. : http://www.naic.edu/~astro/sdss5/talks/schtalk_09.pdf
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: Tue Aug 18 04:22:46 2009
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MAY II 1996, PAGE 161

ASTRONOMY & ASTROPHYSICS SUPPLEMENT SERIES Astron. Astrophys. Suppl. Ser. 117, 161-165 (1996)

Understanding radio polarimetry. III. Interpreting the IAU/IEEE definitions of the Stokes parameters
J.P. Hamaker and J.D. Bregman
Netherlands Foundation for Research in Astronomy, Postbus 2, 7990 AA Dwingeloo, The Netherlands Received September 26; accepted October 22, 1995 Abstract. -- In two companion papers (Paper I, Hamaker et al. 1996; Paper II, Sault et al. 1996), a new theory of radio-interferometric polarimetry and its application to the calibration of interferometer arrays are presented. To complete our study of radio polarimetry, we examine here the definition of the Stokes parameters adopted by Commission 40 of the IAU (1974) and the way this definition works out in the mathematical equations. Using the formalism of Paper I, we give a simplified derivation of the frequently-cited `black-box' formula originally derived by Morris et al. (1964). We show that their original version is in error in the sign of Stokes V , the correct sign being that given by Weiler (1973) and Thompson et al. (1986). Key words: methods: analytical -- methods: data analysis -- techniques: interferometers -- techniques: polarimeters -- polarization

1. Introduction In a companion paper (Hamaker et al. 1996, Paper I) we have presented a theory that describes the operation of a polarimetric radio interferometer in terms of the properties of its constituent elements and in doing so unifies the heretofore disjoint realms of radio and optical polarimetry. In a second paper (Sault et al., Paper II) we apply this theory along with theorems borrowed from optical polarimetry to the problem of calibrating an interferometer array such as an aperture-synthesis telescope. In practical applications, the theory must be supplemented by precise definitions of the coordinate frames and the Stokes parameters that are used. This problem was first addressed by the Institute of Radio Engineers in 1942; the most recent version of their definition was published in 1969 (IEEE 1969). For radio-astronomical applications, the IAU (1974) endorses the IEEE standard, supplementing it with definitions of the Cartesian coordinate frame shown in Fig. 1 and of the sign of the Stokes parameter V . Most published work on actual polarimetric interferometer observations infers the source's Stokes-parameter brightness distributions from a formula derived by Morris et al. (1964). Weiler (1973) rederives their result, agreeing except for the sign of Stokes V . Thompson et al. (1987)
Send offprint requests to : J.P. Hamaker, jph@nfra.nl

include his version in their textbook, even though they suggest in their wording that they agree with Morris et al. Clearly the situation needs to be clarified; starting from a complete interpretation of the definitions, we are in a good position to do so. We shall show Weiler's version indeed to be the correct one.

2. The Stokes parameters in a single p oint in the field The definition of the Stokes parameters most frequently found in the literature is in terms of the auto- and crosscorrelations of the x and y components of the oscillating electrical field vectors in a Cartesian frame whose z axis is along the direction of propagation. Following the notation of Paper I, we represent the components of the electric field by their time-varying complex amplitudes ex (t), ey (t). The Stokes parameters are then customarily defined by (e.g. Born & Wolf; Thompson et al. 1986):

U = 2 < |ex||ey | cos > V = 2 < |ex||ey | sin >

Q = < |ex |2 - |ey |2 >

I = < |ex |2 + |ey |2 >

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Rcvr1_2 1420 MHz BRD0 3C286 3-AUG-2003
0.10







Fractional Polarization

0.05

0.00

-0.05

-0.10

X-Y XY YX -50 0 Position Angle [deg] 50

DELTAG = -0.040 0.016 PSI = -16.0 4.8 ALPHA = +0.2 2.4 EPSILON = 0.004 +0.004 PHI = +162.1 50.3 QSRC = -0.040 0.006 USRC = -0.085 0.006 POLSRC = +0.094 0.000 PASRC (**UNCORRECTED FOR MASTRO**) = -57.7 0.0 NR GOOD POINTS: X-Y = 41 XY = 42 YX = 42 / 42 SCAN 26 --------------------------------------------------- Mueller Matrix: 1.0000 -0.0198 -0.0085 -0.0198 1.0000 0.0002 -0.0074 -0.0021 0.9612 0.0050 -0.0073 -0.2760 0.0026 0.0075 0.2760 0.9611


Rcvr8_10 10000 MHz BRD0 SP 3C138 12-JAN-2003
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LETTERS
An 84-mG magnetic field in a galaxy at redshift z 5 0.692
Arthur M. Wolfe1, Regina A. Jorgenson1, Timothy Robishaw2, Carl Heiles2 & Jason X. Prochaska3
The magnetic field pervading our Galaxy is a crucial constituent of the interstellar medium: it mediates the dynamics of interstellar clouds, the energy density of cosmic rays, and the formation of stars1. The field associated with ionized interstellar gas has been determined through observations of pulsars in our Galaxy. Radiofrequency measurements of pulse dispersion and the rotation of the plane of linear polarization, that is, Faraday rotation, yield an average value for the magnetic field of B < 3 mG (ref. 2). The possible detection of Faraday rotation of linearly polarized photons emitted by high-redshift quasars3 suggests similar magnetic fields are present in foreground galaxies with redshifts z . 1. As Faraday rotation alone, however, determines neither the magnitude nor observations of the 21-cm absorption line show that the gas layer must extend across more than 0.030 to explain the difference between the velocity centroids of the fringe amplitude and phase-shift spectra9 (although the data are consistent with a magnetic field coherence length of less than 200 pc, the resulting gradient in magnetic pressure would produce velocity differences exceeding the shift of ,3 km s21 across 200 pc detected by very-long-baseline interferometry). By contrast, the transverse dimensions of radio beams subtended at neutral interstellar clouds in the Galaxy are typically less than 1 pc. Second, this field is at least an order of magnitude stronger than the 6-mG average of magnetic fields inferred from Zeeman splitting for such clouds4. We obtained further information about conditions in the absorb-



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Vol 455 | 2 October 2008 | doi:10.1038/nature07264

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NATURE | Vol 455 | 2 October 2008

LETTERS

715 in DLA-3C286, the magnetized gas cannot be confined by its selfgravity. Therefore, self-consistent magnetostatic configurations are ruled out unless the contribution of stars to S exceeds ,350M[ pc22. Although this is larger than the 50M[ pc22 surface density perpendicular to the solar neighbourhood, such surface densities are common in the central regions of galaxies. In fact, high surface densities of stars probably confine the highly magnetized gas
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Figure 1 | Line-depth spectra of Stokes parameters. Data acquired in 12.6 hours of on-source integration with the GBT radio antenna. Because the GBT feeds detect only orthogonal, linearly polarized signals, whereas Zeeman splitting requires measuring circular polarization to construct V(n), we generated V(n) by cross-correlation techniques23. The velocity v 5 0 km s21 corresponds to z 5 0.6921526. a, Line-depth function DI(n) ; (I(n)2Ic(n))/Ic(n). Here I(n) ; s0 1 s90, with sh the power measured in linear-polarization position angle h, corresponds to the total intensity spectrum, and I (n) is a model fit to the I(n) continuum.

Figure 2 | HIRES velocity profiles for dominant low-ionization states of abundant elements in the 21-cm absorber in the direction of quasar 3C 286. Spectral resolution is Dv 5 7.0 km s21 and the average signal-to-noise ratio per 2.1-km-s21 pixel is about 30:1. The bold dashed vertical line denotes the velocity centroid of the single-dish 21-cm absorption feature and the faint dashed vertical lines denotes the velocity centroid of the resonance line shown in the figure. Our least-squares fit of Voigt profiles (red) to the data (black) yields ionic column densities as well as the redshift centroid and velocity dispersion shown in Table 1 (lower and upper green horizontal lines refer to zero and unit normalized fluxes, respectively). Because refractory elements such as Fe and Cr can be depleted onto dust grains25, we used the volatile elements Si and Zn to derive a logarithmic metal abundance with respect to solar abundances of [M/H] 5 21.30. The depletion ratios [Fe/Si] and [Cr/Zn] were then used to derive a conservative upper limit on the logarithmic dust-to-gas ratio relative to Galactic values of [D/G] , 21.8.

DV (n)

in the nuclear rings of barred spirals. These exhibit total field strengths of ,100 mG, inferred by assuming equipartition of magnetic and cosmic-ray energy densities1. However, because the rings


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