On the Origin of Asymmetries in Bilateral Supernova Remnants
Astronomy & Astrophysics manuscript no. sorlando˙6045 c© ESO 2018
November 21, 2018
On the origin of asymmetries in bilateral supernova remnants
S. Orlando1,2, F. Bocchino1,2, F. Reale3,1,2, G. Peres3,1,2 and O. Petruk4,5
1 INAF - Osservatorio Astronomico di Palermo “G.S. Vaiana”, Piazza del Parlamento 1, 90134 Palermo, Italy
2 Consorzio COMETA, via Santa Sofia 64, 95123 Catania, Italy
3 Dip. di Scienze Fisiche & Astronomiche, Univ. di Palermo, Piazza del Parlamento 1, 90134 Palermo, Italy
4 Institute for Applied Problems in Mechanics and Mathematics, Naukova St. 3-b Lviv 79060, Ukraine
5 Astronomical Observatory, National University, Kyryla and Methodia St. 8 Lviv 79008, Ukraine
Received ; accepted
Abstract
Aims. We investigate whether the morphology of bilateral supernova remnants (BSNRs) observed in the radio band is determined mainly either
by a non-uniform interstellar medium (ISM) or by a non-uniform ambient magnetic field.
Methods. We perform 3-D MHD simulations of a spherical SNR shock propagating through a magnetized ISM. Two cases of shock propagation
are considered: 1) through a gradient of ambient density with a uniform ambient magnetic field; 2) through a homogeneous medium with a
gradient of ambient magnetic field strength. From the simulations, we synthesize the synchrotron radio emission, making different assumptions
about the details of acceleration and injection of relativistic electrons.
Results. We find that asymmetric BSNRs are produced if the line-of-sight is not aligned with the gradient of ambient plasma density or with
the gradient of ambient magnetic field strength. We derive useful parameters to quantify the degree of asymmetry of the remnants that may
provide a powerful diagnostic of the microphysics of strong shock waves through the comparison between models and observations.
Conclusions. BSNRs with two radio limbs of different brightness can be explained if a gradient of ambient density or, most likely, of ambient
magnetic field strength is perpendicular to the radio limbs. BSNRs with converging similar radio arcs can be explained if the gradient runs
between the two arcs.
Key words. Magnetohydrodynamics (MHD) – Shock waves – ISM: supernova remnants – ISM: magnetic fields – Radio continuum: ISM
1. Introduction
It is widely accepted that the structure and the chemical abun-
dances of the interstellar medium (ISM) are strongly influ-
enced by supernova (SN) explosions and by their remnants
(SNRs). However, the details of the interaction between SNR
shock fronts and ISM depend, in principle, on many factors,
among which the multiple-phase structure of the medium, its
density and temperature, the intensity and direction of the am-
bient magnetic fields. These factors are not easily determined
and this somewhat hampers our detailed understanding of the
complex ISM.
The bilateral supernova remnants (BSNRs, Gaensler 1998;
also called ”barrel-shaped,” Kesteven & Caswell 1987, or
”bipolar”, Fulbright & Reynolds 1990) are considered a bench-
mark for the study of large scale SNR-ISM interactions, since
no small scale effect like encounters with ISM clouds seems
to be relevant. The BSNRs are characterized by two opposed
radio-bright limbs separated by a region of low surface bright-
ness. In general, the remnants appear asymmetric, distorted and
Send offprint requests to: S. Orlando,
e-mail: orlando@astropa.inaf.it
elongated with respect to the shape and surface brightness of
the two opposed limbs. In most (but not all) of the BSNRs
the symmetry axis is parallel to the galactic plane, and this has
been interpreted as a difficulty for “intrinsic” models, e.g. mod-
els based on SN jets, rather than for “extrinsic” models, e.g.
models based on properties of the surrounding galactic medium
(Gaensler 1998).
In spite of the interest around BSNRs, a satisfactory and
complete model which explains the observed morphology and
the origin of the asymmetries does not exist. The galactic
medium is supposed to be stratified along the lines of con-
stant galactic latitude, and characterized by a large-scale am-
bient magnetic field with field lines probably mostly aligned
with the galactic plane. The magnetic field plays a three-fold
role: first, a magnetic tension and a gradient of the magnetic
field strength is present where the field is perpendicular to the
shock velocity leading to a compression of the plasma; sec-
ond, cosmic ray acceleration is most rapid where the field lines
are perpendicular to the shock speed (Jokipii 1987, Ostrowski
1988); third, the electron injection could be favored where
the magnetic field is parallel to the shock speed (Ellison et al.
1995). Gaensler (1998) notes that magnetic models (i.e. those
http://arxiv.org/abs/0704.0890v1
2 S. Orlando et al.: On the origin of asymmetries in bilateral supernova remnants
considering uniform ISM and ordered magnetic field) cannot
explain the asymmetric morphology of most BSNRs, and in-
vokes a dynamical model based on pre-existing ISM inhomo-
geneities, e.g. large-scale density gradients, tunnels, cavities.
Unfortunately, the predictions of these ad-hoc models have
consisted so far of a qualitative estimate of the BSNRs mor-
phology, with no real estimates of the ISM density interacting
with the shock. Moreover, most likely also non-uniform ambi-
ent magnetic fields may cause asymmetries in BSNRs, without
the need to assume ad-hoc density ISM structures. Two main
aspects of the nature of BSNRs, therefore, remain unexplored:
how and under which physical conditions do the asymmetries
originate in BSNRs? What is more effective in determining the
morphology and the asymmetries of this class of SNRs, the
ambient magnetic field or the non-uniform ISM?
Answering such questions at an adequate level requires
detailed physical modeling, high-level numerical implementa-
tions and extensive simulations. Our purpose here is to inves-
tigate whether the morphology of BSNR observed in the ra-
dio band could be mainly determined by the propagation of
the shock through a non-uniform ISM or, rather, across a non-
uniform ambient magnetic field. To this end, we model the
propagation of a shock generated by an SN explosion in the
magnetized non-uniform ISM with detailed numerical MHD
simulations, considering two complementary cases of shock
propagation: 1) through a gradient of ambient density with a
uniform ambient magnetic field; 2) through a homogeneous
isothermal medium with a gradient of ambient magnetic field
strength.
In Sect. 2 we describe the MHD model, the numerical
setup, and the synthesis of synchrotron emission; in Sect. 3 we
analyze the effects the environment has on the radio emission
of the remnant; finally in Sect. 4 and 5 we discuss the results
and draw our conclusions.
2. Model
2.1. Magnetohydrodynamic modeling
We model the propagation of an SN shock front through a
magnetized ambient medium. The model includes no radia-
tive cooling, no thermal conduction, no eventual magnetic field
amplification and no effects on shock dynamics due to back-
reaction of accelerated cosmic rays. The shock propagation
is modeled by solving numerically the time-dependent ideal
MHD equations of mass, momentum, and energy conservation
in a 3-D cartesian coordinate system (x, y, z):
+ ∇ · (ρu) = 0 , (1)
+ ∇ · (ρuu − BB) + ∇P∗ = 0 , (2)
+ ∇ · [u(ρE + P∗) − B(u · B)] = 0 , (3)
+ ∇ · (uB − Bu) = 0 , (4)
where
P∗ = P +
, E = ǫ +
|u|2 +
are the total pressure (thermal pressure, P, and magnetic pres-
sure) and the total gas energy (internal energy, ǫ, kinetic energy,
and magnetic energy) respectively, t is the time, ρ = µmHnH is
the mass density, µ = 1.3 is the mean atomic mass (assuming
cosmic abundances), mH is the mass of the hydrogen atom, nH
is the hydrogen number density, u is the gas velocity, T is the
temperature, and B is the magnetic field. We use the ideal gas
law, P = (γ − 1)ρǫ, where γ = 5/3 is the adiabatic index. The
simulations are performed using the flash code (Fryxell et al.
2000), an adaptive mesh refinement multiphysics code for as-
trophysical plasmas.
As initial conditions, we adopted the model profiles of
Truelove & McKee (1999), assuming a spherical remnant with
radius r0snr = 4 pc and with total energy E0 = 1.5 × 10
51 erg,
originating from a progenitor star with mass of 15 Msun,
and propagating through an unperturbed magnetohydrostatic
medium. The initial total energy is partitioned so that 1/4 of
the SN energy is contained in thermal energy, and the other
3/4 in kinetic energy. The explosion is at the center (x, y, z) =
(0, 0, 0) of the computational domain which extends between
−30 and 30 pc in all directions. At the coarsest resolution,
the adaptive mesh algorithm used in the flash code (paramesh;
MacNeice et al. 2000) uniformly covers the 3-D computational
domain with a mesh of 83 blocks, each with 83 cells. We al-
low for 3 levels of refinement, with resolution increasing twice
at each refinement level. The refinement criterion adopted
(Löhner 1987) follows the changes in density and temperature.
This grid configuration yields an effective resolution of ≈ 0.1
pc at the finest level, corresponding to an equivalent uniform
mesh of 5123 grid points. We assume zero-gradient conditions
at all boundaries.
We follow the expansion of the remnant for 22 kyrs, consid-
ering two sets of simulations: 1) through a gradient of ambient
density with a uniform ambient magnetic field; or 2) through
a homogeneous isothermal medium with a gradient of ambi-
ent magnetic field strength. Table 1 summarizes the physical
parameters characterizing the simulations.
In the first set of simulations, the ambient magnetic field
is assumed uniform with strength B = 1 µG and oriented
parallel to the x axis. The ambient medium is modeled with
an exponential density stratification along the x or the z direc-
tion (i.e. parallel or perpendicular to the B field) of the form:
n(ξ) = n0 + ni exp(−ξ/h) (where ξ is, respectively, x or z) with
n0 = 0.05 cm
−3 and ni = 0.2 cm
−3, and where h (set either to
25 pc or to 10 pc) is the density scale length. This configuration
has been used by e.g. Hnatyk & Petruk (1999) to describe the
SNR expansion in an environment with a molecular cloud. Our
choice leads to a density variation of a factor ∼ 6 or ∼ 60, re-
spectively, along the x or the z direction over the spatial domain
considered (60 pc in total). The temperature of the unperturbed
ISM is T = 104 K at ξ = 0 and is determined by pressure bal-
ance elsewhere. The adopted values of T = 104 K, n = 0.25
cm−3 and B = 1 µG at ξ = 0, outside the remnant, lead to
S. Orlando et al.: On the origin of asymmetries in bilateral supernova remnants 3
Table 1. Relevant initial parameters of the simulations: n0 and
ni are particle number densities of the stratified unperturbed
ISM (see text), h is the density scale length, and (x, y, z) are
the coordinates of the magnetic dipole moment. The ambient
medium is either uniform or with an exponential density strat-
ification along the x or the z direction (x−strat. and z−strat.,
respectively); the ambient magnetic field is uniform or dipolar
with the dipole oriented along the x axis and located at (x, y, z).
ISM n0 ni h B (x, y, z)
cm−3 cm−3 pc pc
GZ1 z−strat. 0.05 0.2 25 uniform -
GZ2 z−strat. 0.05 0.2 10 uniform -
GX1 x−strat. 0.05 0.2 25 uniform -
GX2 x−strat. 0.05 0.2 10 uniform -
DZ1 uniform 0.25 - - z−strat. (0, 0,−100)
DZ2 uniform 0.25 - - z−strat. (0, 0,−50)
DX1 uniform 0.25 - - x−strat. (−100, 0, 0)
DX2 uniform 0.25 - - x−strat. (−50, 0, 0)
β ∼ 17 (where β = P/(B2/8π) is the ratio of thermal to mag-
netic pressure) a typical order of magnitude of β in the diffuse
regions of the ISM (Mac Low & Klessen 2004).
In the second set of simulations, the unperturbed ambient
medium is uniform with temperature T = 104 K and parti-
cle number density n = 0.25 cm−3. The ambient magnetic
field, B , is assumed to be dipolar. This idealized situation is
adopted here mainly to ensure magnetostaticity of the non-
uniform field. The dipole is oriented parallel to the x axis and
located on the z axis (x = y = 0) either at z = −100 pc or at
z = −50 pc; alternatively the dipole is located on the x axis
(y = z = 0) either at x = −100 pc or at x = −50 pc (as shown
in Fig. 1). In both configurations, the field strength varies by a
factor ∼ 6 (z or x = −100 pc) or ∼ 60 (z or x = −50 pc) over
60 pc: in the first case in the direction perpendicular to the av-
erage ambient field 〈B〉, whereas in the second case parallel to
〈B〉. In all the cases, the initial magnetic field strength is set to
B = 1 µG at the center of the SN explosion (x = y = z = 0).
Note that the transition time from adiabatic to radiative
phase for a SNR is (e.g. Blondin et al. 1998; Petruk 2005)
ttr = 2.84 × 10
4 E4/1751 n
−9/17
ism yr , (5)
where E51 = E0/(10
51 erg) and nism is the particle number den-
sity of the ISM. In our set of simulations, runs GZ2 and GX2
present the lowest values of the transition time, namely ttr ≈ 25
kyr. Since we follow the expansion of the remnant for 22 kyrs,
our modeled SNRs are in the adiabatic phase.
2.2. Nonthermal electrons and synchrotron emission
We synthesize the radio emission from the remnant, assum-
ing that it is only due to synchrotron radiation from relativistic
electrons distributed with a power law spectrum N(E) = KE−ζ ,
where E is the electron energy, N(E) is the number of elec-
trons per unit volume with arbitrary directions of motion and
with energies in the interval [E, E + dE], K is the normaliza-
tion of the electron distribution, and ζ is the power law index.
Figure 1. 2-D sections in the (x, z) plane of the initial mass
density distribution and initial configuration of the unperturbed
dipolar ambient magnetic field in two cases: dipole moment lo-
cated on the z axis (DZ1, left panel), or on the x axis (DX1,
right panel). The initial remnant is at the center of the domain.
Black lines are magnetic field lines.
Following Ginzburg & Syrovatskii (1965), the radio emissivity
can be expressed as:
i(ν) = C1KB
, (6)
where C1 is a constant, B⊥ is the component of the magnetic
field perpendicular to the line-of-sight (LoS), ν is the frequency
of the radiation, α = (ζ − 1)/2 is the synchrotron spectral index
(assumed to be uniform everywhere and taken as 0.5 as ob-
served in many BSNRs). To compute the total radio intensity
(Stokes parameter I) at a given frequency ν0, we integrate the
emissivity i(ν0) along the LoS:
I(ν0) =
i(ν0) dl , (7)
where dl is the increment along the LoS.
The normalization of the electron distribution Ks in Eq.
6 (index “s” refers to the immediately post-shock values) de-
pends on the injection efficiency (the fraction of electrons
that move into the cosmic-ray pool). Unfortunately, it is un-
known how the injection efficiency evolves in time. On theo-
retical grounds, Ks is expected to vary with the shock velocity
Vsh(t) and, in case of inhomogeneous ISM, with the immedi-
ately post-shock value of mass density, ρs; let us assume that
approximately Ks ∝ ρsVsh(t)
−b. Reynolds (1998) considered
three empirical alternatives for b as a free parameter, namely,
b = 0,−1,−2. Petruk & Bandiera (2006) showed that one can
expect b > 0 and its value could be b ≈ 4. Stronger shocks
are more successful in accelerating particles. To be accelerated
effectively, a particle should obtain in each Fermi cycle larger
increase in momentum, which is proportional to the shock ve-
locity. Negative b reflects an expectation that injection effi-
ciency may behave in a way similar to acceleration efficiency:
stronger shocks might inject particles more effectively. In con-
trast, positive b represents a different point of view: efficien-
cies of injection and acceleration may have opposite depen-
dencies on the shock velocity. Stronger shock produces higher
turbulence which is expected to prevent more thermal particles
to recross the shock from downstream to upstream and to be,
4 S. Orlando et al.: On the origin of asymmetries in bilateral supernova remnants
therefore, injected. Since the picture of injection is quite un-
clear from both theoretical and observational points of view,
we do not pay attention to the physical motivations of the value
of b. Instead, our goal is to see how different trends in evolu-
tion of injection efficiency may affect the visible morphology
of SNRs. Such understanding could be useful for future obser-
vational tests on the value of b.
We found, in agreement with Reynolds (1998), that the
value of b does not affect the main features of the sur-
face brightness distribution if SNR evolves in uniform ISM.
Therefore we use the value b = 0 to produce the SNR images in
models with uniform ISM (models DZ1, DZ2, DX1, and DX2).
In cases where non-uniformity of ISM causes variation of the
shock velocity in SNR (models GZ1, GZ2, GX1, and GX2),
we calculate images for b = −2, 0, 2. We follow the model of
Reynolds (1998) in description of the post-shock evolution of
relativistic electrons. Adopting this approach and considering
that ζ = 2 (being α = 0.5, see above), one obtains that (see
Appendix A)
K(a, t)
Ks(R, t)
P(a, t)
P(R, t)
)−b/2 (
ρo(a)
ρo(R)
)−(b+1)/3 (
ρ(a, t)
ρ(R, t)
)5b/6+4/3
where a is the lagrangian coordinate, R is the shock radius, ρ is
the gas density, P is the gas pressure, and the index “o” refers
to the pre-shock values. It is important to note that this formula
accounts for variation of injection efficiency caused by the non-
uniformity of ISM.
The electron injection efficiency may also vary with the
obliquity angle between the external magnetic field and the
shock normal, φBn. The numerical simulations suggest that in-
jection efficiency is larger for parallel shocks, i.e. where the
magnetic field is parallel to the shock speed (obliquity an-
gle close to zero; Ellison et al. 1995). However, it has been
shown (Fulbright & Reynolds 1990) that models with injec-
tion strongly favoring parallel shocks produce SNR maps that
do not resemble any known objects (it is also claimed that
injection is more efficient where the magnetic field is per-
pendicular to the shock speed; Jokipii 1987). On the other
hand, comparison of known SNRs morphologies with model
SNR images calculated for different strengths of the injec-
tion efficiency dependence on obliquity suggests that the in-
jection efficiency in real SNRs could not depend on obliquity
(Petruk, in preparation). In such an unclear situation, we con-
sider the three cases: quasi-parallel, quasi-perpendicular, and
isotropic injection models. Following Fulbright & Reynolds
(1990), we model quasi-parallel injection by multiplying the
normalization of the electron distribution K by cos2 φBn2 (see
also Leckband et al. 1989), where φBn2 is the angle between the
shock normal and the post-shock magnetic field1. By analogy
with the quasi-parallel case, we model quasi-perpendicular in-
jection by multiplying K by sin2 φBn2.
1 For a shock compression ratio of 4 (the shock Mach number is≫
10 in all directions during the whole evolution in each of our simula-
tions), the obliquity angle between the external magnetic field and the
shock normal, φBn, is related to φBn2 by sin
φBn2 = (cot
2 φBn/16+1)
(e.g. Fulbright & Reynolds 1990).
An important point is the degree of ordering of magnetic
field downstream of the shock. Radio polarization observation
of a number of SNRs (e.g. Tycho Dickel et al. 1991, SN1006
Reynolds & Gilmore 1993) show the low degree of polariza-
tion, 10-15% (in case of ordered magnetic field the value ex-
pected is about 70%; Fulbright & Reynolds 1990), indicating
highly disordered magnetic field. Thus we calculate the syn-
chrotron images of SNR for two opposite cases. First, since our
MHD code gives us the three components of magnetic field,
we are able to calculate images with ordered magnetic field.
Second, we introduce the procedure of the magnetic field dis-
ordering (with randomly oriented magnetic field vector with
the same magnitude in each point) and then synthesize the ra-
dio maps. In models which have a disordered magnetic field,
we use the post-shock magnetic field before disordering to cal-
culate the angle φBn2; as discussed by Fulbright & Reynolds
(1990), this corresponds to assume that the disordering process
takes place over a longer time-scale than the electron injection,
occurring in the close proximity of the shock. Since we found
that the asymmetries induced by gradients either of ambient
plasma density or of ambient magnetic field strength are not
significantly affected by the degree of ordering of the magnetic
field downstream of the shock, in the following we will focus
on the models with disordered magnetic field.
The goal of this paper is to look whether non-uniform ISM
or non-uniform magnetic field can produce asymmetries on
BSNRs morphology. In order to clearly see the role of these
two factors in determining the morphology of BSNRs, we use
some simplifying assumptions about electron kinetic and be-
havior of magnetic field in vicinity of the shock front. Our
calculations are performed in the test-particle limit, i.e. they
ignores the energy in cosmic rays. In particular, we do not con-
sider possible amplification of magnetic field by the cosmic-ray
streaming instability (Lucek & Bell 2000, Bell & Lucek 2001).
We expect that the main features of the modeled SNR morphol-
ogy will not change if this process is independent of obliquity
angle. If future investigations show undoubtedly that magnetic
field amplification varies strongly with obliquity, the role of
this effect in producing BSNRs have to be additionally studied.
3. Results
In all the models examined, we found the typical evolution
of adiabatic SNRs expanding through an organized ambient
magnetic field (see Balsara et al. 2001 and references therein):
the fast expansion of the shock front with temperatures of few
millions degrees, and the development of Richtmyer-Meshkov
(R-M) instability, as the forward and reverse shocks progress
through the ISM and ejecta, respectively (see Kane et al. 1999).
As examples, Fig. 2 shows 2-D sections in the (x, z) plane of
the distributions of mass density and of magnetic field strength
for the models GZ2, DZ2, and DX2 at t = 18 kyrs. The in-
ner shell is dominated by the R-M instability that causes the
plasma mixing and the magnetic field amplification. In the in-
ner shell, the magnetic field shows a turbulent structure with
preferentially radial components around the R-M fingers (see
Fig. 3). Note that, some authors have invoked the R-M insta-
bilities to explain the dominant radial magnetic field observed
S. Orlando et al.: On the origin of asymmetries in bilateral supernova remnants 5
Figure 2. 2-D sections in the (x, z) plane of the mass density
distribution (left panels), in log scale, and of the distribution
of the magnetic-field strength (right panels), in log scale, in
the simulations GZ2 (upper panels), DZ2 (middle panels), and
DX2 (lower panels) at t = 18 kyrs. The box in the upper left
panel marks the region shown in Fig. 3.
in the inner shell of SNRs (e.g. Jun & Norman 1996); however,
in our simulations, the radial tendency is observed well inside
the remnant and not immediately behind the shock as inferred
from observations.
We found that, throughout the expansion, the shape of the
remnant is not appreciably distorted by the ambient magnetic
field because, for the values of explosion energy and ambi-
ent field strength (typical of SNRs) used in our simulations,
the kinetic energy of the shock is many orders of magnitude
larger than the energy density in the ambient B field (see also
Mineshige & Shibata 1990). The shape of the remnant does
not differ visually from a sphere also in the cases with den-
sity stratification of the ambient medium2 as it is shown by
Hnatyk & Petruk (1999).
The radio emission of the evolved remnants is character-
ized by an incomplete shell morphology when the viewing an-
gle is not aligned with the direction of the average ambient
2 In these cases, the remnant appears shifted toward the low den-
sity region; see upper panels in Fig. 2 (see also Dohm-Palmer & Jones
1996).
Figure 3. Close-up view of the region marked with a box in
Fig. 2. The dark fingers mark the R-M instability. The mag-
netic field is described by the superimposed arrows the length
of which is proportional to the magnitude of the field vector.
magnetic field (cf. Fulbright & Reynolds 1990); in general, the
radio emission shows an axis of symmetry with low levels of
emission along it, and two bright limbs (arcs) on either side
(see also Gaensler 1998). This morphology is very similar to
that observed in BSNRs.
3.1. Obliquity angle dependence
For each of the models listed in Table 1, we synthesized the
synchrotron radio emission, considering each of the three cases
of variation of electron injection efficiency with shock obliq-
uity: quasi-parallel, quasi-perpendicular, and isotropic particle
injection. As an example, Fig. 4 shows the synchrotron radio
emission synthesized from the uniform ISM model DZ1 with
randomized internal magnetic field at t = 18 kyrs in each of
the three cases. We recall that for these uniform density cases,
we have adopted an injection efficiency independent from the
shock speed (b = 0, Sect. 2.2). All images are maps of total in-
tensity normalized to the maximum intensity of each map and
have a resolution of 400 beams per remnant diameter (DSNR).
The images are derived when the LoS is parallel to the average
direction of the unperturbed ambient magnetic field 〈B〉 (LoS
aligned with the x axis), or perpendicular both to 〈B〉 and to
the gradient of field strength (LoS along y), or parallel to the
gradient of field strength (LoS along z).
The different particle injection models produce images that
can differ considerably in appearance. In particular, the quasi-
parallel case leads to morphologies of the remnant not repro-
duced by the other two cases: a center-brightened SNR when
the LoS is aligned with x (top left panel in Fig. 4), a BSNR
with two bright arcs slanted and converging on the side where
B field strength is higher when the LoS is along y (top center
6 S. Orlando et al.: On the origin of asymmetries in bilateral supernova remnants
Figure 4. Synchrotron radio emission (normalized to the maximum of each panel), at t = 18 kyrs, synthesized from model DZ1
assuming b = 0 (see text) and randomized internal magnetic field, when the LoS is aligned with the x (left), y (center), or z (right)
axis. The figure shows the quasi-parallel (top), isotropic (middle), and quasi-perpendicular (bottom) particle injection cases. The
color scale is linear and is given by the bar on the right. The directions of the average unperturbed ambient magnetic field, 〈B〉,
and of the magnetic field strength gradient, ∇|B |, are shown in the upper left and lower right corners of each panel, respectively.
panel), and a remnant with two symmetric bright spots located
between the center and the border of the remnant when the LoS
is along z (top right panel). Neither the center-brightened rem-
nant nor the double peak structure, showing no structure de-
scribable as a shell, seems to be observed in SNRs3. We found
analogous morphologies in all the models listed in Table 1, con-
sidering the quasi-parallel case. As extensively discussed by
Fulbright & Reynolds (1990) for models with uniform ambient
magnetic field and b = −2, we also conclude that the quasi-
parallel case leads to radio images unlike any observed SNR
(see also Kesteven & Caswell 1987).
3 Excluding filled center and composite SNRs, but these are due to
energy input from a central pulsar.
The isotropic case leads to remnant’s morphologies simi-
lar to those produced in the quasi-perpendicular case although
the latter case shows deeper minima in the radio emission than
the first one. When the LoS is aligned with x (middle left and
bottom left panels in Fig. 4) or with y (middle center and bot-
tom center panels), the remnants have one bright arc on the side
where the B strength is higher. When the LoS is aligned with z
(middle right and bottom right panels), the remnants have two
opposed arcs that appear perfectly symmetric. We found that
the isotropic and quasi-perpendicular cases lead to morpholo-
gies of the remnants similar to those observed.
S. Orlando et al.: On the origin of asymmetries in bilateral supernova remnants 7
Figure 5. Presentation as in Fig. 4 for model GZ1 with randomized internal magnetic field, assuming quasi-perpendicular particle
injection and b = −2 (top panels), b = 0 (middle) and b = 2 (bottom). The directions of the average unperturbed ambient magnetic
field, 〈B〉, and of the ambient plasma density gradient, ∇ρ, are shown in the upper left and lower right corners of each panel,
respectively.
3.2. Non-uniform ISM: dependence from parameter b
For models describing the SNR expansion through a non-
uniform ISM (models GZ1, GZ2, GX1, GX2), we derived the
synthetic radio maps considering three alternatives for the de-
pendence of the injection efficiency on the shock speed, namely
b = −2, 0, 2 (see Sect. 2.2). As an example, Fig. 5 shows the
synthetic maps derived from model GZ1 with randomized in-
ternal magnetic field, assuming quasi-perpendicular particle in-
jection, and considering b = −2 (top panels), b = 0 (middle)
and b = 2 (bottom).
When the LoS is not aligned with the density gradient, the
radio images show asymmetric morphologies of the remnants.
In this case, the main effect of varying b is to change the de-
gree of asymmetry observed in the radio maps. In the example
shown in Fig. 5, the density gradient is aligned with the z axis
and asymmetric morphologies are produced when the LoS is
aligned with x (left panels) or with y (center panels). In all the
cases, the remnant is brighter where the mass density is higher.
On the other hand, the degree of asymmetry increases with in-
creasing value of b.
The reason of such behavior consists in the balance be-
tween the roles of the shock velocity and of density in chang-
ing the injection efficiency. Consider, as an example, the top
left panel in Fig. 5: the increase of the shock velocity on the
north (due to fall of the ambient density) leads to an increase of
the brightness there (due to rise of the injection efficiency) that
partially balances the increase of the brightness on the south
due to higher density of ISM. On the other hand, for the model
shown in the bottom left panel in Fig. 5, the fraction of accel-
8 S. Orlando et al.: On the origin of asymmetries in bilateral supernova remnants
Figure 6. Synchrotron radio emission (normalized to the maximum of each panel), at t = 18 kyrs, synthesized from models
assuming a gradient of ambient plasma density (panels A and D; with b = 2) or of ambient magnetic field strength (panels B
and E; with b = 0) when the LoS is aligned with the y axis. All the models assume quasi-perpendicular particle injection. The
directions of the average unperturbed ambient magnetic field, 〈B〉, and of the plasma density or magnetic field strength gradient,
are shown in the upper left and lower right corners of each panel, respectively. The right panels show two examples of radio maps
(data adapted from Whiteoak & Green 1996 and Gaensler 1998; the arrows point in the north direction) collected for the SNRs
G338.1+0.4 (panel C) and G296.5+10.0 (panel F). The color scale is linear and is given by the bar on the right.
erated electrons increases on the south due to both the rise of
density and the decrease of the shock velocity.
When the LoS is aligned with the density gradient, the radio
images are symmetric. In the example shown in Fig. 5, this
corresponds to the maps derived when the LoS is along z (right
panels); the remnants are characterized by two opposed arcs
with identical surface brightness.
3.3. Morphology
Fig. 6 shows the radio emission maps, at a time of 18 kyrs, syn-
thesized from models with a gradient of ambient plasma den-
sity (panels A and D; assuming b = 2) and of ambient B field
strength (panels B and E; assuming b = 0). All the models as-
sume quasi-perpendicular particle injection (the isotropic case
produces radio maps with similar morphologies and the quasi-
parallel case is discussed later) and randomized internal mag-
netic field. The viewing angle is perpendicular both to the av-
erage direction of the unperturbed ambient magnetic field, 〈B〉,
(direct along the x axis) and to the gradients of density or field
strength (direct either along z, panels A and B, or x, panels D
and E). The right panels show, as examples, the radio maps of
the SNRs G338.1+04 (panel C, data from Whiteoak & Green
1996) and G296.5+10.0 (panel F, from Gaensler 1998).
In the quasi-perpendicular case discussed here, the max-
imum synchrotron emissivity is reached where the magnetic
field is strongly compressed. This configuration has been re-
ferred as “equatorial belt” (e.g. Rothenflug et al. 2004); 〈B〉
runs between the two opposed arcs (along the x axis). We found
that, when the density or the magnetic field strength gradient is
perpendicular to the field itself, the morphology of the radio
map strongly depends on the viewing angle. In these cases, the
two opposed arcs appear perfectly symmetric when the LoS
is aligned with the gradient (see, for instance, the right pan-
els in Fig. 5), otherwise the two arcs can have very different
radio brightness, leading to strongly asymmetric BSNRs (see
panels A and B in Fig. 6). In the former case (LoS aligned
with the gradient), the remnant is characterized by two axes of
symmetry: one between the two symmetric arcs and the other
perpendicular to the two. In models with strong magnetic field
S. Orlando et al.: On the origin of asymmetries in bilateral supernova remnants 9
strength gradients (DZ2; B varies by a factor ∼ 60 over 60 pc),
we found that the radio images are center-brightened when the
LoS is aligned with the gradient (figure not reported). The fact
that center-brightened remnants are not observed suggests that
the external B varies moderately in the neighborhood of the
remnants.
In case of asymmetry, the gradient is always perpendicular
to the arcs, and the brightest arc is located where either mag-
netic field strength or plasma density is higher (see panels A
and B in Fig. 6), since the synchrotron emission depends on
the plasma density, on the pressure, and on the field strength
(see Eqs. 6 and 8); in this case, there is only one axis of sym-
metry oriented along the density or B gradient. When the LoS
is parallel to 〈B〉 (along x in our models), the radio maps show
a shell structure with a maximum intensity located where mag-
netic field strength or plasma density is higher (see left pan-
els in Fig. 4 for isotropic and quasi-perpendicular cases and
left panels in Fig. 5). Our simulations show that, when the
density or the magnetic field strength gradient is perpendicu-
lar to the field itself, remnants with a monopolar morphology
can be observed at LoS not aligned with the gradient (see also
Reynolds & Fulbright 1990). Examples of observed monopolar
remnants are G338.1+0.4 (see panel C in Fig. 6) or G327.4+1.0
or G341.9-0.3.
When the density or B field strength gradient is parallel to
〈B〉 (panels D and E in Fig. 6) and the LoS lies in the plane per-
pendicular to 〈B〉, the morphology of the radio map does not
depend on the viewing angle and the two opposed arcs have
the same radio brightness. In these cases, however, there is only
one axis of symmetry and the two arcs appear slanted and con-
verging on the side where field strength or plasma density is
higher; again, the symmetry axis is aligned with the density
or B strength gradient. Examples of this kind of objects are
G296.5+10.0 (see panel F in Fig. 6) or G332.4-004 or SN1006
(which is, however, much younger than the simulated SNRs).
When the external magnetic field is parallel to the LoS, because
the system is symmetric about the magnetic field, the remnant
is axially symmetric and the radio maps show a complete radio
shell at constant intensity.
In the quasi-parallel case, 〈B〉 runs across the arcs. This
configuration has been referred as “polar caps” and it has been
invoked for the SN1006 remnant (Rothenflug et al. 2004). The
quasi-parallel case, apart from the center-brightened morphol-
ogy discussed in Sect. 3.1, can also produce remnant morpholo-
gies similar to those shown in Fig. 6. As examples, Fig. 7
shows the radio emission maps obtained in the cases dis-
cussed in Fig. 6 but assuming quasi-parallel instead of quasi-
perpendicular particle injection. Again, the viewing angle is
perpendicular both to 〈B〉 (direct along the x axis) and to the
gradients of density or field strength (direct either along z, pan-
els A and B, or x, panels C and D). In the quasi-parallel case,
remnants with a bright radio limb are produced if the gradi-
ent of ambient density or of ambient B field strength is par-
allel to 〈B〉 (instead of perpendicular to 〈B〉 as in the quasi-
perpendicular case), whereas slanting similar radio arcs are ob-
tained if the gradient is perpendicular to 〈B〉 (instead of parallel
as in the quasi-perpendicular case).
4. Discussion
Our simulations show that asymmetric BSNRs are explained
if the ambient medium is characterized by gradients either of
density or of ambient magnetic field strength: the two opposed
arcs have different surface brightness if the gradient runs across
the arcs (see panels A and B in Fig. 6, and panels C and D in
Fig. 7), whereas the two arcs appear slanted and converging
on one side if the gradient runs between them (see panels D
and E in Fig. 6 and panels A and B in Fig. 7). In all the cases
(including the three alternatives for the particle injection), the
symmetry axis of the remnant is always aligned with the gradi-
From the radio maps, we derived the azimuthal intensity
profiles: we first find the point on the map where the intensity
is maximum; then the contour of points at the same distance
from the center of the remnant as the point of maximum in-
tensity defines the azimuthal radio intensity profile. Following
Fulbright & Reynolds (1990), we quantify the degree of “bipo-
larity” of the remnants by using the so-called azimuthal inten-
sity ratio A, i.e. the ratio of maximum to minimum intensity
derived from the azimuthal intensity profiles. In addition, we
quantify the degree of asymmetry of the BSNRs by using a
measure we call the azimuthal intensity ratio Rmax ≥ 1, i.e.
the ratio of the maxima of intensity of the two limbs as de-
rived from the azimuthal intensity profiles, and the azimuthal
distance θD, i.e. the distance in deg of the two maxima. In the
case of symmetric BSNRs, Rmax = 1 and θD = 180
o. As al-
ready noted by Fulbright & Reynolds (1990), the parameter A
depends on the spatial resolution of the radio maps and on the
aspect angle (i.e. the angle between the LoS and the unper-
turbed magnetic field); moreover we note that, in real observa-
tions, the measure of A gives a lower limit to its real value if the
background is not accurately taken into account. On the other
hand, the parameters Rmax and θD have a much less critical de-
pendency on these factors and, therefore, they may provide a
more robust diagnostic in the comparison between models and
observations.
Fig. 8 shows the values of A, Rmax, and θD derived for all
the cases examined in this paper, considering the LoS aligned
with the y axis, and radio maps with a resolution of 25 beams
per remnant diameter4 (DSNR). Note that, our choice of the
LoS aligned with y (aspect angle φ = 90o) implies that the
values of A in Fig. 8 are upper limits, being A maximum at
φ = 90o and minimum at φ = 0o (see Fulbright & Reynolds
1990). The three models of particle injection (isotropic, quasi-
perpendicular and quasi-parallel) lead to different values of A.
In the isotropic and quasi-perpendicular cases, most of the val-
ues of A range between 5 and 20 (for model DX2, A is even
larger than 100); in the quasi-parallel case, the values of A are
larger than 500.
We found that, in general, a gradient of the ambient mag-
netic field strength leads to remnant morphologies similar to
those induced by a gradient of plasma density (compare, for
instance, panel A with B and panel D with E in Fig. 6). On the
other hand, if b < 0 in GX and GZ models, ambient B field
4 After the radio maps are calculated, they are convolved with a
gaussian function with σ corresponding to the required resolution.
10 S. Orlando et al.: On the origin of asymmetries in bilateral supernova remnants
Figure 7. Presentation as in Fig. 6, as-
suming quasi-parallel instead of quasi-
perpendicular particle injection.
gradients are more effective in determining the morphology of
asymmetric BSNRs. This is seen in a more quantitative form
in Fig. 8. DX and DZ models give Rmax values higher and θD
values lower than GX and GZ models with b < 0: a modest
gradient of the magnetic field (models DX1 and DZ1) gives
a value of Rmax higher or θD lower than the two models with
strong density gradients (models GX2 and GZ2) and b < 0.
Fig. 8 also shows that, in models with a density gradient,
the degree of asymmetry of the remnant increases with increas-
ing value of b; the GX and GZ models with b > 0 give values
of Rmax and θD comparable with (or, in the case of Rmax, even
larger of) those derived from DX and DZ models. In the case of
quasi-parallel particle injection for remnants with converging
similar arcs, it is necessary a strong gradient of density perpen-
dicular to B and b ≥ 0 (compare models GZ1 and GZ2 in the
lower panel in Fig. 8) to give values of θD comparable to those
obtained with a moderate gradient of ambient B field strength
perpendicular to B (see model DZ1 in Fig. 8).
In order to compare our model predictions with observa-
tions of real BSNRs, we have selected 11 SNR shells which
show one or two clear lobes of emission in archive total in-
tensity radio images, separated by a region of minima. We
have discarded all those cases in which several point-like or
extended sources appear superimposed to the bright limbs, or
other cases in which the location of maximum or minimum
emission around the shell is difficult to derive. Unlike other lists
of BSNRs published in the literature (e.g. Kesteven & Caswell
1987; Fulbright & Reynolds 1990; Gaensler 1998), here we fo-
cus on a reliable measure of the parameters A, Rmax and θD; we
avoid, therefore, patchy and irregular limbs, as in the case of
G320.4-01.2 of Gaensler (1998). Moreover, we are obviously
not discarding remnants which have constraints on A, Rmax or
θD (e.g. Fulbright & Reynolds 1990 considered only cases with
Rmax < 2), and we are considering remnants observed with a
resolution greater than 10 beams per remnant diameter. Since
in our models we follow the remnant evolution during the adi-
abatic phase, we also need to discard objects that are clearly
in the radiative phase. Unfortunately, for most of the objects
selected, there is no indication of their evolutionary stage in lit-
erature. Assuming that the remnant expands in a medium with
particle number density nism <∼ 0.3 cm
−3, the shock radius de-
rived from the Sedov solution at time ttr (i.e. at the transition
time from the adiabatic to the radiative phase; see Eq. 5) is
rtr = 19 E
−7/17
35 pc , (9)
where we have assumed that E51 = 1.5. Therefore, we only
considered remnants with radius rsnr < 35 pc (i.e. with size <
70 pc) that are, most likely, in the adiabatic phase. Our list does
not pretend to be complete or representative of the class, and
it is compiled to derive the observed values of the parameters
A, Rmax and θD with the lowest uncertainties. For this reason,
we have considered remnants for which a total intensity radio
image in digital format is available. Actually, in most of the
cases, we have used the 843 MHz data of the MOST supernova
remnant catalogue (Whiteoak & Green 1996).
S. Orlando et al.: On the origin of asymmetries in bilateral supernova remnants 11
Figure 8. Azimuthal intensity ratio A (i.e.
the ratio of maximum to minimum inten-
sity around the shell of emission - see text;
upper panel), azimuthal intensity ratio Rmax
(i.e. the ratio of the maxima of intensity
of the two limbs around the shell; middle
panel), and azimuthal distance θD (i.e. the
distance in deg of the two maxima of in-
tensity around the shell; lower panel) for
all the cases examined, considering the LoS
aligned with the y axis and a spatial res-
olution of 25 beams per remnant diame-
ter, DSNR. Black crosses: isotropic; red tri-
angles: quasi-perpendicular; blue diamonds:
quasi-parallel.
Our list is reported in Table 2. We have separated evolved
and young SNRs. While the young SNRs listed in Table 2 have
very reliable measurement of A, Rmax and θD and a good record
of literature, making them very good candidate to test the diag-
nostic power of our model, we stress that the models we are
considering in this paper are focused on evolved SNRs; we
leave the discussion about young SNRs to a separate work. For
each object in Table 2, we show the apparent size, the distance
(from dedicated studies where possible, otherwise from the re-
vised Σ − D relation of Case & Bhattacharya 1998; see their
paper for caveats on usage of the Σ − D relation to derive SNR
distance), the real size, the resolution of the observation, and
the parameters A, Rmax, and θD we have introduced here.
Table 2 shows that most of the 11 remnants have A ≤ 10,
i.e. values consistent with those derived in Fig. 8 for the three
alternatives for the particle injection (recall that the values
shown in the figure have to be considered as upper limits).
Four remnants show high values of A (10 < A < 100) that
are difficult to explain in terms of the isotropic or the quasi-
perpendicular injection models with b < 0 unless the remnant
expands through a non-uniform ambient magnetic field (see
models DX2, and DZ2 in Fig. 8). In the light of these consid-
erations, we cannot exclude a priori any of the three alternative
models for the particle injection.
Four of the 11 objects in Table 2 show values of Rmax ≥ 2,
pointing out that, in these objects, the bipolar morphology is
asymmetric with the two radio limbs differing significantly in
intensity. An example of this kind of remnants is G338.1+0.4
(see panel C in Fig. 6). In the light of our results, its morphol-
ogy can be explained if a gradient of ambient density or of
ambient magnetic field strength is either perpendicular to the
average ambient magnetic field, 〈B〉, in the isotropic and quasi-
perpendicular cases or parallel to 〈B〉 in the quasi-parallel case.
It is worth noting that reveling such a gradient from the obser-
vations may be a powerful diagnostic to discriminate among
the alternative particle injection models, producing real ad-
vances in the understanding of the nonthermal physics of strong
shock waves.
An extreme example of a monopolar remnant with a bright
radio limb is G327.4+1.0 whose value of Rmax is larger than
10. Fig. 8 shows that high values of Rmax can be easily ex-
plained as due to non-uniform ambient magnetic field strength
or to non-uniform ambient density if b > 0. We suggest that the
morphology of G327.4+1.0 may give some hints on the value
of b (and, therefore, on the dependence of the injection effi-
ciency on the shock velocity) if the observations show that the
asymmetry is due to a non-uniform ambient medium through
which the remnant expands.
12 S. Orlando et al.: On the origin of asymmetries in bilateral supernova remnants
Table 2. List of barrel-shaped SNR shells for which a measurement of A, Rmax, and θD is presented for comparison with our
models.
Remnanta Flux size d size Res.b A Rmax θD Ref./notes
Jy arcmin kpc pc beams/DSNR deg
Evolved Remnants
G296.5+10.0 48 90 × 65 2.1 55 × 40 108 > 11 1.2 85 1
G299.6-0.5 1.1 13 × 13 18.1 68 18 6 2 160 2
G304.6+0.1 18 8 × 8 7.9 18 11 20 1.5 120 3
G327.4+1.0 2.1 14 × 13 13.9 56 19 > 10 > 10 ND 2,4
G332.0+0.2 8.9 12 × 12 < 20 < 70 17 5 1 145 2,7
G338.1+0.4 3.8 16 × 14 9.9 46 × 40 21 3 2 > 120 2
G341.9-0.3 2.7 7 × 7 14.0 28 10 8 3 170 2
G346.6-0.2 8.7 11 × 10 8.2 26 × 23 15 2 1.1 110 2,7
G351.7+0.8 11 18 × 14 6.7 35 × 27 22 2 1.6 130 2
Young Remnants
G327.6+14.6 19 30 × 30 2.2 19 × 19 42 22 1 127 5
G332.4-0.4 34 11 × 10 3.1 10 × 9 15 7 1.6 98 6
References and notes. - (1) A.k.a. PKS 1209-51/52. Age: 3–20 kyrs, Roger et al. (1988). Distance from Giacani et al. (2000). (2) Distance
derived by Case & Bhattacharya (1998) using a revised Σ−D relation. (3) Distance from Caswell et al. (1975). (4) This shell has only one limb
(“monopolar” according to the definition of Fulbright & Reynolds 1990). A and Rmax are lower limits and no θD is derived. (5) A.k.a. SN1006.
Distance from Winkler et al. (2003). (6) A.k.a. RCW103. Distance from Reynoso et al. (2004). (7) Two maxima have been found in one lobe.
θD is the average of the two.
a All the data are from the MOST supernova remnant catalogue (Whiteoak & Green 1996), except where noted.
b Spatial resolution of the observation in beams per remnant diameter.
In Table 2, six of the 11 remnants (including the two young
remnants SN1006 and RCW103) have values of θD < 140
pointing out that, in these objects, the two bright radio limbs
appear slanted and converging on one side. An example of this
class of objects is G296.5+10.0 (a.k.a PKS 1209-51/52) shown
in panel F in Fig. 6. In this case, the value of θD ∼ 85
o de-
rived from the observations may be easily explained as due
to a gradient of magnetic field strength either parallel to 〈B〉
in the isotropic and quasi-perpendicular cases or perpendicu-
lar to 〈B〉 in the quasi-parallel case. Models with a gradient
of ambient density cannot explain the low values of θD found
for G296.5+10.0 unless the gradients are strong (the density
should change by a factor 60 over 60 pc) and the dependence
of the injection efficiency on the shock velocity gives5 b ≥ 2.
5. Conclusions
Our findings have significant implications on the diagnostics
and lead to several useful conclusions:
1. The three different particle injection models (namely,
quasi-parallel, quasi-perpendicular and isotropic dependence
of injection efficiency from shock obliquity) can produce con-
siderably different images (see Fig. 4). The isotropic and quasi-
perpendicular cases lead to radio images similar to those ob-
5 Large positive values of b do not necessarily mean an increas-
ing fraction of shock energy going into relativistic particles as the
shock slows down because decelerating shock accelerates particles to
smaller Emax , namely the maximum energy at which the electrons are
accelerated.
served. The parallel-case may produce radio images unlike any
observed SNR (center-brightened or with a double-peak struc-
ture not describable as a shell). This is in agreement with the
findings of Fulbright & Reynolds (1990).
2. In models with gradients of the ambient density, the
dependence of the injection efficiency on the shock velocity
(through the parameter b defined in Sect. 2.2) affects the degree
of asymmetry of the radio images: the asymmetry increases
with increasing value of b.
3. Small variations of the ambient magnetic field lead to
significant asymmetries in the morphology of BSNRs (see
Figs. 6 and 7). Therefore, we conclude that the close similar-
ity of the radio brightness of the opposed limbs of a BSNR
(i.e. Rmax ≈ 1 and θD ≈ 180
o) is evidence of uniform ambient
B field where the remnant expands.
4. Variations of the ambient density lead to asymmetries
of the remnant with extent comparable to that caused by non-
uniform ambient magnetic field if b = 2.
5. Strongly asymmetric BSNRs (i.e. Rmax ≫ 1 or θD ≪
180o) imply either moderate variations of B or strong (moder-
ate) variations of the ISM density if b < 2 (b ≥ 2) as in the
case, for instance, of interaction with a giant molecular cloud.
6. BSNRs with different intensities of the emission of the
radio arcs (i.e. Rmax > 1) can be produced by models with a
gradient of density or of magnetic field strength perpendicular
to the arc (upper panels in Fig. 6 and lower panels in Fig. 7),
and the brightest arc is in the region of higher plasma density
or higher magnetic field strength.
S. Orlando et al.: On the origin of asymmetries in bilateral supernova remnants 13
7. Remnants with two slanting similar arcs (i.e. θD < 180
can be produced by models with a gradient of density or of
magnetic field strength running centered between the two arcs
(lower panels in Fig. 6 and upper panels in Fig. 7), and the
region of convergence is where either the plasma density or the
magnetic field strength is higher.
8. In all the cases examined, the symmetry axis of the rem-
nant is always aligned with the gradient of density or of mag-
netic field.
We found that the degree of ordering of the magnetic field
downstream of the shock does not affect significantly the asym-
metries induced by gradients either of ambient plasma density
or of ambient magnetic field strength; thus our conclusions, de-
rived in the case of disordered magnetic field, do not change in
the case of ordered magnetic field.
We defined useful model parameters to quantify the degree
of asymmetry of the remnants. These parameters may provide a
powerful diagnostic in the comparison between models and ob-
servations, as we have shown in a few cases drawn from a ran-
domly selected sample of BSNRs presented in Table 2. For in-
stance, if the density of the external medium is known by other
means (e.g. thermal X-rays, HI and Co maps, etc.), BSNRs can
be very useful to investigate the variation of the efficiency of
electron injection with the angle between the shock normal and
the ambient magnetic field or to investigate the dependence of
the injection efficiency from the shock velocity. Alternatively,
BSNRs can be used as probes to trace the local configuration
of the galactic magnetic field if the dependence of the injection
efficiency from the obliquity is known.
It is worth emphasizing that our model follows the evolu-
tion of the remnant during the adiabatic phase and, therefore,
their applicability is limited to this evolutionary stage. In the
radiative phase, the high degree of compression suggested by
radiative shocks leads to increase of the radio brightness due
to compression of ambient magnetic field and electrons. Since
our model neglects the radiative cooling it is limited to rela-
tively small compression ratios and, therefore, it is not able to
simulate this mechanism of limb brightening.
It will be interesting to expand the present study, consider-
ing the detailed comparison of model results with observations.
This may lead to a major advance in the study of interactions
between the magnetized ISM and the whole galactic SNR pop-
ulation (not only BSNRs), since the mechanisms at work in the
BSNRs are also valid for SNRs of more complex morphology.
Acknowledgements. We thank the referee for constructive and help-
ful criticism. The software used in this work was in part devel-
oped by the DOE-supported ASC/Alliance Center for Astrophysical
Thermonuclear Flashes at the University of Chicago. The simula-
tions have been executed at CINECA (Bologna, Italy) in the frame-
work of the INAF-CINECA agreement. This work was supported by
Ministero dell’Istruzione, dell’Università e della Ricerca, by Istituto
Nazionale di Astrofisica, and by Agenzia Spaziale Italiana (ASI/INAF
I/023/05/0).
Appendix A: Derivation of Eq. (8)
We follow Reynolds (1998) in the description of the evolution of elec-
tron distribution. His approach is extended here to the possibility to
deal with non-uniform ISM (cf. Petruk 2006). Fluid element a ≡ R(ti)
was shocked at time ti, where R is the radius of the shock, and a is the
Lagrangian coordinate. At that time the electron distribution on the
shock was
N(Ei , a, ti) = Ks(a, ti)E
i , (A.1)
where Ei is the electron energy at time ti, Ks is the normalization of
the electron distribution immediately after the shock (in the following,
index “s” refers to the immediately post-shock values), and ζ is the
power law index. Since we are interested in radio emission, we have
to account for only energy losses of electrons due to the adiabatic
expansion (Reynolds 1998):
, (A.2)
where ρ is the mass density, so the energy varies as
E = Ei
ρ(a, t)
ρs(a, ti)
. (A.3)
The conservation law for the number of particles per unit volume per
unit energy interval
N(E, a, t) = N(Ei, a, ti)
a2 da dEi
σr2 dr dE
, (A.4)
where σ is the shock compression ratio and r is the Eulerian coor-
dinate, together with the continuity equation ρo(a)a
2da = ρ(a, t)r2dr
(index “o” refers to the pre-shock values) and the derivative
ρ(a, t)
ρs(a, ti)
)−1/3
, (A.5)
implies that downstream
N(E, a, t) = Ks(a, ti)E
ρ(a, t)
ρs(a, ti)
)(ζ+2)/3
. (A.6)
If Ks ∝ ρsVsh(t)
−b, where Vsh(t) is the shock velocity and ρs is the
immediately post-shock value of density, then
Ks(a, ti) = Ks(R, t)
ρo(a)
ρo(R)
Vsh(t)
Vsh(ti)
. (A.7)
Therefore, the distribution of relativistic electrons follows
K(a, t)
Ks(R, t)
N(E, a, t)
N(E,R, t)
ρo(a)
ρo(R)
Vsh(t)
Vsh(ti)
ρ(a, t)
ρs(a, ti)
)(ζ+2)/3
. (A.8)
Now we can substitute Eq. A.8 with the ratio of the shock velocities
which comes from the expression (Hnatyk & Petruk 1999)
P(a, t)
Ps(R, t)
ρo(a)
ρo(R)
)−2/3 (
Vsh(ti)
Vsh(t)
ρ(a, t)
ρs(R, t)
. (A.9)
Thus, finally
K(a, t)
Ks(R, t)
P(a, t)
Ps(R, t)
)−b/2 (
ρo(a)
ρo(R)
)−(b+ζ−1)/3 (
ρ(a, t)
ρs(R, t)
)5b/6+(ζ+2)/3
. (A.10)
This formula may easily be used to calculate the profile of K(a) for
known P(a) and ρ(a) in the case of the radial flow of fluid. In the case
when mixing is allowed, the position R should correspond to the same
part of the shock which was at a at time ti.
14 S. Orlando et al.: On the origin of asymmetries in bilateral supernova remnants
References
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	Introduction
	Model
	Magnetohydrodynamic modeling
	Nonthermal electrons and synchrotron emission
	Results
	Obliquity angle dependence
	Non-uniform ISM: dependence from parameter b
	Morphology
	Discussion
	Conclusions
	Derivation of Eq. (8)
