contour statistics, depolarization canals and interstellar turbulence
DESCRIPTION
Contour statistics, depolarization canals and interstellar turbulence. Anvar Shukurov School of Mathematics and Statistics, Newcastle, U.K. Synchrotron emission in interstellar medium. Total intensity Polarized intensity + polarization angle . +. = 0. Polarization and depolarization. - PowerPoint PPT PresentationTRANSCRIPT
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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.
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Synchrotron emission in interstellar medium
Total intensity
Polarized intensity
+ polarization angle
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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
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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
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Gaensler et al., ApJ, 549, 959, 2001. ATCA, = 1.38 GHz ( = 21.7 cm), W = 90” 70”.
Narrow, elongated regions of zero polarized intensity
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Abrupt change in by /2 across a canal
Haverkorn et al. A&A 2000
P
Gaensler et al., ApJ, 549, 959, 2001
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Position and appearance depend on the wavelength
Haverkorn et al., AA, 403, 1031, 2003Westerbork, = 341-375 MHz, W = 5’
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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’
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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
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Complex polarization ( // line of sight)
= synchrotron emissivity, B = magnetic field, = wavelength,
n = thermal electron number density, Q, U, I = Stokes parameters
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Fractional polarization p, polarization angle and Faraday rotation measure RM:
Faraday depth to distance z:
Faraday depth:
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Differential Faraday rotation
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Uniform slab, thickness 2h, F = 2KnBzh2:
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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?
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The problem of overshoots
• A random function F(x).
• What is the mean separation of positions xi such that F(xi) = Fc (= const) ?
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f (F) = the probability density of F;f (F, F' ) = the joint probability density of F and
F' = dF/dx;
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Great simplification: Gaussian random functions(and F a GRF!)
F(x) and F'(x) are statistically independent,
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Contours of a random function in 2D
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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
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Contours around high peaks
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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)
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Mean separation of canals (Shukurov & Berkhuijsen MN 2003)
lT 0.6 pc at L = 1 kpc Re(RM) = (l0/lT)2 104105
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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