Efficient cosmic ray acceleration, hydrodynamics, and Self-consistent Thermal X-ray Emission applied to SNR RX J1713.7-3946

Abstract

We model the broad-band emission from SNR RX J1713.7-3946 including, for the
first time, a consistent calculation of thermal X-ray emission together with
non-thermal emission in a nonlinear diffusive shock acceleration (DSA) model.
Our model tracks the evolution of the SNR including the plasma ionization state
between the forward shock and the contact discontinuity. We use a plasma
emissivity code to predict the thermal X-ray emission spectrum assuming the
initially cold electrons are heated either by Coulomb collisions with the shock
heated protons (the slowest possible heating), or come into instant
equilibration with the protons. For either electron heating model, electrons
reach >10^7 K rapidly and the X-ray line emission near 1 keV is more than 10
times as luminous as the underlying thermal bremsstrahlung continuum. Since
recent Suzaku observations show no detectable line emission, this places strong
constraints on the unshocked ambient medium density and on the relativistic
electron to proton ratio. For the uniform circumstellar medium (CSM) models we
consider, the low densities and high relativistic electron to proton ratios
required to match the Suzaku X-ray observations definitively rule out
pion-decay as the emission process producing GeV-TeV photons. We show that
leptonic models, where inverse-Compton scattering against the cosmic background
radiation dominates the GeV-TeV emission, produce better fits to the broad-band
thermal and non-thermal observations in a uniform CSM.

Full text

Page 1

arXiv:1001.1932v2 [astro-ph.HE] 1 Feb 2010
ApJ in press
Preprint typeset using LATEX style emulateapj v. 05/04/06
EFFICIENT COSMIC RAY ACCELERATION, HYDRODYNAMICS, AND SELF-CONSISTENT THERMAL
X-RAY EMISSION APPLIED TO SNR RX J1713.7-3946
Donald C. Ellison,1Daniel J. Patnaude,2Patrick Slane,2and John Raymond2
ApJ in press
ABSTRACT
We model the broad-band emission from SNR RX J1713.7-3946 including, for the first time, a
consistent calculation of thermal X-ray emission together with non-thermal emission in a nonlinear
diffusive shock acceleration (DSA) model. Our model tracks the evolution of the SNR including the
plasma ionization state between the forward shock and the contact discontinuity. We use a plasma
emissivity code to predict the thermal X-ray emission spectrum assuming the initially cold electrons
are heated either by Coulomb collisions with the shock heated protons (the slowest possible heating),
or come into instant equilibration with the protons. For either electron heating model, electrons reach
? 107K rapidly and the X-ray line emission near 1 keV is more than 10 times as luminous as the
underlying thermal bremsstrahlung continuum. Since recent Suzaku observations show no detectable
line emission, this places strong constraints on the unshocked ambient medium density and on the
relativistic electron to proton ratio. For the uniform circumstellar medium (CSM) models we consider,
the low densities and high relativistic electron to proton ratios required to match the Suzaku X-ray
observations definitively rule out pion-decay as the emission process producing GeV-TeV photons. We
show that leptonic models, where inverse-Compton scattering against the cosmic background radiation
dominates the GeV-TeV emission, produce better fits to the broad-band thermal and non-thermal
observations in a uniform CSM.
Subject headings: acceleration of particles, shock waves, ISM: cosmic rays, ISM: supernova remnants,
magnetic fields, turbulence
1. INTRODUCTION
The supernova remnant RX J1713.7-3946 (G347.3-0.5)
has been detected at photon energies ranging from radio
to TeV γ-rays. The GeV-TeV detections in particular
make this SNR an important test-bed for models of
particle acceleration in astrophysical shocks, and a large
number of fits to the data have been presented with an
array of environmental and particle acceleration param-
eters. Invariably, parameters are found that allow good
fits to the non-thermal observations (or some sub-set
of the observations). A critical question for cosmic-ray
(CR) origin concerns the production of the GeV-TeV γ-
rays. Are these γ-rays primarily from inverse-Compton
(IC) emission from relativistic electrons, or pion-decay
emission from the interaction of relativistic hadrons with
the ambient medium? Models with good fits to the TeV
emission with either inverse-Compton or pion-decay
have been presented (e.g., Porter, Moskalenko & Strong
2006;Berezhko & V¨ olk2008;
Morlino, Amato & Blasi 2009; Zirakashvili & Aharonian
2009; Yamazaki, Kohri & Katagiri 2009), and strong
but conflicting claims for or against one or the other
scenario, based on broad-band continuum observations,
have been made (e.g., Katz & Waxman 2008; Plaga
2008; Berezhko & V¨ olk 2009). We find that it is hard
to discriminate on the basis of continuum emission
alone, but that thermal X-ray line emission can easily
differentiate between IC and pion-decay models because
pion-decay requires a high proton number density, np,
Tanaka et al.2008;
1Physics
Box 8202, Raleigh, NC 27695, U.S.A.; don ellison@ncsu.edu,
avladim@ncsu.edu
2Smithsonian Astrophysical Observatory, MS-3, 60 Garden
Street, Cambridge, MA 02138, USA
Department,NorthCarolinaStateUniversity,
and the thermal emission scales as n2
Until now, fits to the broad-band emission that incor-
porate nonlinear diffusive shock acceleration (DSA) have
not accurately accounted for the thermal X-ray emis-
sion that might be present. We do this here for a SNR
evolving in a uniform circumstellar medium (CSM) with
no density enhancements as might occur with a pre-SN
dense shell, nearby molecular cloud, etc. We find that the
lack of observed thermal line emission eliminates pion-
decay as the source of TeV emission in models with uni-
form circumstellar media.
The essential elements of our CR-hydro-NEI model
have been presented in Ellison & Cassam-Chena¨ ı (2005);
Ellison et al. (2007); Patnaude, Ellison & Slane (2009),
and references therein. We couple a one-dimensional hy-
drodynamic simulation of an evolving SNR with nonlin-
ear diffusive shock acceleration. The ionization struc-
ture, free electron number density, and electron tem-
perature in the evolving interaction region between the
forward shock (FS) and contact discontinuity (CD)
are determined with a self-consistent treatment of the
nonequilibrium ionization (NEI). We couple our com-
puted nonequilibrium ionization fractions of heavy ele-
ments to an updated version of the Raymond & Smith
(1977) plasma emissivity code to compute the thermal
X-ray emission.
Simultaneously, the shock accelerated, non-thermal
electron and proton spectra are calculated, evolved, and
used to determine the synchrotron, IC, non-thermal
bremsstrahlung, and pion-decay emission from the SNR.
We therefore obtain, for the first time, consistent thermal
and non-thermal emission in an evolving SNR.
p.
2. MODEL

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Any reasonably complete broad-band model of a SNR
has a host of parameters. SNR RX J1713.7-3946 is no
exception and in this paper we do not present a full pa-
rameter search. Instead, we concentrate on three essen-
tial coupled components: (1) the SNR hydrodynamics,
(2) nonlinear DSA, and (3) non-equilibrium ionization.
Following the majority of work on SNR RX J1713.7-
3946, we assume an age tSNR≃ 1600yr, and a distance
DSNR ≃ 1kpc. Using the observed angular size, DSNR
implies a forward shock radius RFS≃ 8.7pc. While SNR
RX J1713.7-3946 is believed to be a core-collapse SN, we
again follow the majority of work on this remnant and
assume a uniform CSM with constant proton number
density, np, and constant unshocked magnetic field, B0.
We will present models where a pre-SN wind is assumed
in future work. Besides np and B0, the following en-
vironmental parameters are required to model the SNR
evolution: the SN explosion energy, ESN, the ejecta mass,
Mej(we assume an exponential mass distribution for the
ejecta), and the temperature of the unshocked CSM, T0.
We show models with two sets of parameters. In the
“hadronic”model, the parameters are such that the GeV-
TeV emission is dominated by pion-decay, while in the
“leptonic” model, IC produces the GeV-TeV emission. In
both cases, the parameters are chosen to simultaneously
match the HESS TeV observations (Aharonian et al.
2007) and the Suzaku X-ray continuum (Tanaka et al.
2008).The hadronic and leptonic names refer to the
particles, protons or electrons, mainly responsible for
the GeV-TeV emission. As we show below, both models
place the majority of the accelerated particle energy in
protons, not electrons.
We include an amplification factor, Bamp, for the
shocked magnetic field.In our simple ad hoc model
of magnetic field amplification (MFA), the compressed
magnetic field immediately behind the shock is in-
creased by a factor Bamp.
stream field is then evolved in the downstream re-
gion as described in Ellison et al.
more self-consistent models of MFA see, for exam-
ple, Vladimirov, Ellison & Bykov (2006); Caprioli et al.
(2008); Vladimirov, Bykov, & Ellison (2009).
To model the nonthermal radiation, we need additional
parameters for nonlinear DSA.3These are the accelera-
tion efficiency, EDSA (i.e., the instantaneous fraction of
shock ram kinetic energy flux placed in superthermal pro-
tons), the relativistic electron to relativistic proton ratio,
Kep,4the maximum energy the protons obtain Emax
a factor, αcut, characterizing the shape of the turnover
region around Emax
p
. We determine Emax
The amplified down-
(2007).For
p
, and
p
by limiting the
3The model of nonlinear DSA we use here is based on the semi-
analytic model developed by Blasi, Gabici & Vannoni (2005) and
Amato & Blasi (2006). In our implementation, we fix the accel-
eration efficiency rather than the injection fraction as is done by
Blasi and co-workers. While this difference may have important
consequences during the early stages of the SNR evolution when
the FS Mach number is extremely large (see Berezhko & Ellison
1999), it makes no significant difference to the integrated spectra
at the later times we show here.
4Note that Kep sets the post-shock relativistic electron density
given the post-shock relativistic proton density. The post-shock
thermal electron density, which determines the bremsstrahlung
continuum and the X-ray line emission, is set by the densities and
ionization states of the post-shock hydrogen and heavier elements.
The model parameters Kep and np are independent.
acceleration when the acceleration time matches the SNR
age or when the upstream diffusion length matches some
fraction, fsk, of the shock radius, whichever comes first.5
The factor αcutsmoothes the particle spectrum around
Emax
p
mimicking the effects of particle escape (see, for ex-
ample, Zirakashvili & Ptuskin 2008). The above param-
eters are fully defined in Ellison, Decourchelle & Ballet
(2004) and Ellison & Cassam-Chena¨ ı (2005).
ficiency of DSA has been directly measured at the
quasi-parallel Earth bow shock with EDSA
(Ellison, Moebius & Paschmann 1990).
idence,based on particular models,
the efficiency in some young SNRs,
some regions of the FS, can be 50% or more (e.g.,
V¨ olk, Berezhko & Ksenofontov 2003; Warren et al. 2005;
Helder et al. 2009).
For the thermal X-ray emission, we assume cosmic
abundances and compare two extremes for heating the
initially cold electrons.The slowest possible heating
is from Coulomb collisions and the fastest is instant
equilibration between electrons and protons, presum-
ably produced by wave-particle interactions. For shock
speeds above ∼ 1000km s−1, it has been suggested that
electrons are heated very rapidly to kT ∼ 0.3keV by
lower hybrid waves, after which continued heating to
kT ∼ 1keV proceeds through Coulomb collisions (e.g.,
Ghavamian, Laming & Rakowski 2007).
show below, Coulomb collisions alone rapidly heat the
gas to ∼ 0.3keV, any difference between lower hybrid
wave heating and Coulomb heating would only be impor-
tant for UV and optical lines, so pure Coulomb models
are appropriate for the X-ray emission.
It is important to note that, in our CR-hydro-NEI
model for the interaction region between the CD and FS,
including X-ray line emission only requires two additional
assumptions. One is the CSM elemental abundance and
the other is the electron heating model. For Type Ia SNe,
and a wide range of low-to-moderate mass core-collapse
SNe, it is reasonable to assume solar abundances for
the CSM (e.g., Chiosi & Maeder 1986; Kudritzki & Puls
2000). The two heating extremes we consider cover all
likely possibilities.
The ef-
? 0.25
Indirect ev-
suggests that
at least in
Since, as we
3. RESULTS
For our leptonic model we assume np = 0.05cm−3,
B0 = 3µG, EDSA = 0.25, Kep = 2×10−2, Bamp = 1,
fsk = 0.1, and αcut = 1.
np= 0.2cm−3, B0= 2µG, EDSA= 0.5, Kep= 7×10−4,
Bamp = 5, fsk = 0.05, and αcut = 1. In both mod-
els, the values for ESN and Mej are varied with np to
obtain RFS ∼ 8 − 10pc at tSNR = 1600yr. Thus, for
the leptonic model, ESN= 1×1051erg and Mej= 3M⊙,
while the hadronic model uses ESN = 2×1051erg and
Mej= 1.4M⊙.6For a particular np, other combinations
of ESNand Mejgiving RFS≃ 8 − 10pc at 1600yr yield
similar results. In all cases, we assume T0= 104K.7
For the hadronic model,
5The diffusion length in the FS precursor is determined assum-
ing “Bohm diffusion,” where a particle’s mean free path is on the
order of its gyroradius.
6The value Mej= 1.4M⊙is not meant to imply that we believe
SNR RX J1713.7-3946 originated from a Type Ia supernova.
7As long as T0? 106K, the unshocked temperature only weakly
influences our results.

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Fig. 1.— The top four panels show the free electron density,
ne, the temperature, the ionization age, net, and the magnetic
field in a parcel of gas first shocked at 200yr. In panels (A) and
(C), the solid curves are for Coulomb equilibration and the dashed
curves (barely visible) are for instant equilibration. In panel (B),
the dashed curve is the proton temperature and the solid curve
the electron temperature assuming Coulomb equilibration.
dotted red curve in (B) shows the equal electron and proton tem-
peratures assuming instant equilibration. In the bottom panel, the
solid curves show the total emitted flux, at tSNR = 1600yr, per
arbitrary unit mass, in the band 1 − 2 keV from parcels of gas
shocked at previous times. The dashed curves in the bottom panel
show the corresponding flux from the bremsstrahlung continuum.
The parameters are for our hadronic model with np= 0.2cm−3.
The
At the end of the simulation, we obtain for the
leptonic (hadronic) model:
RFS ≃ 9.3 (8.8)pc; the forward shock speed VFS ≃
3000 (2300)km s−1; the magnetic field immediately be-
hind the FS B2≃ 10 (36)µG; the overall FS compression
ratio Rtot ≃ 4.6 (5.6), the subshock compression ratio
Rsub ≃ 3.98(3.86), the fraction of SN explosion energy
placed in CR ions ≃ 0.13 (0.4), and the mass swept up
by the FS ≃ 6(19)M⊙.
In Fig. 1 we illustrate the properties of our CR-hydro-
NEI model by following particular parcels of plasma. In
the top four panels we show, for our hadronic model, the
free electron number density, ne, the electron and proton
temperatures, the ionization parameter or age, net (t is
the forward shock radius
Fig. 2.— The top two panels show fits to the Suzaku RX J1713.7-
3946 observations with our hadronic model for both Coulomb and
instant temperature equilibration but ignoring the X-line emission.
The blue (heavy wt.) solid curve is the synchrotron continuum,
the black solid curve is the thermal emission (only lines above
10−4MeV are included), and the dotted curve is the underlying
bremsstrahlung continuum. The observed emission would be the
sum (not shown in the top two panels) of the solid black and blue
curves. The bottom panel shows the leptonic model (with Coulomb
equilibration) where parameters have been chosen to be consistent
with the Suzaku observations.For the hadronic model, the ra-
diation intensity is multiplied by 0.95 to match the observations.
For the leptonic model, a normalization factor of 0.2 is required to
match the observations. We note that the Suzaku data have been
adjusted for interstellar extinction so no extinction is applied to
the model in this plot.
the time since the parcel was shocked), and the magnetic
field in a parcel of plasma that is overtaken by the FS at
200yr. The red dotted curve in panel (B) gives the tem-
perature assuming instant equilibration. Even though
Te/Tp ? 0.1 throughout the simulation for Coulomb
equilibration, Te approaches 107K (∼ 850eV) rapidly
before leveling out.
In the bottom panel of Figure 1, we plot the thermal
X-ray emission between 1 and 2 keV, for both instant
and Coulomb equilibration, at the end of the simulation
for parcels of plasma shocked at previous times. The
dashed curves are the continuum emission between 1 and
2 keV and the total emission (solid curves), including
lines, stands well above this regardless of the electron
equilibration. As the left end of the bottom panel shows,
at tSNR≃ 1600yr, plasma that was shocked ? 200 years
earlier is sufficiently ionized to produce a substantial flux
in lines regardless of the electron equilibration.

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In the top two panels of Figure 2 we compare
our hadronic model to Suzaku observations of J1713
(Tanaka et al. 2008) for Coulomb (top panel) and in-
stant equilibration (middle panel). The Suzaku observa-
tions have been adjusted for interstellar extinction and
all model parameters are the same as in Fig. 1. For our
hadronic model, we have chosen parameters that result
in pion-decay dominating the GeV-TeV emission, i.e., np
must be above some limit and Kepmust be below some
limit for this to be the case. Figure 2 makes it clear,
however, that the X-ray line emission is much stronger
in the hadronic model than can be accommodated by ob-
servations. This is true for Coulomb equilibration even
though the bremsstrahlung continuum remains well be-
low the Suzaku observations. The only way to lower this
emission relative to the synchrotron continuum would be
to increase Kep or to decrease np to values that would
then no longer reproduce the observed gamma-ray emis-
sion. This is true regardless of the electron equilibration.
We note that lowering np in uniform CSM models re-
quires lowering ESNto maintain RFS∼ 8 − 10pc.
We are unable to find any set of parameters that gives
pion-decay dominating the TeV emission without pro-
ducing emission lines around 1 keV that are inconsistent
with the Suzaku observations.
In the bottom panel of Figure 2 we show our leptonic
model where we have chosen parameters to be consis-
tent with the smooth Suzaku observations. In addition
to the parameters discussed already, we have arbitrar-
ily adjusted the overall normalization of both models to
match the observations. The hadronic model has been
multiplied by 0.95 and the leptonic model by 0.2. Nor-
malization values < 1 might correspond, observationally,
to a partially complete shell morphology for the SNR,
or possibly some reduction in the DSA injection and/or
acceleration efficiency over some fraction of the SNR sur-
face (e.g., Berezhko & V¨ olk 2008).
In Figure 3, we show our best fit hadronic and lep-
tonic models, folded through the Suzaku XIS instrument
response.8For both models, we simulated 20ks obser-
vations of the entire SNR with no background subtrac-
tion, assuming a Galactic column density nH = 7.9 ×
1021cm−2. When compared to the Suzaku observations
(cf., Figs. 10 or 11 in Tanaka et al. 2008), it is clear that
Suzaku would have detected lines as strong as those pro-
duced in our hadronic model had they been present.
In Fig. 4, we show broad-band fits to radio, Suzaku,
preliminary Fermi-LAT, and HESS observations of RX
J1713.7-3946. The hadronic and leptonic models both
produce reasonable fits if the thermal X-ray line emission
is ignored. When the thermal X-rays are considered, the
hadronic model is excluded. Only the cosmic microwave
background is used to determine the IC emission.
It is important to note in considering Figs. 2 and 4 that
equally good fits to the continuum observations can be
obtained with different parameter combinations. This,
and the fact that the various models that have been
applied to RX J1713.7-3946 differ in details, accounts
for the relatively small differences in parameters we ob-
tain compared to those obtained by other modelers (e.g.,
Berezhko & V¨ olk 2008; Morlino, Amato & Blasi 2009).
8
Responsematricesareavailableat
http://heasarc.nasa.gov/docs/suzaku/prop tools/xis mat.html.
Fig. 3.— Simulated Suzaku XIS spectra of RX J1713.3-3946.
In the top panel, the best fit hadronic model is shown, with np =
0.2cm−3, while in the bottom panel, the best fit leptonic model
is shown, with np = 0.05cm−3. In both panels, the blue curve is
the contribution from the thermal X-ray emission, while the red
curve is the contribution from synchrotron emission. The spectra
correspond to a simulated 20 ks observation and are normalized to
match the unabsorbed 1.0 - 10.0 keV flux of 7.65 ×10−10erg cm−2
s−1found by Tanaka et al. (2009). In these simulated observations,
we assume a Galactic nH= 7.9×1021cm−2.
However, consistency with the thermal X-ray line emis-
sion forces the CSM density down and Kep up so no
set of parameters can be found that result in pion-decay
dominating the GeV-TeV emission.
Characteristically of efficient DSA, the CR-hydro-NEI
model produces an overall shock compression, Rtot> 4,
and a subshock compression, Rsub < 4.
less, even with 50% efficiency (EDSA = 0.5), Rsub re-
mains large enough for electrons temperatures to be high
enough for strong line production.
The only factor we see that could lower the thermal
emission substantially in a uniform CSM model, is the
abundance. If the CSM is nearly devoid of heavy ele-
ments, thermal line emission will be suppressed. Deple-
tion onto dust will cut down C, Mg, Si and Fe, but it
will not affect the O lines, which are the brightest in the
model, or N or Ne. Furthermore, a substantial fraction
of the dust is destroyed once net becomes a few times
1010s-cm−3, so some of the refractory elements would
be liberated (e.g., Williams et al. 2006). One does not
expect really severe depletion in the low density uniform
medium, but there could be significant dust in a red giant
wind.
It is also possible that the progenitor was a Wolf-Rayet
Neverthe-

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Fig. 4.— Broad-band fits to radio (Acero et al. 2009), Suzaku
(Tanaka et al. 2008), preliminary Fermi-LAT (Funk et al. 2009),
and HESS observations (Aharonian et al. 2007) of RX J1713.7-
3946. The top panel is our hadronic model and the bottom panel
is our leptonic model. In both cases, the blue curve is synchrotron,
the black is pion-decay, the red is IC, and the dotted is non-thermal
bremsstrahlung. The dashed black curve is the sum including the
X-ray line emission. As in Fig. 2, a normalization factor of 0.95
(0.2) has been applied to the hadronic (leptonic) model.
Fig. 5.— Phase-space distribution functions multiplied by p4for
our hadronic (black solid curves) and leptonic (red dashed curves)
models. These are integrated spectra at the end of the simulation
in cgs units and represent the total material swept-up by the FS.
(WR) star, and this could give anomalous abundances
(e.g., Crowther 2007). Conversion of H to He reduces
the number of electrons, weakening the line emission per
unit mass by as much as a factor of two. However, WC
and WO stars show much larger overabundances of O
and Ne, which produce the strongest lines in the spectra,
so the lines would be strongly enhanced. In WN stars,
carbon and oxygen have been converted to nitrogen. The
O lines would be weakened, and the N VII line at 500
eV would be luminous but badly attenuated. The Ne
IX and X lines at 922 and 1022 eV would then be the
strongest in the spectrum at 0.5 to 1 times the strengths
predicted. Thus, even in the case of a progenitor wind
with anomalous abundances, we would still expect to see
strong line emission in the swept-up CSM, and this would
be present in the Suzaku observations.
4. DISCUSSION AND CONCLUSIONS
While several authors have proposed that emis-
sion lines could be undetectable in J1713 because of
low shock temperatures or time-dependent ionization
(e.g., Drury et al. 2009; Morlino, Amato & Blasi 2009;
Berezhko & V¨ olk 2009), we find that a SNR with prop-
erties typically ascribed to J1713, expanding in a uni-
form CSM with solar elemental abundances, will produce
strong X-ray emission lines when electron equilibration
and non-equilibrium ionization are taken into account.
This places constraints on the CSM density, np, and on
the relativistic electron to proton ratio, Kep, to be con-
sistent with Suzaku observations which show a smooth
X-ray synchrotron continuum with no lines.
While particular values of np and Kep will depend
somewhat on details of various DSA and SNR models, in
any uniform CSM model the CSM must have a relatively
low density and the electron to proton ratio of shock ac-
celerated particles must be relatively high in order to pro-
duce a satisfactory fit to the Suzaku data. Models where
pion-decay produces the observed TeV emission require
densities that are too high and values of Kepthat are too
low to be consistent with the Suzaku observations. We
note that we have actually only computed a lower limit
to the line emission since we have not included line emis-
sion from the ejecta material heated by the reverse shock.
If emission from a RS had been included, our conclusion
that pion-decay is excluded could only be strengthened.
Apart from minor differences, our fit to the broad-
band spectrum (bottom panel Fig. 4) is consistent
with others (e.g., Porter, Moskalenko & Strong 2006;
Morlino, Amato & Blasi 2009) where IC dominated the
TeV emission. Our results differ substantially from the
conjecture made by Drury et al. (2009) that the post-
shock temperature can be reduced below X-ray emit-
ting temperatures in strong shocks. The conclusions of
Drury et al. (2009) are based on scaling arguments in
the limit of extremely high sonic and Alfv´ en Mach num-
bers where the acceleration efficiency approaches 100%.
In this case, the subshock may become weak enough
to limit heating to the values Drury et al. (2009) sug-
gest. However, Mach numbers as high as assumed in
the Drury et al. (2009) scalings are not obtained for rea-
sonable ambient magnetic fields and other parameters
normally assigned to RX J1713.7-3946. When nonlinear
effects are fully taken into account for J1713 parameters
(see also, Morlino, Amato & Blasi 2009), and for accel-