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A&A
Volume 596, December 2016
Article Number A105
Number of page(s) 15
Section Interstellar and circumstellar matter
DOI https://doi.org/10.1051/0004-6361/201628636
Published online 12 December 2016

© ESO, 2016

1. Introduction

Interstellar magnetic fields are tied to the interstellar gas. Together with cosmic rays they form a dynamical system that is an important (but debated) facet of the physics of galaxies. Magnetic fields play a pivotal role because they control the density and distribution of cosmic rays, and they act on the dynamics if the gas. Much of the physics involved in this interplay is encoded in the structure of interstellar magnetic fields. Observations of synchrotron emission and its polarization, as well as Faraday rotation and dust polarization, provide the means to characterize the structure of magnetic fields within galaxies (Haverkorn 2015; Lazarian & Pogosyan 2016; Beck 2016).

Since dust grains are mixed with interstellar gas, dust polarization data are well suited to investigate the physical coupling between the gas dynamics and the magnetic field structure, in other words to characterize magnetohydrodynamical (MHD) turbulence in the interstellar medium (ISM; Brandenburg & Lazarian 2013; Falceta-Gonçalves et al. 2014). Anisotropic dust grains tend to align with their longer axes perpendicular to the local magnetic field, and thus their emission is polarized perpendicular to the magnetic field projection on the plane of the sky (POS). The polarization fraction, p, the ratio between the polarized and total intensities of dust thermal emission, depends on the dust polarization properties and the grain alignment efficiency, but also on the structure of the magnetic field (Lazarian 2007). Thus, information on the magnetic field structure is encoded in the Stokes Q and U maps, as well as in the polarization angle ψ and fraction p.

For a long time, observations of dust polarization from the diffuse ISM were limited to stellar polarization data available for a discrete set of lines of sight (LOS; Heiles 2000). The Planck1 data opened a new perspective on this topic. For the first time, we have maps of the dust polarization in emission over the full sky (Planck Collaboration I 2016). The Planck maps greatly supersede, in sensitivity and statistical power, the data available from earlier ground-based and balloon-borne observations (e.g. Benoît et al. 2004; Ponthieu et al. 2005; Ward-Thompson et al. 2009; Koch et al. 2010; Poidevin et al. 2014; Matthews et al. 2014).

Several studies have already used the Planck data to investigate the link between the dust polarization maps and the structure of the Galactic magnetic field (GMF). Planck Collaboration Int. XIX (2015) presented the first analysis of the polarized sky as seen at 353 GHz (the most sensitive Planck channel for polarized thermal dust emission), focusing on the statistics of p and ψ. The comparison with synthetic polarized emission maps, computed from simulations of anisotropic MHD turbulence, shows that the turbulent structure of the GMF is able to reproduce the main statistical properties of p and ψ in nearby molecular clouds (Planck Collaboration Int. XX 2015). This comparison shows that the mean orientation of the GMF with respect to the LOS plays a major role in the quantitative analysis of these statistical properties. An important result is that in the diffuse ISM, the filamentary structure of matter is observed to be statistically aligned with the GMF (McClure-Griffiths et al. 2006; Clark et al. 2014; Planck Collaboration Int. XXXII 2016; Kalberla et al. 2016).

The spatial structure of the polarization angle has been characterized in Planck Collaboration Int. XIX (2015) using the angle dispersion function \hbox{$\mathcal{S}$} (see Eq. (1) in Hildebrand et al. 2009; and Eq. (6) in Planck Collaboration Int. XIX 2015). The map of \hbox{$\mathcal{S}$} highlights long, narrow structures of high \hbox{$\mathcal{S}$} that trace abrupt changes of ψ at the interfaces between extended areas within which the polarization angle is ordered. Falgarone et al. (2015) found a correlation between the structures in \hbox{$\mathcal{S}$} and large velocity shears in incompressible magnetized turbulence. The structures seen in the Planck data bear a morphological resemblance to features associated with Faraday rotation in gradient maps of polarized synchrotron emission at 1.4 and 2.3 GHz (Gaensler et al. 2011; Iacobelli et al. 2014), which have been related to fluctuations in the GMF and in the ionized gas density in MHD turbulence (Burkhart et al. 2012). Filamentary structures in rotation measure synthesis maps from LOFAR (the Low-Frequency Array) data (Jelić et al. 2015) have been shown to be correlated with the GMF orientation inferred from the Planck dust polarization (Zaroubi et al. 2015). At microwave frequencies, the dust polarization has been demonstrated to be correlated with synchrotron polarization, free from Faraday rotation (Planck Collaboration Int. XXII 2015; Choi & Page 2015). Both emission processes trace the same GMF, but the correlation is not one-to-one due to the difference in the distribution of dust and relativistic electrons in the Galaxy. Jaffe et al. (2013) and Planck Collaboration Int. XLII (2016) described the difficulties faced when trying to reproduce the Planck dust polarization data with existing models of the large-scale GMF (Jaffe et al. 2010; Sun & Reich 2010; Jansson & Farrar 2012), which are mainly constrained by synchrotron emission and Faraday rotation measures of extragalactic radio sources.

thumbnail Fig. 1

Orthographic projections centred on the south Galactic pole of the Planck dust emission intensity, D353 (left), and the Stokes Q353 (centre) and U353 (right) maps, at 353 GHz. A grid of Galactic coordinates is included, labelled in degrees. East is on the left of the maps and the west on the right. We note that the U353 map is in the HEALPix (or CMB) polarization convention, which corresponds to −U353 in the IAU convention.

The GMF structure is also relevant for the modelling of polarized Galactic foregrounds in analyses of the CMB. Thermal emission from Galactic dust is the main polarized foreground at frequencies above 100 GHz (Planck Collaboration X 2016). Planck Collaboration Int. XXX (2016) presented the polarized dust angular power spectra CℓEE\hbox{$C_\ell^{EE}$} and CℓBB\hbox{$C_\ell^{BB}$}, providing cosmologists with a characterization of the dust foreground to CMB polarization. Planck Collaboration Int. XXXVIII (2016) showed that the correlation between the filamentary structure of matter and the GMF orientation may account for the E and B asymmetry, as well as the TE correlation, reported in the analysis of the power spectra of the Planck 353 GHz polarization maps.

Within this broad context, the motivations and objectives of this paper are twofold. First, we extend the analysis of the Planck dust polarization maps to the high Galactic latitudes in the southern sky. This part of the sky is of specific relevance to on-going and future CMB polarization observations performed from Antarctica and Chile (e.g. Errard et al. 2016). It was masked in the Planck Collaboration Int. XIX (2015) analysis because of residual systematic errors in the data. The polarization maps at 353 GHz (Planck Collaboration I 2016; Planck Collaboration VIII 2016) that have been made publicly available by the Planck consortium2 are now suitable for such an analysis. Second, we introduce a modelling framework that relates the dust polarization to the GMF structure, its mean orientation, and a statistical description of its random (turbulent) component. This framework is also a step towards a modelling tool for the dust polarization, which may be used to assess component-separation methods in the analysis of CMB polarization (e.g. Planck Collaboration IX 2016; Planck Collaboration X 2016).

Our data analysis procedure focuses on the southern Galactic cap, the cleanest part of the sky that is directly relevant to CMB observations, in particular those carried out with ground-based telescopes from Antarctica and Chile3. This is also the part of the sky where the LOS through the Galaxy is the shortest, and hence is the region best suited to characterize the turbulent component of the GMF.

The paper is organized as follows. We present the Planck data in Sect. 2. Section 3 introduces our model of the GMF structure in the solar neighbourhood and in Sect. 4 we estimate the mean orientation of the GMF in the solar neighbourhood. In Sect. 5, we characterize the turbulent component of the GMF. The data analysis is based on a phenomenological model that we discuss in Sect. 6, which also contains our future perspectives. The paper’s results are summarized in Sect. 7. The approximations made to compute the Stokes parameters are presented in Appendix A.

2. Data and conventions

We first introduce the data that we will use, discussing the conventions assumed in the analysis of polarization, and presenting the polarization parameters determined around the south Galactic pole.

2.1. Description of the data

The Planck satellite observed the polarized sky in seven frequency bands from 30 to 353 GHz (Planck Collaboration I 2014). In this paper, we only use the data from the High Frequency Instrument (HFI, Lamarre et al. 2010) at the highest frequency, 353 GHz, where the dust emission is the brightest.

We use the publicly available 353 GHz Stokes Q and U (hereafter, Q353 and U353) maps (central and right panels in Fig. 1) and the associated noise maps made with the five independent consecutive sky surveys of the Planck cryogenic mission. We refer to publications by the Planck Collaboration for details of the processing of HFI data, including mapmaking, photometric calibration, and photometric uncertainties (Planck Collaboration I 2016; Planck Collaboration VII 2016; Planck Collaboration VIII 2016). The Q353 and U353 maps are corrected for spectral leakage as described in Planck Collaboration VIII (2016). For the dust total intensity at 353 GHz we use the model map, D353, derived from a modified blackbody fit to the Planck data at ν ≥ 353 GHz, and IRAS at λ = 100 μm (Planck Collaboration XI 2014, left panel in Fig. 1). The data used in this fit are corrected for zodiacal emission and CMB anisotropies. D353 has a also a lower noise than the corresponding 353 GHz Stokes IPlanck map. The Q353 and U353 maps are initially constructed with an effective beamsize of 4.́8, and D353 at 5′. The three maps are in HEALPix format4 with a pixelization Nside = 2048. To increase the signal-to-noise ratio at high Galactic latitudes below −60°, we smooth the three maps to 1° resolution using a Gaussian approximation to the Planck beam. We reduce the HEALPix resolution to Nside = 128 (30.́1 pixels) after smoothing. For the polarization maps, we apply the ismoothing routine of HEALPix, which decomposes the Q and U maps into E and B maps, applies Gaussian smoothing in harmonic space, and transforms the smoothed E and B back into Q and U maps at Nside = 128 resolution.

2.2. Applied conventions in polarization

In terms of Q353, U353, and D353, the quantities p and ψ, are defined as p=Q3532+U3532D353,ψ=12 tan-1(−U353,Q353),\begin{eqnarray} \label{eq:observables} &&p = \frac{\sqrt{Q_{353}^2+U_{353}^2}}{D_{353}},\nonumber \\ && \psi= \frac{1}{2}\,{\rm tan}^{-1}\left(-U_{353},Q_{353}\right), \end{eqnarray}(1)where the minus sign in ψ is needed to change the HEALPix-format maps (or “COSMO convention” for the FITS keyword POLCONV) into the International Astronomical Union (IAU) convention for ψ, measured from the local direction to the north Galactic pole with increasing positive values towards the east. Moreover, in this paper we use the version of the inverse tangent function with two signed arguments to resolve the π ambiguity (ψ corresponds to orientations not to directions).

When considering dust polarization, the Stokes parameters for linear polarization are integral quantities of the optical depth (see Appendix A and Planck Collaboration Int. XX 2015). An empirical expression for p is p=p0 F cos2γ,\begin{equation} p = p_{0}\,F \, \cos^2{\gamma}, \label{eq:p_obs} \end{equation}(2)where γ is the angle between the mean orientation of the GMF and the POS. Therefore, the projection factor, cos2γ, carries information on the orientation of the GMF with respect to the POS. In particular, dust polarization vanishes where the GMF points directly towards or away from the observer. Hereafter, p0 = pdustR is the effective dust polarization fraction, which combines the intrinsic polarization fraction of dust grains pdust (the ratio between the polarization and average cross-sections of dust, as defined in Planck Collaboration Int. XX 2015) and R, the Rayleigh reduction factor (related to the degree of dust grain alignment with the GMF; Greenberg 1968; Lee & Draine 1985). The factor R is equal to 1 for perfect grain alignment. The factor F accounts for the depolarization due to variations of the GMF orientation along the LOS and within the beam.

thumbnail Fig. 2

PlanckD353 (same as in the left panel of Fig. 1) with the “drapery” pattern, orthogonal to the polarization orientation, produced with the LIC algorithm. The part of the sky at b< −60° has been highlighted in colour in this figure.

2.3. Polarization parameters of the southern Galactic cap

Planck Collaboration Int. XIX (2015) characterized the polarized sky at 353 GHz at low and intermediate Galactic latitudes. Now, with the maps released in early 2015 (Planck Collaboration I 2016), we can extend this analysis to the high Galactic latitudes of the southern sky. In this work, we focus on the region around the south Galactic pole (Galactic latitude b < −60°), which is well suited to study emission from dust in the diffuse ISM, and directly relevant to study the dust foreground for CMB polarization.

We compute p and ψ from the Stokes parameters in Fig. 1 at a resolution of 1°. Because of the square of Q and U, and the contribution from noise, p cannot be computed directly from Eq. (1) at high Galactic latitudes where the Planck signal-to-noise is low. A number of algorithms have been proposed (e.g. Montier et al. 2015) to derive unbiased estimates of p; here, we use the pMAS estimator presented in Plaszczynski et al. (2014).

thumbnail Fig. 3

Normalized distributions of the polarization fraction from the pMAS debiased estimator (see text). The black distribution shows pMAS over the whole sky. The green distribution shows pMAS at b< −60°. The green-shaded area represents the 1σ error on pMAS at high latitude.

Figure 2 shows a map of the Planck dust emission intensity, D353, with the drapery pattern of ψ, rotated by π/ 2, produced with the linear integral convolution (LIC) algorithm (Cabral & Leedom 1993) as in Planck Collaboration Int. XXXV (2016) and Planck Collaboration I (2016). This map reveals a high degree of order in ψ at b < −60° (blue region). Figure 3 shows the normalized distributions of the polarization fraction from the pMAS unbiased estimator, over the whole sky (in black) and at b < −60° (in green). We also show the uncertainty on pMAS at high latitude (green-shaded area) due to the error on the zero level of D353 as estimated by Planck Collaboration XI (2014). Both distributions indicate a wide range of pMAS values up to 25%. The main difference at low pMAS values is likely caused by depolarization from LOS variations of the GMF orientation closer to the Galactic plane (Planck Collaboration Int. XIX 2015).

How do we explain the high pMAS values and the observed dispersion in the distribution? As we will show, the GMF structure in the solar neighbourhood is essential to consider when answering this question.

3. Model framework

The polarization of thermal dust emission results from the alignment of elongated grains with respect to the GMF (Stein 1966; Hildebrand 1988). Within the hypothesis that grain polarization properties, including alignment, are homogeneous, the structure of the dust polarization sky reflects the structure of the GMF combined with that of matter. Throughout the paper, we assume that this hypothesis applies to the diffuse ISM, where radiative torques provide a mechanism to efficiently align grains (Dolginov & Mitrofanov 1976; Hoang & Lazarian 2014; Andersson et al. 2015). Our data modelling focuses on the structure of the GMF. This section describes the model framework (Sect. 3.1) and how we proceed to fit it to the data (Sect. 3.2).

3.1. Magnetic field modelling

We now introduce the framework we use to model the GMF structure within the solar neighbourhood. The integral equations of the Stokes I, Q and U parameters are recalled in Appendix A.

We follow earlier works (e.g. Chandrasekhar & Fermi 1953; Hildebrand et al. 2009), expressing the GMF (B) as the sum of its mean (B0) and turbulent (Bt) components: B=B0+Bt.\begin{equation} \vec{B}=\vec{B}_0+\vec{B}_{\rm t}. \label{eq:Bfield} \end{equation}(3)We introduce and discuss the assumptions we make about each of these two components.

Our model aims at describing dust polarization towards the southern Galactic cap at Galactic latitudes b ≤ −60°. We focus on the solar neighbourhood and thereby ignore the structure of the GMF on Galaxy-wide scales. We also ignore the change of its orientation from the disk to the halo (Haverkorn 2015) because dust emission arises mainly from a thin disk. The dust scale height is not measured in the solar neighbourhood, but modelling of the dust emission from the Milky Way indicates that the dust scale height at the solar distance to the Galactic centre is approximately 200 pc (Drimmel & Spergel 2001). Observations of the edge-on spiral galaxy NGC 891, a galaxy analogous to the Milky Way, give a comparable scale height of around 150 pc (Bocchio et al. 2016). These estimates are in agreement with the scale height of the neutral atomic gas in the Milky Way, inferred from H i observations (Dickey & Lockman 1990; Kalberla et al. 2007). Hence, we assume that the vector B0 has a fixed orientation, which represents the mean orientation of the GMF in the solar neighbourhood.

Radio observations of synchrotron emission and polarization reveal a wealth of structures down to pc and sub-pc scales (e.g. Reich et al. 2004; Gaensler et al. 2011; Iacobelli et al. 2013, 2014), such as filaments, canals, lenses, and rings, which carry valuable information about Bt (Fletcher & Shukurov 2006). Heiles (1995) and Haverkorn (2015) reviewed observations that characterize this random component, concluding that it has a strength of about 5 μG, comparable to that of B0. Jones et al. (1992) reached a similar conclusion from stellar polarization data.

The turbulent component of the GMF is significant. To take it into account, we follow earlier works (e.g. Waelkens et al. 2009; Fauvet et al. 2011), modelling each component of the Bt vector with Gaussian realizations. To model dust polarization over the celestial sphere, earlier studies (e.g. Miville-Deschênes et al. 2008; Fauvet et al. 2011; O’Dea et al. 2012) computed independent realizations of the components of Bt for each LOS. This approach ignores the angular coherence of Bt over the sky, which, however, is essential to match the correlated patterns seen in the Planck maps of the dust p and ψ (Planck Collaboration Int. XIX 2015). Because of this, we use a different method. We model Bt with Gaussian realizations on the celestial sphere, computed for an angular power spectrum Cℓ scaling as a power-law ℓαM for ℓ ≥ 2. The amplitude of the spectrum is parametrized by the ratio fM between the standard deviation of | Bt | and | B0 |.

Our spectrum does not have a low ℓ cut-off, which would represent the scale of energy injection of the turbulent energy cascade. Here since we compare the model and the data over a field with an angular extent of 60° (about 1 radian), we implicitly assume that the injection scale is larger than, or comparable to, the scale height of the dust emission (approximately 200 pc, Drimmel & Spergel 2001). The scale of the warm ionized medium (WIM) is larger (about 1−1.5 kpc, Gaensler et al. 2008), but the WIM is not a major component of the dust emission from the diffuse ISM (Planck Collaboration Int. XVII 2014). The range of distances involved in the modelling of dust polarization at high Galactic latitudes is small because there is little interstellar matter within the local bubble, i.e. within 50–100 pc of the Sun (Lallement et al. 2014). The local bubble may extend to larger distances towards the Galactic poles, but this possibility is not well constrained by existing data. In any case, it is reasonable to assume that most of the dust emission at high Galactic latitudes arises from a limited range of distances, which sets a rough correspondence between angles and physical scales in our model.

To compute the Stokes parameters, we approximate the integration along the line of sight (LOS) with a sum over a set of N polarization layers with independent realizations of Bt. The layers are a phenomenological means to represent the variation of Bt along the LOS. Our modelling of Bt is continuous over the celestial sphere, while we use a set of independent orientations along the LOS. At first sight, this may be considered as physically inconsistent. However, in Sect. 6, we relate the polarization layers to the density structure and to the correlation length of Bt along the LOS. Our modelling does not take into account explicitly the density structure of matter along the LOS; the source function (presented in Eqs. (A.1b) and (A.1c)) is assumed to be constant along the LOS. It also ignores the alignment observed between the filamentary structure of the diffuse ISM and the magnetic field.

3.2. Data fitting in three steps: A, B, and C

Our model has six parameters: the two coordinates defining the orientation of B0; fM quantifying the dispersion of B around B0; the number of layers, N; the index αM; and the effective polarization fraction of dust emission, p0. The parameters are not all fitted simultaneously because they are connected to the data in different ways. The coordinates of B0 relate to the large-scale patterns in the Q353 and U353 maps and they do not depend on the other parameters. The triad of parameters fM, N, and αM describe statistical properties of the polarization maps. We determine fM, N, and p0 simultaneously by fitting the 1-point statistics of both ψ and p. To constrain αM it is necessary to use 2-point statistics (i.e. power spectra); this is not done in this paper, but will be the specific topic of a future paper.

In the following two sections, we present three steps in our data-fitting, labelled steps A, B, and C. Step A only takes into account the mean field B0. In Sect. 4, we determine the orientation of B0 by fitting the regular patterns seen in the PlanckQ353 and U353 maps shown in Fig. 1. The other two steps involve both B0 and Bt, as required to reproduce the 1-point statistics of ψ and p. In step B (Sect. 5), Bt is computed from random realizations on the sphere. In this step, the depolarization due to changes in the orientation of Bt along the LOS is accounted for with an F factor in Eq. (2) that is uniform over the sky. This simplifying assumption is often made in analysing polarization data. Step C in Sect. 5.3 is an extension of step B, where we introduce variations of the F factor over the sky by summing Stokes parameters over N polarization layers along the LOS.

4. Mean orientation of the magnetic field

In this step A of our data modelling, we determine the orientation of the mean field B0, ignoring Bt.

thumbnail Fig. 4

Mollweide (top) and orthographic (bottom) projections of the model Stokes parameters, qA (left) and uA (right), for a uniform direction of the GMF towards (l0,b0) = (80°,0°); these are roughly the values inferred from starlight polarization (Heiles 1996). The orthographic projections are centred on the Galactic poles. Galactic coordinates in degrees are shown on all plots.

4.1. Description of step A

We show that the ordered magnetic field produces well-defined polarization patterns in the Q353 and U353 maps, resulting from the variation across the observed region of the angle between the LOS and the ordered field.

Given a Cartesian reference frame xyz, each point on the sphere can be identified by a pair of angular coordinates, hereafter the Galactic longitude and latitude, l and b. The reference frame is chosen to be centred at the observer with ẑ = (0,0,1) pointing towards the north Galactic pole, \hbox{$\vec{\hat{x}}=(1,0,0)$} towards the Galactic centre, and ŷ = (0,1,0) towards positive Galactic longitude.

We define the uniform direction of B0 through the unit vector \hbox{$\vec{\hat{B}}_{0}$}, which depends on the pair of coordinates (l0, b0) as \hbox{$\vec{\hat{B}}_{0}=(\cos{l_0}\,\cos{b_0},\,\sin{l_0}\,\cos{b_0},\,\sin{b_0})$}. We define the generic LOS unit vector \hbox{$\vec{\hat{r}}$} as (cosl cosb, sinl cosb, sinb) on a full-sky HEALPix grid.

Combining \hbox{$\vec{\hat{r}}$} and \hbox{$\vec{\hat{B}}_{0}$}, we can derive the POS component of \hbox{$\vec{\hat{B}}_{0}$}, \hbox{$\vec{\hat{B}}_{0 \perp}$}, as B̂0⊥=B̂0−B̂0∥=B̂0−(B̂0·r̂)r̂,\begin{equation} \vec{\hat{B}}_{0 \perp} = \vec{\hat{B}}_0 - \vec{\hat{B}}_{0 \parallel} = \vec{\hat{B}}_0 - (\vec{\hat{B}}_0 \cdot \vec{\hat{r}}) \vec{\hat{r}}, \label{eq:bperp} \end{equation}(4)where \hbox{$\vec{\hat{B}}_{0 \parallel}$} is the component of \hbox{$\vec{\hat{B}}_0$} along \hbox{$\vec{\hat{r}}$}. In order to define the ψ and γ angles for a given \hbox{$\vec{\hat{r}}$}, we need to derive the north and east directions, tangential to the sphere, which correspond to n̂=(r̂×ẑ)×r̂|(r̂×ẑ)×r̂|,ê=−r̂×n̂|r̂×n̂|,\begin{eqnarray} \label{eq:endirections} && \vec{\hat{n}}=\frac{(\vec{\hat{r}} \times \vec{\hat{z}}) \times \vec{\hat{r}}}{|(\vec{\hat{r}} \times \vec{\hat{z}}) \times \vec{\hat{r}}|}, \nonumber \\ & & \vec{\hat{e}} = \frac{-\vec{\hat{r}} \times \vec{\hat{n}}}{|\vec{\hat{r}} \times \vec{\hat{n}}|}, \end{eqnarray}(5)respectively. The polarization angle is perpendicular to that between \hbox{$\vec{\hat{B}}_{0\perp}$} and \hbox{$\vec{\hat{n}}$}, and γ the angle between \hbox{$\vec{\hat{B}}_0$} and \hbox{$\vec{\hat{B}}_{0\perp}$}. From Eqs. (4) and (5), we derive ψA=arccos(B̂0⊥·n̂|B̂0⊥|)+90◦,cos2γA=1−|B̂0·r̂|2,\begin{eqnarray} \label{eq:psigammam} && \psi_{\rm A}=\arccos{\left(\frac{\vec{\hat{B}}_{0\perp} \cdot \vec{\hat{n}}}{|\vec{\hat{B}}_{0\perp}|}\right) } + 90^\circ, \nonumber \\ && \cos^2{\gamma_{\rm A}}=1-|\vec{\hat{B}}_0 \cdot \vec{\hat{r}}|^2, \end{eqnarray}(6)where the subscript “A” stands for step A, and the sign of arccos is imposed by the sign of \hbox{$\vec{\hat{B}}_{0\perp} \cdot \vec{\hat{e}}$}.

Using Eqs. (1) and (2), we can produce an analytical expressions for the modelled Stokes parameters normalized to the total intensity times p0F, qA and uA, as follows: qA=cos2γAcos2ψA;uA=−cos2γAsin2ψA.\begin{eqnarray} \label{eq:stokesA} && q_{\rm A} = \cos^2{\gamma_{\rm A}}\cos{2\psi_{\rm A}}; \nonumber \\ && u_{\rm A} =- \cos^2{\gamma_{\rm A}}\sin{2\psi_{\rm A}}. \end{eqnarray}(7)We stress that qA and uA only show patterns generated by projection effects. For illustration, in Fig. 4 we present maps of qA and uA for a uniform direction of the GMF towards (l0,b0) = (80°,0°), roughly the direction inferred from starlight polarization data (Heiles 1996). We note that the total intensity of dust emission also depends on the GMF geometry (Planck Collaboration Int. XX 2015). However, as detailed in Appendix A, this is a small effect that does not alter our results.

4.2. Fitting step A to the Planck data

At first glance, the “butterfly” patterns in the Q353 and U353 maps around the south Galactic pole in Fig. 1 resemble those produced with step A in Fig. 4. In order to find the orientation of \hbox{$\vec{\hat{B}}_0$} that best fits the data, we explore the space of Galactic coordinates for (l0,b0), spanning Galactic longitudes between 0° and 180°, and latitudes between −90° and 90°. From Eqs. (1), (2), and (7), we simultaneously fit step A to Q353 and U353 with the corresponding errors, as Q353=p0,AqAD353,U353=p0,AuAD353,\begin{eqnarray} \label{eq:stokesAfit} && Q_{353}= p_{0{\rm,A}}q_{\rm A}D_{353},\nonumber \\ && U_{353}= p_{0{\rm,A}}u_{\rm A}D_{353}, \end{eqnarray}(8)where the factor p0,A represents an average of the product p0F in Eq. (2) over the region where we perform the fit. For each (l0,b0) pair we perform a linear fit to determine p0,A. The fit is carried out for the southern polar cap at b< −60°, after masking the most intense localized structures around the south Galactic pole, as shown in Fig. 5. To remove these regions from the analysis, we fit a Gaussian profile to the histogram of pixel values of D353 below b = −60°. We then mask all pixels with \hbox{$D_{353}> \bar{x}_D + 4\,\sigma_{\bar{x}_D}$}, where \hbox{$\bar{x}_D$} and σx̅D\hbox{$\sigma_{\bar{x}_D}$} are the mean and the standard deviation of the Gaussian fit.

thumbnail Fig. 5

Same as in the left panel of Fig. 1, but now highlighting the b < −60° region, excluding the brightest clouds (in grey in the image), which has been used to fit the orientation of the mean magnetic field in step A of our model fit.

thumbnail Fig. 6

Step A: orthographic projections of qA (left) and uA (right) centred on the south Galactic pole, for the best-fit direction of the uniform GMF towards (l0,b0) = (70°,24°). The sky at b> −60° is masked here.

The fit is done over an area of 2652 deg2, corresponding to 2652 independent data beams. Since the number of parameters is 3, the number of degrees of freedom, Nd.o.f., is large. We find a best-fit direction of the mean GMF towards Galactic coordinates l0 = 70° ± 5° and b0 = 24° ± 5°. The value of p0,A corresponding to this direction is (12 ± 1)%, which corresponds to the peak of the distribution of pMAS in this area (see Fig. 3). The statistical errors are small but there are significant uncertainties on the three parameters from residual, uncorrected, systematic effects in the data. We quote these uncertainties, which we estimated repeating the fit on maps produced with ten different subsets of the data (Planck Collaboration 2015). We notice that, because of the 180° ambiguity in the definition of ψ, the opposite direction (l0 + π,−b0) is an equivalent solution of our fit. However, the chosen solution is the closest to the mean GMF direction derived from observations of pulsars in the solar neighbourhood (Rand & Kulkarni 1989; Ferrière 2015), which, unlike dust polarization are sensitive to the sign of the GMF. Our determination of l0 is in agreement with earlier values derived from starlight polarization (e.g. Heiles 1996). The positive value of b0 is consistent with the positive sign of the median value of rotation measures derived from observations of extragalactic radio sources in the direction of the southern Galactic cap (Taylor et al. 2009; Mao et al. 2010). For illustration, we show the best-fit model maps of qA and uA around the south pole in Fig. 6.

We note that the obtained value of p0,A is a substantial fraction of the maximum p (>18%) reported in Planck Collaboration Int. XIX (2015) at intermediate Galactic latitudes. We also stress that this value of p0,A is only a lower limit to the effective dust polarization fraction because step A does not take into account any depolarizing effects along the LOS, associated with variations of the GMF orientation.

5. Turbulent component of the magnetic field

The Planck maps show structures in polarization on a wide range of scales (Fig. 1), not accounted for by the single field orientation of step A, which we associate with the turbulent component of the magnetic field Bt. In Sects. 5.1 and 5.2, Bt is assumed to vary only across the sky (step B), while in Sect. 5.3, we take into account its variations both across the sky and along the LOS (step C).

5.1. Step B: dispersion of the polarization angle

thumbnail Fig. 7

Orthographic projections centred on the south Galactic pole of Q353R\hbox{$\StokesQ^{R}_{353}$} (left) and U353R\hbox{$\StokesU^{R}_{353}$} (right), the Stokes parameters in a reference frame rotated with respect to the best-fit direction of the uniform component of the GMF towards (l0,b0) = (70°,24°). The sky for the masked b> −60° region appears in grey.

thumbnail Fig. 8

Results of step B. Top: histogram of ψR, the polarization angle inferred from the Stokes parameters rotated with respect to the best-fit uniform direction of the GMF (Q353R\hbox{$\StokesQ^{R}_{353}$} and U353R\hbox{$\StokesU^{R}_{353}$}), over the southern Galactic cap (black dots). The error bars represent the Poisson noise within each bin of the histogram. The green line represents the mean of the step B results for fM = 0.4 over 20 different realizations. The green shaded regions correspond to ± 1σ (light green) and ± 2σ (dark green) variations of the model. Bottom: histogram of p2 obtained when combining the Year 1 and Year 2 maps (black dots). The error bars here represent the Poisson noise within each bin of the histogram. Step B is now shown in blue. The dashed vertical line corresponds to a value of the polarization fraction of 12%.

In Sect. 4.2, we found that the best-fit orientation of B0 in step A is given by (l0,b0) = (70°,24°). We can now obtain maps of the corresponding normalized Stokes parameters, u0A and q0A, as well as a map of the associated polarization angle ψ0A=12tan-1(−u0A,q0A).\begin{equation} \psi_{0\rm A}=\frac{1}{2}{\rm tan}^{-1}\left(-u_{0 \rm A},q_{0 \rm A}\right). \label{eq:psiA} \end{equation}(9)The angle ψ0A allows us to rotate, at each point on the sky, the reference direction used to compute the Stokes parameters (Q353, U353). With this new reference, the qA map in Fig. 6 would be that of cos2γA, and uA would be null (see Eq. (7)). To obtain the rotated values Q353R\hbox{$\StokesQ^{R}_{353}$} and U353R\hbox{$\StokesU^{R}_{353}$}, we apply to the data the following rotation matrix (e.g. Delabrouille et al. 2009): (Q353RU353R)=(cos2ψ0Asin2ψ0A−sin2ψ0Acos2ψ0A)(Q353U353)·\begin{equation} \left(\begin{array}{c} Q^{\rm R}_{353}\\ U^{\rm R}_{353} \end{array}\right)=\left(\begin{array}{ccc} \cos{2 \psi_{0\rm A}} & \sin{2 \psi_{0\rm A}} \\ -\sin{2 \psi_{0\rm A}} & \cos{2 \psi_{0\rm A}}\end{array}\right) \left(\begin{array}{c} Q_{353}\\ U_{353} \end{array}\right)\cdot \label{eq:rotpsi} \end{equation}(10)The maps of Q353R\hbox{$\StokesQ^{R}_{353}$} and U353R\hbox{$\StokesU^{R}_{353}$} are shown in Fig. 7, where the butterfly patterns, caused by the uniform component of the GMF, are now removed by the change of reference. The polarization angle that can be derived from Q353R\hbox{$\StokesQ^{R}_{353}$} and U353R\hbox{$\StokesU^{R}_{353}$} as ψR=12tan-1(−U353R,Q353R),\begin{equation} \psi_{\rm R}=\frac{1}{2}{\rm tan}^{-1}(-U^{\rm R}_{353},Q^{\rm R}_{353} ), \label{eq:psirot} \end{equation}(11)represents the dispersion of B⊥ around B0 ⊥. The histogram of ψR for b< −60°, shown in the top panel of Fig. 8 (black dots with Poisson noise as error bars), has a 1σ dispersion of 12°.

To characterize Bt, it is necessary to account for projection effects (Falceta-Gonçalves et al. 2008; Planck Collaboration Int. XXXII 2016). Planck Collaboration Int. XXXII (2016) describes a geometric model, which we use in this paper to characterize the 3D dispersion of B with respect to B0, given the histogram of ψR. Each component of Bt is obtained with an independent realization of a Gaussian field with an angular power spectrum equal to a power law of index αM, for multipoles ℓ ≥ 2. The degree of alignment between B and B0 is parameterized by fM, which represents the ratio between the strengths of the turbulent and mean components of the GMF.

In the top panel of Fig. 8, we show that for fM = 0.4 the model reproduces the histogram of ψR fairly well. We computed 20 different Gaussian realizations to take into account the statistical variance of the model. The green line represents the average of the 20 realizations, whereas the green shaded regions are the ± 1σ (light) and ± 2σ (dark) variations of the model. In these calculations, as in Planck Collaboration Int. XXXII (2016), the spectral index αM has a value of −1.5. This specific choice does not impact the distribution of ψR, or that of p. However, we note that the variance of the histogram, i.e. the dispersion of histogram values between independent realizations, increases for decreasing values of αM.

5.2. Step B: histogram of the polarization fraction

We showed that the structure of the GMF on the sphere allows us to reproduce ψR over the southern Galactic cap. Here, we characterize the distribution of p at b< −60° and we show that step B is not sufficient to describe the data.

As already discussed above, the noise bias on p represents an intrinsic problem. To circumvent it, we compute unbiased values of p2 by multiplying Stokes parameters from subsets of the data. Doing this, instead of using pMAS as in Sect. 2.3, gives us control over the level of noise in the data, as we now demonstrate. We use the year-maps (denoted by the indices “Y1” and “Y2”), which have uncorrelated instrumental noise, and compute p2 as p2=Q353Y1Q353Y2+U353Y1U353Y2(D353)2·\begin{equation} p^2=\frac{Q^{\rm Y1}_{353}Q^{\rm Y2}_{353} +U^{\rm Y1}_{353}U^{\rm Y2}_{353}}{(D_{353})^2}\cdot \label{eq:p2d} \end{equation}(12)We also estimate p2 from the DetSet maps (made from different subsets of detectors, see Planck Collaboration 2015), and we find good agreement between the two estimates using distinct subsets of the data.

In order to model p2, we make use of the results obtained from fitting steps A and B to the data. Given B0, pointing towards (l0,b0) = (70°,24°), we add Bt to it with normalization parameter fM = 0.4. In doing so, we now produce the two variables qB and uB, as qA and uA in Eq. (7), where now the angles take into account the turbulent component of the GMF. We then make realizations of the Planck statistical noise (nQi and nUi, with i = 1,2), and, as in Eq. (8), we produce two pairs of independent samples of modelled Stokes Q and U as QMi=p0qBD353+nQi,UMi=p0uBD353+nUi,\begin{eqnarray} \label{eq:stokesnoise} && Q_{{\rm M}i}= p_0q_{\rm B}D_{353} + n_{Qi},\nonumber \\ & &U_{{\rm M}i}= p_0u_{\rm B}D_{353} + n_{Ui}, \end{eqnarray}(13)in which i = 1,2 and p0 = 12%. Thus, the modelled p2 results from pM2=QM1QM2+UM1UM2(D353)2·\begin{equation} p_{\rm M}^2=\frac{Q_{\rm M1}Q_{\rm M2}+U_{\rm M1}U_{\rm M2}}{(D_{353})^2}\cdot \label{eq:p2m} \end{equation}(14)In the bottom panel of Fig. 8, we show the comparison between the histograms of p2 for the data (black dots) and for the model. In particular, we present the average over 20 realizations of step B (blue line) and the corresponding ± 1σ (light blue shaded region) and ± 2σ (dark blue) variations. The dashed vertical line refers to the value of p0 = 12%. We notice that our modelling of p2 seems to appropriately take into account the data noise since it nicely fits the negative p2 values, which result from noise in the combination of the individual year maps.

thumbnail Fig. 9

Cartoon illustrating, for step C, the integration of qC along the LOS, with four distinct polarization layers for the same value of fM and the same mean orientation of the GMF. Each map in this cartoon is a realization of the model.

thumbnail Fig. 10

Results of step C. Maps of χtot2\hbox{$\chi^2_{\rm tot}$} from the fit of step C to the data for p0 = 26% (left), for N = 7 (centre), and for fM = 0.9 (right). The three maps show in colours and with contours the quantity log10(χtot2)\hbox{${\rm log}_{10} (\chi^2_{\rm tot})$}.

However, from Fig. 8 it is clear that our description of the GMF structure using step B does not provide a satisfactory characterization of the distribution of p2. The data show a more prominent peak in the distribution towards very low p2 values than seen in the model, for which the histogram peaks near the value of p0. Moreover, the large dispersion in the data, also found by Planck Collaboration Int. XIX (2015) at intermediate Galactic latitudes, produces a long tail in the distribution towards high values of p2, which is not reproduced by the model.

5.3. Step C: line-of-sight depolarization

Now we consider the effect of depolarization, associated with variations of the GMF orientation along the LOS. This additional step is essential to account for the dispersion of p and correctly estimate the amplitude of the turbulent component of the GMF with respect to its mean component because the dispersion of the polarization angle is reduced by averaging along the LOS (Myers & Goodman 1991; Jones et al. 1992; Houde et al. 2009).

Figure 9 illustrates step C with a simple cartoon. In order to account for the LOS integration that characterizes the polarization data, we produce N distinct maps of qB,i and uB,i (with i from 1 to N), for a common, but freely varying value of fM, while fixing αM = −1.5 (as in step B), and for the best-fit orientation of B0 obtained with step A. The Gaussian realizations of Bt are different for each layer. All layers have the same B0 but an independent Bt in Eq. (3). Then, we model the LOS depolarization by averaging the Stokes parameters over the N layers as follows: qC=∑i=1NqB,iN;uC=∑i=1NuB,iN·\begin{eqnarray} \label{eq:intQU} && q_{\rm C}= \frac{\sum^N_{i=1} q_{{\rm B},i}}{N};\nonumber \\ && u_{\rm C}= \frac{\sum^N_{i=1} u_{{\rm B},i}}{N}\cdot \end{eqnarray}(15)We follow the same procedure as in Sects. 5.1 and 5.2, with qB and uB replaced by qC and uC, to obtain model distributions of p2 and ψR.

Given αM, the modelled distributions of p2 and ψR depend on three main parameters, namely p0, fM, and N. We fit the data exploring the parameter spaces of p0 between 15% and 40% with steps of 1%, of fM between 0.2 and 1.8 with steps of 0.1, and of N between 1 and 17 with steps of 1. The distributions of p2 and ψR have about 200 bins each. For each triad of parameters we compute maps of the reduced χ2 for the combined p2 and ψR fit, using χtot2=χp22+χψR2,\begin{equation} \chi^2_{\rm tot}=\chi^2_{p^2}+\chi^2_{\psi_{\rm R}}, \label{eq:chi2tot} \end{equation}(16)where in computing the χ2 distributions we fit the data with the mean of the 20 realizations, and we add their dispersion in quadrature to the error bar of the observations. Fitting the distribution of ψR between −40° and 40° (where most of the data lie), we obtain a best fit for a minimum χtot2\hbox{$\chi^2_{\rm tot}$} of 2.8, for p0 = 26%, fM = 0.9, and N = 7. In Fig. 10 we show three maps of χtot2\hbox{$\chi^2_{\rm tot}$}; each one corresponds to the parameter space for two parameters given the best-fit value of the third one. The χtot2\hbox{$\chi^2_{\rm tot}$} maps reveal some correlation among the three parameters. The variance of each model among the 20 different realizations represents the dominant uncertainty of the fit, and it is correlated between the bins of the histogram. Repeating the χ2-minimization for each one of the 20 realizations, the fit constrains the range of values for the main parameters to 0.8 <fM< 1, 5 <N< 9, and 23% <p0< 29%. Step C generates a mean value of the depolarization factor F that is about 0.5, and thus leads to an estimate of p0 twice larger than in step A. The best-fit value of (26 ± 3)% is comparable with the maximum value of the observed reported in Planck Collaboration Int. XIX (2015).

As in Fig. 8, the histograms of p2 and ψR for the best-fit triad are shown in the bottom and top panels of Fig. 11, respectively. The top panel of Fig. 11 shows that if we consider a few (N ≃ 7) independent polarization layers along the LOS, this provides us with an estimate of fM that is closer to equality between the turbulent and mean components of the GMF than for step B (for which N = 1, see Sect. 5.1). A value of fM = 0.9 with N = 1 would generate a much broader distribution of ψR than the observed one. The bottom panel of Fig. 11 shows that step C, unlike step B, can reproduce the histogram of p2 quite well. The combination of a small number of independent polarization layers along the LOS produces the large dispersion in p2 that is observed in the data.

thumbnail Fig. 11

Results of step C. This is the same as in Fig. 8, but with the model histograms now corresponding to step C with fM = 0.9, N = 7, and p0 = 26% (dashed-vertical line).

thumbnail Fig. 12

Model histograms of pC2\hbox{$p^2_{\rm C}$} obtained around the south Galactic pole from step C, using fM = 0.9, and with the value of N varying from 1 (dark blue) to 100 (dark red).

Our results show that, in order to reproduce the p2 distribution seen in the data, only a small number of polarization layers is needed. In Fig. 12, we present the effect of changing N on the distribution of p2 obtained with step C as pC2=qC2+uC2\hbox{$p^2_{\rm C}=q^2_{\rm C}+u^2_{\rm C}$}. In this case noise is not added and we fix l0 = 70°, b0 = 24°, and fM = 0.9, but vary N from 1 (dark blue line) to 100 (dark red line). The figure shows that for an increasing number of layers because of the central-limit theorem, the model distributions tend to rapidly converge towards a low p2 value, without the broad dispersion observed in the data. For large values of N, the width of the p2 distribution is dominated by the projection factor, cos2γ, in Eq. (7). Note that the histogram of pC2\hbox{$p^2_{\rm C}$} for N = 1 is not directly comparable with the modelled p2 distribution in Fig. 8 because it does not include noise.

6. Discussion

We have presented a phenomenological model that is able to describe the 1-point statistics of p and ψ for the Planck dust polarization data around the south Galactic pole, using a few parameters to describe the uniform and turbulent components of the GMF. We stress that our model is not entirely physical and certainly not unique. We made several assumptions, including: a single orientation of the mean field B0; a uniform ratio fM of the turbulent to mean strengths of the GMF along the LOS; a fixed value for the number of polarization layers, N, independent of the total dust intensity (unlike what was considered by Jones et al. 1992); and isotropy of the turbulent component, Bt. These assumptions restrict us from fitting the data over a larger portion of the sky than the southern Galactic cap. For the time being, we limit our study to this sky area. We now discuss the interpretation of our model in relation to the ISM physics and we present future perspectives on the modelling.

6.1. Density structure of the ISM

Our description of the turbulent component of the GMF along the LOS is based on a finite number of independent layers, rather than on a continuous variation computed from the power spectrum of the GMF, as was included in some earlier models (e.g. Miville-Deschênes et al. 2008; O’Dea et al. 2012; Planck Collaboration Int. XLII 2016). The density structure of the diffuse ISM provides one argument in favour of this approximation.

If we are in practice observing a finite number of localized density structures from the cold neutral medium (CNM) along the LOS, then the discretization of the GMF orientation is appropriate. Such structures appear as extended features on the sky in dust emission maps, with a power-law power spectrum. This statement is exemplified by the images and the power spectrum analysis of the dust emission from the Polaris cloud in Miville-Deschênes et al. (2010). The superposition of such clouds fits with our model, where the angular correlation is described with a continuous power spectrum, different from our ansatz for the radial correlation.

As shown for the diffuse ISM (Clark et al. 2014; Planck Collaboration Int. XXXII 2016; Planck Collaboration Int. XXXVIII 2016; Kalberla et al. 2016), the GMF orientation is correlated with the structure of matter as traced by H i or dust emission. Our modelling does not include the density structure of the ISM, nor does it include the correlation between matter and the magnetic field orientation; however, the polarization layers could phenomenologically represent distinct matter structures along the LOS. In this interpretation the GMF orientations are not completely uncorrelated. Although each CNM structure has a different turbulent component of the GMF, they share the same mean component. This correlation between the values of ψ of individual structures and those measured for the background emission in their surroundings is in fact observed in the Planck data (Planck Collaboration Int. XXXII 2016).

Observations of H i in absorption and emission have shown that, in the solar neighbourhood, about 60% of all H i arises from the warm neutral medium (WNM) and gas that is out of thermal equilibrium (Heiles & Troland 2003). Moreover, the diffuse ISM also includes the WIM, which accounts for about 25% of the gas column density (Reynolds 1989). These diffuse and warm components of the ISM are expected to contribute to the dust emission observed at high Galactic latitudes, both in intensity and in polarization. This contribution, which may be dominant, cannot be described by a small number of localized structures. For such media, the layers acquire a physical meaning if their spacing corresponds approximately to the correlation length of the turbulent component of the GMF.

6.2. Correlation length of the magnetic field

thumbnail Fig. 13

Model histograms of p2 normalized to unity with p0, obtained for a continuously varying GMF orientation along the LOS, with fM = 0.9 for several values of α, between 0 (black curve) and −3 (yellow curve). To facilitate the comparison of the histogram of (p/p0)2 with that in Fig. 12, we have used the same bin width (0.01) to compute both histograms.

In their modelling of dust polarization in molecular clouds, Myers & Goodman (1991) and Planck Collaboration Int. XXXV (2016) introduced a correlation length that is associated with the coupling scale through collisions between ions and neutrals. For the Planck data relating to the diffuse ISM, we propose a different interpretation. Following Eilek (1989), we derive the correlation length of the turbulent component of the GMF (lc) from the 2-point auto-correlation function, CB, of each of the three components of Bt: ∫CB(s) ds=lc σB2,\begin{equation} \int C_{B}(s) \, \mathrm{d} s = l_{\rm c} \, \sigma_{B}^2, \label{eq:eilek} \end{equation}(17)where s is the lag of CB along one given direction and σB is the dispersion of Bi. In this framework, the number of correlation lengths along the LOS is Nc = L/lc, where L is the effective extent of matter along the LOS. We compute CB from Gaussian realizations of Bi for power-law spectra5, and, from there, Nc integrating Eq. (17) up to the lag where CB = 0. Nc depends on the spectral index α of the power spectrum of the components of Bt. We find values of Nc of 16, 10, 6, and 5 for spectral indices of the power law spectrum α = −1.5, −2, −2.5, and −3, respectively.

We can now compute the Stokes parameters for this continuous description of Bt and the mean orientation, B0 (determined in Sect. 4), through the integral equations described in Appendix A, for several values of α using a constant source function as in step C. The Gaussian realizations and the integrals are computed over 1024 of points along each LOS at b< −60°. In this approach, used earlier by Miville-Deschênes et al. (2008), O’Dea et al. (2012), there is no correlation of Bt between nearby pixels on the sky. Hence, we cannot produce realistic images but we do sample the 1-point distribution of p2.

The histograms of p2 (normalized to unity with p0) are presented in Fig. 13 for several values of α, with fM = 0.9 and no data noise. We use the same binning as in Fig. 12 to allow for a direct comparison between the two sets of histograms. The continuous description of Bt matches the standard deviation of ψR measured in the Planck data for α ≃ −3. However, the corresponding histogram of (p/p0)2 in Fig. 13 is narrower than the one for N = 7 in Fig. 12, which fits the data better. We conclude that the number of polarization layers may be interpreted as the number of effective modes contributing to the variations of the orientation of Bt along the LOS within the WNM and WIM. From this view point, the low value of N derived from the data fit reflects the steepness of the power spectrum of Bt; however, this interpretation does not fully account for the data because it ignores the density structure of the diffuse ISM (i.e. the CNM).

6.3. Future perspectives

We now briefly outline a few future directions that could be taken to extend our data analysis and modelling.

We have started to investigate the impact of the GMF structure on the statistics of the polarization parameters. In an upcoming paper we will use the model presented in this work to reproduce the dust polarization power spectra measured by Planck (Planck Collaboration Int. XXX 2016) and constrain the value of αM, the value of which is left open in this paper. Another future project will be to introduce the density structure and its correlation with the orientation of the GMF within each polarization layer. Such a study will enable us to assess the respective contributions of the density and the GMF structure to the statistics of the dust polarization data.

In the present work, we have aimed at providing a phenomenological method to relate the dust polarization at high Galactic latitudes to the structure of the GMF. We want to stress the simplicity of our approach, which allowed us to fit the large-scale patterns of the polarization sky and the one point statistics of ψ and p, measured towards the southern Galactic cap, with a few parameters. In a future paper we will extend the present study to a larger fraction of the high-latitude sky on both hemispheres to assess the ability of our model to describe the Planck353 GHz dust polarization data over a larger area than the one considered in this study.

Our work is a main step towards a model which may be used to compute realizations of the dust polarization sky that fit the E, B and TE power spectra of dust polarization spectra reported in Planck Collaboration Int. XXX (2016) for the multipoles range 30 <ℓ< 300. Such simulated maps will be useful to assess the accuracy of component-separation methods in present and future CMB experiments designed to search the B mode CMB polarization expected from primordial gravity waves (BICEP2/Keck Array and Planck Collaborations 2015).

Planck Collaboration Int. XXXVIII (2016) and Clark et al. (2015) associated the asymmetry between EE and BB power spectra of dust polarization (i.e. CℓBB≃0.5 CℓEE\hbox{$C_\ell^{BB}\simeq 0.5 \, C_\ell^{EE}$}, Planck Collaboration Int. XXX 2016) with the correlation between the structure of the GMF and the distribution of interstellar matter. Future models will need to take this correlation into account in order to realistically assess the accuracy to which, for a given experiment, dust and CMB polarization can be separated.

7. Summary

We have analysed the Planck maps of the Stokes parameters at high Galactic latitudes over the sky area b < −60°, which is well suited for describing the Galactic magnetic field (GMF) structure in the diffuse interstellar medium (ISM), and is directly relevant for cosmic microwave background (CMB) studies. We characterized the structure of the Stokes parameter maps at 353 GHz, as well as the statistics of the polarization fraction p and angle ψ. We presented simple geometrical models, which relate the data to the structure of the GMF in the solar neighbourhood. Combining models of the turbulent and ordered components of the GMF, we have reproduced the patterns of the Stokes Q and U maps at large angular scales, as well as the histograms of p and ψ. The main results of the paper are listed below.

  • We find that the histogram of p of the southern Galactic cap has a similar dispersion as that measured over the whole sky, although with a smaller depolarization, caused by line-of-sight (LOS) variations of the GMF orientation, on and near the Galactic plane.

  • The Stokes Q and U maps show regular patterns at large scales, which we associate with the mean orientation of the GMF in the solar neighbourhood. We build a geometric model and find a mean orientation towards Galactic coordinates (l0,b0) = (70°,24°), compatible with previous estimates. The fit also provides us with the average value of p at b ≤ −60°, which is (12 ± 1)%.

  • By means of a simple description of the turbulent component of the GMF (Gaussian and isotropic), we manage to account for both the dispersion of ψ and the histogram of p. The effect of depolarization caused by the GMF fluctuations along the LOS is introduced through an approximation where the integrals along the LOS are replaced by a discrete sum over only a few independent polarization layers. This approach successfully reproduces the p and ψ distributions using N ≃ 4–9 layers.

  • The integration along the LOS generates a mean depolarization factor that is about 0.5 and thus leads to an estimate of p0 about two times greater than the average value of p. The best-fit value of the effective polarization of dust, which combines the intrinsic polarization of dust grains and their degree of alignment with the GMF, is (26 ± 3)%.

  • Our description of the turbulent component of the GMF corresponds to a rough equality between the turbulent and mean strengths of the GMF. The same conclusion was reached from modelling the dispersion of polarization angles measured for CNM filamentary structures by Planck Collaboration Int. XXXII (2016). We extend this to the diffuse ISM observed in the high latitude sky, which comprises both WNM and CNM gas.

The present study represents the first step towards the characterization of the magnetized properties of the diffuse ISM by means of the Planck data. We argue that both the density structure and the effective correlation length of the GMF contribute to the large dispersion of p observed in the data. This can be further investigated using MHD numerical simulations. The next step in our modelling of dust polarization at high Galactic latitudes will be to fit the E and B power spectra. This will constrain the spectral index of the GMF power spectrum, providing information on the turbulent energy cascade in the diffuse ISM. It is also a required step before using our model to compute simulated maps for assessing component-separation methods in CMB polarization projects.

1

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 and led by Principal Investigators from France and Italy, telescope reflectors provided through a collaboration between ESA and a scientific consortium led and funded by Denmark, and additional contributions from NASA (USA).

5

To a good approximation, σB2−CB\hbox{$\sigma_{B}^2 - C_{B}$} can be fitted with a power law of the lag s.

Acknowledgments

The Planck Collaboration acknowledges the support of: ESA; CNES, and CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MINECO, 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); ERC and PRACE (EU). A description of the Planck Collaboration and a list of its members, indicating which technical or scientific activities they have been involved in, can be found at http://www.cosmos.esa.int/web/planck/planckcollaboration. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement No. 267934.

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Appendix A: Approximations for dust polarization

In this Appendix, we detail the approximations made to model the Stokes parameters for linear polarization from dust emission. For the sake of clarity, we recall the integral equations of the Stokes parameters I, Q, and U from Planck Collaboration Int. XX (2015): I=∫Sν e−τν[1−p0(cos2γ−23)]dτν;Q=∫p0 Sν e−τνcos(2ψ)cos2γ dτν;U=∫p0 Sν e−τνsin(2ψ)cos2γ dτν.\appendix \setcounter{section}{1} % subequation 6390 0 \begin{eqnarray} \label{eq:simulated_I} \StokesI &=&\int S_\nu\,e^{-\tau_\nu}\left[1-p_{0} \left(\cos^2\polangsky-\frac{2}{3}\right)\right]\mathrm{d}\tau_\nu;\\ \label{eq:simulated_Q} \StokesQ&=&\int p_{0}\,S_\nu\,e^{-\tau_\nu}\cos\left(2\psi\right) \cos^2\polangsky\,\mathrm{d}\tau_\nu;\\ \label{eq:simulated_U} \StokesU&=&\int p_{0}\,S_\nu\,e^{-\tau_\nu}\sin\left(2\psi\right) \cos^2\polangsky\,\mathrm{d}\tau_\nu. \end{eqnarray}Here τν is the optical depth and Sν is the source function of dust emission, while p0 and the angles (ψ,γ) are the same as in Sect. 2.2.

We make two additional points: (1) in order to relate p as shown in Eq. (2) to the mean orientation of the GMF with respect to the POS (the angle γ), we need to assume that all parameters in Eqs. (A.1a)–(A.1c) are roughly uniform along the LOS; and (2) the total intensity in Eq. (A.1a) also depends on the GMF orientation through the angle γ. However, throughout our modelling procedure, we neglect this dependence.

In general the corrections to Stokes I caused by the GMF geometry are small, ranging roughly range between −7% and + 13% for p0 ≃ 20% (Planck Collaboration Int. XIX 2015). In our study, we focus on a region of the sky where the depolarization produced by cos2γ is small (cos2γ is mostly close to unity over the southern Galactic cap). Hence, in our study, the correction to Eq. (A.1a) would always be negative and less than 10%. Thus, in Sect. 5.3 we might estimate a value of p0 slightly greater than the true value that we would have obtained by modelling the GMF correction for Stokes I. In practice Eq. (8) in Sect. 4.2 would change as follows: Q353=p0qA1−p0(cos2γ−23)D353;U353=p0uA1−p0(cos2γ−23)D353.\appendix \setcounter{section}{1} \begin{eqnarray} \label{eq:stokesAfitcor} && Q_{353}= \frac{p_0q_{\rm A}}{1-p_0(\cos^2\gamma - \frac{2}{3})}D_{353};\nonumber \\ && U_{353}= \frac{p_0u_{\rm A}}{1-p_0(\cos^2\gamma - \frac{2}{3})}D_{353}. \end{eqnarray}(A.2)The fits of steps A, B, and C would then not be linear in p0 anymore, substantially complicating the fit. We argue that, considering the overall approximations (analytical and astrophysical) of our models, the GMF geometry in Stokes I is a minor issue.

All Figures

thumbnail Fig. 1

Orthographic projections centred on the south Galactic pole of the Planck dust emission intensity, D353 (left), and the Stokes Q353 (centre) and U353 (right) maps, at 353 GHz. A grid of Galactic coordinates is included, labelled in degrees. East is on the left of the maps and the west on the right. We note that the U353 map is in the HEALPix (or CMB) polarization convention, which corresponds to −U353 in the IAU convention.
In the text
thumbnail Fig. 2

PlanckD353 (same as in the left panel of Fig. 1) with the “drapery” pattern, orthogonal to the polarization orientation, produced with the LIC algorithm. The part of the sky at b< −60° has been highlighted in colour in this figure.

In the text
thumbnail Fig. 3

Normalized distributions of the polarization fraction from the pMAS debiased estimator (see text). The black distribution shows pMAS over the whole sky. The green distribution shows pMAS at b< −60°. The green-shaded area represents the 1σ error on pMAS at high latitude.
In the text
thumbnail Fig. 4

Mollweide (top) and orthographic (bottom) projections of the model Stokes parameters, qA (left) and uA (right), for a uniform direction of the GMF towards (l0,b0) = (80°,0°); these are roughly the values inferred from starlight polarization (Heiles 1996). The orthographic projections are centred on the Galactic poles. Galactic coordinates in degrees are shown on all plots.
In the text
thumbnail Fig. 5

Same as in the left panel of Fig. 1, but now highlighting the b < −60° region, excluding the brightest clouds (in grey in the image), which has been used to fit the orientation of the mean magnetic field in step A of our model fit.
In the text
thumbnail Fig. 6

Step A: orthographic projections of qA (left) and uA (right) centred on the south Galactic pole, for the best-fit direction of the uniform GMF towards (l0,b0) = (70°,24°). The sky at b> −60° is masked here.
In the text
thumbnail Fig. 7

Orthographic projections centred on the south Galactic pole of Q353R\hbox{$\StokesQ^{R}_{353}$} (left) and U353R\hbox{$\StokesU^{R}_{353}$} (right), the Stokes parameters in a reference frame rotated with respect to the best-fit direction of the uniform component of the GMF towards (l0,b0) = (70°,24°). The sky for the masked b> −60° region appears in grey.
In the text
thumbnail Fig. 8

Results of step B. Top: histogram of ψR, the polarization angle inferred from the Stokes parameters rotated with respect to the best-fit uniform direction of the GMF (Q353R\hbox{$\StokesQ^{R}_{353}$} and U353R\hbox{$\StokesU^{R}_{353}$}), over the southern Galactic cap (black dots). The error bars represent the Poisson noise within each bin of the histogram. The green line represents the mean of the step B results for fM = 0.4 over 20 different realizations. The green shaded regions correspond to ± 1σ (light green) and ± 2σ (dark green) variations of the model. Bottom: histogram of p2 obtained when combining the Year 1 and Year 2 maps (black dots). The error bars here represent the Poisson noise within each bin of the histogram. Step B is now shown in blue. The dashed vertical line corresponds to a value of the polarization fraction of 12%.
In the text
thumbnail Fig. 9

Cartoon illustrating, for step C, the integration of qC along the LOS, with four distinct polarization layers for the same value of fM and the same mean orientation of the GMF. Each map in this cartoon is a realization of the model.
In the text
thumbnail Fig. 10

Results of step C. Maps of χtot2\hbox{$\chi^2_{\rm tot}$} from the fit of step C to the data for p0 = 26% (left), for N = 7 (centre), and for fM = 0.9 (right). The three maps show in colours and with contours the quantity log10(χtot2)\hbox{${\rm log}_{10} (\chi^2_{\rm tot})$}.
In the text
thumbnail Fig. 11

Results of step C. This is the same as in Fig. 8, but with the model histograms now corresponding to step C with fM = 0.9, N = 7, and p0 = 26% (dashed-vertical line).
In the text
thumbnail Fig. 12

Model histograms of pC2\hbox{$p^2_{\rm C}$} obtained around the south Galactic pole from step C, using fM = 0.9, and with the value of N varying from 1 (dark blue) to 100 (dark red).
In the text
thumbnail Fig. 13

Model histograms of p2 normalized to unity with p0, obtained for a continuously varying GMF orientation along the LOS, with fM = 0.9 for several values of α, between 0 (black curve) and −3 (yellow curve). To facilitate the comparison of the histogram of (p/p0)2 with that in Fig. 12, we have used the same bin width (0.01) to compute both histograms.
In the text

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