COSMIC RAY ACCELERATION
and
TRANSPORT
LECTURE I
Pasquale Blasi
INAF/Arcetri Astrophysical Observatory
4th School on Cosmic Rays and Astrophysics UFABC - Santo André - São Paulo – Brazil
Lecture 1 - plan
• Short historical introduction to CRs
• Some observational data
• Basics of Cosmic Ray Transport
– Interaction of particles and waves: why particles diffuse
– Diffusion Model and Leaky Box model
– Bases of the Supernova paradigm for the origin of CR
Lecture 2 - plan
• Particle Acceleration
–
–
–
–
–
Second order Fermi particle acceleration
First order Fermi acceleration at non-relativistic shocks
Bell’s approach
Transport equation approach
The limitations of the test-particle approach
• Do charged particles act on the waves?
– Simple arguments for wave growth
Lecture 3 – plan
(research oriented)
• Modern aspects of diffusive shock
acceleration
– DSA as a non-linear problem
– The SNR paradigm for the origin of CRs
• Magnetic field amplification
• Maximum energy of accelerated particles
• Balmer dominated shocks
• Transport of CR in the Galaxy
– Chemical composition
– Anisotropy
Early History of Cosmic Rays
Ionized by what?
• 1895: X-rays (Roengten)
• 1896: Radioactivity (Becquerel)
• But ionization remained, though to a
lesser extent, when the electroscope
was inserted in a lead or water cavity
Victor F. Hess: the 1912 flight
+
Wulf Electroscope
(1909)
+
6am August 7, 1912
Aussig, Austria
+
COSMIC Rays
The Spectrum of Cosmic Rays
Knee
s
140 GeV
2.5 TeV
2nd knee? Dip/Ankle
20 TeV 100 TeV 450 TeV
GZK?
The Chemical Composition of
Cosmic Rays
 int
1

 few Myr
n gasc  spall
Unstable Elements
Simpson and Garcia-Munoz 1988
τ 10 Be  1.5 106 yr
Age of Cosmic
Rays about
10-15 million years
Balloon flights
For Cosmic Rays
Laboratory
Experiment
PROPAGATION OF COSMIC RAYS
 DISC
300 pc

 3000years
(1/3)c
15 kpc
 150,000 years
(1/3)c
PROPAGATION TIME ALONG
THE ARMS OF THE GALAXY
3 kpc

 30,000 years
(1/3)c
PROPAGATION TIME IN THE
HALO
 GAL 
 HALO
PROPAGATION TIME IN
THE DISC
ALL THESE TIME SCALES ARE EXCEEDINGLY SHORT TO BE
MADE COMPATIBLE WITH THE ABUNDANCE OF LIGHT
ELEMENTS
DIFFUSIVE
PROPAGATION
A qualitative look at the diffusive
propagation of CR
If  is the mean distance between two scattering centers, then the time
necessary for a particle to travel a distance R is
 diff
  R 2
R2
    
1
c/3 
c
3
Mean distance between
Scattering centers
From the measured abundance of light elements and from the decay time of

Unstable
elements we know that the diffusion time on scales of about 1 kpc
Must be about 5 million years. It immediately follows that
λ~ 1 pc
D 
1
c (5 -10) 1028 cm2 s1
3
Diffusion
Coefficient
The Leaky Box Model
The diffusion of CR can be described through an equation
similar to that of Heat transfer
n
r
2
 D(E) n  q(E, r )
t
H
Ignorance of
Diffusion +
Assumption of
stationarity
n
n
D 2 q
q
H
 esc (E)

n(E)  q(E) esc (E)

Leakage
injection
H2
 esc (E) 
D(E)
Since D(E) grows with E the observed
spectrum n(E) is always steeper
than the injected spectrum q(E)
Primary/Primary and
Secondary/Primary ratios (CREAM
2008)
Dependence of the Diffusion Coefficient
on energy – a leaky box approach
qsec (E)  nprim (E) Y  ngas c
nsec (E)  qsec (E) conf (E)
Secondary
x(E)
  Y n gas c  conf ( E ) 
Primary
x nucl
xnucl  50 g cm-2
x(E) n gas mp c  conf (E)
From the previous plot we see that at low energies P/S ~ 0.1 which implies
X(E) ~ 5 g cm-2
As a function of energy:

D(E)  (1/ X ( E))  E
 ~ 0.5
Electrons (and positrons)
Leaky Box with Energy Losses
When the propagating particles are electrons, energy losses may become
important:
n(E)
n(E)

 q(E)
 esc (E)  loss (E)
 loss (E) 
E
1

(dE / dt) E
dE dE  dE 
       E 2
dt dt syn dt ICS

n(E) 
q(E)
1
1

 esc (E)  loss (E)
E   if escape dominates (low E)


E  1 if losses dominate (high E)
Positron ratio
Positrons are only produced
as secondary products:
p p    X
     
   e   e   
While CR propagate from
their sources to Earth
throughout the Galaxy
Quick look at the positron excess
PRIMARY PROTONS:
nCR (E)  N CR (E) R  esc (E)E - E
PRIMARY ELECTRONS: (b= d
for diffusion, b=1
ne (E)  Ne (E) R Min esc (E), loss (E)  E  e E 
SECONDARY POSITRONS INJECTION:
q (E')dE' nCR (E)dE nH  pp c E
- -
SECONDARY POSITRONS EQUILIBRIUM:
n (E)  q (E) Min esc (E), loss (E)  E    
n
 E (  e )
ne
CANNOT GROW!
POSSIBLE EXPLANATIONS OF THE
PAMELA EXCESS
- SUBTLETIES OF PROPAGATION
(Shaviv et al. 2009)
- REACCELERATION OF SECONDARY PAIRS IN SNR
(Blasi 2009, Blasi&Serpico2009, Alhers et al. 2009)
- PULSARS
(Hooper, Blasi & Serpico 2008, Grasso et al. 2009,
BUT see pre-PAMELA work from Bueshing et al. 2008)
COSMIC RAY TRANSPORT:
Basic Concepts
CHARGED PARTICLES
IN A MAGNETIC FIELD
DIFFUSIVE PARTICLE
ACCELERATION
COSMIC RAY
PROPAGATION IN THE
GALAXY AND OUTSIDE
Charged Particles in a regular B-field


dp
 v 
 q E   B
dt
c


In the absence of an electric field one obtains
the well known solution:
p z  Constant
vx  V0 cos[ t]
v y  V0 sin[ t]
LARMOR FREQUENCY
q B0

mc
A few remarks…
• THE MAGNETIC FIELD DOES NOT CHANGE
PARTICLE ENERGY -> NO ACCELERATION
BY B FIELDS
• A RELATIVISTIC PARTICLE MOVES IN THE z
DIRECTION ON AVERAGE AT c/3
Motion of a charged particle in a
random magnetic field
 Bx

B0
z
By

dp
dt
δB  B0
 
δB ┴ B0


 q  (B0  δB)

v
c
THIS CHANGES ONLY
THE X AND Y COMPONENTS
OF THE MOMENTUM
THIS TERM CHANGES
ONLY THE DIRECTION
OF PZ=Pμ
SITTING IN THE REFERENCE FRAME OF THE THE WAVE, THERE
IS NO ELECTRIC FIELD…AND IF THE WAVE IS SLOW COMPARED
WITH THE PARTICLE (THIS IS GENERALLY THE CASE) THEN THE
WAVE IS STATIONARY AND Z=vμt
RATE OF CHANGE OF THE PITCH ANGLE IN TIME
Diffusive motion
ONE CAN EASILY SHOW THAT
BUT:
d
0
dt
Many waves
IN GENERAL ONE DOES NOT HAVE A SINGLE WAVE BUT RATHER
A POWER SPECTRUM:
THEREFORE INTEGRATING OVER ALL OF THEM:
OR IN A MORE IMMEDIATE FORMALISM:
ΔμΔμ
π
 Ω (1- μ 2 )kresF(kres )
Δt
2
k res
Ω

vμ
RESONANCE!!!
DIFFUSION COEFFICIENT
THE RANDOM CHANGE OF THE PITCH ANGLE IS
DESCRIBED BY A DIFFUSION COEFFICIENT
ΔθΔθ
π
Dμμ 
 Ωk resF(kres )
Δt
4
FRACTIONAL
POWER (δB/B0)2
=G(kres)
THE DEFLECTION ANGLE CHANGES BY ORDER UNITY
IN A TIME:
PATHLENGTH FOR DIFFUSION ~ vτ
1
τ
Ω G(k res )
2
ΔzΔz
v
 v2 τ 
Δt
Ω G(k res )
SPATIAL DIFFUSION COEFF.
PARTICLE SCATTERING
• EACH TIME THAT A RESONANCE OCCURS THE
PARTICLE CHANGES PITCH ANGLE BY Δθ~δB/B
WITH A RANDOM SIGN
• THE RESONANCE OCCURS ONLY FOR RIGHT
HAND POLARIZED WAVES IF THE PARTICLES
MOVES TO THE RIGHT (AND VICEVERSA)
• THE RESONANCE CONDITION TELLS US THAT 1) IF
k<<1/rL PARTICLES SURF ADIABATICALLY AND 2)
IF k>>1/rL PARTICLES HARDLY FEEL THE WAVES
The Diffusion Equation
In its simplest version, the diffusion of CR from a source
can be described through an equation similar to that of
Heat transfer
n
r
r
 D(E, r )n  q(E, r )
t
The Green function of this partial differential equation is
simple to calculate if D(E,r)=D(E):

r r 2 

r r
1
(r  r ')
(t', r ';t, r ) 
exp

3/2
 4D(E)(t  t')
4D(E)(t  t')
t> t'
H
Rd
disc
2h
Halo
Particle escape
In general: Rd > H >> h
ASSUMPTIONS:
1.Instantaneous injection of particles in a point in the disc
2.Infinitely thin disc, h  0 and infinitely extended disc, Rd
3. Free escape of the particles from above and below the halo n(z  H , r, E)  0
In order to fulfill this boundary condition the correct Green function is
r r
(t', r ';t, r ) 
 (x  x')2 (y  y')2 
1
exp

3/2
4D(E)

4

D(E)





 (z  zn' )2 
 
 (-1) exp 4D(E)
 
n=-
+
n
 > 0 z'n  (1)n z'2nH
Contribution of many sources

N(E) 
nCR (E)   d  dr 2 r
(z  0,r  0, x  y  0)
2
 Rd
0
0
Rd
Integral in tau is analytical
N(E) 
nCR (E) 
2 D(E) R d

1
 (1)  ds
n
n
0
1
s2  2n(H / Rd )
2
s= r/R d
In the limit H/Rd<<1
N(E) 
H
N(E)  H2
nCR (E) 

 N(E)/D(E)
2
2 D(E) Rd R d
2H Rd D(E)
Diffusion
time
You can compare this result with the less fundamental leaky box model
The Diffusion Model is not fully
equivalent to the Leaky Box Model
Diffusion
CR primary
CR electrons
(with dominant
Losses)

Leaky Box
N(E)  H
nCR (E) 
2 D(E) R2d
ne (E) 
N(E)  H
nCR (E) 
2 D(E) R2d
N e (E) 
 (E)
2 loss
2  D(E) loss (E) R d


E
 1
  
2 2
ne (E) 
N e (E) 
 loss (E)
2
2 H Rd

Similar situation for nuclei when spallation dominates
 E  1

The Supernova remnant paradigm in numbers
Let us assume that the rate of SN in the Galaxy is R and each produces a
power law spectrum of protons N(E)=K (E/E0)-g and we take E0~m~1 GeV
ECR 

dEN (E)E 
K
 CR ESN  K  (  2)CR ESN
 2
and energies are taken to be normalized to E0.
Order 1051 erg
The observed spectrum of protons at Earth is
and taking D(E)~(r/3GV)d where r is the rigidity:
and comparing with the observed spectrum
Relatively large efficiencies
required
A curiosity
Life on Earth is based on CNO elements, as well as heavier elements such as
Fe (your blood is red!)
All of these elements are formed ONLY in stars and liberated into space by the
explosion of supernovae…
But supernovae are usually formed in regions of star formation, or molecular
Clouds…
These clouds form by gravitational collapse…BUT their gravitational collapse
time would be too short to form stars in the first place…
UNLESS… the clouds are very weakly ionized (remember the electroscopes?)
and this allows magnetic fields in the ISM to oppose and slow down collapse
COSMIC RAYS, produced in SN explosions, also create the conditions for the
stars to be created and later explode