Planck intermediate results. XX. Comparison of polarized thermal emission from Galactic dust with simulations of MHD turbulence

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Volume 576 (April 2015)

A&A, 576 (2015) A105

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Issue		A&A Volume 576, April 2015


Article Number		A105
Number of page(s)		27
Section		Interstellar and circumstellar matter
DOI		https://doi.org/10.1051/0004-6361/201424086
Published online		13 April 2015

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Appendix A: Additional figures

In the main body of the paper, we showed maps and plots for the Chamaeleon-Musca and Ophiuchus fields. In this appendix we show similar figures for the remaining eight fields, in the same order as in Tables 1 and 2. We first show maps similar to Fig. 3 (Figs. A.1 to A.8), then distribution functions of p and N_H similar to Fig. 4 (Figs. A.9 to A.16), and finally distribution functions of and p similar to Fig. 7 (Figs. A.17 to A.24).

Fig. A.1

Same as Fig. 3, but for the Polaris Flare field. Top: total intensity at 353 GHz. Middle: polarization fraction p, column density N_H (contours in units of 10²¹ cm^-2), and magnetic orientation (bars). Bottom: angle dispersion function with lag δ = 16′ (see Sect. 2.5) with contours and bars identical to the middle row. Note that contours levels are different from those of Fig. 3.

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	Fig. A.2 Same as Fig. 3, but for the Taurus field.
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	Fig. A.3 Same as Fig. 3, but for the Orion field.
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	Fig. A.4 Same as Fig. 3, but for the Microscopium field.
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	Fig. A.5 Same as Fig. 3, but for the Pisces field.
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	Fig. A.6 Same as Fig. 3, but for the Perseus field.
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	Fig. A.7 Same as Fig. 3, but for the Ara field.
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	Fig. A.8 Same as Fig. 3, but for the Pavo field.
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Fig. A.9

Same as Fig. 4, but for the Polaris Flare field. Two-dimensional distribution function of polarization fraction p and column density N_H. The distribution function is presented in logarithmic colour scale and includes only points for which p/σ_p> 3. The dashed red line corresponds to the absolute maximum polarization fraction p_max and the solid red curves show the upper and lower envelopes of p as functions of N_H. The solid black line is a linear fit to the decrease of the maximum polarization fraction with column density at the high end of N_H (see Table 2 for the fitting ranges and fit parameters).

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	Fig. A.10 Same as Fig. 4, but for the Taurus field.
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	Fig. A.11 Same as Fig. 4, but for the Orion field.
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	Fig. A.12 Same as Fig. 4, but for the Microscopium field. Note that the ranges in N_H and p are different from Fig. 4, and that no fit is performed.
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	Fig. A.13 Same as Fig. 4, but for the Pisces field. Note that the ranges in N_H and p are different from Fig. 4, and that no fit is performed.
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	Fig. A.14 Same as Fig. 4, but for the Perseus field. Note that the ranges in N_H and p are different from Fig. 4, and that no fit is performed.
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	Fig. A.15 Same as Fig. 4, but for the Ara field. Note that the ranges in N_H and p are different from Fig. 4, and that no fit is performed.
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	Fig. A.16 Same as Fig. 4, but for the Pavo field. Note that the ranges in N_H and p are different from Fig. 4, and that no fit is performed.
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Fig. A.17

Same as Fig. 7, but for the Polaris Flare field. Two-dimensional distribution function of and polarization fraction p. The angle dispersion function is computed at a lag δ = 16′. Only pixels for which p/σ_p> 3 are retained. The dashed grey line is the large-scale fit (with FWHM = 1° and ) , the solid black line shows the mean for each bin in p (the bin size is Δlog (p) = 0.008) and the dashed black line is a linear fit of that curve in log-log space, restricted to bins in p which contain at least 1% of the total number of points (so about 150 points per bin).

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	Fig. A.18 Same as Fig. 7, but for the Taurus field.
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	Fig. A.19 Same as Fig. 7, but for the Orion field.
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	Fig. A.20 Same as Fig. 7, but for the Microscopium field. Note that the range in p is different from Fig. 7.
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	Fig. A.21 Same as Fig. 7, but for the Pisces field. Note that the range in p is different from Fig. 7.
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	Fig. A.22 Same as Fig. 7, but for the Perseus field. Note that the range in p is different from Fig. 7.
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	Fig. A.23 Same as Fig. 7, but for the Ara field. Note that the range in p is different from Fig. 7.
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	Fig. A.24 Same as Fig. 7, but for the Pavo field. Note that the range in p is different from Fig. 7.
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Appendix B: Derivation of the Stokes parameters for emission

The derivation of the Stokes equations Eqs. (5)–(7), as presented by Wardle & Königl (1990) based upon Lee & Draine (1985), considers the extinction cross sections C_∥ and C_⊥ for light that is polarized parallel or perpendicular to the grain symmetry axis, and distinguishes oblate and prolate grains. Say that at each point M on the line of sight we define a reference frame (Mx₀y₀z₀) such that z₀ points to the observer, and the local magnetic field B is in the (My₀z₀) plane. With β the angle between B and the angular momentum J of a rotating grain at M, and γ the angle between B and the plane of the sky, as defined in Fig. 14, Lee & Draine (1985) give, for oblate grains and for prolate grains For spherical grains, all these cross-sections are of course equal, C_x₀ = C_y₀ = C_⊥ = C_∥. The expressions for the Stokes parameters in terms of the cross-sections are where the average ⟨ ... ⟩ is performed on the possible angles β. The equivalent expressions given by Wardle & Königl (1990) are incorrect in omitting the factor 1 / 2 (it is easily checked that our expressions match the expected form of I in the case of spherical grains, and of P/I in the case of fully polarizing grains: 100% polarization when C_y₀ = 0.

Computation of the sums and differences of C_x₀ and C_y₀ for both grain geometries leads to the same expressions for the Stokes parameters where we have introduced the average cross-section (B.11)and the polarization cross section These expressions match those in Martin (1972), Martin (1974), Martin (1975), and Draine & Fraisse (2009); those adopted by Lee & Draine (1985) are a factor 2 larger. The parameter p₀ is then given by (B.14)with R a Rayleigh reduction factor accounting for the chosen form of imperfect alignment (Lee & Draine 1985). Writing the equations for I, Q and U using the optical depth τ_ν (which is small in the submillimetre) in place of the physical position s on the line of sight, one is led to Eqs. (5)–(7).

The intrinsic polarization fraction is easily computed for both grain geometries:

© ESO, 2015

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