1. Contour statistics, depolarization canals and interstellar turbulence Anvar Shukurov School of Mathematics and Statistics, Newcastle, U.K.

2. Synchrotron emission in interstellar medium Total intensity Polarized intensity + polarization angle 

3. Polarization and depolarization P = p e 2i , complex polarization, p = P/I p : degree of polarization (fraction of the radiation flux that is polarized) ;  : polarization angle Depolarization: superposition of two polarized waves,  1 =  2 +  /2  P 1 + P 2 = 0  Faraday rotation:  =  0 + RM  2 Faraday rotation can depolarize radiation += 0

4. Depolarization canals in radio maps of the Milky Way  Narrow, elongated regions of zero polarized intensity  Abrupt change in  by  /2 across a canal  Position and appearance depend on the wavelength  No counterparts in total emission

5. Gaensler et al., ApJ, 549, 959, 2001. ATCA, = 1.38 GHz ( = 21.7 cm), W = 90”  70”.  Narrow, elongated regions of zero polarized intensity

6.  Abrupt change in  by  /2 across a canal Haverkorn et al. A&A 2000  P Gaensler et al., ApJ, 549, 959, 2001   

7.  Position and appearance depend on the wavelength Haverkorn et al., AA, 403, 1031, 2003 Westerbork, = 341-375 MHz, W = 5’

8.  No counterparts in total emission Uyaniker et al., A&A Suppl, 138, 31, 1999. Effelsberg, 1.4 GHz, W = 9.35’

9. No counterparts in I  propagation effects (not produced by any gas filaments or sheets) Sensitivity to  Faraday depolarization Potentially rich source of information on ISM

10. Complex polarization ( // line of sight)  = synchrotron emissivity, B = magnetic field, = wavelength, n = thermal electron number density, Q, U, I = Stokes parameters

11. Fractional polarization p, polarization angle  and Faraday rotation measure RM : Faraday depth to distance z : Faraday depth:

12. Differential Faraday rotation

13. Uniform slab, thickness 2h, F = 2KnB z h 2 :

14. Implications Canals: | F | =  n  | RM | = F / (2 2 )=  n/(2 2 )  Canals are contours of RM(x), an observable quantity F(x) & RM  Gaussian random functions What is the mean separation of contours of a (Gaussian) random function?

15. The problem of overshoots A random function F(x). What is the mean separation of positions x i such that F(x i ) = F c (= const) ?

16.  f (F) = the probability density of F;  f (F, F' ) = the joint probability density of F and F' = dF/dx; 

17. Great simplification: Gaussian random functions (and F  a GRF!) F(x) and F'(x) are statistically independent,

19. Contours of a random function in 2D

20. Useful references Sveshnikov A. A., 1966, Applied Methods in the Theory of Random Functions (Pergamon Press: Oxford) Vanmarcke E., 1983, Random Fields: Analysis and Synthesis (MIT Press: Cambridge, Mass.) Longuet-Higgins M. S., 1957, Phil. Trans. R. Soc. London, Ser. A, 249, 321 Ryden, 1988, ApJ, 333, L41 Ryden et al., 1989, ApJ, 340, 647

21. Contours around high peaks

22. Tend to be closed curves (around x = 0). F (0) =  F, >> 1;  F (0) = 0. For a Gaussian random function, i.e., the mean profile F ( r ) around a high peak follows the autocorrelation function (Peebles, 1984, ApJ 277, 470; Bardeen et al., 1986, ApJ 304, 15)

23. Mean separation of canals (Shukurov & Berkhuijsen MN 2003) l T  0.6 pc at L = 1 kpc  Re (RM) = (l 0 /l T ) 2  10 4  10 5

24. Conclusions The nature of depolarization canals seems to be understood. They are sensitive to important physical parameters of the ISM (autocorrelation function of RM). New tool for the studies of the ISM turbulence: contour statistics (contours of RM, I, P, ….) Details in: Fletcher & Shukurov, astro-ph/0602536