Pasquale BlasiINAF/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 2nd knee? Dip/Ankle GZK?

140 GeV 2.5 TeV 20 TeV 100 TeV 450 TeVs

The Chemical Composition of Cosmic Rays

Myr few cn

1

spallgasint

Unstable ElementsSimpson and Garcia-Munoz 1988

Balloon flights For Cosmic Rays

LaboratoryExperiment

yr101.5 τ 6Be10

Age of Cosmic Rays about 10-15 million years

PROPAGATION OF COSMIC RAYS

years 3000(1/3)c

pc 300DISC

years 000,150(1/3)c

kpc 15GAL

years 000,30(1/3)c

kpc 3HALO

PROPAGATION TIME IN THE DISC

PROPAGATION TIME ALONGTHE ARMS OF THE GALAXY

PROPAGATION TIME IN THEHALO

ALL THESE TIME SCALES ARE EXCEEDINGLY SHORT TO BEMADE COMPATIBLE WITH THE ABUNDANCE OF LIGHT ELEMENTS

DIFFUSIVE PROPAGATION

A qualitative look at the diffusive propagation of CR

€

diff = λ

c/3

⎛

⎝ ⎜

⎞

⎠ ⎟R

λ

⎛

⎝ ⎜

⎞

⎠ ⎟2

= R2

13

cλ

pc 1 ~λ

If is the mean distance between two scattering centers, then the time necessary for a particle to travel a distance R is

Mean distance betweenScattering centers

From the measured abundance of light elements and from the decay time ofUnstable elements we know that the diffusion time on scales of about 1 kpc Must be about 5 million years. It immediately follows that

€

D = 1

3c λ = (5 -10) ×1028 cm2s−1

DiffusionCoefficient

The Leaky Box ModelThe diffusion of CR can be described through an equation similar to that of Heat transfer

€

∂n

∂t+ D(E)∇2n = q(E,

r r )

Ignorance of Diffusion +Assumption of stationarity

H €

Dn

H 2= q⇒

n

τ esc(E)= q

€

n(E) = q(E)τ esc(E) τ esc(E) =H 2

D(E)

Since D(E) grows with E the observed spectrum n(E) is always

steeper than the injected spectrum q(E)

Since D(E) grows with E the observed spectrum n(E) is always

steeper than the injected spectrum q(E)

injectionLeakage

Primary/Primary and Secondary/Primary ratios

(CREAM 2008)

Dependence of the Diffusion Coefficient on energy – a leaky box

approach

€

qsec(E) = nprim (E) Y σ ngas c

€

nsec(E) = qsec(E)τ conf (E)

cm g 50 x x

x(E) )( c n Y

Primary

Secondary 2-nucl

nuclconfgas E

(E) n x(E) conf gas cmp

From the previous plot we see that at low energies P/S ~ 0.1 which impliesX(E) ~ 5 g cm-2

As a function of energy: 0.5 ~ E ))(/1( D(E) EX

Electrons (and positrons)

Leaky Box with Energy Losses

€

n(E)

τ esc(E)−

n(E)

τ loss (E)= q(E) τ loss (E) =

E

(dE /dt)∝

1

E

When the propagating particles are electrons, energy losses may become important:

€

dE

dt=

dE

dt

⎛

⎝ ⎜

⎞

⎠ ⎟syn

+dE

dt

⎛

⎝ ⎜

⎞

⎠ ⎟ICS

∝ E2

€

n(E) =q(E)

1τ esc(E)

+1

τ loss (E)

=

€

E−γ −δ if escape dominates (low E)

E−γ −1 if losses dominate (high E)

Positron ratioPositrons are only producedas secondary products:

While CR propagate fromtheir sources to Earth throughout the Galaxy

€

p + p → π ± + X

π ± → μ ± +ν μ

μ ± → e± +ν e +ν μ

Quick look at the positron excess

PRIMARY PROTONS:

€

nCR (E) = NCR (E) R τ esc(E)∝E -γ E−δ

PRIMARY ELECTRONS: (for diffusion,for losses)

€

ne (E) = Ne(E) R Min τ esc(E),τ loss(E)[ ]∝ E−γ e E−β

SECONDARY POSITRONS INJECTION:

SECONDARY POSITRONS EQUILIBRIUM:

€

q+(E')dE'= nCR(E)dE nH σ pp c∝E -γ -δ

€

n+(E) = q+(E) Min τ esc(E),τ loss(E)[ ]∝ E−γ −δ −β

€

n+

ne

∝ E−(γ −γ e )−δ CANNOT GROW!

- 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 PARTICLESIN A MAGNETIC FIELD

DIFFUSIVE PARTICLEACCELERATION

COSMIC RAY PROPAGATION IN THEGALAXY AND OUTSIDE

Charged Particles in a regular B-field

B

c

vEq

dt

pd

In the absence of an electric field one obtains the well known solution:

Constantpz t]cos[ Vv 0x

t]sin[ Vv 0y c m

B q 0

LARMOR FREQUENCY

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

0B Bδ0B

xB

yBz

0BδB ┴

)BδB( q 0cv

dtpd

THIS CHANGES ONLYTHE X AND Y COMPONENTSOF THE MOMENTUM

THIS TERM CHANGESONLY THE DIRECTIONOF 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:

0dt

d

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:

)F(k)kμ-(1 Ω2

π

Δt

ΔμΔμ resres

2vμ

Ωk res

RESONANCE!!!

DIFFUSION COEFFICIENT

)F(kΩk4

π

Δt

ΔθΔθD resresμμ

)G(k Ω

1τ

res

THE RANDOM CHANGE OF THE PITCH ANGLE ISDESCRIBED BY A DIFFUSION COEFFICIENT

FRACTIONAL POWER (δB/B0)2

=G(kres)

THE DEFLECTION ANGLE CHANGES BY ORDER UNITYIN A TIME:

PATHLENGTH FOR DIFFUSION ~ vτ

)G(k Ω

v τv

Δt

ΔzΔz

res

22

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 EquationIn its simplest version, the diffusion of CR from a sourcecan be described through an equation similar to that ofHeat transfer

€

∂n

∂t+∇ D(E,

r r )∇n[ ] = q(E,

r r )

The Green function of this partial differential equation issimple to calculate if D(E,r)=D(E):

€

ℑ(t ',r r ';t,

r r ) =

1

4πD(E)(t − t ' )[ ]3/2 exp −

(r r −

r r ' )2

4D(E)(t − t ' )

⎡

⎣ ⎢

⎤

⎦ ⎥ t > t'

H

2h

Rd

disc

HaloParticle escapeIn general: Rd > H >> h

ASSUMPTIONS:1.Instantaneous injection of particles in a point in the disc2.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

€

ℑ(t ',r r ';t,

r r ) =

1

4πD(E)τ[ ]3/2 exp −

(x − x' )2 +(y − y' )2

4D(E)τ

⎡

⎣ ⎢

⎤

⎦ ⎥ (-1)n exp −

(z − zn' )2

4D(E)τ

⎡

⎣ ⎢

⎤

⎦ ⎥

n=-∞

+∞

∑ τ > 0 zn' = (−1)n z'+2nH

Contribution of many sources

€

nCR (E) = dτ0

∞

∫ dr0

Rd

∫ 2π r N(E) ℜ

π Rd2

ℑ (z = 0,r = 0, x = y = 0)

Integral in tau is analytical

€

nCR(E) =N(E) ℜ

2π D(E) Rd

(−1)n

n=−∞

+∞

∑ ds 1

s2 + 2n(H / Rd )[ ]2

0

1

∫ s = r/Rd

In the limit H/Rd<<1

€

nCR(E) =N(E) ℜ

2π D(E) Rd

H

Rd

= N(E) ℜ2Hπ Rd

2

H2

D(E) ∝ N(E)/D(E)

Diffusiontime

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 Leaky Box

CR primary

CR electrons(with dominant Losses)€

nCR(E) =N(E) ℜ H

2π D(E) Rd2

€

nCR(E) =N(E) ℜ H

2π D(E) Rd2

€

ne(E) =Ne(E) ℜ

2π H Rd2

τ loss (E)

€

ne(E) =Ne(E) ℜ

2π D(E)τ loss (E) Rd2

τ loss (E)

€

∝ E−γ −

δ

2−

1

2

€

∝ E−γ −1

Similar situation for nuclei when spallation dominates

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)- and we take E0~m~1 GeV

and energies are taken to be normalized to E0.

The observed spectrum of protons at Earth isand taking D(E)~(/3GV)where is the rigidity:

and comparing with the observed spectrum

€

ECR = dEN(E)E =K

γ − 2∫ = ξCRESN ⇒ K = (γ − 2)ξCR ESN

Order 1051 erg

Relatively large efficienciesrequired

A curiosityLife 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 theexplosion of supernovae…

But supernovae are usually formed in regions of star formation, or molecular Clouds…

These clouds form by gravitational collapse…BUT their gravitational collapsetime 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 thestars to be created and later explode