Studying turbulence from polarized synchrotron emission with multi-frequency measurement Hyeseung Lee1 with Jungyeon Cho1, A. Lazarian2 1Chungnam Nation University, South Korea 2University of Wisconsin-Madison, USA EANAM 2016, Beijing, China

Motivation 1 MHD Turbulence BK workshop 2016

Magnetohydrodynamic Turbulence Motivation 1 Magnetic reconnection star formation Cosmic rays Magnetohydrodynamic Turbulence   Density Velocity Magnetic Field Magnetic Field Synchrotron emission PDF Faraday rotation Power spectrum Structure function  BK workshop 2016

Magnetohydrodynamic Turbulence Motivation 1 Magnetic reconnection star formation Cosmic rays Magnetohydrodynamic Turbulence   Density Velocity Magnetic Field Magnetic Field Synchrotron emission PDF Faraday rotation Power spectrum Structure function  BK workshop 2016

Magnetohydrodynamic Turbulence Motivation 1 Magnetic reconnection star formation Cosmic rays Magnetohydrodynamic Turbulence   Density Velocity Magnetic Field Magnetic Field Synchrotron emission PDF Faraday rotation Power spectrum Structure function  BK workshop 2016

Method –Data 2-1 Synthetic Data B0 = 0 N3=5123 In Fourier space where kmax= N/2 (N=resolution) N3=5123 In Fourier space 2 |A(k)|2k-m m=11/3 for Kolmogorov (Cho&Lazarian 2010) Spectrum of magnetic field follows a Kolmogorov spectrum EANAM 2016, Beijing, China

Method –Data 2-2 Synthetic Data B0 = 0 where kmax= N/2 (N=resolution) N3=5123 Turbulence Data : based on a 3rd order accurate hybrid non-osciallatory (ENO) scheme in a periodic box of size 2π (Cho & Lazarian 2002) MA = v/VA ~ 0.7 MS = v/a ~ 0.7 EANAM 2016, Beijing, China

Method : Polarization from synchrotron rad. 2-3 Polarized intensity observed at a 2D position X on the plane of the sky at wavelength λ z Intrinsic polarization defined by the Stokes parameters Q and U : Pj = Qj + iUj Faraday rotation measure EANAM 2016, Beijing, China

shell-integrated 1D spectrum Ring-integrated 1D spectrum Statistics – Power spectrum 2-4 shell-integrated 1D spectrum for a 3D variable Ring-integrated 1D spectrum for a 2D variable ky Ky Kx k+1 k kx kz EANAM 2016, Beijing, China

Result 1 – spectral index of EED () 3-1 The variations of the spectral index of relativistic electron energy distribution change the amplitude of the fluctuations, but not the spectral slope of the synchrotron power spectrum. (Lazarian&Pogosyan2012)  = 1.5 ~ 4.0  = 2.0 EED (Electron Energy Distribution) : N(E)dE=N0E-γdE EANAM 2016, Beijing, China

3-3 Result 2 - synchrotron radiation & Faraday rotation in code unit fluctuations in F.R. measure Faraday depolarization effect synchrotron emission F.R. effect (Lazarian&Pogosyan 2016) dP/dλ2 is also useful to recover the statistics of MHD turbulence! EANAM 2016, Beijing, China

Interferometric method 4-1 ★ number of baselines noise Telescope resolution (Configuration of arrays determines wave-vectors in Fourier space and observations directly give Fourier amplitude at the wave-vectors. These wave-numbers produce power spectrum of polarization.) NBASE=30 S/N=1/5 θFWHM=3’ KNAG 2016

4-2 Results 3 – using MHD turbulence data θFWHM=3’ , NBASE = 30 , S/N = 1/5 EANAM 2016, Beijing, China

Summary 5 Our numerical results show that we can study MHD turbulence through polarized synchrotron emission. This study can be performed in the presence of Faraday rotation and depolarization caused by turbulent magnetic field, in the settings when only Faraday rotation is responsible for the polarization fluctuations, in the presence of effects of finite beamsize, noise, and a few baselines  Our present study paves the way for the successful use of spectrum with observational data. EANAM 2016, Beijing, China

statistical description : anisotropy In progress Structure function Quadrupole moment 2-nd order structure function  R|| R⊥ Ii(X)-Ii (X+R) Quadrupole ratio <Bx> ~ 1.0   x z y B0 EANAM 2016, Beijing, China

Mode decoupling : Alfven, fast, slow In progress EANAM 2016, Beijing, China

Mode decoupling : Alfven, fast, slow In progress Alfven Fast Slow B0 EANAM 2016, Beijing, China

Polarization from spatially separated medium In progress LOS x y <By> ~ 1.0 z <Bx> ~ 1.0 B0 B0 EANAM 2016, Beijing, China

Thank you for your attention! Any questions?

statistical description : power spectrum 2-0 real-space distribution of v(r), b(r), ρ(r), … Fourier transform || Amplitude (S)  + + + wave number (k ∝1/λ)  || Assuming that we have a signal, real-space distribution of quantitie, that is formed by the combination of numerous waves with amplitude. If these waves extend from – infinity to + infinity, the Fourier transform of this signal yields a number of pairs of real, even functions with corresponding amplitudes as depicted in the figure. The integrand Fourier transform S^2 can be interpreted as a function describing the energy contained in the signal at the wave-number, k. This distribution of power into wave-number components is power spectrum. The Kolmogorov spectrum is well-known power spectrum. E(k) k [Hz] Power spectrum : E(k) e.g) E(k) ~ k5/3 (Kolmogorov spectrum) EANAM 2016, Beijing, China

Method –Data 2-1 Synthetic Data B0 = 0 N3=5123 where kmax= N/2 (N=resolution) N3=5123 k-5/3 for magnetic field k-1 for density EANAM 2016, Beijing, China

Fixed intrinsic synchrotron emission (Q/I=1, U/I=0) Result 2 - synchrotron radiation or Faraday rotation 3-2 Effect of Faraday rotation Fixed intrinsic synchrotron emission (Q/I=1, U/I=0) Pj = Qj + iUj P1 P2 P3 P4 xn-4 xn-2 xn-1 xn λ~1 P1= P2= P3= P4 P1 Φ1 Φ2 Φ3 Faraday depolarization is significant when K < Bñ, and it is insignificant when K > Bñ When lambda^2 ~ Kmax / B ∣ñ, the polarized emission from each grid point becomes completely uncorrelated and the emission from each grid point contributes randomly,6 which makes the spectrum proportional to K. P2 Φ2 Φ3 Φ1≠ Φ2≠ Φ3 EANAM 2016, Beijing, China

Uniform Faraday roatation (ne(z)=1, Bz(z)=1) Result 2 - synchrotron radiation or Faraday rotation 3-2 Effect of synchrotron emission Uniform Faraday roatation (ne(z)=1, Bz(z)=1) P1 P2 P3 P4 xn-4 xn-2 xn-1 xn small λ P1≠ P2≠ P3≠ P4 Φ1 P1 Φ2 Φ3 (Configuration of arrays determines wave-vectors in Fourier space and observations directly give Fourier amplitude at the wave-vectors. These wave-numbers produce power spectrum of polarization.) Φ2 Φ3 P2 small-K large-K Faraday depolarization effect negligible Φ1= Φ2= Φ3 EANAM 2016, Beijing, China

Interferometric method 4-0 ★ Telescope resolution noise number of baselines We can obtain spectrum in Fourier space for certain wave-vectors through interferometric observations! Ky Kx (Configuration of arrays determines wave-vectors in Fourier space and observations directly give Fourier amplitude at the wave-vectors. These wave-numbers produce power spectrum of polarization.) KNAG 2016

4-1 ★ Result 3 – effect of telescope resolution θFWHM=3’ ★ EANAM 2016, Beijing, China

Result 3 – number of baselines 4-2 ★ number of baselines Ky Kx NBASE=30 EANAM 2016, Beijing, China

Result 3 – effect of noise 4-3 ★ noise number of baselines Ky Kx S/N=1/5 EANAM 2016, Beijing, China

2-nd order structure function  R  I(X)-I(X+R)      R R  R   R