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To model the interstellar polarization one needs to calculate the forward-transmitted radiation for an ensemble of non-spherical aligned dust grains. This procedure consists of two steps: 1) computations of the extinction cross-sections for two polarization modes and 2) averaging of cross-sections for given particles size and orientation distributions. Although the average cross-sections should be compared with observations, behaviour of the polarization cross-sections and alignment mechanisms are often analysed separately.
Let non-polarized stellar radiation passes through a dusty cloud
with a homogeneous magnetic field. As follows from observations and
theoretical considerations (Dolginov et al., [1979]),
the magnetic field determines the direction of alignment of dust grains.
The angle between the line of sight and the magnetic field is
(
).
The linear polarization
produced by a rotating spheroidal particle
of same size is
Note that the problem of grain alignment is one of the most difficult in the physics of cosmic dust. Here, the interaction of solid particles with gas, radiation and magnetic field is closely connected. Davies and Greenstein ([1951]) assumed that Fe atoms embedded in dielectric particles gave them paramagnetic properties and opened the possibility of interaction with a weak interstellar magnetic field. The required orientation arises as a result of the effect of paramagnetic relaxation of thermally rotating grains. The Davies-Greenstein mechanism was further developed by Jones and Spitzer ([1967]) who obtained expressions for the distribution of angular momentum. In the simplest case, it is
Rotation is an important factor of any grain alignment mechanism.
The faster it is the more effective the grain alignment should be.
The Davies-Greenstein mechanism considers thermally rotating
grains. Purcell ([1979]) suggested a mechanism
of supra-thermal spin alignment
(SSA; ``pinwheel'' mechanism)
where the grains were spun up to very high velocities
as a result of the desorption of H molecules from their surfaces.
In this case, the alignment function is described by
Eq. (9) but the parameter
is
The development and current status of the major alignment mechanisms and principal physical processes forming their basis are reviewed by Roberge ([1996]), Lazarian et al. ([1997]) and Lazarian ([2000]). Unfortunately, the astrophysical significance of different alignment mechanisms remains unclear. This is connected, in particular, with very rough theoretical estimates of the polarization efficiency when instead of an alignment function like that given by Eq. (9) the Rayleigh reduction factor (see Greenberg, [1968]) is used (e.g., Lazarian et al., [1997]).
The circular polarization is proportional to the product
For simplicity, non-rotating particles of the same orientation
are frequently considered. In this case of ``picket fence'' (PF)
orientation, there are no integrals over angles
and
in Eq. (8).
The polarization degree is proportional to the polarization cross-section
, where
.
The dichroic polarization efficiency is defined
by the ratio of the polarization cross-section (factor) to the
extinction one
A more complicated case is the perfect rotational (2D) orientation
(or perfect Davies-Greenstein orientation, PDG)
when the major axis of a non-spherical particle always lies in the same plane.
For the 2D orientation, integration is performed over the spin
angle only.
This gives for prolate spheroids
As a result, the expected polarization will be determined by: