Download Kinetic Simulations of Particle Acceleration at Astrophysical Shocks Damiano Caprioli

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Kinetic Simulations of
Particle Acceleration at
Astrophysical Shocks
Damiano Caprioli
Princeton University
in collaboration with A. Spitkovsky
1
Collisionless shocks
Mediated by collective electromagnetic interactions
Sources of non-thermal particles and emission
Reproducible in laboratory
2
A universal acceleration mechanism
Fermi mechanism (Fermi, 1954): random scattering leads to energy gain
In shocks, particles gain energy at any reflection (Blandford & Ostriker; Bell; Axford
et al.; 1978): Diffusive Shock Acceleration (DSA)
!
!
!
! Downstream (post-shock)
Upstream (pre-shock)
∝
2
Test-particle
squeezed
between
converging
flows
-𝛼
DSA produces power-laws N(p) 4𝜋p p in momentum, depending on the
compression ratio R=ρd/ρu only. For strong shocks: 𝛼=4
4Ms2
R= 2
Ms + 3
3R
↵=
R 1
3
Acceleration from first principles
Full Particle-In-Cell approach
(…, Spitkovsky 2008, Niemiec+2008, Stroman+2009, Riquelme
& Spitkovsky 2010, Sironi & Spitkovsky 2011, Park+2012,2015,
Niemiec+2012, Guo+2014, DC+15…)
Define electromagnetic field on a grid
Move particles via Lorentz force
Evolve fields via Maxwell equations
Computationally very challenging!
Hybrid approach:
Fluid electrons - Kinetic protons
(Winske & Omidi; Lipatov 2002; Giacalone et al.; Gargaté &
Spitkovsky 2012, DC & Spitkovsky 2013-2015,…)
massless electrons for more macroscopical
time/length scales
4
Hybrid simulations of collisionless shocks
Reflecting wall
DENSITY + PARTICLES
Upstream Flow
Shock propagation
dHybrid code (Gargaté et al, 2007)
Out of plane B FIELD
Initial B field
5
Spectrum evolution
Diffusive Shock
Acceleration:
f(p) p-4
∝
4𝜋p2f(p)dp=f(E)dE
!
Downstream Spectrum
Maxwellian
85% Energy
∝
f(E)∝E
f(E) E-2 (relativ.)
-1.5 (non
rel.)
Non-thermal Tail
15% Energy
DC & Spitkovsky, 2014a
6
CR-driven instability
DC & Spitkovsky, 2013
7
3D simulations of a parallel shock
DC &
Spitkovsky,
2014a
8
−1.5
0
Parallel vs Oblique shocks
1000
2000
3000
4000
5000
x [c/ p]
6000
7000
8000
9000
MA=10
1
10
Thermal
M=10
0
10
ϑ=0
ϑ = 20
ϑ = 30
ϑ = 45
ϑ = 50
ϑ = 60
ϑ = 80
Non-Thermal
−1
E f(E)
10
.
c
c
A
rift
−2
10
B0
D
k
c
ho
−3
10
S
−4
10
−3
−2
10
15
10000
−1
10
0
10
10
E/Esh
e
i
1v
s
ffu10
Di
.
c
c
A
2
3
10
10
Each point is
a simulation
with about
9
10 particles
DC & Spitkovsky, 2014
Efficiency (%)
M= 5
M=10
10
𝜗
Vsh
M=30
M=50
5
DC & Spitkovsky, 2014
0
0
10
20
30
40
θ (deg)
50
60
70
80
9
n/n0 (t = 200ωc−1)
6
2000
y[c/ωp ]
1500
4
1000
2
500
0
2000
4000
6000
8000
10000
12000
14000
0
16000
x[c/ωp ]
Btot /B0 (t = 200ωc−1 )
30
2000
25
y[c/ωp ]
1500
20
1000
15
10
500
5
0
2000
4000
6000
8000
10000
12000
14000
16000
1
x[c/ωp ]
y[c/ωp ]
Bx
(t = 200ωc−1)
2000
10
1500
5
1000
0
500
−5
0
2000
4000
6000
8000
10000
12000
14000
16000
x[c/ωp ]
y[c/ωp ]
By
(t = 200ωc−1)
2000
10
1500
5
1000
0
500
−5
0
2000
4000
6000
8000
10000
12000
14000
16000
−10
x[c/ωp ]
y[c/ωp ]
Bz
(t = 200ωc−1)
2000
10
1500
5
1000
0
500
−5
0
2000
4000
6000
8000
10000
x[c/ωp ]
12000
14000
16000
Hybrid
simulations of a
very strong
shock: M=100
−10
!
Total 𝜹B/B
larger than 10 in
the precursor!
!
Very expensive
to study in the
hybrid limit
10
8
Caprioli & Spitkovsky
Dependence on inclination and MA
20
M = 100
M = 80
15
5
10
15
20
25
30
Btot /B0 (t = 200ωc−1 )
M = 30
Btot /B0
1
10
y[c/ωp ]
⃗0
B
5
100
1000
1500
2000
2500
3000
3500
4000
4500
5000
0
0
x[c/ωp ]
M = 20
Angle dependence due to ion
injection (DC, Pop, Spitkovsky 2015)
M = 10
500
300
100
200
300
400
500
x − xsh [c/ωp ]
y[c/ωp ]
500
300
1500
2000
2500
3000
3500
4000
4500
50
5000
300
100
1500
2000
2500
3000
3500
4000
4500
5000
y[c/ωp ]
500
⌧
30
20
10
300
0
10
100
1500
2000
2500
3000
3500
4000
4500
x[c/ωp ]
y[c/ωp ]
500
300
100
1500
2000
2500
3000
3500
4000
4500
3500
4000
4500
3500
4000
4500
x[c/ωp ]
y[c/ωp ]
500
300
100
1000
1500
2000
2500
3000
x[c/ωp ]
y[c/ωp ]
500
300
100
1000
900
1000
Simulations
x[c/ωp ]
1000
800
40
⟨Btot /B0 ⟩2
y[c/ωp ]
500
1000
700
⟨Btot /B0 ⟩2 ∝ MA
x[c/ωp ]
1000
600
60
100
1000
M = 50
1500
2000
2500
3000
x[c/ωp ]
20
30
40
50
60
Btot
B0
70
2
up
⇡ 3⇠cr MA
80
90
100
MA
5000
Figure 6. Top panel : Magnetic field profile immediately upstream of the shock, for different Mach numbers as in the legend, at t = 100ω
The profile is calculated by averaging over 200c/ωp in the transverse size and over 20ωc−1 in time, in order to smoothen the time and sp
fluctuations due to the Bottom panel : Total magnetic field amplification factor in the precursor, averaged over a distance ∆x = 10M c
ahead of the shock, as a function of the Alfvénic Mach number (red symbols). The dashed line ⟨Btot /B0 ⟩2 ∝ MA is consistent with
prediction of resonant streaming instability (see text for details). A color figure is available in the online journal.
More B amplification for
stronger (higher MA) shocks!
where Pw and Pcr are the pressure (along x) in magnetic
field and in CRs, and M̃A = (1 + 1/r)MA is the Alfvénic
5000
Mach number in the shock reference frame (r ≈ 4 for
a strong shock, thereby typically M̃A ≃ 1.25MA); We
have also introduced the transverse
(self-generated) com!
2
ponent of the field, B⊥ (x) = By (x) + Bz2 (x).
Assuming isotropy in the self-generated magnetic field,
2
Btot
2
2
one has B⊥
= 23 Btot
, and in turn Pw ≈ 12π
. Dividing
5000
2
both members of eq. 1 by ρũ , where ũ is the fluid velocity int the shock frame, and introducing the normalized
(xsh )
CR pressure at the shock position ξcr = Pcrρũ
, one
2
finally gets
"
#2
Btot
≈ 3ξcr M̃A .
(2)
B0 sh
5000
The actual value of ξcr can be derived by measuring the
Mach numbers considered here, it varies between 10 a
15% at t = 200ωc−1 (also see figure 3 in Paper I). Qu
remarkably, if we pose ξcr = 0.15, eq. 2 provides a v
good fitting to the amplification factors inferred from
simulations (dashed line in figure 6).
The extrapolation of the presented results to hig
Mach numbers according to eq. 2 is consistent with
hypothesis that CR-induced instabilities can account
the effective magnetic field amplification inferred at
blast waves of young SNRs, even with moderate CR
celeration efficiencies of about 10–20%.
It would be tempting to conclude that resonant strea
ing instability is the almost effective channel throu
which the CR current amplify the pre-existing magn
field, but there are some caveats. The non-reson
streaming instability (Bell 2004, 2005) is predicted to
the fastest to grow, and it might saturate on time-sca
11 re
shorter than the advection time in the precursor:
nant (and also long-wavelength modes, see Bykov et
In agreement with the prediction
of CR-driven streaming instability
3D simulations
Simulations of ion acceleration at shocks: DSA efficiency
16
Caprioli & Spitkovsky
17
Self-generated B
Bz /B0
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
Log10[Ef(E)](t = 200ωc−1)
2
Log10(E/Esh )
1.5
1
0.5
0
−0.5
−1
400
𝜗=0o
500
600
700
x[c/ωp ]
800
900
1000
500
600
700
x[c/ωp ]
800
900
1000
500
600
700
x[c/ωp ]
800
900
1000
ϑ = 0 deg
2
Log10(E/Esh )
1.5
Bz /B0
1
0.5
0
−0.5
−1
400
ϑ = 45 deg
𝜗=45o
Log10(E/Esh )
1
0.5
0
−0.5
−1
Bz /B0
400
1
10
ϑ = 80 deg
ϑ = 45 deg
ϑ = 0 deg
0
10
−1
Ef(E)
10
−2
10
−3
10
𝜗=80o
Post-shock ion spectrum
−4
10
ϑ = 80 deg
−2
10
−1
10
0
10
E/Esh
1
10
12
2
10
– 27 –
SN 1006: a parallel
accelerator
– 24 –
X-ray emission
(red=thermal
white=synchrotron)
Magnetic field
amplification and
particle acceleration
where the shock is
parallel
Inclination of B
wrt to the
shock normal
Reynoso et al. 2013
(a) Magnetic vectors
Polarization
(low=turbulent
high=ordered)
13
y[
High-beta plasmas
4
2
The Alfvènic Mach # controls magnetic field amplification
The (magneto-)sonic Mach #
−3
−2.5
−2
−1.5
−1
0
1000
controls
−0.5
1500 dynamics
2000 and 2500
3000
shock
CR
spectrum
x[c/ωp ]
0
8
M s = 5; M A = 5
M s = 5; M A = 30
M s = 30; MA = 30
Log10[Ef(E)](t = 130ωc−1)
Magnetic fields
1
are amplified also
0
in high-β plasmas!
−1
6
1000
2000
3000
x[c/ωp ]
Btot /B0
Log10(E/Esh )
2
4
2
4000
5000
0
6000
1000
1
10
1500
2000
2500
x[c/ωp ]
3000
3500
β=2
β = 60
0
10
CR spectra agree with
DSA prediction
−1
10
Ef(E)
1
−2
10
−3
10
(steeper than p-4 for r<4)
−4
10
−2
10
−1
10
0
1
10
10
E/Esh
2
10
3
10
14
Electron/ion acceleration
Full PIC simulations: Tristan-MP code (Park, DC, Spitkovsky 2015, PRL)
M=20, vsh=0.1c, quasi-parallel shock
Electrons are accelerated: electron/proton ratio is a few %
Density
Ions
Electrons
Momentum spectrum
Energy spectrum
Self-generated B field
15
Electron acceleration
o
PIC simulations of oblique shocks (𝜗=60 ): DC, Park, Spitkovsky, in prep.
Only electrons are accelerated!
Density
Density
Ions
Self-generated B field
DSA spectra
Electrons
Btot/B0
~2-10% in energy
~1% in number
Stay tuned for scalings
with M, β, Vsh,…
16
Electron acceleration
Planetary bow shocks
(Earth, Venus, Saturn,…)
In situ measurements:
Geotail, Venus Orbiter, Polar,
SoHO, WIND, Cluster, …
Radio relics in galaxy clusters
(sausage, toothbrush,…)
CIZA J2242.8+5301
Extended polarized structures
Fermi-LAT limits on 𝛾-ray
emission: constrain e/p ratio!
17
CRAFT: a Cosmic-Ray Fast Analytic Tool
(Caprioli et al. 2009-2015, to be publicly released soon)
Iterative solution of the CR transport equation:
!
Injection
!
!
u
PB + Pcr
!
!
CR distribution
function
!
Mass+momentum
conservation eqs.
Pcr
Magnetic turbulence transport eq.
Very fast: a few seconds on a laptop (vs days on clusters)
Embeds microphysics from kinetic simulations into (M)HD
13.0
Tycho: a clear-cutSuzaku
hadronic accelerator
Log!Ν FΝ " #Jy Hz$
12.5
G. Morlino and D. Caprioli: Strong evidences of hadron acceleration in Tycho’s Supernova Remnant
G. Morlino and D. Caprioli: Strong evidences of hadron acceleration in Tycho’s Supernova Remnant
12.0
13.5
FermiLAT
d D. Caprioli: Strong evidences of hadron acceleration in Tycho’s Supernova Remnant
lerated electrons is calculated
n only the cosmic microwave
t also the Galactic background
hotons produced by the local
11.5
13.0
1
12.511.0
Suzaku
Radio
12.010.5
some observational features,
e strongly affected by the past
ny reliable calculation has to
volution. In this paper we use
heory, but we couple this theution of the remnant provided
We divide the SNR evolution
me that for each time step the
like has been done in Caprioli,
r, as showed by Caprioli et al.
e-dependent approaches return
elativistic shocks.
r kinetic model with the multiof Tycho from the radio to the
dial profile of X-ray and radio
existing data of Tycho’s SNR
efficient acceleration of CR nuat a fraction around 12 per cent
en converted in CRs. Such an
ed magnetic field of ∼ 300µG,
asured X-ray rim thickness. In
field enhances the velocity of
cing the effective compression
s, whose spectrum turns out to
important consequence of this
s to fit the observed gamma-ray
V band, as due to neutral pion
ork it is not possible to explain
without violating many other
FermiLAT
11.5
Fig. 1. Radio image of the Tycho’s remnant at 1.5 GHz in linear
color scale. Image credit: NRAO/VLA Archive Survey, (c) 2005-2007
AUI/NRAO.
10.0
11.0
VERITAS
3
Vsh !"10 km!s#
20
10.5 9.5
10
B2 !"100 ΜG#
Radio
5
Ξcr #100
10
15
2
10.0
Rsh !pc
1
10
20
9.5
50
100
200
VERITAS
E2 F#E$ !eV%cm2 %s"
Log!Ν FΝ " #Jy Hz$
al reaction of the field reduces
a and affects the prediction for
Caprioli et al., 2009). A cruhe scattering centers, which is
to the shock speed, but could
the magnetic field is ampliov, 2006; Caprioli et al., 2009;
hen this occurs, the total comd particles may be appreciably
of accelerated particles may be
Log!Ν" #Hz$
20
Total
FermiLAT
VERITAS
Π0 " ΓΓ
0.1
ICIR
0.01
25
ICCMB
brem
ICGal
Age $yr%
Fig. 2. Time evolution of shock radius R , shock velocity V , magnetic
Fig. 6. Spatiallyfield
integrated
distribution
curves show synchrotron
and pion
immediately behind thespectral
shock B and CR acceleration
efficiency
10 energy
15 of Tycho. The20
25 emission, thermal electron bremsstrahlung
108
109
1010
1011
1012
1013
ξ = P /ρ V .
decay as calculated within our model (see text for details). The experimental data are, respectively: radio from Reynolds & Ellison (1992); X-rays
lows: in §2 we summarize the
from Suzaku (courtesy of Toru Tamagawa), GeV gamma-rays
from Fermi-LAT (Giordano et al. 2012) and TeV gamma-rays from VERITAS
ar particle acceleration and our a set of parameters has been showed to be suitable for describLog!Ν" #Hz$
E !eV"
In §3 we outline the macro- ing the evolution of the FS position and velocity for a type Ia
etparamal. 2011).
Both given
Fermi-LAT
VERITAS data include only statistical error at 1σ.
,(Acciari
in order to fix the free
SNR: the parametrization
in table 7 of Trueloveand
& Mc Kee
widely discuss the comparison (1999) in fact differs from the exact numerical solution of about
Gamma-ray
emission
observed by
Fermi-LAT
by VERITAS in
compared
withsupernova
spectral energy
distribution produced by pion decay (dotFig.
Spatiallyspecintegrated
of Tycho. The curves show synchrotron emission, thermal electron bremsstrahlung
and pion
or the 6.
multi-wavelength
G. Morlino
andFig.
D.11.
Caprioli:
Strong
evidence
for
hadron and
acceleration
Tycho’s
remnant
3 per cent spectral
typically, and energy
of 7 per centdistribution
at most. Such a solution,
fferent energy band separately.
sh
sh
2
cr
cr
0
2
sh
av
Brightness #Jy%arcmin2 $
of the shock evolution during the ejecta-dominated stage.
The circumstellar medium is taken as homogeneous with
proton number density n0 = 0.3 cm−3 and temperature T 0 =
104 K. Following the conclusion of Tian & Leahy (2011), we
assume that the remnant expands into the uniform interstellar
medium (ISM) without interacting with any MC. With these parameters, the reference value for the beginning of the Sedov-
Radio profile " 1.5 GHz
13.0
spherical symmetry, which is somehow expected just because
he northeastern region is brighter than the rest of the remnant.
712.8
612.6
Account for spectra,
SNR hydrodynamics,
and morphology
while for Fermi-LAT the systematic uncertainties are comparable or even larger than the statistical error especially for the lowest energy bins due
to difficulties in evaluating the galactic background (see
Fig. 3 in Giordano et al., 2011, and the related discussion).
13.0
X!ray profile " 1 keV
Log$Ν FΝ % "Jy Hz#
1.5
by following the analytic preMc Kee (1999). More precisely,
rgy ES N = 1051 erg and one
structure function is taken as
uelove & Mc Kee, 1999). Such
2
Brightness
!Hz!sr#
Log!Ν "erg!s!cm
FΝ " #Jy Hz$
which doesour
not include
explicitly
possible
of the CR The experimental data are, respectivley: radio from Reynolds & Ellison (1992); X-raysdashed line), relativistic bremsstrahlung (dot-dot-dashed) and ICS computed for three different photon fields: CMB (dashed), Galactic background
decay as calculated within
model
(seethe
text
forrole
details).
pressure in the SNR evolution, is still expect to hold for moderately
acceleration
efficiencies, (below
10 per
rom Suzaku (courtesy
ofsmall
Toru
Tamagawa)
GeVabout
gamma-rays
from Fermi-LAT (Giordano et al., 2011)
and TeV gamma-rays from VERITAS(dotted) and IR photons produced by local warm dust (solid). The thick solid line is the sum of all the contributions. Both Fermi-LAT and
cent). We checked a posteriori that the efficiency needed to
813.0
VERITAS data points include only statistical errors at13.5
1σ. For VERITAS data the systematic error is found to be ∼ 30% (Acciari et al., 2011),
Acciari et al., 2011).fit Both
Fermi-LAT
and
VERITAS
data include only statistical error at 1 σ.
observations
does not require
a more
complex treatment
12.5
Log!Ν FΝ " #Jy Hz$
background, we are left with ICS on the IR background due to idence of the fact that SNRs do accelerate protons, at least up to
3
12.8
local dust as the only viable candidate. However,
as predicted energies of about 500 TeV. The proton acceleration efficiency is
12.0
1.0
5
2
, corresponding to converting in CRs
by standard ICS theory and as showed in Fig. 11, the expected found to be ∼ 0.06ρ0 V sh
12.4
12.6
photon spectrum below the cut-off is typically flatter than par- a fraction of about 12 per cent of the kinetic energy density
3
ent electrons’ one, and more precisely is ∝ ν−1.6 11.5
for an electron 12 ρ0 V sh
. As estimated for instance in §3 of the review by Hillas
4
Another subtle but interesting difference is that the emis−2.2
12.4
spectrum
∝
E
,
clearly
at
odds
with
Fermi-LAT
data
in
the
(2005),
such a value is consistent with the hypothesis that SNRs
12.2
sion peaks slightly more inwards than in our model; as a conGeV
range.
are
the
sources
of Galactic CRs, provided that the residence time
<
<
11.0
sequence,
0.5 also the emission detected in the region 0.6 ∼ r/R sh ∼
3
12.2
Another point worth noticing is that the ICS on the CMB in the Milky Way scales with ∼ E −1/3 .
0.8 is found to be a bit larger than the theoretical prediction.
12.0
radiation is sensitive to the steepening of the total10.5
electron specThis difference might have different explanations. The most ob12.0
2
trum
above
∼100
GeV
(Fig.
4)
due
to
the
synchrotron
losses
vious, and already mentioned, is the possible deviation from the
particles undergo while being advected downstream, while for
11.8
spherical symmetry. Another possibility is given by placing the
11.8
the ICS on the IR+optical background the onset10.0
of the Klein1
CD in 0.0
a different position: if one assumed the CD to be located
8
10
12
14
16
18
20
Nishina
regime
(above
Ee ≈ 7 TeV
for photons of 1 eV) does
0.0
0.2
0.4
0.6
0.8
1.0
1.2
It is important to remember that the actual CRs produced by
0.94
0.95
0.96
0.97
0.98
0.99
1.00
18.0
18.2
18.4
18.618.2
18.8 18.4
18.0
18.6
18.8
closer to the center (i.e. if one took the CD/FS ratio to be a few
not allow us to probe significantly the steep region of the elec- a single SNR is given
by the
convolution over time of different
Log$Ν%
"Hz#
per cent smaller), the theoretical prediction would
Log!Ν" #Hz$
R%Rsh nicely fit the
R!R
tron
sh spectrum.
contributions with non trivial spectra, and namely the flux of
Log!Ν"
#Hz$
data. However, we can not forget that this explanation would be
In other words, ICS on the CMB radiation
low and particle
escaping
the remnant
upstream
during the
SedovFig. is
10.tooSynchrotron
emission
calculated
byfrom
assuming
constant
downFig. 8. X-ray emission due to synchrotron (dashed line) and to synatFig.
odds7.with
the findings
of Warren
et al.radio
(2005),
who estimated
cannot
be
boosted
by invoking
apoints
larger electron density, while Taylor stages and the bulk of particles released in the ISM at the
Fig.
9.
Projected
X-ray
emission
at
1
keV.
The
Chandra
data
Surface
brightness
of the
emission
at 1.5 GHz
as
a
funcchrotron
plus
thermal
bremsstrahlung
(solid
line).
Data
from
the
Suzaku
Fig.
8.
X-ray
emission
due
to
synchrotron
(dashed
line)
and
to
synstream
magnetic
field
equal
to
100
(dotted
line),
200
(dashed
line),
and
he position of the CD to be more towards the forward shock,
on
IR and/or
optical
background,
which might as well be SNR’s death (Caprioli, Blasi & Amato, 2009; Caprioli, Amato
are
from (Cassam-Chenaï
et al. 2007, ICS
see
their
Fig. 15).
The
solid
line
tion
of
the
radius
(data
as
in
Fig.
1).
The
thin
solid
line
represents
the
telescope
(courtesy
of Toru
Tamagawa).
chrotron
plus
thermal
bremsstrahlung
(solid
line).
Data
from
the
Suzaku
300 µG
(solid
line).&The
normalization
the electron
spectrumspectrum
is takenof
namely around 0.93R sh.
enhanced with respect to the mean Galactic
value,
cannot
Blasi,
2010a). In thisofrespect,
the instantaneous
the projected radial profile of locally
synchrotron
emission convolved
projected radial profile computed from our model using Eq. (16), while shows
−3
provide a spectral slope in agreement with both
and
to beFermi-LAT
Kep = 1.6
× 10accelerated
for all particles
the curves.
telescope (courtesy of Toru Tamagawa).
in Tycho, which is inferred to be as steep
(assumed
to be 0.5 arcsec).
the thick solid line corresponds to the same profile convoluted with a with the Chandra point spread functionVERITAS
data.
as ∝ E −2.2 , provides a hint of the fact that SNRs can indeed
We are therefore forced to conclude that the present multi- produce rather steep CR spectra as required to account for the
Gaussian with a PSF of 15 arcsec.
wavelength analysis of Tycho’s emission represents the best ev- ∝ E −2.7 diffuse spectrum of Galactic CRs (Caprioli, 2011b).
A final comment on the radio profile concerns the effects of 4.2. X-ray emission
Hadron acc. eff. ~10%
Protons up to 0.5 PeV
Only two free parameters: injection efficiency and electron/proton ratio
he non-linear Landau damping in the determination of the mag-
19
nonthermal
et field
al. 2005).
A detailed
the9lineradio data the magnetic field at the shock has to be !200 µG,
electrons
in a(Warren
magnetic
as large
as ∼300 model
µG. InofFig.
Conclusions
15%
Acceleration at shocks can be
efficient: >15%
85%
Btot
(t = 500ωc−1 )
1000
y[c/ωp ]
CRs amplify the B field via
500
streaming instability
Ion DSA efficient at parallel,
strong shocks
Electron DSA efficient at
oblique shocks
15
2000
Efficiency (%)
0
1500
2500
3000
3500
4000
x[c/ωp ]
4500
Ions
10
5000
5500
M= 5
M=10
M=30
M=50
5
0
0
10
20
30
40
θ (deg)
50
60
70
80
𝜗=60o
20