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Fitting the observed dependence includes
calculations of the polarization cross-sections
averaged over particles orientation and size distribution,
their normalization and the comparison of obtained dependencies
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,
,
and the width of the polarization curve
related to the parameter
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
on particle parameters.
shifts to longer wavelengths
(
decreases) if the particle size
grows,
the aspect ratio
becomes smaller and the inclination
(angle
) increases (prolate spheroids) or
decreases (oblate spheroids). The use of more absorbing particles
(Fig. 16, upper left panel) also leads to a shift
of
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.
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].
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 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 with the particle parameters
are similar to those discussed in Sect. 2.1:
the ratio grows with an increase of
and
(
) 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
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
for five stars
in the wavelength range
-
.
In the visible and near UV (
-
),
these dependencies are quite similar for all stars and
their variations with wavelength may be approximated as
.
The dashed segment in Fig. 18 represents this
``observational'' dependence.
The angular dependence of the polarization factors and polarization efficiencies
gives a possibility of concluding that there should exist anticorrelation
in variations of and
.
The smaller
(or
) is, the smaller the polarization efficiency
and the larger
should be.
Such a tendency can be displayed as a relation between
and
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
and small
can be attributed to the small values of
the angle
, i.e. here the magnetic field is nearly parallel to
the line of sight.