Particle Acceleration At Astrophysical Shocks

1
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
2
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

3
A Phenomenological Introduction Some Open
Problems in Cosmic Ray Astrophysics
4
All Particle Spectrum of Cosmic Rays
Knee 2nd Knee Dip GZK?
5
Kascade data
Hoerandel Astro-ph 0508014
Proton knee
Helium knee
Iron
6
A lot of uncertainties depending upon the
simulations used for the interpretation of the
chemical composition
?
7
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
Berezinsky, Gazizov Grigorieva 2003 Aloisio,
Berezinsky, Blasi, Gazizov Grigorieva 2005
THE TWISTED ANKLE THE TRANSITION TAKES PLACE
THROUGH A SECOND KNEE AND A DIP WHICH ARE
PERFECTLY INTERPRETED AS PARTICLE PHYSICS
FEATURES OF BETHE-HEITLER PAIR PRODUCTION.
8
The Ankle and The Twisted Ankle
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The Twisted Ankle Model
2nd knee
Dip
Galactic CRs (mainly Iron)
Depends on the IG magnetic field
10
The Elusive GZK feature...
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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
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Small Scale Anisotropies
PB De Marco 2003
10-5 Mpc-3
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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)
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Common lore on the origin of the bulk of CRs

• Cosmic Rays are accelerated at SNRs
• The acceleration mechanism is Fermi at the 1st
  order at the SN shock
• If so, we should see the gammas from production
  and decay of neutral pions
• Diffusion in the ISM steepens the spectrum
  Injection Q(E) -gt n(E)Q(E)/D(E)

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Reminder of the basics of Particle
Acceleration at Shocks
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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 return probability
tending to zero! The problem is that in the
relativistic case the assumption of isotropy of
the function f loses its validity.
SPECTRAL SLOPE
18
The Diffusion-Convection Equation A more formal
approach
ADIABATIC COMPRESSION OR SHOCK COMPRESSION
INJECTION TERM
ADVECTION WITH THE FLUID
SPATIAL DIFFUSION
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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 For a typical SNR the
maximum energy comes out as FRACTIONS OF GeV !!!
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A second CRUCIAL issue the energetics
About 10 of the Kinetic energy in the SN must be
converted To CRs
Summary
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)
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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
22
Particle Acceleration in the Non Linear
Regime Shock Modification
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Why Did We think About This?

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

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• 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

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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)
Very Flat Spectra at high energy
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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 Non Linear
Shock Acceleration
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
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
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Going Non Linear Part II
Coping with the Self-Generation of Magnetic
field by the Accelerated Particles
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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 Of the instability
C VA
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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)
There is a mode with an imaginary part of the
frequency CRs excite Alfven Waves resonantly
and the growth rate is found to be
37
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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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 PB (2006)
40
Magnetic and CR Energy as functions of the
Distance from the Shock Front
Amato PB 2006
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Super-Bohm Diffusion
Amato PB 2006
Spectra Slopes
Diffusion Coefficient
42
Recent Developments in The Self-Generation of
Magnetic Scattering
43
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 used to follow the field
amplification when the linear theory starts to
fail
Bell 2004
For typical parameters of a SNR we get dB/B300
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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.
52
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

53
Possible Observational Evidence for Amplified
Magnetic Fields
Volk, Berezhko Ksenofontov (2005)
Rim 1 Rim 2
Lower Fields
240 µG
360 ?G
But may be we are seeing the scale of the Damping
(Pohl et al. 2005) ???
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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