4.1. Fitting with Serkowski’s Law

The dependence between optical wavelength and interstellar linear polarization was first noted by ^{Gehrels (1960)}. Then ^{Coyne, Gehrels & Serkowski (1974)} and ^{Serkowski, Mathewson & Ford (1975)} showed that it was an effective tool to investigate the different properties of the interstellar medium. ^{Serkowski (1973)} found that this dependence is well represented by the following empirical relation:

where P_λ is the percent polarization at wavelength λ, P_λmax is the maximum polarization which occurs at λ_max, and K is a constant for all stars. If the polarization is produced by aligned interstellar dust particles, we assume that the observed data (in terms of wavelength within the bands BV RI) will follow equation (1) and each star will have a P_λmax and a λ_max values. To perform the fitting, we adopted K = 1.66λ_max + 0.01 (^{Whittet et al. 1992}).

The parameters of Serkowski’s law are related to the properties of the intervening interstellar medium. P_λmax is related to the column density of the medium and degree of alignment of the interstellar grains and λ_max is related to the average size of the polarizing grains. The individual P_λmax, λ_max values, together with the star identification from CLB, are listed in Table 2, excluding star #339 because only data in the filter V are available.

Table 2 also shows the σ₁ parameter (the unit weight error of the fit, Column 4) which quantifies the departure of the data from the standard Serkowski law. This parameter is given by:

where m is the number of colors and r_λ = P_λ − P_max exp(−K ln²(λ_max/λ)).

In our scheme, if a star exhibits σ₁ > 1.70, the measured polarization may have a component of non-interstellar origin (intrinsic component). A second criterion to test the origin of the polarization is the λ_max value. The stars found to display a value of this parameter lower than the average corresponding to the interstellar medium (0.55 μm; ^{Serkowski et al. 1975}) are candidates to have an intrinsic component of polarization (^{Orsatti et al. 1998}, ^{Vergne et al. 2016}). And one last criterion is to make a visual check of the polarization behavior and the angle in the four filters for the complete sample of stars, observing if the polarization behaves according to what was predicted by Serkowski and if the angle shows variations due to a possible rotation.

Figures 4a, b show P_λmax versus λ_max (Panel a) and P_λmax versus σ₁ (Panel b), respectively. The full line represents the limit values for both parameters to consider a likely candidate for intrinsic polarization. A few stars of the sample have, in Panels a and b, suspicious λ_max and σ₁ values and only eight of them have values much higher than the established levels for these parameters. In Table 2 these stars are indicated with a star symbol in Column 1.

Fig. 4 (a) λ_max vs. P_λmax. Full circles are the member stars, open circles are stars with unknown membership and open triangles are stars classified as non-members. (b) σ₁ vs. P_λmax. The symbols are the same as those used in Panel (a). 

Stars #114, #122, #139, #255, #342 have extreme values of σ₁ and stars #119, #137, #147 have a λ_max value of less than 0.55 μm. To complete the analysis we plot in Figures 5a, b polarization and angle versus the wavelength for each of the eight stars mentioned, including the fit of the law of Serkowski with a full line. The stars that present a suspicious σ₁ parameter show in general good adjustment to Serkowski. A different case is star #139, which according to ^{Fitzgerald et al. 1979} it is a B8IV(II/III), showing a rotation of the measured polarimetry in the B filter, a result that is fully expected for stars with extended atmospheres. Considering the three stars with the lowest values of λ_max, the member star #119 displays a bad adjustment to Serkowski with a polarization that decreases with the wavelength. Also rotation is observed with variations in the polarization angle of up to about 15º. For this star only photometric data (CLB) are available in the literature, but this type of behavior in polarization and in orientation angle was found in other studies for the case of binary systems (^{Vergne et al. 2016}).

Fig. 5 The plot displays both the multicolor polarization percentage and the polarization angle dependence on wavelength for some of the stars with features of intrinsic polarization. 

In Figure 5 star #260 is also included, although it does not show suspicious λ_max and σ₁ values, but its polarization does not show a good fit to Serkowski and the variation of the angle shows that there is rotation.

4.2. Stokes Plane

The individual Stokes parameters, Q_V = P_V cos (2θ_V) and U_V = P_V sin(2θ_V), can be used as a good criterion to identify members and non-members of a cluster, particularly when the field stars have colors that are similar to those of cluster members (^{Martínez et al. 2004}, ^{Vergne et al. 2007}, ^{Feinstein et al. 2008}). They also allow us to study the nature of the dust properties of the ISM, the ICM (intra-cluster medium) and the magnetic field associated with them. As the polarization of the light from an star depends on the degree of alignment of the dust grains located in front of the star, the member stars are expected to be grouped on the QU-plane, while the non-member stars should be scattered in this plane.

In particular, NGC 5662 is not an easy cluster for memberships assignment, due to its extension, and so it is contaminated by a large number of field stars. Added to the difficulty of determining memberships, the interstellar reddening varies across the cluster (CLB). ^{King (1980)} calculated the probabilities of membership, based on relative proper motions in relation to the observed V magnitude. Later, CLB determined memberships by the position in the color-color and color-magnitude diagrams of a total of 237 stars in the field of NGC 5662. They predicted that 88% of the observed stars had a higher probability than 50% of membership.

Figure 6a shows the Stokes plane in the V -band, Q_V -U_V, for the totality of the sample stars, distinguishing members (full circles), non-members (open triangle) and unknown membership stars (open circles). The GP is drawn with a full line and the solar neighborhood is shown with an open square box at the Q_V =0.0, U_V =0.0 coordinates. One would expect the polarization angles of the members to be similar, but not those of the foreground or background non-member stars. In the present figure there is no such clear trend: all stars lie widely scattered on the plane, not showing any evident grouping. To have a clearer idea of how the polarizing dust is affecting the stars of the region and how it is distributed, we made different plots of the QU-plane for the member and probable member stars (Figure 6b) and the non-member stars (Figure 6c).

Fig. 6 Q_V vs U_V Stokes parameters for the V -bandpass. (a) Plot for all the stars. (b) Plot for the member stars and the stars with unknown membership. Three different groups were indicated because of their location on the sky (see Text). (c) Plot of the non-members stars. The full line is the inclination of the Galactic Plane, the open square box indicates the solar vicinity and the symbols are as in Figure 4. 

Figure 6b shows quite a dispersion and no strong concentration as is expected for member stars on the QU-Plane. In spite of this, if we study it in greater detail we notice that the stars are grouped according to their location on the sky plane (Figure 1). Stars placed within the box A are located on the sky in the South-West zone, being the most polarized group of the sample with a mean polarization and mean orientation of P_V = 3.0 % ± 0.104 and θ_V =98º.1 ± 0.75, respectively. This group is composed of nine stars: eight members (#237, #238, #240, #241, #339, #342, #343 and #345) and one probable member (# 240). Star #122, a probable member, is located on the edge of the box of Group A and with polarimetric parameters similar to the stars of this group, but its projection is on the sky places in the central region of the cluster. Also, its location on the QU-plane is not reliable because its σ1 parameter exceeds our criterion limit, so probably it has an intrinsic component in its polarization, although in Figure 5 this fact is not reflected.

The six stars of Group B, all members, are located on the North-West (#255, #256, #259, #261, #262, #362). They have lower polarizations than Group A, with mean values of P_V = 1.83% ± 0.112 and θ_V = 96º.8 ± 1.368.

Group C has an intermediate polarization and orientation of the polarization vectors, more than 10º with respect to Groups A and B. Its stars are located in the North of the cluster with a mean polarization and a orientation of P_V = 2.33 %± 0.086 and θ_V = 83º .2 ± 0.794. This group is composed of six members (#119, #132, #139, #146, #150, #151) and two probable members (#144, #152). Star #119 was not considered to determine the average polarization and orientation of the group because it shows signs of intrinsic polarization through the parameter λ_max.

Stars #106, #136 and #143 are quite aligned to the GP, and may be probable non-member stars with colors similar to the stars of the cluster; their polarization and angle are very different to the rest of the member stars, and show no evidence of an intrinsic polarization component.

Stars #111 and #260 deserve special consideration because both appear very scattered on the QU-plane with respect to the rest of stars. Star #111 is the most polarized in the sample, 4.2%, and with a similar orientation to Group A. According to our estimates it would be farther away than the cluster, about 930 pc, explaining the high polarization. Due to its location on the CLB’s C-C diagram this star is likely a non-member. Star #260 is located on the sky plane over the North-West zone of the cluster, as the Group B stars, but it is more polarized. Its position in the QU-plane can be explained by the existence of a intrinsic polarization component (Figure 5). The rest of the member stars are shown in the central zone of the QU-plane (Figure 6b); their projections on the plane of the sky are located in a central band from East to West.

Figure 6c shows the QU-plane of non-member stars, where some apparent grouping of the stars is observed. There is a group with values of Q_V between −2 and −3 and values of U_V between 0.6 and −0.5, and a second group located in a range of −1.8 and −3 for Q_V and −1.2 and −1.4 for U_V. There are also two stars isolated from the rest, #104 and #140. In general, both groups of stars show no tendency in their projections on the sky plane. It is also remarkable that in the second stellar group mentioned, none of the stars belong to the central zone but are located at the edges of the cluster field (see Figure 1). Concerning star #104, its polarization vector is aligned to the GP, and together with the member stars #106, #136 and #143 is projected on the plane of the sky at the eastern edge. Finally, star #140, of spectral type B7 V with an estimated excess E_B−V = 0.45 mag, is probably a star farther away than the cluster, at a distance of more than 900 pc. This value is compatible with the observed polarization of P_V =3.31%. So, it is probable that it is a background star polarized by denser dust clouds than those that are polarizing cluster members. Its projection on the plane of the sky places it to the East, very close to the last four stars mentioned in the previous paragraph.

The QU-plane of the cluster stars and field stars in the region make us think that they are being polarized by several clouds of dust and that each of them is affecting different areas of the cluster and its surroundings.

If the polarizing dust in the region of NGC 5662 is distributed as we estimate, the QU-plane, in this case, would not be an efficient polarimetric tool to establish memberships.

4.3. Polarization Efficiency

The ratio of P_λmax to the color excess E_B−V is called the polarization efficiency. This amount in a given direction depends on the characteristics of the dust grains, size, shape and composition of the particles, and the projection of the alignment by the magnetic field. These characteristics and the magnetic field could change along the line of sight to the cluster.

Figure 7 shows the polarization efficiency for the stars on the region of NGC 5662. The empirical limit to the efficiency of the ISM, given by ^{Serkowski et al.(1975)} P_λmax = 9 E_B−V, is represented by a continuous line in the figure. The polarization efficiency of the ISM in general is given on average by P_λmax = 5 E_B−V, according to the plots of ^{Serkowski et al. (1975)}. This is shown by a dashed line; the dotted line represents the average polarization efficiency of the ISM estimated by ^{Fosalba el al. (2002)}, being valid for E_B−V < 1.0 mag. Excesses E_B−V of the stars were obtained from CLB and from the relation between spectral type and color indices following ^{Schmidt-Kaler (1982)}.

Fig. 7 Plot P_λmax vs. E_B−V for stars in the direction of NGC 5662. The solid line shows the empirical upper limit for the polarization efficiency (P _V ≃ 9E_B−V), the dotted line P_λmax = 3.5 EB-V0.8 represents the average efficiency estimated by ^{Fosalba et al. (2002)}, and the observed normal efficiency P_λmax = 5 E_B−V is represented by a dashed line. Symbols are as in Figure 4. 

This figure shows that most of the stars identified as members by CLB lie on the full line corresponding to efficiency 9 (empirical limit), and only a few of them, with an excess less (0.2 mag.) than the average excess, are located below this upper limit between the lines of efficiency 5 and 9.

The polarization efficiency of NGC 5662 is much higher than that found by us for other clusters and for those studied by other authors (^{Eswaraiah et al. 2011}, ^{Pandey et al. 2013}). This high efficiency may be due to different factors: several member stars could present an intrinsic component in their polarization, but this is not the case because there are few stars of the cluster with this type of behavior. Another explanation may be that we are looking at the fully transverse polarization vectors as found by ^{Vrba, Marraco & Strom (1992)} in the region of Chamaeleon 1 and 2.

Concerning non-member stars we observed groups with different efficiencies. A group has an average excess of 0.2 mag. and its mean efficiency is of 12.6% mag⁻¹, exceeding the empirical limit and none of these stars show any evidence of intrinsic polarization. A second group of non-members are located between the lines of efficiency 9 and 5, with an average excess and efficiency of 0.4 mag and 6.7 % mag⁻¹ respectively. This value of efficiency is similar to that of members already mentioned with smaller excesses. A third group of nonmember stars has a normal efficiency of about five, #124, #158, #214, #253, which exhibit an average excess of 0.57 mag and efficiency of 4.97% ⁻¹.

Finally, stars #104 and #107 are located near the estimate of ^{Fosalva et al. (2002)}. According to their polarizations we can assume that their excesses may have been over estimated.

Considering only the non-member stars for which we have been able to estimate distances from magnitudes and colors (^{Schmidt-Kaler 1982}), it could be said that the first group consists of foreground stars with distances between 150 and 400 pc, the 2nd group would be located between 400 and 800 pc (near the cluster) and the 3rd group would be background stars, behind the cluster.

Clearly the non-member stars are being polarized by dust with different polarimetric characteristics; this is why observed data show different efficiencies. This result also emerges from the QU-plot. The fact that the second group of non-member stars show excesses greater than the cluster, while its stars are closer to the observer and with a lower efficiency, makes us think that we are facing a patchy distribution of dust clouds along the way to the cluster, which have also a variable density. According to our estimates the first dust component would be located at a distance of less than 200 pc, which would polarize the first group of non-member stars and also the cluster.

A second component of dust with variable density would be located at a distance of approximately 500 pc, affecting the second group of non-member stars and having some windows of transparency that allow us to observe stars in the cluster without noticeable light extinction. This would explain the difference in efficiency between the cluster and the 2nd group of non-member stars, and also the difference in their excesses. Finally, the cluster would be located at a distance of approximately 800 to 900 pc, coinciding with the distance estimated by CLB. The nonmember stars belonging to the background would polarized by a component of dust with normal efficiency (P_λmax /E_B−V = 5).

We would be identifying at least two components of dust between the observer and the cluster, and a component behind it.

^{Voshchinnikov, Il’in & Das (2016)} found for the stellar clusters that the polarization of their stars is mainly caused by dust in the diffuse medium in the line of sight. In addition, these authors observed what they call a open cluster pattern, in the λ_max vs. efficiency plot, which is made up of a set of points around a mean polarization efficiency of 9% mag⁻¹ with λ_max between 0.5-0.6 μm.

In Figure 8, we show a plot similar to that of ^{Voshchinnikov et al. (2016)} for members and probable members of NGC 5662, noting that most of the stars tend to concentrate around the point of efficiency 9 and λ_max of 0.55 μm, but with a decrease of the efficiency for greater values of λ_max. In particular, for most of the stars that follow this trend (#152, #259, #260, #261, #262, #265) their projections on the plane of the sky (Figure 1) are on the North. This may be the result of the presence of a dark cloud next to this zone of the cluster, and we would be observing a combination of open cluster pattern and dark cloud pattern according to ^{Voshchinnikov et al.(2016)}.

Fig. 8 P_λmax /E_B−V vs. E_B−V for the stars with excesses E_B−V, excluding the non-members stars. Symbols are similar to those in Figure 4. 

The image in the V band of the cluster and its surroundings does not show any dark area on the North. According to the average value of the λ_max of these last six stars (0.68 μm), this confirms that they are being polarized by dust whose grains have a larger size than the dust that polarizes the light coming from the other stars.

4.4. Dust Distribution and Absorption Behavior

The degree of polarization of a star is not a continuous function of the distance due to the fact that dust is located mostly in layers between the observer and the star. If the stellar light encounters a dust layer, a sudden jump in the trend of the polarization is observed and the number of jumps is associated with the number of dust components in the path to the stars.

On the other hand, the alignment of the dust grains depends on its chemical composition. The interaction of the dust grains with the magnetic field will be different if the same grains are composed of pure iron or iron plus other elements (magnetic alignment). ^{Voshchinnikov et al. (2012)} found that there is a weak correlation between the polarization and [Fe/H]_d or [Mg/H]_d and a strong correlation between polarization and [Si/H]_d. Their results coincide with the assumption of ^{Mathis (1986)} that only the silicate grains contribute to the observed polarization. Another important finding is that dust grains of all types contribute to A_V, and hence to E_B−V, whereas only silicates are responsible for the observed polarization.

Figures 9a and b show the evolution of the absorption and the polarization, respectively, for the observed stars for which we have been able to estimate distances in the region of NGC 5662.

Fig. 9 (a) A_V vs. distance (pc) in the area of NGC 5662. The full lines show the trend of the absorption from the Sun to the NGC 5662. (b) P_V vs. distance (pc). Symbols are similar to those in Figure 4. 

Concerning the trend of the absorption (Figure 9a), the plot shows the same jumps that we have previously discussed, an increase of the extinction up to about ≈ 200 pc and 500 pc. The cluster is located at a distance greater than 700 pc, having an average absorption between 0.9 and 1.0 mag. Therefore, the existence of the two layers of dust in front of the cluster is shown in Figure 9a. The trend of the absorption behind the cluster shows an abrupt growth, affecting the background stars. This result coincides with the behavior observed by ^{Neckel & Klare (1980, Figure 6i-179)} in the region, but shows more detail.

The polarization (Figure 9b) does not show a trend as clear as the absorption. One possible explanation, according to the work of ^{Voshchinnikov et al. (2012)}, is that the dust grains are not composed of silicates. Therefore, the polarization would not be strongly affected.