NonNon--Linear Theory of Particle Linear Theory of Particle Acceleration at Astrophysical Acceleration at Astrophysical

ShocksShocks

Pasquale Pasquale BlasiBlasiINAF/Arcetri Astrophysical Observatory, INAF/Arcetri Astrophysical Observatory,

FirenzeFirenze, Italy, Italy

TeV Workshop, Madison, August 2006

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,u∫Return Probability from UP=1

( ) 1 < µ d µ µ P µ d , 0d∫Return Probability from DOWN<1

( )1

3loglog

1

00 −

≈−⎟⎟⎠

⎞⎜⎜⎝

⎛−−

rGP=γ

ppN=EN

γP Total Return Probability from DOWNG Fractional Energy Gain per cycler 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 21 = µ d µ+u f =Φ 220out −−∫

( ) ( )² u+1 21 = µ d µ+u f =Φ 20020i ∫

( )( ) 2

2

2 41² 1² 1 u

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 34 uu=

E∆E=G −

SPECTRAL SLOPE

( )( ) 1

3

34

41 log log

log

21

2

−≈

−

−−

ruu

u=GP=γ return

The DiffusionThe Diffusion--Convection Equation: A more formal approachConvection Equation: A more formal approach

( ) 21

1

221

10

3

43 uu

u

pp

πpN

uuu=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+pfp

dxdu+

xfu

xfD

x=tf

∂∂

∂∂

−⎥⎦⎤

⎢⎣⎡

∂∂

∂∂

∂∂

31

SPATIAL DIFFUSION

ADVECTION WITH THEFLUID

ADIABATIC COMPRESSIONOR SHOCK COMPRESSION

INJECTION TERM

The Need for a NonThe Need for a Non--Linear TheoryLinear Theory

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

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

Cosmic Ray Modified Shock Waves

Self-Generation of MagneticField and Magnetic Scattering

Particle Acceleration in the

Non Linear Regime: Shock Modification

Going Non Linear: Part I

Why Did We think About This?Why Did We think About This?

Divergent Energy Spectrum Divergent Energy Spectrum

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

tandtand at fixed efficiency of acceleration there are at fixed efficiency of acceleration there are values of values of PmaxPmax for which the integral exceeds 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

The Basic Physics of Modified Shocks

Und

istu

rbed

Med

ium

Shoc

k Fr

ont v

subshock Precursor

( ) ( ) 00 uρ=xuxρ

( ) ( ) ( ) ( )xP+xP+xuxρ=P+uρ CRgg,2

02

00

( )tp,x,Q+pfp

dxdu+

xfu

xfD

x=

tf

∂∂

∂∂

−⎥⎦

⎤⎢⎣

⎡∂∂

∂∂

∂∂

31

Conservation of Mass

Conservation of MomentumEquation of DiffusionConvection for theAccelerated Particles

Main Predictions of Particle Acceleration Main Predictions of Particle Acceleration at Cosmic Ray Modified Shocksat Cosmic Ray Modified 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 The Total Compression Factor may well exceed 4. The Compression factor at the Compression factor at the subshocksubshock is <4is <4

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

The Precursor, together with Diffusion Coefficient The Precursor, together with Diffusion Coefficient increasing with pincreasing with p--> > NON POWER LAW SPECTRA!!!NON POWER LAW SPECTRA!!!Softer at low energy and harder at high energySofter at low energy and harder at high 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=pmax 10 5 10× 10050,10,0 =M

IncreasingPmax

Rankine

-Hugoniot

The suppressed heating might have already been detected (Hughes, Rakowski & Decourchelle (2000))

Going Non Linear: Part II

Coping with the Self-Generation of Magnetic field by the Accelerated Particles

The Classical Bell (1978) The Classical Bell (1978) -- LagageLagage--CesarskyCesarsky (1983) Approach(1983) Approach

Basic Assumptions:1. The Spectrum is a power law2. The pressure contributed by CR's is relatively small3. 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 B0 B1

The equation of motion of a particle in this field is

( )10 B+BvcZe=

dtpd rrrr

×

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=

dtdµ / cos

10

12/12

−−

( ) ( )( )pFpcr

vλ=pD L

31

31

≈

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

Maximum Energy Maximum Energy a la a la LagageLagage--CesarskyCesarsky

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=

uEDτ µGeV

shockacc

21000

162 3.310 −−≈

We recall that the knee in the CR spectrum is at 101066 GeVGeV 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=xFu+

tF

−∂∂

∂∂

Wave growth HERE IS THE CRUCIAL PART!

Wave damping

Bell 1978

( ) ( )( )pFpcr

vλ=pD L

31

31

≈

Standard calculation of the Streaming Instability (Achterberg 1983)

LR,χ+=ωkc 12

22

( ) ( )⎥⎦

⎤⎢⎣

⎡∂∂

⎟⎠

⎞⎜⎝

⎛ −∂∂

−−

∫∫ µfµ

ωkv

p+

pf

kvΩ'±ωpvµpµddp

ωeπ=χ LR, 114 2222

µ

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

zPCR∂

∂= Av σ

Maximum Level of Turbulent Self- Generated Field

σF=xFu+

tF

∂∂

∂∂

Stationarity

zP

v zF u CR

A ∂∂

=∂∂

Integrating

12 220

2>>=

uP

MBB CR

A ρδ Breaking of Linear

Theory…

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

Non Linear DSA with SelfNon Linear DSA with Self--Generated Generated AlfvenicAlfvenic turbulence turbulence (Amato & PB 2006)(Amato & PB 2006)

We Generalized the previous formalism We Generalized the previous formalism to include the 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 eqeq. . for the for the selfself--generated Wavesgenerated WavesWe 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 and Spectra of Accelerated Particles and Slopes as functions of momentumSlopes as functions of momentum

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

Amato & PB 2006

SuperSuper--BohmBohm DiffusionDiffusion

Amato & PB 2006

Spectra

Slopes

DiffusionCoefficient

How do we look for NL Effects in DSA?How do we look for NL Effects in DSA?

Curvature in the radiation spectra (electrons in the field Curvature in the radiation spectra (electrons in the field of protons) of protons) –– (indications of this in the IR(indications of this in the IR--radio spectra of radio spectra of SNRsSNRs by by Reynolds)Reynolds)

Amplification in the magnetic field at the shock Amplification in the magnetic field at the shock (seen in (seen in Chandra observations of the rims of Chandra observations of the rims of SNRsSNRs shocks)shocks)

Heat Suppression downstream Heat Suppression downstream (detection claimed by (detection claimed by Hughes, Rakowski & Decourchelle 2000)

All these elements are suggestive of very efficient CR All these elements are suggestive of very efficient CR acceleration in acceleration in SNRsSNRs shocks. BUT similar effects may be shocks. BUT similar effects may be expected in all accelerators with shock frontsexpected in all accelerators with shock fronts

A few notes on NLDSAA few notes on NLDSAThe spectrum “observed” at the source through nonThe spectrum “observed” at the source through non--thermal radiation may not be the spectrum of thermal radiation may not be the spectrum of CR’sCR’sThe spectrum at the source is most likely concave though a The spectrum at the source is most likely concave though a convolution with convolution with ppmaxmax(t(t) has never been carried out) has never been carried outThe spectrum seen outside (upstream infinity) is peculiar The spectrum seen outside (upstream infinity) is peculiar of NLof NL--DSA:DSA:

pmax

E

E

F(E) F(E)

CONCLUSIONSCONCLUSIONSParticle Acceleration at shocks occurs in the nonParticle Acceleration at shocks occurs in the non--linear linear regime: these are NOT just corrections, but rather the regime: these are NOT just corrections, but rather the reason why the mechanism works in the first placereason why the mechanism works in the first place

The efficiency of acceleration is quite highThe efficiency of acceleration is quite high

Concave spectra, heating suppression and amplified Concave spectra, heating suppression and amplified magnetic fields are the main symptoms of NLmagnetic fields are the main symptoms of NL--DSADSA

More work is needed to relate more strictly these More work is needed to relate more strictly these complex calculations to the phenomenology of CR complex calculations to the phenomenology of CR acceleratorsaccelerators

The future developments will have to deal with the The future developments will have to deal with the crucial issue of magnetic field amplification in the fully crucial issue of magnetic field amplification in the fully nonnon--linear regime, and the generalization to relativistic linear regime, and the generalization to relativistic shocksshocks

Exact Solution for Particle Acceleration at Modified ShocksExact Solution for Particle Acceleration at Modified Shocksfor Arbitrary Diffusion Coefficientsfor Arbitrary Diffusion Coefficients

Basic Equations( ) ( ) 00uρ=xuxρ

( ) ( ) ( ) ( ) 0200

2g,cg P+uρ=xP+xP+xuxρ

( ) ( ) ( ) ( ) ( ) ( ) ( )tp,x,Q+pxp,fp

dxdu+

xxp,fxu

xxp,fxp,D

x=

txp,f

∂∂

∂∂

−⎥⎦

⎤⎢⎣

⎡∂

∂∂∂

∂∂

31

( ) ( ) ( )xp,fpvpdpπ=xPc3

34∫

( ) ( ) ( ) ( )( )⎥⎦

⎤⎢⎣

⎡ −≈ ∫ p,x'D

x'u'dxpqpfpx,f3

exp0 ( )pdfd

=pqlnln 0−

( ) ( )( )

( ) ⎥⎦

⎤⎢⎣

⎡−

−⎟⎟⎠

⎞⎜⎜⎝

⎛− ∫ 1

3exp

413

30

0 p'URp'UR

p'dp'

πpηn

pURR

=pftot

tot

injtot

totDISTRIBUTION FUNCTION AT THESHOCK

Amato & Blasi (2005)

( ) 232/1 1

34 ξ

sb eξRπ

=η −− Recipe for Injection

It is useful to introduce the equations in dimensionless form:

( ) ( ) ( ) ( ) ( )200

20

20

111

uρxP

=xξxUγM

xUγM

+=xξ cc

γc

−−−

( ) ( ) ( ) ( )( ) ( ) ( )

( ) 00

3200

3exp3

4upqxp,D=xx

x'xx'U'dxpfpvpdp

uρπ=xξ p

pc

⎥⎥⎦

⎤

⎢⎢⎣

⎡− ∫∫

One should not forget that the solution still depends on f0 which in turn depends on the function

( ) ( ) ( )( )( )⎥

⎥⎦

⎤

⎢⎢⎣

⎡− ∫∫ x'x

x'U'dxxx

xUdx=pUpp

exp12

Differentiating with respect to x we get ...

p

F(p)

INJECTEDPARTICLES

( )( ) ( ) ( ) ( )

( )

( ) ( ) ( )( )

,

x'xx'U'dxpfpvpdp

x'xx'U'dxpfpv

xxpdp

=xλ

p

pp

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

∫∫

∫∫

exp

exp1

03

03

( ) ( ) ( )xUxξxλ=dxdξ

cc −

The function ξc(x) has always the right boundary conditions at the shock and at infinity but ONLY for the right solution the pressure at the shock is that obtained with the f0 that is obtained iteratively (NON LINEARITY)