Astroparticle Physics : Fermi’s Theories of
Shock Acceleration - II
Outline:
Pratik Majumdar
SINP, Kolkata
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Shock Acceleration
Fermi’s theory of 2nd and 1st
order acceleration
Application to simple cases
Post MSc lectures, SINP, December 2012
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
Acceleration of
Cosmic Rays
No. of particles
Man-made accelerators
Energy
Post MSc lectures, SINP, December 2012
Electromotive Acceleration
Post MSc lectures, SINP, December 2012
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)
Random
B-Field
SNR
Max. Energy about 1015 eV
Efficiency ~ 10%, needed for
CR from SNR
No. of particles
Power law spectrum
Emax
Energy
Predicts a E-2.0
spectrum
Post MSc lectures, SINP, December 2012
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
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
Shock Conditions Solutions
Solve for shock in perfect gas.
• Enthalpy =
•
•
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Shock Conditions Contd….
For monatomic gas : show that
=
4
Heating of gas takes place
Post MSc lectures, SINP, December 2012
Post MSc lectures, SINP, December 2012
Post MSc lectures, SINP, December 2012
Acceleration theory, Contd…
•
Probability of escape : Pesc after k
encounters, so prob of remaining in the
source : (1 – Pesc )k
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So, no. of encounters needed to reach E
Post MSc lectures, SINP, December 2012
Acceleration Theory Contd…
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No. of particles accelerated to energies >
E after k interactions :
Post MSc lectures, SINP, December 2012
Post MSc lectures, SINP, December 2012
What did we learn ???
•
•
Now
frpm
Pesc = Tcycle/Tesc
acceleration /escape : prob per encounter of escape
acceleration region
After acceleration for time t,
•
n
•
Higher energy particles take longer time
to accelerate
max
= t/Tcycle, So, E ≤ E0 (1+∈)t/Tcycle
Let’s apply these to some specific
astrophysical cases……
Post MSc lectures, SINP, December 2012
Post MSc lectures, SINP, December 2012
Post MSc lectures, SINP, December 2012
Post MSc lectures, SINP, December 2012
Post MSc lectures, SINP, December 2012
Post MSc lectures, SINP, December 2012
Post MSc lectures, SINP, December 2012
Post MSc lectures, SINP, December 2012
Fermi 1st Order Acceleration
Post MSc lectures, SINP, December 2012
Fermi 1st Order Acceleration
Post MSc lectures, SINP, December 2012
Post MSc lectures, SINP, December 2012
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
•
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Upstream
Up -> Down and vice versa : increase in energy, increment of same order
Post MSc lectures, SINP, December 2012
An example : SNR explosion
• Frequency of SuperNovae explosions:
f = 1 SN / (30 - 70) 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 1024 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…)
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]
Backup Slides
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 + E/E = 1 + 4V/3c
P = 1 – U/c ……  N(E)dE ~ E-2dE
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Summer Lectures, DESY, August 25 Berlin
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
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Acceleration rate is dE/dt = xE/Tcycle
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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
Rate of Acceleration
Post MSc lectures, SINP, December 2012
SNR Parameters
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Mean ejecta speed : v = (2ESN/Mej)1/2
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Radius swept away : R = (3Mej/4P)1/3
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Sweep time : t0 = R/v
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ISM density :  = 1.4mpnh
Post MSc lectures, SINP, December 2012
Post MSc lectures, SINP, December 2012
Post MSc lectures, SINP, December 2012
How does the spectrum look like ?
Post MSc lectures, SINP, December 2012