The method proposed in this paper to determine relative position and photometry of the components of a binary system consists of calculating ratios of PDFs: the twofold PDF of the double star speckle pattern computed for a space-lag equal to the star separation is divided by the twofold PDF of an unresolved star, computed for the same space-lag . Alternatively, and in a case of lack of a good reference star, the twofold PDF of the binary speckle pattern itself, computed for a space-lag perpendicular to the star separation, may also be used as reference. The result, which is described in the text as the Q (or ) function, clearly evidences the region of the ( ) plane where . This procedure, in some aspects, solves the problem of the PI technique emphasized in the introduction of this paper, i.e. the fact that it is a non-linear approach for which there is no simple separation between functions of the object and of the speckle pattern. This pseudo-linear result was clearly illustrated in Fig. 1 (click here). Moreover, the use of a radial integration gives directly the value of with no ambiguity on relative position of components.
Other representations may be considered to emphasize the dissymetry of the twofold PDF. For example, we have noticed that the ratio of P to its transpose quantity gave results similar to . For the sake of conciseness, these results are not reported here. In any case, since ratios are taken, a problem may arise when the twofold PDF used as a reference is equal to zero. This is not a major problem, but rather the effect of insufficient statistics in terms of number of samples. This problem could also be resolved if smoothed versions of twofold PDFs are used.
The ratio approach, even though we seek to obtain the linear relation discussed above, remains fully empirical. The question may arise about the meaning of these twofold PDFs ratios in terms of theory of probability and statistics. The ratio may be considered as the measure of some distance between probabilities, one bearing the information about the double star embedded in the speckle pattern, and the other being relevant to the PSF only. This approach is used in empirical hypothesis testing; however, the use of a ratio is not a common measure of distance (Allen 1990). Attempts were made to use differences of PDFs instead of ratios (Lyon 1993), but the results were found to be less attractive than the present ones.
Another possibility of a theoretical meaning for the ratio of PDFs is to refer to entropy and the information given by PDFs. The information associated with an event is equal to minus the logarithm of the probability of that event. The ratio we perform may be therefore linked to the difference between the information that comes from the law of probability of the intensity of a binary star speckle pattern, and that of an unresolved star. A deeper development of this approach, that we will not further develop here, would lead to the use of some Kullback-Leibler representation (Taupin 1988), of the form .
An alternative to the present method is to deal with CFs instead of PDFs. The division of the PDFs corresponds, in the Fourier space, to a deconvolution of the CFs. Surprisingly, we found that a division of the CFs leads to a similar result, since the CF computed for the binary (and for its separation) presents a characteristic ridge as well, which is also tremendously enhanced by dividing it by the CF of the PSF. This is an interesting behavior that we plan to study later.
The points discussed above are interesting problems of probability theory and signal processing, and will be developed elsewhere.
Several developments of the method are possible. A first one consists of the treatment of low-light level data. In this case, as discussed by Sultani et al. (1995), the PDF suffers a Poisson-Mandel transform that must be inverted. However, preliminary checks made on simulated data have shown that the information about was already clearly visible in the ratio of low-light level PDFs. Another development is the extension of the procedure multiple stars. The analysis of triple stars speckle patterns is currently under processing and the results will be given in a near future.