolution of model A around the best time ($t=1.47$ Gyr):
projected gas density (top row); emission-weighted temperature (bottom row).
The white contours represent total projected mass. Figure 3: Density and
temperature maps of the A1758N-only simulation, i.e. in the absence of A1758S.
The white contours represent total projected mass. Figure 4: Model A compared
to the higher-velocity variants A1 and A2, at the best time: projected gas
density (left column); emission-weighted temperature (right column). The white
contours represent total projected mass.

To explore the possible effects of different velocities, we create two
variants of the default scenario A. These variants are model A1, with an
initial velocity of 632 km s-1, and model A2, with an initial velocity of 1391
km s-1. These are the initial relative velocities of A1758S, and they point
towards the centre of mass of A1758N. Because of the higher velocities, the
initial condition of A1758S naturally needs to be placed farther away. The
initial separations are thus 3836 kpc for model A1, and 4795 kpc for model A2.

Figure 5: Comparison of all models (A to E) at the best time: projected gas
density (top row); emission-weighted temperature (bottom row). The white
contours represent total projected mass.

A comparison between the models A, A1 and A2 is shown in Fig. 4. Projected gas
density and emission-weighted temperature are shown only at the best time for
the three models. Looking at the intermediate region between the A1758N and
A1758S clusters, it is clear that the compression of the gas increases in the
higher velocity models. The density enhancement that was subtle in model A
becomes slightly more noticeable in A1 and much more so in A2. Likewise, the
temperature enhancement also becomes more pronounced in models A1 and A2, with
the temperature of the shocked gas increasing progressively from $\sim$10 to
$\sim$12 to $\sim$15 keV in models A, A1 and A2, respectively. Nevertheless,
the original shock fronts around A1758N itself remain quite similar to the
default model A.

It should be noted that the velocities quoted earlier refer to the initial
conditions. In model A, the A1758S cluster was accelerated from rest to 1718
km s-1. This is the final relative velocity (i.e., at the best moment).
Similarly, the final relative velocities of models A1 and A2 are,
respectively, 1714 km s-1 and 2668 km s-1. In the case of A1, the A1758S
cluster had to cover a longer distance; but, starting with a non-zero
velocity, it spent most of its trajectory with higher velocities than in model
A. This led to cumulatively more gas compression, although at the specific
instant $t=1.47$ Gyr the instantaneous velocities in A and A1 happen to be
similar. For A2, on the other hand, the final velocity is substantially
higher.

###  3.2 Other scenarios

Having established the properties of the more straightforward radial scenario
A, in this section we will compare it to the remaining scenarios B–E.

Fig. 5 shows the results for models A–E, displaying the projected gas density
and the emission-weighted temperature. Only the best moment is shown. In all
cases, A1758S has approximately the same position, $\sim$2.2 Mpc from A1758N –
as can be seen both in the density and in the temperature maps because its
position coincides with the peak of projected mass in the south, i.e. the
center of the countour lines in Fig. 5. The central temperature of A1758S is
$\sim$6 keV.

Regarding the internal structure of A1758N, one can discern some differences
in the panels of Fig. 5. Although the global morphological features are
roughly preserved in all cases, the density maps do exhibit some variation
depending on the model. This can be understood as a natural consequence of the
additional gravitational influence of A1758S. The approach of A1758S will
inevitably perturb the original NW-NE orbit to some degree.

Likewise, the overall temperature structure of A1758N is perturbed as well –
more evidently so than the density maps. The shock fronts in such a
configuration are understood to propagate with a higher velocity than the
velocity of the cluster themselves (Springel & Farrar, 2007, M15). For this
reason, the shape and extent of those A1758N shock fronts are somewhat more
time-sensitive than other mechanical features. Thus, a relatively small
gravitational perturbation might alter the exact orbit of the NW-NE pair in
such a way as to slightly delay or anticipate the initial propagation of the
shock front. This perturbation would shift the gas density and DM peaks by a
small extent, but would change the global shape of the shock fronts more
noticeably.

In any case, in this paper we do not aim to refine the internal details of the
A1758N model to accommodate the new perturbation. Rather, we assume that the
A1758N remains qualitatively similar and focus on the intermediate region
between the clusters. In the following, a description of each scenario B–E is
presented individually.

In scenario B, the A1758S cluster moves tangentially at first, and then is
slightly bent due to the gravitational attraction of A1758N. By the time
A1758S reaches its desired position, its velocity is 2191 km s-1. The
resulting gas compression is perpendicular to the motion of A1758S and is seen
as something akin to proper bridge, appearing to extend from one cluster to
the other (density B panel of Fig. 5). Consequently, the heated gas in the
temperature is also oriented in the same perpendicular direction. This shocked
gas modifies (and partially merges with) the southwestern edges of the NW
shock front. The shape of the model B shock is qualitatively dissimilar from
that of model A, as they were indeed meant to be as orthogonal to one another
as possible, thus exploring the extremes of two possible regimes.
Nevertheless, the values of the temperatures in the shocked regions of A and B
are roughly in the same range. Since A1758S comes from the east in this model,
it spends most of its approach in the vicinity of NE, or at least on its
general side. Thus, the NE shock front is the most perturbed and altered with
respect to the default model.

In scenario C, the A1758S cluster starts above the plane, coming down. Its
initial velocity vector is purely vertical, but the gravitational attraction
of A1758N naturally adds a velocity component towards the northern structure.
In a sense, the resulting model C is a three-dimensional version of the orbit
of model A, in which the A1758S is also coming down while it moves towards
A1758N. However, the final motion is still predominantly along the line of
sight. The final velocity is 2606 km s-1, but the $v_{z}$ component is 2368 km
s-1. The projection of the resulting shock is less straightforward to envision
in such a case. (In fact, projections of complicated three-dimensional
temperature structures tend to be quite non-intuitive in simulated cluster
mergers; Machado et al. 2022). The C temperature panel of Fig. 5 indicates the
general morphology of the intermediate shock is closer to A than to B, as
would be expected. The velocities are such that the gas was heated to $\sim$15
keV. In the C density panel of Fig. 5, the supposed compressed gas is
practically not noticeable. This might be understood if the $xy$ plane is
though of as a projection of an inclined shock whose major contribution is
parallel, nor perpendicular to it. In this configuration, an overdensity ahead
of the incoming cluster would be diluted in projection.

In scenario D, the A1758S cluster starts at a moment shortly prior to the
apocentric passage, and then turns for an incoming orbit with generally
oriented velocity. This scenario can be considered as a general intermediate
version between A (radial) and B (tangential) – but coming from the other
side, i.e. from the western side. This is the reason why the D temperature
panel of Fig. 5 vaguely resembles a flipped version of B. In this case, the NE
was not affected as in B; and the NW proved more resilient, since it was more
to the north, i.e. farther away from the perturber than NE was in model B. The
resulting shock of scenario D also has maximum temperatures of about $\sim$15
keV. The bridge in the D density panel of Fig. 5 is more complex, as would be
expected from an orbit with more varied velocity components, including a point
of return in the orbit. Still, this resulting density bridge can also be
regarded qualitatively as a flipped version of model B. The final velocity of
model D is 1864 km s-1.

In scenario E, the A1758S cluster starts directly above the centre of mass of
the A1758N cluster, with an outwards velocity. As it moves radially outward,
it also gains a downward velocity component, as it is pulled by the
gravitational attraction of A1758N. It is interesting to see that the NW and
NE shock fronts look more developed in the E temperature panel of Fig. 5 than
in any of the other scenarios. As the initial condition of A1758S was placed
instantaneously above A1758N, initially A1758S attracts both NW and NE towards
itself, thus bringing them together slightly earlier, and precipitating the
NW-NE collision to occur sooner by a short amount of time. The consequence in
the density map is not great. But the NW and NE shock fronts look like a later
version of the original model. Focusing on the intermediate region between the
northern and southern clusters, the E density panel of Fig. 5 exhibits only a
very tenuous irregular filament which is seen in other models, but no
compression per se. This is understandable, since the A1758S cluster is moving
outwards. In the E temperature panel of Fig. 5, the gas below $\sim$12 keV
would seem to be more related to disturbances in the outer edges of the
northern fronts, than to the motion of A1758S itself. The velocity of A1758S
in this scenario is 1895 km s-1. The predominant component is vertical, with
$v_{z}=-1779$ km s-1. In the $xy$ plane, the velocity component of 655 km s-1
points away from A1758N. In this sense, model E is the exception among the
five studied scenarios: it is the only outgoing one.

A direct side-by-side comparison between simulated and observed temperature
maps would be challenging. Instead, we give here a brief description of the
approximate temperature values encountered in observations. In Schellenberger
et al. (2019) temperatures in excess of 13 keV are seen in the region slightly
to the south of A1758N (their figure 3). This is generally consistent with our
models A–D, which display temperatures peaking at roughly 15 keV in the
intermediate region between A1758N and A1758S. The possible exception to this
agreement would be model E, where the temperature is slightly lower (Fig. 5).
In Botteon et al. (2018), temperatures in excess of 13 keV are also seen in
the intermediate region (at least in the low signal-to-noise temperature map
of their figure A2). This is again generally consistent with the temperature
maps of our models A–D (Fig. 5), but a more quantitative comparison will be
made in the next subsection.

###  3.3 Shocks

Figure 6: Illustrative example of the regions used to measure the temperature
between the clusters, following Botteon et al. (2018). For convenience, this
figure is drawn with a different position angle with respect to the other
figures in the paper.

In this section, we investigate how the temperature structure might help
interpret the meaning of the possible shocks.

First, we quantify the temperature profile in a manner that is inspired by the
so-called transversal profile measured by Botteon et al. (2018); their figure
9. In order to do so, we define five rectangular regions, as illustrated in
the example of Fig. 6. For convenience when drawing the rectangles, the
snapshots were rotated around the $z$ axis. Regions 1 and 5 have dimensions
574 kpc $\times$ 910 kpc; regions 2 to 4 have dimensions 385 kpc $\times$ 652
kpc. Within each region, we take the mean and the standard deviation of the
emission-weighted projected temperature. This is done for each of the five
scenarios. In each scenario, the set of regions is actually shifted sideways
by a given arbitrary amount along the horizontal axis of Fig. 6. This is done
in a way to have the highest temperatures land within regions 2 or 3, as is
the case in the observations of Botteon et al. (2018). This freedom is
justified, since the morphology of the temperature structure varies from model
to model, and thus it would be difficult to demand an absolute positioning of
the presumed shocked gas. This measurement can be understood both as a rough
attempt at observational comparison, but also as a theoretical
characterisation of the profiles of each scenario.

Figure 7: Temperatures measured within regions such as in Fig. 6. The black
points are taken from Botteon et al. (2018). Within each region, the points
are slightly shifted sideways, for clarity.

The resulting transversal temperature profiles are shown in Fig. 7. The black
circles in Fig. 7 are temperature values taken from Botteon et al. (2018). The
other symbols correspond to our scenarios A–E. Within each region, the points
were shifted for clarity. The global result of these measurements is that,
even with a wide variety of morphologies, the simulated temperatures fall in
the same approximate ranges as the observational results – in some cases, with
a borderline agreement of more than 1$\sigma$. However, the peak temperature
(i.e. the temperature increase from region 1 to 2, followed by a decline) is
more pronounced in the observation than in the simulations. In particular,
scenario E shows no suggestion of a peak. In view of these simulated profiles,
it seems difficult to argue for a meaningful equatorial shock (Ha et al.,
2018) in our scenarios. Thus, we refrain from estimating Mach numbers under
such assumption.

Instead, if we understand the heated gas in the intermediate region as shocks
moving ahead of the incoming A1758S, we may estimate their Mach numbers in the
usual ways. The assumption here is that the direction of the propagation of
the shock front is perpendicular to the surface of temperature discontinuity
(or rather, its projection onto the $xy$ plane). For example, in scenarios A
and C, the direction of propagation of the shock front would be radially
towards A1758N. In scenarios B and D, it would be perpendicular to the
predominant direction of the heated gas.

Table 2: Mach numbers estimated via the temperature jump ($\mathcal{M}_{kT}$) and via the shock velocity ($\mathcal{M}_{v}$). model | $\mathcal{M}_{kT}$ | $\mathcal{M}_{v}$
---|---|---
A | 1.8–1.9 | 1.2
A1 | 2.0–2.1 | 1.2
A2 | 2.1–2.2 | 1.5
B | 1.6–1.7 | 1.4
C | 1.7–1.9 | 1.1
D | 1.7–1.8 | 1.9

One method to estimate the Mach number is using the amplitude of the
temperature discontinuity:

$\frac{T_{2}}{T_{1}}=\frac{5\mathcal{M}_{kT}^{4}+14\mathcal{M}_{kT}^{2}-3}{16\mathcal{M}_{kT}^{2}},$
(1)

where $T_{1}$ is the pre-shock temperature and $T_{2}$ is the post-shock
temperature (Landau & Lifshitz, 1959). The Mach number estimated in this way
is $\mathcal{M}_{kT}$; this is usually employed in observations but can also
be applied to simulations. The other method consists in measuring the velocity
of the shock front directly, from successive snapshots of the simulation. The
location of the shock front is determined by the temperature discontinuity.
Its advancement as a function of time gives the velocity $v_{s}$. However, the
upstream gas may not be stationary (Springel & Farrar, 2007; Machado & Lima
Neto, 2013). Rather, this gas ahead of the shock may have a velocity $u$ (in
the same rest frame). Thus, the shock encounters the pre-shocked gas with an
effective velocity of $v_{s}-u$. For this reason, the Mach number computed via
the shock velocity is:

$\mathcal{M}_{v}=\frac{v_{s}-u}{c_{s}},$ (2)

where $c_{s}^{2}=\frac{\gamma kT}{\mu m_{p}}$ is the sound speed of the pre-
shock gas. In simulations, the estimates of Mach number via the temperature
jump ($\mathcal{M}_{kT}$) or via the shock velocity ($\mathcal{M}_{v}$) will
generally coincide very well, if the merger configuration is that of a clean
textbook shock, such as the ‘Bullet Cluster’ (Clowe et al., 2006) itself. The
Mach numbers estimated for our scenarios are presented in Table 2, for both
methods. They generally do not agree, with the temperature method almost
always overestimating the Mach number, in comparison to the velocity method.
An estimated confidence range is offered for the temperature method, due to
the uncertainty in the choice of the region corresponding to the pre-shock
gas. In fact, the temperature method is expected to be quite unreliable in
this context, due to the complex temperature structure in the pre-shock
region. Indeed, even though the so-called pre-shock has not yet been reached
by the A1758S-induced shock, it is far from an undisturbed gas. On the
contrary, it is a region quite perturbed by the A1758N-induced disturbances.
The velocity methods would tend to be more reliable, but even considering the
$u$ correction, it is still a complex configuration quite different from an
idealised discontinuity. Scenario E is omitted from the Mach number analysis.
Since that is an outgoing configuration, there would be no justification to
interpret the hot gas in the intermediate region as a shock front. Indeed,
visual inspection of the time evolution suggests that the hot region
corresponds do the edges of the A1758N shocks dissipating as they expand.
Overall, it is only possible to conclude that the Mach numbers of our
scenarios may be in the approximate range 1–2, if the velocity method is to be
relied upon.

###  3.4 X-ray emission

Finally, we resort to producing mocks of the X-ray emission in an attempt to
make a more direct connection with observations.

In order to produce realistic X-ray mocks, we employ pyXSIM222https://hea-
www.cfa.harvard.edu/~jzuhone/pyxsim (ZuHone & Hallman, 2016), which is a
Python package for simulating X-ray emission from astrophysical sources, based
on the PHOX algorithm (Biffi et al., 2012, 2013). The use of pyXSIM is
analogous to the procedures followed in Doubrawa et al. (2020) or Moura et al.
(2021). A fixed metallicity of 0.3 Z⊙ is assumed, as well as a hydrogen column
density of $n_{\rm H}=1.03\times 10^{20}\,{\rm cm}^{-2}$ in the direction of
A1758. The photons are projected along the line of sight, and the response
from the detector is taken into account; in this case Chandra ACIS-I, with an
exposure time of 150 ks (as in Botteon et al., 2018) and an energy band of
0.5–7.0 keV.

Figure 8: X-ray mocks for all models (from A to E). The colours represent
simulated X-ray counts smoothed with an adaptive kernel (see text), in
logarithm scale. The bottom right panel shows the exposure-map corrected
mosaic image of Chandra exposures, also adaptively smoothed.

The simulated images are equivalent to the exposure-map corrected images from
real Chadra data, corrected from vignetting effects. The effect of CCD gaps of
the simulated ACIS-I exposures were minimised by simulating a dithered
observation. These mock images were further processed by using an adaptively
smoothing, using a Gaussian kernel that increases in order to have a preset
significance. We have used the csmooth tool from the ciao package from the
Chandra X-ray Center (CXC)333https://cxc.cfa.harvard.edu/ciao/.

The resulting X-ray mocks are shown in Fig. 8 for models A–E. Contours of
projected mass are not shown in Fig. 8, to allow for a clearer inspection of
subtle morphological features. Moreover, the location of the X-rays peaks with
respect to the projected mass is qualitatively the same as in the case of gas
density, which can be consulted in the previous figures. The main result of
Fig. 8 is that a weak bridge in the intermediate region between A1758S and
A1758N can be seen in the smoothed mock images, with somewhat different
morphologies. The presence of these X-ray bridges shows that the mild over-
densities seen in the gas density maps were indeed translated into the more
realistic synthetic X-ray images. The fine-tuning of the internal sctructure
of A1758N itself is not the main focus of the present paper. Yet, it is
interesting to notice in Fig. 8 that model B exhibits perhaps the best
morphological agreement with the Chandra observation.

In order to compare our simulated images with the actual observations, we have
downloaded the publicly available Chandra observations, 5 ACIS-I (totalling
176.6 ks) and one ACIS-S (59 ks), with Obs Ids: 2213, 5772, 7710, 13997,
15538, and 15540. We have reprocessed all these exposures following the
standard procedure using the chandra_repro tool. Then, we have created a
single broad band (0.5–7.0 keV), exposure-map corrected image by merging all 6
exposures with the tool merge_obs. Finally, as we did with the mock images, we
have adaptively smoothed the Chandra image, which is shown in the bottom right
panel of Fig. 8. The bridge between the North and South components is clearly
seen.

To further compare simulations with observation, we measured profiles of X-ray
surface brightness across the region of the bridge. This is similar but not
identical to the measurements in Botteon et al. (2020). We defined 4 slices in
a direction perpendicular to the line connecting A1758N and A1758S. These
slices are shown in Fig. 9 with respect to the observed Chandra image. Along
each given slice, the X-ray surface brightness was measured within 15
successive rectangular regions, each of size 153 kpc $\times$ 251 kpc, with
point sources masked. The ratio was computed between the excess in each region
with respect to the average surface brightness of that entire slice. In this
definition, if the bridge were absent, the profiles would be consistent with
noise around the value of zero. This procedure was applied with the same
techniques for the observed Chandra image, as well as to the simulated models
A–E.

Figure 9: Slices used to measure the X-ray surface brightness overlaid with
the Chandra exposure map corrected image. Each slice is composed of 15
rectangular regions. Each region is $36^{\prime\prime}\times
59^{\prime\prime}$ (or 153 kpc $\times$ 251 kpc), rotated by $15^{\circ}$
counterclockwise. Red circles represent masked regions around point sources.
Figure 10: Profiles of relative X-ray surface brightness measured along the
slices defined in Fig. 9. The black points correspond to measurements from the
Chandra image. The color lines represent the equivalent measurements using the
mock images produces for the simulated models A–E. The ‘$x$’ axis corresponds
to the directions along the slices seen in Fig. 9; in each case, $x=0$ means
the central bin of that slice. Note the different vertical scales in the righ-
hand axes. Figure 11: Bridge widths obtained from the FWHM of the X-ray
surface brightness profiles of Fig. 10. Within each slice, the points are
artificially shifted sideways, for clarity.

The resulting relative profiles of X-ray surface brightness are shown in Fig.
10, where the observational data (points) are compared to the mocks of each
simulated model (lines). The peaks in the observational profiles are
comparable to the ones measured in Botteon et al. (2020). Peaks are detectable
in all the 4 slices, but peaks 1 and 4 are somewhat higher, because those
slices are closer to A1758N and A1758S, respectively. Thus the profiles of
slices 1 and 4 are partially capturing contributions from the clusters
themselves. On the other hand, the two intermediate peaks 2 and 3 are
comparatively lower, since those slices cross the lower density regions
between the two clusters. For this reason, slices 2 and 3 are the most
meaningful when trying to characterize the bridge.

Regarding the mocks, the relative profiles of X-ray surface brightness are
shown in each of the 5 panels of Fig. 10. The fact that peaks are clearly
obtained in all cases indicates that the simulated X-ray bridge is in fact
detectable (as was already seen morphologically in Fig. 8). However, the
amplitudes of the simulated peaks do not match precisely the amplitudes of the
observed peaks. For a qualitative comparison to be made, the vertical ranges
of Fig. 10 were not kept fixed. This means that the excess X-ray emission is
present in all simulations, but the absolute contrast seems to be
overestimated in the simulations, by factors of typically $\sim$2–3 in models
A, B and C; and as much as a factor of $\sim$4 in models D and E. These
results might be sensitive to the arbitrary definition of the slices.

Nevertheless, we may consider the relative heights of the peaks in each slice
of Fig. 10. We find that model B is the only one where the observed pattern is
recovered: peaks 2 and 3 are slightly lower than both peaks 1 and 4. For
models A and C, the first peak is indeed the highest of the four slices, but
peak 4 is not meaningfully higher than the intermediate ones. Models D and E
fail at this comparison.

Even if the absolute values of the peaks are not directly comparable, their
widths give us some relevant estimation of how physically broad the bridges
are. We obtained (both for observation and simulations) the full width at half
maximum (FWHM) of each profile. Errors bars were estimated by considering the
FWHM obtained allowing $\pm 3\sigma$ from the data points. The resulting
widths are shown in Fig. 11. One noticeable feature of this comparison is that
all models fail at reproducing the width of slice 4. This discrepancy was
already perceptible looking at the slices 4 in Fig. 10, where the observed
profile is clearly wider than all the simulated ones. Slice 4 presumably
encompasses relevant contributions from the X-ray emission in the vicinity of
A1758S. The fact that all simulations underestimate the width of the X-ray
contrast in that slice indicates the limitations of representing A1758S by one
single structure.

Focusing on the remaining slices in Fig. 11, we find that model B seems to be
the one that more closely approaches the Chandra values – specially bearing in
mind that regions 2 and 3 are the most meaningful to characterize the bridge
region itself. Model A would be the next best match, since it is at least
marginally consistent with three slices (1, 2 and 3). Model C is marginally
consistent with two slices (1 and 2). Model D fails at all slices, while model
E matches only slice 1 and none of the more relevant intermediate slices.

##  4 Discussion

A1758 is a complex galaxy cluster, composed of two major structures, A1758N
and A1758S, each of which seems to be undergoing its own internal binary
merger. The internal structure of A1758N had been modelled by dedicated
hydrodynamical $N$-body simulations previously in M15 and MO17. The presence
of A1758S had been ignored in those earlier works, because there was no strong
evidence of interaction at the time. With the recent confirmation by Botteon
et al. (2018, 2020) of a radio bridge connecting A1758N and A1758S, it became
necessary to consider the arrival of A1758S.

Theoretically, simulations generally predict strong compression in the early
stages of a merger. Thus it becomes relevant to pose the question of whether
A1758S could be falling towards A1758N without inducing very obvious
enhancement in X-ray surface brightness between the two clusters. In this
paper, we aimed to address this question.

We investigated five scenarios: radial approach (A), tangential approach (B),
vertical approach (C), post apocentric (D) and outgoing (E). The motivation
for choosing this set of orbits was to attempt to cover a wide variety of
possible configurations. In a sense, models A, B and C somehow represent an
orthogonal basis of initial velocities in the three linearly independent
directions. Model D acts as a kind of generic or intermediate configuration,
albeit confined to the $xy$ plane. Finally, model E is the exception, being
the sole outgoing model. It should be noted that the directions of the final
velocities are naturally different from the initial ones, in general. The
difficulty in finding suitable models is the fine-tuning necessary to achieve
the synchrony between the arrival of A1758S at its observed position, and the
best moment of the already-established internal A1758N model.

Within each proposed scenario, we offer in this paper only one instance of
such an orbit, without sweeping the parameter space of possible velocities. If
one particular model fails to meet the observational constraints, this does
not mean that that entire family of models is automatically ruled out, because
alternate versions with different velocities could in principle reconcile the
simulation with the observation. Conversely, if one given model is shown to be
satisfactory, this is sufficient to claim that that is a plausible scenario.

Additionally, in the present paper, we have not yet delved into the internal
structure of A1758S itself. Rather, we focused on the interaction between
A1758S as a whole and A1758N. Similarly we took the final MO17 model of A1758N
to be the fiducial one. We commented on some of the minor morphological
impacts that the arrival of A1758S induce on the NW and NE shock fronts, but
without attempting to re-refine the A1758N model itself to accommodate for the
new perturbation.

Another important caveat is that in the present exploration we have not
attempted to satisfy the line-of-sight relative velocity between A1758N and
A1758S, which is estimated to be of approximately $-1200$ km s-1 (Monteiro-
Oliveira et al., 2017; Bianconi et al., 2020). Models C and E are the only
ones which, by construction, have nonzero velocities along the line of sight.
They end the simulation with $-2368$ km s-1 and $-1779$ km s-1, respectively.

The temperatures of the shocked gas between the clusters were found to be in
the same approximate ranges as in the observations. This is an indication that
the choices of initial velocities are not only plausible, but also
specifically consistent with the known observational constraints. However, we
did not interpret that temperature structure as an equatorial shock (Ha et
al., 2018). Instead, we understood that the shock fronts in the incoming
models are associated with the compressed gas ahead of A1758S. We estimated
the Mach numbers directly from the advancing positions of the temperature
discontinuity in successive snapshots, and obtained values in the range
$\sim$1–2. This method uses the velocity of the shock front but also takes
into account the correction due to the infalling pre-shock gas (Springel &
Farrar, 2007). The attempt to estimate the same Mach numbers using the method
of temperature jumps revealed that this methods might be too unreliable in
this case, due to the complex nature of the temperature structure in the
surrounding gas. That gas has already been severely perturbed by the shock
fronts emanating from the internal A1758N collision.

Shock-heating of the gas is a naturally expected phenomenon in pre-merging
system, commonly predicted in simulations (e.g. Takizawa, 1999), and also
observed in many pre-merging clusters (e.g. Paterno-Mahler et al., 2014;
Akamatsu et al., 2016, 2017, among many others). If the projected separation
between the two clusters is larger than the sum of their individual virial
radii, temperature enhancements are generally not observed (e.g. Andrade-
Santos et al., 2015; Wegner et al., 2017). In the case of A1758, it is
conceivable that the intervening gas, having been previously perturbed by the
internal A1758N collision, may be more prone to heating than it would
otherwise be, if A1758N were a relaxed cluster. It is also interesting to note
that if the edges of the expanding shock fronts of NW and NE were to be
interpreted as an equatorial shock of their own, then they might contribute to
boost the incoming shock due to the arrival of A1758S. In other words, an
equatorial shock emanating from the A1758N merger axis would be parallel to
the shock front of the incoming A1758S.

We produced X-ray mock images from the simulations. These mocks revealed that
a bridge of X-ray emission is indeed predicted by the simulations. By
measuring transversal profiles, we were able to quantify the excess in X-ray
surface brightness in the region between A1758N and A1758S. The amplitude of
this excess is somewhat overestimated in the simulations by factors of
$\sim$2–3 in the best cases (models A, B and C); and by factors of as much as
$\sim$4 in the other models. This means that the X-ray excess is seen more
sharply in the simulations, when compared to the relatively attenuate profiles
seen in the Chandra data. A factor of $\sim$2–3 does not seem excessively
discrepant, especially bearing in mind the highly idealised nature of these
simulations, as well as the uncertainties in the processing of the
observational data.

Nevertheless, in the X-ray surface brightness profiles, the relative heights
of the peaks revealed additional results. Model B proved to be the only one
which systematically follows the same pattern as the observational data, in
having the two lowest peaks in the intermediate region.

Furthermore, the peaks in the X-ray surface brightness profiles were used to
estimate the physical width of the bridge region. Model B was the one that
most approached the observationally determined widths, followed by models A
and C. Finally, regarding the X-ray mocks, it is interesting to notice that in
the end model B produced the best-matching internal morphology of A1758N
itself, even though that feature was not deliberately selected for.

An additional caveat is to highlight that magnetic fields and relativistic
electrons were not taken into account in these simulations and we have not
calculated radio emissions. The analyses of this paper were restricted to
theoretical hydrodynamical quantities (temperatures and densities) and to
synthetic X-ray observations. The simulations predict slightly overestimated
enhancements in X-ray surface brightness. Only the thermal contribution was
taken into account. Thus, our simulated X-ray emissions should be regarded as
lower limits, since the possible inverse Compton contribution (Brunetti &
Vazza, 2020) was not included. The possible bridges we attempted to
investigate were bridges of gas density or bridges of (bremsstrahlung) X-ray
emission. We did not investigate bridges of radio emission, nor X-ray due to
inverse Compton emission.

Also, for simplicity, the A1758N model was kept on the plane of the sky, since
this proved sufficient to attain a suitable model in M15. In other words, with
the orbit of NW/NE kept on the plane of the sky, this gave rise to an adequate
explanation of the internal A1758N merger. This does not mean that non-zero
inclinations could be ruled out. In the five scenarios we chose to put forth
in this paper, the initial velocities of A1758S were set, for simplicity,
either fully on the $xy$ plane, or fully along $z$. But due to the
gravitational attraction of A1758N, the models with initially vertical
velocities ended up with generically three-dimensional velocities, although in
the final configuration the predominant component was still along the line of
sight (away from the observer in model C, and towards the observer in model
E). In any case, it would be difficult to argue that the scenarios presented
here are making quantitative predictions, for example, regarding specific
values of velocity in the line of sight. Within each scenario, we offered one
possible example in that regime, but without systematically sampling the
parameter space.

##  5 Conclusions

The main conclusions of this paper may be summarised as follows:

  * •

Densities: The proposed incoming scenarios are all generally consistent with
mild gas overdensities in the intermediate region between A1758N and A1758S.

  * •

Temperatures: The compression due to the incoming A1758S gives rise to shock-
heated gas with a temperature range comparable to observations, in all the
incoming models. Simulations suggest Mach numbers of roughly 1–2.

  * •

X-rays: All the scenarios exhibit detectable X-ray excess in the intermediate
region between the two clusters. The best simulations overestimate the
amplitude of the X-ray excess by a factor of $\sim$2–3.

  * •

Best scenarios: The model of tangential approach (B) proved to be the one that
most closely matches the features of the X-ray surface brightness profiles.
Models of radial (A) and vertical (C) approach were shown to be marginally
compatible as well.

  * •

Disfavoured scenarios: The outgoing scenario (E) is generally disfavoured for
failing to meet observational constraints, as well as the post-apocentric
approach (D).

New spectroscopic data on A1758S should enable a refined modelling taking into
account the internal structure of A1758S itself. Simultaneously, future
refinements of these simulations should aim to introduce a line-of-sight
velocity component into the current best models to evaluate how they are
impacted. Future observations with the next-generation X-ray observatory
Athena (Nandra et al., 2013) might probe the bridge region more deeply. If
A1758N and A1758S are to merge in the future, as seems likely from the
simulations presented here, then the resulting object will be among the most
massive known in the Universe.

## acknowledgments

We thank the anonymous referee for the helpful suggestions which improved the
results of the paper. The authors acknowledge the National Laboratory for
Scientific Computing (LNCC/MCTI, Brazil) for providing HPC resources of the
SDumont supercomputer, which have contributed to the research results reported
within this paper. REGM acknowledges support from the Brazilian agency
Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) through
grants 303426/2018-7 and 307205/2021-5. REGM acknowledges support from
Fundação de Apoio à Ciência, Tecnologia e Inovação do Paraná through grant
18.148.096-3 – NAPI Fenômenos Extremos do Universo. RPA acknowledges the
financial support from UTFPR. GBLN acknowledges support from CNPq through
grant 303130/2019-9.

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