COSMIC RAY ACCELERATION

COSMIC RAY ACCELERATION

The cosmic Ray energy spectrum with a good

approximation has a power law form:

 = K E

How can we explain this form:

FERMI idea (original) MODEL

[Fermi Acceleration of second Order) Idea of Fermi:

CR acceleration is a stochastic process

Ensemble of many “events” in each of which a particle gains only a small amount of energy proportional to E:

Process is stopped with probability P_esc Two parameters , P_esc

 E =  E

FERMI ORIGINAL MODEL:

General structure:

Single acceleration event:

Particle with Energy E

in the event it gains an energy proportional to E

 E =  E

The events are iterated with probability

1-P

The iteration is stopped with probability

P

Two parameters  , P

Discrete Spectrum of the toy model

Differential Spectrum slope 

Integral Spectrum slope 

Collision with

Moving Plasma Clouds

in the Galaxy

E

E







Collisions with a

Macroscopic Object Moving with velocity

v

v E

E



E

 relative to the cloud particle is assumed

ultra-relativistic p = E

 velocity of the cloud

 ~ 4/3 ²

Spectrum too soft

Spectrum dependent on

details of cloud kinematics!!

MODIFICATION of the original FERMI Model

ACCELERATION at SHOCK FRONTS

FERMI 1

ORDER

ACCELERATION

SHOCK in a fluid

Surface of Discontinuity in the Thermodynamics quantities

(Density, Temperature, Velocity)

Trinity Test (1945)

CAS A

(1667)

Unshocked material  at rest

Piston Shock

Front

Unshocked material

Fluid at Rest

Shock

Discontinuity

shocked material

More dense

Higher Temperature v < v_shock

v

Unshocked material Shock

Discontinuity at rest

shocked material

More dense

Higher Temperature v < v_shock

v

v

Shock Rest Frame

Downstream 

STRONG SHOCK 

Unshocked  material shocked  material 

Unshocked material  at rest

Compression

factor r ratio of specific heats

STRONG SHOCK

Unshocked material Shocked material

Shock Rest Frame

Kinematics Relation at the Shock Rankine Huguniot Relations

Conservation of MASS (number of Particles), MOMENTUM,

ENERGY

U

, 

U

, 

Upstream Downstream

E

E

From equations of shocks in fluid:



/ 

= ( 

+1)M

/(( 

-1)M

+2)



=c

/c

=5/3 (gas monoatomici)

M = 

/c

(c

velocita' del suono)

Numero di Mach

The energy spectrum of particle accelerated near Shock Waves has a UNIVERSAL FORM.

The (integral) spectral index is close to the injection one as deduced from local measurement corrected with escape time .

Additional material

Temperature Ratio

Compression 2/ 1

M = Mach Number = v₁/v_sound

M = Mach Number = v₁/v_sound Temperature T 2/T 1