1. Acceleration of Cosmic Rays
Pankaj Jain
IIT Kanpur

2. spectrum Flux: number ------------------- m2 sr s GeV Knee at 1015 eV
The spectrum steepens after the knee,
Ankle at eV,
perhaps an indication of a change over from galactic to extragalactic origin

3. Spectrum
KNEE
ANKLE

4. Spectral Index  Flux 1/E = 2.7 1015 eV<E<109 eV
At higher energies spectrum again becomes steeper ( > 4)

5. 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

6. 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

7. Origin These massive objects could be: Topological defects
Super heavy particles
Primordial black holes

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

9. Our astrophysical neighbourhood
Distance 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

10. The Milky Way

11. Distribution of galaxies in our neighbourhood
CFA Survey 1986

12. 2dF Galaxy Redshift Survey
3D location of
galaxies

13. 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 Nuclei
Gamma Ray Bursts

14. Acceleration Mechanisms
Fermi acceleration
Betatron acceleration: acceleration due to time varying electromagnetic fields

15. 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 wave

16. 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

17. 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 limit
If its mass exceeds this limit, it collapses and explodes into a supernova

18. Supernova Explosion 0.1 sec 0.5 sec 2 hours
Brightens by 100 million times
months

19. 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

20. Brightest supernova observed SN1006
visible in day time
3 times size of Venus
Intensity comparable to Moon

21. Remnant of the Supernova explosion seen in China in AD1054 (Crab Nebula)
also observable in day time
Expansion:
angular size increasing at rate 1.6’’ per 10 years

24. 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

25. 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

26. particles may gain energy by scattering on astrophysical structures
Magnetic
cloud

27. Fermi Acceleration: simple example
Mass   x y
S: observer U

28. Fermi Acceleration: simple exampley’
In S’: vi’ = v
vf’ = -v v
Elastic scattering -v x y
S: observer U
In S: vi = v – U
vf = - v – U= - vi - 2U
 net gain in speed

29. 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

30. This principle used in Voyager

31. 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

32. Relativistic calculation
Frame S:
angle of incidence = 
Ei = E
Pi = P U  x y
S: observer

33. Relativistic calculation
Frame S’:
pix’ = (Px + E U/c2)
Ei’ = (E + UPx)
Pfx’ = - Pix’
Ef’ = Ei’ S’ x’ y’
Pix’
“Mirror”
-Pix’ U
Frame S:
Ef = (Ef’ – UPfx’)

34. Relativistic calculation cont’
Gain in energy per scattering:
Particles will gain or loose energy depending on the angle of incidence

35. Fermi Acceleration
Lets 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

36. Fermi Acceleration per collision
Charged particles are accelerated by repeatedly scattering from some astrophysical structures
per collision
U = speed of structure
v = speed of particle
We need average E/E over many collisions

37. Fermi Acceleration v Head on collisions are more probable U
Prob of collision  v + U cos v
Head on collisions are more probable  U

38. 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 = E =1+(8/3) (U/c)2
Let P = probability the particle remains at the site after one collision, depends on the time of escape from the site
After k collisions we have N = N0 Pk particles
with energy E  E0 k
N( E) = const Eln P/ln 
dN = N(E) dE = const Eln P/ln  1

39. 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

40. Achieved by Fermi First order mechanism
It would be nice to have a process where the particles gain energy in each encounter.
In this case
Achieved by Fermi First order mechanism

41. Reference:
High Energy Astrophysics by
Malcolm S. Longair

42. First order Fermi acceleration
Particles accelerated by strong shocks generated
By supernova explosion
strong shock:
shock speed >> upstream sound speed
(104 Km/s) (10 Km/s)
upstream
downstream
shock US
assume that a flux of high energy particles exist both upstream and downstream

43. First order Fermi acceleration
Shock front
downstream
V2, 2
upstream
V1, 1
In shock frame US
V2=|US|/4
V1=|US|
1V1 = 2V2
2/1=(+1)/( 1) = 4 for strong shocks
 = 5/3 monoatomic or fully ionized gas
V2 = V1/4

44. First order Fermi acceleration
downstream frame
3|US|/4
isotropic
upstream 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

45. First order Fermi acceleration
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

46. First order Fermi acceleration
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

47. Velocity of particle v  c E = Pc Px = (E/c) cos
U = 3US/4
upstream frame
E, Px
vc
downstream
upstream
In downstream frame:
E’ = (E + PxU)
U<< c,   1
Velocity of particle v  c
E = Pc
Px = (E/c) cos
E’ = E + (E/c) U cos

48. 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

49. Now the important point is that the situation is exactly identical for a particle crossing the shock from downstream to upstream
3|US|/4
downstream
upstream

50. First order Fermi acceleration
For each crossing:
U = 3US/4
For each round trip
 = E/E0 = 1+ 4U/3c

51. Prob. for particle to remain at site
P = 1  Pesc
Let N = number density of particles
flux of particles crossing the shock from either direction = Nc/4
in downstream particles are removed at rate Nv2 = NUs/4
Fraction of particles lost per cycle = Us/c = Pesc
P = 1  Us/c

52. ln P = ln (1Us/c) =  Us/c
ln  = ln (1+4U/3c) = 4U/3c = Us/c
ln P/ln  = 1
N(E) dE  E2 dE
We get a power law with exponent 2
We get a universal exponent. However we get 2 instead of 2.7

53. The spectral index might become steeper if we take into account:
Loss of energy
leakage from our galaxy

54. Leakage from galaxy Leaky box modell N(E) = observed flux Steady state
Time of escape from galaxy
Source spectrum
N(E) = observed flux
Steady state

55. Acceleration up to KNEE (1015 eV)
N(E) = Q(E) x tesc (E)
Time of escape from galaxy
Observed flux
Source spectrum
tesc(E)  E Q(E)  1/E2
 0.6 – 0.7
 Spectral index = 2.6 – 2.7

56. 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

57. 30 P Fe

58. 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

59. 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

60. 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

61. At E < 1018.5 eV (ankle), the cosmic rays are believed to originate inside the milky way
At E > eV, their origin is probably extragalactic, beyond the milky way

62. Spectrum
KNEE
ANKLE

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

64. Acceleration beyond 1020 eV
We 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

65. The likely sites include Gamma Ray Bursts Active Galactic Nuclei
The basic limitation comes from the magnitude of the magnetic field required to confine high energy particles in a given region.
The likely sites include
Gamma Ray Bursts
Active Galactic Nuclei
Colliding galaxies
Neutron Star
M.Boratav
GRB
Protons (100 EeV)
Protons (1 ZeV)
Active Galaxies
Colliding Galaxies

66. Active Galactic Nuclei

67. 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)

68. A possible acceleration site associated with shocks formed by colliding galaxies

69. 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

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

71. Pulsar

72. Chandra Associates Pulsar and Historic Supernova (386 AD) witnessed by Chinese Astronomers
Constellation: Sagittarius
X-ray image

73. Beyond 1020 eV p +  2.7K  +  p + 0 n + +
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 + +

74. Beyond 1020 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)

75. Spectrum (AUGER)
Yamamoto et al 2007

76. 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

77. 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

78. 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 …

79. 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