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

2. 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

3. Observation and regions chosen We used the 1 Ms observation of U. Hwang et al. ( Astrophys. J. 615, L117-L120, 2004 ) dim areas (low statistics) In Out Region 1

4. Spectral analysis: power-law steepens going inside Region 1 - in Region 1 - out  ph      Region 8 - in    Region 8 - out    Region 2 - in Region 2 - out  ph   ph       ph 

5. 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 0.3 – 2.0 keV 3.0 – 6.0 keV 6.0 – 10 keV Actual data w  E -1/2

6. The model (by DL and ML) Diffusion of particles, advection ( V adv = 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 =  D B = 1/3*  cr g Injection of particles with a power-law dist Only downstream emission considered Shock compression ratio of 4

7. p - power-law index of the injected electron distribution L dif / R - ratio, diffusion length to the radius of SNR L adv / L dif - ratio, advection length to diffusion length - relative importance of these two processes Found L dif / R and L adv / L dif from the fitting to the data, which allowed for estimation of both the magnetic field and the diffusion coefficient for each filament Parameters of the model In the model, diffusion causes: Spectral hardening going outward Filament widths depend weakly on energy

8. Region 1 - outRegion 1 - in Results Implemented the model in Xspec  satisfactory fits    /  ~ 0.45

9. Results Average magnetic field ~ 40  G (20 – 115  G ) Results consistent with Bohm diffusion:  ~ 1.4 aver (0.05 – 11.0, most around 0.1 ) { D = 1/3 *  (mc 3 /qB)  } p ~ 4.5 Electron

10. Estimation of turbulence For arbitrary shock obliquity angle the diffusion coefficient is Where Since we are considering then :

11. 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

12. 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,  ~ 0.05 – 11.0 Magnetic fields in filaments range from 20  G to 115  G Moderately strong turbulence ~ 0.2 -0.4 in the regions of the filaments with low  ≤ 0.15 Next step: Understand the implications of our results for cosmic ray acceleration

13. Additional slides

14. 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. The model

15. 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)

16. 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

17. 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

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

19. 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.

20. 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

21. 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 = V*t sync = V/  B 2  and  = ( / L ) 1/2 for a delta-profile

22. 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

23. C 1 = L dif /R: controls the width of the projected profile (estimated by adjusting this width, defined at the 20% intensity level). (~0.02) C 2 = L adv (1keV) /L dif : determines the difference between the X-ray photon indexes in and out (values from 2 to 8). Does not affect the width

24. Proper motion of X-ray filaments Patnaude & Fesen, 2009 V sh = 5200 +- 500 km/s

25. Maximum e energy:  acc ~  synch Assuming D =  D B and isotropic diffusion (integrated directions contribute)