Download Particle Acceleration at Quasi

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.