Transcript Blasi-2 - 4th School on Cosmic Rays and Astrophysics

COSMIC RAY ACCELERATION
and
TRANSPORT
LECTURE 2
Pasquale Blasi
INAF/Arcetri Astrophysical Observatory
4th School on Cosmic Rays and Astrophysics UFABC - Santo André - São Paulo – Brazil
Acceleration of charged particles
The presence of non-thermal particles is deduced in a myriad of situations
in Nature (from the solar wind to the AGNs, from SNRs to GRBs, from pulsars
to mQSO)
PARTICLE
ACCELERATION
BUT usually (through not always) in the same regions there is evidence for
Thermalized plasmas, therefore the questions arises
WHICH PROCESSES DETERMINE WHETHER A PARTICLE IS
GOING TO BE ACCELERATED OR RATHER BE THERMALIZED ?
Acceleration of charged particles
All acceleration processes we are aware of are of electro-magnetic
nature – but magnetic fields DO NOT MAKE WORK on charged particles
WHAT IS THE ORIGIN OF THE ELECTRIC
FIELDS THAT PRODUCE ACCELERATION?
ACCELERATION MECHANISMS ARE CLASSIFIED ACCORDING WITH
THE ORIGIN OF THE ELECTRIC FIELDS
REGULAR
STOCHASTIC
ACCELERATION
ACCELERATION
REGULAR ACCELERATION
Large mean scale electric fields are produced on some spatial scale Lreg
r
E 0
DIFFICULT TO CREATE NET ELECTRIC FIELD IN ASTROPHYSICS
BECAUSE OF HIGH CONDUCTIVITY, BUT SOME EXCEPTIONS:
Unipolar Inductor

Magnetic reconnection
STOCHASTIC ACCELERATION
Most astrophysical acceleration processes belong to this class
r
E 0
r2
E 0
The stochastic electric field may result from random fluctuations on
a typical scale Lst but with random orientations so that on average the
field vanishes.
 reg
Tmax  ZeEreg Lreg
If both regular and stochastic acceleration occur:
Lreg 
1/2
 ZeEst Lst    ZeEst Lst Lreg 
 Lst 
1/2
st
max
T
2nd order Fermi Acceleration (Fermi, 1949)
E '  γEi (1  βμ)
E f  γ 2 Ei (1  βμ)(1  βμ ' )
E
E
E
E
1

2
'
1
2
2
d

'

(1
)(1
+

')
1


(1-  ) 1

-1
1
2 E
  d' (1  )
2
E
-1
1

PROBABILITY OF
ENCOUNTER
4 2
 
3
'
LOSSES AND GAINS
ARE PRESENT BUT DO
NOT COMPENSATE
EXACTLY
WHY WOULD MAGNETIC CLOUDS ACCELERATE PARTICLES?
WHERE ARE THE ELECTRIC FIELDS?
In the Fermi example the electric fields are induced by the motion of
the magnetized moving clouds
In reality we need to go back to our example of motion of a charged
particles in a group of Alfven waves…what if we do not sit in the
reference frame of the waves?
mv  m(v  vw )  p  m vw  
As usual:
p  0
pp
2 
 mvw 
t
t

1 v

t
3 Dzz
2
pp
1 p2 2
2 1 v
D pp 
 mvw 

vw
t
3 Dzz 3 Dzz
2
Where you should recall that:
Therefore:
The time for diffusion
 in momentum space is then:
 v 
Dzz
Dzz  v 
p
 pp 
 3 2  3 2    3 zz     zz
vw
v vw 
 Dpp
vw 
2
1 2
1 v2
Dzz  v  zz 
3
3 G
2
2
DIFFUSION IN SPACE IMPLIES THAT A (2nd
ORDER) DIFFUSION IN MOMENTUM TAKES
PLACE (ACCELERATION)
A PRIMER ON SHOCK WAVES
For σ~10-25 cm2 and density n~1 cm-3 the typical interaction length
is ~3 Mpc >> than the typical size of astrophysical objects and even
Larger than the Galaxy!!!
COLLISIONLESS SHOCKS
UPSTREAM
DOWNSTREAM
ρ

  ρu 
t
x

u     u 2  Pgas
t
x

0
-∞
U1
+∞
U2

  1 2 Pgas 
  1 3 Pgasu 
 u 
    u 
t  2
  1
x  2
  1 
STATIONARY SHOCKS
ρ2
4M 2
 2
ρ1 M  3
p2 5 2 1
 M 
p1 4
4
 10 2 2  2 2

M

M

2



T2  3
3  3


2
T1
8 
 M
3 
4
M→∞
M→∞
M→∞
6 1u12
p2 
8
3
T2  mu12
16
SHOCK WAVES ARE MAINLY HEATING MACHINES!
BOUNCING BETWEEN
APPROACHING MIRRORS
UPSTREAM
DOWNSTREAM
0
-∞
U1
+∞
U2
TOTAL FLUX
1
N
Nv
J   dΩ vμ 
4π
4
0
V=U1-U2>0 Relative velocity
INITIAL ENERGY DOWNS: E
E d  E(1 - V )
-1< μ <0
E u  E(1 - V )(1  V ' )
0< μ’ <1
ANvμ
P(  )dμ 
dμ  2 μd
Nv
4


E
4
'
'
'
  d 2  d 2 (1  V )(1  V )  1  (U1  U 2 )
E
3
0
1
1
0
FIRST ORDER
A FEW IMPORTANT POINTS:
I. There are no configurations that lead to losses
II. The mean energy gain is now first order in V
III. The energy gain is basically independent of any detail
on how particles scatter back and forth!
RETURN PROBABILITIES AND SPECTRUM
OF ACCELERATED PARTICLES
UPSTREAM
DOWNSTREAM
0
-∞
U1
+∞
U2
1
1
in   d f 0 (u2   )  (1  u2 ) 2
2
u 2
 out
u2
1
  d f 0 (u2   )  (1  u2 ) 2
2
1
Return Probability from Downstream
out 1  u2 
Pd 

 1  4u2
2
in 1  u2 
2
HIGH PROBABILITY OF RETURN FROM DOWNSTREAM
BUT TENDS TO ZERO FOR HIGH U2
ENERGY GAIN:
 4 
Ek 1  1  V  Ek
 3 
E0 → E1 → E2 → --- → EK=[1+(4/3)V]K E0
 EK
ln 
 E0

4


  K ln 1  U1  U 2 
 3


N0 → N1=N0*Pret → --- → NK=N0*PretK
 NK
ln 
 N0

  K ln 1  4U 2 

Putting these two expressions together we get:
 NK 
ln 

N
 0 
K
ln 1  4U 2 
 EK 
ln  
 E0 
 4

ln 1  (U1  U 2 )
 3

Therefore:
 EK
N ( E K )  N 0 
 E0




3

r 1
U1
r
U2
THE SLOPE OF THE DIFFERENTIAL SPECTRUM WILL
BE γ+1=(r+2)/(r-1) → 2 FOR r→4 (STRONG SHOCK)
THE TRANSPORT EQUATION APPROACH
f
  f 
f 1 du f
=
D u
+
p
+ Q  x, p, t 

t x  x 
x 3 dx p
UP
DOWN
Integrating around the shock:
df 0  p 
 f   f  1


D

D
+
u

u
p
+Q0  p =0

 

2
1
dp
 x  2  x 1 3
0-
-∞
U1
0+
+∞ Integrating from upstr. infinity to 0-:
U2
 f 
 D  =u1 f 0
 x 1
and requiring homogeneity downstream:
df 0
3
u1 f 0  Q0 
p =
dp u2  u1
THE TRANSPORT EQUATION APPROACH
INTEGRATION OF THIS SIMPLE EQUATION GIVES:
3u1 N inj 
f 0  p =
2 
u1  u2 4πpinj

 3u1
p  u1  u2
pinj 
NOTE THAT THIS IS IN P
SPACE NAMELY
N(p)dp=4π p2 f(p)dp
Therefore the slope is
3r/(r-1)
1. THE SPECTRUM OF ACCELERATED PARTICLES IS A POWER
LAW EXTENDING TO INFINITE MOMENTA
2. THE SLOPE DEPENDS UNIQUELY ON THE COMPRESSION
FACTOR AND IS INDEPENDENT OF THE DIFFUSION
PROPERTIES
3. INJECTION IS TREATED AS A FREE PARAMETER WHICH
DETERMINES THE NORMALIZATION
TEST PARTICLE SPECTRUM
SOME IMPORTANT COMMENTS
 THE STATIONARY PROBLEM DOES NOT ALLOW TO HAVE A
MAX MOMENTUM!
 THE NORMALIZATION IS ARBITRARY THEREFORE THERE IS
NO CONTROL ON THE AMOUNT OF ENERGY IN CR
 AND YET IT HAS BEEN OBTAINED IN THE TEST PARTICLE
APPROXIMATION
 THE SOLUTION DOES NOT DEPEND ON WHAT IS THE
MECHANISM THAT CAUSES PARTICLES TO BOUNCE BACK
AND FORTH
 FOR STRONG SHOCKS THE SPECTRUM IS UNIVERSAL AND
CLOSE TO E-2
 IT HAS BEEN IMPLICITELY ASSUMED THAT WHATEVER
SCATTERS THE PARTICLES IS AT REST (OR SLOW) IN THE
FLUID FRAME
A FREE ESCAPE BOUNDARY CONDITION
UP
DOWN
THE ESCAPE OF PARTICLES AT
X=X0 CAN BE SIMULATED BY
TAKING
f ( x0 , p)  0
x0
THIS REFLECTS IN AN EXP CUTOFF AT SOME
MAX MOMENTUM
ESCAPE FLUX TOWARDS UPSTREAM INFINITY!!!
ESCAPE FLUX IN TEST
PARTICLE THEORY
FOR D(E) PROPORTIONAL
TO E (BOHM DIFFUSION):
pMAX
r 1

p*
3r
SOME FOOD FOR THOUGHT
 WHAT DETERMINES THE MAX MOMENTUM IN
REALITY?
 IF THE RETURN PROBABILITY FROM UPSTREAM IS
UNITY, WHAT ARE COSMIC RAYS MADE OF?
 ARE WE SURE THAT THE 10-20% EFFICIENCY WE
NEED FOR SNR TO BE THE SOURCES OF
GALACTIC CR ARE STILL COMPATIBLE WITH THE
TEST PARTICLE REGIME?
MAXIMUM MOMENTUM OF
ACCELERATED PARTICLES
THE ACCELERATION TIME IS GIVEN BY:
 acc
3  D1 ( E ) D2 ( E ) 




U 1  U 2  U1
U2 
AND SHOULD BE COMPARED WITH THE AGE OF THE
ACCELERATOR, FOR INSTANCE A SUPERNOVA REMNANT
AS AN ESTIMATE:
 acc   age
Emax
IF THE SHOCK IS PROPAGATING IN THE ISM ONE
WOULD BE TEMPTED TO ASSUME D(E)=Dgal(E)
 E 
Dgal ( E )  A

 GeV 

WHERE TYPICALLY:
A=(1-10) 1027 cm2/s
α=0.3-0.5
Emax, GeV  0.31u 
2
8 1000

/ A27 
1/ 
FOR ALL CHOICES OF PARAMETERS THE MAX
ENERGY OBTAINED IN THIS WAY IS FRACTIONS
OF GeV, THEREFORE IRRELEVANT !!!
…BUT IT WOULD BE HIGHER IF D(E) WERE MUCH
SMALLER…CAN IT HAPPEN?
DIFFERENT PHASES OF A SNR
THERE IS AN INITIAL PERIOD DURING WHICH THE SHELL OF
THE SN EXPANDS FREELY (FREE EXPANSION PHASE -BALLISTIC
MOTION):
MASS OF THE EJECTA: Mej
TOTAL KINETIC ENERGY: E51
FREE EXPANSION VELOCITY:
Vs 
2E ej
M ej
-1/2
 109 E1/2
M
51
ej, cm/s
BUT THE SHOCK SWEEPS THE MATERIAL IN FRONT OF IT
AND AT SOME POINT IT ACCUMULATES ENOUGH MATERIAL
TO SLOW DOWN THE EXPANDING SHELL:
SEDOV PHASE:
TSedov  300 E
-1/2
51
n
-1/3
M
5/6

years
The sound speed in the ISM is about 106 cm/s
Mach number  100 - 1000
STRONG
SHOCK
Simple implications
During free expansion the shock fron moves with constant speed
Therefore its position scales with t
The diffusion front moves proportional to t1/2
During the free expansion phase the particles are not allowed to
Leave the acceleration box, which is the reason why the maximum
Energy increases
During the Sedov-Taylor expansion the radius of the blast waves
Grows as t2/5, slower than the diffusion front
THE MAXIMUM ENERGY OF ACCELERATED PARTICLES DECREASES
WITH TIME
THIS IS THE PHASE DURING WHICH THE PARTICLES CAN POSSIBLY
BECOME COSMIC RAYS
OVERLAP OF ESCAPE FLUXES: A
SIMPLE ESTIMATE
E MAX (t)  t

1 3
dE max
2
EQ(E)dE  Fesc (t) Vs 4 R sh
dE  t1/2 dE  E -1 dE
2
dt
BE VERY CAREFUL…THIS IS JUST A WAY TO SHOW HOW YOU
GET ROUGHLY A POWER LAW BUT SUMMING NON-POWER LAWS.
MORE DETAILED CALC’S SHOW DEPARTURES FROM THIS SIMPLE
TREND