Spectral analysis of
non-thermal filaments
in Cas A
Miguel Araya
D. Lomiashvili, C. Chang, M. Lyutikov, W. Cui
Department of Physics, Purdue University
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
We used the 1 Ms
observation of U. Hwang
et al. (Astrophys. J. 615,
L117-L120, 2004)
Observation and regions
chosen
Region 1
Out
In
dim areas (low statistics)
Spectral analysis: power-law steepens going inside
Region 1 - in
Gph = 2.49
Region 8 - in
+0.11
-0.09
G = 2.59
Region 1 - out
Gph = 2.41
+0.15
-0.13
Region 8 - out
G = 2.16
Region 2 - in
Gph = 2.49
+0.09
-0.08
Region 2 - out
Gph = 2.16
+0.20
-0.16
+0.11
-0.10
+0.11
-0.10
• When diffusion is NOT considered
– The width of the filaments strongly depends on the
observed frequency w a E-1/2
– Very small difference between “inner” and “outter”
photon indices is obtained
Actual data
0.3 – 2.0 keV
3.0 – 6.0 keV
6.0 – 10 keV
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 = 1/3*hcrg
•Injection of particles with a power-law dist
•Only downstream emission considered
•Shock compression ratio of 4
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
Results
•Implemented the model in Xspec
Region 1 - in

satisfactory fits
Region 1 - out
c2 / u ~ 0.45
c2 / u ~ 0.45
Results
•Average magnetic field ~ 40 mG
(20 – 115 mG)
•Results consistent with Bohm diffusion:
h ~ 1.4 aver
•p ~ 4.5
•Electron
(0.05 – 11.0, most around 0.1)
{ D = 1/3 *h (mc3/qB) g }
Estimation of turbulence
•For arbitrary shock obliquity angle
the diffusion coefficient is
Where
Since we are considering
then :
Constraints on the shock structure and estimation of turbulence
• Constraints on
values of
and the turbulence level from the estimated
• Constraining the turbulence level is possible without adopting any
particular orientation only in the case of low
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
filaments with low h ≤ 0.15
~ 0.2 -0.4 in the regions of the
•Next step: Understand the implications of our results for cosmic ray
acceleration
Additional slides
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.
Processes neglected
• Adiabatic losses
(losses due to the expansion of the SNR)
The expansion isn’t 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)
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
• 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
• As a rough estimate of turbulence level we can use
perpendicular diffusion coefficient :
• Thus , for our estimated avg. value of
and turbulence is strong
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.
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
Model with NO diffusion
• The model with advection and
synchrotron losses only
– The width of the filament strongly depends on
the observed frequency
Width = V*tsync = V/bB2g and g = (n/nL)1/2 for a delta-profile
– Very small difference between “inner” and
“outer” spectral indices
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
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
Proper motion of X-ray filaments
Patnaude & Fesen, 2009
Vsh = 5200 +- 500 km/s
Maximum e energy:
tacc ~ tsynch
Assuming D = hDB and isotropic diffusion (integrated
directions contribute)