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2. Imaging a binary star by using PDFs


A binary system, for which none of the stars is individually resolved by the telescope, is the most simple object that can be considered for image reconstruction. Its perfect image is made of two points of intensities I1 and I2 , separated by a vector of position tex2html_wrap_inline1927 corresponding to the angular separation.

Let us denote as tex2html_wrap_inline1929 the instantaneous monochromatic speckle pattern produced at the focus of the telescope by a point-source (i.e. a single star unresolved by the telescope, or a reference star). tex2html_wrap_inline1929 is therefore the point-spread function (PSF) if one considers a unit mean intensity. Assuming isoplanatism, the observed binary star speckle pattern tex2html_wrap_inline1933 can be written as:


where: tex2html_wrap_inline1935, and I0 is the intensity of the binary system corresponding to its overall magnitude.

The relevant information for the imaging of the binary is contained in the three parameters I1 , I2 and tex2html_wrap_inline1927, or equivalently in the three parameters I0 , tex2html_wrap_inline1947 and tex2html_wrap_inline1927. Unless very accurate photometry is performed, we cannot access the absolute value of I0 , so the imaging parameters to retrieve are tex2html_wrap_inline1947 and tex2html_wrap_inline1927. Whereas tex2html_wrap_inline1927 (or equivalently tex2html_wrap_inline1959) and the value (greater or not than 1) of tex2html_wrap_inline1947 give a point in the orbit of the binary, an accurate value of tex2html_wrap_inline1947 leads to relative photometry of the system. The object of our analysis will be therefore to obtain with no ambiguity tex2html_wrap_inline1927 and tex2html_wrap_inline1947. Let us now show how an analysis of the PDFs can achieve this goal.

Let us first denote tex2html_wrap_inline1969 the intensity value taken by tex2html_wrap_inline1971 and tex2html_wrap_inline1973 that of tex2html_wrap_inline1975, where tex2html_wrap_inline1971 describes the intensity distribution in the speckle pattern at a position tex2html_wrap_inline1979, and tex2html_wrap_inline1981 is a space-lag. As we assume stationarity in space, the second-order statistics of tex2html_wrap_inline1971 are completely defined (Lee 1960) by the twofold PDF tex2html_wrap_inline1985.

The quantity tex2html_wrap_inline1987 measures the probability that tex2html_wrap_inline1971 has an intensity value lying in the elementary interval tex2html_wrap_inline1991 while tex2html_wrap_inline1975, of the same speckle pattern, has an intensity value lying in the interval tex2html_wrap_inline1995.

As discussed by Aime et al. (1990), there is a strong difference between twofold PDFs of speckle patterns produced by a point-source and a binary star. For a given value of tex2html_wrap_inline1981, the observed PDFs appear as joint occurrence histograms of the discretized values tex2html_wrap_inline1969 and tex2html_wrap_inline1973, and can be represented as gray-level images. As we shall see in what follows, the twofold PDF of a point-source has an overall symmetrical structure in tex2html_wrap_inline1969 and tex2html_wrap_inline1973 whatever the value of tex2html_wrap_inline1981. Whereas for tex2html_wrap_inline1981 close to the star separation vector tex2html_wrap_inline1927 of the binary, the corresponding twofold PDF of the double star speckle pattern has an arrow-head shape with a trend towards a direction tex2html_wrap_inline2013. There is a unique relationship between the shape of the twofold PDF and tex2html_wrap_inline1947.

Carbillet et al. (1996a) presented a calibration procedure that uses a parametric approach leading to an estimation of the two parameters tex2html_wrap_inline1927 and tex2html_wrap_inline1947 from one-dimensional near-infrared speckle data. We present here a new approach that is found to give better results for two-dimensional visible speckle data. The separation d (modulus of tex2html_wrap_inline1927) and the position angle PA (with a tex2html_wrap_inline2025 quadrant indetermination) need within the present framework to be determined by the by now classical power spectrum analysis and visibility function calculus of Labeyrie's technique. We will now focus on the most accurate way possible of determining tex2html_wrap_inline1947 (and the absolute quadrant) by using an analysis of the PDF's slices computed for tex2html_wrap_inline2029 (or equivalently tex2html_wrap_inline2031).

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