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Linear polarization: wavelength dependence

Fitting the observed dependence $P(\lambda)$ includes calculations of the polarization cross-sections $\langle C_{\rm pol}(\lambda) \rangle$ averaged over particles orientation and size distribution, their normalization and the comparison of obtained dependencies $P(\lambda)/P_{\rm max}(\lambda_{\rm max}/\lambda)$ with Serkowski's curve representing observational data for a chosen star. Here, only two parameters of the observed polarization are considered: the wavelength at which the polarization reaches a maximum, $\lambda_{\rm max}$, and the width of the polarization curve $W$ related to the parameter $K$ of Serkowski's curve (Eq. (3.43) in Voshchinnikov [2002]. Thus, the absolute value of polarization is ignored.

Variations of the polarization factors with wavelength are shown in Fig. 16 for homogeneous spheroidal particles with PF orientation. The Figure illustrates how changes of the chemical composition, particle size, shape and orientation influence the polarization factors. The distinction between prolate and oblate spheroids is also clearly seen. In particular, oblate particles polarize radiation more efficiently than prolate ones. From Fig. 16, it is possible to estimate approximately the dependence of $\lambda_{\rm max}$ on particle parameters. $\lambda_{\rm max}$ shifts to longer wavelengths ($\lambda^{-1}$ decreases) if the particle size $r_V$ grows, the aspect ratio $a/b$ becomes smaller and the inclination (angle $\alpha$) increases (prolate spheroids) or decreases (oblate spheroids). The use of more absorbing particles (Fig. 16, upper left panel) also leads to a shift of $\lambda_{\rm max}$ to the IR wavelengths. Note that the maximum polarization occurs at shorter wavelengths for oblate spheroids in comparison with prolate spheroids having the same parameters.

Figure 16: Wavelength dependence of the polarization factors for homogeneous spheroids consisting of astrosil and amorphous carbon. The effects of variations of particle size, shape and orientation is illustrated.

As a result, one can easily find an ensemble of particles with a combination of parameters which reproduce the position of the maximum on the polarization curve. Unfortunately, to adjust another parameter -- the width of polarization curve -- is a more difficult problem: the theoretical curves are narrower than the observed ones. This is seen from Fig. 17 where the normalized polarization factors from Fig. 16 are compared with Serkowski's curves calculated using Eqs. (3.41) and (3.45) from [2002].

Figure: Normalized polarization factors for homogeneous spheroids consisting of astrosil and amorphous carbon as a function of $\lambda_{\rm max}/\lambda$. Serkowski's curves were calculated using Eqs. (3.41) and (3.45) from [2002].

The curve with $\lambda_{\rm max}=0.55\,{\mu}\rm {m}$ is noticeably wider than the theoretical curve. Only by increasing $\lambda_{\rm max}$ (say up to 1${\mu}\rm {m}$ which leads to the decrease of $W$) can the width of the polarization curve be reproduced. However, the position of the maximum for theoretical factors occurs at smaller wavelengths than 1${\mu}\rm {m}$ (Fig. 16).

Il'in and Henning ([2002]) have performed extensive calculations of interstellar polarization for polydisperse ensembles of spheroidal particles with a shape distribution. They found that the profile of the polarization curve mainly determined by oblate spheroids was always narrower than the observed one. Prolate spheroids alone can produce the polarization curves nearly as wide as observed but they are very blueshifted compared with the observed curves. Note also that the increase of $W$ occurs if the direction of grain alignment deviates from normal to the line of sight (Voshchinnikov et al., [1986]; Voshchinnikov, [1989]; Il'in and Henning, [2002]).

Very likely, it is better to compare the theoretical polarization efficiencies with observations than normalized polarization given by Serkowski's curve. This allows one to include extinction into consideration and comprehend grain alignment efficiency in a full manner. The wavelength dependencies of the polarization efficiency for particles with the same parameters as in Fig. 16 are plotted in Fig. 18. Variations of the ratio $P/\tau$ with the particle parameters are similar to those discussed in Sect. 2.1: the ratio grows with an increase of $a/b$ and $\alpha$ ($\Omega$) and is usually larger for oblate spheroids than for prolate ones. The use of more absorbing particles (e.g. of amorphous carbon) leads to an increase of polarization in the red part of the spectrum ( , Fig. 18, upper left panel). The maximum of the polarization efficiency strongly shifts with a change of particle radius (Fig. 18, upper right panel) but approximately corresponds to the close values of $x_V$ as is shown in Fig. 11.

The presentation of the observational data in the form of the wavelength dependence of the polarization efficiency is traditionally not performed. However, Whittet ([1996b]) in Fig. 9 shows the normalized dependence $P(\lambda)/A(\lambda)\cdot A_{\rm V}/P_{\rm max}$ for five stars in the wavelength range $\lambda^{-1} \approx 0.2$- $8\,{\mu}\rm {m}^{-1}$. In the visible and near UV ( $\lambda^{-1} \approx 1.8$- $3\,{\mu}\rm {m}^{-1}$), these dependencies are quite similar for all stars and their variations with wavelength may be approximated as $\lambda^{1.60}$. The dashed segment in Fig. 18 represents this ``observational'' dependence.

Figure 18: Wavelength dependence of the polarization efficiency for homogeneous spheroids consisting of astrosil and amorphous carbon. The effects of variations of the particle size, shape and orientation is illustrated. The dashed segment shows the approximate wavelength dependence of $P/\tau$ at optical wavelengths for five stars as reported by Whittet ([1996b]).

The angular dependence of the polarization factors and polarization efficiencies gives a possibility of concluding that there should exist anticorrelation in variations of $P/\tau$ and $\lambda_{\rm max}$. The smaller $\alpha$ (or $\Omega$) is, the smaller the polarization efficiency and the larger $\lambda_{\rm max}$ should be. Such a tendency can be displayed as a relation between $P_{\rm max}/E({\rm B-V})$ and $\lambda_{\rm max}$ for stars located in different galactic directions and seen through single dust clouds. Apparently, such a dependence exists for stars with distances kpc located in the galactic plane (see Voshchinnikov, [1989]). It can be used for diagnostics of the inclination of the magnetic field to the line of sight where the directions with large $\lambda_{\rm max}$ and small $P_{\rm max}/E({\rm B-V})$ can be attributed to the small values of the angle $\Omega$, i.e. here the magnetic field is nearly parallel to the line of sight.