Particle Acceleration At Astrophysical ShocksParticle Acceleration At Astrophysical Shocks

And Implications for The Origin of Cosmic RaysAnd Implications for The Origin of Cosmic Rays

Pasquale BlasiArcetri 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?

Proton

knee

Helium

kneeIron

Kascade data Hoerandel Astro-ph 0508014

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

THE TRANSITION TAKES PLACE THROUGH A SECOND KNEE AND A DIP WHICH ARE

PERFECTLY INTERPRETED AS PARTICLE PHYSICS FEATURES OF BETHE-HEITLER

PAIR PRODUCTION.

Berezinsky, Gazizov & Grigorieva 2003

Aloisio, Berezinsky, Blasi, Gazizov & Grigorieva 2005

The Ankle and The Twisted The Ankle and The Twisted AnkleAnkle

The Twisted Ankle The Twisted Ankle ModelModel

2nd knee

Dip

Depends on theIG magnetic field

Galactic CR’s (mainly Iron)

The Elusive GZK feature...The Elusive GZK feature...

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

The GZK feature and its statistical volatility... The GZK feature and its statistical volatility...

Small Scale Anisotropies Small Scale Anisotropies PB & De Marco 2003

10-5 Mpc-3

Statistical Volatility of the SSA signalStatistical 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 Common lore on the origin of of the bulk of CRsthe bulk of CRsCosmic Rays are accelerated at SNR’sCosmic Rays are accelerated at SNR’s

The acceleration mechanism is Fermi at The acceleration mechanism is Fermi at the 1the 1stst order at the SN shock order at the SN shock

If so, we should see the gamma’s from If so, we should see the gamma’s from production and decay of neutral pionsproduction and decay of neutral pions

Diffusion in the ISM steepens the Diffusion in the ISM steepens the spectrum: Injection Q(E) -> spectrum: Injection Q(E) -> n(E)~Q(E)/D(E)n(E)~Q(E)/D(E)

Reminder of the basics of Particle Acceleration at Shocks

xx=0

TestParticle

The particleis always advected back to the shock

The particle may either diffuse backto the shock or be advected downstream

First Order Fermi Acceleration: a PrimerFirst Order Fermi Acceleration: a Primer

1 = μ d μ μP μ d 0,uReturn Probability from UP=1

1 < μ d μ μ P μ d , 0dReturn Probability from DOWN<1

1

3

log

log1

00

rG

P=γ

p

pN=EN

γP 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 Down At zero order the distribution of (relativistic)particles downstream is isotropic: f(µ)=f0

Return Probability = Escaping Flux/Entering Flux

² u1 2

1 = μ d μ+u f =Φ 220out

² u+1 2

1 = μ d μ+u f =Φ 20020i

2

2

2 41² 1

² 1u

u+

u=Preturn

Newtonian Limit

Close to unity

for u2<<1!

The extrapolation of this equation to the relativistic case would givea return probability tending to zero!The problem is that in the relativistic case the assumption of isotropyof the function f loses its validity.

21 3

4 uu=

E

ΔE=G

SPECTRAL SLOPE

1

3

34

41 log

log

log

21

2

ruu

u=

G

P=γ return

The Diffusion-Convection Equation: A more formal approachThe Diffusion-Convection Equation: A more formal approach

21

1

221

10

3

4

3 uuu

p

p

πp

N

uu

u=pf

injinj

inj

The solution is a powerlaw in momentum

The slope depends ONLYon the compression ratio(not on the diffusion coef.)

Injection momentum andefficiency are free param.

tp,x,Q+p

fp

dx

du+

x

fu

x

fD

x=

t

f

3

1

SPATIAL DIFFUSION

ADVECTION WITH THEFLUID

ADIABATIC COMPRESSIONOR SHOCK COMPRESSION

INJECTION TERM

A CRUCIAL ISSUE: the maximum energymaximum 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 !!!For a typical SNR the maximum energy comes out as FRACTIONS OF GeV !!!

0.60.310 3 29 αE=ED αGeV

2

2

1

1

21

3

u

ED+

u

ED

uu=Eτacc

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

cmeV/ 1 V

1

SN Rate

E

3

Diff

SNR

About 10% of theKinetic energy in theSN must be convertedTo 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)

The Need for a Non-Linear The Need for a Non-Linear TheoryTheory

• The relatively Large Efficiency may The relatively Large Efficiency may break the Test Particle break the Test Particle Approximation…What happens Approximation…What happens then?then?

• Non Linear effects must be invoked Non Linear effects must be invoked to enhance the acceleration to enhance the acceleration efficiency (problem with Emax) efficiency (problem with Emax)

Cosmic Ray Modified Shock Waves

Self-Generation of MagneticField and Magnetic Scattering

Particle Acceleration in the

Non Linear Regime: Shock Modification

Why Did We think About Why Did We think About This?This? Divergent Energy Spectrum Divergent Energy Spectrum

At Fixed energy crossing the shock front At Fixed energy crossing the shock front ρρuu2 2

tand at fixed efficiency of acceleration there are tand at fixed efficiency of acceleration there are values of Pmax for which the integral exceeds values of Pmax for which the integral exceeds ρρuu2 2 (absurd!)(absurd!)

minmaxCR EEEENdE=E /ln

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 Approaches to Particle Acceleration at Acceleration at Modified ShocksModified Shocks Two-Fluid ModelsTwo-Fluid Models The background plasma and the CRs are treated as two The background plasma and the CRs are treated as two

separate fluids. These thermodynamical Models do not separate fluids. These thermodynamical Models do not provide any info on the Particle Spectrumprovide any info on the Particle Spectrum

Kinetic ApproachesKinetic Approaches The exact transport equation for CRs and the coservation The exact transport equation for CRs and the coservation

eqs for the plasma are solved. These Models provide eqs for the plasma are solved. These Models provide everything and contain the Two-fluid modelseverything and contain the Two-fluid models

Numerical and Monte Carlo ApproachesNumerical and Monte Carlo Approaches Equations are solved with numerical integratorsEquations are solved with numerical integrators. . Particles Particles

are shot at the shock and followed while they diffuse and are shot at the shock and followed while they diffuse and modify the shockmodify the shock

The Basic Physics of Modified Shocks

Un

dis

turb

ed

Med

ium

Sh

ock F

ron

t v

subshock Precursor

00 uρ=xuxρ

xP+xP+xuxρ=P+uρ CRgg,2

02

00

tp,x,Q+p

fp

dx

du+

x

fu

x

fD

x=

t

f

3

1

Conservation of Mass

Conservation of Momentum

Equation of DiffusionConvection for theAccelerated Particles

Main Predictions of Particle Main Predictions of Particle Acceleration Acceleration at Cosmic Ray at Cosmic Ray Modified ShocksModified Shocks

• Formation of a Precursor in the Upstream plasmaFormation of a Precursor in the Upstream plasma

• The Total Compression Factor may well exceed 4. The Total Compression Factor may well exceed 4. The Compression factor at the subshock is <4The Compression factor at the subshock is <4

• Energy Conservation implies that the Shock is Energy Conservation implies that the Shock is less efficient in heating the gas downstreamless efficient in heating the gas downstream

• The Precursor, together with Diffusion The Precursor, together with Diffusion Coefficient increasing with p-> Coefficient increasing with p-> NON POWER LAW NON POWER LAW SPECTRA!!!SPECTRA!!! Softer at low energy and harder at Softer at low energy and harder at high energyhigh energy

Spectra at Modified ShocksSpectra at Modified Shocks

Very Flat Spectra at high energyAmato and PB (2005)

Efficiency of Acceleration (PB, Gabici & Vannoni

(2005))

Note that only thisFlux ends up DOWNSTREAM!!!

This escapes out To UPSTREAM

Suppression of Gas Heating

PB, Gabici & Vannoni (2005)

cm=p max 10 3

cm=p max 10 5 10 10050,10,0 =M

Increasing

Pmax

Ranki

ne-H

ugon

iot

The suppressed heating might have already been

detected (Hughes, Rakowski & Decourchelle (2000))

Mach number M0 Rsub Rtot CR fracCR 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

Summary of Results on Efficiency of Non Linear Shock AccelerationSummary of Results on Efficiency of Non Linear Shock Acceleration

Amato & PB (2005)Be Aware that the Damping Of waves to the thermal plasmaWas neglected (on purpose) here!When this effect is introducedThe 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) ApproachThe 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.

C VA

Excitation

Of the

instability

Pitch angle scattering and Spatial DiffusionPitch angle scattering and Spatial Diffusion

The Alfven waves can be imagined as small perturbations on top of a

background B-field

B B0B1

The equation of motion of a particle in this field is

10 B+Bvc

Ze=

dt

pd

In the reference frame of the waves, the momentum of the particle remains

unchanged in module but changes in direction due to the perturbation:

mcγZeB=Ωψ+tkvμΩpc

BμZev=

dt

dμ/ cos

10

12/12

pF

pcrvλ=pD L

3

1

3

1

2)/())(( BBkkpF The Diffusion coeff reducesTo the Bohm Diffusion for Strong Turbulence F(p)~1

Maximum Energy Maximum Energy a la a la Lagage-CesarskyLagage-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

secuBE=

u

EDτ μGeV

shockacc

21000

162

3.310

We recall that the knee in the CR spectrum is at 101066 GeV 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?

ΓFσF=x

Fu+

t

F

Wave growth HERE IS THE CRUCIAL PART!

Wave damping

Bell 1978

pF

pcrvλ=pD L

3

1

3

1

Standard calculation of the Streaming Instability (Achterberg 1983)

LR,χ+=ω

kc1

2

22

μ

fμ

ω

kv

p+

p

f

kvΩ'±ω

pvμpμddp

ω

eπ=χ LR, 114 2222

There is a mode with an imaginary part of the frequency: CR’s excite AlfvenWaves resonantly and the growth rate is found to be:

z

PCR

Av

Maximum Level of Turbulent Self- Generated Field

σF=x

Fu+

t

F

Stationarity

z

Pv

z

F u CR

A

Integrating

1222

0

2

u

PM

B

B CRA

Breaking of LinearTheory…

For typical parameters of a SNR one has δB/B~20.

Non Linear DSA with Self-Non Linear DSA with Self-Generated Generated

Alfvenic turbulence Alfvenic turbulence (Amato & PB (Amato & PB 2006)2006)We Generalized the previous We Generalized the previous

formalism to include the formalism to include the PrecursorPrecursor!!We Solved the Equations for a CR We Solved the Equations for a CR

Modified Shock Modified Shock togethertogether with the with the eq. for the eq. for the self-generated Wavesself-generated Waves

We have for the first time a We have for the first time a Diffusion Coefficient as an outputDiffusion Coefficient as an output of of the calculationthe calculation

Spectra of Accelerated Particles Spectra of Accelerated Particles and and Slopes as functions of Slopes as functions of momentum momentum

Amato & PB (2006)

Magnetic and CR Energy as Magnetic and CR Energy as functions of the Distance from functions of the Distance from the Shock Frontthe Shock Front

Amato & PB 2006

Super-BohmSuper-Bohm Diffusion Diffusion

Amato & PB 2006

Spectra

Slopes DiffusionCoefficient

Recent Developments inThe Self-Generation of Magnetic Scattering

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

UPSTREAM

SHOCK

B

SHOCK

B

NON LINEAR AMPLIFICATION OF THE NON LINEAR AMPLIFICATION OF THE UPSTREAM MAGNETIC FIELD RevisitedUPSTREAM MAGNETIC FIELD Revisited

22 // ρucvPcvδB sCRs

Generation of Magnetic Turbulence near Collisionless ShocksGeneration of Magnetic Turbulence near Collisionless Shocks

Upstream

vs

SHOCKIn 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

sCRCR ven=J

retCRtot J+J=J

Assumption: all accelerated particles are protons

The total current must satisfy the Maxwell Equation

retCR J+Jc

π=B

4 CRret JB

π

c=J

4

Clearly at the zero order Jret = JCR

Next job: write the equation of motion of the fluid, which now feels a force:

BJ=F ret

BJBBπ

cP=

t

uρBJ+P=

t

uρ CRret

4

In addition we have the Equation for conservation of mass

and the induction equation:

uρ=t

ρ

Bu=t

B

t

B

c=E

1

SUMMARIZING:

BJBBπ

cP=

t

uρ CR

4

Bu=t

B

uρ=t

ρ

Linearization of these eqs in Fourier Space leads to 7 eqs.For 8 unknowns.

In order to close the system weNeed 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)

Many pages later one gets the DISPERSION RELATION for the allowed perturbations.

B B

J J 0

p

fp

x

fv

t

f

0

CRCR

)1(cρ

kBJ vk -ω

0

0CR2 22A

There is a purely growing Non Alfvenic, NonResonant, CR driven mode if k vA < ω0

2

σ)(1cρ

kBJω

0

0CR20

Bell 2004

Re(Re(ωω) and Im() and Im(ωω) as functions of k) as functions of k

Saturation of the GrowthSaturation of the Growth

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

CRret JBπ

c=J

4CRJ

cB

4

Bell 2004

Hybrid Simulations used to follow the field amplificationHybrid Simulations used to follow the field amplification

when the linear theory starts to failwhen the linear theory starts to fail

2CR2

A20

2

ρu

P

c

uM

B

δB For typical parameters of a

SNR we get δB/B~300

Some Shortcomings of this ResultSome 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 How do we look for NL Effects in DSA?DSA?

• Curvature in the radiation spectra Curvature in the radiation spectra (elentrons in the field of protons)(elentrons in the field of protons)

• Amplification in the magnetic field at the Amplification in the magnetic field at the shockshock

• Heat Suppression downstreamHeat Suppression downstream

Possible Observational Evidence for Amplified Magnetic FieldsPossible Observational Evidence for Amplified Magnetic Fields

240 µG

360 G

Volk, Berezhko & Ksenofontov (2005)

Rim 1

Rim 2

Lower

Fields

But may be we are seeing the scale of theDamping (Pohl et al. 2005) ???

CONCLUSIONSCONCLUSIONS

Crucial steps ahead are being done in the understanding of Crucial steps ahead are being done in the understanding of the origin of CRs through measurements of spectra and the origin of CRs through measurements of spectra and composition of different componentscomposition of different components

We understood the if shock acceleration is the mechanism We understood the if shock acceleration is the mechanism of acceleration, NON LINEAR effects have to be taken into of acceleration, NON LINEAR effects have to be taken into account. These effects are NOT just CORRECTIONS.account. These effects are NOT just CORRECTIONS.

The CR reaction enhances the acceleration efficiency, The CR reaction enhances the acceleration efficiency, hardens spectra at HE, suppressed heatinghardens spectra at HE, suppressed heating

The maximum energies observed in galactic CRs can be The maximum energies observed in galactic CRs can be understood only through efficient NL self generation of understood only through efficient NL self generation of turbulent fieldsturbulent fields