Acceleration of Cosmic Rays Pankaj Jain IIT Kanpur

Acceleration of Cosmic Rays

Pankaj JainIIT Kanpur

spectrumspectrum

3/1 E

7.2/1 E

Knee at 1015 eV

The spectrum steepens after the knee,

Ankle at 1018.5 eV, perhaps an indication of a change over from galactic to extragalactic origin

Flux: number ------------------- m2 sr s GeV

SpectrumSpectrum

KNEE ANKLE

Spectral Index

Flux 1/E

= 2.7 1015 eV<E<109 eV = 3.1 1018.5 eV>E >1015 eV 2.7 E>1018.5 eV

At higher energies spectrum again becomes steeper ( > 4)

Origin

The high energy cosmic rays probably arise due to acceleration of charged particles at some astrophysical sites

supernova shock waves Active galactic nuclei gamma ray bursts pulsars galaxy mergers

Bottom up model

Origin

Alternatively very massive objects might decay in our galaxy and produce the entire spectrum of high energy cosmic rays

The massive objects would be a relic from early universe with mass

M > 1015 mass of proton

Top Down Model

Origin

These massive objects could be:

Topological defectsSuper heavy particlesPrimordial black holes

Origin

Here we shall focus on the Bottom Up Model where the particles are accelerated at some astrophysical sites

Our astrophysical neighbourhoodDistance to nearest star 1.3 pc (4 light years)

Milky way disk diameter 30 Kpc

Galaxies also arrange themselves in Groups or clusters

These further organize in super clusters, size about 100 Mpc

Beyond this universe is isotropic and homogeneous

The Milky WayThe Milky Way

CFA Survey 1986

Distribution of galaxies in our neighbourhood

2dF Galaxy Redshift Survey

3D location of230 000 galaxies

As we go to distances larger than 100 Mpc we enter the regime of cosmology

z = v/c = H0 d (Hubble Law)

z 1 at distances of order 1 Gpc

As we go to large z (or distance), the Universe looks very different. It has much higher population of exotic objects like

Active Galactic NucleiGamma Ray Bursts

Acceleration Mechanisms

Fermi acceleration

Betatron acceleration: acceleration due to time varying electromagnetic fields

Fermi Acceleration: Basic Idea

Charged particles are accelerated by repeatedly scattering from some astrophysical structures

Ex: Supernova shock waves, magnetic field irregularities

At each scattering particles gain a small amount of energy

Particles are confined to the acceleration site by magnetic field

Shock w

ave

Supernova Explosion

Stars more massive than 3 solar masses end their life in a supernova explosion.

This happens when either C or O is ignited in the core. The ignition is explosive and blows up the entire star.

For more massive stars the core becomes dominated by iron. Explosion occurs due to collapse of the iron core. The core becomes a neutron star or a black hole

Supernova Explosion

Supernova explosions also happen in binary star systems

In this case one of the stars accretes or captures matter from its binary partner, becomes unstable and explodes

For example, the star may be a white dwarf MWD < 1.44 Solar Mass Chandrasekhar limitIf its mass exceeds this limit, it collapses and explodes into a supernova

0.1 sec

0.5 sec

2 hours

months

Supernova Explosion

Brightens by 100 million times

Supernova Explosion, Aftermath

The explosion sends out matter into interstellar space at very high speed, exceeding the sound speed in the medium, leading to a strong shock wave

For sufficiently massive star the core becomes a pulsar (neutron star) or a black hole

Brightest supernova observed SN1006

visible in day time

3 times size of Venus

Intensity comparable to Moon

Remnant of the Supernova explosion seen in China in AD1054 (Crab Nebula)

Expansion:angular size increasing at rate 1.6’’ per 10 years

also observable in day time

Magnetic Fields in Astrophysics

Magnetic fields are associated with almost all astrophysical sites

Our galaxy has a magnetic field of mean strength 3 G

The field is turbulent

Magnetic Fields in Astrophysics

Cosmic rays are confined at the astrophysical sites by magnetic field

They may also scatter on the magnetic field irregularities and gain or loose energy

Magnetic cloud

U

particles may gain energy by scattering on astrophysical structures

Fermi Acceleration: simple example

U

x

y

S: observer

Mass

Fermi Acceleration: simple example

U

v

-v

x

y

S: observer

S’

x’

y’In S’: vi’ = v vf’ = -v

Elastic scattering

In S: vi = v – U vf = - v – U= - vi - 2U net gain in speed

simple example cont’

Gain in energy per scattering:

E = Ef – Ei = (1/2) m (v+U)2 – (1/2) m (v-U)2

= 2mvU = 2mvi U + 2m U2

This principle used in Voyager

particles move at speed close to the speed of light.

Hence we need to make a relativistic calculation at oblique angles

v c (velocity of light)U << c

Frame S: angle of incidence = Ei = EPi = P

Relativistic calculation

U

x

y

S: observer

Relativistic calculation

U

Pix’

S’

x’

y’

“Mirror”

Frame S’: pix’ = (Px + E U/c2) Ei’ = (E + UPx)

Pfx’ = - Pix

’

Ef’ = Ei

’

Frame S: Ef = (Ef

’ – UPfx’)

22 /1

1

cU

-Pix’

Relativistic calculation cont’

Gain in energy per scattering:

2

2

2 2cos2cU

cUv

EEE

i

if

Particles will gain or loose energy depending on the angle of incidence

Fermi AccelerationLets assume that initially N0 charged particles with mean energy E0 per particle are confined in the accelerating region by magnetic field.

they undergo repeated interactions with the magnetic clouds

Fermi Acceleration

2

2 2cos2

cU

cUv

EE

Charged particles are accelerated by repeatedly scattering from some astrophysical structures

U = speed of structure v = speed of particle

per collision

We need average E/E over many collisions

Fermi AccelerationProb of collision v + U cos

2

38

cU

EE

U

Head on collisions are more probable

v

Lets assume that initially N0 charged particles with mean energy E0 per particle are confined in the accelerating region by magnetic field

After one collision E = E0 =1+(8/3) (U/c)2

Let P = probability the particle remains in the site after one collision, depends on the time of escape from the site

After k collisions we have N = N0 Pk particleswith energy E E0 k

N( E) = const Eln P/ln

dN = N(E) dE = const Eln P/ln 1

N(E) dE = const Eln P/ln 1 E

We have obtained a power law, as desired

However it depends on details of the accelerating site such as P, . We see the same in all directions

Also the mean energy gain per collision U2

Second order Fermi acceleration

The process is very slow

The particle might escape before achieving required energy or energy losses might become very significant

It would be nice to have a process where the particles gain energy in each encounter.

In this case

cU

EE

Achieved by Fermi First order mechanism

Reference:

High Energy Astrophysics

by

Malcolm S. Longair

First order Fermi acceleration

Particles accelerated by strong shocks generatedBy supernova explosion

strong shock:shock speed >> upstream sound speed (104 Km/s) (10 Km/s)

US

upstreamdownstream

assume that a flux of high energy particles exist both upstream and downstream

shoc

k


downstreamV2, 2

upstreamV1, 1

Shock

front

US V1=|US|V2=|US|/4

1V1 = 2V2

2/1=(+1)/( 1) = 4 for strong shocks

= 5/3 monoatomic or fully ionized gas

V2 = V1/4

In shock frame

The particle velocities are isotropic both upstream and downstream in their local frames.

High energy particles are repeatedly brought to the shock front where they undergo acceleration at each crossing

3|US|/4 3|US|/4isotropic isotropic


downstream frame

upstream frame

Consider high energy particles crossing the shock from upstream to downstream

The particles hardly notice the shock

Downstream medium approaches the particles at speed

U = 3US/4

3|US|/4 isotropic


The particles undergo repeated scattering on magnetic irregularities and become isotropic in downstream medium

Let’s determine the energy of the particle in the frame in which the downstream particles are isotropic


In downstream frame:

E’ = (E + PxU)U<< c, 1Velocity of particle v cE = Pc Px = (E/c) cosE’ = E + (E/c) U cos

U = 3US/4vc

upstreamdownstream

upstream frame E, Px

coscU

EE

We next average this over from 0 to /2

Rate at which particles approach the shock cos

Number of particles at angle sin d =d cos

Prob. of particle to arrive at shock at angle

P() d = 2 cos dcos

1)(2/

0

dP

cU

cUPd

EE

32cos)(

2/

0

Now the important point is that the situation is exactly identical for a particle crossing the shock from downstream to upstream

3|US|/4 upstreamdownstream

For each crossing:

U = 3US/4


cU

EE

32

cU

EE

34

For each round trip

= E/E0 = 1+ 4U/3c

Prob. for particle to remain at site

P = 1 Pesc

Let N = number density of particlesflux of particles crossing the shock from either direction = Nc/4

in downstream particles are removed at rate Nv2 = NUs/4Fraction of particles lost per cycle = Us/c = Pesc

P = 1 Us/c

ln P = ln (1Us/c) = Us/c

ln = ln (1+4U/3c) = 4U/3c = Us/c

ln P/ln = 1 N(E) dE E2 dE

We get a power law with exponent 2

We get a universal exponent. However we get 2 instead of 2.7

The spectral index might become steeper if we take into account:

Loss of energy

leakage from our galaxy

Leakage from galaxy

Leaky box modell

Steady state

Source spectrumTime of escape from galaxy

N(E) = observed flux

Acceleration up to KNEE (1015 eV)

N(E) = Q(E) x tesc (E)

Observed fluxSource spectrum

tesc(E) E Q(E) 1/E2

0.6 – 0.7 Spectral index = 2.6 – 2.7

Time of escape from galaxy

Numbers

Typical increase in energy in each crossing of the shock wave = 1 %

Acceleration phase lasts about 105 years

Typical energies that can be achieved are

105 GeV/nucleon

Hence heavier nuclei can achieve higher energies

30

P

Fe

Acceleration beyond KNEE

Furthermore beyond the KNEE milky way magnetic fields may not be able to confine protons

However they can confine heavier nuclei, which may also be accelerated by supernova shock waves

Hence the composition becomes heavier beyond the KNEE

Evidence for supernova acceleration

If high energy particles originate at supernova remnants, then we should also observe gamma rays from these sites

Gamma rays are produced by interaction with other particles.

These gamma rays have been seen and partially confirm the model

Do supernovae produce enough energy to account for cosmic ray energy?

K.E. per supernova 1051 ergs

about 3 supernovae per century

release energy at rate 1042 ergs per sec in the milky way

This is enough to power cosmic rays if 15% goes into these particles

• At E < 1018.5 eV (ankle), the cosmic rays are believed to originate inside the milky way

• At E > 1018.5 eV, their origin is probably extragalactic, beyond the milky way

SpectrumSpectrum

KNEE ANKLE

• At E > 1020 eV, it is very difficulty to find an astrophysical site which can accelerate particles to such high energies

Acceleration beyond 1020 eVWe need to find source which is able to

confine particles at such high energies

Let B = magnetic field, L = size of the region Z = charge on particle

Emax = ZBL

M.BoratavThe basic limitation comes from the magnitude of the magnetic field required to confine high energy particles in a given region.

The likely sites includeGamma Ray BurstsActive Galactic Nuclei

Neutron Star

GRB Protons (100 EeV)

Protons (1 ZeV)Active Galaxies

Colliding Galaxies

Active Galactic Nuclei

Active Galactic Nuclei

The core (quasar) contains a massive black hole which may accelerate particles to very high energies The jets may be beaming towards us (Blazar)

A possible acceleration site associated with shocks formed by colliding galaxies

Time varying magnetic fields

• Some objects such as pulsars have very strong magnetic fields

• As the object rotates the magnetic field changes with time

• This can create very large electric field which can accelerate particles very quickly

• However normally these magnetic fields occur in dense regions, where the particles may also loose considerable energy

Pulsar emits electromagnetic radiation in a cone surrounding the magnetic field dipole axis

Pulsar

Chandra Associates Pulsar and Historic Supernova (386 AD) witnessed by Chinese Astronomers

X-ray image

Constellation: Sagittarius

Beyond 10Beyond 102020 eV eV

It is expected that ultra high energy cosmic rays are either protons, nuclei or photons.

However all of these particles loose significant energy while propagating over cosmological distances at E > 1020 eV.

Protons loose energy by collisions with CMBR

p + 2.7K + p + 0

n + +

Beyond 10Beyond 102020 eV eV

Photons (pair production on background photons), Nuclei (photo-disintegration) are also attenuated. Hence either the source of these particles is within 100 Mpc or

spectrum should strongly decay at E>1020 eV

(GZK cutoff, Greisen-Zatsepin-Kuzmin, 1966)

Spectrum (AUGER)

Yamamoto et al 2007

Conclusions

Supernova shocks in our galaxy are the most likely sites for acceleration upto 1018 eV

The acceleration probably occurs through first order Fermi mechanism

Beyond 1018 eV, the cosmic rays probably originate outside our galaxy

Beyond 1020 eV, there are very few sites which can accelerate particles to such high energies

The KNEE

Furthermore we should observe an anisotropy in the arrival directions of cosmic rays.

This is because we have more supernovas in the center of the galaxy

We are located far from the galactic center

The KNEE

The knee, however, cannot be explained easily

If the protons cannot be accelerated beyond the KNEE and/or

cannot be confined by the galactic magnetic fields, then we might expect an exponential decay.

This would later meet a harder spectrum (=2.7) due to heavier nucleus, such a Helium …

The KNEE

Hence we should see more cosmic rays from the galactic center in comparison to the opposite direction.

Such an anisotropy has not been seen

Hence the cause of KNEE is not understood

Documents

Acceleration of Cosmic Rays Pankaj Jain IIT Kanpur