Issue 
A&A
Volume 576, April 2015



Article Number  A105  
Number of page(s)  27  
Section  Interstellar and circumstellar matter  
DOI  https://doi.org/10.1051/00046361/201424086  
Published online  13 April 2015 
Planck intermediate results. XX. Comparison of polarized thermal emission from Galactic dust with simulations of MHD turbulence^{⋆}
^{1}
APC, AstroParticule et Cosmologie, Université Paris Diderot,
CNRS/IN2P3, CEA/lrfu, Observatoire de Paris, Sorbonne Paris Cité,
10 rue Alice Domon et Léonie
Duquet, 75205
Paris Cedex 13,
France
^{2}
African Institute for Mathematical Sciences,
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Muizenberg, Cape
Town, South Africa
^{3}
Agenzia Spaziale Italiana Science Data Center,
via del Politecnico snc,
00133
Roma,
Italy
^{4}
Agenzia Spaziale Italiana, Viale Liegi 26, 00198
Roma,
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^{5}
Astrophysics Group, Cavendish Laboratory, University of
Cambridge, J J Thomson
Avenue, Cambridge
CB3 0HE,
UK
^{6}
Astrophysics & Cosmology Research Unit, School of
Mathematics, Statistics & Computer Science, University of
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Africa
^{7}
Atacama Large Millimeter/submillimeter Array, ALMA Santiago
Central Offices, Alonso de
Cordova 3107, Vitacura, Casilla
763 0355
Santiago,
Chile
^{8}
CITA, University of Toronto, 60 St. George St., Toronto, ON
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Canada
^{9}
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^{10}
California Institute of Technology,
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^{11}
Centro de Estudios de Física del Cosmos de Aragón
(CEFCA), Plaza San Juan, 1,
planta 2, 44001
Teruel,
Spain
^{12}
Computational Cosmology Center, Lawrence Berkeley National
Laboratory, Berkeley, California, USA
^{13}
DSM/Irfu/SPP, CEASaclay, 91191
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France
^{14}
DTU Space, National Space Institute, Technical University of
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327, 2800
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^{15}
Département de Physique Théorique, Université de
Genève, 24 quai E.
Ansermet, 1211
Genève 4,
Switzerland
^{16}
Departamento de Física Fundamental, Facultad de
Ciencias,Universidad de Salamanca, 37008
Salamanca,
Spain
^{17}
Departamento de Física, Universidad de Oviedo,
Avda. Calvo Sotelo s/n,
33007
Oviedo,
Spain
^{18}
Department of Astronomy and Astrophysics, University of
Toronto, 50 Saint George
Street, Toronto, Ontario, Canada
^{19}
Department of Astrophysics/IMAPP, Radboud University
Nijmegen, PO Box
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Netherlands
^{20}
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^{21}
Department of Physics and Astronomy, Dana and David Dornsife
College of Letter, Arts and Sciences, University of Southern
California, Los
Angeles, CA
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^{22}
Department of Physics and Astronomy, University College
London, London
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UK
^{23}
Department of Physics, Florida State
University, Keen Physics
Building, 77 Chieftan Way, Tallahassee, Florida, USA
^{24}
Department of Physics, Gustaf Hällströmin katu 2a,
University of Helsinki, 00014
Helsinki,
Finland
^{25}
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Princeton, New Jersey,
USA
^{26}
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California, Santa
Barbara, California, USA
^{27}
Department of Physics, University of Illinois at
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Green Street, Urbana, Illinois, USA
^{28}
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degli Studi di Padova, via
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Sapienza, P.le A. Moro
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Dipartimento di Fisica, Università degli Studi di
Milano, via Celoria,
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Milano,
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^{32}
Dipartimento di Fisica, Università degli Studi di
Trieste, via A. Valerio
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^{33}
Dipartimento di Fisica, Università di Roma Tor
Vergata, via della Ricerca
Scientifica, 1, 00133
Roma,
Italy
^{34}
Discovery Center, Niels Bohr Institute,
Blegdamsvej 17,
2100
Copenhagen,
Denmark
^{35}
European Southern Observatory, ESO Vitacura, Alonso de Cordova 3107, Vitacura,
Casilla
19001
Santiago,
Chile
^{36}
European Space Agency, ESAC, Planck Science Office, Camino
bajo del Castillo, s/n,
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Cañada, Madrid, Spain
^{37}
European Space Agency, ESTEC, Keplerlaan 1, 2201 AZ
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Netherlands
^{38}
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^{40}
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^{41}
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^{42}
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via G.B. Tiepolo 11,
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INAF/IASF Bologna, via Gobetti 101, 40129
Bologna,
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^{45}
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Bologna,
Italy
^{46}
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Sapienza, P.le Aldo Moro
2, 00185,
Roma,
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^{47}
INFN/National Institute for Nuclear Physics,
via Valerio 2,
34127
Trieste,
Italy
^{48}
Imperial College London, Astrophysics group, Blackett
Laboratory, Prince Consort
Road, London,
SW7 2AZ,
UK
^{49}
Infrared Processing and Analysis Center, California
Institute of Technology, Pasadena, CA
91125,
USA
^{50}
Institut d’Astrophysique Spatiale, CNRS (UMR 8617),
Université ParisSud 11, Bâtiment 121, 91405
Orsay,
France
^{51}
Institut d’Astrophysique de Paris, CNRS (UMR
7095), 98bis boulevard
Arago, 75014
Paris,
France
^{52}
Institute for Space Sciences, 077125
BucharestMagurale,
Romania
^{53}
Institute of Astronomy, University of
Cambridge, Madingley
Road, Cambridge
CB3 0HA,
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^{54}
Institute of Theoretical Astrophysics, University of
Oslo, Blindern, 0315
Oslo,
Norway
^{55}
Instituto de Física de Cantabria (CSICUniversidad de
Cantabria), Avda. de los
Castros s/n, Santander, Spain
^{56}
Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove
Drive, Pasadena, California, USA
^{57}
Jodrell Bank Centre for Astrophysics, Alan Turing Building,
School of Physics and Astronomy, The University of Manchester,
Oxford Road, Manchester, M13 9PL,
UK
^{58}
Kavli Institute for Cosmology Cambridge,
Madingley Road,
Cambridge, CB3 0HA,
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^{59}
LAL, UniversitéParisSud, CNRS/IN2P3, 91898
Orsay,
France
^{60}
LERMA, CNRS, Observatoire de Paris,
61 avenue de
l’Observatoire, 75014
Paris,
France
^{61}
Laboratoire AIM, IRFU/Service d’Astrophysique – CEA/DSM –
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^{62}
Laboratoire Traitement et Communication de l’Information,
CNRS (UMR 5141) and Télécom ParisTech, 46 rue Barrault, 75634
Paris Cedex 13,
France
^{63}
Laboratoire de Physique Subatomique et de Cosmologie,
Université Joseph Fourier Grenoble I, CNRS/IN2P3, Institut National
Polytechnique de Grenoble, 53
rue des Martyrs, 38026
Grenoble Cedex,
France
^{64}
Laboratoire de Physique Théorique, Université ParisSud 11
& CNRS, Bâtiment
210, 91405
Orsay,
France
^{65}
Lawrence Berkeley National Laboratory,
Berkeley, California,
USA
^{66}
MaxPlanckInstitut für Astrophysik,
KarlSchwarzschildStr. 1,
85741
Garching,
Germany
^{67}
National University of Ireland, Department of Experimental
Physics, Maynooth, Co.
Kildare, Ireland
^{68}
Niels Bohr Institute, Blegdamsvej 17, 2100
Copenhagen,
Denmark
^{69}
Observational Cosmology, Mail Stop 36717, California
Institute of Technology, Pasadena, CA
91125,
USA
^{70}
SISSA, Astrophysics Sector, via Bonomea 265, 34136
Trieste,
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^{71}
School of Physics and Astronomy, Cardiff
University, Queens Buildings,
The Parade, Cardiff, CF24
3AA, UK
^{72}
Space Sciences Laboratory, University of
California, Berkeley, California, USA
^{73}
Special Astrophysical Observatory, Russian Academy of
Sciences, Nizhnij Arkhyz,
Zelenchukskiy region, KarachaiCherkessian Republic, 369167,
Russia
^{74}
SubDepartment of Astrophysics, University of
Oxford, Keble
Road, Oxford
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UK
^{75}
UPMC Univ Paris 06, UMR 7095, 98bis boulevard Arago, 75014, Paris, France
^{76}
Université de Toulous, UPSOMP, IRAP,
31028
Toulouse Cedex 4,
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^{77}
University of Granada, Departamento de Física Teórica y del
Cosmos, Facultad de Ciencias, 18071
Granada,
Spain
^{78}
University of Granada, Instituto Carlos I de Física Teórica
y Computacional, 18071
Granada,
Spain
^{79}
Warsaw University Observatory, Aleje Ujazdowskie 4, 00478
Warszawa,
Poland
Received: 28 April 2014
Accepted: 4 September 2014
Polarized emission observed by Planck HFI at 353 GHz towards a sample of nearby fields is presented, focusing on the statistics of polarization fractions p and angles ψ. The polarization fractions and column densities in these nearby fields are representative of the range of values obtained over the whole sky. We find that: (i) the largest polarization fractions are reached in the most diffuse fields; (ii) the maximum polarization fraction p_{max} decreases with column density N_{H} in the more opaque fields with N_{H}> 10^{21} cm^{2}; and (iii) the polarization fraction along a given line of sight is correlated with the local spatial coherence of the polarization angle. These observations are compared to polarized emission maps computed in simulations of anisotropic magnetohydrodynamical turbulence in which we assume a uniform intrinsic polarization fraction of the dust grains. We find that an estimate of this parameter may be recovered from the maximum polarization fraction p_{max} in diffuse regions where the magnetic field is ordered on large scales and perpendicular to the line of sight. This emphasizes the impact of anisotropies of the magnetic field on the emerging polarization signal. The decrease of the maximum polarization fraction with column density in nearby molecular clouds is well reproduced in the simulations, indicating that it is essentially due to the turbulent structure of the magnetic field: an accumulation of variously polarized structures along the line of sight leads to such an anticorrelation. In the simulations, polarization fractions are also found to anticorrelate with the angle dispersion function 𝒮. However, the dispersion of the polarization angle for a given polarization fraction is found to be larger in the simulations than in the observations, suggesting a shortcoming in the physical content of these numerical models. In summary, we find that the turbulent structure of the magnetic field is able to reproduce the main statistical properties of the dust polarization as observed in a variety of nearby clouds, dense cores excluded, and that the largescale field orientation with respect to the line of sight plays a major role in the quantitative analysis of these statistical properties.
Key words: ISM: general / dust, extinction / ISM: magnetic fields / ISM: clouds / infrared: ISM / submillimeter: ISM
Appendices are available in electronic form at http://www.aanda.org
© ESO, 2015
1. Introduction
Planck^{1} (Tauber et al. 2010; Planck Collaboration I 2011) is the third generation spacemission aimed at mapping the anisotropies of the cosmic microwave background (CMB). With its unprecedented sensitivity and large spectral coverage (nine channels from 30 GHz to 857 GHz) it has provided exquisite maps of that relic radiation (Planck Collaboration I 2014). With its polarimetric capabilities up to 353 GHz, Planck will also provide clues on the physics of the early Universe, by measuring the CMB polarization. However, dominant foreground emission is also partially polarized, masking the primordial signal. In the range of the High Frequency Instrument (HFI, Lamarre et al. 2010), from 100 GHz to 857 GHz, the main contribution to the observed radiation, besides point sources, is thermal emission from dust grains.
The angular momenta of aspherical and spinning grains tend to align with the local magnetic field, although the details of how this alignment proceeds are still the subject of study: see for instance Andersson (2012) for a review on observational constraints regarding grain alignment with respect to current dust models. Submillimetre thermal dust emission is therefore polarized and represents a powerful tool to study interstellar magnetic fields and dust properties. Ideally, we would like to know where in interstellar clouds, and with what efficiency the dust emission and extinction is polarized. This would allow us to use polarization data to infer the spatial structure of the magnetic field. There is an extensive literature on this topic based on observations of starlight polarization, which have been interpreted from two different viewpoints, i.e., grain alignement and magnetic field structure, without achieving a clear understanding of the respective roles of these processes in accounting for variations of polarization across the sky. A number of papers (e.g., Pereyra & Magalhães 2007; Alves et al. 2008; Marchwinski et al. 2012) use the data to infer the magnetic field strength using the ChandreskarFermi method (Chandrasekhar & Fermi 1953). Other papers focus on the observed decrease of polarization fraction p with N_{H} to interpret the data as a decrease of the dust alignment efficiency in dense clouds (Lazarian et al. 1997; Whittet et al. 2008; Chapman et al. 2011).
Magnetohydrodynamical (MHD) simulations provide a theoretical framework to consider both aspects in the interpretation of polarization datasets. Ostriker et al. (2001) were among the first to present simulated polarization maps from MHD simulations, for comparison with data and to study the field structure beyond the simple ChandrasekharFermi method. FalcetaGonçalves et al. (2008) used a similar technique to study the effect of the Alfvénic Mach number, while Pelkonen et al. (2009) added to this approach the modelling of the alignment process by radiative torques (Hoang & Lazarian 2008).
Planck has mapped the polarized dust emission with great sensitivity and resolution (Planck Collaboration Int. XIX 2014), allowing us to characterize spatial variations of dust polarization and compare data with MHD simulations with unprecedented statistics. This paper is the second in a series of four dealing with a first presentation of the Planck polarized thermal emission from Galactic dust. The other three are the following: Planck Collaboration Int. XIX (2014) describes the polarized dust emission at 353 GHz as seen by Planck over the whole sky and shows in particular that the maximum polarization fraction p_{max} at a given total gas column density N_{H} decreases as N_{H} increases, and that there is an anticorrelation between polarization fractions p and angle dispersion functions , an effect which has also been seen with starlight polarization data (Hatano et al. 2013). Planck Collaboration Int. XXI (2014) compares polarized thermal emission from dust at 353 GHz to polarization in extinction in the visible towards a sample of stars. Finally, Planck Collaboration Int. XXII (2014) discusses the variation of polarized thermal emission from dust with frequency, from 70 to 353 GHz. Both Planck Collaboration Int. XXI (2014) and Planck Collaboration Int. XXII (2014) aim at providing constraints for models of interstellar dust.
In this paper, we use Planck polarization data at 353 GHz to present statistics of polarization fractions and angles in nearby interstellar clouds seen outside the Galactic plane. We then compare the Planck results with simulated observations of polarized thermal dust emission at 353 GHz built from a threedimensional MHD simulation of the formation of a molecular cloud within colliding flows (Hennebelle et al. 2008).
In these simulated observations, we work under the assumption that the optical properties and the intrinsic polarization fraction of dust grains are constant. At this stage we do not aim at testing models of grain alignment. In this picture, it is expected that the polarization fraction should be maximal when the magnetic field is in the plane of the sky and should, in this case, yield valuable information on the intrinsic polarization fraction. That is why we first focus on the decrease of the maximum value of p, rather than its mean or median values, with increasing column density. We then consider the correlation between polarization fractions and local measures of the dispersion in polarization angles, as it is expected that larger angular dispersions should lower the observed polarization fraction.
The paper is organized as follows. Section 2 describes the Planck data used and the statistics drawn from them in the selected regions. Section 3 presents simulated polarized emission observations based on an MHD simulation of interstellar turbulence and compares their statistical properties with those found towards similar fields in the Planck data. Conclusions are given in Sect. 4. Appendix A presents supplementary figures, and Appendix B details the derivation of the equations yielding the Stokes parameters for dust emission.
2. Planck observations of polarized dust emission
2.1. Planck allsky data postprocessing
The data processing of Planck HFI is presented in Planck Collaboration VI (2014), Planck Collaboration VII (2014), Planck Collaboration VIII (2014), Planck Collaboration IX (2014), and Planck Collaboration X (2014). The specifics of the data processing in terms of polarization are given in Planck Collaboration Int. XIX (2014). We use the same Planck data set as that presented in Planck Collaboration Int. XIX (2014), i.e., full 5survey HFI mission data for Stokes I, Q, and U at 353 GHz (which is the Planck channel offering the best signaltonoise ratio for dust polarization) from the “DR3” internal data release. Bandpass mismatch between individual elements of a pair of polarization sensitive bolometers (PSBs) is corrected using inflight measurements for the dust emission but not for the negligible CO J = 3→2 emission (Planck Collaboration IX 2014). From the total intensity map we subtract the offset I_{offset} = 0.0887 MJy sr^{1} to set the Galactic zero level at 353 GHz (Planck Collaboration XI 2014). Note that this value includes the cosmic infrared background (CIB) monopole and is slightly different from the one given in Planck Collaboration XI (2014), as the maps are not the same (full mission vs. nominal mission). We do not correct for zodiacal light emission, nor for the residual dipole identified by Planck Collaboration XI (2014) at 353 GHz. CMB and CIB fluctuations are ignored, since the regions selected in this study are outside the CMBCIB mask described in Planck Collaboration Int. XIX (2014), so the polarized emission there is dominated by the dust.
The Planck polarization and intensity data that we use in this analysis have been generated in exactly the same manner as the data publicly released in March 2013 and described in Planck Collaboration I (2014) and associated papers. Note, however, that the publicly available data include only temperature maps based on the first two surveys. Planck Collaboration XVI (2014) shows the very good consistency of cosmological models derived from intensity only with polarization data at small scales (high CMB multipoles). However, as detailed in Planck Collaboration VI (2014; see their Fig. 27), the 2013 polarization data are known to be affected by systematic effects at low multipoles which were not yet fully corrected, and thus these data were not used for cosmology^{2}. We have been careful to check that the Galactic science results in this paper are robust with respect to these systematics^{3}.
Fig. 1 Locations of the selected nearby fields. The background map represents optical depth τ_{353} at 353 GHz on a logarithmic scale, at 5′ resolution (Planck Collaboration XI 2014). The map uses a Mollweide projection in Galactic coordinates, with (l,b) = (0°,0°) at the centre. 
Locations and properties of the selected fields.
We focus in this paper on the polarization fractions p and the polarization angles ψ derived from the Stokes I, Q, and U maps obtained by Planck at 353 GHz and at an angular resolution of 15′. In the absence of noise, p and ψ are defined by (1)and (2)Note that ψ is here defined in the HEALPix^{4} convention (Górski et al. 2005), which means that angles are counted positively clockwise from the northsouth direction. Working in that convention instead of the IAU one, which is anticlockwise (Planck Collaboration Int. XXI 2014), has however no impact on the results presented here. Additionally, since we work on ratios of Stokes parameters, no colour correction is necessary.
When (possibly correlated) noise affects the Stokes parameters, the polarization fraction computed directly using Eq. (1) is biased. We call this one the “naïve” estimator of p, but various methods have been devised to correct for the bias (Montier et al. 2014a), and their respective efficiencies are compared in Montier et al. (2014b). Among them is the modified asymptotic (MAS) estimator introduced by Plaszczynski et al. (2014), which is computed from the naive estimator and the noise covariance matrix pertaining to Q and U. Another estimator of the polarization fraction and angle is the Bayesian estimator described in Montier et al. (2014a) and Planck Collaboration Int. XIX (2014), which has the advantage of taking into account the full noise covariance matrix in I, Q and U, and also taking into account the uncertainty on the zerolevel offset for I. In the rest of this paper, except where noted, the maps of polarization fraction p and polarization angle ψ at 353 GHz refer to these Bayesian estimators. The Bayesian method also provides maps of the polarization fraction and angle uncertainties, σ_{p} and σ_{ψ}.
For the total hydrogen column density map N_{H}, we use a conversion from the optical depth at 353 GHz, τ_{353}, derived from Planck Collaboration XI (2014): for N_{H} ≳ 2 × 10^{21} cm^{2}, the dust opacity is approximately constant, with σ_{353} = τ_{353}/N_{H} ≃ 1.2 × 10^{26} cm^{2}. We are aware that this conversion is crude, with possible variations in dust opacity of the order of 20% to 25%, but our findings do not critically depend on that calibration.
All of the maps used in this study have a HEALPix resolution N_{side} = 1024.
2.2. Overview of the statistics of polarized emission in various fields
We have selected ten regions, each 12° × 12° in size, that are highlighted in Fig. 1 and whose locations are given in Table 1. These are the same as some of the individual regions mentioned in Planck Collaboration Int. XIX (2014). All of these fields are outside the Galactic plane and probe nearby interstellar material, but they exhibit very different physical conditions, from the diffuse, turbulent ISM with little to no starforming activity (Polaris Flare), to selfgravitating, starforming clouds (Orion). They also differ in terms of polarized emission. Some diffuse regions have high polarization fractions (e.g., Pavo), while some have low polarization fractions (e.g., Polaris Flare). This variety of conditions in terms of polarization fraction and gas content is emphasized in Fig. 2, which shows the distribution of p and N_{H} in these regions, compared with the largescale distribution shown in Planck Collaboration Int. XIX (2014). The latter is represented by its upper and lower envelopes, computed from the 0.01% and 99.99% percentiles of the p distribution within each bin in column density. All the envelopes of twodimensional distribution functions shown in this paper are computed in this fashion. Note that to facilitate the comparison with Planck Collaboration Int. XIX (2014), Fig. 2 uses maps at 1° resolution. In the rest of the paper, as already stated, we use 15′ resolution maps.
Fig. 2 Twodimensional distribution functions of polarization fraction p and column density N_{H} in the fields highlighted in Fig. 1. Top: Polaris Flare (magenta), Taurus (red), ChamaeleonMusca (black), Microscopium (blue), and Pisces (green). Bottom: Ophiuchus (magenta), Pavo (red), Ara (black), Perseus (blue), and Orion (green). On both panels, the solid red lines show the upper and lower envelopes (see text) of the largescale distribution of p and N_{H}, clipped below N_{H} = 2 × 10^{20} cm^{2}, while the dashed red lines correspond to p = 0 and the maximum value p = 0.198 (i.e., 19.8%) quoted in Planck Collaboration Int. XIX (2014). 
It appears that for column densities between a few times 10^{20} cm^{2} and a few times 10^{22} cm^{2}, the selected fields probe most of the range of polarization fractions observed over the whole sky in this range of column densities. The diffuse Polaris Flare field shows low polarization, while high polarization fractions are reached at similar column densities in the ChamaeleonMusca complex, which, being closer to the Galactic plane, is threaded by the largescale Galactic magnetic field. Another notable feature of Fig. 2 is the fact that in regions with the largest column densities (Taurus, Orion, and Ophiuchus) the maximum polarization fraction decreases with increasing N_{H}, and that the slopes are comparable to the largescale trend.
In the following, we perform statistical analyses of the polarization data in these nearby fields by simply selecting HEALPix pixels whose centres fall within the region of interest, directly from the largescale maps. Only pixels for which p/σ_{p}> 3 are retained. This threshold is a reasonable value above which the polarization signaltonoise ratio is properly estimated (Montier et al. 2014b). Note that some of the fields in Table 1 are quite diffuse (e.g., Pavo), so that the dynamic range in column densities is too small to exhibit a significant relationship between p_{max} and N_{H}. These diffuse fields are therefore discarded in the later analysis.
We also build local maps of polarized emission using gnomic projections of the HEALPix maps. These are shown in the middle row panels of Fig. 3 for the Ophiuchus and ChamaeleonMusca fields. Similar figures for all other fields are given in Appendix A. On all these maps, which share the same scale, we show the polarization fractions p at 353 GHz (colour scale) overlaid with contours of the total gas column density and bars of constant length giving the orientation of the apparent projection of the magnetic field on the plane of the sky. These are built by rotating the 353 GHz polarization bars by 90° so as to recover the average magnetic field orientation in the plane of the sky. In the rest of the paper, we will refer to the rotated polarization bars as the magnetic orientation bars. Note that although they are plotted once every few pixels only, to improve visibility, each of these bars represents the orientation at the given pixel. In other words, beyond the 15′ smoothing performed on the Stokes maps, no further averaging is done to plot the orientation bars on Fig. 3 and similar plots.
The largescale structure of the Galactic magnetic field appears clearly (see e.g., the top part of the ChamaeleonMusca field, Fig. 3). There is also a strong correlation between the coherence of the polarization orientation and the level of polarization fraction, in the sense that more ordered regions have higher polarization fractions. This feature, which is already seen at 1° resolution in Planck Collaboration Int. XIX (2014), is discussed later on in Sect. 2.5.
A final qualitative aspect of these maps is that regions with higher column densities tend to be less polarized than their surroundings. An example of this effect can be seen in the ChamaeleonMusca field (Fig. 3, center right panel) near (l,b) = (301°, −9°), where p ≃ 10%, while it is surrounded by more diffuse material with p ≃ 15%. A future paper (Planck Collaboration Int. XIII 2014) will discuss in more detail the structure of the polarized thermal emission with respect to the morphology of the clouds themselves.
Fig. 3 Maps of the Ophiuchus and ChamaeleonMusca fields. Left: Ophiuchus field. Right: ChamaeleonMusca field. Top: total intensity at 353 GHz. Middle: polarization fraction p, column density N_{H} (contours in units of 10^{21} cm^{2}), and magnetic orientation (bars, see text). Bottom: angle dispersion function with lag δ = 16′ (see Sect. 2.5) with contours and bars identical to the middle row. In all maps, the 15′ beam is shown in the lowerleft corner. 
Polarization statistics in the selected fields.
2.3. Maximum polarization fraction
We give in Table 2 the maximum polarization fractions p_{max} in all the selected fields. Note that for the most diffuse fields Microscopium, Pisces, Perseus, Ara, and Pavo, the quoted values should be taken with caution, since most pixels in these regions have , which corresponds roughly to , and therefore the effect of the (uncertain) zerolevel offset on the polarization fraction p may not be negligible.
In the less diffuse fields, the values of p_{max} are noticeably larger than those found in the same fields at 1° resolution in Planck Collaboration Int. XIX (2014)^{5}, which shows the strong effect of spatial resolution on polarization measurements. The uncertainties σ_{pmax} on the maximum polarization fractions, listed in Table 2, are derived from the various sources of uncertainty involved.
First, the noise properties on the Stokes parameters I, Q, and U in each pixel are described in the data by the noise covariance matrices, which are input in the Bayesian method of Montier et al. (2014a) and Planck Collaboration Int. XIX (2014), and lead to a map of the uncertainty σ_{p} on the polarization fraction. This includes the 0.0068 MJy sr^{1} uncertainty on the zerolevel offset. We then compute the difference σ_{pmax,p} between the maximum polarization fractions found in the maps of p − σ_{p} and p + σ_{p}.
Second, there is a part of the uncertainty related to the method used to debias the data (Montier et al. 2014a). We have computed the standard deviation σ_{pmax,d} of the maximum polarization fractions obtained in each field when using the “naïve” , modified asymptotic (MAS, Plaszczynski et al. 2014) and Bayesian estimators of p.
Third, we have computed the standard deviation σ_{pmax,s} of the maximum polarization fractions obtained in each field when considering subsets of the data, namely halfring maps (one half of each stable pointing period) and detector set maps (one half of the detectors).
The final uncertainty quoted in Table 2 is then given by the quadratic sum (3)It should be noted that the last contribution is usually the dominant one in the selected fields, and that the uncertainty related to the debiasing method is much smaller than the other two.
Fig. 4 Twodimensional distribution function of polarization fraction p and column density N_{H}. Top: Ophiuchus field. Bottom: ChamaeleonMusca field. The distribution functions are presented in logarithmic colour scale and include only points for which p/σ_{p}> 3. The dashed red lines correspond to the absolute maximum polarization fractions 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). 
2.4. Polarization fraction vs. column density
We show the distributions of p and N_{H} for the Ophiuchus and ChamaeleonMusca fields in Fig. 4 and for all other fields in Appendix A. The decrease in maximum polarization fraction p_{max} at higher column densities is apparent for all fields, above a given threshold in N_{H} that depends on the field and is of the order of 10^{21} to 3 × 10^{21} cm^{2}, corresponding to visual extinctions A_{V} ≃ 0.6 to 1.7, for the fields that are not too diffuse (Polaris Flare, Taurus, Orion, ChamaeleonMusca, and Ophiuchus). Below this threshold, the polarization fraction may be related to the background more than to the clouds themselves. To quantify the decrease in maximum polarization fraction p_{max} with increasing N_{H}, we consider the upper envelope of the distribution of p and N_{H}, computed as described in Sect. 2.2, and fit this curve with a function , restricted to a range of column densities that depends on the field considered (see Table 2). Note that we perform this fit for the above five fields only, for which there is a large enough dynamic range in column density.
Results of these fits are shown as solid black lines on each panel of Fig. 4, and values of the slopes m and intercepts c are listed in Table 2. Uncertainties on these parameters are derived in the same way as for the maximum polarization fractions p_{max} in the previous section. The slopes m range between −0.068 for Orion and −0.140 for Taurus, and regions exhibiting stronger column density peaks (e.g., Orion) tend to have shallower slopes than more diffuse molecular clouds (e.g., Polaris Flare).
As mentioned before, the pixels selected for plotting Fig. 4 and performing the fits are those for which the polarization signaltonoise ratio is p/σ_{p}> 3. We have checked that modifying this threshold does not change our results, as can be seen in Fig. 5, which shows the same as the top panel of Fig. 4 but with a signaltonoise ratio threshold p/σ_{p}> 10. The effect of that stricter selection is to remove points below the original lower envelope, but leaves the upper envelope unchanged. Consequently, both the absolute maximum polarization fraction p_{max} and the slope of the decrease of p_{max} at the high end of column densities are quite robust.
2.5. Polarization angle coherence vs. polarization fraction
We show in the bottom row panels of Fig. 3 the maps of the angle dispersion functions for the Ophiuchus and ChamaeleonMusca fields. Similar maps for all other fields are shown in Appendix A. We recall that this function, defined in Planck Collaboration Int. XIX (2014), is (4)where the sum extends over pixels whose distances from the central pixel r are between δ/ 2 and 3δ/ 2. Here they are computed at a lag δ = 16′, comparable to the size of the beam’s FWHM. One can readily see filamentary structures that correspond to regions where the polarization angle is less ordered or where it changes abruptly. These filaments are already noted at 1° resolution in Planck Collaboration Int. XIX (2014) over several degrees. These regions of large angular dispersions correspond to regions of low polarization fraction, as can be seen for instance by comparing the middle and bottom row panels of Fig. 3.
When increasing the value of the lag δ, we obtain maps of such as that presented in Fig. 6 for the Ophiuchus field at δ = 34′ (approximately twice the FWHM). It appears that the overall value of increases with lag, as already noted in Hildebrand et al. (2009) and Planck Collaboration Int. XIX (2014). However, since has an upper limit of 90°, this means that the anticorrelation with p (see below) will flatten out at large lags. Note however that a completely random sample yields (Planck Collaboration Int. XIX 2014). Values larger than this are few, but they do exist, as can be seen on the maps of in Figs. 3 and 6. They may be linked to sharp boundaries between two wellordered regions: for instance, the angle dispersion function at the interface between two halfplanes with orthogonal magnetic orientations is .
Fig. 6 Map of for the Ophiuchus field computed at δ = 34′. Contours are the same as in the map at δ = 16′ (bottom left panel of Fig. 3). 
Fig. 7 Twodimensional distribution function of and polarization fraction p for the Ophiuchus field. 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 largescale 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 loglog space, restricted to bins in p which contain at least 1% of the total number of points (so about 150 points per bin). 
To confirm the visual impression that the spatial coherence of the polarization angle is anticorrelated with the polarization fraction, we show the distribution function of these two quantities for the Ophiuchus and ChamaeleonMusca fields in Figs. 7 and 8, respectively, and for all other fields in Appendix A.
The largescale anticorrelation seen in Planck Collaboration Int. XIX (2014) at 1° resolution and δ = 30′ is also present when using a lag close to the beam size. With , we find it to be , where is measured in degrees. Since in this case the ratio δ/ FWHM is the same as for our higher resolution maps (FWHM = 15′ and δ = 16′), we compare the anticorrelations found in the selected fields to this law. Note that the slope −0.75 is similar to the value −0.834 quoted in Planck Collaboration Int. XIX (2014), but the intercept is larger (−0.06 vs. −0.504). This points to a global increase of at larger δ/ FWHM values, which we interpret as a decorrelation of polarization angles at larger lags.
The distributions of p and in the various fields considered show an anticorrelation very similar to the largescale trend, with slopes and intercepts of the fits through the data points that are very close to the largescale fit values. When increasing the lag at the same resolution, however, increases and the anticorrelation with p flattens out, as can be seen in Fig. 9. The linear fits for the individual fields are listed in Table 2. The uncertainties on the parameters m′ and c′ are the quadratic sums of uncertainties obtained in three ways: (i) by performing the linear regression using the three estimators of p, i.e., the “naïve”, MAS and Bayesian ones; (ii) by using halfring maps and detector set maps; (iii) via a MonteCarlo simulation using the maps of polarization fraction uncertainty σ_{p} and angle dispersion function uncertainty (Planck Collaboration Int. XIX 2014).
3. Simulations of polarized emission
3.1. Simulations of MHD turbulence
We aim to compare the observed polarization statistics in the selected fields to predictions built on the results of a numerical simulation of MHD turbulence. This simulation is described in detail in Hennebelle et al. (2008)^{6}. It follows the formation of clumps of dense and cold gas (cold neutral medium, CNM) out of magnetized warm neutral atomic gas (warm neutral medium, WNM) in an open box of 50 pc on each side, without reaching the stage when cold cores of column density larger than 2 × 10^{22} cm^{2} form.
The simulation cube initially contains a uniform distribution of WNM with density n_{H} = 1 cm^{3} and temperature T = 8000 K, and two converging flows of that same gas are injected from opposing faces along the x axis with a velocity ΔV_{x} ≃ 40 km s^{1} relative to each other. Spatial modulations of the velocity are imposed on the incoming flows, with amplitudes relative to the mean flow of about unity and a periodicity of about 10 pc. Periodic boundary conditions are applied on the remaining four faces. The total mass contained in the cube continuously increases with time. The magnetic field’s initial direction is along that of the incoming flows, and its intensity is about 5 μG, consistent with observational values at these densities (Crutcher et al. 2010). There is therefore a largescale anisotropic component of the magnetic field throughout the simulation, as well as a turbulent component linked to the velocity perturbations imposed on the converging flows.
These flows collide near the midplane, where the combined effects of cooling and selfgravity eventually lead to the formation of dense (n_{H}> 100 cm^{3}) clumps of cold gas (T of the order of 10–50 K; Hennebelle & Audit 2007). To follow that condensation, the grid is adaptively refined, with an effective (maximum) resolution of 0.05 pc.
In this paper, we select a cubic subset (18 pc × 18 pc × 18 pc) of the density and magnetic field in the simulation snapshot timed at t = 10.9 Myr, which corresponds to an evolved state of the simulation, given the crossing time t_{c} ≃ 2.4 Myr. The structures present in the simulation are due to the collision of the incoming flows and not to a pure gravitational collapse, since the initial freefall time is t_{ff} ≃ 44 Myr. However, some of the densest structures (n_{H}> 10^{4} cm^{3}) may have had time to collapse.
The chosen subset is located near the midplane, so that the influence of boundary conditions is minimal. It contains approximately 3200 M_{⊙} of gas; its physical properties are listed in Table 3, and the distribution functions of total gas density n_{H} and magnetic field components B_{x}, B_{y}, B_{z} are shown in Figs. 10 and 11, respectively. The standard deviations are very similar for all three magnetic field components, but only the x component has a significant mean value, which shows that the mean magnetic field within the cube is approximatively aligned with the x axis, that is with the incoming flows.
Fig. 10 Distribution function of the total gas density n_{H} in the selected subset of the simulation, with cell sizes 0.1 pc × 0.1 pc × 0.1 pc. The solid red line shows the mean value ⟨n_{H}⟩ = 17 cm^{3} and the solid blue line the median value . 
Physical properties of the subset of the simulation.
We would like to stress here that the MHD simulations we use for comparison with the Planck polarization data do not faithfully reproduce the whole range of densities and column densities spanned by the cloud sample of Table 1, i.e., from diffuse molecular clouds (Polaris Flare) to massive starforming clouds (Orion). However, as shown in Table 1, only a few percent of the pixels (at most 5.4% in Orion) have column densities larger than 10^{22} cm^{2} in these fields, the regions of star formation filling only a small fraction of the area in each field. The MHD simulations with their broad range of densities (Fig. 10) and column densities reaching^{7}N_{H} = 1.6 × 10^{22} cm^{2} are therefore representative of the dynamics of the bulk of the gas. Together with their anisotropy, due to the largescale magnetic field pervading the cube, these simulations are particularly well suited to analyse the polarization properties of nearby molecular clouds immersed in their low density and largescale environment.
To compute simulated polarization fractions p, the local gas density n_{H} and magnetic field components B_{x}, B_{y}, B_{z} are extracted from the simulation and interpolated on a regular grid at the nexttohighest spatial resolution available, so that pixel sizes are approximately Δx = 0.1 pc. These cubes are used in the following section to build simulated polarized emission maps. However, they are first rotated around the y axis, as sketched out in Fig. 13, to explore the full range of possible angles between the mean magnetic field and the line of sight, and therefore to test the effects of the largescale magnetic field’s anisotropy. The viewing angle α introduced in Fig. 13 is such that the mean magnetic field is approximately in the plane of the sky for α = 0°, and along the line of sight for α = 90°.
Fig. 11 Distribution functions of the components of the magnetic field, B_{x} (blue), B_{y} (green), and B_{z} (red), in the selected subset of the simulation, with cell sizes 0.1 pc × 0.1 pc × 0.1 pc. 
Fig. 12 Distribution functions of the total gas column density N_{H} in the selected subset of the simulation, using viewing angles α = 0° (red) and α = 90° (blue). These distribution functions are computed after convolution with the 15′ beam. 
Fig. 13 Sketch of the rotation of the simulation subset. 
Fig. 14 Definition of angles. Here the line of sight is along the z′ axis (see Fig. 13), γ is the angle the magnetic field B makes with the plane of the sky, φ is the local polarization angle, and χ is the position angle of the plane of the sky projection B_{⊥}, both in the HEALPix convention, so counted positively clockwise from the northsouth direction, while the IAU convention is anticlockwise (Planck Collaboration Int. XIX 2014). 
Fig. 15 Simulated Planck maps. Top: total gas column density. Middle: polarization fraction and angle, with contours being the column density at values indicated in units of 10^{21} cm^{2}, and the bars indicate magnetic orientation. Bottom: angle dispersion function at lag δ = 16′, with the same contours and bars as in the middle row. Left: viewing angle α = 0°. Right: viewing angle α = 90°. In each row, the same colour scale is used. In the lower left corner of each plot (yellow circle) is the 15′ FWHM beam. 
3.2. Simulated Planck observations
We build simulated Stokes I, Q, and U maps by integrating along the line of sight (z′ in Fig. 13) through the rotated simulation cube, following the method in Wardle & Königl (1990), Fiege & Pudritz (2000), Pelkonen et al. (2009), and Padovani et al. (2012). Because of a number of inconsistencies in the literature, we give the correct derivation in Appendix B, drawing on the works of Lee & Draine (1985) and Wardle & Königl (1990). This results in: Here p_{0} is a polarization fraction parameter related to the intrinsic polarization fraction (see Eq. (8) and Appendix B), γ is the angle that the local magnetic field makes with the plane of the sky, and φ is the local polarization angle in the HEALPix convention. This angle differs by 90° from the angle χ of the plane of the sky projection of the magnetic field, as defined in Fig. 14, and should not be confused with the actual polarization angle ψ. These angles are equal (φ = ψ) only for a uniform magnetic field along the line of sight.
Note that the corrective term in Eq. (5) is incorrectly written in Fiege & Pudritz (2000), Gonçalves et al. (2005), Pelkonen et al. (2009), and Padovani et al. (2012), with p_{0}/ 2 instead of p_{0}.
The hypotheses made here, besides the absence of background radiation, are that p_{0} = 0.2 is uniform, that the source function S_{ν} = B_{ν}(T_{d}) is that of a blackbody with an assumed uniform dust temperature T_{d} = 18 K, and that since we are working at 353 GHz the optical depth is simply given by dτ_{ν} = σ_{353}n_{H} dz′. We use the value σ_{353} = 1.2 × 10^{26} cm^{2} (see Sect. 2.1), and n_{H} is the total gas density in the simulation. Given the maximum gas column density in the simulation subset computed over all possible viewing angles α, N_{H,max} = 1.6 × 10^{22} cm^{2}, the maximum optical depth at 353 GHz using this conversion factor is τ_{max} = 1.9 × 10^{4}, so we may safely neglect optical depth effects and take e^{−τν} = 1 in the I, Q, and U integrals. We are aware (Planck Collaboration XI 2014) that the opacity actually varies with N_{H}, but the variation is at most a factor of 3 from the value assumed here, so the optical depth is in any case much lower than unity. Moreover, the choice of the conversion factor has no impact on the simulated maps of polarization fractions and angles, provided that a constant value is assumed along each line of sight.
We note that the dense cores that exist in our simulated cube are only weakly shielded from the ambient UV radiation field. Indeed, the mean column density through the cube is about 10^{21} cm^{2} (corresponding to A_{V} ≃ 0.6), which is comparable to the values in the simulation of Pelkonen et al. (2009), but over a much larger volume (18 pc box compared to less than 1 pc); the bulk of the gas is therefore more fragmented and radiation penetrates more easily (Levrier et al. 2012). That is why we take a uniform parameter p_{0}.
The maps of Stokes parameters are placed at a distance of D = 100 pc and convolved with a circular 15′ FWHM Gaussian beam (corresponding to a physical size 0.44 pc). The resulting field of view is a little less than 10° across, which is comparable to the selected Planck fields, and small enough that separate smoothing of Stokes I, Q, and U is not an issue (see Appendix A of Planck Collaboration Int. XIX 2014). Maps of polarization fractions and angles are then built from these convolved Stokes parameter maps using Eqs. (1), (2) for consistency with the Planck data. Let us stress that ψ is defined in the HEALPix convention, which means that it is counted positively clockwise from the northsouth direction, and not in the IAU convention (anticlockwise).
Figure 15 (middle row) shows the maps of polarization fraction p and magnetic orientation in these simulated observations, when integrating along the mean magnetic field (α = 90°), and perpendicular to it (α = 0°). The largescale component of the magnetic field is clearly visible in several regions, for instance in the lower right corner of the α = 0° case: it leads to longrange coherence in the polarization angle, which correlates with the highest polarization fractions and lowest column densities. Conversely, when integrating along the direction of the largescale field (α = 90°, right column), p is on average much lower, and no such longrange ordering of χ is visible, although some local correlations are present. These effects are expected from the vectorial nature of the polarization: with the magnetic field more or less aligned with the line of sight, only its transverse fluctuations lead to a signal in polarization, and these fluctuations are isotropic in the plane of the sky, so they cancel out in the integration (along the line of sight and also through beam dilution). This correlation between p and spatial coherence of the polarization angle is discussed later on (Sect. 3.4).
Statistics of simulated maps of the polarization fraction (maximum, mean and standard deviation) are shown as a function of the viewing angle α in Fig. 16. We find the maximum polarization fraction to be p_{max} ≃ 0.14–0.21 (depending on the viewing angle α). On some lines of sight, in the most tenuous parts of the map integrated perpendicularly to the largescale B (e.g., in the lower right corner of the map in the α = 0° case), p_{max} almost reaches the theoretical maximum value possible, which is the intrinsic polarization fraction, (8)obtained when the medium is homogeneous and the magnetic field is uniform and parallel to the plane of the sky (γ = 0°). Figure 16 emphasizes the importance of the magnetic field geometry on the measured p_{max}, as that value varies by about 40% over the range of viewing angles.
Fig. 16 Statistics of polarization fractions in the simulated Planck observations as a function of viewing angle α (see Fig. 13). The solid blue line shows p_{max}, the solid black line shows the mean p, and the solid green line shows the value of p for the most diffuse lines of sight in the map. The dashed black line marks the polarization fraction parameter p_{0} and the dashed green line gives the theoretical polarization fraction in the case where the density and magnetic field are homogeneous and the latter makes an angle α with the plane of the sky (see text). The grey region shows the ± 1σ spread around the mean p. 
Fig. 17 Twodimensional distribution functions of polarization fractions and logarithmic column densities in the simulated Planck observations. Top: viewing angle α = 0°. Middle: viewing angle α = 45°. Bottom: viewing angle α = 90°. The dashed red horizontal lines and the solid red and black lines are the same as in Fig. 4. The fits to the upper envelopes are performed for N_{H}> 10^{21} cm^{2}. 
Fig. 18 Comparison between the distributions of the polarization fractions p and logarithmic column densities in the simulations (colour scale, all viewing angles combined, with upper and lower envelopes in solid red lines) and those of the observations in the selected fields (solid black lines). Note that the latter are restricted to N_{H}> 10^{21} cm^{2}. Dashed lines are linear fits of the form on the distributions’ upper envelopes, restricted to a common range of column densities 2 × 10^{21} cm^{2}<N_{H} < 2 × 10^{22} cm^{2}. 
Fig. 19 Twodimensional distribution function of log (p) and in the simulated observations for δ = 16′ and α = 0°. The solid black curve represents the evolution of the mean per bin of log (p). A linear fit is performed, restricted to bins in log (p) that contain at least 1% of the total number of points. This fit is shown as the dashed black line. The dashed grey line is the largescale fit presented in Sect. 2.5. 
Fig. 20 Slopes m′(top) and intercepts c′(bottom) of the linear fits to the distribution of log (p) and in the simulated observations, as a function of viewing angle α. The lag is δ = 16′. The dashed blue lines indicate the values for the largescale fit presented in Sect. 2.5, the dashed red lines represent the average slope and intercept over the range of α, and the grey areas indicate ± 1σ around the mean, with the standard deviation σ computed statistically over all angles. 
Fig. 21 Distribution functions of polarization fraction p and densityweighted mean of cos^{2}γ (γ is the angle of the magnetic field with respect to the plane of the sky, see Fig. 14) along the line of sight z′ in the simulation cube. Top: viewing angle α = 0°. Bottom: viewing angle α = 90°. The solid black lines show the mean values per bin of p. 
3.3. Polarization fraction vs. column density
We show in Fig. 17 the joint distribution function of polarization fractions p and total gas column densities N_{H} in the simulated observations when integrating along both directions used in Fig. 15, and in the intermediate case α = 45°. The most striking feature of the plots in Fig. 17 is the different behaviour at low column densities N_{H} < 10^{20} cm^{2}. Along these lines of sight, the density is essentially uniform, with n_{H} of about 2 cm^{3}, so the computed polarization is entirely due to magnetic field geometry; when we integrate with α = 0° the mean magnetic field is almost in the plane of the sky, γ ≃ 0°, and polarized emission is at its highest, while when we integrate with α = 90°, then the ordered field is almost along the line of sight, so γ ≃ 90° and no polarized emission appears. In fact, for each value of α, polarization fractions observed towards the most diffuse lines of sight are well reproduced by the formula for a homogeneous medium, easily derived from Eqs. (5)–(7), (9)as can be seen in Fig. 16. We may therefore only derive the polarization fraction parameter p_{0} from the maximum observed value p_{max} if the angle between the magnetic field and the plane of the sky is known, which is a strong assumption.
The second striking feature of Fig. 17 is the decrease of the maximum polarization fraction with increasing column density, as observed in the data. The same linear fit yields slopes that span values from −0.025 (for α = 80°) to −0.15 (for α = −15°), the latter being comparable to those found in the data for the selected fields.
For a global comparison between simulations and observations, we show in Fig. 18 the distribution of p and N_{H} for all the simulated fields, with their upper and lower envelopes, together with the envelope for the selected sky fields. Linear fits to the distributions’ upper envelopes are performed, restricted to a common range of column densities 2 × 10^{21} cm^{2}<N_{H} < 2 × 10^{22} cm^{2}. They yield similar values in terms of both slopes (m = −0.109 for simulations, compared to m = −0.113 for the selected fields) and intercepts (c = 2.52 for simulations, compared to c = 2.59 for the selected fields). Note that the “ripple” pattern in the density plot at low N_{H} is due to the sampling in viewing angles α, and is a signature of the decrease of p with viewing angle for the most diffuse lines of sight, as already noted in Fig. 16.
3.4. Polarization angle coherence vs. polarization fraction
The angle dispersion function is computed from the simulated ψ maps, using a lag δ = 16′, as we did for the data. We first note that the mean angle dispersion function is larger when the largescale magnetic field is oriented along the line of sight, with for α = 0° and for α = 90°, a result that is consistent with the findings of FalcetaGonçalves et al. (2008). Maps of (for the α = 0° and α = 90° cases) can be seen in the lower row panels of Fig. 15, exhibiting filamentary patterns similar to those found in observations. These filaments of high also correspond to regions where the polarization angle rotates on small scales, and are correlated with regions of low polarization fraction p (compare with the middle row panels of Fig. 15). This anticorrelation is clearly seen in distribution functions of log (p) and , as shown in Fig. 19 for the α = 0° case. A linear fit to the mean per bin of log (p) is performed, restricted to bins which contain at least 1% of the total number of points and limited to p<p_{0} to avoid the most diffuse lines of sight. The slope and intercept of the anticorrelation observed in the data are fairly well reproduced (m′ = −1.0 ± 0.3 and c′ = 0.02 ± 0.34 over the range of α, compared to m′ = −0.75 and c′ = −0.06 in observations) with steeper slopes for viewing angles α ≃ 0° and shallower slopes for viewing angles α ≃ 90° (see Fig. 20). However, since the slopes in simulations are generally steeper than what is observed, but with very similar intercepts at p = 1, the angle dispersion function in simulations is globally higher than in observations for a given polarization fraction.
This result suggests that, in the simulations, the angle dispersion function is too large for a given polarization fraction, i.e., that the magnetic field is too tangled. Since the physical processes one can think of to reduce the field’s tangling (e.g., larger field intensity with respect to turbulence or partial ionneutral decoupling) would also affect p, we propose that this difference comes from the lack of power in the low frequency modes of the simulated turbulence, as illustrated by the fact that the power spectra of the velocity and magnetic field components flatten out at small wavenumber k. In reality, molecular clouds are organized in a selfsimilar structure over a broad range of scales and that is therefore not properly reproduced in the simulations we used. In short, the largescale fluctuations of the magnetic field are closer to random in simulations than in reality.
3.5. Statistics on the magnetic field fluctuations in the simulations
Fig. 22 Distribution functions of polarization fraction p and densityweighted mean of sinχ (χ is the position angle of the projection of the magnetic field in the plane of the sky, see Fig. 14) along the line of sight z′ in the simulation cube. Top: viewing angle α = 0°. Bottom: viewing angle α = 90°. The solid black lines show the mean values per bin of p. 
Fig. 23 Distribution functions of polarization fraction p and densityweighted standard deviation of cos^{2}γ along the line of sight z′ in the simulation cube. Top: viewing angle α = 0°. Bottom: viewing angle α = 90°. The solid black lines show the mean values per bin of p. 
Fig. 24 Distribution functions of angle dispersion function and densityweighted mean of cos^{2}γ in the simulation cube. Top: viewing angle α = 0°. Bottom: viewing angle α = 90°. The solid black lines show the mean values per bin of . 
We investigate here the possible causes of the variations in the polarization fraction p and the dispersion of the polarization angle in the simulations, i.e., what are the respective roles of the field tangling and the orientation of the largescale field in the variations of p and . To quantify these roles, we compute the average and dispersion along the line of sight of both cos^{2}γ and sinχ (see Fig. 14 for the definition of angles). These quantities are computed for different viewing angles. In the following, we write the magnetic field as B = B_{0} + ΔB, where B_{0} is the largescale ordered field and ΔB is the fluctuating part of B.
The role of the average values of the angles γ and χ along the line of sight is illustrated in Figs. 21 and 22. First, the role of the largescale field B_{0} is clear: the largest values of p are obtained when B_{0} is viewed in the plane of the sky (α ≃ 0°). The largest p values are obtained when the average of cos^{2}γ along the line of sight stays close to unity. In that case, the field perturbations are such that they keep the field close to the plane of the sky, on average, hence the large p. The same effect is visible in the top panel of Fig. 22 where the largest polarization fractions are obtained for average values of χ close to 90°.
However, even in this configuration (α = 0°), small values of p are obtained. The fraction of low p values is clearly larger when the largescale field is viewed along the line of sight (α = 90°). The remarkable feature visible in Fig. 21 (bottom panel) is the proportionality of p_{max} with the average of cos^{2}γ: the smaller this average, the closer γ is to 90°, therefore the closer the field is aligned with the line of sight, and the smaller the resulting value of p_{max}. One also sees in Fig. 21 that reaches much smaller values when B_{0} is along the line of sight (bottom panel), producing lower values of p than in the case where B_{0} is in the plane of the sky (top panel).
We note, interestingly, that the same effect is not visible in Fig. 22, which displays the line of sight average of sinχ versus p: there is no such upper value of p that would scale with the average of sinχ because this fluctuation of the field direction is measured in the plane of the sky and does not affect the maximal polarization fraction that can be obtained. Instead, when B_{0} is along the line of sight for instance, the scatter of ⟨ sinχ ⟩ along the line of sight is the largest and the resulting values of p are low.
Figure 23 also illustrates the effect of the field tangling: the larger the dispersion of cos^{2}γ along the line of sight (and the larger the scatter of this dispersion), the smaller p is. Obviously, when the line of sight is dominated by the largescale field, the scatter is the lowest.
Figure 24 shows the joint distribution of the average of cos^{2}γ and , where one recognizes the role of the largescale field when α = 0°: the lowest values of are obtained when γ stays close to 0°, meaning that the field is more or less in the plane of the sky. Clearly, the largest values of are obtained when the influence of the largescale field is minimized (α = 90°, bottom panel).
4. Conclusions
To summarize, the maximum polarization fraction p_{max} observed towards the sample of nearby fields selected in this study is reached in the most diffuse fields. The largescale decrease of p_{max} with increasing N_{H} is seen in the individual fields considered here, as soon as N_{H}> 10^{21} cm^{2}. This trend is fairly well reproduced by numerical simulations of anisotropic MHD turbulence, even assuming uniform dust temperatures and grain alignment efficiencies in the gas weakly shielded from the UV radiation. The polarization of thermal dust emission observed by Planck towards these regions is essentially related to the geometry of the magnetic field and in particular to its orientation at large scales with respect to the line of sight. We do not discuss the evolution of polarization fractions at large column densities N_{H}> 3 × 10^{22} cm^{2}, for which the MHD simulation considered is not suitable. It is clear, however, that additional processes must be at work to achieve the change of slope in the p_{max} vs. relation observed towards the most opaque lines of sight. This change is probably related to variations in the properties of dust alignment, as pointed out by Soler et al. (2013). We also find that polarization fractions observed by Planck towards these nearby regions correlate well with the local coherence of the polarization angle, which is measured using the angle dispersion function . This correlation is also found in simulations, with slopes that are very close to observational values. In simulations, however, values of for a given polarization fraction are globally too high compared to observations, which points to a possible limitation of the specific MHD simulation used.
Planck (http://www.esa.int/Planck) is a project of the European Space Agency (ESA) with instruments provided by two scientific consortia funded by ESA member states (in particular the lead countries France and Italy), with contributions from NASA (USA) and telescope reflectors provided by a collaboration between ESA and a scientific consortium led and funded by Denmark.
The errorbars we quote include uncertainties associated with residual systematics as estimated by repeating the analysis on different subsets of the data. We have also checked our data analysis on the latest version of the maps available to the consortium to check that the results we find are consistent within the errorbars quoted in this paper.
http://healpix.jpl.nasa.gov. See in particular the latest version of the HEALPix primer, available at http://healpix.jpl.nasa.gov/pdf/intro.pdf
It was performed with the RAMSES code (Teyssier 2002; Fromang et al. 2006), whose adaptive mesh refinement capabilities allow for a locally high spatial sampling. It is freely available via the STARFORMAT project, http://starformat.obspm.fr/. To be precise, it is the Fiducial run under the tab Colliding flow simulation.
Acknowledgments
The development of Planck has been supported by: ESA; CNES and CNRS/INSUIN2P3INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN, JA and RES (Spain); Tekes, AoF and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); and PRACE (EU). A description of the Planck Collaboration and a list of its members, including the technical or scientific activities in which they have been involved, can be found at http://www.sciops.esa.int/index.php?project=planck&page=Planck_Collaboration. Some of the results in this paper have been derived using the HEALPix package. The authors would like to thank Charles Beichman for his careful reading of the manuscript and useful comments. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/20072013) / ERC grant agreement No. 267934.
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Online material
Appendix A: Additional figures
In the main body of the paper, we showed maps and plots for the ChamaeleonMusca 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 10^{21} 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. 
Fig. A.9 Same as Fig. 4, but for the Polaris Flare field. Twodimensional 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). 
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. 
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. 
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. 
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. 
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. 
Fig. A.17 Same as Fig. 7, but for the Polaris Flare field. Twodimensional 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 largescale 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 loglog space, restricted to bins in p which contain at least 1% of the total number of points (so about 150 points per bin). 
Fig. A.20 Same as Fig. 7, but for the Microscopium field. Note that the range in p is different from Fig. 7. 
Fig. A.21 Same as Fig. 7, but for the Pisces field. Note that the range in p is different from Fig. 7. 
Fig. A.22 Same as Fig. 7, but for the Perseus field. Note that the range in p is different from Fig. 7. 
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 (Mx_{0}y_{0}z_{0}) such that z_{0} points to the observer, and the local magnetic field B is in the (My_{0}z_{0}) 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 crosssections are of course equal, C_{x0} = C_{y0} = C_{⊥} = C_{∥}. The expressions for the Stokes parameters in terms of the crosssections 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 crosssection (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 p_{0} 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:
All Tables
All Figures
Fig. 1 Locations of the selected nearby fields. The background map represents optical depth τ_{353} at 353 GHz on a logarithmic scale, at 5′ resolution (Planck Collaboration XI 2014). The map uses a Mollweide projection in Galactic coordinates, with (l,b) = (0°,0°) at the centre. 

In the text 
Fig. 2 Twodimensional distribution functions of polarization fraction p and column density N_{H} in the fields highlighted in Fig. 1. Top: Polaris Flare (magenta), Taurus (red), ChamaeleonMusca (black), Microscopium (blue), and Pisces (green). Bottom: Ophiuchus (magenta), Pavo (red), Ara (black), Perseus (blue), and Orion (green). On both panels, the solid red lines show the upper and lower envelopes (see text) of the largescale distribution of p and N_{H}, clipped below N_{H} = 2 × 10^{20} cm^{2}, while the dashed red lines correspond to p = 0 and the maximum value p = 0.198 (i.e., 19.8%) quoted in Planck Collaboration Int. XIX (2014). 

In the text 
Fig. 3 Maps of the Ophiuchus and ChamaeleonMusca fields. Left: Ophiuchus field. Right: ChamaeleonMusca field. Top: total intensity at 353 GHz. Middle: polarization fraction p, column density N_{H} (contours in units of 10^{21} cm^{2}), and magnetic orientation (bars, see text). Bottom: angle dispersion function with lag δ = 16′ (see Sect. 2.5) with contours and bars identical to the middle row. In all maps, the 15′ beam is shown in the lowerleft corner. 

In the text 
Fig. 4 Twodimensional distribution function of polarization fraction p and column density N_{H}. Top: Ophiuchus field. Bottom: ChamaeleonMusca field. The distribution functions are presented in logarithmic colour scale and include only points for which p/σ_{p}> 3. The dashed red lines correspond to the absolute maximum polarization fractions 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). 

In the text 
Fig. 5 Same as the top panel of Fig. 4, but using only pixels for which p/σ_{p}> 10. 

In the text 
Fig. 6 Map of for the Ophiuchus field computed at δ = 34′. Contours are the same as in the map at δ = 16′ (bottom left panel of Fig. 3). 

In the text 
Fig. 7 Twodimensional distribution function of and polarization fraction p for the Ophiuchus field. 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 largescale 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 loglog space, restricted to bins in p which contain at least 1% of the total number of points (so about 150 points per bin). 

In the text 
Fig. 8 Same as Fig. 7, but for the ChamaeleonMusca field. 

In the text 
Fig. 9 Same as Fig. 7, but for a lag δ = 34′. 

In the text 
Fig. 10 Distribution function of the total gas density n_{H} in the selected subset of the simulation, with cell sizes 0.1 pc × 0.1 pc × 0.1 pc. The solid red line shows the mean value ⟨n_{H}⟩ = 17 cm^{3} and the solid blue line the median value . 

In the text 
Fig. 11 Distribution functions of the components of the magnetic field, B_{x} (blue), B_{y} (green), and B_{z} (red), in the selected subset of the simulation, with cell sizes 0.1 pc × 0.1 pc × 0.1 pc. 

In the text 
Fig. 12 Distribution functions of the total gas column density N_{H} in the selected subset of the simulation, using viewing angles α = 0° (red) and α = 90° (blue). These distribution functions are computed after convolution with the 15′ beam. 

In the text 
Fig. 13 Sketch of the rotation of the simulation subset. 

In the text 
Fig. 14 Definition of angles. Here the line of sight is along the z′ axis (see Fig. 13), γ is the angle the magnetic field B makes with the plane of the sky, φ is the local polarization angle, and χ is the position angle of the plane of the sky projection B_{⊥}, both in the HEALPix convention, so counted positively clockwise from the northsouth direction, while the IAU convention is anticlockwise (Planck Collaboration Int. XIX 2014). 

In the text 
Fig. 15 Simulated Planck maps. Top: total gas column density. Middle: polarization fraction and angle, with contours being the column density at values indicated in units of 10^{21} cm^{2}, and the bars indicate magnetic orientation. Bottom: angle dispersion function at lag δ = 16′, with the same contours and bars as in the middle row. Left: viewing angle α = 0°. Right: viewing angle α = 90°. In each row, the same colour scale is used. In the lower left corner of each plot (yellow circle) is the 15′ FWHM beam. 

In the text 
Fig. 16 Statistics of polarization fractions in the simulated Planck observations as a function of viewing angle α (see Fig. 13). The solid blue line shows p_{max}, the solid black line shows the mean p, and the solid green line shows the value of p for the most diffuse lines of sight in the map. The dashed black line marks the polarization fraction parameter p_{0} and the dashed green line gives the theoretical polarization fraction in the case where the density and magnetic field are homogeneous and the latter makes an angle α with the plane of the sky (see text). The grey region shows the ± 1σ spread around the mean p. 

In the text 
Fig. 17 Twodimensional distribution functions of polarization fractions and logarithmic column densities in the simulated Planck observations. Top: viewing angle α = 0°. Middle: viewing angle α = 45°. Bottom: viewing angle α = 90°. The dashed red horizontal lines and the solid red and black lines are the same as in Fig. 4. The fits to the upper envelopes are performed for N_{H}> 10^{21} cm^{2}. 

In the text 
Fig. 18 Comparison between the distributions of the polarization fractions p and logarithmic column densities in the simulations (colour scale, all viewing angles combined, with upper and lower envelopes in solid red lines) and those of the observations in the selected fields (solid black lines). Note that the latter are restricted to N_{H}> 10^{21} cm^{2}. Dashed lines are linear fits of the form on the distributions’ upper envelopes, restricted to a common range of column densities 2 × 10^{21} cm^{2}<N_{H} < 2 × 10^{22} cm^{2}. 

In the text 
Fig. 19 Twodimensional distribution function of log (p) and in the simulated observations for δ = 16′ and α = 0°. The solid black curve represents the evolution of the mean per bin of log (p). A linear fit is performed, restricted to bins in log (p) that contain at least 1% of the total number of points. This fit is shown as the dashed black line. The dashed grey line is the largescale fit presented in Sect. 2.5. 

In the text 
Fig. 20 Slopes m′(top) and intercepts c′(bottom) of the linear fits to the distribution of log (p) and in the simulated observations, as a function of viewing angle α. The lag is δ = 16′. The dashed blue lines indicate the values for the largescale fit presented in Sect. 2.5, the dashed red lines represent the average slope and intercept over the range of α, and the grey areas indicate ± 1σ around the mean, with the standard deviation σ computed statistically over all angles. 

In the text 
Fig. 21 Distribution functions of polarization fraction p and densityweighted mean of cos^{2}γ (γ is the angle of the magnetic field with respect to the plane of the sky, see Fig. 14) along the line of sight z′ in the simulation cube. Top: viewing angle α = 0°. Bottom: viewing angle α = 90°. The solid black lines show the mean values per bin of p. 

In the text 
Fig. 22 Distribution functions of polarization fraction p and densityweighted mean of sinχ (χ is the position angle of the projection of the magnetic field in the plane of the sky, see Fig. 14) along the line of sight z′ in the simulation cube. Top: viewing angle α = 0°. Bottom: viewing angle α = 90°. The solid black lines show the mean values per bin of p. 

In the text 
Fig. 23 Distribution functions of polarization fraction p and densityweighted standard deviation of cos^{2}γ along the line of sight z′ in the simulation cube. Top: viewing angle α = 0°. Bottom: viewing angle α = 90°. The solid black lines show the mean values per bin of p. 

In the text 
Fig. 24 Distribution functions of angle dispersion function and densityweighted mean of cos^{2}γ in the simulation cube. Top: viewing angle α = 0°. Bottom: viewing angle α = 90°. The solid black lines show the mean values per bin of . 

In the text 
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 10^{21} 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. 

In the text 
Fig. A.2 Same as Fig. 3, but for the Taurus field. 

In the text 
Fig. A.3 Same as Fig. 3, but for the Orion field. 

In the text 
Fig. A.4 Same as Fig. 3, but for the Microscopium field. 

In the text 
Fig. A.5 Same as Fig. 3, but for the Pisces field. 

In the text 
Fig. A.6 Same as Fig. 3, but for the Perseus field. 

In the text 
Fig. A.7 Same as Fig. 3, but for the Ara field. 

In the text 
Fig. A.8 Same as Fig. 3, but for the Pavo field. 

In the text 
Fig. A.9 Same as Fig. 4, but for the Polaris Flare field. Twodimensional 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). 

In the text 
Fig. A.10 Same as Fig. 4, but for the Taurus field. 

In the text 
Fig. A.11 Same as Fig. 4, but for the Orion field. 

In the text 
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. 

In the text 
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. 

In the text 
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. 

In the text 
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. 

In the text 
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. 

In the text 
Fig. A.17 Same as Fig. 7, but for the Polaris Flare field. Twodimensional 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 largescale 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 loglog space, restricted to bins in p which contain at least 1% of the total number of points (so about 150 points per bin). 

In the text 
Fig. A.18 Same as Fig. 7, but for the Taurus field. 

In the text 
Fig. A.19 Same as Fig. 7, but for the Orion field. 

In the text 
Fig. A.20 Same as Fig. 7, but for the Microscopium field. Note that the range in p is different from Fig. 7. 

In the text 
Fig. A.21 Same as Fig. 7, but for the Pisces field. Note that the range in p is different from Fig. 7. 

In the text 
Fig. A.22 Same as Fig. 7, but for the Perseus field. Note that the range in p is different from Fig. 7. 

In the text 
Fig. A.23 Same as Fig. 7, but for the Ara field. Note that the range in p is different from Fig. 7. 

In the text 
Fig. A.24 Same as Fig. 7, but for the Pavo field. Note that the range in p is different from Fig. 7. 

In the text 
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