Acceleration of Very High Energy Cosmic Rays

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Acceleration of Very High Energy Cosmic Rays

• Pasquale Blasi
• INAF/Arcetri Astrophysical Observatory

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First Order Fermi Acceleration a Primer
x0
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
Return Probability from UP1
Return Probability from DOWNlt1
P Total Return Probability from DOWN G
Fractional Energy Gain per cycle r
Compression factor at the shock
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The Return Probability and Energy Gain for
Non-Relativistic Shocks
At zero order the distribution of
(relativistic) particles downstream is isotropic
f(µ)f0
Up Down
Return Probability Escaping Flux/Entering Flux
Close to unity for u2ltlt1!
Newtonian Limit
The extrapolation of this equation to the
relativistic case would give a vanishing return
probability! ANISOTROPY!!!
SPECTRAL SLOPE
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The Need for a Non-Linear Theory

• The relatively Large Efficiency may break the
  Test Particle ApproximationWhat happens then?
• Non Linear effects must be invoked to enhance the
  acceleration efficiency (problem with Emax)

Cosmic Ray Modified Shock
Waves Self-Generation of Magnetic Field and
Magnetic Scattering
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• Going Non Linear Part I

Particle Acceleration in the Non Linear
Regime Shock Modification
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Why Did We Even think About This?

• Divergent Energy Spectrum
  
  
• At Fixed energy crossing the shock front ?u2
  and at fixed efficiency of acceleration there are
  values of Pmax for which the integral exceeds ?u2
  (absurd!)
• Escape at highest energy particles ? 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

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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

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The Basic Physics of Modified Shocks
v
Undisturbed Medium
Shock Front
subshock Precursor
Conservation of Mass
Conservation of Momentum
Equation of Diffusion Convection for
the Accelerated Particles
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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 lt4
• Energy Conservation implies that the Shock is
  less efficient in heating the gas downstream
• The Precursor, together with Diffusion
  Coefficient increasing with p-gt NON POWER LAW
  SPECTRA!!! Softer at low energy and harder at
  high energy

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Spectra at Modified Shocks
Amato and PB (2005)
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Efficiency of Acceleration (PB, Gabici Vannoni
(2005))
This escapes out To UPSTREAM
Note that only this Flux ends up DOWNSTREAM!!!
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Suppression of Gas Heating
Rankine-Hugoniot
Increasing Pmax
The suppressed heating might have already been
detected (Hughes, Rakowski Decourchelle (2000))
PB, Gabici Vannoni (2005)
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Summary of Results on Efficiency of NL-DSA
Mach number M0 Rsub Rtot CR frac
Pinj/mc ?
4 3.19 3.57
0.1 0.035 3.4 10-4
10 3.413 6.57 0.47
0.02 3.7 10-4
50 3.27 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
DAMPING can reduce these Efficienciessee later
Amato PB (2005)
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Going Non Linear Part II
Coping with the Self-Generation of Magnetic
field by the Accelerated Particles
Through Streaming Instability
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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
In the reference frame of the waves, the momentum
of the particle remains unchanged in module but
changes in direction due to the perturbation
The Diffusion coeff reduces To the Bohm Diffusion
for Strong Turbulence F(p)1
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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. 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
106 GeV and the ankle at 3 109 GeV. The problem
of accelerating CR's to useful energies remains...
BUT what generates the necessary turbulence
anyway?
Wave growth HERE IS THE CRUCIAL PART!
Bell 1978
Wave damping
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Standard calculation of the Streaming Instability
(Achterberg 1983)
Dispersion Relation of Excited waves
There is a mode with an imaginary part of the
frequency CRs excite Alfven Waves resonantly
and the growth rate is found to be
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Maximum Level of Turbulent Self- Generated Field
Stationarity
Integrating
Breaking of Linear Theory
For typical parameters of a SNR one has dB/B20.
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CAUTION Very naïve derivation! Many
complications arise in NL-DSA and NOT ALL
of them lead to enhancement of Emax
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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

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Spectra of Accelerated Particles and Slopes as
functions of momentum
Amato and PB 2006
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Magnetic and CR Energy as functions of the
Distance from the Shock Front
Amato PB 2006
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Self-Generated Diffusion
Amato PB 2006
Spectra Slopes
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The role of turbulent heating

• Alfven heating
• Acoustic instability

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NON LINEAR AMPLIFICATION OF THE UPSTREAM
MAGNETIC FIELD Revisited
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
Lucek Bell 2003 Bell
2004
B
UPSTREAM
B
SHOCK
SHOCK
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Generation of Magnetic Turbulence near
Collisionless Shocks
Assumption all accelerated particles are protons
SHOCK
In the Reference frame of the UPSTREAM FLUID, the
accelerated particles look as an incoming
current The plasma is forced by the high
conductivity to remain quasi-neutral, which
produces a return current such that the total
current is
Upstream
vs
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The total current must satisfy the Maxwell
Equation
Clearly at the zero order Jret JCR
Next job write the equation of motion of the
fluid, which now feels a force
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In addition we have the Equation for conservation
of mass and the induction equation
SUMMARIZING
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
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Perturbations of the VLASOV EQUATION give the
missing equations which defines The conductivity
s (Complex Number) Many pages later one gets
the DISPERSION RELATION for the allowed
perturbations.
There is a purely growing Non Alfvenic,
Non Resonant, CR driven mode if k vA lt ?02
Bell 2004
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Re(?) and Im(?) as functions of k
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Saturation of the Growth

• The growth of the mode is expected when the
  return current (namely the driving term) vanishes

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Hybrid Simulations (PICMHD) used to follow the
field Amplification when the linear theory
starts to fail
Bell 2004
For typical parameters of a SNR dB/B300
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Possible Observational Evidence for Amplified
Magnetic Fields
Volk, Berezhko Ksenofontov (2005)
Rim 1 Rim 2
Lower Fields
240 µG
360 ?G

• Add to all this, the numerous
• Predictions of Non-linear Diffusive
• Shock Acceleration!!!
• Curved spectra
• Heating suppression downstream
• B-field amplification

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General lesson to be learned
The level of magnetic turbulence which is
relevant for the acceleration of particles is
usually the self-generated one, which may
correspond to dB/Bgtgt1 and correspondingly larger
maximum energies. THIS IS TRUE FOR ALL
ACCELERATORS and should be kept in mind in
producing Hillas-like plotsTHE RELEVANT B-FIELD
MIGHT HAVE NOTHING TO DO WITH THE
AMBIENT FIELD. In the vicinity of the
accelerator the dynamical reaction of the
accelerated particles may not be negligible!
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How do we look for NL Effects in DSA?

• Curvature in the radiation spectra (electrons in
  the field of protons) (indications of this in
  the IR-radio spectra of SNRs by Reynolds)
• Amplification in the magnetic field at the shock
  (seen in Chandra observations of the rims of SNRs
  shocks)
• Heat Suppression downstream (detection claimed by
  Hughes, Rakowski Decourchelle 2000)
• All these elements are suggestive of very
  efficient CR acceleration in SNRs shocks. BUT
  similar effects may be expected in all
  accelerators with shock fronts

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Some General points
1. At upstream infinity there is no escape with
the exception of pmax(t)
F(E)
F(E)
E
E
pmax
Some care is needed when concluding for instance
that the gamma Rays from the GC might be the
signature of diffusion of particles From the
sources.
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The spectrum from a source of CRs
p4 N(p)
Trapped Flux Escaping Flux at
Upstream Infinity
p
N(p)
p
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Peculiar Aspects of Particle Acceleration at
relativistic shocks
we find that p2 (1/3) ?2 and ?2
1/3
Taubs relations
No equipartition ultrarelativistic
pressureless
For a homogeneous magnetic field downstream NO
PARTICLE is expected to return back to the
shock. The turbulent structure of the B-field
is crucial!
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Reflection at the relativistic mirror works ONLY
at the first interaction. After that
E
4?2 E

• IF THE MAGNETIC FIELD IS ONLY
• INSIDE THE PLASMOIDS,THEY EXPAND
• UNDER THE EFFECT OF MAGNETIC
• PRESSURE?
• 2. THEY FORM COLLISIONLESS BOW
• SHOCKS, WELL KNOWN TO BE FORMED
• FOR INSTANCE AROUND PWNae?
• 3. INJECTION OF SUPRATHERMAL
• PARTICLES THROUGH THE BOW SHOCK

?
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Universal or non-universal
Most calculations of particle acceleration at
relativistic shocks lead to the so-called
UNIVERSAL SPECTRUM This important result is
obtained by solving the same equation
ASSUMPTIONS the scattering is diffusive and in
the so-called Small Pitch Angle Scattering
(SPAS) regime (Dµµ is constant ONLY for the
isotropic case)
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The transport equation in the most general case is
Vietri 2003 PB Vietri 2005
Arbitrary Scattering Function
For instance, in the case of Large Angle
Scattering
Universal
PB Vietri 2005
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More non-universality
Lemoine et al. 2006
Anisotropic Turbulence
Isotropic turbulence
Again, one ends up again in a situation of
quasi-coherent perpendicular Field downstream
PARTICLES DO NOT RETURN ? SPECTRUM STEEPENS
Caveat what if the field upstream does not have
a coherence length? (e.g. Scale invariant
spectrum 1/k)
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Conclusions

• Particle acceleration at astrophysical shocks
  still has major open aspects (Non linear theory)
• A determination of the max momentum of
  accelerated particles is still missing in the NL
  approach
• The issue of WHICH sources should be faced after
  these (crucial) aspects are clarified (e.g. naïve
  arguments rule out SNR, detail ones do not)