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Appendix A: Uncertainties for polarization quantities

We start with the values of the Stokes parameters, I, Q, and U, and the noise covariance matrix, [ C ], at the position of each star. By definition, the uncertainties σ_QS of Q_S and σ_US of U_S are calculated from the variances and , respectively.

The variance of the polarized intensity (without bias correction) is (A.1)from which we derive the S / N of the polarization intensity, P/σ_P. For the uncertainty in the position angle ψ_S we use the approximate formula in Eq. (5), which is appropriate because we always require the S/N to be larger than 3¹⁶.

The uncertainty of the projected polarization fraction Q_S/I_S has the following dependence: (A.2)and the same holds for U_S/I_S. The covariance of Q_S/I_S with U_S/I_S follows: (A.3)The uncertainties σ_QS/IS and σ_US/IS, used for plotting only, are derived simply from the variances C_Q/I,Q/I and C_U/I,U/I.

When the data are smoothed, with the method described in Appendix A of Planck Collaboration Int. XIX (2015)¹⁷, the formulae hold substituting the smoothed Stokes parameters and the elements of the corresponding covariance matrix, [].

Appendix B: Extinction catalogues

Appendix B.1: Extinction and colour excess catalogues

Using stellar atmosphere models, Fitzpatrick & Massa (2007) provide well-determined E(B − V) and A_V measurements for 328 stars, 14 of which could be identified in the Heiles (2000) polarization catalogue via their catalogue identifiers (HD – Henry Draper, BD – Bonner Durchmusterung, CD – Cordoba Durchmusterung, or CPD – Cape Photographic Durchmusterung identifiers). This was the basis for what we refer to as the “FM07 sample”. Similarly, from the Valencic et al. (2004) and Wegner (2002, 2003) extinction catalogues (derived with the more standard technique based on “unreddened” reference stars), we generated the VA04 and WE23 samples; we note that we have removed stars in common with previously-defined samples, with the same order in priority of the samples. These three samples all contain measurements of both A_V and E(B − V), providing an estimate of R_V, a useful diagnostic of the diffuse ISM where R_V is close to 3.1.

The Savage et al. (1985) catalogue provides measures of E(B − V) to 1415 stars, 1085 of which were identified in the Heiles (2000) catalogue. Lacking a measure of R_V, we assumed the standard value for the diffuse ISM, R_V = 3.1; its uncertainty δR_V = 0.4¹⁸ adds another uncertainty to our estimate of τ_V. Again removing stars in common with previous samples, we built the SA85 sample.

Appendix B.2: Deriving E(B–V) from the Kharchenko & Roeser star catalogue

	Fig. B.1 Correlation between our derived E(B − V) from the Kharchenko & Roeser (2009) catalogue with that for common stars in the FM07, VA04, and WE23 samples. Only data with S / N on E(B − V) larger than 3 are presented. The red line is a 1:1 correlation, while the black line is a fit.
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As described below, using the high-quality photometry in the all-sky catalogue of Kharchenko & Roeser (2009) we were able to derive E(B − V) and its uncertainty for more than 3000 stars present in the Heiles (2000) catalogue. Stars absent from other samples then form the KR09 sample. As for the SA85 catalogue, we assumed R_V = 3.1 ± 0.4 for all stars.

The Kharchenko & Roeser (2009) catalogue (SIMBAD reference code I/280 B) used for the derivation of E(B − V) is a compilation of space and ground-based observational data for more than 2.5 million stars. The catalogued data include, among others, B and V magnitudes in the Johnson system, K magnitude, HD number, and the spectral type and luminosity class of the star. TOPCAT¹⁹ (Taylor 2005) was used to cross-match the polarization (Heiles 2000) and extinction (Kharchenko & Roeser 2009) catalogues with the HD number as the first-order criterion. Where two or more stars are identified with the same number, the one for which the visual magnitude is closest to that in the polarization catalogue was retained. For the rest of the catalogue, a coordinate-match criterion with a 2′′ radius was applied, using the visual magnitude to choose between candidates if necessary.

We have used the intrinsic colors (B − V)₀ derived by Fitzgerald (1970) for different spectral classifications. The colour excess for each of star was calculated according to E(B − V) = (B − V)_KR09 − (B − V)₀, using the B and V magnitudes and the corresponding intrinsic colour deduced from the spectral classification in the Kharchenko & Roeser (2009) catalogue. Following Savage et al. (1985), Be stars and stars with spectral types B8 and B9 were removed.

A few hundred stars overlapping the FM07, VA04, WE23, or SA85 samples allowed us to check the quality of our derivation of E(B − V). Figure B.1 reveals a good correlation with the other samples, with our KR09 E(B − V) tending to underestimate the reddening to the star by about 9%. This small discrepancy has only a small impact on the derived R_{S /V} (the KR09 sample represents one third of our sample, see Table 1) and does not affect R_P/p.

Appendix C: Robustness tests

From the joint fits and bootstrap analysis of uncertainties in Sects. 6.1 and 6.2 we found R_{S /V} = 4.1 ± 0.2 and R_P/p = (5.3 ± 0.2) MJy sr^-1. In this appendix we investigate the robustness of these polarization ratios with respect to the selection criteria defining the sample, the data used, the region analyzed, and the methodology. We derive each time their mean values and uncertainties.

As a potential drawback owing to its simplicity, R_P/p involves systematic dependences on parameters such as the ambient radiation field, the submillimetre opacity of aligned grains, and the presence of a background beyond the star. Because P_S and p_V are proportional to the column density (of polarizing dust) probed with their respective observations in the submillimetre and visible, the correlation presented in Fig. 7 is potentially biased (an overestimate) if there is systematically a background beyond the stars selected (Fig. 1). Thanks to the normalization of P_S by I_S and p_V by τ_V, such dependences are weakened²⁰ in the analysis of R_{S /V}.

Appendix C.1: Selection criteria

We explore the effects of varying the limits of the four selection criteria presented in Sect. 5. We do this one criterion at a time, with the others unchanged. We also examine other alternatives for defining the sample.

S/N.

The accuracy of the polarization degree in extinction data is not a limiting factor because the mean S / N is about 10 for the selected stars. Asking for a S / N threshold higher than 3 (Sect. 5.1) for P_S and for A_V could bias our estimates of R_{S /V}, which is proportional to these quantities. It would also exclude many diffuse regions where such a high S/N cannot be achieved at 5′ resolution. Nevertheless, we find no significant variation of the polarization ratios when imposing S/N> 1 (268 stars, R_{S /V} = 4.1 ± 0.1, R_P/p = (5.3 ± 0.2) MJy sr^-1) or S/N> 10 (68 stars, R_{S /V} = 4.3 ± 0.2, R_P/p = (5.4 ± 0.2) MJy sr^-1).

Diffuse ISM.

The E(B − V)_S criterion (Eq. (7) in Sect. 5.2) is responsible for the removal of lines of sight toward denser environments or toward the Galactic plane. Ignoring this criterion so that these stars are included gives R_{S /V} = 4.0 ± 0.2 and R_P/p = (5.7 ± 0.2) MJy sr^-1, for 284 stars. On the other hand, we can be more strict in our selection by imposing lower E(B − V)_S. A limit rather than our reference criterion 0.8 has no effect. With even lower column densities ( and 0.3), we get R_{S /V} = 4.7 ± 0.2 and R_{S /V} = 4.7 ± 0.4, for 82 and 42 stars, respectively. While an increase in R_{S /V} with decreasing column density could arise through the inverse dependence of R_{S /V} on I_S (Eq. (1)), the evidence is not strong. Changes in R_P/p are not significant: (5.7 ± 0.2) MJy sr^-1 and (5.6 ± 0.4) MJy sr^-1, respectively.

We can restrict our sample to lines of sight where the ratio of the total to selective extinction, R_V, is close to 3.1, its characteristic value for the diffuse ISM (e.g., Fitzpatrick 2004). Specifically, we exclude those lines of sight where R_V was not measured (SA85, KR09), and impose 2.6 <R_V< 3.6. Our sample is then reduced to 69 stars and gives R_{S /V} = 4.0 ± 0.2 and R_P/p = (5.2 ± 0.2) MJy sr^-1.

A similar selection can be made on the basis of the wavelength corresponding to the peak of the polarization curve in extinction, λ_max, as taken from Serkowski et al. (1975). Imposing 0.5 μm <λ_max< 0.6 μm, we find R_{S /V} = 4.0 ± 0.3 and R_P/p = (5.1 ± 0.3) MJy sr^-1, for 34 stars.

Column density ratio.

The polarization ratio R_{S /V} is, by construction, proportional to E(B − V) and could therefore anticorrelate with R_τS. However, we do not find such dependance when varying the upper limit of R_τS from 1.2 (R_{S /V} = 4.2 ± 0.2, 121 stars) to 1.8 (R_{S /V} = 4.1 ± 0.1, 231 stars). Going beyond the limits where angles agree, with upper limits of 2.0 and 3.0, yields the same value 4.1 ± 0.2 for the polarization ratio, for 251 and 284 stars, respectively. We also tested other proxies to estimate the total column density observed by Planck. Replacing the dust optical depth at 353 GHz, used to derive E(B − V)_S, by the Hi 21 cm emission or the dust radiance (the total power emitted by dust, Planck Collaboration XI 2014) did not affect our polarization ratios significantly.

Orthogonality.

If we become more restrictive in our selection based on the difference between position angles (Eq. (11) in Sect. 5.4), by requiring a 1σ agreement our sample shrinks to 112 stars. The quality of the fit is preserved, as expected: R_{S /V} = 4.2 ± 0.2 and R_P/p = (5.5 ± 0.2) MJy sr^-1.

Galactic height.

As mentioned in Sect. 5.3, the Galactic height of the star can play a role, similar to that of the column density ratio R_τS, in selecting lines of sight with a low probability of background emission. Still requiring R_τS< 1.6 and then selecting on H> 100 pc, H> 150 pc, and H> 200 pc, we find the same polarization ratios, R_{S /V} = 4.1 ± 0.1 for 136, 91 and 53 stars, respectively. Selecting on H> 100 pc without selecting on the column density ratio, we obtain R_{S /V} = 4.0 ± 0.2 (172 stars). Results for R_P/p are also similar, with an average R_P/p = (5.2 ± 0.1) MJy sr^-1.

	Fig. C.1 Planck line of sight dust temperature Tdust (Planck Collaboration XI 2014) and the column density to the star, E(B − V), for the independent samples.
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Thermal dust temperature.

The dust temperature T_dust from Planck Collaboration XI (2014) characterizes the spectral energy distribution of the combined emission I_S from all dust components in the column of dust along the line of sight and has no direct connection to the sample selection. Figure C.1 shows the distribution in the T_dust – column density (as measured by E(B − V)) plane. The ranges of E(B − V) and T_dust are considerable for each sample. In this plane, there is a band showing a slight anti-correlation of T_dust and E(B − V); there are also several lines of sight with T_dust ≃ 21 K but with a range of E(B − V).

We looked for any dependencies of R_{S /V}, R_P/p, and I_S/A_V on T_dust, for data with and without the correction for leakage of intensity into polarization. In Fig. C.2 the data were binned in T_dust, each bin containing the same number of stars. The blue curve in the right panel of Fig. C.2 shows the relative change arising from the expected increase in I_S (but not A_V) when T_dust increases. It has been fit to the data in the vertical direction. A similar trend in R_P/p arising from P_S would be expected in the middle panel if the subset of grains that are polarizing had the same temperature as characterized the total emission. This appears to be consistent with the corrected data. Under the same hypothesis, the trend for R_{S /V} would be flat in the left panel. This too appears to be consistent with the corrected data.

Appendix C.2: Data used

Here we test the sensitivity of our results to the choice of extinction catalogues and to the smoothing and original processing of the Planck data. The main sources of data uncertainty are, for the extinction data, the measure of A_V and, for the polarized emission data, the instrumental systematics related to the correction for the leakage of intensity to polarization (Sect. 3.1).

Extinction catalogues.

One important source of uncertainty in R_{S /V} (not R_P/p) is the measure of the dust extinction in the visible, A_V. In Table C.1 we summarize our results for each catalogue taken separately, independently of the others (i.e., we do not remove common stars). Although the catalogues neither share all of the same stars nor have the same extinction data for stars in common, the estimates obtained for R_{S /V} and I_S/A_V are compatible. R_P/p is independent of A_V, therefore of any extinction samples. Its variations among catalogues helps to constrain its statistical uncertainty, here less than 0.2 MJy sr^-1.

Fig. C.2 Left: mean RS /V as a function of the mean Tdust, each plotted with the standard deviation, in bins of equal number. Two versions of the Planck data have been used: with (black) and without (red) correction for leakage of intensity into polarization. Middle: the same, but for RP/p. Right: the same, but for IS/AV. The blue curves, motivated in the right panel, show the expected response of the ratios to an increase in Tdust, according to a simple model in which the subset of grains that are polarizing had the same temperature as characterized the total emission (see text).

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Smoothing.

In order to increase the S / N of the Planck polarization data and increase the quality of the correlations in Q and U, we chose to smooth our maps with a 5′ FWHM Gaussian, making the effective map resolution 7′. Because Q and U are algebraic quantities derived from a polarization pseudo-vector, the smoothing of polarization maps statistically tends to diminish the polarization intensity P_S, which would propagate into the polarization ratios R_{S /V} and R_P/p. Using the raw (not smoothed) data, or data smoothed with a beam of 3′ and 8′ (keeping the same sample as was selected using data smoothed with a 5′ beam to allow for an unbiased comparison), we find R_{S /V} = 4.0 ± 0.2, 4.1 ± 0.2, 4.2 ± 0.2, and R_P/p = (5.2 ± 0.3) MJy sr^-1, (5.3 ± 0.2) MJy sr^-1, (5.4 ± 0.2) MJy sr^-1), respectively.

We note that the mean values of R_{S /V} and R_P/p could still be underestimated owing to depolarization in the Planck beam, which does not have a counterpart in the visible measurement (see Fig. 1); however, this effect should be small and within the uncertainties.

Zodiacal emission.

Removing zodiacal emission, or not, in deriving I_S (Sect. 3.1) does not affect our result, with R_{S /V} = 4.1 ± 0.2 in both cases.

Contamination by CMB polarization at 353 GHz.

Following the approach in Planck Collaboration Int. XXII (2015), we can remove the CMB patterns in intensity and polarization from the 353 GHz maps by subtracting the 100 GHz (I,Q,U) maps in CMB thermodynamic temperature units. This method unfortunately adds noise to the 353 GHz maps, and is therefore used only as a check. It also has the drawback of subtracting a fraction of dust emission that is present in the 100 GHz channel. However, dust is then subtracted both in intensity and in polarization, though perhaps not proportionally to the polarization fraction at 353 GHz (however, this is a second-order effect). With this 100 GHz-subtracted version of Q and U, we obtain R_{S /V} = 4.1 ± 0.2 and R_P/p = (5.2 ± 0.2) MJy sr^-1, for 203 stars.

Leakage correction.

A small correction for leakage of intensity into polarization has been applied to the Planck polarization data used here (see Sect. 3.1). While this correction is imperfect, the alternative of ignoring this correction leaves systematic errors in the data. For the version of the data not corrected for leakage, we obtain figures similar to Fig. 7, for 196 selected stars, with Pearson correlation coefficients − 0.94 and − 0.95 and and 2.56, for R_{S /V} and R_P/p, respectively. Running the bootstrap analysis we find R_{S /V} = 4.3 ± 0.2 and R_P/p = (5.5 ± 0.2) MJy sr^-1, systematic changes of + 0.2 and + 0.2 MJy sr^-1 compared to our values for data corrected for leakage. Therefore, the correction of the March 2013 Planck polarization data for this leakage is a significant source of systematic uncertainty in R_{S /V} and R_P/p, though perhaps the uncertainty is not as much as 0.2 or of the same sign.

Appendix C.3: Region analyzed

The polarization ratios that we derived are an average over the sky. Here we examine the ratios for spatial subsets of the data.

Galactic hemisphere or latitude.

We find no significant variation of the polarization ratios between the two hemispheres: R_{S /V} = 4.0 ± 0.3 and R_P/p = (5.1 ± 0.3) MJy sr^-1 for the northern Galactic hemisphere, and R_{S /V} = 4.2 ± 0.1) and R_P/p = (5.4 ± 0.2) MJy sr^-1 for the southern.

Polarization ratios might depend on the latitude of stars if that were indicative of different potential backgrounds. Selecting high latitude stars from both hemispheres (| b | > 6°, 95 stars) to limit the presence of backgrounds, we find R_{S /V} = 4.2 ± 0.3, R_P/p = (5.2 ± 0.3) MJy sr^-1, and I_S/A_V = (1.24 ± 0.07) MJy sr^-1. For low latitude stars (, 111 stars), R_{S /V} = 4.1 ± 0.1, R_P/p = (5.3 ± 0.2) MJy sr^-1, and I_S/A_V = (1.25 ± 0.04) MJy sr^-1, with no indication of any contamination by backgrounds.

Selected regions on the sky.

Table C.2 presents the polarization ratios for three regions, among them the Fan which contains almost one third of our selected stars; these results are close to the overall average. If we select all stars except those from the Fan, we find R_{S /V} = 4.2 ± 0.2 and R_P/p = (5.4 ± 0.2) MJy sr^-1. Table C.2 also presents the polarization ratios for two other regions in the local ISM where stars in our sample are more concentrated (see Fig. 5): the Aquila Rift, and the Ara region (Planck Collaboration Int. XIX 2015). Taking into account the uncertainties, we conclude that our total sample is not biased by any particular region and there is no evidence for spatial variations.

Appendix C.4: Correlation plots in P

Fig. C.3 Left: correlation of debiased polarization fraction in emission with that in extinction (Pearson coefficient 0.74). Right: correlation of the debiased polarized emission (in MJy sr-1) with starlight polarization degree (Pearson coefficient 0.87). The range corresponds to one quadrant in Fig. 7. The fits are forced to go through the origin and have slopes RS /V = 4.18 ± 0.04 () and RP/p = (5.41 ± 0.04) MJy sr-1 (), respectively.

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In Sect. 6.1 we derived R_{S /V} and R_P/p using joint correlation plots in Q and U rather than in the biased quantity P. However, our selection of P_S and p_V with S/N> 3 implies that the bias should not be too significant and it is possible to debias P at least statistically (the Modified Asymptotic debiasing method of Plaszczynski et al. 2014 was used; see also references in Sect. 6.1). This is confirmed by the correlation plot in Fig. C.3 for debiased polarization fractions (which are almost identical to those for the original data). The data in the submillimetre and visible show a fairly good correlation, though, compared with Fig. 7, have a smaller dynamic range and a smaller Pearson correlation coefficient. We fit the slopes, forcing the fit to go through the origin unlike for the fits in Fig. 7. Whether with debiased data or not, the polarization ratios (from bootstrapping) are essentially identical: R_{S /V} = 4.2 ± 0.1 and R_P/p = (5.4 ± 0.1) MJy sr^-1, and also the same as found in the preferred analysis in Fig. 7.

Appendix D: R_S/V and its relationship to the maximum observed polarization fractions

For a given dust model, including the grain shape, the maximum polarization fraction that can be observed corresponds to the ideal case of optimal dust alignment: the magnetic field lies in the plane of the sky, has the same orientation (position angle) along the line of sight, and the dust alignment efficiency with respect to the field is perfect. The maximum p_V/τ_V ≃ 3% observed in extinction (corresponding to , Serkowski et al. 1975), is supposed to be close to this ideal case (Draine & Fraisse 2009).For our selected sample of lines of sight, Fig. 6 (left panel) shows this classical envelope and the corresponding envelope (P_S/I_S = 3% × R_{S /V} = 12.9%) transferred to emission (right panel).

We have also investigated the upper envelope that might be derived independently from the emission data. At a resolution of 1°Planck HFI has revealed regions with P_S/I_S greater than 20% (Planck Collaboration Int. XIX 2015), albeit for only a very small fraction (0.001, their Fig. 18) of lines of sight, toward local diffuse clouds. This value, which is already a high envelope, might have been even larger were it at the finer resolution of starlight measurements. Combined with the 3% limit from stars, this would apparently imply R_{S /V}> 6, significantly higher than our mean value R_{S /V} = 4.2.

However, for statistical reasons, these two estimates of R_{S /V} cannot be compared straightforwardly: Planck statistics are based on (almost) full-sky data, while those of Serkowski et al. (1975) are based on less than 300 stars. A more consistent statistical comparison can be sought. Analyzing Fig. 9 of Serkowski et al. (1975) and limiting our analysis to stars

satisfying 0.15 <E(B − V) < 0.8 as in our selection criterion (Sect. 5.2), the upper envelope is approximately the 96% percentile of p_V/E(B − V) (in this interval of E(B − V), 8 stars out of about 200 lie above this envelope). The corresponding 96% percentile of P_S/I_S in our selected sample is P_S/I_S = 14.5% (8 stars out of 206 above that line). As a complement, we can obtain an estimate of the 96% percentile of the full Planck map by smoothing the 353 GHz maps with a Gaussian of 5′ and selecting those pixels with 0.15 <E(B − V)_S< 0.8. The 96% percentile of P_S/I_S (after debiasing) in this sample of over 10⁷ pixels is found to be 13.2%. Combining these estimates based on consistent percentiles implies R_{S /V} in the range 4.2−4.6, compatible with our direct, and more rigorous, result.

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