Nonlinear Model

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Non-linear Model of Diffusive Shock Acceleration
of Cosmic Rays in the Presence of Self-Consistent
Generation of Stochastic Magnetic Fields PhD
Thesis Research Plan PhD candidate Andrey
Vladimirov Research advisor Prof. Donald C.
Ellison, NCSU Advisory committee Profs. Steven
Reynolds, PY, NCSU James Selgrade, MA,
NCSU Albert Young, PY, NCSU this work is
done in collaboration with Prof. Andrei Bykov,
Ioffe Physical-Technical Institute, St.
Petersburg, Russia North Carolina State
University January 16, 2008
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Plan of Presentation - Background 1) Cosmic
Rays 2) Diffusive Shock Acceleration 3)
Magnetic Turbulence in Shocks - What we do 1)
Monte Carlo Simulation 2) Magnetic Field
Amplification Module - Completed and Planned
Work - Conclusion
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Cosmic Rays (CR) extremely energetic (109-1019
eV) charged particles protons (86),
alpha-particles (13), heavier nuclei and
electrons. CR flux detected at Earth is very
isotropic (lt 510-4 anisotropy) can't see the
source.
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Spectrum of CR energies detected at Earth is a
superposition of spectra of all sources, strongly
modified by propagation through the
Galaxy. Source spectrum is unknown, and our work
is intended to model it. Picture credit
http//astroparticle.uchicago.edu/
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Cas A in radio by NRAO/AUI
RXJ1713.7-3946 in gamma rays by H.E.S.S.
SN 1006 in x-ray by Chandra
Although CRs can't be detected coming from a
source, information about them can be inferred
from broadband electromagnetic radiation
observations (radio, X-ray, gamma ray ranges
produced by synchrotron, bremsstrahlung, inverse
Compton, pion decay emission). The above images
are supernova remnants (SNRs).
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Fermi-I mechanism (also known as Diffusive
Shock Acceleration, DSA) was proposed as a model
for production of CRs at SNR shocks (blast waves)
by Axford, W. I. Leer, E. Skadron, G. The
Acceleration of Cosmic Rays by Shock Waves,
1977, ICRC Krymskii, G. F. A regular
mechanism for accelerating charged particles at
the shock front, 06/1977 (76?) in DoSSR Bell,
A. R. The acceleration of cosmic rays in shock
fronts, 01/1978 in MNRAS Blandford, R. D. and
Ostriker, J. P. Particle acceleration by
astrophysical shocks, 04/1978 in ApJ
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How shocks accelerate particles (simple analogy)
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How shocks accelerate particles (DSA)
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Note what makes the particle scatter and bounce
back is not collisions with other particles, but
motion in a tangled magnetic field. DSA occurs
only in collisionless shocks, where
particle-particle collisions are negligible
compared to the effect of the magnetic field
force on the particles. Collisionless shocks
have not yet been modeled successfully in the
laboratory, because large spatial scales and
extremely low vacuum are required, so DSA can
only occur in space. such experiments were
attempted in the Rutherford Appleton laboratory
in the UK using a powerful laser called Vulcan,
but did not succeed.
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When accelerated particles bear a negligible
amount of energy, their spectrum can be
calculated analytically (as done in the 1977-1978
papers). We call it a 'test-particle case'.
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If the accelerated particles carry enough energy
to feed back on the shocked flow, the problem
becomes much more complicated (we call it 'the
non-linear case'), and the spectrum gains a
concave shape.
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There is observational evidence indicating that
in nature high Mach number collisionless shocks
accelerate particles so efficiently, that the
non-linear shock modification is
inevitable. Ellison, D. C. Moebius, E.
Paschmann, G. Particle injection and
acceleration at earth's bow shock - Comparison of
upstream and downstream events, 1990 in
ApJ Warren, J. et al. Cosmic-Ray Acceleration
at the Forward Shock in Tycho's Supernova
Remnant Evidence from Chandra X-Ray
Observations, 2005 in ApJ
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Knowing how the shock (and the particle spectrum)
is modified by the acceleration is extremely
important for interpreting SNR observations.
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Shocks and cosmic rays are widespread and play an
important role in the universe, therefore
studying them has numerous astrophysical
applications. An important hot topic is
cosmology Enßlin, T. A. Pfrommer, C.
Springel, V. Jubelgas, M. Cosmic ray physics
in calculations of cosmological structure
formation, 10/2007 in AA
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In order for DSA to operate, there must be strong
magnetic turbulence in the shock vicinity. Is
the turbulence in the interstellar medium into
which a SNR shock expands sufficient for DSA? Or
must the turbulence be produced by the shock
itself? The latter scenario is supported by
observations Achterberg, Blandford and
Reynolds Evidence for enchanced MHD turbulence
outside sharp-rimmed supernova remnants, 1993 in
AA
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The most favored model now is that the turbulent
magnetic field is produced by some efficient,
non-linear plasma instability, which transfers
energy from CRs to the magnetic field
fluctuations. Axford, Leer and Skadron 1977,
ICRC paper turbulence generation (linear regime
DBltltB0) McKenzie, Voelk 1982 in AA (magnetic
field amplification to saturation at DBB0) Bell
Lucek 2001 in MNRAS (magnetic field
amplification with DBgtgtB0, based on recent
observations of high magnetic fields)
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Reasons for assuming an instability for magnetic
field generation - Interstellar magnetic fields
are estimated to be 1-3 ?G, but recent
observations report fields in SNR shocks up to
1000 ?G (!) (e.g., Bamba et al., 2003 in ApJ) -
Extremely rapid X-ray emission variation was
recently reported, suggesting a very efficient
in-situ production of the magnetic turbulence
(Uchiyama et al., 2007 in Nature) -
Instabilities of magnetic fields in plasmas in
presence of energetic particles are known to
exist and have a sufficient growth rate (in the
linear regime) (e.g., Bell Lucek, 2001, in
MNRAS) - Assuming strong amplification of
magnetic fields at SNR shocks helps solve the
problem of accelerating CRs up to the 'knee'.
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Note on observations The extremely high
magnetic fields claimed in recent observations
are the cornerstone of our theoretical
research. The evidence for the high magnetic
fields comes from 3 observational phenomena of
SNR electromagnetic emission 1) Extremely thin
X-ray filaments seen at the blast wave
location 2) Comparison between radio and
gamma-ray part of emission spectra 3) Detailed
fits to emission spectra These observations may
have other theoretical explanations than very
high magnetic fields. They are being actively
discussed in the literature.
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Note on theory Magnetic turbulence is known to
be responsible for particle confinement near the
shock and the subsequent CR production through
DSA. Now we are assuming that, in turn, magnetic
turbulence itself is produced by the CR particles
through a plasma instability. Because both
processes are highly non-linear, an analytical
derivation of a self-consistent solution for the
shock structure and CR energy distribution is
impossible. Numerical models have to be developed.
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Existing approaches to modeling shocks with
efficient DSA (in the 'non-linear case')
include Semi-analytical approximate solutions
(e.g., Malkov, Blasi) Monte Carlo simulation
(e.g., Ellison) Particle-in-cell simulations
(e.g., Spitkovsky)
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Monte Carlo method is an excellent compromise
between semi-analytical and particle-in-cell
simulations because - The crucial quantity
the number of particles that get accelerated (so
called particle injection rate) is not
parameterised, as in most semi-analytical models.
It depends solely on the assumed particle
scattering strength (i. e., diffusion
coefficient) - It is technically easier to
include assumed physical processes (plasma
heating, particle escape, radiative energy
losses, etc.) into the Monte Carlo simulation
than into the semi-analytical models - Is a lot
faster than the PIC simulations and allows models
with a realistic dynamical range, which PIC
simulations are not likely to achieve in the
foreseeable future.
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How Monte Carlo works
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How Monte Carlo works 1. Assume
some flow speed u(x), calculate particle
acceleration
x
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How Monte Carlo works 2. Too many
particles accelerated to conserve momentum.
Choose a smoother u(x).
Flow speed, u(x)
Particle distribution, p4f(p)
Momentum flux, Fp(x)
Coordinate, x
Coordinate, x
Particle momentum, p
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How Monte Carlo works 3. With the
new u(x) we get a different particle spectrum,
but now too few particles got accelerated.
Flow speed, u(x)
Particle distribution, p4f(p)
Momentum flux, Fp(x)
Coordinate, x
Coordinate, x
Particle momentum, p
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How Monte Carlo works 4. Refine
u(x) and keep iterating on it until...
Flow speed, u(x)
Particle distribution, p4f(p)
Momentum flux, Fp(x)
Coordinate, x
Coordinate, x
Particle momentum, p
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How Monte Carlo works 5. Until the
derived solution conserves momentum. This is the
physically feasible shock structure and particle
distribution.
Flow speed, u(x)
Particle distribution, p4f(p)
Momentum flux, Fp(x)
Coordinate, x
Coordinate, x
Particle momentum, p
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• Limitations of the Monte Carlo simulation
• - Our simulation doesn't contain magnetic field
  structure. Rather, it uses a prescription for the
  particle mean free paths (or diffusion
  coefficient), which comes from theoretical
  calculations based on the properties of the
  assumed magnetic fields.
• - Monte Carlo describes a steady-state 1D shock
  wave and doesn't account for possible deviation
  of a shock from the quasi-stationary state.
• - We follow acceleration of protons, not
  electrons. Protons determine the structure of the
  shock, but electrons produce the observable
  synchrotron radiation. Basic uncertainties in how
  electrons are injected and accelerated compared
  to protons remains a longstanding problem in
  shock acceleration science.

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What about the magnetic field generation?
- Various theoretical models have been proposed
in literature. Cosmic ray streaming instability
has been assumed to produce Alfven waves that can
efficiently scatter particles.
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Magnetic Field Amplification issues - How the
instability evolves when field fluctuations
become large is not easily described
analytically - When CR streaming is strong
(which may be the case in SNR shocks), the
generated waves differ from Alfvenic (Bell 2004,
AA) - Other instabilities may be contributing
to the magnetic field growth. - Bottom line
it's not easy to describe wave growth
analytically.
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• Our description of MFA
• The CR streaming instability in the simplest case
  can be described as follows
• Here Umag(x) is the local magnetic turbulence
  energy density, Pcr(x) is the local CR pressure
  and va is the Alfven speed. The magnetic energy
  density growth rate is determined by the CR
  pressure gradient.

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Our description of MFA - We generalized the
above equation to include turbulence advection,
resonant interaction of particles with different
turbulence wavelengths, growth of magnetic field
due to plasma compression, and other
effects These equations reproduce the
streaming instability behavior when DBltB0, and
their extrapolation to DBgtgtB0 is hypothesised.
Magnetic Turbulence Energy Density
Pressure of CRs resonating with k
Plasma flow speed
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Our parameterization of MFA... - Allows us to
investigate how magnetic field generation
influences the shock structure. We can predict
the effect of amplified magnetic fields on
observable quantities like the maximum particle
energy, shock compression ratio, thermal gas
temperature. - Lays the groundwork for
implementing more realistic models of magnetic
field growth.
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Results with MFA (from VEB 2006 in ApJ)
We predicted amplification of B to several
hundreds of microgauss (in self-consistent
solution), greater amount of amplification of
initially lower field and an increase in the
maximum particle energy smaller than the field
amplification factor.
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Research opportunities - We already can make
predictions about the value and the structure of
stochastic magnetic field at shocks, about the
maximum particle momentum produced by SNR shocks
and about the efficiency of DSA with MFA. - We
would like to be able to account for an important
process expected in MFA turbulence dissipation
in shock precursor (e.g., Bell 2004 in AA).
Dissipation of the amplified turbulence not only
decreases the value of the amplified field, but
also heats the thermal gas, which must strongly
affect particle injection rate into the
acceleration process.
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Turbulence dissipation
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Research opportunities - We would like to
calculate the diffusion coefficient more
precisely than the current assumption of Bohm
diffusion The above assumes that for
particles of all energies the complete spectrum
of magnetic fluctuations acts as a wave with DB
B0 (for particle scattering purposes).
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Refining the diffusion coefficient Low
energy particles have small gyroradii and feel
all parts of the spectrum as a uniform field.
High energy particles have longer gyroradii, and
to them most magnetic fluctuations are
short-scale and affect their motion in a
different way. Our refinement of diffusion
coefficient must account for that. It is crucial
for determining the maximum CR particle energy.
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Research opportunities - We would like to
account for the effect of particles streaming
away from the shock (escaping particles) at the
upstream free escape boundary. It may be
important for the maximum CR particle energy and
for the value of the amplified field. (Bykov
and Toptygin 2005 in AstL Blandford and Funk
2007 at GLAST symposium)
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Similar models Blasi, P. and Amato, E.
Theory of nonlinear particle acceleration at
shocks and self-generation of the magnetic
field, 2007 in (arXiv) BA developed a
semi-analytic model with a similar assumption for
magnetic field amplification. Their model runs
faster than the MC and allows for a large dynamic
range without computational difficulties, it also
allows a time-dependent calculation. However,
particle injection in BA's model is less
realistic than in MC, and they don't simulate the
case with size-limited maximum energy of CR
particles.
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Similar models Diamond, P. H. Malkov, M. A.
Dynamics of Mesoscale Magnetic Field in
Diffusive Shock Acceleration, 2007 in ApJ DM
investigate an alternative possibility for
magnetic field growth modulational or decay
instability of Alfven waves (as opposed to the
resonant CR instability), which leads to
generation of longer wavelength magnetic field
fluctuations and may be able to explain
acceleration to the ankle of the CR spectrum at
1018 eV. They haven't incorporated their
magnetic field growth into a shock model yet, but
Malkov had a semi-analytical model, results of
which (without MFA) were in an agreement with the
results of our Monte Carlo code.
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Work done 1981 2004 DE developed and
thoroughly tested against analytical predictions
and observations the MC code without magnetic
field amplification. 2005 2006 AV developed
the system of equations parameterizing the
magnetic field growth, designed a numerical
integrator to solve them and incorporated it into
the MC code. 2006 The above results were
published (Vladimirov, Ellison and Bykov in ApJ,
2006).
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Work done 2007 AV developed a version of the
MC code that was more suitable for the magnetic
turbulence amplification problem, incorporated
the magnetic field amplification into it and
tested against the analytical predictions and the
old version. The new version of the MC code runs
faster and allows the simulation of shocks with
realistic dynamic ranges. For instance, in our
comment on the Uchiyama et al. 2007 paper in
Nature we were able to illustrate our point about
the maximum CR energy with results produced by
the new MC code. 2008 The above mentioned
response is our second paper Ellison and
Vladimirov, ApJL, 2008 (to be publised in 3 days).
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Work done 2007 2008 AV developed a
parameterised description of turbulence
dissipation and thermal plasma heating, it was
incorporated into the MC code. A paper on these
results is in preparation and will be submitted
to ApJ.
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Planned work Spring 2008 AV, Bykov and DE
will finish and submit the paper with results on
generated turbulence dissipation in shocks. Our
results will provide insight on how magnetic
field generation and dissipation affects particle
injection into the acceleration process in
DSA. Spring-Summer 2008 We will develop a
parameterization of the diffusion coefficient
that is more realistic than the Bohm diffusion
currently used. With this done, we will be able
to investigate the effect of magnetic turbulence
generation on the maximal energy of the
accelerated particles.
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Planned work Summer-Fall 2008 We will publish
the results of our model with the refined
diffusion coefficient Fall-Winter 2008 We will
investigate the possibility of including into the
model the effect of particles leaving the system
at the upstream free escape boundary. This may be
important for maximum energy of the accelerated
particles. We may write and submit a paper with
these results before AV's graduation. Winter
2008 Thesis defence.
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Summary - Magnetic field amplification in DSA
is a new, important and actively investigated
problem in astrophysics. - We model DSA with a
Monte Carlo simulation, introducing into it a
parameterization of magnetic turbulence
amplification. - Apart from extremely limited PIC
simulations, our code is the first model to
describe injection and acceleration of particles,
non-linear shock modification and strong magnetic
field amplification consistently with each
other. - We are planning to improve the current
model by including turbulence dissipation,
diffusion coefficient refinement and, probably,
the effect of escaping particles. - AV's PhD
thesis will be based on 4-5 refereed publications
(including the 2 already published papers), which
describe our methods and results.
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Non-linear Model of Diffusive Shock Acceleration
of Cosmic Rays in the Presence of Self-Consistent
Generation of Stochastic Magnetic Fields PhD
Thesis Research Plan PhD candidate Andrey
Vladimirov Research advisor prof. Donald C.
Ellison, NCSU Advisory committee profs. Steven
Reynolds, PY, NCSU James Selgrade, MA,
NCSU Albert Young, PY, NCSU this work is
done in collaboration with prof. Andrei Bykov,
Ioffe Physical-Technical Institute, St.
Petersburg, Russia North Carolina State
University January 16, 2008