Figure 1 shows the reference system of Cartesian coordinates (x, y, z) centered on the star O. (Oz) is the rotation axis and the plane (Oxz) includes the line of sight. A point M of the envelope is characterized by three spherical coordinates () in this system and by () in the system (, , ) related to the sight direction () and where () coincides with (Oy).
|Figure 1: Each point M of the scattering envelope is characterized by three spherical coordinates (r, , ) in the Cartesian system (x, y, z) centered on the star O. (Oz) is the axis of rotation and the plane (Oxz) includes the line of sight. M is also defined by (r, , ) in the system (, , ) related to the sight direction (). () is coincident with (Oy) and i denotes the axial inclination. The scattering angle at M is|
The angle between OM and the line of sight () is the
scattering angle . From Fig. 1
and by denoting i the axial inclination
(i.e. the angle ) we obtain:
Characteristically the scattered intensity is maximal along the axis of the incident radiation (on-axis scattering) and is minimal perpendicularly to this direction (off-axis scattering).
By convention, the local degree of polarization at M is defined by the difference between
the maximal and minimal intensities normalized to the sum of these intensities. Hence:
The direction of polarization is perpendicular to the scattering plane defined by OM and (). Thus it lies along (where denotes the vectorial product). In the plane (O) the polarization is normal to OM and therefore tangential to the envelope.
We describe the polarization state at M0 (which is partially polarized along the angular direction ) by a four-element Stokes vector (Collett 1993):
The first component of is the total intensity of
the radiation. It equals in natural
light (Eq. 6). If the scattering envelope is now observed through a linear polarizer, the Stokes
vector after the polarizer is found from Eq. (6) by a linear transformation, as follows:
For and we obtain two polarized intensity maps. They correspond to a linear polarizer parallel then perpendicular to () respectively. They are denoted by the subscripts // and respectively. The intensity maps are given by the first components of the resulting Stokes vectors (Eq. 7):
A stellar interferometer yields a measurement of the amplitude and the phase of the object
spatial coherence function at the spatial frequency (where B
is the interferometric baseline
and the wavelength). Furthermore, we call "visibility"
the modulus of the Fourier Transform
of this intensity distribution at the spatial frequency normalized to the value at the zero
frequency (Françon 1966):
For convenience we assume that the baseline lies along (). Each intensity map provides a polarized visibility . Assuming the baseline along () leads to three other visibilities . For spherical envelopes these visibilities satisfy and and changing the baseline is useless. But for non axi-symmetric envelopes, each baseline provides independent visibilities which enable to constrain the envelope geometry (See Sects. 4.3 and 5.5).
Considering the star and the envelope as uniform disks we can fit each visibility curve
versus spatial frequency with the function:
Each polarized visibility thus provides an equivalent angular diameter (, , ). is larger than and is between (Eq. 9).
To study the effect of polarization on visibility we define the degree of polarized visibility
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