diffusive shock acceleration of high-energy cosmic rays

“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 JokipiiUniversity of Arizona


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


Actual Particle Orbits – Including Scattering

Decker, 1988


Diffusive Shock Acceleration

• Start with the cosmic-ray transport equation

advection diffusion drift energy change


Diffusive Shock Acceleration

• Solve the cosmic-ray transport equation for the following geometry


• 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


The observed quite-time cosmic-ray spectrum


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


Acceleration Time in Diffusive Shock Acceleration

• The timescale to accelerate particles to a momentum p (from a much smaller momentum p0) is given by

Where


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.


Acceleration Rate as a Function of Shock-Normal Angle:(assumes the billiard-ball approximation)

Jokipii, 1987


Maximum Energy Assumes Sedov solution for SNR blast wave

Bohm Diffusion

Perpendicular Shock(Hard-sphere scattering)


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


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


This leads to a general expression for the validity of diffusive shock acceleration

Diffusive shock acceleration is applicable


Special Cases of the general limit:


The case of hard-sphere scattering

Classical-scattering theory gives

For most astrophysical applications

Thus, using classical-scattering theory, we find


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 !


Test-particle simulations using synthesized magnetic turbulence (Giacalone and Jokipii, ApJ, 1999 + one extra point)


The case of field-line random walk

Thus, for a perpendicular shock, we find

→ The same as for a parallel shock !


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:


Test-particle simulations using synthesized magnetic turbulence (Giacalone and Jokipii, ApJ, 1999 + one extra point)


• 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


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


More results from test-particle simulations

Effect of Turbulence Amplitude Time dependent case


More results from test-particle simulations


• 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


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


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.


1D Hybrid Simulation of a Parallel Shock

Downstream Energy Spectra


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


Simulated Magnetic Field Bz


Energetic-Particle Intensity --

Zoomed-in View


Large vs. Small Domain-Size Comparison


vs. Shock Comparison


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.

diffusive shock acceleration of high-energy cosmic rays

Documents

function of shock

larger maximum energy

maximum energy attainable

shock compression ratio

acceleration rate

upper energy limits

general limit

highestenergy cosmic