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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:
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,
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
• 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