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Astroparticle Physics : Fermi’s Theories of Shock Acceleration - II Outline: Pratik Majumdar SINP, Kolkata 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 = • • Post MSc lectures, SINP, December 2012 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 • 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 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 • Post MSc lectures, SINP, December 2012 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/(4/3R3)= 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 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 • 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 • 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 SNR Parameters • Mean ejecta speed : v = (2ESN/Mej)1/2 • Radius swept away : R = (3Mej/4P)1/3 • Sweep time : t0 = R/v • 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