Transcript The Transport of Cosmic Rays

Pre-existing Turbulence,
Magnetic Fields and
Particle Acceleration
at a
Supernova Blast Wave
J. R. Jokipii
University of Arizona
Presented at the meeting: Kinetic Modeling of
Astrophysical Plasmas, Krakow, October 8, 2008
Outline of talk
• Supernova blast waves and cosmic rays.
• Observational evidence for enhanced magnetic
field behind the shock.
• What enhances the magnetic field? Bell
instability?
• Enhanced B from pre-existing turbulence.
• Acceleration to the “knee” in the CR spectrum:
results from a new global model which explicitly
includes the magnetic-field angle show
encouraging agreement with observations.
Galactic cosmic-rays and SNR’s
• The spectrum, up to the “knee” at ~ 3x1015
eV, is probably the result of diffusive shock
acceleration at supernovae blast waves.
• Lagage and Cesarsky (1983) estimated
the maximum energy to be less than 1014
eV/charge
– They assumed Bohm diffusion, a
nominal IS magnetic field and a planar
parallel shock (I will show later that the
geometry is very important).
• However, it has been shown long ago that
a higher maximum energy is attained for a
larger upstream magnetic field or a quasiperpendicular shock (or both).
• Consider, first, the magnetic field
magnitude.
Observational constraints on the magnetic-field magnitude
Berezhko et al., 2004
SN 1006: Bamba et al, 2003
• Berezhko et al. (2004) compared X-ray observations of a
supernova remnant with a model of shock acceleration of
electrons (Ee ~ 100 TeV) including synchrotron losses and
concluded that the observations could only result if the remnant
magnetic field were very strong (B > 100μG)
This analysis only constrains B behind the shock!
What enhances B in the remnant?
• Bell and Lucek (2001) proposed that a cosmic-ray
current upstream of the shock produces a Jcr xB
force on the plasma which drives an instability which
then results in a large magnetic-field amplification.
• Berezhko et al., 2004 used this mechanism to
explain their large field stating that “There is no
alternative process, without ad hoc-assumptions, in
the literature, or a new one, which we could
reasonably imagine, that would amplify the MF in a
collisionless shock without particle acceleration”.
• Nonetheless, Joe Giacalone and I have found such
an alternative process – the interaction of preexisting, large-scale turbulence with the shock.
Motivation for considering upstream turbulence
Consider in situ observations
In the solar wind.
Although energetic particles are often
associated with shocks, The
observations nearly always have
anomalies.
Single-spacecraft observations have
seen these anomalies for decades.
But they were not well understood.
We have suggested that they are the
result of fluctuations or turbulence.
Multiple in situ observations of the
same shock have also shown this.
In situ spacecraft observations of
energetic protons at a propagating shock.
(from Scholer, JGR, 88, 1977, 1983)
Neugebauer and Giacalone (2006),
using multiple, nearby spacecraft,
studied the energetic particles
associated with propagating shock
waves in the solar wind.
They found that for the same shock,
the characteristics of the energetic
particles varied with distance along the
shock face. The coherence scale of
these random variations was the same
as that of ambient, solar-wind
turbulence!
They concluded that the propagation of
the shock through the pre-existing
turbulence produced large variations in
the shock, which affect the acceleration
of particles.
Hybrid Simulations of a perpendicular shock moving
through a turbulent magnetized plasma
Magnetic field
Density of
Energetic Particles
IGPP 2007, Honolulu
Giacalone, 2005
This is similar to what seen in images (from Berezhko and Volk 2005)
The interplanetary density turbulence
spectrum (Goldstein and Sicsoe, 1972).
Note: (± n/n)2 is of order unity. This is a advected
spectrum, f $ wavenumber
The interstellar medium is also turbulent!
Again, the turbulence is of large amplitude:
(± n/n)2 is of order unity.
The interplanetary density turbulence
spectrum (Goldstein and Sicsoe, 1972).
Note: (± n/n)2 is of order unity. This is a advected
spectrum, f $ wavenumber
The interstellar medium is also turbulent!
Again, the turbulence is of large amplitude:
(± n/n)2 is of order unity.
We Suggest that this Large-Scale pre-existing interstellar
turbulence can account for many observations at a SN shock
Tycho supernova
remnant in X-Rays.
• Large scale, upstream interstellar turbulence causes temporal and
spatial fluctuations in the shock. and in the downstream flow.
These are probably present at all interstellar shocks.
• We have addressed a number of the phenomena, and in particular,
the magnetic field, quantitatively (Giacalone & Jokipii, Ap. J., 663,
L41, 2007). We have found large magnetic-field amplification for
strong shocks caused by the upstream turbulence, not cosmic rays.
(See also Sironi and Goodman, 2007)
We used 2-D MHD simulations
of a strong shock moving through
pre-existing turbulence
12L
Density
c
• The figure shows the density in a
numerical simulation of a shock wave
moving into a turbulent MHD plasma
• Approach:
– We solve the MHD equations for a
fluid reflected off of a rigid wall
– Shock moves from right to left
– The upstream medium contains
turbulent density fluctuations
• We impose log-normal statistics, and a
Kolmogorov spectrum
• The fluctuations are continually injected
at the upstream boundary.
• We are currently working on the more-difficult
3-D version.
)
8Lc
4Lc
0
0
2Lc
4Lc
6Lc
Results for a very high Alfven Mach number.
Interpretation:
The enhanced B downstream of a shock moving through density
turbulence (without cosmic-ray excited waves!) is due primarily to
the induced downstream vorticity and is quite robust.
The growth time of the fluctuations in the downstream flow may be
estimated to be
 ¼ U1/Lc
If rsh is the ratio of densities or flow velocities at the shock,
we have for the spatial scale of magnetic-field amplification:
 ¼ U1/rsh  ¼ Lc/(rsh Us}
For typical parameters, this gives
 ¼ parsec.
Note that Ellison and Blondin (2001) assume r > 4 (due to efficient
particle acceleration). If this is the case, the distance above may be
shorter.
Particle Acceleration
• Our turbulent amplification does not increase B
upstream of the shock, as does the Bell
mechanism.
• Hence, since particle acceleration depends
primarily on the upstream field, we cannot reach
the knee at a quasi-parallel or Bohm shock.
• However, it is well-established that a planar
perpendicular shock can get to the knee using the
approach of Lagage and Cesarsky.
The magnetic-field shock-normal angle
• A SNR blast waves moves into a B
with a preferred direction
– The angle between B and shock
normal varies from 90 deg to 0 deg.
• The physics of acceleration at
parallel and perpendicular shocks is
different
Parallel shocks  slow
Perpendicular shocks  fast
(K┴ < K║)
• for a given time interval or size of shock, a perpendicular shock
will yield a larger maximum energy than a parallel shock.
For a modified Planar Sedov Blast wave, we have the maximum energies:
Parallel shock
Perpendicular shock
The acceleration time depends on  and ? << k
The rate of change of the maximum energy is much larger for quasi-perpendicular
shocks.
Hence, for any given situation, a perpendicular shock will yield a larger maximum
energy than a parallel shock.
So, reaching the knee is not a problem.
But the shock is not planar.
We should consider a spherical shock, with a varying magnetic-field angle.
B?
or
Relation to the Observed Morphology
of Supernova Remnants
• Rothenflug, et al, in 2004, pointed out that
observations rule out a “barrel” shape and
suggested that that consequently ruled out
acceleration at the perpendicular shock.
• However, it is by no means clear that
acceleration at the perpendicular shock leads to
a barrel shape. Need global modeling.
• We have looked into this problem (Giacalone,
Jokipii, Kota, Bobik, to be submitted after this
meeting.)
The Parker Transport Equation:
)
Diffusion
)
)
Convection w. plasma
Grad & Curvature Drift
) Energy change – shock accel
) Source
Solve Parker’s equation in
this situation including the
shock, with a variable
shock-normal angle,
This gives diffusive shock
acceleration in this geometry
We have solved Parker’s transport equation for a spherical,
modified Sedov blast wave propagating in a uniform magnetic field. We
used two completely independent and different codes—finite-different
ADI and stochastic integration and found the same results.
For reasons of computational economy, we started the particles at
the shock at an energy of 3 x 1012 eV, with injection rate constant with
shock radius and angle. Reducing the injection where the shock was
quasi-perpendicular did not change the results qualitatively.
We considered various values of the ratio of perpendicular to parallel
diffusion ranging from .01 to 1, with parallel mean free paths at the
highest energy in the range from 2 to 10, in a nominal 5 x 10-6 gauss
interstellar magnetic field. We considered a range of downstream
magnetic field values to account for possible downstream magnetic-field
amplification.
The following results are quite robust. We find that acceleration to
the knee is possible. Moreover, and unexpectedly, the accelerated
particles, although being accelerated at the equator, migrate to the polar
regions, where the shock is quasi-parallel.
Our interpretation of the simulations is that the accelerated particles tend to
stay on the original field line, which stays behind as the shock moves out. The
observed radiation comes from these lower-energy particles.
Most of the results presented here are from the stochastic integration
code, which has a wider dynamical range and allows particles to be tracked
Histogram of the starting shock-normal angle for two final energies.
Energy spectrum summed over all particles.
Conclusions
• Large-scale, broadband upstream turbulence plays a significant role in
astrophysical shocks and the transport/acceleration of energetic particles.
• Pre-existing fluid density fluctuations affect significantly the downstream
magnetic field. In a very strong shock, large field amplification may occur.
This may help explain observations of magnetic-field enhancements at
supernova blast waves. This is a robust result.
• Acceleration of cosmic rays at perpendicular shocks is a possibility. This
would enable acceleration to the knee without upstream magnetic field
amplification. The Rothenflug observational constraints can be satisfied,
with significant acceleration at the perpendicular shock.
• This is a possible alternative scenario to the Bell upstream magnetic-field
amplification scenario.