# The Large Magellanic Cloud’s $\sim 30$ Kiloparsec Bow Shock and its Impact
on the Circumgalactic Medium

David J. Setton  Department of Astrophysical Sciences, Princeton University, 4
Ivy Lane, Princeton, NJ 08544, USA  Department of Physics and Astronomy and
PITT PACC, University of Pittsburgh, Pittsburgh, PA 15260, USA  Gurtina Besla
Steward Observatory, University of Arizona, Tucson, AZ 85705  Ekta Patel
Department of Astronomy, University of California, Berkeley, 501 Campbell
Hall, Berkeley, CA, 94720, USA  Department of Physics and Astronomy,
University of Utah, 115 South 1400 East, Salt Lake City, Utah 84112, USA
Cameron Hummels TAPIR, California Institute of Technology, Pasadena, CA 91125,
USA  Yong Zheng Department of Physics, Applied Physics and Astronomy,
Rensselaer Polytechnic Institute, Troy, NY 12180  Evan Schneider Department of
Physics and Astronomy and PITT PACC, University of Pittsburgh, Pittsburgh, PA
15260, USA  Munier Salem Stripe, 354 Oyster Point Blvd South San Francisco,
94080

###### Abstract

The interaction between the supersonic motion of the Large Magellanic Cloud
(LMC) and the Circumgalactic Medium (CGM) is expected to result in a bow shock
that leads the LMC’s gaseous disk. In this letter, we use hydrodynamic
simulations of the LMC’s recent infall to predict the extent of this shock and
its effect on the Milky Way’s (MW) CGM. The simulations clearly predict the
existence of an asymmetric shock with a present day stand-off radius of $\sim
6.7$ kpc and a transverse diameter of $\sim 30$ kpc. Over the past 500 Myr,
$\sim 8\%$ of the MW’s CGM in the southern hemisphere should have interacted
with the shock front. This interaction may have had the effect of smoothing
over inhomogeneities and increasing mixing in the MW CGM. We find
observational evidence of the existence of the bow shock in recent $H\alpha$
maps of the LMC, providing a potential explanation for the envelope of ionized
gas surrounding the LMC. Furthermore, the interaction of the bow shock with
the MW CGM may also explain observations of ionized gas surrounding the
Magellanic Stream. Using recent orbital histories of MW satellites, we find
that many satellites have likely interacted with the LMC shock. Additionally,
the dwarf galaxy Ret2 is currently sitting inside the shock, which may impact
the interpretation of reported gamma ray excess in Ret2. This work highlights
how bow shocks associated with infalling satellites are an under-explored, yet
potentially very important dynamical mixing process in the circumgalactic and
intracluster media.

Galaxies: hydrodynamical simulations – Galaxies: Large Magellanic Cloud –
Galaxies: Circumgalactic Medium

††journal: ApJ Letters††software: Astropy (Astropy Collaboration et al.,
2022), Matplotlib (Hunter, 2007), yt (Turk et al., 2011), Enzo Bryan et al.
(2014)††thanks: E-mail<EMAIL_ADDRESS>Brinson Prize
Fellow††thanks: Hubble Fellow

##  1 Introduction

The Circumgalactic Medium (CGM) plays a crucial role in the evolution of
galaxies (e.g., Tumlinson et al., 2011; Putman et al., 2012; Tumlinson et al.,
2017; Péroux & Howk, 2020). An understanding of this multi-phase medium is
essential to modeling the mechanisms that replenish and deplete the gas
reservoir in the star forming interstellar medium over a galaxy’s lifetime.
The CGM is a massive reservoir containing enriched material produced in star
formation and subsequent supernovae explosion (e.g., Peeples et al., 2014).
However, there is still considerable uncertainty in our understanding of the
baryon cycle in galaxies, especially because there is strong evidence that the
CGM at $z=0$ is still out of hydrostatic equilibrium (e.g., Werk et al., 2014;
Conroy et al., 2021; Lochhaas et al., 2023). As such, understanding the multi-
phase structure of the CGM and the sources of mixing is necessary to
understand the flow of gas in and out of the CGM.

Much of the focus on the state of the CGM has been on inflows and outflows as
the predominant mechanisms that influence its multi-phase structure.
Simulations predict that at z=0 the CGM of galaxies is still being supplied
with low-metallicity gas from the intergalactic medium (e.g., Stern et al.,
2020). Outflows of the enriched galactic interstellar medium have been
observed in galaxies, driven by star formation (e.g., Tremonti et al., 2007;
Rubin et al., 2014; Diamond-Stanic et al., 2021) and active galactic nuclei,
which may also serve to mechanically heat the existing CGM (see references in
Alexander & Hickox, 2012; Kormendy & Ho, 2013). Additionally, the accretion of
cool CGM gas enriches a galaxy’s interstellar medium, as has been observed in
star forming galaxies (e.g., Rubin et al., 2012).

While much of the observational work on the CGM has focused on gas entering
via either outflows from the host galaxy or accretion from the IGM,
simulations have shown that in MW mass halos as much as $\sim 20\%$ of the gas
mass of the CGM is expected to have originated in satellite galaxies (Hafen et
al., 2019). In this study, we examine the impact of satellite galaxies on the
structure and dynamics of the host galaxy’s CGM, rather than its contribution
to the mass budget.

Cosmological simulations show that the dominant building block of MW mass
halos ($\sim 10^{12}$ M⊙) are subhalos that harbor $\sim$10% of the host mass
(Stewart et al., 2008). Indeed, the MW’s most massive satellite galaxy, the
Large Magellanic Cloud (LMC), is believed to have had a halo mass of
$M_{\mathrm{halo}}\sim 10^{11}$ M⊙ at infall (Besla et al., 2012). Thus, we
consider the specific case of the LMC, which is just past its pericentric
approach to the MW (Besla et al., 2010; Kallivayalil et al., 2013).
Consequently, the LMC is moving very quickly, at 320 km/s relative to the MW
(Kallivayalil et al., 2013). This speed implies that the LMC is moving
supersonically through the MW’s CGM and should generate a bow shock. It has
been previously suggested that the LMC hosts a bow shock that may impact its
star formation history (de Boer et al., 1998). Here, we use detailed
hydrodynamical simulations from Salem et al. (2015) to predict the existence,
morphology and kinematics of the LMC’s bow shock using the known 3D velocity
vector of the LMC.

There is clear evidence for interaction between the LMC and the CGM of the MW
in the form of the Magellanic Stream, a gas structure that trails behind the
Magellanic Clouds (Mathewson et al., 1974). The origin of the Stream has been
attributed to a combination of tidal forces and ram pressure stripping of the
LMC’s interstellar medium (D’Onghia & Fox, 2016). Ram pressure stripping will
result in the truncation of the LMC’s gas disk. Indeed its gas disk is
observed to be significantly smaller than its stellar disk (radius of 6 kpc
vs. $\sim$ 18 kpc, see Mackey et al. 2018). Hydrodynamic simulations of the
impact of ram pressure on the LMC’s gaseous disk have reproduced the observed
truncation of the LMC’s gas disk (Salem et al., 2015), proving that the LMC is
hydrodynamically interacting with the MW’s CGM.

In this letter, we further examine the Salem et al. (2015) simulations to
study the impact of the supersonic motion of the LMC on the MW CGM itself. We
posit that the resulting bow shock plays a key role in mixing the CGM as the
$\sim 30$-kpc shock swept through the southern hemisphere during the LMC’s
first infall. We predict the shape, jump conditions, and physical extent of
this shock and discuss the consequences of this structure to our understanding
of the CGM and relationship with dwarf satellites of the Milky Way.

This letter is laid out as follows. In Section 2, we describe the updated run
of the fiducial simulation from Salem et al. (2015) that we use in this work.
In Section 4, we present our predictions for the physical properties of the
LMC bow shock. In Section 5, we estimate the observability of this shock.
Finally, in Section 6, we discuss the influence that the LMC’s bow shock may
have had on: 1) mixing of the MW’s CGM; 2) the ISM of several of Milky Way’s
other dwarf satellite galaxies; and 3) the interpretation of observed ionized
gas associated with the Magellanic System. Unless otherwise specified,
throughout this work “CGM” refers to the circumgalactic medium of the Milky
Way.

Figure 1: Number density of all gas (CGM and LMC disk gas; left) and only hot
CGM gas (T$>10^{6}$ K, right) projected along our line of sight towards the
LMC, shown on log and linear scales, respectively, to highlight the dynamic
range of density. The cold gas of the disk dominates the total density. In the
hot gas, a bow shock induced by the galaxy’s supersonic motion accounts for
most of the differences in density. This shock subtends 10s of kiloparsecs in
the simulation, and the column density is $\sim 2$ times larger in the shocked
CGM than it is in the unshocked regions. In both panels, the observed extent
of the LMC stellar disk ($r\sim$18 kpc, see Mackey et al. 2018) is shown as a
dashed white line, illustrating that both the gas disk and the shock front are
located at significantly smaller radii than the stellar disk. The stand-off
radius for the shock is $R_{so}=6.7$ kpc, while the gas disk is truncated at
$\sim$6.2 kpc (Salem et al., 2015).

##  2 Description of the Simulation

To place constraints on the shape and structure of the LMC’s bow shock, we
utilize the same Enzo (Bryan et al., 2014) hydrodynamic simulation as in Salem
et al. (2015). This simulation was performed to compare the effects of ram
pressure stripping on the truncation of the LMC’s gas disk to observations,
thereby constraining the properties of the CGM at the location of the LMC.
Their fiducial simulation, which best matches the disk truncation, is also
very well suited to studying other effects of the LMC’s supersonic motion
through the CGM such as the aforementioned bow shock. While the full details
of the simulation are described in detail in Salem et al. (2015), here we
summarize the key simulation conditions and highlight any differences in our
implementation.

The simulation uses a “wind tunnel” approach by placing the LMC at rest in a
box and modeling the MW CGM as a headwind with evolving velocity and density
structure meant to mirror the actual CGM conditions the LMC passes through
along its orbit. The orbit of the LMC is a first infall scenario, such that
the LMC has not made a previous orbit about the MW as in Kallivayalil et al.
(2013).

In the original simulation, the LMC was placed at the coordinates (20, 20, 20)
kpc in a $60\times 60\times 60$ kpc3 box. In our implementation, we re-ran the
simulation using a $100\times 100\times 100$ kpc3 box, placing the LMC at the
same relative location. The larger box size was chosen to prevent the bow
shock from reaching the edges of the simulation box and interacting with
boundary conditions in the present day simulation snapshot. The resolution of
the simulation is consequently lower by a factor of 1.6 compared to Salem et
al. (2015), but as we are concerned with the response of the larger scale CGM,
the lower resolution does not impact our findings.

The simulation includes self-gravitating gas in the form of the LMC gas disk
and the ambient MW CGM. The simulation does initialize with a small ($5\times
10^{6}\ M_{\odot}$) LMC CGM, but that halo is quickly swept away by the wind
and is thus not assumed to survive the LMC infall. The simulation does not
utilize radiative cooling or star formation feedback. The gravitational
potentials of the LMC stars and dark matter are treated as static potentials.
The LMC dark matter halo is modeled using a spherical profile with
$\rho_{0}=3.4\times 10^{-24}\ \mathrm{g\ cm^{-3}}$, $r_{0}=3.4$ kpc, and
$M\mathrm{(100kpc)}=5\times 10^{10}M_{\odot}$ following Burkert (1995). The
stellar component is modeled as a Plummer-Kuzmin Disk (Miyamoto & Nagai, 1975)
with $M_{\star}=2.9\times 10^{9}M_{\odot}$, $a_{\star}=1.7$ kpc, and
$b_{\star}=0.34$ kpc. The initial gas distribution is set as an exponential
disk with $M_{\mathrm{gas}}=5\times 10^{8}M_{\odot}$, $a_{\mathrm{gas}}=1.7$
kpc and $b_{\mathrm{gas}}=0.34$ kpc (following the stellar distribution), and
$M_{\mathrm{tot}}=7.2\times 10^{8}M_{\odot}$ (Tonnesen & Bryan, 2009). See
Table 1 in Salem et al. (2015) for more details.

Figure 2: Radial profiles (blue) for the density (left), temperature (center),
and velocity (right) along a vector that points from the LMC disk center in
the direction of the LMC’s velocity vector. The red line shows the expected
jump values for the LMC’s density and temperature, assuming a CGM temperature
of 1.19$\times 10^{6}$ K and a velocity of 350 km/s as in the simulation. We
additionally show the stand-off radius, $R_{\mathrm{SO}}$, as calculated by
the algorithm outlined in Section 4.2. The theoretical predictions match the
simulated shock conditions well in density and temperature. Additionally,
there is a clear discontinuity in the velocity structure of the gas in the
shocked region wher the shocked gas is moving at $\sim$100 km/s relative to
the LMC, significantly slower than the ambient CGM. The grey shaded region
shows the extent of the LMC gas disk.

##  3 The Bow Shock of the LMC

Following Salem et al. (2015), in the simulation, gas is accelerated towards
the LMC, and the gas just ahead of the LMC at present day is in fact moving at
a speed of $\sim$350 km/s relative to the LMC center of mass, slightly higher
than the value of 320 km/s reported in Kallivayalil et al. (2013). In order to
be consistent with the simulation, we utilize this value in conjunction with
the CGM conditions to calculate the Mach Number ($M$):

$M=\frac{\lvert\vec{v}\rvert}{c_{s}}=\lvert\vec{v}\rvert\sqrt{\frac{m}{\gamma
kT}}$ (1)

where $\gamma=5/3$, $m$ is the effective mass of hydrogen ($\mu\times
1.67\times 10^{-27}$ kg where $\mu=0.6$), and $k$ is the Boltzmann constant
$1.381\times 10^{-23}\mathrm{\frac{J}{K}}$. In the fiducial model for the CGM
in Salem et al. (2015), the gas temperature of the CGM is 1.19$\times 10^{6}$
K. At this temperature the sound speed is 165 km/s, assuming $\gamma=5/3$;
consequently the LMC in the simulation is moving at Mach $\sim 2.1$. This
temperature assumption is well supported by x-ray observations of O VII and O
VIII that can be well modeled in a hot CGM plasma (Miller & Bregman, 2013,
2015). We note that because the Mach number scales as $T^{-1/2}$, if the CGM
is colder than we assume while still being thermally supported, we would
actually predict a stronger shock by as much as an order of magnitude that
would keep the shocked gas at $T\gtrsim 10^{6}$ K. However, this argument
would not hold in the case of a cooler cosmic ray dominated CGM (e.g., Ji et
al., 2020), where the pressure support would not be primarily thermal. Given
that there is strong observational evidence for CGM temperatures that would
result in the supersonic motion of the LMC, we predict that a bow shock will
lead the system. Throughout the rest of this letter, we quantify the strength
and shape of the shock assuming the fiducial CGM conditions from Salem et al.
(2015).

In Figure 1, we show the simulated gas density of the LMC and CGM projected
along the line of sight. The plots show all gas (LMC disk gas and CGM; left)
and only the hot CGM gas ($>10^{6}$ K; right). See Table 3 in Salem et al.
(2015) for the detailed coordinate transformations that takes the simulated
LMC from the box frame to a line-of-sight frame, where the LMC velocity vector
is aligned with the 3D Galactocentric velocity vector and the disk is inclined
correctly in Cartesian coordinates from our viewing perspective at the
location of the Sun.

It is clear that shocked (higher density and temperature) CGM gas surrounds
the simulated LMC. The projected column of hot gas is $\sim 2$ times larger
than that of the column through undisturbed ambient CGM. Additionally, while
the shock exists in front of the LMC gas disk, it is predominantly located
inside the LMC’s observed stellar disk, which is represented as a white dashed
circle of radius 18.5 kpc (Mackey et al., 2018).

In the following sections, we use this simulation to explore the influence of
the shock on the CGM gas as the LMC falls into the Milky Way’s potential.

Figure 3: The density (left) and temperature (right) in a slice through a
single resolution element of the simulation volume ($\sim 0.5$ kpc), which is
oriented parallel to the velocity vector along the x-axis. The color scale is
chosen such that the density and temperature are saturated for the LMC disk to
highlight the shock structure. The grid of vectors indicate the velocity of
the CGM relative to the LMC. As expected, the magnitude of the velocity at the
shock drops near the boundary, indicating that the shock is traveling with the
LMC (see also Figure 2). The purple points show the shock locations found in
the simulation by our shock finding algorithm. The white line shows the
expected shape for a bow shock around a spherically symmetric object (e.g., a
star, see Mac Low et al., 1991; Cox et al., 2012) with the same stand off
radius as the LMC ($R_{so}=6.7$ kpc). The shapes are similar, but the
orientation of the LMC disk relative to its velocity vector results in an
asymmetric shock.

##  4 Shock Conditions and Shape

###  4.1 Shock Conditions

Due to the complicated nature of hydrodynamic equations, complex behavior such
as the formation of a bow shock around the inclined LMC disk cannot be modeled
analytically. However, the leading edge of a shock can be approximated as one-
dimensional, and therefore can be analyzed to first order using fluid
equations. The Rankine-Hugoniot jump conditions characterize the strength of
the jump in terms of temperature ($T$) and density ($\rho$) ratios based on
the Mach number ($M$), and the heat capacity ratio ($\gamma$).

$\frac{T_{2}}{T_{1}}=\frac{[(\gamma+1)+2\gamma(M_{1}^{2}-1)][(\gamma+1)+(\gamma-1)(M_{1}^{2}-1)]}{(\gamma+1)^{2}M_{1}^{2}}$
(2)

$\frac{\rho_{2}}{\rho_{1}}=\frac{(\gamma+1)M_{1}^{2}}{(\gamma+1)+(\gamma-1)(M_{1}^{2}-1)}$
(3)

To compare the conditions in the simulation to the theoretical jump
conditions, we extract the density, temperature, and pressures of the
simulated gas along a ray which points from the center of the LMC along the
direction of the LMC velocity in the present day simulation slice. In Figure
2, we show the density, temperature, and pressure along this ray. In red, we
show the predicted shock strength from the Rankine-Hugoniot conditions
assuming $\gamma=5/3$, a CGM temperature of 1.19$\times 10^{6}$ K (see Figure
2), and a velocity of 350 km/s (corresponding to a Mach number of $\sim 2.1$),
the present day effective velocity of the LMC in the simulation. The density,
temperature, and pressure jumps in the simulation match the predicted ones
well, demonstrating that the resultant shock in the simulation is consistent
with theoretical predictions. We note that while the simulation does not
utilize radiative cooling, we do not expect this to affect the shock as the
cooling time for the gas is on the order of a 4-40 Gyr (assuming the ambient
CGM temperature and density in the fiducial simulation and a cooling rate of
order $\sim 1-10\times 10^{-23}\ \mathrm{erg\ s^{-1}\ cm^{3}}$, spanning a
wide range of possible metallicities of the gas), much longer than any other
relevant timescales within the simulation. We expect this to be true for the
shocked gas even in the case that the ambient CGM is significantly cooler than
we assume in the simulation, as the higher Mach number in conjunction with
Equation 2 implies that the temperature of the shocked region should be $\geq
10^{6}$ K regardless of the temperature of the ambient CGM.

Additionally, we measure the stand-off radius of the shock front, the distance
from the center of the LMC to the shock front along the direction of the LMC
velocity vector, and compare to empirically motivated lab measurements of
shock shape. Billig (1967) showed that the stand-off radius for a shock
generated around a rigid body, $R_{so}$, takes the form
$R_{so}/R=1+0.143e^{3.24/M}$ where $M$ is the Mach number. While the LMC disk
is neither a rigid body nor is it symmetric, we can approximate its radius as
$R_{\mathrm{eff}}=R_{\mathrm{gas}}\mathrm{sin}\theta$ where $R_{\mathrm{gas}}$
is $\sim 6$ kpc and $\theta$ is the angle between the LMC’s angular momentum
vector and its velocity. Substituting these values along with the Mach number,
2.1, we obtain an estimate for the stand off radius of $\approx 5.75$ kpc.
This is within $\sim 20\%$ of the stand-off radius in the simulation, 6.7 kpc
(see Figure 2). Throughout the rest of this work, we adopt the empirically
derived 6.7 kpc as our fiducial bow shock stand-off radius at present day.

Figure 4: The orbital path of the LMC (gas: grey, stars: yellow) and its
evolving bow shock over the last 500 Myr in the YZ Galactocentric plane.
Because the simulations allow for time evolving conditions of the wind density
and speed that mimics the LMC’s orbit, we can utilize our shock finding
algorithm to identify the location of the shock along the past orbit of the
LMC. The extended shock (green points) has carved out a significant volume of
the CGM in the southern sky over this period of time; we estimate that as much
as $\sim 8\%$ of the gas in the southern hemisphere may have interacted with
the boundary of the shock during this time period. Additionally, we show the
location of the SMC at each timestep (orange star) and the position of Ret 2
at present (blue diamond).

###  4.2 Shock Shape

While the empirical shapes of shocks surrounding rigid, symmetrical bodies are
well characterized (e.g. Mac Low et al., 1991; Cox et al., 2012), the diffuse
nature of the LMC gas as well as the angle between the disk orientation and
infall velocity results in an asymmetric shock shape that cannot be modeled
analytically. In Figure 3, we show slices ($\sim 0.5$ kpc, one resolution
element thick) in density (left) and temperature (right) centered on the LMC
in the present day simulation,. The x-axis in both slices is aligned with the
bulk flow simulation velocity (shown as black arrows), which is the inverse of
the LMC velocity at this time. The shock boundary can be clearly seen in both
density and temperature. However, because of the inclination of the LMC disk
relative to its motion, the shock is not symmetric. This can be clearly seen
in the comparison to the analytic solution for a shock around a rigid body
(Mac Low et al., 1991; Cox et al., 2012):

$y=\sqrt{3}R_{so}\sqrt{1-\frac{x}{R_{so}}},$ (4)

where $R_{so}$ is the stand-off radius, or the radius of the shock along the
velocity vector of the object’s supersonic motion, x is the coordinate aligned
with the velocity of the wind, and y is a vector perpendicular to x.

In Figure 3 we show this empirical shock shape for $R_{so}=6.7$ kpc (white
line). In the regions directly preceding and below the LMC in this slice, the
analytic model matches the shape of the shock quite well. However, the disk
inclination causes the simulated shock to have a wider opening angle in the
regions nearest to the disk, resulting in a strongly asymmetric simulated
shock.

In order to empirically measure the shock’s shape in the simulation, we
automate the finding of the shock by drawing rays in four planes parallel to
the LMC’s velocity from the center of the LMC to the edge of the box at angles
in the plane spanning $-140^{\circ}$ to $140^{\circ}$ where $0^{\circ}$ is
defined as the direction of the LMC’s velocity in the slice. Along each ray,
we define the shock location as the first point from the end of the ray where
the temperature reaches 25% of the predicted jump in temperature from the
Rankine-Hugoniot conditions (see Equation 2), using the velocity of the LMC
and the temperature of the gas to derive the Mach number at that time. This
procedure finds the location of the shock along a line of sight with a high
degree of accuracy, although it can sometimes fail in the early times of the
simulation when a line of sight moves through stripped LMC gas behind the LMC.
The results of this automated shock finding on a present day simulation slice
are also shown in Figure 3 as purple points, which, in contrast with the shape
prescribed for a symmetric shock around a sphere, trace the edge of the shock
extremely well.

In addition to measuring the shape of the shock in the present day slice, we
can also use the simulation to measure the evolution of the shock as a
function of time. In Figure 4 we show the location of the LMC+shock system in
six slices ranging from present day (left) to 500 Myr ago (right), assuming
that the orientation of the disk remains constant throughout the infall for
ease of visualization (though we note that the orientation of the disk is not
expected to change significantly via LMC/SMC interactions during the infall,
see Besla et al. 2012). As the velocity of the LMC increases as it approaches
pericenter, the shock’s opening angle becomes smaller and the stand-off radius
becomes smaller, but even 500 Myr ago the supersonic motion of the LMC clearly
results in a shock that subtends the extent of the LMC. Note that there are a
few measurements of the shock location that are not contiguous with the
otherwise continuous shocks. These tend to result from failures in our shock-
finding algorithm as our ray moves through trailing gas stripped material from
the LMC disk, especially in early simulation slices.

The physical extent of the shock is significantly larger than the LMC gas
disk. The shock extends $\sim 30$ kpc from the center of the LMC in the plane
traverse to the LMC velocity, in contrast with the 6 kpc gas disk.
Additionally, the leading edge of the shock is located well within the LMC’s
stellar disk ($\sim$18 kpc, see Mackey et al. 2018) due to the truncation of
the gas from ram pressure during infall (Salem et al., 2015). We predict that
along the past orbit of the LMC, the MW CGM should be $\sim 2-4$ times as
dense as the ambient CGM at that radius as evidence of the shock’s passage.

##  5 Observability of the Shock

Our simulation predicts that ram pressure stripping and the bow shock will
result in sharp boundaries between the LMC gas disk, the shocked region, and
the ambient CGM. As such, ionized gas probes such as $H\alpha$ emission, which
are strongly sensitive to the density and temperature of the gas, may trace
the different physical conditions in these regions. While it is likely that
$H\alpha$ emission in the star forming regions of cold ISM within LMC will
dominate the total flux in the region and the shock itself is too hot to be a
significant source of $H\alpha$ flux, we propose that interactions between the
shocked CGM gas and cold clouds surrounding the LMC may ionize hydrogen and
result in an extended $H\alpha$ signature that that extends beyond the cold
neutral gas and is systematically moving at the same velocity as the LMC.

Recent measurements of the LMC using the Wisconsin $H\alpha$ Mapper (WHAM)
show exactly such a feature. Smart et al. (2023) measured spatially extended
$H\alpha$ along the direction of the LMC’s leading edge at the systematic
velocity of the LMC (see Figure 7 in that work). In Figure 5, we overlay an
H$\alpha$ intensity map of the LMC obtained from the WHAM survey (Haffner et
al., 2003) that was studied in Smart et al. (2023) on our projected density
map of hot ($T>10^{6}$ K) gas. To do so, we follow the procedure in Smart et
al. (2023) to integrate the WHAM data from $-130$ to $110$ km s-1 in the
velocity reference frame of the LMC (see their Equation 5). We then project
the H$\alpha$ map into an orthographic projection based on Equation 1 in Gaia
Collaboration et al. (2021) (see also Choi et al., 2022) assuming an LMC
distance of 50.1 kpc (Freedman et al., 2001) and center of mass at (RA,
DEC)=(78.76o, $-69.19^{\rm o}$) (van der Marel & Kallivayalil, 2014), which
were the same values used in Salem et al. (2015). We show contours for
intensities of 0.1, 0.5, 5, and 50 Rayleighs with proportionally increasing
line widths. We find that the asymmetric feature along the leading arm is in
excellent agreement with the predicted shock front. While the majority of the
ionized gas is located within the gas disk of the LMC, the $I_{H\alpha}>0.1$ R
contours of the ionized hydrogen clearly extend into the shocked region past
the extent of the neutral gas disk and truncate near the edge of the shock.
While the analysis presented here is purely morphological (as the exact extent
of the shock is sensitive to the temperature of the ambient CGM, which is not
well constrained in Salem et al. 2015 and the distribution of cold clumps in
the region surrounding the LMC is a significant unknown), the alignment of
this asymmetric $H\alpha$ with our predicted shock provides a tantalizing
piece of observational evidence that supports the shock’s existence. A more
detailed prediction from this simulation of the $H\alpha$ brightness within a
shocked medium containing cold molecular clouds is currently underway.

Additionally, our simulations allow us to roughly predict the column density
excess caused by the presence of the shock, assuming an otherwise isotropic
and constant density CGM. To do so, we generate two gas density profiles, one
along a line of sight through our simulation box that points directly through
the shocked medium at the stand off radius ($\sim$ 6 kpc along the LMC’s
velocity vector), and another 15 kpc in front of the LMC, well past the stand
off radius of the shock. Computing the column density over a distance of 60
kpc, we find that the shock-tracing and non-shock tracing column densities are
$\sim 3\times 10^{19}\ \mathrm{cm}^{2}$ and $\sim 2\times 10^{19}\
\mathrm{cm}^{2}$, respectively. As such, we expect that the enhancement in
column density along a line of sight that intersects the leading edge of the
shock should be $\sim 10^{19}\ \mathrm{cm^{2}}$ for the $T\sim 10^{6}$ K
assumed in our simulation. As we show in Figure 1, this enhancement in column
density will not be restricted to the region directly in front of the shock,
but would be observable on scales of $\sim 30$ kpc tracing out the shape of
the shock.

Note, we have estimated the non-thermal emission to be produced by the LMC bow
shock via Fermi processes. To estimate the non-thermal emission produced via
Fermi process in the bow shock of the LMC, we calculated synchrotron radiation
($\lesssim 10^{-12}\ \mathrm{erg/s/cm^{2}/sr}$), emission from inverse Compton
scattering ($\lesssim 10^{-12}\ \mathrm{erg/s/cm^{2}/sr}$), and synchrotron
self-Compton emission ($\lesssim 10^{-14}\ \mathrm{erg/s/cm^{2}/sr}$)
following the methodology outlined by Wang & Loeb (2015). However, we find
that the non-thermal emission produced by the LMC shock will not be strongly
observable at any wavelength.

Figure 5: The simulated LMC-shock system presented in this work (as in Figure
1 for $T>10^{6}$ K) shown with observed $H\alpha$ contours measured with the
Wisconsin $H\alpha$ mapper (Smart et al., 2023). We show integrated $H\alpha$
contours encompassing 0.1, 0.5, 5, and 50 Rayeleigh intensity integrated
between -130 km/s and 110 km/s in the LMC frame of reference. We suggest that
the diffuse $H\alpha$ structure leading the LMC disk may be associated with
the shock, providing a clear explanation for the $H\alpha$ structure that is
more extended and follows a different morphology than the neutral HI gas.

##  6 Discussion

###  6.1 Satellite Bow Shocks as Sources of Mixing and Heating within the CGM

The infall of the LMC has been shown to exert significant perturbations on the
density and kinematics of the MW’s dark matter and stellar halos (Garavito-
Camargo et al., 2019; Conroy et al., 2021), highlighting that massive
satellite galaxies can give rise to non-equilibrium halo structures. We posit
that the LMC bow shock may similarly play a role in the structure of CGM gas,
as the bow shock associated with the LMC can sweep out a significant volume as
it falls into the MW.

In Figure 4, we show the physical extent of the LMC/shock system at six time
steps in relation to the static Milky Way disk, illustrating the path of the
shock over the past 500 Myr. To calculate the volume of gas that may have been
shocked during the LMC infall, we integrate along the LMC path, approximating
the shock as an $r=30$ kpc disk that is oriented perpendicular to the path of
the LMC at all points in the orbit. Over the past 500 Myr, the LMC has
traveled $\sim 160$ kpc, meaning that $\sim 5.5\times 10^{5}\
\mathrm{kpc^{3}}$ of CGM gas will have interacted with the bow shock and
potentially been perturbed during this period. This represents $\sim 8.2\%$ of
the total CGM volume bounded within the r$\sim 140$ kpc hemisphere of the
southern sky that the LMC has occupied over the past 500 Myr.

While the jump in density in temperature induced by the shock is modest given
the CGM conditions in the simulation, we posit that it still may influence the
CGM dynamics in two key ways. Firstly, the shock interaction with small scale
fluctuations in the CGM may provide an important source of turbulent pressure
support to the CGM at large radii. This effect has been observed in
cosmological simulations (see Figure 12 in Lochhaas et al., 2023), and we
posit that it is likely occurring in our own backyard on significant physical
scales. Secondly, if there are existing cool gas structures that the shock
sweeps past, the post-shocked gas will drive a velocity shear between the hot
post-shocked gas and the pre-existing cool gas, which could alter cloud
survival conditions (e.g., Gronke & Oh, 2018; Abruzzo et al., 2023) such that
small cool clouds that could survive in the ambient CGM would be destroyed in
the interaction with the shock. We note that the volume carved out by the
interaction will grow strongly with time, as the LMC is only on its first
infall (Besla et al., 2012) and the shock will continue to sweep out CGM gas
until the galaxies merge.

In other words, the infall of a massive satellite may be a significant driver
of the mixing of dense clouds with more diffuse CGM gas in the CGM,
particularly if the satellite has made multiple orbits (e.g. the case of the
Sagittarius dwarf about the MW and the progenitor of the Giant Southern Stream
about M31). This mechanism for mixing may be even more significant in cluster
environments, where both gas densities and galaxy speeds are higher.

###  6.2 Implications for the origin of ionized gas associated with the
Magellanic System

The Magellanic Clouds are currently contributing $4.87\times 10^{8}$ M⊙ (d/55
kpc$)^{2}$ (Brüns et al., 2005) of neutral HI to the MW’s CGM, where d is the
assumed Galactocentric distance to the gas. This gas is being removed from the
Clouds and forms large-scale gaseous structures, like the Magellanic Bridge
and Stream (Putman et al., 2003; Nidever et al., 2010). But this HI gas is
only $\sim 0.5-2\%$ of the MW’s total CGM mass budget (3-10 $\times 10^{10}$
M⊙; Faerman et al., 2022), assuming a distance to the Magellanic Stream of
d=55 kpc.

There is, however, also a significant ionized gas component associated
spatially and kinematically with the Magellanic Stream ($\sim 2\times 10^{9}$
M⊙ (d/55 kpc)2; Fox et al., 2014), Magellanic Bridge (0.7-1.7 $\times 10^{8}$
M⊙; Barger et al., 2013), and surrounding the LMC (0.6-1.8 $\times 10^{9}$ M⊙;
Smart et al., 2023). It has been posited that much of this gas originates in a
“Magellanic Corona” (Krishnarao et al., 2022) that is being removed through a
combination of ram pressure and tidal stripping (Lucchini et al., 2020).

The ionized gas component surrounding the neutral Magellanic Stream is a large
mass budget that could contribute between 3-10% of the MW’s total CGM mass,
assuming an average d=55 kpc to the Stream. If the Stream is located on
average at a larger distance of $\sim$100 kpc (Besla et al., 2012), the
associated ionized gas could contribute 8-26% of the total CGM mass budget.
This is a non-negligible contribution to the CGM, making it important to
understand the origin of this ionized gas.

In this study, we have illustrated that MW CGM gas ahead of the LMC should be
accelerated to the LMC’s systemic speed and form a bow shock. We posit that
the interactions between this hot, shocked gas and cooler clouds within the
CGM could result in elevated ionization in the region trailing the LMC infall.
This means that the bow shock could explain a significant portion of the
ionized gas observed surrounding the LMC, without appealing to an existing LMC
CGM that resists dispersal by ram pressure (Section 5), providing an
alternative explanation for the ionized gas observed at high ionization states
associated with a “Magellanic Corona”. Indeed, the predicted enhancement in
column density along the the lines of sight through the shock (see Figure 1
and Section 5) are within $\sim$ a factor of 2-3 of the inferred coronal
columns from Figure 3 in Krishnarao et al. (2022), indicating that a bow shock
provides a plausible explanation for a significant amount of the excess
material observed around the LMC.

This scenario further implies that, as the LMC moves through the MW’s CGM, it
would leave a trail of ionized/shocked CGM gas (see Figure 4). In other words,
the observation of ionized gas spatially trailing the past orbit of the Clouds
and moving at speeds consistent with the neutral Stream does not require that
the ionized gas be removed from the Clouds themselves. Rather, this ionized
gas could plausibly result from the interface between the MW’s CGM and the
LMC’s gas disk, or the neutral hydrogen Stream, as both structures move
through the MW’s CGM. The observed line ratios of CIV, OVI, SiIV, etc,
surrounding the Stream (Fox et al., 2010, 2014) strongly point to
interface/mixing layers between the neutral HI in the Stream and the MW CGM.
However, shocks can produce similar ratios (e.g., Wakker et al., 2012),
meaning that it is plausible that some of the observed ionized gas stems
directly from the interaction between the bow shock and the MW CGM.

This idea is important for understanding the total mass budget of the
Magellanic Clouds at infall, which has consequences for our understanding of
the baryon fraction in low mass galaxies. Many models of the Magellanic Stream
require the bulk of its gas mass to originate from the SMC (e.g., Besla et
al., 2012). If we assume that 50% of the ionized + neutral gas budget of the
Magellanic Stream and Bridge originate from the SMC, and that the SMC had an
infall halo mass of order $2\times 10^{10}$ M⊙ (Besla et al., 2010), and
current neutral gas mass of $4\times 10^{8}$ M⊙ (Brüns et al., 2005), the SMC
would need to have a baryon fraction of $\sim$8-23% at infall, depending on
the distance to the Magellanic Stream (55-100 kpc). Such baryon fractions are
much higher than the 3-5% in MW-like galaxies. In the shallower halo
potentials of low mass galaxies, stellar winds should be more efficient,
making baryon fractions even lower, not higher (Besla, 2015). Importantly, the
total ionized gas mass associated with the Stream has been used as an argument
against tidal-driven models for the origin of the Stream (see review in
D’Onghia & Fox, 2016), but if much of that gas is of MW CGM origin, the mass
discrepancies between the models and observations are significantly less of an
issue.

This argument about the origin of the Magellanic corona does not imply that
the LMC did not once possess an ionized CGM. Indeed, cosmological simulations
of isolated LMC mass analogs do possess CGMs (Jahn et al., 2022). Thus, some
component of the present-day ionized gas surrounding the Magellanic Stream
could come from the LMC’s CGM that has been stripped. This is the case in the
simulations of Salem et al. (2015), who initialize their LMC with a low mass
CGM ($5\times 10^{6}$ M⊙) that is almost immediately swept into an ionized
tail owing to ram pressure. Given that the truncated HI disk of the LMC
provides clear evidence of direct impact from ram pressure (Salem et al.,
2015), our simulation suggests that the ionized gas at high ionization states
within 20-30 kpc of the LMC observed by Krishnarao et al. (2022) is unlikely
to be the primordial CGM gas that the LMC possessed at infall.

We note that the initialized LMC halo mass in our simulations is $\sim 3$
orders of magnitude lower than that in recent simulations where the LMC CGM
has survived the infall into the LMC halo as a Magellanic Corona Lucchini et
al. (2020). In these simulations, some of the initial LMC CGM does survive the
interactions with the MW CGM. However, our simulations are designed to match
the observed truncation of the LMC’s gaseous disk Salem et al. (2015). Ram
pressure stripping is dependent on the gas surface density of the infalling
satellite. In order to achieve a truncation radius for the HI disk that is
compatible with observations, the ram pressure faced by the LMC must be
sufficiently strong to remove high gas density material in the disk at
distances of $\sim$5-6 kpc. By definition this ram pressure must be strong
enough to also remove the initial CGM gas, which is always lower density than
the gas disk itself. Indeed in Salem et al. (2015) the simulated low density
LMC CGM is removed by ram pressure before the gas disk is truncated. This
would be true regardless of the LMC halo mass, although the removed CGM may
still be retained at large distances in a more massive LMC model. As a result,
we propose here instead that the ionized gas within 20-30 kpc of the LMC
observed by Krishnarao et al. (2022) is not the primordial LMC CGM (i.e. CGM
the LMC possessed at infall). Rather, this gas is likely a combination of new
outflows from ongoing star formation in the LMC’s disk as well as shocked CGM
gas from the LMC’s bow shock.

Table 1: Visual classification of satellite positions relative to the LMC shock at a given time. All satellites that we find to have been near the shock at any point in the simulation are listed here. A label of $<10$ kpc indicates that a MW satellite is within 10 kpc of any point of the 3D shock boundary at that step in the simulation.  Time | Aqu2 | Car3 | Hor1 | Hor2 | Hyi1 | Hyd2 | Phx2 | Pis2 | Ret2 | Sag2 | SMC | Tuc3 | Tuc4 | Tuc5
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
[Myr Ago] |  |  |  |  |  |  |  |  |  |  |  |  |  |
-500 |  |  $<$10 kpc |  |  |  |  |  |  $<$10 kpc |  |  |  |  |  |
-400 |  |  $<$10 kpc |  |  |  |  $<$10 kpc |  $<$10 kpc |  |  | In shock |  |  |  |
-300 |  |  $<$10 kpc |  |  | In shock |  | In shock |  |  | In shock |  |  |  |
-200 |  $<$10 kpc | In shock |  |  | In shock |  |  $<$10 kpc |  |  |  |  $<$10 kpc |  |  |
-100 |  |  $<$10 kpc |  |  | In shock |  |  |  |  |  $<$10 kpc | In shock |  | In shock |  $<$10 kpc
Present |  |  |  $<$10 kpc |  $<$10 kpc |  $<$10 kpc |  |  |  | In shock |  | In shock |  $<$10 kpc |  |

###  6.3 Interaction of the LMC’s Bow Shock with Satellites

As the LMC’s bow shock intersects with a significant volume of the ambient
CGM, it is also possible that the bow shock has interacted with satellite
galaxies associated with the LMC. To investigate which satellite galaxies may
have interacted with the bow shock, we use orbital histories calculated
following the methodology of Patel et al. (2020), which includes the combined
gravitational influence of the MW, LMC, and SMC.

We adopt their low MW mass (virial mass of $1\times 10^{12}\,M_{\odot}$)
gravitational potential and identical parameters. However, the MW center of
mass is held fixed and does not move in response to the LMC’s passage, since
the MW is also fixed relative to the LMC in the orbit used in the Salem et al.
(2015) simulations. Secondly, we model the LMC’s gravitational potential as a
Plummer sphere with a mass of $1.8\times 10^{11}\,M_{\odot}$ and a Plummer
scale length of $a=20$ kpc. The dynamical friction imparted by the MW and LMC
remain unchanged. All satellites were treated identically to Patel et al.
(2020). They identified six satellite galaxies that may be dynamically
associated to the LMC. These include six ultra-faint dwarfs (Phx2, Ret2, Car2,
Car3, Hor1, Hyi1) and the SMC. We construct orbital histories using the line-
of-sight velocities and distances as listed in Patel et al. (2020). Proper
motions are adopted from $Gaia$ eDR3 as reported in McConnachie & Venn (2020).

We use the defined LMC+shock system at each of the six timesteps in Figure 4
along with the orbital histories of all ultra-faint and classical dwarf
galaxies explored in Patel et al. (2020) to assess whether a given satellite
is near or inside the shock at a given timestep. We flag satellites as “In
shock” if they are located between the 3D shock boundary and the LMC, and
additionally identify galaxies that are within 10 kpc of the 3D shock
boundary. The results of this visual classification are presented in Table 1.
Note that we have not explored the orbital uncertainties resulting from
measurement errors on the 6D phase space properties of each galaxy.

The LMC’s companion galaxy, the SMC, has spent a considerable amount of its
recent history inside the shock and near the boundary (see the star in Figure
4). The SMC is also moving supersonically through the CGM: its speed of $\sim
250$ km/s (Zivick et al., 2018) translates to a Mach number of $\sim$1.5,
assuming the same model for the CGM as adopted in our simulations. Since the
SMC is a gas rich galaxy, it should also produce a bow shock, but that shock
will be weaker and smaller than that of the LMC. Interestingly, since the SMC
is in orbit about the LMC, it is likely that the shocks of the LMC and the SMC
themselves have been interacting over the past $\sim 500$ Myr. The interaction
between bow shocks could provide an additional source of heating to the CGM.
This may be more generalizable in galaxy cluster environments.

We find that Car3 and Hyi1 are the most likely other candidates for extended
periods of interaction with the LMC shock; they appear to have spent $\sim
500$ Myr of their orbits near the shock and have likely crossed the boundary
at least once. Phx2, and Sag2 are also candidates for shock interaction given
the amount of time ($\sim 300$ Myr) they have spent near the LMC.
Additionally, we find that Ret2 is inside the shock in the present day
simulation slice (see Figure 4).

While the jump in density due to the shock is modest, the physical extent of
the shock is significantly larger than that of the LMC’s gas disk.
Interactions with this higher density gas may have had the effect of enhancing
ram pressure on the gas in associated satellite galaxies, expediting the
removal of gas. Ultra-faint dwarfs are largely quenched during the epoch of
reionization (Brown et al., 2014), although there may be some differences in
quenching timescales for the LMC satellites (Sacchi et al., 2021). While these
results imply that ultra-faint dwarfs would not retain significant amounts of
star forming gas at late times, cosmological simulations illustrate that
isolated ultra-faint dwarfs can retain significant reservoirs of ionized gas,
and even neutral gas (depending on their halo mass), after reionization (Jeon
et al., 2017, 2019). Emerick et al. (2016) use hydrodynamic simulations to
illustrate that ram pressure stripping is not very efficient at removing gas
from the inner regions of even ultra-faint dwarfs. Meaning that it is
plausible that ultra-faint dwarfs may retain some gas reservoirs even while in
orbit about the MW. If ultra-faint dwarfs also cross the LMC bow shock
boundary, they would experience higher ram pressure efficiency than average.
This may help to remove all gas from the system, or maybe augment the
ionization state of the gas inside the galaxy.

The idea that ultra-faint dwarfs may interact with or be in proximity to the
LMC’s bow shock raises an interesting connection with indirect dark matter
detection efforts. In particular, we find that the Ret2 ultra-faint dwarf is
currently located inside the LMC’s bow shock. Intriguingly, this galaxy has
been found to harbor a gamma ray excess that has been potentially attributed
as a signal of dark matter annihilation (Hooper & Linden, 2015; Bonnivard et
al., 2015; Geringer-Sameth et al., 2015; Drlica-Wagner et al., 2015). While we
do not find that the gamma ray emission from the shock itself will be
substantial, it could help to augment any gamma ray signal that already exists
in Ret2 owing to dark matter annihilation. Moreover, if Ret2 is interacting
with the shock and harbors any residual gas, the gas could be sufficiently
excited to emit gamma rays. This scenario may help to explain why the signal
is particularly strong in Ret2, versus other similarly massive ultra-faint
dwarfs.

We suggest that the LMC bow shock may be an intriguing complication to the
interpretation of any detection of gamma ray excess in dwarf galaxies that are
in its proximity, making our study of the geometry of the shock a critical
tool to understand all astrophysical origins for a gamma ray excess.

##  7 Conclusions

Using a simulation of the LMC’s gas disk over the past 500 Myr (Salem et al.,
2015), we characterize the size and structure of the bow shock that is
predicted to accompany the LMC as it falls into the MW potential at supersonic
speeds ($M\sim 2.1$). We find that:

  * •

The MW CGM gas leading the LMC is expected to jump in density and temperature
by a factor of 2-3, matching theoretical predictions (see Figure 2). We
characterize the shape of the shock and find that it is asymmetric due to the
mismatch in orientation between the LMC’s disk and velocity vector (see Figure
3).

  * •

We predict that the distance to the shock front along the direction of the LMC
velocity vector should be $\sim 6.7$ kpc with a sharply defined discontinuity.
The shock’s approximate transverse diameter should be $\sim 30$ kpc at present
day.

  * •

The shock swept over a large fraction of the CGM ($\sim 8\%$ of the southern
sky at r$<$140 kpc, see Figure 4). The infall of massive satellites may
present an important and understudied dynamical mixing process in the CGM.

  * •

The morphology of the predicted shock aligns very well with observed H$\alpha$
emission that extends along the LMC leading edge and moves at speeds
consistent with the systemic speed of the LMC (see Figure 7 in Smart et al.
2023). We propose that this occurs owing to interactions between the shocked
MW CGM gas and the cold clouds surrounding the LMC, making this emission a
signature of the existence of the bow shock.

  * •

Given the ram pressure stripping required to truncate the LMC gas disk Salem
et al. (2015), our results suggests that the observed increase in ionized gas
near the LMC (e.g. Krishnarao et al., 2022), is newly formed from stellar
outflows or shocked MW CGM gas, rather than being a primordial, pre-infall LMC
CGM.

  * •

The SMC has spent the past $\sim 200$ Myr inside or near the shock. Because it
is moving supersonically, it is very likely that there have been shock-shock
interactions between the LMC and SMC shocks that could act as source of CGM
heating.

  * •

Many other satellite galaxies may have interacted with the shock over the past
500 Myr. In particular, we find that Car3, Hyi1, Phx2, and Sag2 are likely
candidates for having interacted with the shock, which might have helped to
remove gas from these systems. We further find that at present day, the Ret2
ultra-faint dwarf is in proximity to the shock, which may complicate efforts
to understand the origin of any detected gamma ray excess.

The presence of a massive satellite with a bow shock in the CGM of our own
Galaxy underscores the importance of studying the dynamics of massive
satellites in addition to inflows/outflows to understand the properties of the
multi-phase CGM. In particular, we posit that the accretion of massive
satellites is an important dynamical process that can aid in mixing in the
CGM. Given that most MW-mass galaxies have accreted an LMC-mass subhalo at
some point in the past (Stewart et al., 2008), this work may have relevance
for understanding the CGM of other MW analogs (e.g., Tumlinson et al., 2011;
Lehner et al., 2020). More generally, this study has relevance for a wide
range of environments where galaxies are moving at high speeds through a
gaseous medium, such as cluster jellyfish galaxies (e.g., Poggianti et al.,
2017; Werle et al., 2022).

This study further illustrates how the LMC can change the kinematic properties
of the MW’s CGM, forcing CGM gas to move at speeds similar to its own systemic
velocity. This result has important consequences for the interpretation of the
origin of ionized gas associated with the LMC/SMC system. This gas has thus
far been attributed to gas removed from the Magellanic Clouds, however, we
argued that this gas could simply be CGM gas from the MW with kinematics that
have been altered by the passage of the LMC.

Support for this work was provided by The Brinson Foundation through a Brinson
Prize Fellowship grant. GB acknowledges support from NSF CAREER award
AST-1941096. EP acknowledges financial support provided by NASA through grant
number HST-GO-16628. Support for this work was also provided by NASA through
the NASA Hubble Fellowship grant # HST-HF2-51540.001-A awarded by the Space
Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Incorporated, under NASA contract
NAS5-26555. We thank Yumi Choi for sharing her orthographic projection code,
which makes it possible to directly compare observed H$\alpha$ emission to the
simulation in Figure 5. We also thank Xiawei Wang for assisting with the
calculations related to the observability of the shock. Additionally, we thank
Jess Werk and Mary Putman for providing early comments on the paper. Finally,
we thank Brianna Smart for helping us acquire the WHAM data for the generation
of Figure 5.

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