Interstellar Polarization Survey. II. General Interstellar Medium

The optical polarization data were obtained as a part of the IPS program (see Magalhães et al. 2005; A. M. Magalhães et al. 2023 in preparation), based on observations made at the Observatório do Pico dos Dias/Laboratório Nacional de Astrofísica (LNA; Brazil). The polarimeter, a modified CCD camera, consists of a Savart prism and a rotating half-wave wave plate (see Magalhães et al. 1996 for more details). A remarkable feature of the polarimeter is the simultaneous imaging of both the ordinary and the extraordinary beams, which allows for photon noise–limited observations even under nonphotometric conditions, as well as cancellation of any sky polarization. The polarimeter was mounted onto the Cassegrain focus of the IAG Boller & Chivens 61 cm telescope at the Observatório do Pico dos Dias, operated by LNA in Brazil. The IPS covered different regions of the sky aiming at various distinct science goals, such as open clusters or dense clouds.

The Savart plate allows us to simultaneously image two perpendicularly polarized images of each star in the field. As detailed in Magalhães et al. (1984, 1996), we can write the ratio between the difference and the sum of these image counts for each object in terms of the Stokes parameters Q and U of the object. This difference-to-sum ratio varies sinusoidally as a function of the wave plate position. Typically, a polarization measurement consists of images taken at eight wave plate positions (0°, 225, 45°, 675, 180°, 2025, 225°, and 2475) separated by 225. Four of these positions alternately measure Q and –Q (0°, 45°, 180°, and 225°), and the other four measure U and –U. This averages out any possible effect from inhomogeneities of the wave plate and also removes any contribution of the background to the polarization. The Stokes parameters are then estimated by a least-squares solution of the sinusoidal curve, whose amplitude is defined by Q and U, through the diff/sum points as a function of wave plate position. The accuracy of Q and U (and hence P) is then estimated from the squared differences from each diff/sum point to the fitted curve. One can compare the accuracies thus obtained (the ones quoted in our results) with photon noise estimates as discussed in detail in Magalhães et al. (1984, 1996; A. M. Magalhães et al. 2023 in preparation). The accuracies are compatible with the expected photon noise values.

For this paper, we looked at the 38 "General ISM" fields of view (IPS-GI) observed in the V band, for which the observations were carried out between 2000 April and 2003 August. The Galactic coordinates and number of objects (after applying quality filters; see Section 4.2) of these fields can be found in Table 1. The fields are approximately 03 × 03 in size and mostly located close to the Galactic plane (∣b∣ < 10°). For a total of 41,108 stars, each field contains, on average, about 1000 stars. The fields are primarily centered on stars in the Heiles (2000) catalog. Each field was carefully selected to avoid dense structures and clouds, thus focusing on the diffuse ISM. Some fields were observed more than once using different exposure times, typically ranging from 10 to 30 s per wave plate position. The different exposure times ensure a good covering of a wide magnitude range, achieving high signal-to-noise ratio measurements of P/σ_P > 5 for some faint stars of about V_mag = 18, although, due to the different exposure times, we cannot claim completeness.

The data were reduced using the SOLVEPOL pipeline, developed specifically for imaging polarimetry data of this type (Ramirez et al. 2017). SOLVEPOL uses the astrometry.net software (Lang et al. 2010) to calibrate the astrometry and the Guide Star Catalog version 2.3 (Lasker et al. 2008) for the magnitude calibration. Images were corrected for bias and flat-fielding whenever possible, and if no bias frames and flat fields were available, the raw images were used in the processing. This did not lead to significantly lower-quality results. SOLVEPOL produces tables with polarimetric (degree of linear polarization P, polarization angle θ) and photometric (V-band magnitudes) measurements and their associated uncertainties for each detected star. SOLVEPOL uses the following equations to calculate the degree of polarization and the polarization angle (see Ramirez et al. 2017 for more details):

$$\begin{eqnarray}&&P=\sqrt{{Q}^{2}+{U}^{2}},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\theta =\displaystyle \frac{1}{2}\arctan \displaystyle \frac{U}{Q},\end{eqnarray} \tag{ 2 }$$

in which Q and U are the normalized Stokes parameters, and P refers to the degree of linear polarization as a fraction of the total intensity. We follow the conventions from Ramirez et al. (2017) and refer to that paper for more details. Degrees of polarization will be presented as a percentage throughout the present work.

The output tables were used for further processing. Entries from stars that were observed more than once were merged. The final value is an average of all observations, weighted with the inverse of the squared uncertainties in the Stokes parameters. Observations of standard stars of known polarization were used to determine the correction of the polarization angle to the equatorial coordinate system. To find the instrumental polarization, we used observations of known unpolarized standard stars. We calculated weighted averages of the Stokes parameters Q and U per observing run. The weights were set equal to the inverse of the squares of the uncertainties in Q and U. The Stokes parameters were then used to find the polarization, following the equations above. This led to an average instrumental polarization of 0.07%. This is smaller than the observational uncertainties, as the average observational error in the degree of polarization across the entire data set is approximately 0.3%; thus, we did not correct for instrumental polarization.

In this paper, the averages of the degree of linear polarization P and polarization angle θ are calculated following the conventions from Pereyra & Magalhaes (2007). The polarimetric averages are weighted by the inverse of the squared uncertainty in the Stokes parameters Q and U.

Measurements of the degree of linear polarization are biased at low P/σ_P (Clarke & Stewart 1986). The polarization values in this paper have not been debiased. We instead filtered the data to only include sources with P/σ_P > 5, thus avoiding the need to debias data. However, for verification, we compared our biased degrees of polarization to values that have been debiased using the estimator from Plaszczynski et al. (2014),

$$\begin{eqnarray}&&{P}_{d}=P-{\sigma }_{P}^{2}\displaystyle \frac{1-{e}^{\tfrac{-{P}^{2}}{{\sigma }_{P}^{2}}}}{2P},\end{eqnarray} \tag{ 3 }$$

where P_d is the debiased polarization. Correcting the polarization degrees using this estimator did not lead to significant differences in the values. Differences between the biased and debiased polarization values are of the order of 10⁻³%, well below the typical uncertainties. This is consistent with the findings of Plaszczynski et al. (2014), who found that their estimator becomes indistinguishable from other estimators beyond P/σ_P > 3. We therefore conclude that the P/σ_P filter is an effective limiting criterion in avoiding statistical difficulty. Applied to the full, unfiltered data set, the difference between the biased and debiased values is larger (of the order of 0.1%, on average, the same order of magnitude as the measurement error). This shows the need for debiasing if the data were used without stringent P/σ_P filters. We note that the applied P/σ_P filter may inadvertently remove lower-polarization stars.

Our analysis required each star to have an estimate of its distance and interstellar extinction in addition to its interstellar polarization. Therefore, we cross-correlated our initial polarimetric sample with other catalogs as described below. Each observed field was assigned a unique identifier, and each star in the catalog was matched to a Gaia EDR3 counterpart (Brown et al. 2021) using Topcat (Taylor 2005). Matches are based on position (a 3'' margin, corresponding to 1.5σ_position for IPS) and magnitude (2 mag margin). This margin in magnitude is to account for differences in Gaia's G-band magnitude and IPS's V-band magnitude. Visual inspection of the positions of the stars in each catalog showed that there is little chance of mismatches; most stars are close to only one other potential match. Even if more than one potential match were nearby, inclusion of the magnitude filter often finds the best match. However, uncertain matches will be excluded from further analysis; see also Section 4.2. For 96% of the IPS sources, a Gaia counterpart was found within the given margins. In addition, we added data from the Anders et al. (2022) StarHorse-based catalog, which includes not only distances but also V-band extinctions and many other stellar parameters. The Anders et al. (2022) parameters were matched to each star using the unique Gaia EDR3 ID.

Another auxialiary catalog that provides distances is that of Bailer-Jones et al. (2021), which is also based on Gaia EDR3. However, because we will also be needing other stellar parameters such as extinction, considering the close relationship between extinction and distance, we have decided to use the Anders et al. (2022) catalog. However, for completeness, we include Bailer-Jones et al. (2021) photogeometric distances in the analysis where appropriate to confirm that the differences between the two catalogs have no effect on our conclusions.

Considering the close relation between polarization and the distribution of dust, we must critically assess the quality of the V-band extinction, as A_V is one of the most direct tracers of dust available in the Anders et al. (2022) catalog (the median value is called A_V50 therein). There are multiple reasons for using parameters from Anders et al. (2022). First, the Bayesian algorithm, StarHorse, uses the photometry of multiple surveys (e.g., Gaia EDR3, 2MASS, AllWISE, PanSTARRS1 DR1, and SkyMapper DR2) and precise parallax observations from Gaia EDR3 (Brown et al. 2021), covering the entire sky and increasing the accuracy of its parameters in comparison with past versions of the catalog. Second, due to the location of the IPS-GI fields in the southern sky, some fields of view are covered either poorly or not at all by other available dust maps, such as Marshall et al. (2006), Green et al. (2019), and Lallement et al. (2019). This also includes coverage and precision in distance. Finally, the small size of the fields challenges the resolution of dust maps such as Collaboration et al. (2016); a single pixel, to which a single A_V value is assigned, may contain multiple IPS-GI stars. Additionally, Collaboration et al. (2016) measured the total extinction in emission integrated along the entire Galaxy path length with no regard for variations along the line of sight.

We quantitatively compared the A_V50 values of Anders et al. (2022) to other available extinction measurements from the Marshall et al. (2006) and Green et al. (2019) three-dimensional dust maps. To this end, the Marshall et al. (2006) K-band extinctions were converted to the V band with the relative extinction value A_K /A_V = 0.078 (Wang & Chen 2019, Table 3). The Green et al. (2019) extinctions, which are initially in arbitrary units, were converted to the V band using the method suggested by the author for StarHorse-like data (Equations (30) and (31) from Green et al. 2019). We found that despite the differences between the methods to map the dust extinction and the limitations on distance and sky coverage, the extinction values of the matching sources are in good agreement with each other. The median systematic difference of Green et al. (2019) and Marshall et al. (2006) from the Anders et al. (2022) extinctions is consistent within the 68% confidence interval (between the 16th and 84th percentiles of the difference) out to at least A_V50 ∼ 4 mag. The median systematic difference increases for A_V50 > 4 mag. A comparison with the Collaboration et al. (2016) measurements in intermediate-latitude IPS-GI fields (b > 7°) showed a similar result. We refer the reader to Y. Angarita et al. (2023, in preparation) for a continuation of the analysis of the extinctions in the IPS-GI fields.

In addition to the parameters from IPS-GI, Gaia EDR3, and Anders et al. (2022), we also used the V-band extinction, A_V50, to derive values for the reddening E(B−V) and hydrogen column density N_H using the following equations:

$$\begin{eqnarray}&&E(B-V)=\displaystyle \frac{{A}_{V50}}{3.1},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{N}_{{\rm{H}}}=5.8\times {10}^{21}{\mathrm{cm}}^{-2}E(B-V),\end{eqnarray} \tag{ 5 }$$

where the latter is from Bohlin et al. (1978). We will henceforth refer to the median (50th percentile) A_V50 values from Anders et al. (2022) as A_V . Similarly, we will use dist or d for the median (50th percentile) distance dist_V50 from Anders et al. (2022).

Although large databases of starlight polarization in the diffuse medium are rare, we compared our findings to other available measurements to verify their accuracy. To this end, we compared the IPS-GI polarization values with those from the Heiles (2000) catalog. We used Topcat to match sources common to both catalogs. We found matches within 15 for 31 stars. The comparisons between the polarization degrees and polarization angles are shown in Figure 1.

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Figure 1 shows good agreement between Heiles' catalog and the IPS-GI measurements. Almost all stars are O and B types, and seven are classified variable stars in the General Catalogue of Variable Stars (version 5.1; Samus et al. 2017). In addition, one star is a young stellar object (YSO), and another is a Wolf–Rayet (W-R) star. The variable stars, as well as the YSO and W-R star, are marked red in Figure 1. Many OB stars are known to exhibit polarimetric variability (see, for example, Bjorkman 1994), which could partly explain the difference in measurements between the IPS and Heiles (2000). We note an apparent systemic deviation in seven stars, for which we found a degree of polarization higher than the Heiles measurement. One outlier stands out in the polarization angle comparison. The polarimetric measurements for this star from Heiles' catalog are taken from Hiltner (1956). This star, identified as BD–14 4922, has also been observed polarimetrically as part of the Cikota et al. (2016) analysis of archival data of polarized standard stars. This study, which also includes V-band observations, finds a polarization angle for BD–14 4922 of θ = 50° ± 007. This is in agreement with our own polarimetry of this star, where we found a polarization angle of θ = 49° ± 044. We therefore excluded this star from further comparison of the polarization angles with Heiles (2000). Furthermore, we also excluded all known variable stars from further comparison; see stars circled in red in Figure 1. Variable stars are also expected to show polarimetric variability and are therefore not suitable for a comparison of this type.

We calculated a modified reduced χ² statistic using the following equation:

$$\begin{eqnarray}&&{\chi }_{\mathrm{red}}^{2}=\displaystyle \frac{1}{N}\sum \displaystyle \frac{{X}_{\mathrm{dif},i}^{2}}{{\sigma }_{i}^{2}},\end{eqnarray} \tag{ 6 }$$

where N is the number of observations (22 in the case of polarization degree, 21 for the polarization angle); X_dif is the absolute difference between the observations, either the degree of polarization or the polarization angle; and σ_i is the square root of the sum of the squared errors in both measurements. This leads to ${\chi }_{\mathrm{red},{\text{}}P}^{2}=12.2$ for the degree of polarization P and ${\chi }_{\mathrm{red},\theta }^{2}=4.9$ for the polarization angle θ. To visualize the differences between the measurements of the two catalogs, we created Bland–Altman plots (see Martin Bland & Altman 1986 for more details) for both parameters. These are presented in Figure 2. Following the process outlined in Martin Bland & Altman (1986), we defined a mean value, as well as limits of agreement out to 2σ for both parameters. These plots show that for the degree of polarization P, all stars have measured polarization (taking errors into account) that falls within the 2σ boundaries. For the polarization angle θ, only one star falls outside this limit. Because of the heterogeneous nature of the Heiles (2000) compilation, as well as the varying quality and considerable age of some of the data therein, we are inclined to assign most differences between the catalogs to the uncertainties in the Heiles (2000) agglomeration.

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We have cross-matched the IPS data with two auxiliary catalogs: Gaia EDR3 (Brown et al. 2021) and Anders et al.'s Starhorse-based catalog (Anders et al. 2022). We wish to ensure that the analysis is applied only to the highest-quality data. Therefore, we have applied six selection criteria that were defined based on the degree of linear polarization P and its error σ_P , various quality flags native to the auxiliary catalogs, and the cross-matching between the IPS-GI and Gaia EDR3 catalogs. These will be discussed in more detail below.

The first criterion, based on the degree of linear polarization P and its error σ_P , ensures that all sources under consideration have a high-polarization signal-to-noise ratio: P/σ_P > 5. Furthermore, using this filter, we avoid the statistically problematic region discussed in Clarke & Stewart (1986) and Section 2.3. At high P/σ_P , the Rician bias produced by the positive nature of the polarization is mitigated (see, for example, Simmons & Stewart 1985).

Second, we retain only stars with small absolute polarimetric errors, i.e., σ_P ≤ 0.8%. This filter removes stars with spurious polarimetric measurements.

Third, we filter for spurious Gaia results using the fidelity parameter; see also Rybizki et al. (2022). Applying fidelity >0.5 removes stars with poor astrometric solutions.

Fourth, SH_outflag is an Anders et al. (2022) native parameter, and filtering for outflag = 0000 is recommended by the authors. This removes spurious results caused by unreliable extinctions, overly large distances, and poor StarHorse convergence.

The fifth filter is based on the recommendations of Riello et al. (2021); see Sections 6 and 9.4 therein. The color excess factor described in Riello et al. (2021) can be used to check for inconsistencies in the various photometries, allowing for the exclusion of spurious sources.

Finally, we remove any sources for which there is more than one possible match in the Gaia catalog. Despite cross-matching based on not only spatial position but also magnitude, some stars have multiple potential matches. These stars are removed from further consideration to ensure that only certain matches are used in the analysis.

In sum, the selection criteria are as follows.

• 1.  
  p/σ_p > 5, which ensures that only high-quality polarimetric observations are taken into consideration.
• 2.  
  dp < 0.8%, which removes sources with spurious polarimetric errors.
• 3.  
  Fidelity >0.5, which removes spurious results from Gaia fitting routines; see Rybizki et al. (2022).
• 4.  
  SH_outflag = 0, which removes spurious results from Anders et al. (2022).
• 5.  
  C^*/σ_C < 5, which exclude sources with spurious color excess; see also Riello et al. (2021, Equation (18) therein).
• 6.  
  Only high-certainty matches, which removes results for which the cross-matching between the IPS and Gaia EDR3 catalogs was uncertain.

Applying all six criteria, we retained a subsample of 10,516 sources to study in more detail. On average, each field will contain ∼275 filtered stars.

The filtered subsample shows a lack of low-polarization sources. This appears to be caused by the P/σ_P > 5 filter. This introduced a bias toward stars with higher polarization, inadvertently filtering out low-polarization sources. However, although stringent, this filter ensures that all analysis is applied only to the highest-quality observations. In addition, it removes the need for debiasing, as explained in Section 2.3.

After filtering, a single star remains that exhibits a very high degree of polarization, i.e., P > 10%. This high-polarization measurement may be indicative of intrinsic polarization, but it may also be a result of, for example, a favorable geometry of the magnetic field. This has been shown to lead to an increase in the polarized signal from stars; see, for example, Panopoulou et al. (2019). Further investigation of this individual star, as well as other stars that show extraordinary polarimetry, is beyond the scope of this paper.

Assuming the starlight to be initially unpolarized, all measured polarization is a direct result of starlight passing through the ISM. As the polarization is a function of the location where it is produced, which can be defined by Galactic longitude l, Galactic latitude b, and distance d, it is important to study the spatial distribution of the polarization.

The IPS-GI observations are focused on certain lines of sight in the diffuse ISM, which leads to a very inhomogeneous source distribution across the sky; see Figure 4. Figure 3 shows an example of a typical ordered field (C47; see Table 1) located at l = 3205, b = −12. After applying quality filters (see Section 4.2), this field contains 240 stars. Each star's polarization vector was plotted; its orientation is equal to the Galactic polarization angle. and its length depends on the degree of polarization. The uniformity of the polarization vectors is typical for this data set (see middle panel of Figure 7) and indicative of a dominant large-scale magnetic field even over the full range of distances (see Figure 7, middle and right panels).

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Figure 4 shows the locations of all observed fields within the Milky Way. In addition, the distributions for Galactic longitude, Galactic latitude, and distance in Figure 5 show that the observed fields lie within a longitude range of 220° < l < 50° and that most fields are within 15° latitude from the Galactic plane. Two sparsely populated fields are positioned further above and below the Galactic plane; see annotations in Figure 4. We also indicated the location of example field C47 (see Figure 3). Finally, most stars are located nearby (around d ∼ 2 kpc), with significant numbers out to d = 6 kpc (see Figure 5, right), with certain stars at even higher distances.

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Next, we looked at the distributions of the V-band magnitude and the polarimetric parameters P and θ. Figure 6 (left) shows the distribution of the magnitudes for the filtered survey data; IPS-GI covers a range of magnitudes from roughly 8 to 19. The majority of stars have V < 15. We again note that this survey is not complete in magnitude due to the different integration times of different fields.

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Figure 6 (middle and right) shows the distributions of the degree of polarization P and the angle of polarization θ, respectively. The degree of polarization peaks around 3%, with a notable absence of very low polarization sources (see Section 5.1). Furthermore, some stars show very high (P > 10%) polarization (see Section 5.2). The distribution of polarization angles peaks around 90°, which is expected for a line-of-sight averaged magnetic field that is parallel to the Galactic plane. This is in agreement with observations based on radio polarimetry (see, for example, Haverkorn 2015), as well as observations of external spiral galaxies (Beck 2015). The distribution also shows a second peak around θ = 40° and a somewhat flattened structure around θ = 140°. These features will be discussed in more detail in Section 5.5.

Returning to our example field, C47, we took a closer look at the distributions of the degree of polarization and the polarization angle. Histograms showing the distributions of those parameters are presented in Figure 7. We include a comparison of different distances for this field in Figure 7 (right). These figures quantify the structure seen in Figure 3. The degrees of polarization vary over a relatively wide range, but the polarization angle is highly peaked around an average θ_Gal = 96°.

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Table 1 describes the main properties of the IPS-GI filtered subsample. The weighted averages of the polarimetric parameters P and θ were calculated following conventions from Pereyra & Magalhaes (2007). From this table, we found that some fields have an average orientation that deviates greatly from θ_Gal ∼ 90°, a value expected for an average magnetic field direction in the plane of the Milky Way. Some of these fields account for the features mentioned above, e.g., the histogram of polarization angles in Figure 6 (right). For example, field C4 has an average Galactic polarization angle of θ_Gal = 45°. With 455 stars, this field contributes significantly to the shape of Figure 6 (right). Furthermore, fields C7, C50, and C57 have average polarization angles of θ_Gal = 155°, 132°, and 142°, respectively. These fields explain the apparent flattening of the histogram in Figure 6 (right).

Table 1 has also been summarized in Figure 8, which qualitatively shows the average polarization vectors per field on a sky plot. The background shows the GMF as detected by Planck. The optical polarization vectors are mostly in agreement with an ordered large-scale magnetic field parallel to the Galactic plane, as can be seen in the background. The differences between polarization angles as observed by IPS and Planck are shown in Figure 9, indicating agreement typically within Δθ ∼ 10°. Any turbulent components along the line of sight at different distances are canceled out or diminished. This can only be concluded based on starlight polarization data in conjunction with distance measurements, as Planck and similar submillimeter surveys show the result of an integration along an entire line of sight.

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Due to the nature of the polarization, we expect to see correlations between certain parameters, for example, a decrease of the polarization for higher Galactic latitudes due to the decrease in integrated dust content along those lines of sight. To investigate this, as well as other correlations, the data were divided into nearby (d < 4 kpc) and distant (d > 4 kpc), as well as high (∣b∣ > 15°) and low (∣b∣ < 15°) latitude. The weighted average degrees of polarization and average Galactic polarization angles for these bins are presented in Table 2. The values were calculated following the method of Pereyra & Magalhaes (2007). This first coarse look appeared to reveal two expected correlations. First, the polarization increases as a function of distance. We expected to see this because the polarization of the starlight is caused by dichroic extinction in the ISM. As the starlight travels through more ISM dust, the light becomes more polarized if the magnetic field projection does not change considerably along the line of sight. In the low-latitude subsample of stars, the average polarization increases by 0.7% when comparing nearby and distant stars.

Second, high-latitude stars show less polarization compared to low-latitude stars. The starlight from low-latitude stars passes through more dust than their high-latitude counterparts, thus leading to an increase in average polarization. Indeed, we find that the V-band extinction in the low-latitude sample is much higher than in the high-latitude sample: A_V = 1.7 mag for the low-latitude sources versus A_V = 0.9 mag for the high-latitude stars. Comparing low- and high-latitude nearby stars, the degree of polarization increases by almost a full percentage point. However, it remains important to acknowledge the low number of stars in the high-latitude subsample.

We next present the distributions of Galactic latitude and longitude in conjunction with the polarimetric parameters P and θ on a field-by-field basis, revealing a rich structure, as can be seen in Figures 10 and 11. First, looking at latitude confirmed the findings from Table 2. As can be seen in Figure 10 (left), the highest weighted average degrees of polarization are found in fields located near the center of the Galactic plane at b = 0°. As for the polarization angle in Figure 10 (right), we note that the fields around b = 0° show a wide range of polarization angles centered around θ_Gal ≈ 90°, whereas the two bins above and below the Galactic plane show very low polarization. We note, however, that the fields further away from the plane are sparsely populated.

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We also investigated average values by longitude. Figure 11 (left) shows the weighted average degrees of polarization per field. The polarization appears to be highest in the direction of the Galactic center and its immediate surroundings. This is caused by the higher densities of dust associated with that region (see, for example, Misiriotis et al. 2006; Rezaei Kh et al. 2017). Plotting the average polarization angle per field as in Figure 11 (right) reveals an average polarization angle θ_Gal ∼ 90° throughout the longitude range, with significant outliers above and below that value; see also the vectors in Figure 8. As described in Section 5.5 and Table 1, these fields may be indicative of variations of the magnetic field across the Galactic plane or intervening structures, such as spiral arms throughout the line of sight.

Figure 12 shows the V-band extinction A_V and degree of polarization P as a function of distance for the low-latitude (∣b∣ < 5°) and high-latitude (∣b∣ > 5°) stars. As can be seen in Figure 12 (left), at low latitudes, i.e., in the Galactic plane, we are able to trace dust out to distances of at least 3 kpc. The continuous increase in extinction indicates a significant dust presence out to large distances. The case for the high-latitude stars is very different, where the V-band extinction flattens out beyond a distance of 1 kpc, beyond which we are less sensitive to dust and, by extension, polarization. The relationship between polarization and distance is more complicated; see also Figure 12 (right). Although we are able to trace dust out to large distances, the degree of polarization does not necessarily increase linearly with distance.

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We analyzed polarimetric parameters and distance in more detail. First, we binned the data in bins of 400 pc, limited to a maximum distance of 6 kpc to ensure each bin has a significant population of stars. Figure 13 shows the relevant distributions. Figure 13 (right) shows a stable average position angle θ_Gal ∼ 90° out to d ∼ 6 kpc. This is indicative of a dominant ordered magnetic field. The position angle θ_Gal gives information on the magnetic field along the line of sight out to the distance where there is still a significant amount of dust present. The flattening beyond 3 kpc may be explained by, for example, a change in the large-scale magnetic field leading to depolarization as the signal is integrated along the line of sight. Another possibility is a lack of dust beyond a certain distance. As light passes through magnetic field–aligned dust, we expect the degree of polarization to increase. As we go beyond the dusty thick Galactic disk, we expect the dust content to decrease dramatically, indicated by a flattening of the increase of polarization. However, the precise profile will depend on the line of sight. Bias effects due to incompleteness of (weaker) sources at larger distances and/or signal-to-noise ratio filtering may slightly increase the degree of polarization. However, analysis of this is beyond the scope of the paper.

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To further investigate the correlation between distance and degree of polarization, we applied a polynomial fit to the data presented in Figure 13 (left). The overall trend appears to agree well with a third-degree polynomial. A fit to the data using the distances taken from Anders et al. (2022) finds the following coefficients:

$$\begin{eqnarray}&&P=0.087+1.526d-0.396{d}^{2}+0.033{d}^{3},\end{eqnarray} \tag{ 7 }$$

where P is the degree of polarization in percent, and d is the distance in kiloparsecs. The bottom of Figure 13 (left) shows the residuals for each data point. The third-degree polynomial trend is in agreement with findings from Fosalba et al. (2002) based on Heiles (2000) data; see also the dotted line in Figure 13 (left), although the coefficients found therein differ slightly from those in Equation (7). (Fosalba et al. 2002 found P = 0.13 + 1.81d − 0.47d² + 0.036d³; see Equation (1) therein.) We note that there is no physical basis for applying a third-degree polynomial, and physical information cannot be derived from the coefficients of the fit. Furthermore, we emphasize that the distribution of stars across the sky and the depth of the survey differ greatly between Fosalba et al. (2002) and the IPS-GI. Nonetheless, it is remarkable that we independently found an overall trend that is very similar to Fosalba et al. (2002). We find similar trends regardless of the distance catalog used. The polynomial curve fitted to the Bailer-Jones et al. (2021) distances is in agreement with Equation (7) within the errors on the parameters.

Next, we investigated the correlations between the degree of polarization and various photometric parameters. Figure 14 (left) shows the degree of polarization as a function of the V-band magnitude for the entire sample of stars. Although the density of stars increases greatly toward higher magnitude, peaking at V ∼16, two trends are apparent. First, we note the increase of the error in the degree of polarization toward higher magnitude. Fainter stars appear to have a larger error in the degree of polarization, which is expected. Despite employing longer observing times to include higher-magnitude stars in the sample, a higher error cannot be fully eliminated. Second, the degree of polarization itself appears to increase with higher magnitude. A similar trend is visible in the Heiles (2000) data, although it must be noted that the highest magnitude therein is V ∼13. It is important to note that in this case, the lack of lower-polarization, higher-magnitude sources is partially caused by the filtering, especially the P/σ_P filter that excludes low-polarization stars. As the error in polarization is relatively higher for those sources, they are inadvertently filtered out. Finally, stars observed under less than ideal weather conditions will tend to have larger errors due to, for example, a lower photon count. They would contribute to the spread in accuracy beyond what could be expected from the (unattenuated) magnitudes.

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Figure 14 (right) shows the average polarization per V-band extinction (A_V , taken from Anders et al. 2022) bin. The data were binned to extinction bins of 0.21 mag wide. As the extinction is a reliable tracer of the dust contents along a line of sight, the apparent increase of polarization as a function of that extinction is expected. Figure 12 (left) shows that for the lower-latitude fields specifically, we are able to trace dust to large distances of at least 3 kpc, as is indicated by the continuous increase in extinction with distance shown in the leftmost plot. Although this does not necessarily correspond to an increase in polarization (e.g., depolarization may occur along a line of sight), we are able to probe the larger distances in terms of polarimetry. A detailed analysis of the polarization behavior with extinction will be presented in a forthcoming paper (Y. Angarita et al. 2023, in preparation).

The degree of polarization depends on the dust distribution, dust alignment, and magnetic field direction. Dust rotational alignment with the magnetic field is thought to be close to 100% in diffuse gas (see, e.g., Mathis 1986; Kim & Martin 1995; Panopoulou et al. 2019). Considering only the magnetic field orientation, we expect the degree of polarization to be maximal when the magnetic field is oriented in the plane of the sky and minimal when the magnetic field is along the line of sight. For a uniform magnetic field following the spiral arms, this would produce a maximum degree of polarization in the direction of the Galactic center and decreasing polarization degrees away from the Galactic center. Our data are consistent with this expectation; see Figure 11 (left). However, Figure 11 (left) also displays large variations in polarization degree as a function of longitude, which can be caused by structures in the magnetic field direction and/or variations in the dust distribution along the line of sight. Interstellar turbulence will partially depolarize the radiation on path lengths longer than a few hundred parsecs, leading to a decrease in the degree of polarization. In addition, the presence of spiral arms can significantly distort magnetic field directions, possibly inducing a complex longitude dependence in the polarization degree data (see, e.g., Gómez & Cox 2004).

The observed distribution of degree of polarization centers around 2% (Figure 6, middle), and the average polarization degree is higher than in the Heiles (2000) catalog (Fosalba et al. 2002), which is likely because the IPS-GI fields are more concentrated toward lower Galactic latitudes than the Heiles compilation. In addition, the IPS-GI contains more distant stars, which may show a more strongly polarized signal.

Varying dust distributions will induce differences in polarization degree as a function of Galactic latitude and longitude. Most notably, at high latitudes, dust will only be present in the nearest part of the line of sight; for a field at b ∼ 45°, a dust layer of 150 pc half-thickness (Sparke & Gallagher 2007) would result in dust only along the nearest ∼200 pc. This is apparent in the low degrees of polarization in the two high-latitude fields (Figure 10, left). The variable dust distribution in the Galactic disk, combined with varying magnetic field directions, causes the large scatter in polarization degree in the left panels of Figures 10 and 11.

The observed distribution of polarization angle is surprisingly uniform over each 03 × 03 field, with a clear preferred angle for most of the fields (as, for example, in Figure 3). This is a clear indication that the turbulent component of the magnetic field, thought to have maximum correlation scales up to about a few hundred parsecs (Beck et al. 2016), is averaged out along the (∼kiloparsec) lines of sight toward the IPS-GI stars at low latitudes.

For the two fields at high latitudes, where significant dust only exists in the first few hundred parsecs along a line of sight, contributions from the turbulent magnetic field component will play a more significant role. For fields at intermediate latitudes (∣b∣ ∼ 10°), a uniform dust layer of ∼150 pc half-thickness would end at ∼850 pc distance, which means that some polarization angle scatter due to turbulence may still be there. Closer to the plane, with longer path lengths containing dust, variations in the mean polarization angle are the largest (see Figure 11, right), although interstellar turbulence is expected to play a smaller role in that region due to the long path lengths involved. Variations in mean polarization angle are created by a changing direction of the plane-of-the-sky component of the magnetic field, weighted by the dust distribution. These variations can be due to either larger-scale structures in the magnetic field, such as spiral arms (Gómez & Cox 2004), or small-scale structures, such as dust clouds along the line of sight. In addition, the number of inhomogeneities in the dust distribution is higher close to the Galactic plane, causing additional variations in the mean polarization angle per field. Even though the IPS-GI fields were selected to not have major dense structures along their lines of sight, there are still a number of fields for which dust clouds dominate the polarization signal along the line of sight (M. J. F. Versteeg et al. 2023, in preparation).

Furthermore, in most fields, the field-averaged polarization angle has the same orientation as the magnetic field orientation averaged over the whole line of sight through the Milky Way as inferred by Planck polarized dust data (Figures 8 and 9). As the IPS-GI stars only probe part of this line of sight, the similarity of the angle orientation between IPS-GI and Planck points to a remarkable uniformity of the large-scale magnetic field direction along these lines of sight. However, there are exceptions to this uniformity in polarization angles, as will be discussed in a forthcoming paper.

The dispersion in polarization angle is presented in Table 1, and its dependence on Galactic longitude and latitude is shown in Figure 15. Most angle dispersions are between 5° and 25°, except for a few fields. First, the two high-latitude fields contain so few stars that the calculated angle dispersion is very unreliable. Second, there are five fields with high angle dispersions around 30° (C30, C34, C42, C45, and C56), four of which are located close to the Galactic plane. Field C34 is located at high latitude but contains only nine stars. All fields are characterized by a relatively low number of stars, and fields C30, C42, and C56 also show deviating distributions of polarization degree (Y. Angarita et al. 2023, in preparation). Therefore, it is possible that these fields contain anomalous structures (for example, C56 is located at the edge of an H ii region) or have deviating dust and/or magnetic field properties. However, it should be noted that Medan & Andersson (2019) noted a similar dependence of angle dispersion as a function of Galactic longitude in their study of polarized starlight from magnetized dust in the Local Bubble wall, which they suggested might be connected to the presence of OB associations changing the radiative alignment of the grains.

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For a uniform distribution of homogeneous dust along the line of sight in a homogeneous magnetic field, one would expect a linear increase in the degree of polarization with extinction. For nearby stars (up to an extinction of about ∼1 mag), the IPS-GI stars show this behavior, on average (Figure 14, right). However, at higher extinctions, the relation between degree of polarization and extinction flattens out, indicating a lower increase in polarization with extinction due to depolarization. The depolarization is mostly caused by variable orientations along the line of sight of the plane-of-the-sky magnetic field component due to interstellar turbulence and/or mesoscale structure like spiral arms. We also emphasize that most of the data presented above have distances smaller than 3 kpc. Although high-quality data exist at higher distances, the associated average uncertainties also increase. We are therefore less sensitive to structures that may exist beyond 3 kpc.

We note that our analysis is based on the assumption that the starlight we observed is intrinsically unpolarized. While this may be true for most stars, we cannot eliminate all intrinsically polarized sources through the applied filters. When studying individual stars in this catalog, it is therefore important to keep this in mind. However, considering the uniformity of the parameters and their agreement with the assumption that our stars are intrinsically unpolarized, we do not expect intrinsically polarized stars to significantly influence the overall statistics of the catalog. We do note that stars with deviating polarization angles (see, for example, in Figure 7, right) are candidates for a more detailed analysis of their polarimetric properties.

In addition, it is of the utmost importance to keep in mind the highly fragmented spatial distribution of sources in the plane of the sky. While it is tempting to use the data presented in this paper to draw generalizing conclusions about the ISM and GMF, this must be done with caution. Due to their small size, 03 × 03, the fields may not be representative of the general structure and conditions of the surrounding ISM. Any outliers or standout features presented in the above sections may be falsely interpreted as characteristics of the whole ISM or GMF, while it may be one (dominant) field that allows the average values to deviate from a general trend. Therefore, it remains important to compare the data presented here to other starlight polarization sources such as Heiles (2000), as well as complementary data at other wavelengths (e.g., Planck dust polarization, Abergel et al. 2014, as in Figures 8 and 9 or GPIPS infrared data, Clemens et al. 2020), but this is beyond the scope of this paper. The kind of study presented in this paper could benefit from homogeneous data covering a large area on the sky. Planned polarimetric surveys, such as SouthPol (Magalhães et al. 2012) and Pasiphae (Tassis et al. 2018), will contribute greatly to the understanding of the ISM.