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Post MSc lectures, SINP, December 2012
Pratik MajumdarSINP, Kolkata
Outline: Shock Acceleration Fermi’s theory of 2nd and
1st order acceleration Application to simple cases
Astroparticle Physics : Fermi’s Theories of Shock Acceleration - II
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
Man-made accelerators
No.
of
part
icle
s
Energy
Electromotive Acceleration
Post MSc lectures, SINP, December 2012
Post MSc lectures, SINP, December 2012
Shock acceleration mechanism(by Enrico Fermi)
Predicts a E-2.0 spectrum
Particles (electrons and hadrons) get scattered many times in shock front and gain energy in each cycle (TeV energies several 100 years)
No
. of p
art
icle
s
Energy
Power law spectrum
Max. Energy about 1015 eVEfficiency ~ 10%, needed forCR from SNR
Emax
Random B-Field
SNR
Post MSc lectures, SINP, December 2012
Fermi Acceleration
Stochastic MechanismCharged 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
fronts1st order acceleration
Shocks
Post MSc lectures, SINP, December 2012
Shock wave propagating through aStationary gas at supersonic velocity U
Flow of gas in a reference frame In which the shock front is stationary
Energy flux througha surface normal to v
Momentum flux through shock wave
Shock Conditions Solutions
• Solve for shock in perfect gas. • Enthalpy =
•
Post MSc lectures, SINP, December 2012
Shock Conditions Contd….
Post MSc lectures, SINP, December 2012
For monatomic gas : show that ¿ 4
Heating of gas takes place
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
• So, no. of encounters needed to reach E
Post MSc lectures, SINP, December 2012
Acceleration Theory Contd…
• 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 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
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
Post MSc lectures, SINP, December 2012
Post MSc lectures, SINP, December 2012
Upstream
Up -> Down and vice versa : increase in energy, increment of same order
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 10-24 g/cm3):SN=M/Volume=M/(4/3R3)= 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
Summer Lectures, DESY, August 25 Berlin
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
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
Rate of Acceleration
Post MSc lectures, SINP, December 2012
Post MSc lectures, SINP, December 2012
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
Post MSc lectures, SINP, December 2012
How does the spectrum look like ?
Post MSc lectures, SINP, December 2012