1. 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

2. Motivation 1
MHD Turbulence
BK workshop 2016

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 4-2 Results 3 – using MHD turbulence data
θFWHM=3’ , NBASE = 30 , S/N = 1/5
EANAM 2016, Beijing, China

14. 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

15. 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

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

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

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

19. Thank you for your attention!
Any questions?

21. 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

22. 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

23. 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

24. 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

25. 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

26. 4-1 ★ Result 3 – effect of telescope resolution θFWHM=3’ ★
EANAM 2016, Beijing, China

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

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

29. 2-nd order structure function R 
I(X)-I(X+R)      R R  R   R