1. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 Diffusive Shock Acceleration of High-Energy Cosmic Rays The origin of the very-highest-energy cosmic rays remains one of the most-important unsolved problems in astrophysics. One of the most-important accelerators is the mechanism of diffusive shock acceleration In this talk, we discuss this mechanism and, in particular, the upper energy limits and the importance of the magnetic field angle Joe Giacalone & Randy Jokipii University of Arizona

2. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 The Importance of the Magnetic-Field Angle Acceleration to high energies: –Parallel Shocks Very slow Efficient –Perpendicular Shocks Much faster Also efficient (we point out in this talk that there is no injection problem) New numerical simulations –Hybrid simulations (self consistent) show efficient acceleration of thermal ions by a perpendicular shock

3. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05

5. Actual Particle Orbits – Including Scattering Decker, 1988

6. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 Diffusive Shock Acceleration Start with the cosmic-ray transport equation advection diffusion drift energy change

7. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 Diffusive Shock Acceleration Solve the cosmic-ray transport equation for the following geometry

8. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 The steady-state solution for, for an infinite system, is given by The downstream distribution is power law with a spectral index that depends only on the shock compression ratio! Kennel et al, 1986

9. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 The observed quite-time cosmic-ray spectrum

10. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 What about the maximum energy attainable? The energy is limited either by geometry or by a finite time. Acceleration takes time. The ideal power law energy spectrum is not created instantly. The maximum energy over a given time interval strongly depends on the shock-normal angle Parallel shocks  slow Perpendicular shocks  fast

11. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 Acceleration Time in Diffusive Shock Acceleration The timescale to accelerate particles to a momentum p (from a much smaller momentum p 0 ) is given by Where

12. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 One may conclude: The acceleration time depends on The rate of change of the maximum energy is much larger for perpendicular shocks. Hence, for any given situation, a perpendicular shock will yield a larger maximum energy than a parallel shock. Most discussions of the maximum energy have been limited to a parallel shock, for no good reason.

13. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 Acceleration Rate as a Function of Shock-Normal Angle: (assumes the billiard-ball approximation) Jokipii, 1987

14. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 Maximum Energy Assumes Sedov solution for SNR blast wave Bohm Diffusion Perpendicular Shock (Hard-sphere scattering)

15. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 What about Injection and the limit of diffusive shock acceleration? An often-invoked injection criterion is This assumes, for no good reason, that there is NO motion normal the average magnetic field –In general, particles move normal to the field, and this is important for the injection problem

16. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 It is useful to examine the limits of theory of Diffusive Shock Acceleration Because the distribution should be nearly isotropic, we require that the diffusive streaming anisotropy be small. is the streaming flux where and

17. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 This leads to a general expression for the validity of diffusive shock acceleration Diffusive shock acceleration is applicable

18. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 Special Cases of the general limit:

19. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 Special Cases of the general limit:

20. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 Special Cases of the general limit:

21. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 The case of hard-sphere scattering Classical-scattering theory gives For most astrophysical applications Thus, using classical-scattering theory, we find

22. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 The case of hard-sphere scattering Classical-scattering theory gives For most astrophysical applications Thus, using classical-scattering theory, we find However, classical-scattering theory is NOT a good approximation for perpendicular transport !

23. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 Test-particle simulations using synthesized magnetic turbulence (Giacalone and Jokipii, ApJ, 1999 + one extra point)

24. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 The case of field-line random walk Thus, for a perpendicular shock, we find → The same as for a parallel shock !

25. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 How does the increased diffusion normal to the field effect the acceleration time? The ratio of the acceleration time to that for Bohm diffusion is given by:

26. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 Test-particle simulations using synthesized magnetic turbulence (Giacalone and Jokipii, ApJ, 1999 + one extra point)

27. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 To test these ideas, we have performed test-particle simulations Particles are followed in the synthesized shock fields The magnetic field is composed of a mean, plus a fluctuating component derived from an assumed power spectrum –Maxwell’s equations are satisfied across the shock Assumed geometry in test-particle simulations

28. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 Test-particle simulations Shock moves through a plasma with a magnetic field derived from a power spectrum (e.g. Decker & Vlahos, 1986) Assumed power spectrum

29. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 More results from test-particle simulations Effect of Turbulence Amplitude Time dependent case

30. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 More results from test-particle simulations

31. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 What is the process by which thermal particles become part of the high- energy population? The process is non- linear and requires a kinetic description The Injection Problem

32. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 The Fields Must be Fully Three Dimensional If the fields contain an ignorable coordinate, then the particles are unphysically forced to remain within a single gyroradius from the field line that it started

33. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 The Hybrid Simulation Self-consistent plasma simulation (kinetic ions, MHD fluid electrons, often assumed to be massless) Used to study the structure of shocks, ion distributions, upstream ions, etc., acting on ion length and time scales Can also be used to study the acceleration of thermal ions to high energies.

34. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 1D Hybrid Simulation of a Parallel Shock Downstream Energy Spectra

35. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 New 2D Hybrid Simulations We have performed large 2D simulations (500 × 4000 ) to investigate the effect of long- wavelength magnetic fluctuations on the acceleration of thermal ions at a shock. “Seed” magnetic fluctuations are imposed on the system Particles are tied to field lines, but move normal to the mean field by following meandering lines of force

36. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 Simulated Magnetic Field B z

37. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 Energetic-Particle Intensity -- Zoomed-in View

38. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 Large vs. Small Domain-Size Comparison

39. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 vs. Shock Comparison

40. “Physics at the End of the Galactic Cosmic-Ray Spectrum” Aspen, CO 4/28/05 Conclusions 1.Shocks provide a natural explanation for most cosmic rays. 2.The maximum energies generally quoted for shock acceleration greatly underestimate the maximum energy, as they apply only to parallel shocks. 3.Considerations of quasi-perpendicular shocks lead to much higher possible energies. 4.Perpendicular shocks are as efficient at accelerating thermal particles as are parallel shocks. There is not an “injection” problem for shocks occurring in a wide- variety of astrophysical plasmas.