Particle Acceleration At Astrophysical Shocks
And Implications for The Origin of Cosmic Rays
Pasquale Blasi
Arcetri Astrophysical Observatory, Firenze - Italy
INAF/National Institute for Astrophysics
Outline
1. Phenomenological Introduction: Open Problems in Cosmic Ray Astrophysics
1. Reminder of the basics of particle acceleration at shock waves
2. Maximum energy of accelerated particles and need for self-generated
magnetic fields
3. Non linear effects:
3.1 Shock modification due to accelerated cosmic rays: the failure of the
test particle approach
3.2 Self-generated magnetic turbulence
3.3 Both Ingredients together
4. Recent developments in the investigation of the self-generated field
A Phenomenological Introduction
Some Open Problems in
Cosmic Ray Astrophysics
All Particle Spectrum of Cosmic Rays
Knee
2nd Knee
Dip
GZK?
Kascade data
Hoerandel
Astro-ph 0508014
Proton
knee
Helium
knee
Iron
A lot of uncertainties depending upon the simulations used for the
interpretation of the chemical composition
?
The Transition from Galactic to Extra-Galactic Cosmic Rays
Two Schools of Thought:
THE ANKLE
THE TRANSITION TAKES PLACE AT ENERGY AROUND 1019 eV WHERE THE FADING
GALACTIC COMPONENT IS SUBSTITUTED BY AN EXTRA-GALACTIC COMPONENT
WHICH CORRESPONDS TO A FLAT INJECTION SPECTRUM
THE TWISTED ANKLE
Berezinsky, Gazizov & Grigorieva 2003
Aloisio, Berezinsky, Blasi, Gazizov & Grigorieva 2005
THE TRANSITION TAKES PLACE THROUGH A SECOND KNEE AND A DIP WHICH ARE
PERFECTLY INTERPRETED AS PARTICLE PHYSICS FEATURES OF BETHE-HEITLER
PAIR PRODUCTION.
The Ankle and The Twisted Ankle
The Twisted Ankle Model
2nd knee
Dip
Galactic CR’s
(mainly Iron)
Depends on the
IG magnetic field
The Elusive GZK feature...
The GZK feature and its statistical volatility...
De Marco, PB & Olinto (2005)
It is absolutely reasonable
that with current statistics
there are large uncertainties
on the existence of the
GZK feature
The spectrum at the end is
VERY MODEL DEPENDENT.
We might see a suppression
that is due to Emax rather
to the photopion production
Small Scale Anisotropies
PB & De Marco 2003
10-5 Mpc-3
Statistical Volatility of the SSA signal
De Marco, PB & Olinto 2005
THE SSA AND THE
SPECTRUM OF AGASA
DO NOT APPEAR
COMPATIBLE WITH
EACH OTHER (5 sigma)
Common lore on the origin of
the bulk of CRs
Cosmic Rays are accelerated at SNR’s
The acceleration mechanism is Fermi at
the 1st order at the SN shock
If so, we should see the gamma’s from
production and decay of neutral pions
Diffusion in the ISM steepens the
spectrum: Injection Q(E) ->
n(E)~Q(E)/D(E)
Reminder of the basics of
Particle Acceleration at Shocks
First Order Fermi Acceleration: a Primer
x=0
x
Test
Particle
The particle may
either diffuse back
to the shock or
be advected downstream
The particle
is always advected
back to the shock

 
 d μ P μ μd μ < 1
d μ Pu μ 0, μ d μ = 1
d
Return Probability from UP=1
 p
N  E  = N 0  
 p0 
 γ 1
logP
γ=
logG
0,
Return Probability from DOWN<1
3

r 1
P
Total Return Probability from DOWN
G
Fractional Energy Gain per cycle
r
Compression factor at the shock
The Return Probability and Energy Gain for Non-Relativistic Shocks
Up
At zero order the distribution of (relativistic)
particles downstream is isotropic: f(µ)=f0
Down
Return Probability = Escaping Flux/Entering Flux
1
1  u2  ²
Φout =   f0 u2 + μ d μ =
2
1
Φi = f 0 u 2 + μ 0  d μ 0 = 1 + u 2  ²
2


1  u2  ²
Preturn =
 1  4u2
1 + u2  ²
Close to unity
for u2<<1!
Newtonian Limit
The extrapolation of this equation to the relativistic case would give
a return probability tending to zero!
The problem is that in the relativistic case the assumption of isotropy
of the function f loses its validity.
ΔE
4
u1  u2 
G=
=
E
3
SPECTRAL SLOPE
γ=
log Preturn log 1  4u2 
3
=

4
log G
r 1
u1  u2 
3
The Diffusion-Convection Equation: A more formal approach
ADIABATIC COMPRESSION
OR SHOCK COMPRESSION
f
  f 
f 1 du f
= D   u +
p + Q x, p, t 
t x  x 
x 3 dx p
INJECTION
TERM
ADVECTION
WITH THE
FLUID
SPATIAL DIFFUSION
The solution is a power
law in momentum
3u1 N inj 
f 0  p =
2 
u1  u2 4πpinj

 3u1
p  u1  u2
pinj 
The slope depends ONLY
on the compression ratio
(not on the diffusion coef.)
Injection momentum and
efficiency are free param.
A CRUCIAL ISSUE: the maximum energy of accelerated particles
1. The maximum energy is determined in general by the balance between the
Acceleration time and the shortest between the lifetime of the shock and
the loss time of particles
2. For the ISM, the diffusion coefficient derived from propagation is roughly
α
D E  = 3  10 29 EGeV
3
τ acc E  =
u1  u2
α  0.3  0.6
 D1 E  D2 E  
+


u
u
2
 1

For a typical SNR the maximum energy comes out as FRACTIONS OF GeV !!!
Particle Acceleration at Parallel Collisionless Shocks works ONLY if there is
additional magnetic scattering close to the shock surface!
A second CRUCIAL issue: the energetics

 ESNR
Rate SN

Diff
3
1
 1 eV/ cm
V
Summary
About 10% of the
Kinetic energy in the
SN must be converted
To CRs
1. Particle Acceleration at shocks generates a spectrum E- with ~2
2. We have problems achieving the highest energies observed in galactic
Cosmic Rays
3. The Cosmic Rays must carry about 10% of the total energy of the
SNR shell
4. Where are the gamma rays from pp scattering?
5. Why so few TeV sources?
6. Anisotropy at the highest energies (1016-1017 eV)
The Need for a Non-Linear Theory
• The relatively Large Efficiency may break
the Test Particle Approximation…What
happens then? Cosmic Ray Modified
Shock Waves
• Non Linear effects must be invoked to
enhance the acceleration efficiency
(problem with Emax)
Self-Generation of Magnetic
Field and Magnetic
Scattering
Particle Acceleration in the
Non Linear Regime: Shock Modification
Why Did We think About This?
 Divergent Energy Spectrum
ECR = dEEN E   lnEmax / Emin 

At Fixed energy crossing the shock front ρu2
tand at fixed efficiency of acceleration there are
values of Pmax for which the integral exceeds ρu2
(absurd!)
 If the few highest energy particles
that escape from upstream carry enough
energy, the shock becomes dissipative,
therefore more compressive
 If Enough Energy is channelled to CRs
then the adiabatic index changes from
5/3 to 4/3. Again this enhances the
Shock Compressibility and thereby the
Modification
Approaches to Particle Acceleration at
Modified Shocks
 Two-Fluid Models
The background plasma and the CRs are treated as two separate
fluids. These thermodynamical Models do not provide any info on
the Particle Spectrum
 Kinetic Approaches
The exact transport equation for CRs and the coservation eqs for
the plasma are solved. These Models provide everything and contain
the Two-fluid models
 Numerical and Monte Carlo Approaches
Equations are solved with numerical integrators. Particles are shot
at the shock and followed while they diffuse and modify the shock
ρ x  u x  = ρ0 u0
Undisturbed Medium
Shock Front
The Basic Physics of Modified Shocks
v
subshock
Precursor
Conservation of Mass
ρ0 u02 + Pg,0 = ρ x  u x  2+ Pg  x  + PCR  x 
Conservation of Momentum
Equation of Diffusion
f
  f 
f 1 du f
=
D

u
+
p
+ Q  x, p, t  Convection for the


t
x  x 
x 3 dx p
Accelerated Particles
Main Predictions of Particle Acceleration
at Cosmic Ray Modified Shocks
• Formation of a Precursor in the Upstream plasma
• The Total Compression Factor may well exceed 4. The
Compression factor at the subshock is <4
• Energy Conservation implies that the Shock is less
efficient in heating the gas downstream
• The Precursor, together with Diffusion Coefficient
increasing with p-> NON POWER LAW SPECTRA!!!
Softer at low energy and harder at high energy
Spectra at Modified Shocks
Very Flat Spectra at high energy
Amato and PB (2005)
Efficiency of Acceleration (PB, Gabici & Vannoni (2005))
This escapes out
To UPSTREAM
Note that only this
Flux ends up
DOWNSTREAM!!!
Suppression of Gas Heating
p
max = 10
3
mc
Increasing
Pmax
p max = 5  10 10 m c
PB, Gabici & Vannoni (2005)
M 0 = 10, 50, 100
The suppressed heating might have already been
detected (Hughes, Rakowski & Decourchelle (2000))
Summary of Results on Efficiency of Non Linear Shock Acceleration
Pinj/mc

0.1
0.035
3.4 10-4
6.57
0.47
0.02
3.7 10-4
23.18
0.85
0.005
3.5 10-4
100
3.21 39.76
0.91
0.0032
3.4 10-4
300
3.19 91.06
0.96
0.0014
3.4 10-4
500
3.29 129.57
0.97
0.001
3.4 10-4
Mach number M0
Rsub
Rtot
4
3.19
3.57
10
3.413
50
3.27
CR frac
Amato & PB (2005)
Be Aware that the Damping
Of waves to the thermal plasma
Was neglected (on purpose) here!
When this effect is introduced
The efficiencies clearly decrease BUT still remain quite large
Going Non Linear: Part II
Coping with the Self-Generation of
Magnetic field by the Accelerated Particles
The Classical Bell (1978) - Lagage-Cesarsky (1983) Approach
Basic Assumptions:
1. The Spectrum is a power law
2. The pressure contributed by CR's is relatively small
3. All Accelerated particles are protons
The basic physics is in the so-called streaming instability:
of particles that propagates in a plasma is forced to move at speed smaller
or equal to the Alfven speed, due to the excition of Alfven waves in the medium.
Excitation
C
Of the
instability
VA
Pitch angle scattering and Spatial Diffusion
The Alfven waves can be imagined as small perturbations on top of a
background B-field
The equation of motion of a particle in this field is


dp Ze  
=
v  B0 + B1
dt
c

B B0 B1

In the reference frame of the waves, the momentum of the particle remains
unchanged in module but changes in direction due to the perturbation:


2 1/ 2
dμ Zev 1  μ
=
dt
pc
D p  =
B1
cosΩ  kvμ t + ψ 
1
vλ 
3
1 crL  p
3 F  p
Ω = ZeB0 / mcγ
F ( p(k ))  k(B / B)2
The Diffusion coeff reduces
To the Bohm Diffusion for
Strong Turbulence F(p)~1
Maximum Energy a la Lagage-Cesarsky
In the LC approach the lowest diffusion coefficient, namely the highest energy,
can be achieved when F(p)~1 and the diffusion coefficient is Bohm-like.
1
D p = vλ 
3
1 crL  p
3 F  p
τ acc 
D E 
2
ushock
2
= 3.3106 EGeV B μ1 u1000
sec
For a life-time of the source of the order of 1000 yr, we easily get
Emax ~ 104-5 GeV
We recall that the knee in the CR spectrum is at
at ~3
109 GeV.
106 GeV
and the ankle
The problem of accelerating CR's to useful energies remains...
BUT what generates the necessary turbulence anyway?
Wave growth HERE IS THE CRUCIAL PART!
F
F
+u
= σF  ΓF
t
x
Wave damping
Bell 1978
Standard calculation of the Streaming Instability
c 2k 2
ω2
χ
R, L
4π 2 e 2
=
ω
(Achterberg 1983)
= 1 + χ R, L


p 2 1  μ 2 v  p   f 1  kv
 f 
dp d μ
 μ 
 + 
ω ± Ω'  vk  p p  ω
 μ 
 
There is a mode with an imaginary part of the frequency: CR’s excite Alfven
Waves resonantly and the growth rate is found to be:
  vA
PCR
z
Maximum Level of Turbulent Self- Generated Field
F
F
+u
= σF
t
x
Stationarity
PCR
F
u
 vA
z
z
Integrating
B 2
B02
 2M A
PCR
u
2
 1
Breaking of Linear
Theory…
For typical parameters of a SNR one has δB/B~20.
Non Linear DSA with Self-Generated
Alfvenic turbulence (Amato & PB 2006)
 We
Generalized the previous formalism
to include the Precursor!
 We Solved the Equations for a CR
Modified Shock together with the eq.
for the self-generated Waves
 We have for the first time a
Diffusion Coefficient as an output of
the calculation
Spectra of Accelerated Particles and
Slopes as functions of momentum
Amato & PB (2006)
Magnetic and CR Energy as functions
of the Distance from the Shock Front
Amato & PB 2006
Super-Bohm Diffusion
Amato & PB 2006
Spectra
Slopes
Diffusion
Coefficient
Recent Developments in
The Self-Generation of
Magnetic Scattering
NON LINEAR AMPLIFICATION OF THE
UPSTREAM MAGNETIC FIELD Revisited
B
UPSTREAM
B
Some recent investigations suggest that the generation of waves upstream of
the shock may enhance the value of the magnetic field not only up to the
ambient medium field but in fact up to
δB 2  v s / c PCR  v s / c ρu2
Lucek & Bell 2003
Bell 2004
Generation of Magnetic Turbulence near Collisionless Shocks
Assumption: all accelerated particles are protons
In the Reference frame of the UPSTREAM
Upstream
FLUID, the accelerated particles look as an
incoming current:
vs


J CR = nCRev s
The plasma is forced by the high conductivity
to remain quasi-neutral, which produces a
return current such that the total current is



J tot = J CR + J ret
The total current must satisfy the Maxwell Equation

 4π 

 B =
J CR + J ret
c


 
c
J ret =
  B  J CR
4π
Clearly at the zero order Jret = JCR
Next job: write the equation of motion of the fluid, which now feels a force:
 

F = J ret  B



u
ρ = P + J ret  B
t

 

u
c 
ρ = P  B    B  J CR  B
t
4π


In addition we have the Equation for conservation of mass

ρ
=    ρu 
t
and the induction equation:


1 B
 E = 
c t
SUMMARIZING:

 

u
c 
ρ = P  B    B  J CR  B
t
4π


 
B
=   u B
t


ρ
=    ρu 
t



 
B
=   u B
t


Linearization of these eqs in
Fourier Space leads to 7 eqs.
For 8 unknowns.
In order to close the system we
Need a relation between the
Perturbed CR current and the
Other quantities
Perturbations of the VLASOV EQUATION give the missing equations which defines
The conductivity
σ (Complex Number)
f  f  f
 v   p   0
t
x
p

J CR 
J CR  
B
B0
Many pages later one gets the DISPERSION RELATION for the allowed perturbations.
J CR kB0
ω -k vA  
(1   )
ρ 0c
2
2
2
There is a purely growing Non Alfvenic, Non
Resonant, CR driven mode if k vA < ω02
Bell 2004
2
ω0
J CR kB0

(1  σ)
ρ 0c
Re(ω) and Im(ω) as functions of k
Saturation of the Growth
The growth of the mode is expected when the return
current (namely the driving term) vanishes:

 
c
J ret =
  B  J CR
4π
 4 
  B 
J CR
c
Hybrid Simulations used to follow the field amplification
when the linear theory starts to fail
δB 2
2
B0

2
MA
u PCR
c ρu 2
Bell 2004
For typical parameters of a
SNR we get δB/B~300
Some Shortcomings of this Result
1. Bell (2004) uses the approximation of power law spectra, which we have seen is
NOT VALID for modified shocks, as Bell assumes the shock to be.
2. The calculations also assume spatially constant spectra and diffusion properties,
which again are not applicable.
3. The maximum energy is fixed a priori, but for modified shocks this is not
allowed, because that is the place where most pressure is present.
How do we look for NL Effects in DSA?
• Curvature in the radiation spectra (elentrons in
the field of protons)
• Amplification in the magnetic field at the shock
• Heat Suppression downstream
Possible Observational Evidence for Amplified Magnetic Fields
Volk, Berezhko & Ksenofontov (2005)
Rim 1
Lower
Fields
240 µG
360 G
Rim 2
But may be we are seeing the scale of the
Damping (Pohl et al. 2005) ???
CONCLUSIONS
 Crucial steps ahead are being done in the understanding of the
origin of CRs through measurements of spectra and composition of
different components
 We understood the if shock acceleration is the mechanism of
acceleration, NON LINEAR effects have to be taken into account.
These effects are NOT just CORRECTIONS.
 The CR reaction enhances the acceleration efficiency, hardens
spectra at HE, suppressed heating
 The maximum energies observed in galactic CRs can be understood
only through efficient NL self generation of turbulent fields