Spectral analysis of non-thermal filaments in Cas A

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) Region 1 Out In dim areas (low statistics)

4. Spectral analysis: power-law steepens going inside Region 1 - in Region 8 - in Gph = 2.49 +0.11 -0.09 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

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 w a E-1/2 0.3 – 2.0 keV Actual data 3.0 – 6.0 keV 6.0 – 10 keV

6. 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

7. 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 / RandLadv / Ldiffrom 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

8. Results • Implemented the model in Xspec satisfactory fits Region 1 - in Region 1 - out c2/ u ~ 0.45 c2/ u ~ 0.45

9. 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

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, • 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

13. Additional slides

14. 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.

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*tsync = V/bB2g and g = (n/nL)1/2 for a delta-profile

22. 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 / RandL_adv / L_difwhich will allow us to uniquely estimate the magnetic field and the diffusion coefficient in the region of particular filament

23. 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

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

25. Maximum e energy: tacc ~ tsynch Assuming D = hDB and isotropic diffusion (integrated directions contribute)