Diapositiva 1

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Spectral analysis of non-thermal filaments in Cas
A
Miguel Araya D. Lomiashvili, C. Chang, M.
Lyutikov, W. Cui
Department of Physics, Purdue University
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Supernova Remnants

• Associating Non-thermal filaments
  Shock
• Aging of electrons and filament properties
• (e.g., width at different energies)
• Synchrotron rims modeled as thin
• spherical regions

• Our purpose
• Evaluate role of particle diffusion in the
  shocked plasma (implications for cosmic ray
  acceleration)
• - Estimate the value of the magnetic
  field

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Observation and regions chosen
We used the 1 Ms observation of U. Hwang et al.
(Astrophys. J. 615, L117-L120, 2004)
Region 1
Out
In
dim areas (low statistics)
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Spectral analysis power-law steepens going inside
Region 1 - in
Region 8 - in
0.11 -0.09
Gph 2.49
G 2.59
0.20 -0.16
Region 1 - out
Region 8 - out
Region 2 - in
G 2.16
0.11 -0.10
Gph 2.41
0.15 -0.13
Gph 2.49
0.09 -0.08
Region 2 - out
Gph 2.16
0.11 -0.10
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• When diffusion is NOT considered
• The width of the filaments strongly depends on
  the observed frequency
• Very small difference between inner and
  outter photon indices is obtained

w a E-1/2
0.3 2.0 keV
Actual data
3.0 6.0 keV
6.0 10 keV
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The model (by DL and ML)

• Diffusion of particles, advection (Vadv 1300
  km/s) and
• synchrotron losses in a randomly oriented
  magnetic field B
• Solution to the diffusion-loss
  equation
• Syrovatskii (1959) advection
• Isotropic diffusion assumed - D hDB
  1/3hcrg
• Injection of particles with a power-law dist
• Only downstream emission considered
• Shock compression ratio of 4

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Parameters of the model

• p - power-law index of the injected electron
  distribution
• Ldif / R - ratio, diffusion length
  to the radius of SNR
• Ladv / Ldif - ratio, advection length
  to diffusion length - relative
  importance of these two processes
• Found Ldif / R and Ladv / Ldif from the fitting
  to the data, which allowed for estimation of both
  the magnetic field and the diffusion coefficient
  for each filament

• In the model, diffusion causes
• Spectral hardening going outward
• Filament widths depend weakly on energy

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Results

• Implemented the model in Xspec ?
  satisfactory fits

Region 1 - out
Region 1 - in
c2 / u 0.45
c2 / u 0.45
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Results

• Average magnetic field 40 mG (20 115 mG)
• Results consistent with Bohm diffusion
• h 1.4 aver (0.05 11.0, most
  around 0.1) D 1/3 h (mc3/qB) g
• p 4.5
• Electron

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Estimation of turbulence

• For arbitrary shock obliquity angle the
  diffusion coefficient is
• Where Since we are considering
  then

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Constraints on the shock structure and estimation
of turbulence

• Constraints on and the turbulence level
  from the estimated values of

• Constraining the turbulence level is possible
  without adopting any particular orientation only
  in the case of low

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Summary

• Outward hardening of X-ray spectra has
  systematically been seen in all filaments studied
  
• Width of filaments expected to strongly depend
  on energy when diffusion is not important, but
  diffusion becomes necessary in the model to
  explain the data
• Hardening of spectra is explained by diffusion
  of particles. Data consistent with Bohm-type
  diffusion,
• h
  0.05 11.0
• Magnetic fields in filaments range from 20 mG
  to 115 mG
• Moderately strong turbulence
  0.2 -0.4 in the regions of the filaments with
  low h 0.15
• Next step Understand the implications of our
  results for cosmic ray acceleration

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Additional slides
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The model

• The filaments are thought to be a result of the
  synchrotron radiation from relativistic TeV
  electrons which are probably accelerated at the
  forward shock.
• We assume that the synchrotron radiation losses
  and a diffusion are the dominant processes and an
  evolution of the non-thermal electron
  distribution can be described by solving the
  Diffusion-loss equation with an advection. We
  used the solution given by Syrovatskii (1959) and
  included the advection.

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Processes neglected

• Adiabatic losses (losses due to the expansion of
  the SNR)The expansion isnt fast enough in order
  this process to be important.
• Bremsstrahlung losses (Synchrotron losses
  dominate)
• Acceleration of the particles
• Absorption
• Inverse Compton losses (Synchrotron losses
  dominate)

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About turbulence

• Diffusion is assumed to be Bohm type
  - Bohm diffusion coefficient
  - is what we measure
  (estimate)
• We consider an isotropic diffusion
  - Diffusion coefficient parallel
  to the field
  
  - Diffusion coefficient perpendicular
  to the field
• 
  
• - gyro radius
  - scattering mean free path
  
  
  

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• In the quasi-linear formalism,
  - gyroradius is related to the energy density
  in resonant waves (e.g.,
  Blandford Eichler 1987).
• If is small ( ? 1 ) , in which case the
  turbulence is strong ,
  nonlinear theory should be used.
• as , that is, for strong turbulence,
  the distinction between parallel and
  perpendicular to the turbulent field becomes
  lost. So if we get that is close to 1 then
  our assumption about isotropic diffusion will be
  consistent with theory

REYNOLDS Vol. 493
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• As a rough estimate of turbulence level we can
  use perpendicular diffusion coefficient
• Thus , for our estimated avg. value of and
  turbulence is strong

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Further Approximations

• Strong shock approximation. Compression ratio
  4
• In the calculations we use static magnetic field
  with uniform magnitude although Bohm type
  diffusion requires some level of turbulence.
• We account the emission only from the downstream
  region, since the magnetic field strength is
  weaker upstream and diffusion and advection
  contributions are opposite to each other.
• The delta-function approximation is used while
  calculating the specific synchrotron power
  radiated by each electron.

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Theoretical Model

• Steady state distribution of particles within the
  filament is the result of the following processes
  
• Continuous injection of the particles with
  power-law distribution
• Advection
• Synchrotron losses
• Diffusion of the particles on magnetic
  irregularities

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Model with NO diffusion

• The model with advection and synchrotron losses
  only
• The width of the filament strongly depends on the
  observed frequency
• Very small difference between inner and outer
  spectral indices

Width Vtsync V/bB2g and g (n/nL)1/2 for
a delta-profile
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Parameters of the model

• After nondimensionalizing the solution of
  diffusion-loss equation we get 3 free parameters
  for our model (V_adv we take from the
  measurements of the proper motion of FS)
• p power-law index of the injected electron
  distribution
• L_dif / R - the ratio of the diffusion length
  over the radius of SNR
• L_adv / L_dif - the ratio of the advection length
  over the diffusion length. Shows the relative
  importance of these two processes
• From the fitting of the model with observational
  data we can find L_dif / R and L_adv / L_dif
  which will allow us to uniquely estimate the
  magnetic field and the diffusion coefficient in
  the region of particular filament

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Parameters of the model

• C1 Ldif/R controls the width of the projected
  profile (estimated by adjusting this width,
  defined at the 20 intensity level). (0.02)
• C2 Ladv(1keV)/Ldif determines the difference
  between the X-ray photon indexes in and out
  (values from 2 to 8). Does not affect the width

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Proper motion of X-ray filaments
Patnaude Fesen, 2009
Vsh 5200 - 500 km/s
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Maximum e energy
tacc tsynch
Assuming D hDB and isotropic diffusion
(integrated directions contribute)
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