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3. Theoretical model


3.1. General expressions

Let us first recall the definition of the single-fold characteristic function (CF) tex2html_wrap_inline2033 of tex2html_wrap_inline1971, the intensity at the focus of the telescope that can be either the PSF tex2html_wrap_inline1929 or the binary star speckle pattern tex2html_wrap_inline1933. tex2html_wrap_inline2033 is the complex function of the real variable w defined as:


where the symbol tex2html_wrap_inline2045 denotes the expected value of tex2html_wrap_inline2047, and tex2html_wrap_inline2049 is the single-fold PDF-- and the inverse Fourier transform of tex2html_wrap_inline2033.

By generalizing Eq. (2 (click here)) to two dimensions, we can derive the twofold CF of tex2html_wrap_inline1971:


On substituting tex2html_wrap_inline1929 to tex2html_wrap_inline1971 in the above equation, we directly obtain the twofold CF of the PSF as:


while, if tex2html_wrap_inline1971 represents the binary star speckle pattern tex2html_wrap_inline1933, Eq. (3 (click here)) takes the following form:


In the particular case when tex2html_wrap_inline1981 is equal to the star separation tex2html_wrap_inline1927, this last equation becomes:


As shown by Aime et al. (1993), this expression can be written as a central slice of the threefold CF of tex2html_wrap_inline1929. A much simpler expression can be used if we assume that the separation d is large with respect to the speckle size s , so that tex2html_wrap_inline1929, tex2html_wrap_inline2075 and tex2html_wrap_inline2077 are statistically independent from one another. In that case, assuming that the process is stationary in space, the twofold CF reduces to the product of single-fold CFs of tex2html_wrap_inline1929:


By Fourier-inverting this last equation, it leads to (Aime 1993):


where tex2html_wrap_inline2081 stands for a two-dimensional convolution and tex2html_wrap_inline2083 is the Dirac distribution.

3.2. Gaussian model

Figure 1: Gray-level representation of the theoretical twofold PDF of a binary star computed for tex2html_wrap_inline2085 and tex2html_wrap_inline2087 a), the twofold PDF of a point-source b), and the corresponding Q function c)

We shall now assume that the complex amplitude of the wave at the focus of a large telescope is a circular Gaussian process, i.e. real and imaginary parts of the wave are independent and have identical Gaussian densities. This corresponds to a fully developed speckle pattern. In that case, the intensity of the PSF-- that we defined with mean intensity equal to one --follows the well known negative exponential law:


By substituting this last equation into Eq. (8 (click here)), one obtains the twofold PDF in the normal case (Aime 1993). In the present paper, we shall write this expression as:


where we have underscored the term tex2html_wrap_inline2091 that corresponds to the twofold PDF of the PSF, within the assumption of statistical independence used to deduce Eq. (7 (click here)) from Eq. (6 (click here)). In that case, the twofold PDF of the binary star appears as the product of the twofold PDF of the PSF and a function denoted as tex2html_wrap_inline2093, and defined as the following ratio:


As we shall see in the following, the function tex2html_wrap_inline2093 makes it very easy to recover the value of tex2html_wrap_inline1947. This is illustrated in Fig. 1 (click here) that shows the respective shapes of PB(2) , PS(2) and Q , computed for the Gaussian model. From these gray-level representations, one can immediately note how the information about tex2html_wrap_inline1947, already present in the twofold PDF of the binary, is tremendously enhanced in the Q function.

Let us now describe how the information about tex2html_wrap_inline1947 is present in this function. We can write Eq. (11 (click here)) as:


The shape of this function is mainly given by the first term inside brackets. The quantity tex2html_wrap_inline2111 present in this first exponential divides the tex2html_wrap_inline2113 plane of Q into two regions, with a delimiting ridge of slope tex2html_wrap_inline2013.

3.3. Radial integrations

Figure 2: Plots of the analytical radial integrations for the theoretical twofold PDF of a binary star computed for tex2html_wrap_inline2085 and tex2html_wrap_inline2087 a), the twofold PDF of a point-source b), and the corresponding Q function c). These plots precisely correspond to the gray-level representations shown in Fig. 1 (click here)

An easy way to detect the ridge described previously, and shown in Fig. 1 (click here) for tex2html_wrap_inline2087, is to radially integrat the Q function in the tex2html_wrap_inline2113 plane. Analytically, this operation can be written as:


where: tex2html_wrap_inline2133 is the radial integration of Q , tex2html_wrap_inline2137, tex2html_wrap_inline2139, tex2html_wrap_inline2141 is the maximum value of tex2html_wrap_inline2143, i.e.: tex2html_wrap_inline2145 with tex2html_wrap_inline2147 the actual maximum value of intensity determined by the practical binning. We can first consider the ideal case where tex2html_wrap_inline2149. Then tex2html_wrap_inline2133 becomes:


Here again, the main part of tex2html_wrap_inline2133 comes from the first term, the second one being almost negligible compared to it. Moreover, the quantity tex2html_wrap_inline2155 divides the axis of tex2html_wrap_inline2157 into two regions, causing the relevant behavior:


This is due to the fact that tex2html_wrap_inline2093 rapidly converges to tex2html_wrap_inline2161 for tex2html_wrap_inline2013 (that corresponds to tex2html_wrap_inline2165), and as tex2html_wrap_inline1973 increases. This general behavior will allow us easily to find the exact value of tex2html_wrap_inline1947 by searching for the infinite maximum of tex2html_wrap_inline2133.

In practice, we have to consider that tex2html_wrap_inline2141 has a finite value. In that case the value of tex2html_wrap_inline2133 becomes finite too, but the main figure is kept: tex2html_wrap_inline2133 has a very clear maximum for the right value tex2html_wrap_inline2179 of the intensity ratio of the binary star. Figure 2 (click here) shows tex2html_wrap_inline2133 compared to the radial integrations performed on the twofold PDF of the PSF and on the twofold PDF of a binary star speckle pattern. As one can see from these plots, the maximum of both the radial integrations of PB(2) and Q gives the value of tex2html_wrap_inline1947, but the maximum of Q is tex2html_wrap_inline2191 times higher (for the present case where tex2html_wrap_inline2193) and much better defined.

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