1. Polarization 2005, Orsay, 13/09/2005 Depolarization canals in Milky Way radio maps Anvar Shukurov and Andrew Fletcher School of Mathematics and Statistics, Newcastle, U.K.

2. Outline Observational properties Origin: Differential Faraday rotation Gradients of Faraday rotation across the beam Physics extracted from canals

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

4.  Abrupt change in  by  /2 across a canal Haverkorn et al. 2000  PI Gaensler et al., ApJ, 549, 959, 2001   

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

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

7. No counterparts in I  propagation effects Sensitivity to  Faraday depolarization Potentially rich source of information on ISM

8. Complex polarization ( // l.o.s.)

9. Fractional polarization p, polarization angle  and Faraday rotation measure RM : Potential Faraday rotation:

10. Magneto-ionic layer + synchrotron emission, uniform along the l.o.s., varying across the sky,  =  0 Differential Faraday rotation produces canals

11. Uniform slab, thickness 2h, R = 2KnB z h, F = R 2 : There exists a reference frame in the sky plane where Q (or U ) changes sign across a canal produced by DFR, whereas U (or Q ) does not.

12. Faraday screen: magneto-ionic layer in front of emitting layer, both uniform along the l.o.s., F = R 2 varies across the sky Variation of F across the beam produces canals Discontinuity in F(x),  F =  /2  canals,  =  /2 Continuous variation,  F=  / 2  no canals,  =  /2

13. Canals with a  /2 jump in  can only be produced by discontinuities in F and RM:  x/ D < 0.2 F D = FWHM of a Gaussian beam  F =   2  x x FF

14. Continuous variation,  F =   canals, but with  =  We predict canals, produced in a Faraday screen, without any variation in  across them (i.e., with  F = n  ). Moreover, canals can occur with any  F, if (1)  F = D  F = n  and (2) F ( x ) is continuous

15. Simple model of a Faraday screen Both Q and U change sign across a canal produced in a Faraday screen.

16. Implications: DFR canals Canals: | F | =  n  | RM | =  n/(2 2 )  Canals are contours of RM(x) RM(x):  Gaussian random function, S/N > 1 What is the mean separation of contours of a (Gaussian) random function?

17. The problem of overshoots Consider a random function F(x). What is the mean separation of positions x i such that F(x i ) = F 0 (= const) ? x F F0F0 §9 in A. Sveshnikov, Applied Methods of the Theory of Random Functions, Pergamon, 1966

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

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

21. 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

22. Canals in Faraday screens: tracer of shock fronts Observations: Haverkorn et al., AA, 403, 1031, 2003 Simulations: Haverkorn & Heitsch, AA, 421, 1011, 2004

23. Canals in Faraday screen:  F=  R 2 =(n + 1 / 2 )  Haverkorn et al. (2003):    R = 2.1 rad/m 2 (  = 85 cm)  Shock front, 1D compression: n 2 /n 1 =  , B 2 /B 1 =  , R 2 /R 1 =   2,  R = (   2 -1) R 1     1.3 (M = shock’s Mach number)

24. Distribution function of shocks (Bykov & Toptygin, Ap&SS 138, 341, 1987) PDF of time intervals between passages of M-shocks: Mean separation of shocks M > M 0 in the sky plane:

25. Mean separation of shocks, Haverkorn et al. (2003) M 0 = 1.2, Depth = 600 pc, c s = 10 km/s, f cl = 0.25  L  90' (= 20 pc) (within a factor of 2 of what’s observed) Smaller  larger M 0  larger L

26. Conclusions The nature of depolarization canals seems to be understood. They are sensitive to important physical parameters of the ISM (autocorrelation function of RM or Mach number of shocks). New tool for the studies of ISM turbulence: contour statistics (contours of RM, I, PI, ….) Details in: Fletcher & Shukurov, astro-ph/0510XXXX