IUCAA, Pune, 19/04/2005

Contour statistics, depolarization canalsand interstellar turbulence

Anvar Shukurov

School of Mathematics and Statistics, Newcastle, U.K.

Synchrotron emission in interstellar medium

Total intensity

Polarized intensity

+ polarization angle

Polarization and depolarization

P = p e2i , complex polarization, p = P/Ip: degree of polarization (fraction of the radiation flux that is polarized);: polarization angle

Depolarization: superposition of two polarized waves,

1 = 2 + /2 P1 + P2 = 0

Faraday rotation: = 0 + RM2

Faraday rotation can depolarize radiation

+ = 0

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

Gaensler et al., ApJ, 549, 959, 2001. ATCA, = 1.38 GHz ( = 21.7 cm), W = 90” 70”.

Narrow, elongated regions of zero polarized intensity

Abrupt change in by /2 across a canal

Haverkorn et al. A&A 2000

P

Gaensler et al., ApJ, 549, 959, 2001

Position and appearance depend on the wavelength

Haverkorn et al., AA, 403, 1031, 2003Westerbork, = 341-375 MHz, W = 5’

No counterparts in total emission

Uya

nike

r et

al.,

A&

A S

uppl

, 13

8, 3

1, 1

999.

Eff

elsb

erg,

1.4

GH

z, W

= 9

.35’

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

Complex polarization ( // line of sight)

= synchrotron emissivity, B = magnetic field, = wavelength,

n = thermal electron number density, Q, U, I = Stokes parameters

Fractional polarization p, polarization angle and Faraday rotation measure RM:

Faraday depth to distance z:

Faraday depth:

Differential Faraday rotation

Uniform slab, thickness 2h, F = 2KnBzh2:

Implications

• Canals: |F| = n |RM| = F/(22)= n/(22)

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?

The problem of overshoots

• A random function F(x).

• What is the mean separation of positions xi such that F(xi) = Fc (= const) ?

f (F) = the probability density of F;f (F, F' ) = the joint probability density of F and

F' = dF/dx;

Great simplification: Gaussian random functions(and F a GRF!)

F(x) and F'(x) are statistically independent,

Contours of a random function in 2D

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

Contours around high peaks

Contours around high peaks

• 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)

Mean separation of canals (Shukurov & Berkhuijsen MN 2003)

lT 0.6 pc at L = 1 kpc Re(RM) = (l0/lT)2 104105

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