# Non-thermal radio supernova remnants of exiled Wolf-Rayet stars

D. M.-A. Meyer1, M. Pohl1,2, M. Petrov3 and L. Oskinova1,4
1 Universität Potsdam, Institut für Physik und Astronomie, Karl-Liebknecht-
Strasse 24/25, 14476 Potsdam, Germany
2 DESY Platanenallee 6, 15738 Zeuthen, Germany
3 Max Planck Computing and Data Facility (MPCDF), Gießenbachstrasse 2, D-85748
Garching, Germany
4 Department of Astronomy, Kazan Federal University, Kremlevskaya Str 18,
Kazan, Russia
E-mail<EMAIL_ADDRESS>
(Received; accepted)

###### Abstract

A signification fraction of Galactic massive stars ($\geq 8\,\rm M_{\odot}$)
are ejected from their parent cluster and supersonically sail away through the
interstellar medium (ISM). The winds of these fast-moving stars blow
asymmetric bubbles thus creating a circumstellar environment in which stars
eventually die with a supernova explosion. The morphology of the resulting
remnant is largely governed by the circumstellar medium of the defunct
progenitor star. In this paper, we present 2D magneto-hydrodynamical
simulations investigating the effect of the ISM magnetic field on the shape of
the supernova remnants of a $35\,\mathrm{M}_{\odot}$ star evolving through a
Wolf-Rayet phase and running with velocity $20$ and $40\,\rm km\,\rm s^{-1}$,
respectively. A $7\,\mu\rm G$ ambient magnetic field is sufficient to modify
the properties of the expanding supernova shock front and in particular to
prevent the formation of filamentary structures. Prior to the supernova
explosion, the compressed magnetic field in the circumstellar medium
stabilises the wind/ISM contact discontinuity in the tail of the wind bubble.
A consequence is a reduced mixing efficiency of ejecta and wind materials in
the inner region of the remnant, where the supernova shock wave propagates.
Radiative transfer calculations for synchrotron emission reveal that the non-
thermal radio emission has characteristic features reflecting the asymmetry of
exiled core-collapse supernova remnants from Wolf-Rayet progenitors. Our
models are qualitatively consistent with the radio appearance of several
remnants of high-mass progenitors, namely the bilateral G296.5+10.0 and the
shell-type remnants CTB109 and Kes 17, respectively.

###### keywords:

methods: MHD – radiation mechanisms: non-thermal – stars: massive – ISM:
supernova remnants.

##  1 Introduction

Massive stars are born with masses $M_{\star}\geq 8\,\rm M_{\odot}$. Despite
their rareness, they are of prime importance in the cycle of matter in the
interstellar medium (ISM) of our Galaxy (Langer, 2012). After a relatively
long hydrogen-burning main-sequence phase, they experience a series of
evolutionary phases characterised by abrupt changes in their surface
properties (radius, effective temperature, mass-loss rate and wind velocity).
Those evolutionary phases alternate between hot, possibly eruptive phases of
dilute supersonic winds (Brott et al., 2011) and colder periods of inflated
radius with a dense, slow stellar wind (Ekström et al., 2012). The number and
characteristics of the post-main-sequence phases are, amongst other, a
function of the initial mass, the intrinsic rotation (Yoon & Langer, 2005),
and the chemical composition of the star (Sanyal et al., 2017). These various
winds produce shells in the circumstellar medium that develop instabilities
and eventually collide together (Garcia-Segura et al., 1996; Freyer et al.,
2003, 2006). They chemically enrich the ISM and drive turbulence in it, on
account of the large amount of momentum and energy the winds deposit in the
stellar surroundings. Finally, the majority of massive stars end their life as
core-collapse supernova, whose shock wave expands into their circumstellar
medium (Woosley et al., 2002), shaped by stellar winds and radiation.

Wolf-Rayet stars are an advance evolutionary stage of stars with initial mass
$\geq 20\,\rm M_{\odot}$, generally following a supergiant phase. Their
stellar winds are fast, dense, and chemically enriched in C, N and O elements
(Hamann et al., 2006; Bestenlehner et al., 2014; Sander et al., 2012). The
interaction of fast Wolf-Rayet wind with slower wind material expelled at
previous evolutionary stage results in complex stellar surroundings such as
ring nebulae (Gvaramadze et al., 2010a; Stock & Barlow, 2010; Meyer et al.,
2020) or bipolar bubbles (Gvaramadze et al., 2010b). A fraction of Wolf-Rayet
stars are located at high Galactic latitude (Munoz et al., 2017; Toalá et al.,
2018; Moffat et al., 1998). These fast-moving Wolf-Rayet stars that left their
parent stellar clusters and reached low-density regions of the ISM. There,
they eventually explode as a core-collapse supernova inside the cavity carved
by the stellar wind (Franco et al., 1991; Rozyczka et al., 1993; Dwarkadas,
2007). Wolf-Rayet-evolving stars are therefore ideal progenitor candidates for
core-collapse supernova remnants (Katsuda et al., 2018).

Several mechanisms determine the morphology of the supernova remnants of
massive progenitors. Clumpiness affecting the shock-wave propagation can arise
from wind-wind interaction as observed in the supernova remnant Cas A (van
Veelen et al., 2009). In addition to instabilities directly developing in the
supernova explosion itself (Janka et al., 2016), asymmetries in supernova
remnants may be a direct consequence of interactions between the expanding
shock wave and an anisotropic circumstellar medium. Of prime importance for
the shaping of supernova remnants is the peculiar motion of very high-mass
progenitors moving through the ISM (Meyer et al., 2020). As an example, RWC 86
(Broersen et al., 2014; Gvaramadze et al., 2017) or the Cygnus Loop
(Aschenbach & Leahy, 1999; Fang et al., 2017) reveal features consistent with
the typical characteristics of off-center explosions in massive stellar wind
bubbles, suggesting that they might have been produced by a fast-moving
progenitor, see also Toledo-Roy et al. (2014). All the numerous mechanisms,
that induce deviations from sphericity in supernova shock waves, can operate
in parallel, providing a huge parameter space governing the evolution of core-
collapse supernova remnants. Explanations of their observed morphologies are
subject to degeneracies and alternative scenarios. Runaway Wolf-Rayet stars
constitute therefore the ideal candidates for the production of isolated,
_asymmetric_ core-collapse supernova remnants (Meyer et al., 2015).

The structure and properties of the ISM are also involved in the shaping of
supernova remnants (Ferreira & de Jager, 2008). The ISM has an intrinsic
filamentary, turbulent, and magnetised nature. Its gravito-turbulent
evolution, powered by the formation of massive pre-stellar cores, stellar wind
outflows, and supernova feedback enriching the ISM, drives turbulence in it
and participates in the formation of the next generation of stars. Native ISM
magnetic field is an important player in the evolution of the circumstellar
medium around massive stars. As an example, the internal physics of bow shock
nebulae around runaway stars (Gvaramadze et al., 2014; Meyer et al., 2014; van
Marle et al., 2014; Meyer et al., 2017), as well as the organisation of
supernova remnants (Orlando et al., 2007; Ferreira & de Jager, 2008; Orlando
et al., 2008; Schneiter et al., 2010; Orlando et al., 2012) are partially
determined by the local ambient magnetic field. Furthermore, the local
direction of magnetic field makes thermal conduction anisotropic (Balsara et
al., 2008; Meyer et al., 2017) and can suppress (magneto-)hydrodynamical
instabilities (Viallet & Baty, 2007; van Marle et al., 2014). Importantly, it
has been shown that the magnetisation of the ISM strongly elongates stellar
wind bubbles around static massive stars along the direction of the local
field lines (van Marle et al., 2015). The question is therefore, how important
for the shaping of the supernova remnants are the effects of the ISM
magnetisation as compared to those of the motion of runaway Wolf-Rayet
progenitors?

In this work, we investigate, by means of numerical magneto-hydrodynamical
(MHD) simulations, the effects of a background ISM magnetic field on the
morphological evolution of supernova remnants generated by runaway massive
progenitors. We adopt the standard two-dimensional axisymmetric approach
developed by many authors (Comerón & Kaper, 1998; Mackey et al., 2012; Meyer
et al., 2016). It consists in first modeling the pre-supernova circumstellar
medium of massive progenitors before launching a supernova blastwave in it
(Velázquez et al., 2006; Chiotellis et al., 2012; van Marle et al., 2012;
Meyer et al., 2015). We examine the remnant morphologies and perform radiative
transfer calculations for their non-thermal radio synchrotron emission maps.
The mixing of supernova ejecta, stellar winds and ISM material is also
discussed, comparing models with and without ISM magnetic field. Last, we
discuss these remnants in the context of cosmic-ray acceleration before
comparing them to Galactic supernova remnants from massive progenitors.

Our study is organised as follows. First, we present the numerical methods
used for the MHD simulations of supernova remnants of $35\,\rm M_{\odot}$
runaway massive stars in Section 2. We describe our results for the dynamical
evolution of both the stellar surroundings and the supernova remnant, together
with predictive non-thermal radio synchrotron emission maps of these objects
in Section 3. We analyse therein the effects of the presence of the ISM
magnetic field onto the remnants evolution. Our results are further discussed
in Section 4, and finally, we present our conclusions in Section 5.

##  2 Numerical simulations

This section describes the methods used to perform simulations of the
circumstellar medium of a $35\,\mathrm{M}_{\odot}$ massive star evolving up to
the Wolf-Rayet phase and ending its life in a supernova explosion. We simulate
the stellar surroundings from the zero-age phase of the progenitor to the late
phase of supernova remnant evolution, varying the velocity of the star
relative to the ISM and investigating the role of the ISM magnetic field. The
simulations are used for further radiative-transfer calculations of non-
thermal radio synchrotron emission.

###  2.1 Simulation method for the pre-supernova phase

The pre-supernova circumstellar medium around the progenitor star is the wind-
blown bubble generated by interaction between the stellar wind and the local
ISM. We simulate it as described in Meyer et al. (2020). We first perform 2D
cylindrical, axisymmetric, magneto-hydrodynamics numerical models with a
coordinate system $[z_{\rm min};z_{\rm max}]\times[O;R_{\rm max}]$ which is
mapped with a uniform grid of spatial resolution $R_{\rm max}/N_{\rm R}$. The
stellar wind of the $35\,\mathrm{M}_{\odot}$ star is released at the center of
the domain into a uniformly distributed ISM. A circular wind zone of radius
$20$ cells is filled with the wind density profiles,

$\rho_{w}(r)=\frac{\dot{M}}{4\pi r^{2}v_{\rm w}},$ (1)

where $\dot{M}$ is the wind mass-loss rate at different evolutionary phases
interpolated from a stellar evolutionary track, $r$ is the distance to the
origin of the domain, $O$, and $v_{\rm w}$ is the velocity of the stellar wind
(Comerón & Kaper, 1998; van Marle et al., 2011; van Marle et al., 2014).

In Fig. 1 we show the evolutionary path of the star and its wind, that we use
in the simulations. The stellar mass (panel a, in $\rm M_{\odot}$), the mass-
loss rate (panel b, in $\rm M_{\odot}\,\rm yr^{-1}$), and the terminal wind
velocity (panel c, in $\rm km\,\rm s^{-1}$) are displayed beginning at the age
$3\,\rm Myr$. The wind properties of this zero-age-main-sequence, non-rotating
$35$-$\rm M_{\odot}$ star at Galactic metallicity has been interpolated from
the Geneva library of stellar models calculated with the genec code (Ekström
et al., 2012) by means of the online interface
syclist111https://www.unige.ch/sciences/astro/evolution/en/database/syclist/.
The terminal speed, $v_{\rm w}$, is modified for high effective temperatures
and massive stars using the approximation of Eldridge et al. (2006),

$v_{\rm w}=\sqrt{\beta(T)}v_{\rm
esc}=\sqrt{\beta(T)\frac{2GM_{\star}}{R_{\star}}},$ (2)

where $v_{\rm esc}$ is the escape speed of the star, $R_{\star}$ the stellar
radius, and,

$\beta_{\rm w}(T)=\begin{cases}1.0&\text{if }T\leq 10000\,\rm K,\\\
1.4&\text{if }T\leq 21000\,\rm K,\\\ 2.65&\text{if }T>21000\,\rm K,\\\
\end{cases}$ (3)

a corrective function depending on the temperature $T$.

The star first experiences a rather long main-sequence phase lasting about
$4.8\,\rm Myr$, blowing winds with $\dot{M}\approx 10^{-6.2}\,\rm
M_{\odot}\,\rm yr^{-1}$ and $v_{\rm w}\approx 3000\,\rm km\,\rm s^{-1}$. After
the long main-sequence phase, the star becomes cooler and inflates to become a
red supergiant with mass-loss rate $\dot{M}\approx 10^{-4}\,\rm M_{\odot}\,\rm
yr^{-1}$ and wind speed $v_{\rm w}\approx 50\,\rm km\,\rm s^{-1}$. It finally
evolves to the Wolf-Rayet phase, characterised by both a high mass-loss rate
($\dot{M}\approx 10^{-5.0}\,\rm M_{\odot}\,\rm yr^{-1}$) and a large wind
speed ($v_{\rm w}\approx 1500\,\rm km\,\rm s^{-1}$).

Figure 1: Stellar properties at the end of the main-sequence and during the
post-main-sequence evolution of the $35\,\mathrm{M}_{\odot}$ star. The panels
show the stellar mass (top, panel a), mass-loss rate (middle, panel b), and
wind velocity (bottom, panel c) as function of time (in $\rm Myr$).

To study the circumstellar medium around runaway Wolf-Rayet stars, we
conducted a series of simulations with varying stellar velocities spanning
from $v_{\star}=10$ to $v_{\star}=40\,\rm km\,\rm s^{-1}$. The star moves in
$z$-direction, and we simulate in the frame of the star, ISM gas of number
density $n_{\rm ISM}\approx 0.79\,\rm cm^{-3}$ and temperature $T_{\rm
ISM}\approx 8000\,\rm K$ as in the H ii regions around hot stars. The ISM
material flows in with speed $v_{\star}$ at the boundary $z=z_{\rm max}$.
Outflow boundaries conditions are set at $z=z_{\rm min}$ and $R=R_{\rm max}$,
respectively. Each value of $v_{\star}$ is explored with and without
magnetization of the ISM. The ISM magnetic field direction is parallel to the
$Oz$ axis, as a direct consequence of the simulation geometry, and it is set
to $B_{\mathrm{ISM}}=7\,\mu\rm G$ that is typical value for the warm phase of
the ISM (van Marle et al., 2014; van Marle et al., 2015; Meyer et al., 2017).
The flow of material past the stellar wind is characterised by the Alfvén
speed,

$v_{\rm A}=\sqrt{\frac{\mn@boldsymbol{B}_{\rm ISM}\cdot\mn@boldsymbol{B}_{\rm
ISM}}{4\pi nm_{\rm H}}},$ (4)

that together with the sound speed (Eq. 18) determines the Alfvénic and sonic
Mach number of the stellar wind bubble in the ISM. We list both for each model
in Table 1.

A continuity equation,

$\frac{\partial(\rho Q_{1})}{\partial
t}+\mn@boldsymbol{\nabla}\cdot(\mn@boldsymbol{v}\rho Q_{1})=0,$ (5)

is used to trace the mixing of stellar wind material into the ISM, with $\rho$
the mass density, respectively. Initially, the tracer $Q_{1}$ is set to
$Q_{1}(\mn@boldsymbol{r})=1$ in the wind and to $Q_{1}(\mn@boldsymbol{r})=0$
in the ISM.

###  2.2 Supernova explosion

After establishing the circumstellar medium around the pre-supernova massive
star, we simulate the supernova explosion as a spherically-symmetric shock
wave expanding into the freely-expanding stellar wind of the progenitor. The
supernova-wind interaction then serves as initial condition of a subsequent
two-dimensional calculation of the corresponding remnant (Meyer et al., 2015;
Meyer et al., 2020). The properties of the blastwave are parametrised by the
explosion energy, $E_{\rm ej}=10^{51}\,\rm erg$, and the ejecta mass,

$M_{\rm
ej}=M_{\star}-\int_{t_{\mathrm{ZAMS}}}^{t_{\mathrm{SN}}}\dot{M}(t)~{}dt-
M_{\mathrm{NS}}=11.64\,\rm M_{\odot},$ (6)

where $t_{\mathrm{ZAMS}}$ and $t_{\mathrm{SN}}$ denote the times of zero age
and supernova, respectively, and $M_{\mathrm{NS}}=1.4\,\rm M_{\odot}$ is the
mass of the remnant neutron star left behind the supernova explosion.  Note
that we use the canonical explosion energy typically taken in hydrodynamical
simulations of supernova remnants (van Veelen et al., 2009; van Marle et al.,
2010; van Marle et al., 2012). However, detailed dedicated studies estimate
the energy released throughout the explosion of a core-collapse progenitor to
be rather in the range $E_{\rm ej}=1$$-$$5\times 10^{50}\,\rm erg$ (Smartt,
2009; Janka, 2012; Moriya et al., 2018).  A passive scalar,
$Q_{2}(\mn@boldsymbol{r})$, obeying the continuity equation,

$\frac{\partial(\rho Q_{2})}{\partial
t}+\mn@boldsymbol{\nabla}\cdot(\mn@boldsymbol{v}\rho Q_{2})=0,$ (7)

is used to distinguish supernova ejecta from stellar wind or ISM material, by
setting $Q_{2}(\mn@boldsymbol{r})=1$ in the supernova-ejecta region and
$Q_{2}(\mn@boldsymbol{r})=0$ otherwise.

The supernova shock wave is released into the progenitor’s stellar wind bubble
(Whalen et al., 2008; Zirakashvili & Ptuskin, 2018) by superposing a 1D
blastwave density profile, $\rho(r)$, onto the pre-supernova wind
distribution. We used a typical ejecta profile for the early expansion of a
core-collapse supernovae. It involves a homologuous expansion, $v=r/t$, the
radius of the progenitor star’s core at the time of the supernova,
$r_{\mathrm{core}}$, and the outermost extension $r_{\mathrm{max}}$ of the
blastwave. We start the calculations at

$t_{\rm max}=\frac{r_{\rm max}}{v_{\rm max}},$ (8)

where $v_{\rm max}=30000\,\rm km\,\rm s^{-1}$ is the ejecta velocity at
$r_{\mathrm{max}}$ (van Veelen et al., 2009). The value of $r_{\rm max}$ is
determined by the explosion energy and ejecta mass (Whalen et al., 2008). The
density profile of the ejecta is set as

$\rho(r)=\begin{cases}\rho_{\rm core}(r)&\text{if $r\leq r_{\rm core}$ },\\\
\rho_{\rm max}(r)&\text{if $r_{\rm core}<r<r_{\rm max}$},\\\ \end{cases}$ (9)

where

$\rho_{\rm core}(r)=\frac{1}{4\pi n}\frac{(10E_{\rm
ej}^{n-5})^{-3/2}}{(3M_{\rm ej}^{n-3})^{-5/2}}\frac{1}{t_{\rm max}^{3}},$ (10)

is constant, whereas the ejecta density further out follows a power-law,

$\rho_{\rm max}(r)=\frac{1}{4\pi n}\frac{(10E_{\rm
ej}^{n-5})^{(n-3)/2}}{(3M_{\rm ej}^{n-3})^{(n-5)/2}}\bigg{(}\frac{r}{t_{\rm
max}}\bigg{)}^{-n},$ (11)

with $n=11$ (Chevalier, 1982; Truelove & McKee, 1999).

Table 1:  List of models. The columns indicate the velocity of the star, $v_{\star}$, the grid resolution and size in pc, and the sonic and Alfénic Mach number of the moving star with respect to the ISM. The runs are labelled ”CSM” for the pre-supernova modelling and ”SNR” for the remnant simulations, and likewise ”HD” for hydrodynamics and ”MHD” for magneto-hydrodynamics.  ${\rm{Model}}$ |  $v_{\star}$ ($\rm km\,\rm s^{-1}$) | Grid size | Grid mesh | $\it M$ | $\it M_{\rm A}$
---|---|---|---|---|---
Run-35-MHD-20-CSM | $20$ | $[0;175]\times[-250;100]$ | $2000\times 4000\,\mathrm{cells}$ | $1.0$ | $1.16$
Run-35-HD-20-CSM | $20$ | $[0;175]\times[-250;100]$ | $2000\times 4000\,\mathrm{cells}$ | $1.0$ | $1.16$
Run-35-MHD-40-CSM | $40$ | $[0;150]\times[-300;100]$ | $1500\times 4000\,\mathrm{cells}$ | $2.0$ | $2.32$
Run-35-HD-40-CSM | $40$ | $[0;150]\times[-300;100]$ | $1500\times 4000\,\mathrm{cells}$ | $2.0$ | $2.32$
Run-35-MHD-20-SNR | $20$ | $[0;200]\times[-275;175]$ | $4000\times 9000\,\mathrm{cells}$ | $1.0$ | $1.16$
Run-35-HD-20-SNR | $20$ | $[0;200]\times[-275;175]$ | $4000\times 9000\,\mathrm{cells}$ | $1.0$ | $1.16$
Run-35-MHD-40-SNR | $40$ | $[0;200]\times[-330;170]$ | $~{}~{}4000\times 10000\,\mathrm{cells}$ | $2.0$ | $2.32$
Run-35-HD-40-SNR | $40$ | $[0;200]\times[-330;170]$ | $~{}~{}4000\times 10000\,\mathrm{cells}$ | $2.0$ | $2.32$

Beyond $r_{\mathrm{max}}$, the density profile is that of the freely-expanding
wind as found in the pre-supernova wind bubble simulations. The ejecta speed
at the distance $r_{\rm core}$ from the center of the explosion is (Truelove &
McKee, 1999)

$v_{\rm core}=\bigg{(}\frac{10(n-5)E_{\rm ej}}{3(n-3)M_{\rm
ej}}\bigg{)}^{1/2}.$ (12)

This 1D ejecta-wind interaction solution is mapped onto the 2D domain of the
subsequent simulation. We integrate the equations up to $150\,\rm kyr$ after
the supernova.

###  2.3 Governing equations

The dynamics of a magnetised flow is described by the equations of ideal
magneto-hydrodynamics plus losses and heating by optically-thin radiation,

$\frac{\partial\rho}{\partial
t}+\mn@boldsymbol{\nabla}\cdot\big{(}\rho\mn@boldsymbol{v})=0,$ (13)
$\frac{\partial\mn@boldsymbol{m}}{\partial
t}+\mn@boldsymbol{\nabla}\cdot\Big{(}\mn@boldsymbol{m}{\color[rgb]{0,0,0}\otimes}\mn@boldsymbol{v}+\mn@boldsymbol{B}{\color[rgb]{0,0,0}\otimes}\mn@boldsymbol{B}+\mn@boldsymbol{\hat{\mathbf{I}}}p_{\rm
t}\Big{)}=\mn@boldsymbol{0},$ (14) $\frac{\partial E}{\partial
t}+\mn@boldsymbol{\nabla}\cdot\Big{(}(E+p_{\rm
t})\mn@boldsymbol{v}-\mn@boldsymbol{B}(\mn@boldsymbol{v}\cdot\mn@boldsymbol{B})\Big{)}=\Phi(T,\rho),$
(15)

and,

$\frac{\partial\mn@boldsymbol{B}}{\partial
t}+\mn@boldsymbol{\nabla}\cdot\Big{(}\mn@boldsymbol{v}{\color[rgb]{0,0,0}\otimes}\mn@boldsymbol{B}-\mn@boldsymbol{B}{\color[rgb]{0,0,0}\otimes}\mn@boldsymbol{v}\Big{)}=\mn@boldsymbol{0},$
(16)

with the linear momentum vector, $\mn@boldsymbol{m}=\rho\mn@boldsymbol{v}$,
and the magnetic-field vector, $\mn@boldsymbol{B}$.  The total energy of the
system reads,

$E=\frac{p}{(\gamma-1)}+\frac{\mn@boldsymbol{m}\cdot\mn@boldsymbol{m}}{2\rho}+\frac{\mn@boldsymbol{B}\cdot\mn@boldsymbol{B}}{2},$
(17)

where $\gamma=5/3$ is the adiabatic index for ideal gas and $p$ is the thermal
pressure. The definition of the adiabatic sound speed,

$c_{\rm s}=\sqrt{\frac{\gamma p}{\rho}},$ (18)

closes the system, which we integrate using the so-called eight-wave
algorithm. This second-order unsplit scheme satisfies
$\mn@boldsymbol{\nabla}\cdot\mn@boldsymbol{B}=\mn@boldsymbol{0}$. The time-
march of the algorithm obeys the standard Courant-Friedrich-Levy condition
that is set to $C_{\rm cfl}=0.1$ at the beginning of the simulations.

The source term,

$\mathit{\Phi}(T,\rho)=n_{\mathrm{H}}\Gamma(T)-n^{2}_{\mathrm{H}}\Lambda(T),$
(19)

accounts for optically-thin radiative cooling, $\mathit{\Lambda}(T)$, and
heating, $\mathit{\Gamma}(T)$. The gas temperature is

$T=\mu\frac{m_{\mathrm{H}}}{k_{\rm{B}}}\frac{p}{\rho},$ (20)

where $\mu=0.61$ is the mean molecular weight, $k_{\rm{B}}$ the Boltzmann
constant, and $m_{\rm H}$ the proton mass. The hydrogen number density is
computed as

$n_{\mathrm{H}}=\frac{\rho}{\mu(1+\chi_{\rm He,Z})m_{\mathrm{H}}},$ (21)

with $\chi_{\rm He,Z}$ the mass fraction of all coolants heavier than $\rm H$.
The functions $\Gamma(T)$ and $\Lambda(T)$ are described in details in Meyer
et al. (2017).

Figure 2:  Density rendering of the supernova remnant of a
$35$-$\mathrm{M}_{\odot}$ progenitor moving with $v_{\star}=20\,\rm km\,\rm
s^{-1}$ through uniform ISM of number density $n_{\rm ISM}=0.78\,\rm cm^{-3}$.
Before exploding, the remnant passed through main-sequence, red supergiant and
Wolf-Rayet phases (Ekström et al., 2012). The top panel displays the
hydrodynamical model, whereas the bottom one shows the magneto-hydrodynamical
picture for a $7$-$\mu G$ ambient magnetic field oriented parallel to the
stellar motion in z-direction. Inset boxes highlight the dynamic filamentary
structures developing from ejecta-wind-ISM interactions (left inset) and the
structure of the stellar wind cavity produced by the progenitor’s motion and
located behind the center of the explosion (right inset). The red lines are
iso-temperature contours ($T=10^{6}$ and $10^{7}\,\rm K$), and the blue
contours trace the region with a $10\%$ contribution of supernova ejecta in
number density.

Figure 3:  As Fig. 2, but for the supernova remnant of a progenitor moving
with $40\,\rm km\,\rm s^{-1}$.

###  2.4 Non-thermal synchrotron radio emission

The forward shock of the supernova remnant in particular will accelerate
charged particles, such as electrons, to high energy. In the presence of
magnetic field relativistic electrons produce synchrotron emission that is an
excellent diagnostic (Reynolds, 2008). To permit a comparison with the vast
observational data of non-thermal radio synchrotron emission from supernova
remnants, we produce synthetic emission maps on the basis of our magneto-
hydrodynamical simulations.

Energy losses of GeV-scale electrons are likely negligible, and so we assume
the electron spectrum,

$N(E)=KE^{-s},$ (22)

where $E$ denotes the electron energy and the index, $s=2$, is expected for a
strong shock. Diffusive transport is typically slower than advection in the
GeV band, and so the accelerated electron density follows the gas density and
is in fact proportional to it, if the injection efficiency at the forward
shock is a constant (Drury, 1983b, a).  Amongst the several prescriptions for
the non-thermal synchrotron emission coefficient of a magnetised gas available
in the literature, we choose to make use of that of Jun & Norman (1996). We
refer the reader interested in details regarding to our choice of emission
coefficient within this core-collapse supernova remnant problem in our
Appendix A. Therefore, at a given frequency, $\nu$, the radio synchrotron
emission coefficient reads,

$j_{\rm sync}(\nu)\propto K^{2-s}p^{s-1}B_{\perp}^{(s+1)/2}\nu^{-(s-1)/2},$
(23)

which reduces to

$j_{\rm sync}(\nu)\propto n^{2-s}p^{s-1}B_{\perp}^{(s+1)/2}\nu^{-(s-1)/2},$
(24)

where $p$ is the gas thermal pressure and $B_{\perp}$ is the magnetic-field
component perpendicular to the line of sight.

Let $\vec{l}$ be the unit vector of the observer’s line of sight. Defining the
viewing angle of the observer as $\theta_{\rm obs}=\angle(\vec{l},\vec{B})$,
the total strength of the magnetic field and its perpendicular component are
obtained as

$B_{\perp}=|\vec{B}|\sin(\theta_{\rm obs})$ (25)

and

$|\vec{B}|=\sqrt{B_{\rm R}^{2}+B_{\rm z}^{2}}.$ (26)

Then, at a given frequency the emission coefficient finally reads,

$j_{\rm sync}(\theta_{\rm obs})\propto
n^{2-s}p^{s-1}\Bigg{(}|\vec{B}|\sqrt{1-\Big{(}\frac{\vec{B}\cdot\vec{l}}{|\vec{B}|}\Big{)}^{2}}\Bigg{)}^{(s+1)/2},$
(27)

which we use in our radiative transfer calculations.

For each simulation, we selected snapshots that are representative of the
phases of the supernova remnant evolution, namely at times $6$, $20$ and
$40\,\rm kyr$ after the explosion, respectively. The corresponding density,
temperature, and magnetic-field distributions are first translated from the
two-dimensional cylindrical coordinates system to a three-dimensional
spherical coordinate system ($r$, $\theta$, $\phi$) with $512^{3}$ cells and
the same origin, for which we rotate the cylindrical solution around the
symmetry axis. On each grid zone we pre-calculate the local component of the
magnetic field that is normal to line-of-sight of the observer. Finally, we
use a modified version of the radiative transfer code
RADMC-3D222https://www.ita.uni-
heidelberg.de/$\sim$dullemond/software/radmc-3d/ to perform ray-tracing
integration of the radio synchrotron emission coefficient along a given line-
of-sight with aspect angle $\theta_{\rm obs}$. The non-thermal radio
intensity,

$I=\int_{\rm SNR}j_{\rm sync}(\theta_{\rm obs})dl,$ (28)

is then used to synthesize normalised emission maps.

Figure 4:  Mixing of material in supernova remnants from massive runaway
progenitors. The top row displays results for $v_{\star}=20\,\rm km\,\rm
s^{-1}$ and the bottom panels are for $v_{\star}=40\,\rm km\,\rm s^{-1}$. The
left column shows purely hydrodynamical models while the right column is for
MHD runs. Each remnant is shown at time $40\,\rm kyr$ after the supernova
explosion. Each image gives the ISM fraction ($1-Q_{\mathrm{1}}$) and the
fraction of stellar-wind material ($Q_{\mathrm{1}}$) in the left and right
part, respectively. The black temperature contours have the levels $T=10^{5}$,
$10^{6}$ , $10^{7}\,\rm K$ and the white number density contours stand for
$n=1.0$, $10^{1}$, $10^{2}$, $10^{3}\,\rm cm^{-3}$. The blue contours make the
locations with $10\,\%$ ejecta fraction by number.

##  3 Results

We present in this section the results for the pre- and post-supernova
circumstellar medium of a runaway Galactic $35\,\mathrm{M}_{\odot}$ progenitor
star, investigate how the stellar motion and the magnetisation of the ISM
affect the mixing of materials, and present the evolution of their projected
radio synchrotron emission.

###  3.1 Pre-supernova circumstellar medium of runaway stars

Fig. 2 presents our results for a star moving with $v_{\star}=20\,\rm km\,\rm
s^{-1}$, comparing the hydrodynamical picture (top) with the magneto-
hydrodynamical description (bottom). The red isocontours trace $Q_{1}=0.5$,
which marks the places in the remnant with a 50/50 proportion of ISM and
stellar wind. The wind of the massive star generated an ovoid bubble of size
$\sim 100\,\rm pc$ (Fig. 2a) in which the star is off-centered on account of
stellar motion (cf. Weaver et al., 1977; Meyer et al., 2020). The large-scale
stellar-wind bow shock is organised according to the classical picture of
Weaver et al. (1977), made of an inner termination shock, a contact
discontinuity, and an outer forward shock. They distinguish the expanding
stellar wind, the hot low-density shocked wind, the cold dense ISM gas, and
the ambient medium, respectively. The post-main-sequence wind, i.e. the red
supergiant and Wolf-Rayet materials, are released inside expanding stellar
wind and develop instabilities at the interface separating the cold and hot
gas (Fig. 2).

For a larger speed of the star, $v_{\star}=40\,\rm km\,\rm s^{-1}$, the
spherical symmetry is broken, and the star reaches the forward shock of its
own wind bubble, itself distorted under the effects of stellar motion (Fig.
3a,b). A chimney of unperturbed stellar wind is carved into the layer of
shocked ISM, and the post-main-sequence wind is blown through the tube (red
isocontours). This phenomenon is even more pronounced in the case of a very
fast progenitor star with $v_{\star}=70\,\rm km\,\rm s^{-1}$ (Meyer et al.,
2015). Interestingly, once the star has left its wind bubble, direct wind-ISM
interaction resumes at the distance,

$R_{\rm SO}=\sqrt{\frac{\dot{M}v_{\mathrm{w}}}{4\pi
n_{\mathrm{ISM}}v_{\star}^{2}}},$ (29)

and a new bow shock forms (Baranov et al., 1971). An analytic estimate for the
location of the contact discontinuity of bow shocks reads

$\frac{R({\theta})}{R_{\rm
SO}}=\frac{\sqrt{3(1-\theta)\mathrm{cotan}(\theta)}}{\sin(\theta)},$ (30)

where $\theta$ is the angle to the direction of stellar motion (Wilkin, 1996).
This bow shock is in its turn subject to instabilities (Brighenti & D’Ercole,
1995b, a) and constitutes the location in which the supernova explosion takes
place (Brighenti & D’Ercole, 1994; Chiotellis et al., 2012; Meyer et al.,
2015).

Figure 5:  Comparison between the diffusion timescale, $\Delta_{\rm diff}$,
and the advection timescale, $\Delta_{\rm adv}$, of a particle accelerated at
the forward shock of the supernova remnant in our simulation Run-35-MHD-40-SNR
of a $35$-$\rm M_{\odot}$ progenitor moving with $v_{\star}=40\,\rm km\,\rm
s^{-1}$. The quantity $\Delta_{\rm diff}/\Delta_{\rm adv}$ is plotted as a
function of remnant age for several values of $\eta$ and synchrotron frequency
$\nu_{\rm sync}$, respectively. Values well below unity imply that diffusion
can be neglected in the estimate of radio emission maps.

The interstellar magnetic field has strong impact on the distribution of
shocked ISM, as demonstrated in van Marle et al. (2015). As an example, in
Figs. 2b and 3b one clearly sees that the layer of shocked ISM is puffed up
along the local magnetic field as a result of the damping of Alfvén waves (van
Marle et al., 2015). The effect is weaker for a fast-moving progenitor on
account of the lower ratio of magnetic and ram pressure (Fig. 3b). The entire
interior structure of the wind bubble is elongated when the star moves
quickly. The magnetic-field lines are aligned with the termination shock and
discontinuities, providing additional pressure that modifies the circumstellar
gas dynamics (Figs. 2b). It can also damp instabilities at the contact
discontinuity between hot shocked wind and cold shocked ISM gas, primarily in
the tail of the wind bubble. Our models combine the asymmetry in the stellar
wind bubbles of moving stars (Meyer et al., 2015) with the magnetic-pressure
effect that were previously explored for static stars (van Marle et al.,
2015).

###  3.2 Supernova remnants

Fig. 4 displays the structure of the supernova remnant at time $40\,\rm kyr$
after the explosion. Each panel corresponds to a different simulation, with
density plotted on the left-hand part of the panel and the temperature on the
right-hand part of the panel, respectively. The blue isocontour marks the
regions with 10% abundance of ejecta in number density. The left panels are
derived from hydrodynamical simulations, to be compared with MHD results on
the right. The stellar velocities are $v_{\star}=20\,\rm km\,\rm s^{-1}$ (top)
and $v_{\star}=40\,\rm km\,\rm s^{-1}$ (bottom), respectively.

The shape of the supernova remnants are governed by the distribution of the
pre-supernova circumstellar medium (Meyer et al., 2015; Meyer et al., 2020).
Figs. 4b,d show that the faster the progenitor moves through the ISM, the
sooner the supernova shock wave interacts with the termination shock of the
progenitor’s wind bubble. For smaller $v_{\star}$, the elongated shape of the
MHD wind bubble permits the Wolf-Rayet wind to expands freely into the
unperturbed red supergiant stellar wind and to generate by wind-wind collision
a ring of dense swept-up material (Meyer et al., 2020), inside of which the
blastwave is subsequently released and expands spherically. A similar
situation has been explored for static progenitor in the context of Cas A (van
Veelen et al., 2009). This phenomenon is particularly prominent for small ISM
density, $n_{\rm ISM}\ll 1$, since the radius of the main-sequence wind
termination shock is much larger (van Marle et al., 2015), and so is the
region filled by the last free-streaming wind.

The effects of the ISM magnetic field are also more pronounced for low
progenitor speed, $v_{\star}=20\,\rm km\,\rm s^{-1}$. The supernova remnant
shock wave rapidly interacts with the wind bubble in the progenitor’s
direction of motion, and it is first reverberated towards the center of the
explosion and subsequently channelled into the wind cavity carved during the
main-sequence phase, inducing a hot region hosting a lot of mixing of wind,
ejecta and shocked ISM gas (Fig. 4a). The reflections are different in the MHD
case, where they occur both parallel and normal to the progenitor’s motion, on
account of the tubular shape of the shocked-wind region (Fig. 4b,d). A
different morphology arises for fast-moving progenitors, with a rather unmixed
lobe of shocked ISM ahead of the stellar motion, and a channelled region of
mixed ejecta and wind material in the tail (Figs. 4c,d).

Figure 6:  Normalised projection of the magnetic field perpendicular to the
line of sight, $B_{\perp}/B$, for model Run-35-MHD-40 model with progenitor
speed $v_{\star}=40\,\rm km\,\rm s^{-1}$ at time $80\,\rm kyr$ after the
supernova explosion. The black arrow marks the direction of the observer’s
line-of-sight, making an angle $\theta_{\rm obs}=45^{\circ}$ with the
$z$-axis.

###  3.3 Effect of ISM magnetic field on the remnant’s properties

Magnetic field changes the morphology of the pre-supernova stellar wind
bubble, which will eventually influence the structure of the remnant and the
mixing of material in it. At a magnetized shock, the component of the magnetic
field along the shock normal remains unchanged, and that in the shock plane is
compressed (Shu, 1992). This results in an increased magnetic pressure in the
shocked ISM and, consequently, in an enlargement of the bubble perpendicular
to the direction of motion, which for static wind bubbles has been
demonstrated by van Marle et al. (2015). Further effects are a reduced
compression ratio of the forward shock and a puffing-up of the shocked ISM gas
layer. Although the MHD jump conditions imply that the ISM field is not
compressed ahead of the star, since the cylindrical coordinate system imposes
a parallel field, we can not exclude that this is an artefact. We refer the
reader to the thorough discussion in Meyer et al. (2017). Fully three-
dimensional simulations of both the pre-supernova and remnant phase of moving
massive progenitor star are necessary to address this question in appropriate
detail.

Figure 7:  Normalised maps of radio synchrotron intensity of supernova
remnants with progenitor speed $v_{\star}=20\,\rm km\,\rm s^{-1}$ at times
$6\,\rm kyr$ (top), $40\,\rm kyr$ (middle )and $80\,\rm kyr$ (bottom) after
the supernova explosion, respectively. The viewing angle between the
equatorial plane and the line of sight is $\theta_{\rm obs}=0^{\circ}$ (left),
$\theta_{\rm obs}=45^{\circ}$ (middle), and $\theta_{\rm obs}=90^{\circ}$
(right).

###  3.4 Radio synchrotron emission maps

####  3.4.1 Basic consideration

We perform a back-of-the-envelope estimate of the validity of our assumptions
regarding the radio synchrotron emission. We select one-dimensional cross-
section of the flow variables $\rho$, $p$ and $T$ originating from the center
of the explosion and following the direction of stellar motion, along the axis
of symmetry, $Oz$, in the models, respectively. We then measure the time-
dependent position, $R_{\rm FS}(t)$, and speed, $v_{\rm FS}(t)$, of the
forward shock, using the shock-finder algorithm of the ratpac code
(Telezhinsky et al., 2012, 2013; Bhatt et al., 2020). The radiation synthesis
is based on the assumption that the forward shock accelerates a certain
fraction of particles passing through it, and that the subsequent transport of
radio-emitting electrons in the downstream region is entirely advective.

In our radiation transfer model, diffusion is neglected and we shall now
verify this assumption. Upstream of the forward shock the density of electrons
re-accelerated as cosmic-rays exponentially decreases, $N(E,x)\propto
e^{-x/x_{\rm c}(t)}$ with length scale,

$x_{\rm c}(t)=\frac{D(E)}{v_{\rm FS}(t)},$ (31)

where $D(E)$ the diffusion coefficient and $E$ the energy of the electrons.
For simplicity, we shall assume that the diffusion coefficient is a multiple
of the Bohm limit, $D(E)\approx\eta D_{\rm
Bohm}(E)=\eta\,c/3\,r_{\mathrm{L}}(E)$, where $r_{\mathrm{L}}$ denotes the
Larmor radius of the electrons and $\eta$ is a scalar controlling the . Using
the characteristic frequency of synchrotron radiation, Eq. 31 can be rewritten
as

$\frac{x_{\rm
c}(t)}{\mathrm{pc}}\simeq\frac{\eta}{3000}\sqrt{\frac{\nu_{\mathrm{syn}}}{10\,\mathrm{GHz}}}\left(\frac{v_{\rm
FS}(t)}{3000\,\mathrm{km/s}}\right)^{-1}\left(\frac{B(t)}{10\,\mathrm{\mu
G}}\right)^{-1.5}.$ (32)

It is evident that for reasonably well developed cosmic-ray scattering
($\eta\ll 1000$) the precursor of radio-emitting electrons is tiny compared to
the size of the system and produces a negligible contribution of synchrotron
emission. Note that a significant abundance of magnetic field perpendicular
the shock surface will slow down diffusive transport away from the shock and
hence reduce $\eta$.

Diffusion in the downstream region may transport electrons beyond the
discontinuity to the ejecta region. Within time $t$, advective transport
displaces electrons from the shock by the distance
$\Delta_{\mathrm{adv}}\simeq t\,v_{\rm FS}(t)/4$. Diffusive transport
displaces by $\Delta_{\mathrm{dif}}\simeq\sqrt{D(E)t}$, and so ignorability of
diffusion requires

$t\,v_{\rm FS}(t)/4\gg\sqrt{D(E)t}\quad\Rightarrow\ t\,v_{\rm FS}(t)\gg
16x_{\rm c}(t).$ (33)

With $x_{c}$ as given in Eq. 32 we find that the condition is likely met. Fig.
5 plots $\Delta_{\rm diff}/\Delta_{\rm adv}$ as a function of time for our
simulation model Run-35-MHD-40-SNR and several values of $\eta\leq 10$ and
$\nu_{\rm sync\leq 1\,\rm Ghz}$. To be noted from the figure is that
$\Delta_{\rm diff}/\Delta_{\rm adv}\ll 1$ for times $\leq 200\,\rm kyr$, and
consequently we can ignore cosmic ray diffusion in the production of radio
emission maps.

Figure 8:  As Fig. 7 but here for model Run-35-MHD-40-SNR with progenitor
moving with $40\,\rm km\,\rm s^{-1}$.

####  3.4.2 Non-thermal emission maps

We generate non-thermal radio emission maps from our MHD models of supernova
remnants at representative points of time in the evolution of the supernova
remnants. Using the procedure described in section 2.4 we pre-compute for each
viewing angle, $\theta_{\rm obs}$, the distribution of magnetic field normal
to the line of sight, $B_{\perp}$. An illustrative example based on model
Run-35-MHD-40 at time $80\,\rm kyr$ is given in Fig.6. Then we compute the
radio intensity with our modified version of the radmc3d code. In Fig. 7 we
show intensity maps for supernova remnant ages $6\,\rm kyr$ (top), $20\,\rm
kyr$ (middle )and $80\,\rm kyr$ (bottom) for a progenitor moving with
$v_{\star}=20\,\rm km\,\rm s^{-1}$. We selected three viewing angles to the
equatorial plane, $\theta_{\rm obs}=0^{\circ}$ (left), $\theta_{\rm
obs}=45^{\circ}$ (middle) and $\theta_{\rm obs}=90^{\circ}$ (right) and
normalised the background-substracted maps.  Fig. 8 displays corresponding
radio maps for $v_{\star}=40\,\rm km\,\rm s^{-1}$.

Model Run-35-MHD-20 with progenitor moving at $v_{\star}=20\,\rm km\,\rm
s^{-1}$ traces the expanding supernova blastwave that is distorbed by its
interaction with the circumstellar medium.  Later, $40\,\rm kyr$ after the
explosion, the expanding shock wave has reached the unperturbed ISM and is
fairly bright there. The radio arcs are now larger than at time $20\,\rm kyr$
and the brightest region on the sides have a bilateral morphology. Note that
the maps are background-subtracted, and the region of shocked stellar wind may
be dimmer than the galactic radio background, leaving only the filamentary
arcs prominently visible. For an inclination angle $\theta_{\rm
obs}=45^{\circ}$ the remnants look rounder and more bubbly. One and the same
remnant can appear with bilateral or arced structures depending on
$\theta_{\rm obs}$. For $\theta_{\rm obs}=90^{\circ}$ the observer’s line-of-
sight is aligned with the direction of stellar motion, and the projected
remnant appears as a ring-like structure in the sky, on account of the two-
dimensional nature of the simulations. Since the magnetic field is parallel to
the stellar motion, $B_{\perp}\simeq 0$ in the shocked ISM, and the brightest
emission originates from the regions of mixing, primarily in the reflected
shock wave.

The radio intensity maps of our simulation Run-35-MHD-40 with progenitor speed
$v_{\star}=40\,\rm km\,\rm s^{-1}$ are shown in Fig. 8. The emission are
brighther than in model Run-35-MHD-20 as a result of the faster progenitor
star producing stronger shocks in its supernova remnant. At time $6\,\rm kyr$
after the explosion, the shock wave has already been greatly distorted by the
Wolf-Rayet circumstellar material and has lost sphericity to become as ovoid-
like structure. Later in time, the shock wave adopts a hour-glass-like shape
that appears spherical in the radio map for $\theta_{\rm obs}=45^{\circ}$ and
to a lesser degree at $\theta_{\rm obs}=0^{\circ}$. After $40\,\rm kyr$, the
remnant has an bulb-like morphology which arises from both the shock wave
expansion into the ISM and the channeling of the shock wave into the low-
density cavity of unshocked stellar wind in the tail. The radio intensity peak
shifts to the location where the shock wave intercepts the trail of stellar
wind (Fig. 8d,e). The density in the region of wind-ISM interaction is more
important than in the case of a progenitor moving with $v_{\star}=20\,\rm
km\,\rm s^{-1}$, and so the stabilising effect of the magnetic field inside of
the remnant is reduced. Hence, more ring-like structures appear in the
emission maps, for example in Fig. 8h. We would observe a series of concentric
rings for $\theta_{\rm obs}=90^{\circ}$, each of them corresponding to a ring
in the trail of shocked stellar wind interacting with the channelled supernova
shock wave. There is more variety and complexity in the radio appearance for
faster-moving progenitors.

##  4 Discussion

This section discusses the limitations of our method, compares our results to
earlier results, and further examines our findings in the context of particle
acceleration. Finally, we compare our results with observational data.

###  4.1 Model limitations

As any numerical study, our method suffers from simplifications that limit the
realism of our results. The most obvious one is the cylindrical coordinate
system with rotational invariance, which intrinsically imposes a symmetry axis
to the models. This approach is convenient in the modelling of the
circumstellar medium of massive stars and their subsequent supernova remnants
(Franco et al., 1991; Rozyczka & Tenorio-Tagle, 1995; Comeròn, 1997; Comerón &
Kaper, 1998; van Marle et al., 2005, 2007; Ferreira & de Jager, 2008; van
Marle et al., 2014; Green et al., 2019), at the expense of forcing an
directional alignment of the motion of the progenitor, the local ISM magnetic
field, and progenitor’s axis of rotation. Only fully three-dimensional
simulations permit flexibility in the directional arrangement (e.g. Katushkina
et al., 2017, 2018), but they are far too expensive to permit scanning the
parameter space of remnants from massive progenitors.

Supernova remnants from massive progenitors are multi-phase regions composed
of a warm, magnetized interstellar medium through which the progenitor star
moves, the evolving stellar wind, and a hot component produced by the
interaction between the supernova shock wave and the material of the wind
bubble and the shocked ISM gas. Supernova remnants may be located close to
dense, cold molecular clouds that can further affect their evolution and
modify the gas chemistry.  The pressure of the cosmic rays accelerated in the
supernova remnant (Ferrand et al., 2014), anisotropic heat transfer (Orlando
et al., 2005), photoionizing progenitor radiation, or the turbulence in the
ISM should also be included in the models (Moranchel-Basurto et al., 2017;
Villagran et al., 2020), but that is far beyond the scope of the current study
and may be considered in future work. Last, note that intrinsic dense
molecular (Zhou & Chen, 2011; Zhou et al., 2014, 2016) or low-density
components (Arias et al., 2019; Arias et al., 2019) of the ambient medium are
an additional, in some context an even preponderant element to take into
account in the shaping of core-collapse supernova remnants.

###  4.2 Comparison with previous works

This study extends earlier work beginning with Meyer et al. (2015) on
hydrodynamical models of supernova remnants of runaway stars with initial
$10$, $20$ and $40\,\rm M_{\odot}$, that end their lives as red supergiants
and generate Cygnus-Loop-like nebulae. The second study of this series
explored the appearance of wind nebulae and remnants of a $60$ $\rm M_{\odot}$
progenitor star going through Luminous-Blue-Variable and Wolf-Rayet phases,
with emphasis on the mixing of material inside the remnant Meyer et al.
(2020). What is new and different in the present study is the inclusion of the
ISM magnetic field during both the pre- and the post-supernova phase, together
with the post-processing of radio synchrotron intensity maps. The effect of
ISM magnetisation on the environment of massive stars has been investigated by
van Marle et al. (2015), albeit without distinguishing ejecta from wind and
ISM gas as we do by means of passive scalar tracers. Moreover, our models
include a state-of-art stellar evolutionary model for the wind history of the
$35$ $\rm M_{\odot}$ star that we concentrate on.

###  4.3 Comparison with observations

####  4.3.1 Remnants of core-collapse supernovae

Katsuda et al. (2018) determined several properties, such as the distance and
the progenitor mass, of core-collapse supernova remnants in the Milky Way and
in low-metallicity dwarf galaxies such as the Large and Small Magellanic
Clouds. They found that most of the identified remnants of massive progenitors
in the Galaxy have a zero-age main-sequence mass $\geq 22.5\,\rm M_{\odot}$.
There is a general agreement between predictive stellar evolution models that
the progenitor exploded with such mass, either as a red supergiant or as a
Wolf-Rayet star, although more exotic situations such as blue supergiant
progenitor star exist (Hillebrandt et al., 1987). Note that, in the context of
massive binary systems, the explosion of the component can kick the companion,
producing runaway stars (Lux et al., 2021). Our models explore the Wolf-Rayet
possibility using a zero-age main-sequence $35\,\rm M_{\odot}$ star. As we
concentrate on the evolution of rather older remnants, about $6$$-$$80\,\rm
kyr$ after the explosion, our predictions are applicable to the objects listed
between Kes 79 and W51C in Table 1 of Katsuda et al. (2018).

Our models constitute baseline models to be further tailored to specific
supernova remnants, in particular Kes 79, G350.1-0.3, G292.0+1.8, RX
J1713.7-3946, Kes 79, G290.1-0.8, 3C391, W44, G284.3-1.8 or CTB109.  Note that
C, N and O enriched material, witness of post-main-sequence winds from massive
stars, has been directly observed in the remnant G296.1-0.5 (Castro et al.,
2011), making it an evident candidate of a supernova remnant with Wolf-Rayet
progenitor that is worth exploring numerically with simulations like ours.  As
underlined by Katsuda et al. (2018), the distribution of core-collapse
supernovae in the Galaxy does not fit any initial mass function, which suggest
that there should be many more unidentified remnants our simulations would be
applicable to. Note also that models for core-collapse remnants do not
generally apply to the so-called historical supernova remnants since these are
mostly of type Ia (Green & Stephenson, 2003), except for Cas A (van Veelen et
al., 2009; Zhou et al., 2018) and RWC 86 (Gvaramadze et al., 2017),
respectively.

The space motion of the progenitor is the other fundamental ingredient  of our
simulations, together with the zero-age main-sequence mass. If the massive
progenitor is at rest, then its circumstellar wind bubble remains spherical
(Weaver et al., 1977). The star, and the center of its subsequent explosion
are located at its center (Freyer et al., 2006; Dwarkadas, 2007). A sub-sonic
motion of the progenitor will off-center the remnant with respect to the wind
bubble without changing its overall appearance (Meyer et al., 2020). Hence,
remnants from slowly-moving progenitor star should reflect the spherical
symmetry of their circumstellar medium. However, as in-situ star formation
does not seem to be an obvious route to explain isolated Wolf-Rayet stars
(Gvaramadze et al., 2012; Meyer et al., 2020), static massive stars should
live and die inside of their parent star-formation region, where they would
participate in the regulation of subsequent star formation (Paron et al.,
2009). The feedback from stellar winds and/or jets of other (young) stellar
objects (Bally et al., 2007; Fendt, 2009; Fendt & Sheikhnezami, 2013) will
affect the medium in which massive stars form (Murray et al., 2018), evolve
and die. This should result in a very complex morphology, possibly further
complicated by ISM cavities and enhanced levels of turbulence, in the ambient
medium hosting a huge mix of material in a super bubble in which the supernova
subsequently explodes (van Marle et al., 2012).

####  4.3.2 Comparison with specific objects

Most supernova remnants in the Milky Way lie within $5^{\circ}$ of the
galactic plane. The dilute ISM at high galactic latitudes makes
(circumstellar) shocks weaker, resulting in fainter supernova remnants such
as, for example, the rather evolved radio source G181.1+9.5 (Kothes et al.,
2017). This latitude-dependence of the radio surface brightness of supernova
remnants is known as the $\Sigma$-D relation (Caswell & Lerche, 1979). The
modelled $35\,\rm M_{\odot}$ runaway star moving with $v_{\star}=40\,\rm
km\,\rm s^{-1}$ travels about $220\,\rm pc$ before exploding, about the same
as the height of G181.1+9.5 above the galactic plane ($\approx 250\,\rm pc$).
Dedicated simulations would be highly desirable to explore the differences in
the radio properties between supernova remnants of runaway progenitors in the
Galactic place and those at higher galactic latitudes, as well as the effects
of metallicities.  Our non-thermal radio emission maps authorise a couple of
further, direct comparisons with supernova remnants of massive, evolved
progenitors, namely G296.5+10.0, the shell-type remnants CTB 109, and Kes 17.

First, G296.5+10.0 is a supernova remnant of core-collapse origin, confirmed
by the trace of magnetised wind in which the supernova shock wave expands, and
the presence of a neutron star therein (Harvey-Smith et al., 2010). Radio
observations with the Australia Telescope Compact Arrary at $1.4\,$GHz reveal
a bipolar shape, that is qualitatively similar to those in our model with
velocity $v_{\star}=20\,\rm km\,\rm s^{-1}$, Run-35-MHD-20-SNR, at time
$6\,\rm kyr$, when the shock wave interacts with the stellar-wind bow shock
and accelerates electrons (Fig. 7a). Similarly, Fig. 8a also resembles greatly
the $7\,\rm kyr$ old bilateral supernova remnant G296.5+10.0, implying that
its progenitor might have been a rather fast moving star of mass
$20$$-$$40\,\rm M_{\odot}$. Secondly, the shell-type remnant CTB 109 is the
remnant of a core-collapse supernova remnant of a $30$-$40\rm M_{\odot}$
progenitor, which matches the mass range of the $35$-$\rm M_{\odot}$ stellar
model used in this study. Its age is around $14000\,\rm yr$, see Katsuda et
al. (2018).

According to our results, CTB 109 should be surrounded by a circumstellar
structure, i.e. a red-supergiant wind bubble engulfing a Wolf-Rayet ring, with
which the supernova shock wave interacts, although its overall shape has been
reproduced in the context of a type Ia explosion, i.e. without the presence of
a dense circumstellar wind bubble generated by a massive progenitor (Bolte et
al., 2015). Its opened shell appearance, e.g. as seen with the Canadian
Galactic Plane Survey at $1420\,\rm MHz$ (Kothes & Foster, 2012), is similar
to our model Run-35-MHD-SNR at times $6$-$40\,\rm kyr$ (Fig. 8b,e). Note also
that CTB 109 is a hadronic gamma-ray emitter (Castro et al., 2012). The last
example is the supernova remnant Kes 17 that is less than $40000\,\rm yr$ old
and with a progenitor mass $25$-$30\rm M_{\odot}$ consistent with the Wolf-
Rayet scenario (Katsuda et al., 2018). Its double-arced morphology observed
with the Australian Telescope Array at $20\,\rm cm$ resembles our model
Run-35-MHD-40-SNR at times $40\,\rm kyr$ (Fig. 8c).

##  5 Conclusion

We explore the formation, structure, and radio signatures of supernova
remnants of massive, Wolf-Rayet-evolving supernova progenitors ejected from
their parent cluster and moving through the interstellar medium (ISM) of the
Milky Way. Our study concentrates on the coupled impact of stellar motion and
the magnetisation of the ISM. We perform magneto-hydrodynamical simulations
over the entire stellar lifetime, as they successively evolve through a long
main-sequence phase, a red supergiant and a Wolf-Rayet phase, and eventually
spawn a core-collapse supernova remnant. Numerical models are performed with
the pluto code (Mignone et al., 2007, 2012) by simulating the circumstellar
medium of massive stars, into which we launch a core-collapse supernova shock
wave. We follow its interaction with the stellar surroundings and the local
ambient ISM supported by an organised $7\,\mu\rm G$ magnetic field.
Considering two speeds of the runaway progenitors and running the simulations
up to the oldest evolutionary phase of the supernova remnants, $150\,\rm kyr$
after the explosion, their morphologies, their structures, the mixing of
material happening in them are explored.

The presence of an ISM magnetic field profoundly affects the gas properties.
Prior to the supernova explosion, the compressed magnetic field in the
circumstellar medium stabilises the wind/ISM contact discontinuity in the tail
of the bubble. Indeed, compressed magnetic field in the outer remnant
stabilises and elongates the wind/ISM contact discontinuity of the cavity of
unshocked stellar wind, in which the ejecta is channelled. A consequence is a
reduced mixing efficiency of ejecta and evolved stellar-wind material enriched
in C, N and O elements in the inner region of the remnant, where the supernova
shock wave propagates. Moreover, after the supernova explosion, the density
downstream of the supernova shock front is reduced in our MHD simulations when
it propagates into the pristine ambient medium, on account of the damping of
turbulence. This must influence the acceleration processes of cosmic-ray
electrons and protons in supernova remnants from massive progenitors and will
be investigated in future works (Bhatt et al., 2020). We emphasize the need
for a careful treatment of the gas microphysics to properly simulate young
supernova remnants interacting with circumstellar structures. This
particularly applies to runaway massive progenitors whose supernova shock
front is reverberated towards the center of the explosion, generating a
complex region made of shocks, discontinuities, and filamentary structures, in
which non-thermal particles can be accelerated.

Last, using our modified version of the radiative transfer code radmc/3d
(Dullemond, 2012) we produced synthetic radio-intensity maps showing projected
arcs and filaments that we interpret as a morphological characteristic of
supernova remnants of fast-moving Wolf-Rayet stars.  Our radio predictions are
qualitatively in accordance with the morphology of several core-collapse
remnants, such as the bilateral G296.5+10.0, as well as the shell-type
supernova remnants CTB 109 and Kes 17, identified as originating from
$25$-$40\,M_{\odot}$ progenitors (Katsuda et al., 2018) which might have
undergone a Wolf-Rayet phase.  Our simulations and predictions regarding the
non-thermal emission of supernova remnants from massive progenitors are
relevant for and may be applied to the various galactic and extragalactic
core-collapse remnants (Katsuda et al., 2018).

## Acknowledgements

Authors are grateful to the referee, P. Velazquez, for comments on synchrotron
emission which greatly improved the quality of the manuscript.  The authors
thank Allard Jan van Marle from Ulsan National Institute of Science and
Technology for his kind advices on MHD simulations of the surroundings of
massive stars. The authors acknowledge the North-German Supercomputing
Alliance (HLRN) for providing HPC resources that have contributed to the
research results reported in this paper. M. Petrov acknowledges the Max Planck
Computing and Data Facility (MPCDF) for providing data storage resources and
HPC resources which contributed to test and optimise the pluto code. L.M.O.
acknowledges partial support by the Russian Government Program of Competitive
Growth of Kazan Federal University.

## Data availability

This research made use of the pluto code developed at the University of Torino
by A. Mignone (http://plutocode.ph.unito.it/) and of the radmc-3d code
developed at the University of Heidelberg by C. Dullemond
(https://www.ita.uni-heidelberg.de/$\sim$dullemond/software/radmc-3d/). The
figures have been produced using the Matplotlib plotting library for the
Python programming language (https://matplotlib.org/). The data underlying
this article will be shared on reasonable request to the corresponding author.

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Figure 9:  Normalised maps of radio synchrotron intensity of supernova
remnants with progenitor speed $v_{\star}=20\,\rm km\,\rm s^{-1}$ at time
$6\,\rm kyr$, and calculated using several prescriptions for the non-thermal
emission coefficient. The viewing angle between the equatorial plane and the
line of sight is $\theta_{\rm obs}=45^{\circ}$ (middle).

Figure 10:  Same as Fig. 9 with progenitor speed $v_{\star}=40\,\rm km\,\rm
s^{-1}$ at time $80\,\rm kyr$.

##  Appendix A Emission coefficients for non-thermal synchrotron emission

To the best of our knowledge, three main recipes are available for the
emission coefficient of non-thermal radio synchrotron emission in the context
of supernova remnants (Jun & Norman, 1996; Orlando et al., 2007; Ávila-Aroche
et al., 2020). Considering the electron spectrum in the vicinity of the
shocks,

$N(E)=KE^{-s},$ (34)

where $E$ is the electron energy and $s=2$the index and $K\propto n$. They
read as,

$j_{\rm sync}^{\rm Orlando}(\nu)\propto KB_{\perp}^{(s+1)/2}\nu^{-(s-1)/2},$
(35)

with $\theta_{\rm obs}$ the viewing angle of the observer, $B_{\perp}$ the
magnetic field component perpendicular to the line of sight and $\nu$ the
emission frequency,

$j_{\rm sync}^{\rm Jun}(\nu)\propto
K^{2-s}p^{s-1}B_{\perp}^{(s+1)/2}\nu^{-(s-1)/2},$ (36)

where $p$ is the gas thermal pressure, and

$j_{\rm sync}^{\rm Avila}(\nu)\propto K{\color[rgb]{0,0,0}v^{\prime
2(s-1)}}B_{\perp}^{(s+1)/2}\nu^{-(s-1)/2},$ (37)

where $v^{\prime}$ is the gas velocity in the rest frame of the explosion,
respectively. This diagnostics has been widely used in, e.g. the context of
the core-collapse but also type Ia progenitors such as the historical
supernova remnants Tycho (Moranchel-Basurto et al., 2020) and SN 1006
(Schneiter et al., 2015; Velázquez et al., 2017).

We generate comparative normalised non-thermal radio emission maps from two
selected models of supernova remnants. First, one with a progenitor star
moving rather slowly with velocity $20\,\rm km\,\rm s^{-1}$, and in which the
thermal pressure compares with the ram and magnetic pressures (Fig. 9).
Secondly, a model with a fast-moving progenitor is moving with velocity
$40\,\rm km\,\rm s^{-1}$ and in which the ISM magnetic pressure is dynamically
unimportant (Fig. 10). Fig. 9a reveals the bright radio synchrotron
circumstellar medium of the progenitor, produced by wind-ISM interaction
before the explosion of the massive star, while Fig. 9b,c do not. The recipe
used in Fig. 9a clearly overestimates particle acceleration from the forward
shock of the stellar wind bubble, that is much weaker than the forward shock
of the expanding supernova blastwave. Hence, the emission coefficient in
Orlando et al. (2007) is not the most suitable to our core-collapse remnant
problem. The models calculated with the other emission coefficients do not
permit to select an optimal one for our study (Fig. 9b,c). The emission
coefficient in Eq. 36 of Jun & Norman (1996) has a dependence on the thermal
pressure $p$, implying that it is sensitive to cooling and heating by
optically-thin radiative cooling processes and therefore traces the fast
shocks well. Similarly, the recipe of emission coefficient in Eq. 37 of Ávila-
Aroche et al. (2020) goes as $j_{\rm sync}^{\rm Avila}(\nu)\propto v^{2}$,
that imposes a strong dependence of the chosen frame in which we simulate the
stellar wind bubble and the supernova explosion. Consequently, we decide in
our study to use the recipe of Jun & Norman (1996).