next previous
Up: Can interfero-polarimetry constrain extended


2 Synthetic intensity maps and visibilities in polarized light

2.1 General formalism and basic equations

Thomson scattering occurs when an electromagnetic field impinges upon a free electron. The latter is accelerated and reradiates the incident radiation. This scattered radiation can be observed outside the propagation axis of the incident field. Whatever the polarization state of the incident field, the polarization degree of the scattered radiation varies with its direction (Collett 1993). In the following we assume that the natural (or unpolarized) starlight is scattered by free electrons of the hot envelope only and we do not consider multiple scattering (hypothesis of pure single Thomson scattering).

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 ($r,\theta,\varphi$) in this system and by ($r,\theta^\prime,\varphi^\prime$) in the system (${x^\prime}$, ${y^\prime}$, ${z^\prime}$) related to the sight direction ($O{x^\prime}$) and where ($O{y^\prime}$) coincides with (Oy).

\includegraphics[width=8.5cm]{6394f1.eps}\end{figure} Figure 1: Each point M of the scattering envelope is characterized by three spherical coordinates (r, $\theta$, $\varphi$) 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, $\theta^\prime$, $\varphi^\prime$) in the system (${x^\prime}$, ${y^\prime}$, ${z^\prime}$) related to the sight direction ($O{x^\prime}$). ($O{y^\prime}$) is coincident with (Oy) and i denotes the axial inclination. The scattering angle at M is $\chi$

The angle between OM and the line of sight ($O{x^\prime}$) is the scattering angle $\chi$. From Fig. 1 and by denoting i the axial inclination (i.e. the angle $xO{z^\prime}$) we obtain:
\cos\chi=\sin\theta\cos\varphi\sin i-\cos\theta \cos i=\sin\theta^\prime\cos\varphi^\prime .\end{displaymath} (1)
From the expressions of the transverse components of the incident field and from the motion equations of the free electron, we find the scattered intensity at M (Collett 1993; Brown & McLean 1977):
I(r,\theta,\varphi)=\frac{3\sigma_{\rm T}}{16\pi} n(r,\theta)\frac{1+\cos^2\chi}{r^2} I_0\end{displaymath} (2)
where I0 denotes the isotropic light emitted by O and $n(r,\theta)$ the electron distribution at M. $\sigma_{\rm T}$ is the independent wavelength Thomson cross section (Collett 1993).

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:
p(\theta,\varphi)=\frac{1-{\cos^2\chi}}{1+{\cos^2\chi}}\cdot\end{displaymath} (3)
The degree of polarization does not depend on the radial position r. It equals 0 for on-axis scattering and 1 for off-axis scattering. The scattered radiation is partially polarized for intermediate angles.

The direction of polarization is perpendicular to the scattering plane defined by OM and ($O{x^\prime}$). Thus it lies along $OM\times {O x^\prime}$ (where $\times$ denotes the vectorial product). In the plane (${y^\prime}$O${z^\prime}$) the polarization is normal to OM and therefore tangential to the envelope.

2.2 Intensity maps in polarized light

We aim at obtaining the intensity maps in the plane (${y^\prime}$O${z^\prime}$) for various configurations of the interfero-polarimeter (with and without linear polarizer). For convenience we assume that the two sets of coordinates are identical ($i = 90^\circ$ in Fig. 1) (See discussion in Sect. 5.1). For each point $M_0(0,r_0,\sin\theta_0^\prime,r_0\cos\theta^\prime_0$) of (${y^\prime}$O${z^\prime}$) we compute the intensity and the degree of polarization of the scattered radiation integrated on the sight column. This column lies along ($O{x^\prime}$) and is limited by the envelope dimension ${R_{\rm env}}$ along this direction. The integrated intensity at M0 is given by:
{\bar{I}}(r_0,\theta_0)=\int_{r_0}^{R_{\rm env}}I\left(r,\theta(r),\varphi(r)\right){{\rm d}r}\end{displaymath} (4)
with $\theta(r)={\rm Arc}\cos 
\left( {\frac{r_0\cos\theta_0}{r}}\right)$ and $\varphi(r)={\rm Arc}
\sin \left( {\frac{r_0\sin\theta_0}{r\sin \theta(r))}}\right)$.The integrated degree of polarization is given by:
{\bar{p}}(r_0,\theta_0)=\int^{R_{\rm env}}_{r_0} p\left(\theta(r),\varphi(r)\right){{\rm d}r}\end{displaymath} (5)
By symmetry the direction of polarization at M0 is tangential to the envelope and thus characterized by the angle ${\theta^\prime_0 -\pi/2}$ with respect to ($O{z^\prime}$).

We describe the polarization state at M0 (which is partially polarized along the angular direction ${\theta^\prime_0 -\pi/2}$) by a four-element Stokes vector (Collett 1993):
{{\bf S} (M_0)}&=&\left[(1-{\bar{p}}(M_0)).{{\bf S}_{\rm nat}}\...
 ...{\rm rect}}_{(\theta^\prime_0-\pi/2)}{(M_0)}\right]
where ${{\bf S}_{\rm nat}}=[1;\ 0;\ 0;\ 0]$ is the Stokes vector of natural light and ${\bf S}_{{\rm rect}({\theta^\prime_0 -\pi/2})}(M_0)=\left[1; -\cos(2\theta^\prime_0); 
-\sin (2\theta^\prime_0); 0\right]$ the Stokes vector of totally linearly polarized light along the angular direction ${\theta^\prime_0 -\pi/2}$.

The first component of ${{\bf S} (M_0)}$ is the total intensity of the radiation. It equals ${\bar{I}}(M_0)$ in natural light (Eq. 6). If the scattering envelope is now observed through a linear polarizer, the Stokes vector after the polarizer ${{\bf S}^\prime(M_0)}$ is found from Eq. (6) by a linear transformation, as follows:
{{\bf S}^\prime(M_0)}=[M].{{\bf S}^\prime(M_0)}\end{displaymath} (7)
with [M] the Mueller matrix of the linear polarizer (Collett 1993):

\setlength {\tabcolsep}{0.3mm}
 ...pha)}{\sin(2\alpha)}& \sin^2(2\alpha) & 0\cr
0 & 0 & 0 & 0}\right]\end{eqnarray} (8)
where Pt denotes the transmission of the linear polarizer and $\alpha$ its azimuth.

For $\alpha=0$ and $\alpha=\pi/2$ we obtain two polarized intensity maps. They correspond to a linear polarizer parallel then perpendicular to ($O{z^\prime}$) respectively. They are denoted by the subscripts // and $\perp$ respectively. The intensity maps are given by the first components of the resulting Stokes vectors (Eq. 7):
 ...ft[1+{\bar{p}}(M_0)\cos(2\theta^\prime_0)\right]{\bar{I}}(M_0)\, .\end{eqnarray}

2.3 Visibilities and angular diameters in polarized light

A stellar interferometer yields a measurement of the amplitude and the phase of the object spatial coherence function at the spatial frequency ${B/\lambda}$ (where B is the interferometric baseline and $\lambda$ the wavelength). Furthermore, we call "visibility" the modulus of the Fourier Transform of this intensity distribution at the spatial frequency ${B/\lambda}$ normalized to the value at the zero frequency (Françon 1966):
V({B/\lambda})=\left\vert\frac{\tilde{{\bar{I}}}({B/\lambda})}{\tilde{{\bar{I}}}(0)}\right\vert\end{displaymath} (10)
where V denotes the visibility and $\tilde{}$ the Fourier transform.

For convenience we assume that the baseline lies along ($O{z^\prime}$). Each intensity map provides a polarized visibility $({V_{\rm nat}},{V_{//}},{V_{\perp}})$. Assuming the baseline along ($O{y^\prime}$) leads to three other visibilities $(V^\prime_{\rm nat},V^\prime_{//},V^\prime_\perp)$. For spherical envelopes these visibilities satisfy $V^\prime_{\rm nat}=V_{\rm nat},$ $V^\prime_{//}={V_{\perp}}$ and $V^\prime_\perp={V_{//}}$ 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:
V({B/\lambda})=\left\vert F_*\frac{2J_1({\pi\phi^* B/\lambda...
 ...\rm env} B/\lambda})}{{\pi\phi_{\rm env} B/\lambda}}\right\vert\end{displaymath} (11)
where J1 is the Bessel function of first degree, $\phi^*$ the angular diameter of the star, F* the flux contribution of the star, $F_{\rm env}$ the flux contribution of the envelope and $\phi_{\rm env}$ the angular diameter of the envelope.

Each polarized visibility thus provides an equivalent angular diameter (${\Phi_{\rm nat}}$, ${\Phi_{//}}$, ${\Phi_{\perp}}$). ${\Phi_{\perp}}$ is larger than ${\Phi_{//}}$ and ${\Phi_{\rm nat}}$ is between (Eq. 9).

To study the effect of polarization on visibility we define the degree of polarized visibility PV by:
P_V=\frac{{V_{//}}-{V_{\perp}}}{{V_{\rm nat}}}\cdot\end{displaymath} (12)

next previous
Up: Can interfero-polarimetry constrain extended

Copyright The European Southern Observatory (ESO)