In
Kang (2015)
we calculated the acceleration of cosmicray electrons at weak spherical shocks that are expected to form in the cluster outskirts, and estimated the diffuse synchrotron radiation emitted by those electrons. There we demonstrated that, at decelerating spherical shocks, the volume integrated spectra of both electrons and radiation deviate significantly from the testparticle powerlaws predicted for constant planar shocks, because the shock compression ratio and the flux of inject electrons decrease in time. In this study, we consider spherical blast waves propagating through a constant density core surrounded by an isothermal halo with
ρ
∝
r
^{−n}
in order to explore how the deceleration of the shock affects the radio emission from accelerated electrons. The surface brightness profile and the volumeintegrated radio spectrum of the model shocks are calculated by assuming a ribbonlike shock surface on a spherical shell and the associated downstream region of relativistic electrons. If the postshock magnetic field strength is about 0.7 or 7
µ
G, at the shock age of ∼ 50 Myr, the volumeintegrated radio spectrum steepens gradually with the spectral index from
α
_{inj}
to
α
_{inj}
+ 0.5 over 0.1–10 GHz, where
α
_{inj}
is the injection index at the shock position expected from the diffusive shock acceleration theory. Such gradual steepening could explain the curved radio spectrum of the radio relic in cluster A2266, which was interpreted as a broken powerlaw by
Trasatti et al. (2015)
, if the relic shock is young enough so that the break frequency is around 1 GHz.
1. INTRODUCTION
Cosmological hydrodynamic simulations have shown that shock waves may form due to supersonic flows in the baryonic intracluster medium (ICM) during the formation of the large scale structure in the Universe (e.g.,
Ryu et al. 2003
;
Kang et al. 2007
;
Vazza et al. 2009
;
Skillman et al. 2011
). The time evolution of such shocks in numerical simulations (e.g.,
Vazza et al. 2012
) indicates that shock surfaces behave like spherical bubbles blowing out from the cluster center during major episodes of mergers or infalls from adjacent filaments. Shock surfaces could last for the order of the dynamical time scale of clusters,
t
_{dyn}
∼ 1 Gyr. This implies that some cosmological shock waves are associated with merger induced outflows, and hence shock bubbles are likely to expand into the cluster outskirts where the density profile decreases with radial distance from the cluster center.
Observational evidence for shock bubbles can be found in the socalled “radio relics” detected in the outskirts of galaxy clusters, which are interpreted as diffuse synchrotron emitting structures containing relativistic electrons accelerated at weak ICM shocks (
M_{s}
∼ 2 − 4) (e.g.,
van Weeren et al. 2010
,
2011
;
Nuza et al. 2012
;
Feretti et al. 2012
;
Brüggen et al. 2012
;
Brunetti & Jones 2014
). Some radio relics, for instance, the “Sausage relic” in cluster CIZA J2242.8+5301 and the “Toothbrush” relic in cluster 1RXS J0603.3+4214, have thin arclike shapes of ∼ 50 kpc in width and ∼ 1 − 2 Mpc in length (
van Weeren et al. 2010
,
2012
). They could be represented as a ribbonlike structure on a spherical shell and the associated downstream volume of radioemitting electrons, projected onto the sky plane (e.g.,
van Weeren et al. 2010
;
Kang et al. 2012
).
As shown in
Figure 1
, the viewing depth of a relic shock structure can be parameterized by the extension angle 2
ψ
, while the physical width of the postshock volume of radiating electrons is mainly determined by the advection length, ∆
l
_{adv}
≈
u
_{2}
·
t
_{age}
, for low energy electrons or the cooling length, ∆
l
_{cool}
(
γ_{e}
) ≈
u
_{2}
·
t
_{rad}
(
γ_{e}
) for high energy electrons. Here
u
_{2}
is the flow speed behind the shock and
t
_{rad}
(
γ_{e}
) is the radiative cooling time scale for electrons with Lorentz factor,
γ_{e}
. It remains largely unknown how such a ribbonlike structure can be formed by the spherical outflows in galaxy clusters. We note, however, a recent study by
Shimwell et al. (2015)
who suggested that the
uniform
arclike shape of some radio relics may trace the underlying region of preexisting seed electrons remaining from an old radio lobe.
Left: Schematic diagram showing a spherical shock surface (red) and the postshock electron distribution (blue) behind the shock. The shock surface is modeled as a Mpclong ribbon on the spherical surface with r_{s}, whose viewing depth can be defined by the extension angle 2ψ. The width of the postshock electron distribution behind the shock is determined by ∆l(γ_{e}) ≈ u_{2} · min[t_{age},t_{rad}(γ_{e})]. The peach surface faces the observer. The surface brightness is calculated by integrating the volume emissivity j_{ν} (d) along the path length, h, at a given line of sight (see Equation 4). Here R is the distance behind the projected shock edge in the plane of the sky and d = r_{s} − r is the distance of a shell behind the shock, so (r_{s} − d)^{2} = (r_{s} − R)^{2} + h^{2}. Right: Radio image projected onto the sky plane, if viewed from the top as indicated here. The surface brightness is uniform along the thin arclike shape as in the case of the Sausage relic in CIZA J2242.8+5301 (van Weeren et al. 2010), while it varies with the downstream distance, R(kpc), as shown in Figures 2 and 4.
According to diffusive shock acceleration (DSA) theory, cosmicray (CR) particles can be generated via Fermi 1st order process at collisionless shocks (
Bell 1978
;
Drury 1983
;
Malkov & Drury 2001
). In the
testparticle
regime, the CR energy spectrum at the position of the shock has a powerlaw form,
N
(
E
) ∝
E
^{−s}
, where
s
= (
σ
+ 2)/(
σ
− 1) and
σ
=
ρ
_{2}
/
ρ
_{1}
is the shock compression ratio. Hereafter, we use the sub scripts ‘1’ and ‘2’ to denote the conditions upstream and downstream of shock, respectively. Then the synchrotron radiation spectrum due to these CR electrons has a powerlaw form of
j_{ν}
(
x_{s}
) ∝
ν
−
^{αinj}
, where
is the injection index at the shock. Moreover, the volumeintegrated synchrotron spectrum downstream of a planar shock becomes a simple powerlaw of
J_{ν}
∝
ν
^{−Aν}
with
A_{ν}
=
α
_{inj}
+0.5 above a break frequency, since electrons cool via synchrotron and inverseCompton (iC) losses behind the shock (e.g.,
Kang 2011
). Such spectral characteristics are commonly used to infer the shock Mach number of observed radio relics (e.g.,
van Weeren et al. 2010
;
Stroe et al. 2014
)
In
Kang (2015)
(Paper I), we calculated the electron acceleration at spherical shocks similar to SedovTaylor blast waves with
M_{s}
∼ 2.5 − 4.5, which expand into a hot
uniform
ICM. We found that the electron energy spectrum at the shock location reaches a steady state defined by the
instantaneous
shock parameters. Hence the spatially resolved, synchrotron radiation spectra at the shock can be described properly by the testparticle DSA predictions for steady planar shocks. However, the volume integrated spectra of both electrons and radiation evolve differently from those of planar shocks and exhibit some nonlinear signatures, depending on the timedependence of the shock parameters. For instance, the shock compression ratio
σ
and the injection flux of CR electrons decrease, as the shock expands and slows down, resulting in some curvatures in both electron and radiation spectra.
Magnetic fields play key roles in DSA at collisionless shocks and control the synchrotron cooling and emission by relativistic electrons. The observed magnetic field strength is found to decrease from ∼ 1 − 10
µ
G in the core region to ∼ 0.1 − 1
µ
G in the periphery of clusters (
Feretti et al. 2012
). On the other hand, it is well established that magnetic fields can be ampli fied via resonant and nonresonant instabilities induced by CR protons streaming upstream of strong shocks (
Bell 1978
;
Lucek & Bell 2000
;
Bell 2004
). Recently
Caprioli & Spitkovsky (2014)
have shown that the magnetic field amplification (MFA) factor scales with the Alfvénic Mach number,
M_{A}
, and the CR proton acceleration efficiency as <
δB
/
B
>
^{2}
∼ 3
M_{A}
Here
δB
is the turbulent magnetic fields perpendicular to the mean background magnetic fields,
ρ
_{1}
is the upstream gas density, and
P
_{cr,2}
is the downstream CR pressure. For typical cluster shocks with 2
M_{s}
5 and 10
M_{A}
25 (
Ryu et al. 2003
), the MFA factor due to the streaming stabilities is expected be rather small but not negligible, <
δB
/
B
>
^{2}
∼ 0.3 − 3. However, it has not yet been fully understood how magnetic fields may be amplified both upstream and downstream of a weak shock in high beta ICM plasmas with
β_{p}
=
P_{g}
/
P_{B}
∼ 100.
Therefore, in Paper I, we considered several models with decaying postshock magnetic fields, in which the downstream magnetic field,
B_{d}
(
r
), decreases behind the shock with a scale height of 100150 kpc. Varying
B_{d}
(
r
) profiles were found to have little impact on the spatial distribution of the electron energy spectrum, because the baseline cooling rate of electrons is set by iC scattering off the cosmic background photons. Further, electrons in a broad range of
γ_{e}
contribute to
j_{ν}
(
r
), which smooths out any spatial variations. We note, however, that the
B
^{2}
dependence of the synchrotron emissivity (
j_{ν}
∝
N_{e}B
^{2}
) can become significant in some cases. Moreover, any nonlinear features due to the spatial variations of
N_{e}
(
r
,
γ_{e}
) and
B_{d}
(
r
) are mostly averaged out, leaving only subtle signatures in the volume integrated spectrum,
J_{ν}
.
For the case with a constant background density (e.g.,
MF13
model in Paper 1), the shock speed decreases approximately as
u_{s}
∝
t
^{−3/5}
. In fact, it has not been examined, through cosmological hydrodynamic simulations, whether these shock bubbles would accelerate or decelerate as they propagate through the cluster outskirts. In this study, we have performed additional DSA simulations, in which the initial Sedov blast wave travels through a background medium with
ρ_{u}
∝
r
^{−n}
, where
r
is the radial distance from the cluster center and
n
= 2 − 4. This effectively mimics a blast wave that expands into a constantdensity core surrounded by an isothermal halo with a decreasing pressure pro file. In these new runs, the spherical shock decelerates much more slowly than
u_{s}
∝
t
^{−3/5}
, and so the nonlinear effects due to the deceleration of the shock speed are expected to be reduced compared to the uniform density models considered in Paper I.
We have also calculated the projected surface brightness,
I_{ν}
, which depends on the three dimensional structure of the shock surface and the viewing direction. Because of the curvature in the structure of the model shock, synchrotron emissions from downstream electrons with different ages contribute to the surface brightness along a given lineofsight (i.e., projection effects). So the observed spatial profile of
I_{ν}
(
R
) is calculated by assuming the geometrical configuration described in
Figure 1
. We note that this model with a ribbonlike shock surface gave rise to radio flux pro files,
S_{ν}
(
R
), that were consistent with those of the the Sausage relic and the double relics in ZwCl0008.8+5215 (e.g.,
van Weeren et al. 2010
,
2011
;
Kang et al. 2012
).
In paper I, we demonstrated that the spectral index of the volumeintegrated spectral index increases gradually from
A_{ν}
=
α
_{inj}
to
A_{ν}
=
α
_{inj}
+ 0.5 over a broad frequency range, ∼ (0.1 − 10)
ν
_{br}
, where the break frequency is
ν
_{br}
∼ 0.4 GHz at the shock age of about 50 Myr for postshock magnetic fields
B
_{2}
∼ 0.7 or 7
µ
G (see Equation 9). Here, we explore whether such a transition can explain the broken powerlaw spectra observed in the radio relic in A2256 (
Trasatti et al. 2015
).
In the next section we describe the numerical calculations. The DSA simulation results of blast wave models with different postshock magnetic field profiles and with different background density profiles will be discussed in Section 3. A brief summary will be given in Section 4.
2. NUMERICAL CALCULATIONS
 2.1. Basic Equations
In order to calculate DSA of CR electrons at spherical shocks, we have solved the timedependent diffusionconvection equation for the pitchangleaveraged phase space distribution function for CR electrons,
f_{e}
(
r, p, t
) =
g_{e}
(
r, p, t
)
p
^{−4}
, in the onedimensional (1D) spherically symmetric geometry:
where
u
(
r, t
) is the flow velocity,
y
= ln(
p/m_{e}c
),
m_{e}
is the electron mass,
c
is the speed of light, and
D
(
r, p
) is the spatial diffusion coefficient (
Skilling 1975
). We adopt a Bohmlike diffusion coefficient with a weaker nonrelativistic momentum dependence
The cooling term
b
(
p
) = −
dp/dt
accounts for electron synchrotron and iC losses, and the cooling time scale for electrons is given as
(e.g.,
Kang 2011
). The ‘effective’ magnetic field strength
takes account for radiative losses due to both synchrotron and iC processes, where
B
_{rad}
= 3.24
µ
G(1+
z
)
^{2}
corresponds to the cosmic background radiation at redshift
z
(
Schlickeiser 2002
). In this study, we set
z
= 0.2 as a reference epoch and so
B
_{rad}
= 4.67
µ
G.
Here we adopt a simple phenomenological injection model, in which the particles with
p
p
_{inj}
≈ 5.34
m_{p}u_{s}
(
t
)/
σ
are allowed to cross the shock front and get injected into the CR populations (
Kang et al. 2002
). For spherically expanding shocks considered here, the injection momentum decreases in time as the shock decelerates. Since we are concerned with synchrotron emission from electrons alone, so long as we do not care about the exact units of flux, we can set the electron to proton ratio
K_{e/p}
= 1 in the simulations. More detailed discussion on the preacceleration and injection of electrons can be found in Paper I.
At weak shocks in the testparticle limit, the CR feedback becomes negligible and the background flow,
u
(
r, t
), is governed by the
standard
gasdynamic conservation equations in 1D spherical coordinates (
Kang & Jones 2006
). The testparticle version of CRASH (CosmicRay Amr SHock) code in a comoving spherical grid was used.
 2.2. Simulation Setup
For the initial shock structure, we adopt a SedovTaylor similarity solution propagating in a uniform ICM with the following parameters: the ICM density,
n
_{H,1}
= 10
^{−3 }
cm
^{−3}
, the ICM temperature,
T
_{1}
= 5 × 10
^{7}
K, the initial shock radius,
r_{s,i}
= 0.78Mpc, and the initial shock speed,
u_{s,i}
= 4.5 × 10
^{3}
km s
^{−1}
with the sonic Mach number,
M_{s,i}
= 4.3 at the onset of the simulations. The shock parameters change in time as the spherical shock expands out, depending on the upstream conditions in the cluster outskirts.
 2.3. Synchrotron Radiation
The synchrotron emissivity,
j_{ν}
(
r
), at each shell is calculated (in units of erg cm
^{−3}
s
^{−1}
Hz
^{−1}
str
^{−1}
), using the electron distribution function,
f_{e}
(
r, p, t
), and the magnetic field profile,
B
(
r, t
), from the DSA simulations. Then the radio intensity or surface brightness,
I_{ν}
(erg cm
^{−2}
s
^{−1}
Hz
^{−1}
str
^{−1}
), is calculated by integrating
j_{ν}
along the path length,
h
, as shown in
Figure 1
:
Here
R
is the distance behind the projected shock edge in the plane of the sky and
d
=
r_{s}
−
r
is the distance of a shell behind the shock, where (
r_{s}
−
d
)
^{2}
= (
r_{s}
−
R
)
^{2}
+
h
^{2}
. The extension angle
ψ
= 10
^{◦}
is assumed in this study.
The volume integrated emissivity,
J_{ν}
=
∫
j_{ν}
(
r
)
dV
, is calculated by integrating
j_{ν}
over the downstream volume specified with
r_{s}
and
ψ
, as shown in
Figure 1
.
The spectral indices of
j_{ν}
(
d
),
J_{ν}
, and
I_{ν}
(
R
) are defined as follows:
estimated between
ν_{i}
and
ν
_{i+1}
. We chose the following four frequencies at the source,
ν
_{1}
= 240 MHz,
ν
_{2}
= 600 MHz,
ν
_{3}
= 1.4 GHz, and
ν
_{4}
= 3.0 GHz in
Figures 2
and
4
. Then the redshifted frequency for objects at a redshift
z
is
ν
_{obs}
,
i
=
ν_{i}
/(1 +
z
).
Spherical shock models with three different magnetic field profiles: MF1, MF2, and MF3. The results are shown at t_{age} = 47 Myr, when r_{s} = 0.96 Mpc, u_{s} = 3.4 × 10^{3} km s^{−1} and M_{s} = 3.2. In the upper two rows, spatial distributions of the synchrotron emissivity, ν_{i}j_{νi} (r), and its spectral indices, α_{νi−νi+1(r)} are plotted as a function of the postshock distance d from the shock surface. In the lower two rows, the intensity ν_{i}I_{νi} (R) and its spectral indices, B_{νi−νi+1} (R) are plotted as a function of the projected distance R from the shock. The frequency is ν_{i} = 240 MHz (black solid line), 600 MHz (red dotted), 1.4 GHz (blue dashed), and 3.0 GHz for i = 1, 2, 3, and 4. The downstream volume of radioemitting electrons is assumed to have the shape as illustrated in Figure 1 with ψ = 10^{◦} . Note that j_{ν} and I_{ν} are plotted in arbitrary units.
Spherical shock models with different background density and magnetic field profiles: BD1 (MF1), BD2, BD2b, and BD3b (from top to bottom rows). See Table 1 for the model parameters. Spatial distributions of the intensity νI_{ν} are shown at three shock ages, t_{1} = 13 Myr, t_{2} = 33 Myr, and t_{3} = 67 Myr. The frequency is ν_{1} = 240 MHz (black solid line), ν_{2} = 600 MHz (red dotted), and ν_{1} = 1.4 GHz (blue dashed). The downstream volume of radioemitting electrons is assumed to have the shape as illustrated in Figure 1 with ψ = 10^{◦} . The intensity, νI_{ν} · X, is plotted in an arbitrary unit, where the numerical factor, 1 ≤ X ≤ 5, is adopted in order to plot the quantities in the linear scale.
3. DSA SIMULATION RESULTS
 3.1. Shocks with Different Magnetic Field Profiles
As in the previous study of
Kang et al. (2012)
, we adopt the postshock magnetic field strength
B
_{2}
∼ 7
µ
G in order to model the observed width of the Sausage relic (∆
l
∼ 55 kpc) as the cooling length of
γ_{e}
∼ 10
^{4}
electrons (∼
u
_{2}
t
_{rad}
), while the preshock magnetic field strength is chosen to be
B
_{1}
∼ 2 − 3
µ
G. Since
B
_{1}
cannot be constrained directly from observations, it is adjusted so that
B
_{2}
becomes about 7
µ
G after considering MFA or compression of the perpendicular components of magnetic fields across the shock. However, we will show in the next section that the observed width can be modeled with a weaker magnetic field of
B
_{2}
∼ 0.7
µ
G (see the discussion below Equation 9).
We consider several models whose characteristics are summarized in
Table 1
. As described in Paper I, we adopt the following downstream magnetic field profile,
B_{d}
(
r
), in
MF
models:

MF1:B1= 2µG &B2= 7µG .

MF2:&Bd(r) =B2· (P(r)/P2)1/2forr

MF3:&Bd(r) =B2· (ρ(r)/ρ2) forr
For these models, the upstream density and temperature are assumed to be uniform.
^{a} In fact the models MF1 and BD1 are identical.
In
Figure 2
, we compare these models at the shock age of 47 Myr. The evolution of the shock, i.e.,
r_{s}
(
t
) ∝
t
^{2/5}
and
u_{s}
(
t
) ∝
t
^{−3/5}
, is identical in these three models. Synchrotron emissivity scales with the electron energy spectrum and the magnetic field strength as
j_{ν}
∝
N_{e}
(
γ_{e}
)
B
^{2}
. Of course, the evolution of
N_{e}
(
γ_{e}
) in each
MF
model also depends on the assumed pro file of
B_{d}
(
r
) through DSA and synchrotron cooling. In
MF1
model where
B_{d}
(
r
) =
B
_{2}
, the effects of shock deceleration are dramatically revealed in the spatial distribution of
j_{ν}
(
d
) (at the low frequency of 240MHz). It increases downstream because of higher shock compression and higher electron injection flux at the earlier epoch. On the other hand, at the higher frequency of 1.4 GHz,
j_{ν}
(
d
) decreases behind the shock due to the fast synchrotron/iC cooling of high energy electrons.
The downstream increase of
j_{ν}
(
d
) at low frequencies is softened in the models with decaying postshock magnetic fields, compared to
MF1
model, because of
B
^{2}
dependence of the synchrotron emissivity. In
MF3
model, in which the downstream magnetic field decreases with the gas density behind the shock, the spatial distributions of
j_{ν}
(
d
) at all three frequencies decrease downstream. Thus the downstream distribution of
j_{ν}
(
d
) depends on the shock speed evolution,
u_{s}
(
t
), and the postshock
B_{d}
(
r
) as well as the chosen frequency. In Paper I, we indicated that signatures imprinted on synchrotron emission,
j_{ν}
(
d
), and its volume integrated spectrum,
J_{ν}
, due to different postshock magnetic field profiles could be rather subtle to detect. That expectation arose because we examined the behavior of log
j_{ν}
(
r
) in paper I (see Figure 6 there), while we plot
νj_{ν}
(
d
) in
Figure 2
.
As can be seen in the lower two rows in
Figure 2
, the surface brightness profile,
I_{ν}
(
R
), is affected by projection effects as well as the evolution of
u_{s}
(
t
) and the spatial variation of
B_{d}
(
r
). For instance, the gradual increase of
I_{ν}
just behind the shock up to
R
_{inf}
≈ 15 kpc is due to the increase of the path length, and its inflection point,
R
_{inf}
=
r_{s}
(1 − cos
ψ
), depends on the value of
ψ
. Beyond the inflection point, the path length decreases but
j_{ν}
(
d
) may increase or decrease depending on
u_{s}
(
t
) and
B_{d}
(
r
), resulting in a range of spatial profiles of
I_{ν}
(
R
).
Figure 2
also demonstrates that the spectral indices,
α
_{νi−νi+1}
(
d
) and
B
_{νi−νi+1}
(
R
), at all three frequencies decrease behind the shock and do not show significant variations among the different
MF
models other than faster steepening of both indices for more rapidly decaying
B_{d}
(
r
) profiles.
In summary, at SedovTaylor type spherical shocks decelerating with
u_{s}
∝
t
^{−3/5}
, the energy spectrum,
N_{e}
(
r
,
γ_{e}
), of low energy electrons increases downstream behind the shock. As a result, the spatial distributions of the radio emissivity,
j_{ν}
(
d
), and the surface brightness,
I_{ν}
(
R
), at low radio frequencies (< 1 GHz) could depend significantly on the postshock magnetic field profile.
At high radio frequencies, such dependence becomes relatively weaker, because the width of the postshock spatial distribution of high energy electrons is much narrower. So, the magnetic field profile far downstream has less influence on synchrotron emission. On the other hand, the spectral indices,
α_{ν}
and
B_{ν}
, are relatively insensitive to these variations.
 3.2. Shocks with Different Background Density Profile
In
BD
models, we assume that the initial blast wave propagates into an isothermal halo with a different density profile for
r
>
r_{s,i}
:

BD1:ρu(r) =ρ0·

BD2:ρu(r) =ρ0· (r/rs,i)−2.

BD3:ρu(r) =ρ0· (r/rs,i)−4.
The upstream temperature is
T
_{1}
= 5×10
^{7}
K, and
B
_{1}
= 2
µ
G and
B
_{2}
= 7
µ
G for these models. In fact
MF1
and
BD1
models are identical. We also consider
BD2b
and
BD3b
models, in which the downstream magnetic field profile is the same as
MF2
, i.e.,
In the socall beta model for isothermal ICMs, the gas distribution can be represented by
ρ
(
r
) ∝
r
^{−3β}
in the outskirts of galaxy clusters (
Sarazin 1988
). So
BD2
model corresponds to the beta model with
β
≈ 2/3, which is consistent with typical Xray brightness profile of observed Xray clusters. Recall that in our simulations the spherical blast wave propagating through a uniform ICM is adopted for the initial conditions. So we are effectively considering a spherical blast wave that propagates first in a constantdensity core (i.e.,
ρ
=
ρ
_{0}
for
r
<
r_{s,i}
) and then expands into a surrounding isothermal halo with
ρ
∝
r
^{−n}
(for
r
>
r_{s,i}
).
With the different background density (or gas pressure) profile, the shock speed evolves differently as it expands outward.
Figure 3
shows how the shock radius,
r_{s}
(
t
), shock speed,
u_{s}
(
t
), the sonic Mach number,
M_{s}
(
t
), and the DSA spectral index at the shock position,
α
_{inj}
, vary in time in
BD
models. As expected, the shock decelerates much more slowly if the background pressure decreases outward as in
BD2
and
BD3
models. In
BD3
model, for instance, the shock speed decreases less than 10% during 100 Myr. Hence these
BD
models allow us to explore the dependence of radio spectral properties for a range of the time evolution of the shock.
Evolution of spherical shock models with different background density profile, BD1 (black solid lines), BD2 (red dotted), and BD3 (blue dashed) are shown: shock radius, r_{s}, shock speed, u_{s}, sonic Mach number, M_{s}, and the DSA spectral index at the shock, α_{inj}.
Figure 4
compares the surface brightness profile,
I_{ν}
(
R
), at
ν
_{1}
= 240 MHz (black solid lines),
ν
_{2}
= 600 MHz (red dotted), and
ν
_{3}
= 1.4 GHz (blue dashed) in models with different
ρ_{u}
and
B_{d}
. In
BD1
model, the shock parameters are:
r_{s}
= 0.83Mpc,
u_{s}
= 4.1 × 10
^{3}
km s
^{−1}
and
M_{s}
= 3.9 at
t
_{1}
= 13 Myr;
r_{s}
= 0.91Mpc,
u_{s}
= 3.6 × 10
^{3}
km s
^{−1}
and
M_{s}
= 3.4 at
t
_{2}
= 33 Myr;
r_{s}
= 1.0Mpc,
u_{s}
= 3.1 × 10
^{3}
km s
^{−1}
and
M_{s}
= 2.9 at
t
_{3}
= 67 Myr.
The width of the radio structure at low frequencies, ∆
l
_{adv}
∼
t
_{age}
u_{s}
/
σ
, increases with the shock age, while the amplitude of the surface brightness decreases in time (from left to right in
Figure 4
). For high frequencies (> 1 GHz), however, the width asymptotes to the cooling length, ∆
l
_{cool}
∼
t
_{rad}
(
γ_{e}
)
u_{s}
/
σ
. So the postshock magnetic field strength could be inferred from
t
_{rad}
(
γ_{e}
) with high frequency observations. Since the synchrotron emission from monoenergetic electrons with
γ_{e}
in the postshock magnetic field of
B
_{2}
peaks at
ν
_{peak}
≈ 0.3(3
eB
_{2}
/4
πm_{e}c
)
≈ 0.88 GHz(
B
_{2}
/7
µ
G)(
γ_{e}
/10
^{4}
)
^{2}
, according to Equation (3), the cooling time scale of the electrons that emit mostly at
ν
_{peak}
becomes
where
Here note that the quantity, (
B
_{e,2}
/8.4
µ
G)
^{−2}
(
B
_{2}
/7
µ
G)
^{1/2}
, has an identical value for two values of
B
_{2}
, e.g., 0.7 and 7
µ
G. So the downstream cooling length of the relic shocks becomes degenerate for the two possible values of
B
_{2}
.
At low frequencies (< 1 GHz) the surface brightness is affected by the evolution of
u_{s}
(
t
) and the spatial variation of
B_{d}
(
r
) as well as the projection effects, resulting in a wide range of spatial profiles of
I_{ν}
(
R
). This suggests that, if the radio surface brightness can be spatially resolved at several radio frequencies over ∼ (0.1 − 10) GHz, we may extract the time evolution of
u_{s}
(
t)
as well as the shock age from low frequency observations. So it would be useful to compare multifrequency radio observations with the intensity modeling that accounts for the shock evolution, magnetic field profile and projection effects.
The comparison of
I_{ν}
(
R
) of
MF1
and
MF2
models in
Figure 2
demonstrates that the decaying postshock magnetic field could lessen the shock deceleration signatures in the surface brightness profile. If we compare
I_{ν}
(
R
) of
BD2
and
BD2b
models in
Figure 4
, on the other hand, the difference in their profiles is much smaller than that between
MF1
and
MF2
models. This is because the the shock deceleration effects and the ensuing downstream increase of
N_{e}
(
r
,
γ_{e}
) are relatively milder in
BD2
and
BD2b
models.
In
Figure 5
, we compare the volume integrated spectrum,
J_{ν}
, and its spectral index,
A_{ν}
, at three different shock ages,
t
_{age}
= 13 Myr (black solid), 33 Myr (red dotted), and 67 Myr (blue dashed) for the same set of models shown in
Figure 4
.
Same models shown as in Figure 4. Volume integrated emissivity, J_{ν} , and its spectral index, A_{ν}, are shown at three different shock age, t_{1} = 13 Myr (black solid lines), t_{2} = 33 Myr (red dotted lines), and t_{3} = 67 Myr (blue dashed lines).
For the testparticle powerlaw at steady planar shocks, the radio index
A_{ν}
is the same as
α
_{inj}
for
at the source. For
ν
>
ν
_{br}
,
A_{ν}
is expected to steepen to
α
_{inj }
+ 0.5 due to synchrotron/iC cooling of electrons. Again note that the quantity, (
B
_{e,2}
/8.4
µ
G)
^{−4}
(
B
_{2}
/7
µ
G) can have the same value for two possible value of
B
_{2}
, and so the break frequency at a given shock age becomes identical as well. The observed (redshifted) break frequency corresponds to
ν
_{br,obs}
=
ν
_{br}
/(1 +
z
). If the shock age is
t
_{age}
∼ 67 Myr and
B
_{2}
∼ 0.7 or 7
µ
G, for example, the observed break frequency becomes about 120 MHz for objects at
z
= 0.2. The smallest possible index is
α
_{inj}
= 0.5 at strong shocks, so
A_{ν}
=
α
_{inj}
+ 0.5 ≥ 1.0 above the break frequency.
As can be seen in
Figure 5
, the transition from
α
_{inj}
to
α
_{inj}
+ 0.5 occurs gradually over about two orders of magnitude in frequency range. If the shock is 3070 Myr old for the shock parameters considered here, the volumeintegrated radio spectrum is expected to steepen gradually from 100 MHz to 10 GHz, instead of a sharp broken powerlaw with the break frequency at
ν
_{br,obs}
. Such gradual steepening may explain why the volumeintegrated radio spectra of some observed relic shocks might be interpreted as a broken powerlaw with
A_{ν}
< 1.0 at low frequencies.
For example, according to a recent observation of the relic in A2256, the observed spectral index is
between 351 and 1369 MHz and increases to
between 1369 and 10450 MHz (
Trasatti et al. 2015
). Similar indices were obtained for our
J_{ν}
between 355 and 1413 MHz, and between 1413 MHz and 10 GHz, using the DSA simulation results. The best fits are obtained for
BD2b
and
BD3b
models. In
BD2b
model, at
t
_{age}
= 33 Myr (red dotted line in
Figure 5
), these spectral indices are
and
In
BD3b
model, again at
t
_{age}
= 33 Myr, they are
and
Thus we may explain the curvature in the integrated spectrum of the observed radio relics around 1GHz by the electron population radiatively cooling behind a cluster shock, if we assume that the age of the shock (or the electron acceleration duration) is about 30 Myr, which is relatively young compared to the dynamical time scales of typical clusters.
In the case of
BD3b model
, the spectral index behaves very similarly to that of a plane shock case (see
Figure 4
of Paper 1), since the shock speed is more or less constant in time. Departures from the predictions for the testparticle planar shock are the most severe in
BD1
(
MF1
) model, while it becomes relatively milder in
BD2
and
BD3
models with decreasing halo density profiles.
As shown in Paper I, any variations in the spatial distributions of
f_{e}
(
r
) and
B_{d}
(
r
) are averaged in the volume integrated quantities such as
J_{ν}
. So signatures imprinted on the volumeintegrated emission due to different
B_{d}
(
r
) (e.g., between
BD2
and
BD2b
models) would be too subtle to detect.
4. SUMMARY
We have performed timedependent DSA simulations for cosmicray (CR) electrons at decelerating spherical shocks with parameters relevant for weak cluster shocks:
u_{s}
≈ (3.0 − 4.5) × 10
^{3}
km s
^{−1}
and
M_{s}
≈ 3.0−4.3. Several models with different postshock magnetic field profiles (
MF13
) and different upstream gas density profiles (
BD13
) were considered as summarized in
Table 1
. Using the synchrotron emissivity,
j_{ν}
(
r
), calculated from the CR electron energy spectra at these model shocks, the radio surface brightness pro file,
I_{ν}
(
R
), and the volume integrated spectrum,
J_{ν}
, were estimated by assuming a ribbonlike shock structure described in
Figure 1
.
At low frequencies (< 1 GHz) the surface brightness is affected by the evolution of
u_{s}
(
t
) and the spatial variation of
B_{d}
(
r
) as well as projection effects, resulting in a wide range of spatial profiles of
I_{ν}
(
R
) (see
Figures 2
and
4
). At high frequencies (> 1 GHz), for a given geometrical structure of the shock, such dependences become relatively weaker, because the width of the postshock spatial distribution of high energy electrons is much narrower and so the magnetic field profile far downstream has less influence on synchrotron emission.
For low frequency observations, the width of radio relics increases with the shock age as ∆
l
_{adv}
∼
t
_{age}
u_{s}
/
σ
, while it asymptotes to the cooling length, ∆
l
_{cool}
∼
t
_{rad}
(
γ_{e}
)
u_{s}
/
σ
, for high frequencies. If the surface brightness can be spatially resolved at multifrequency observations over ∼ (0.1 − 10) GHz, we may extract significant information about the time evolution of
u_{s}
(
t
), the shock age,
t
_{age}
, and the postshock magnetic field strength,
B_{d}
(
r
), through the detail modeling of DSA and projection effects. For instance, one may infer two possible values of the postshock magnetic field strength from the observed width of radio relics, provided that the geometrical structure (e.g.,
ψ
) of the radio structure is known. The spectral index of
I_{ν}
(
R
), however, behaves rather similarly in all the models considered here.
If the postshock magnetic field strength is about 0.7 or 7
µ
G, at the shock age of ∼ 30 Myr, the volume integrated radio spectrum has a break frequency,
ν
_{br}
∼ 1 GHz, and steepens gradually with the spectral index from
α
_{inj}
to
α
_{inj}
+ 0.5 over the frequency range of 0.1 10 GHz (see
Figure 5
). Thus, we suggest that such a curved spectrum could explain the observed spectrum of the relic in cluster A2256 (
Trasatti et al. 2015
).
Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2012R1A1B3001065 and 2014R1A1A2057940).
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