2 An overview of the problem

The radiation field across the stellar disk can be described by the stokes parameters, which give the unpolarized and polarized intensity. These are usually denoted by I, Q, U, and V (these parameters are discussed more fully in Clarke & Grainger 1971). Here we shall take the circular polarization V to be zero. U and Q give the state of linear polarization of the radiation. Q is obtained by measuring the difference in intensity in two perpendicular directions, and U by measuring the difference when the polarimiter is rotated through $45^\circ$. The degree of linear polarization may simply be written as (Q² +U²)^1/2/I and the position angle as $\frac{1}{2} \arctan {{U}/{Q}}$. Under rotation of axes by $\psi$, Q and U simply transform as $U_\mathrm{new} = U_\mathrm{old} \cos 2\psi - Q_\mathrm{old} \sin 2 \psi$ and $Q_\mathrm{new} = U_\mathrm{old} \sin 2\psi + Q_\mathrm{old} \cos 2\psi$. I is invariant (as is V).

We will assume that the light emerging from the stellar disk is partially linearly polarized perpendicular to the radial direction. One can calculate the total flux from the eclipsed star by integrating the intensity I over the visible part of the disc. Similarly, to obtain the total polarized flux one simply integrates the Stokes parameter Q over the same part of the disc, introducing a rotation factor to transform the polarized intensity at each point into the appropriate reference frame.  

$$F_\mathrm{I}(t) = \frac{1}{R^2} \relax \int\!\!\int\relax ... ...)r \, \relax {\rm d}\relax r \relax {\rm d}\relax \phi \relax$$ (1)

 

$$F_\mathrm{Q}(t) = \frac{1}{R^2} \relax \int\!\!\int\relax ... ...}r \,\relax {\rm d}\relax r \relax {\rm d}\relax \phi \relax$$ (2)

 

$$F_\mathrm{U}(t) = \frac{1}{R^2} \relax \int\!\!\int\relax ... ...r \,\relax {\rm d}\relax r \relax {\rm d}\relax \phi \relax$$ (3)

where $(r, \phi)$ are polar coordinates on the stellar disk, R is the distance from the observer to the star, and $A(r,\phi ,t )$ is zero for an occulted point on the disk, 1 otherwise.

In this paper we address the problem of extracting I(r) and Q(r) from measurements of $F_\mathrm{I}$, $F_\mathrm{Q}$ and $F_\mathrm{U}$during an eclipse. One might attempt to do this by modelling I(r) and Q(r) for the stellar atmosphere, performing the integrals above and using the data to fit the free parameters in the stellar atmosphere model. But the problem must be treated more carefully. It is a general property of inverse problems, such as the present case, that a very large range of source models are consistent with a given data set. This is essentially due to the smoothing properties of the kernel (here, the occultation function A). Consequently, a forward-modelling approach can give apparently accurate, but actually highly misleading, results. A thorough treatment of inverse problems in astronomy can be found in Craig & Brown (1986); here we will simply state that meaningful results can only be obtained by controlling the instabilities caused by discrete, noisy data. We have used the Backus-Gilbert inversion technique to achieve this control. The principal advantage of the Backus-Gilbert method from our point of view is that it allows us a qualitative understanding of, and thus an explicit quantitative control over, the compromise between bias and stability in our inversion.