Transcript Non-Linear Theory of Particle Acceleration at Astrophysical Shocks Pasquale Blasi INAF/Arcetri Astrophysical Observatory, Firenze, Italy TeV Workshop, Madison, August 2006

Non-Linear Theory of Particle
Acceleration at Astrophysical
Shocks
Pasquale Blasi
INAF/Arcetri Astrophysical Observatory,
Firenze, Italy
TeV Workshop, Madison, August 2006
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 Pu μ 0, μ d μ = 1
Return Probability from UP=1
 p
N  E  = N 0  
 p0 
 γ 1
logP
γ=
logG
 
d μ Pd μ 0 , μ d μ < 1
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 u 2 + μ  d μ =
2
1
Φi = f0 u 2 + μ0  d μ0 = 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
3u1 N inj  p  u1  u2
f0  p =
2 

u1  u2 4πpinj
p
inj


The slope depends ONLY
on the compression ratio
(not on the diffusion coef.)
Injection momentum and
efficiency are free param.
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
Going Non Linear: Part I
Particle Acceleration in the
Non Linear Regime: Shock Modification
Why Did We think About This?
 Divergent Energy Spectrum
ECR = dEEN E   lnE max / E min 

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
ρ x  u x  = ρ0 u0
Undisturbed Medium
Shock Front
The Basic Physics of Modified Shocks
v
subshock
Precursor
Conservation of Mass
ρ0u02 + 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))
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 / mc γ
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
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
How do we look for NL Effects in DSA?
 Curvature in the radiation spectra (electrons in the field
of protons) –
Reynolds)
(indications of this in the IR-radio spectra of SNRs by
 Amplification in the magnetic field at the shock
Chandra observations of the rims of SNRs shocks)
 Heat Suppression downstream
Rakowski & Decourchelle 2000)
(seen in
(detection claimed by Hughes,
 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
A few notes on NLDSA
 The spectrum “observed” at the source through non-
thermal radiation may not be the spectrum of CR’s
 The spectrum at the source is most likely concave though a
convolution with pmax(t) has never been carried out
 The spectrum seen outside (upstream infinity) is peculiar
of NL-DSA:
F(E)
F(E)
E
pmax
E
CONCLUSIONS
 Particle Acceleration at shocks occurs in the non-linear
regime: these are NOT just corrections, but rather the
reason why the mechanism works in the first place
 The efficiency of acceleration is quite high
 Concave spectra, heating suppression and amplified
magnetic fields are the main symptoms of NL-DSA
 More work is needed to relate more strictly these
complex calculations to the phenomenology of CR
accelerators
 The future developments will have to deal with the
crucial issue of magnetic field amplification in the fully
non-linear regime, and the generalization to relativistic
shocks
Exact Solution for Particle Acceleration at Modified Shocks
for Arbitrary Diffusion Coefficients
Amato & Blasi (2005)
Basic Equations
ρ x u x  = ρ0 u0
ρ x u x 2 + Pg  x  + Pc  x  = ρ0 u02 + Pg,0
f  p, x   
f  p, x  
f  p, x  1 du f  p, x 




=
D p, x
u x
+
p
+ Q  x, p,t 


t
x 
x 
x
3 dx
p
4π
Pc  x  =
dpp 3v p f  p, x 
3

  q p 
u x' 
f  x, p   f 0  p exp 
dx'



3
D
x'
,
p




 ηn0

3 Rtot

f 0  p  = 
exp


3
 Rtot U  p   1  4πpinj


dlnf 0
q p  =
dlnp
dp' 3 Rtot U  p' 

p' Rtot U  p'  1 
DISTRIBUTION
FUNCTION AT THE
SHOCK
F(p)
η=
Rsb  1ξ
1/ 2
4
3π
3 ξ 2
e
INJECTED
PARTICLES
Recipe for Injection
p
It is useful to introduce the equations in dimensionless form:
ξc x = 1 +
ξc x =
1
γM 02
 U x 
4π
3 ρ0 u02

1
γ


U
x
2
γM 0
ξc x =
Pc  x 
ρ0 u02

U  x' 
dpp v  p f 0  pexp  dx'

x p  x' 


3
x p x =
3 D p, x 
q pu0
One should not forget that the solution still depends on f0 which in turn
depends on the function
U  p = dxU  x 

2

U  x' 
exp  dx'

x p x
x p  x' 

1

Differentiating with respect to x we get ...
dξ c
=  λ x ξ c  x U  x 
dx
λ x  =


U  x' 
dp p
v  p  f 0  p exp   dx'

x p x
x p  x' 

,

U  x' 
3
dp p v  p  f 0  p exp   dx'

x p  x' 

3


1

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)