Let us first recall the definition of the single-fold characteristic function (CF) of , the intensity at the focus of the telescope that can be either the PSF or the binary star speckle pattern . is the complex function of the real variable w defined as:
where the symbol denotes the expected value of , and is the single-fold PDF-- and the inverse Fourier transform of .
By generalizing Eq. (2 (click here)) to two dimensions, we can derive the twofold CF of :
On substituting to in the above equation, we directly obtain the twofold CF of the PSF as:
while, if represents the binary star speckle pattern , Eq. (3 (click here)) takes the following form:
In the particular case when is equal to the star separation , 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 . 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 , and 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 :
By Fourier-inverting this last equation, it leads to (Aime 1993):
where stands for a two-dimensional convolution and is the Dirac distribution.
Figure 1: Gray-level representation of the theoretical twofold PDF of a binary star computed for and 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 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 , and defined as the following ratio:
As we shall see in the following, the function makes it very easy to recover the value of . 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 , already present in the twofold PDF of the binary, is tremendously enhanced in the Q function.
Let us now describe how the information about 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 present in this first exponential divides the plane of Q into two regions, with a delimiting ridge of slope .
Figure 2: Plots of the analytical radial integrations for the theoretical twofold PDF of a binary star computed for and 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 , is to radially integrat the Q function in the plane. Analytically, this operation can be written as:
where: is the radial integration of Q , , , is the maximum value of , i.e.: with the actual maximum value of intensity determined by the practical binning. We can first consider the ideal case where . Then becomes:
Here again, the main part of comes from the first term, the second one being almost negligible compared to it. Moreover, the quantity divides the axis of into two regions, causing the relevant behavior:
This is due to the fact that rapidly converges to for (that corresponds to ), and as increases. This general behavior will allow us easily to find the exact value of by searching for the infinite maximum of .
In practice, we have to consider that has a finite value. In that case the value of becomes finite too, but the main figure is kept: has a very clear maximum for the right value of the intensity ratio of the binary star. Figure 2 (click here) shows 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 , but the maximum of Q is times higher (for the present case where ) and much better defined.