Pasquale BlasiINAF/Arcetri Astrophysical Observatory

4th School on Cosmic Rays and Astrophysics UFABC - Santo André - São Paulo – Brazil

Acceleration of charged particlesThe 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 QSO)

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 particlesAll 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

ACCELERATIO

N

STOCHASTIC

ACCELERATIO

N

REGULAR ACCELERATIONLarge mean scale electric fields are produced on some spatial scale Lreg

€

rE ≠ 0

DIFFICULT TO CREATE NET ELECTRIC FIELD IN ASTROPHYSICS

BECAUSE OF HIGH CONDUCTIVITY, BUT SOME EXCEPTIONS:

Unipolar Inductor Magnetic

reconnection

STOCHASTIC ACCELERATION

€

rE = 0

r E 2 ≠ 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.

If both regular and stochastic acceleration occur:

€

Tmaxreg ≈ ZeEreg Lreg

€

Tmaxst ≈ ZeEst Lst

Lreg

Lst

⎛

⎝ ⎜

⎞

⎠ ⎟

1/2

= ZeEst Lst Lreg( )1/2

Most astrophysical acceleration processes belong to this class

2nd order Fermi Acceleration (Fermi, 1949)

βμ)(1γEE i'

)βμβμ)(1(1EγE 'i

2f

€

ΔE

E μ '

=1

2dμ ' γ 2

-1

1

∫ (1- βμ )(1+ βμ ' ) −1= γ 2(1- βμ ) −1

€

PROBABILITY OFENCOUNTER

LOSSES AND GAINSARE PRESENT BUT DONOT COMPENSATE EXACTLY

€

ΔE

E μ

= dμ ' 1

2(1− βμ )γ 2

-1

1

∫ ΔE

E μ '

=4

3β 2

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?

€

mvμγ → m(v − vw )μγ ⇒ Δp = m vw γ Δμ

As usual:

€

Δp = 0

€

ΔpΔp

Δt≈ mvwγ( )

2 ΔμΔμ

Δt

Where you should recall that:

€

ΔμΔμΔt

=1

3

v2

Dzz

Therefore:

€

Dpp =ΔpΔp

Δt≈ mvwγ( )

2 1

3

v2

Dzz

=1

3

p2

Dzz

vw2

The time for diffusion in momentum space is then:

€

τ pp =p2

Dpp

= 3Dzz

vw2

= 3Dzz

v2

v

vw

⎛

⎝ ⎜

⎞

⎠ ⎟

2

= 3τ zz

v

vw

⎛

⎝ ⎜

⎞

⎠ ⎟

2

>> τ zz

€

Dzz =1

3v2τ zz =

1

3

v2

ΩGDIFFUSION IN SPACE IMPLIES THAT A (2nd

ORDER) DIFFUSION IN MOMENTUM TAKES

PLACE (ACCELERATION)

A PRIMER ON SHOCK WAVES

ρuxt

ρ

gasPux

ut

2

For σ~10-25 cm2 and density n~1 cm-3 the typical interaction lengthis ~3 Mpc >> than the typical size of astrophysical objects and evenLarger than the Galaxy!!!

COLLISIONLESS SHOCKS

UPSTREAM DOWNSTREAM

U1 U2

-∞ 0 +∞

12

1

12

1 32

uP

ux

Pu

tgasgas

STATIONARY SHOCKS

3M

4M

ρ

ρ2

2

1

2

4

1M

4

5

p

p 2

1

2

2

22

1

2

38

232

32

310

M

MM

T

T

4

8

6 211

2

up

212 16

3muT

M→∞

M→∞

M→∞

SHOCK WAVES ARE MAINLY HEATING MACHINES!

BOUNCING BETWEEN APPROACHING MIRRORS

)V-E(1Ed μ

4

Nvvμ

4π

NdΩJ

1

0

UPSTREAM DOWNSTREAM

U1 U2

-∞ 0 +∞

V=U1-U2>0 Relative velocityINITIAL ENERGY DOWNS: E

)'V1)(V-E(1Eu μμ

-1< μ <0

0< μ’ <1

TOTAL FLUX

μμ μddμNv

ANvμdμP 2

4

)(

)(3

41)1)(1(22 21

''1

0

0

1

' UUVVddE

E

Δ

μμμμμμ

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

2220

1

)1(2

1)(

2

uufdu

out

μμ

2220

1

)1(2

1)(

2

uufdu

in

μμUPSTREAM DOWNSTREAM

U1 U2

-∞ 0 +∞

Return Probability from Downstream

22

2

22 41

1

1u

u

uP

in

outd

HIGH PROBABILITY OF RETURN FROM DOWNSTREAM BUT TENDS TO ZERO FOR HIGH U2

ENERGY GAIN: kk EVE

3

411

E0 → E1 → E2 → --- → EK=[1+(4/3)V]K E0

21

0 3

41ln ln UUK

E

EK

N0 → N1=N0*Pret → --- → NK=N0*PretK

20

41ln ln UKN

NK

Putting these two expressions together we get:

Therefore:

)(

34

1ln

ln

41ln

ln

21

0

2

0

UU

EE

U

NN

K

KK

00)(

E

ENEN K

K 1

3

r

2

1

U

Ur

THE SLOPE OF THE DIFFERENTIAL SPECTRUM WILLBE γ+1=(r+2)/(r-1) → 2 FOR r→4 (STRONG SHOCK)

THE TRANSPORT EQUATION APPROACH

tp,x,Q+p

fp

dx

du+

x

fu

x

fD

x=

t

f

3

1 tp,x,Q+p

fp

dx

du+

x

fu

x

fD

x=

t

f

3

1

UP DOWN

U1 U2

-∞ 0- 0+ +∞

03

10

012

12

=p+Qdp

pdfpuu+

x

fD

x

fD

Integrating around the shock:

Integrating from upstr. infinity to 0-:

and requiring homogeneity downstream:

011

f=ux

fD

00112

0 3Qfu

uu=

dp

dfp

THE TRANSPORT EQUATION APPROACH

21

1

221

10

3

4

3 uuu

p

p

πp

N

uu

u=pf

injinj

inj

INTEGRATION OF THIS SIMPLE EQUATION GIVES:

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

NOTE THAT THIS IS IN PSPACE NAMELY

N(p)dp=4π p2 f(p)dpTherefore the slope is

3r/(r-1)

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

x0

THE ESCAPE OF PARTICLES ATX=X0 CAN BE SIMULATED BYTAKING

0),( 0 pxf

THIS REFLECTS IN AN EXP CUTOFF AT SOMEMAX MOMENTUM

ESCAPE FLUX TOWARDS UPSTREAM INFINITY!!!

ESCAPE FLUX IN TEST PARTICLE THEORY

FOR D(E) PROPORTIONAL TO E (BOHM DIFFUSION):

*3

1p

r

rpMAX

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

2

2

1

1

21

)()(3

U

ED

U

ED

UUaccτ

THE ACCELERATION TIME IS GIVEN BY:

AND SHOULD BE COMPARED WITH THE AGE OF THE ACCELERATOR, FOR INSTANCE A SUPERNOVA REMNANT

AS AN ESTIMATE:

ageacc ττ Emax

IF THE SHOCK IS PROPAGATING IN THE ISM ONEWOULD BE TEMPTED TO ASSUME D(E)=Dgal(E)

WHERE TYPICALLY:A=(1-10) 1027 cm2/sα=0.3-0.5

GeV

EAEDgal )(

€

Emax,GeV ≈ 0.31u82τ 1000 / A27( )

1/α

FOR ALL CHOICES OF PARAMETERS THE MAX ENERGY OBTAINED IN THIS WAY IS FRACTIONSFRACTIONSOF GeVOF GeV, THEREFORE IRRELEVANT !!!

…BUT IT WOULD BE HIGHER IF D(E) WERE MUCH SMALLER…CAN IT HAPPEN?

DIFFERENT PHASES OF A SNR

cm/s M E 10M

2E V 1/2-

ej,1/251

9

ej

ejs

MASS OF THE EJECTA: Mej

FREE EXPANSION VELOCITY:

TOTAL KINETIC ENERGY: E51

THERE IS AN INITIAL PERIOD DURING WHICH THE SHELL OF

THE SN EXPANDS FREELY (FREE EXPANSION PHASE -BALLISTIC

MOTION):

SEDOV PHASE:SEDOV PHASE: years M n E 300 T 5/6-1/3-1/251Sedov

The sound speed in the ISM is about 106 cm/s

1000 - 100 number Mach STRONGSTRONGSHOCKSHOCK

BUT THE SHOCK SWEEPS THE MATERIAL IN FRONT OF IT

AND AT SOME POINT IT ACCUMULATES ENOUGH MATERIAL

TO SLOW DOWN THE EXPANDING SHELL:

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

€

EMAX (t)∝ t−α

€

EQ(E)dE ≈ Fesc(t) 1

2ρVs

3 4πRsh2

dEmax

dtdE∝ t1/2 dE∝ E-1 dE

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