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The Acceleration of Charged Particles by Perpendicular Shocks Joe Giacalone University of Arizona With thanks to my colleagues Randy Jokipii, Jozsef Kota, Fan Guo, and Federico Fraschetti Accelerating Cosmic Ray Comprehension, Princeton, April 14 2015 The importance of the magnetic-field angle for particle acceleration Individual proton trajectories near shocks moving through turbulent magnetic fields (Decker, 1988; see also Decker & Vlahos, 1986) Quasi-parallel shock Quasi-perpendicular shock Increasing Particle Energy Slower Acceleration case More rapid acceleration Position relative to the shock Decker, 1988 Collective behavior • Quantitative calculations for the distribution of many charged particles can be performed by assuming a quasiisotropic distribution and solving the cosmic-ray transport equation (Parker, 1965). – One gets the well-known results from diffusive shock acceleration (power law with spectral index depending on plasma density compression) • Acceleration time / maximum energy: – τacc = κnn/U1 – Because perp. shocks accelerate particles more rapidly. For a given shock age, higher maximum energies occur at perp. shocks. Results from test-particle simulations similar to those of Decker & Vlahos (1986) Time dependent calculation Fields: B(x)=Bb(x)+dB(x) Bb(x) is the background field which includes the jump at the shock Giacalone, 2005a dB(x) is the fluctuating part (determined from an assumed power spectrum) including jump in transverse components at the shock. The “injection problem” • An often-invoked injection (lower limit) criterion is • Note that as θBn approaches 90o, the injection speed becomes very large. • This assumes, for no good reason, that there is NO motion normal the average magnetic field • This expression has led to a widely held misconception that perpendicular shocks are inefficient accelerators of particles Observations at nearly perpendicular shocks show large enhancements of ions with speeds well below the injection speed. There does not seem to be an observed “injection problem” Termination shock (nearly perpendicular) Courtesy Rob Decker ACE observations of an interplanetary shock θBn = 89o The “injection problem” (cont.) • Considering the limits of applicability of diffusive shock acceleration leads to a modification of the injection criterion where KA = (1/3)wrG Giacalone & Jokipii, 1999 • Which, for a perp. shock this becomes Limit for a perpendicular shock • In many realistic cases there is little or no dependence of the injection speed with shock-normal angle Which was derived previously by Jokipii, 1991 The same as for a parallel shock Test-particle simulations using synthesized magnetic turbulence (Giacalone and Jokipii, ApJ, 1999 + one extra point) First 3D hybrid simulations of perpendicular shocks to study injection/acceleration of thermal particles (Giacalone & Ellison, 2000) Box: 150 x 10 x 10 c/ωp No significant acceleration Effect of Simulation Dimensions Hybrid simulations of Perpendicular Shocks For perpendicular shock simulations, it is important to : Rigid wall (a) include large-scale magnetic fluctuations – large simulation boxes are required (b) perform them in at least 2 spatial dimensions to get particle motion normal to the mean magnetic field through large-scale field-line meandering -despite the fact that particles are stuck on field lines* (unless ad-hoc scattering is used) * In 1- and 2-D electromagnetic fields, even very dynamic and turbulent ones, charged particles are un-physically tied to magnetic lines of force. Thus, 3D is ultimately needed to lift this un-physical restriction on particle motion, but is computationally very difficult MUST include large-scale turbulence Downstream energy spectrum – two different simulation box sizes are shown 2D hybrid simulations of a perpendicular shock, including largescale turbulence reveal that highenergy tails are formed from an initially Maxwellian distribution. Even thermal plasma is accelerated efficiently Giacalone, 2005 Magnetic field Density of Energetic Particles Giacalone, 2005 Back to test-particle simulations in synthesized fields near a shock Assumed geometry Effect of excluding large-scale fluctuations Downstream distribution function (steady state) parallel shock P(k) k kCrG,inj kCrG,inj = 1 = 0.1 = 0.001 = 0.0001 Non-uniform injection at perp. shock (on avg.) arising from large-scale turbulence • Locations of particles (projected onto the plane normal to the unit shock normal) that eventually become energetic particles, when they first encountered the shock, Giacalone & Jokipii, ApJ, 2009 The local shock-normal angle at the release point (initial local θBn) kCrG,inj = 0.0001 (full • Black, all particles • Red, all particles that eventually ended up with an energy exceeding 100 times the initial energy • Blue dashed line, the θBn associated with the winj=U1secθBn criterion spectrum is used) Spherical shock moving through pre-existing turbulence Energy release Rsh Lturb Lturb = the turbulence coherence scale (~ 4 pc for ISM) early times (Rsh < Lturb) Small variability in local shocknormal angle along shock Later times (Rsh > Lturb) larger variability along shock Results from test-particle orbit integrations in synthesized magnetic turbulence advecting into a PLANAR shock Downstream spectra At later times, the perp. part of the shock can accelerate particles from low energies because of affect of fieldline meandering tsh = 1000 years Vsh = 109 cm/s Rsh = 10 parsec Lc = 4 parsec σ2 = B02 B0=3µG n2/n1=4 Vinj = Vsh tmax=5 yrs. The dashed line is the prediction of DSA in the strongshock limit. The blue histogram is another realization Results from test-particle orbit integrations in synthesized magnetic turbulence advecting into a PLANAR shock Downstream spectra At early times, the shock (in the perp. region) can still accelerate particles, but there may be considerable time variability tsh = 100 years Vsh = 109 cm/s Rsh = 1 parsec Lc = 4 parsec σ2 = B02 B0=3µG n2/n1=4 Vinj = Vsh tmax=5 yrs. Different colors represent different realizations • • Preliminary results from a testparticle orbit integration using synthesized magnetic turbulence, for a spherical shock The shock is assumed to move with constant speed of 104 km/s (not Sedov) • • Particles followed for 5 years. Plotted (2 lower panels) is the energy vs. time for each particle after it’s 5-year trajectory • The middle panel assumed the particles originate at the blast wave, at the blast wave’s equator, at a single azimuthal angle (0o) • The bottom panel is the same except that particles are released in azimuth over the interval -5o < φ < 5o θBn φ=0 Energy (keV) -5o < φ = 5o Energy (keV) Years From Start of Blast Wave Conclusions • Particles may be efficiently accelerated from low energies at perpendicular shock waves. – There is NO INJECTION PROBLEM! • “Patchiness” in the particle intensity along the shock at low energies results from effects due to the variation in the local shock-normal angle as the shock moves through large-scale magnetic turbulence. • However, even if particles are released where the local shock normal angle is 90o, it may be accelerated to high energy. • (and, as known previously) perpendicular shocks are faster particle accelerators than parallel shocks Extra slides Quantitative predictions of Diffusive Shock Acceleration can be obtained by solving the cosmic-ray transport equation (Parker, 1965) (which assumes isotropy) advection diffusion drift energy change When applied to a simple, planar shock-like discontinuity, the resulting distribution has the form Thus, strong shocks produce flatter energy spectra than weaker shocks. Acceleration Time in Diffusive Shock Acceleration • The time to accelerate particles to an energy E from a smaller energy E0 is given by where Drury, 1983 Forman & Drury, 1985 Is the diffusion coefficient normal to the shock front. θBn is the angle between the shock normal and magnetic field. Because perp. shocks accelerate particles more rapidly. For a given shock age, higher maximum energies occur at perp. shocks. 2D hybrid simulations of a perpendicular shock, including large-scale turbulence reveal that high-energy tails are formed from an initially Maxwellian distribution. Even thermal plasma is accelerated efficiently Magnetic field Density of Energetic Particles Downstream energy spectrum – two different simulation box sizes are shown Giacalone, 2005 Test-particle simulations in synthesized fields near a shock • We integrate the equations of motion for an ensemble of particles in a kinematically prescribed field • The upstream magnetic field is a combination of a background, average magnetic field that makes an angle, θBn relative to the shock normal, and a random component given by a discrete sum of individual waves Assumed geometry • Satisfies Maxwell’s equations • The amplitudes A(kn) are determined from a power spectrum Assumed power spectrum Typical particle trajectories Typical fields profile Downstream field (including fluctuating component) is related to upstream field through characteristics solution to the induction equation and assumed plasma flow velocity quasi-parallel shock quasi-perp. shock How does the increased diffusion normal to the field effect the acceleration time (and hence, the maximum energy)? • The ratio of the acceleration time to that for Bohm diffusion is given by: • This may exceed unity at high energies (depending on the parameters of the turbulence). Acceleration at a perpendicular shock can be much faster than at a parallel shock with Bohm diffusion.