1. Astroparticle Physics : Fermi’s Theories of Shock Acceleration - II
Outline:
Shock Acceleration
Fermi’s theory of 2nd and 1st order acceleration
Application to simple cases
Pratik Majumdar
SINP, Kolkata
Parlero’ telescopio MAGIC come esempio di “ground based experiment” per la gamma-astronomia.
Da quando fu osservata la prima sorgente nel 1989 (wipple) i Ttelescopi cherenkov hanno raffinato tecnologie e metodologie. Ora gli IACT sono un maturo strumento di misura e osservazione per l’astronomia gamma delle alte energie
Siamo giunti alla realizzazione di telescopi cherenkov di seconda generazione. Magic tra questi sara’ quello con la piu’ bassa soglia in energia
Esistono altri progetti di esperimento “ground based” non imaging cherenkov telescopes come solar array (celeste) o rivelatori di particelle in alta quota (ARGO) ma, almenno finora, non hanno dimostrato di avere un ottimale strategia di reiezione del fondo.
Post MSc lectures, SINP, December 2012
M. Mariotti Padova

2. Reading Materials Longair : High Energy Astrophysics
T. Stanev : High Energy Cosmic Rays
T. Gaisser : Particle Physics and Cosmic rays
Many review articles on the subject
Post MSc lectures, SINP, December 2012
M. Mariotti Padova

3. Acceleration of Cosmic Rays Man-made accelerators Energy
No. of particles
Energy
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4. Electromotive Acceleration
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5. Shock acceleration mechanism (by Enrico Fermi)
Particles (electrons and hadrons) get scattered many times in shock front and gain energy in each cycle (TeV energies  several 100 years)
Power law spectrum
SNR
Random B-Field
Emax
No. of particles
Max. Energy about 1015 eV
Efficiency ~ 10%, needed for
CR from SNR
Energy
Predicts a E spectrum
Post MSc lectures, SINP, December 2012
M. Mariotti Padova

6. Fermi Acceleration Stochastic Mechanism
Charged particles collide with clouds in ISM and
are reflected from irregularities
in galactic magnetic field
2nd order
Charged particles can be accelerated to high energies in astrophysical shock fronts
1st order acceleration
Post MSc lectures, SINP, December 2012
M. Mariotti Padova

7. Shocks Shock wave propagating through a
Stationary gas at supersonic velocity U
Flow of gas in a reference frame
In which the shock front is stationary
Energy flux through
a surface normal to v
Momentum flux through shock wave
Post MSc lectures, SINP, December 2012

8. Shock Conditions Solutions
Solve for shock in perfect gas.
Enthalpy =
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9. Shock Conditions Contd….
For monatomic gas : show that = 4
Heating of gas takes place
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12. Acceleration theory, Contd…
Probability of escape : Pesc after k encounters, so prob of remaining in the source : (1 – Pesc )k
So, no. of encounters needed to reach E
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13. Acceleration Theory Contd…
No. of particles accelerated to energies > E after k interactions :
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15. What did we learn ??? Now Pesc = Tcycle/Tesc
acceleration /escape : prob per encounter of escape frpm acceleration region
After acceleration for time t,
n max = t/Tcycle, So, E ≤ E0 (1+∈)t/Tcycle
Higher energy particles take longer time to accelerate
Let’s apply these to some specific astrophysical cases……
Post MSc lectures, SINP, December 2012

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23. Fermi 1st Order Acceleration
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24. Fermi 1st Order Acceleration
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26. Basic Phenomenology of Acceleration
Let us consider strong shock : SNR exploding into a medium.
HE particles front and behind the shock
Shock velocity << velocity of particles
Particles get scattered : vel. Distribution become isotropic on either side of shock
Post MSc lectures, SINP, December 2012

27. Upstream
Up -> Down and vice versa : increase in energy, increment of same order
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28. An example : SNR explosion
Frequency of SuperNovae explosions:
f = 1 SN / ( ) yr
Typical Kinetic energy of the ejected mass:
K ~ 1051 erg
Typical mass:
M = 10 solar masses (20  1033 g)
Power emitted in the form of kinetic energy:
W = K  f = 1051 erg /(30  3  107 s) ~1042 erg/s
Speed of shock front:
V = (2K/M)1/2 ~ 3  108 cm/s
One can roughly assume that the shock front gets “extinguished” when the mass of the ejecta reach a density equal to the average interstellar density (IG ~ 1 p/cm3 = 1.6 10-24 g/cm3):
SN=M/Volume=M/(4/3R3)= IG Volume R ~ 1.4 1019 cm ~ 5 pc
How long does it take to extinguish the accelerator (time of free expansion of the ejecta)?
Tacc = R/V ~ 1000 years (NB these are just “typical” order of magnitude numbers…)

29. Maximum energy from shock acceleration
The energy gain per collision (cfr. Exercise):
E = 4/3 V/c E =   E with ~10-2
If we know the typical time in between two consecutive collisions Tcycle, the maximum number of possible collision is on average:
Ncollisions = Tacc/Tcycle with Tacc as calculated before
And the maximum reachable energy is:
Emax = E  Ncollisions =   E  Tacc/Tcycle
Tcycle (or also the shock front crossing-time) depends on the shock velocity V and the mean free path for magnetic scattering of the particles s
Tcycle = s / V
On the other hand, acceleration can only continue if the particle if confined, that means that s < gyroradius (E/ZeB, Ze charge of the particle, B magnetic field ~ 3 G), which leads to:
Emax ~ 4/3 ZeB/c V2 Tacc ~ 300 Z [TeV]

30. Backup Slides

31. Energy Spectrum : Simplified Case
E = bkE0, average energy of particles after k collisions
N = PkN0, Pk is prob. that the particle remains within the acceleration region after k collisions
 N/No = (E/E0)lnP/lnb
N(E)dE = const. E -1+ lnP/lnb dE
b = 1 + DE/E = 1 + 4V/3c
P = 1 – U/c ……  N(E)dE ~ E-2dE
Summer Lectures, DESY, August 25 Berlin
M. Mariotti Padova

32. Rate of acceleration
For a large plane shock, rate of encounters is given by projection of an isotropic cosmic ray flux on plane shock front. Can be shown : cρCR/4
Acceleration rate is dE/dt = xE/Tcycle
It can be shown :
Tcycle = 4/c (k1/u1 + k2/u2)
u1 = fluid velocity (-ve) relative to shock front, for downstream region : average the residence time of those particles which do not escape.
Post MSc lectures, SINP, December 2012

33. Rate of Acceleration
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34. SNR Parameters Mean ejecta speed : v = (2ESN/Mej)1/2
Radius swept away : R = (3Mej/4Pr)1/3
Sweep time : t0 = R/v
ISM density : r = 1.4mpnh
Post MSc lectures, SINP, December 2012
M. Mariotti Padova

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37. How does the spectrum look like ?
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