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5. Numerical simulations


We assumed in writing the equations in the previous sections that tex2html_wrap_inline2273, i.e. the separation between the components d is large with respect to the speckle size s . In practice, the interesting point for observations is when tex2html_wrap_inline2279, since the aim of every speckle imaging technique is to reach as close as possible the diffraction-limited resolution of the telescope. Then, in order to complete the theoretical study and test the validity and limits of it for practical speckle observations, we chose to make several numerical simulations with different values of d and tex2html_wrap_inline1947. We decided not to report all these simulations in this paper but just the most interesting ones, i.e. for a separation tex2html_wrap_inline2285, and for three different relevant values of tex2html_wrap_inline1947.

5.1. Practical implementation of the method

The functions Q and tex2html_wrap_inline2201 are respectively obtained by dividing the twofold PDF of the binary star computed for tex2html_wrap_inline2029 by that of a point-source, and by that of the binary computed for tex2html_wrap_inline2217. In order to avoid zero divisions during this operation, we made use of an iterating algorithm based on Van Cittert (1931) and already applied to speckle data by Cruzalèbes et al. (1996). The output estimate of this algorithm perfectly converges to the solution of the normal division after an infinite number of iterations. Let be tex2html_wrap_inline2297. If tex2html_wrap_inline2299 the calculation of A may rapidly diverge. One estimate of A can then be:


Because tex2html_wrap_inline2305, it is easy to demonstrate that tex2html_wrap_inline2307. In the case where the denominator C becomes smaller than the limit for which the machine cannot see the difference between 1 and 1+C , we found that it is typically sufficient to perform about ten iterations to estimate the ratio. In the other case, we simply calculated the ratio by normal division.

We also computed the quantity Q-QT to enhance the relevant ridge, where QT is the transpose quantity of Q . The radial integration of this quantity tex2html_wrap_inline2319 is related to IQ by:


So its ideal expression (i.e. when tex2html_wrap_inline2149) follows:


In the present case tex2html_wrap_inline2141 is obviously finite and the general behavior of tex2html_wrap_inline2327 is to have a maximum for the right value of tex2html_wrap_inline1947, like tex2html_wrap_inline2133, but a minimum too for tex2html_wrap_inline2333. For such a quantity the extrema are better defined. A second interest is that it could stand out better between a value of tex2html_wrap_inline1947 close to but greater than 1 and a value of tex2html_wrap_inline1947 close to but smaller than 1.

In practice, to estimate tex2html_wrap_inline2339, we did not directly make use of Q . We computed tex2html_wrap_inline2343 in order to have two different estimates (as PS(2) is determined experimentally) of the intensity ratio when analyzing the quantity tex2html_wrap_inline2347: one corresponding to the maximum of the radial integration, and one to the minimum. The output values of tex2html_wrap_inline2157 are then averaged from these two estimates, together with the corresponding errors.

In addition, and in order to get rid of the effect of statistical fluctuations and to keep only the most significant features, we also smoothed the Q , tex2html_wrap_inline2201, tex2html_wrap_inline2347 and tex2html_wrap_inline2357 estimates by convolving them by a tex2html_wrap_inline2359 unit valued filter.

We consider only the part of the computed quantities Q , tex2html_wrap_inline2201, tex2html_wrap_inline2347 and tex2html_wrap_inline2357 where the signal-to-noise ratio is the best, i.e. where there is a significant number of events in the twofold PDFs of the reference and the binary star. This typically corresponds, in our present case, to an extraction of tex2html_wrap_inline2369 pixels near the origin for PDFs computed with a sample of the intensity of 256 levels.

5.2. General case

Figure 4: Top, first row: logarithmic gray-level representation of the twofold PDFs computed for the simulation made for tex2html_wrap_inline2087 and (dx,dy)=(+3,+4) . Top, second row: linear gray-level representation of the corresponding Q , tex2html_wrap_inline2347, tex2html_wrap_inline2201 and tex2html_wrap_inline2357 functions. The white squares show the tex2html_wrap_inline2369 extraction zone used for the computation of the radial integrations. Bottom: extraction of Q , tex2html_wrap_inline2201, tex2html_wrap_inline2347 and tex2html_wrap_inline2357, together with their respective radial integrations. Its extrema are outlined by dashed lines

The simulation work presented in this subsection made use of two data sets (one for the binary star and one for the point-source), each made of 100 speckle frames of tex2html_wrap_inline2395pixels, simulated with the following parameters:

In Fig. 4 (click here) we represent the PDFs obtained for the point-source for tex2html_wrap_inline2029, and for the binary for tex2html_wrap_inline2029 and for tex2html_wrap_inline2217. The functions Q and tex2html_wrap_inline2201 are deduced from this, and represented together with their radial integrations. The functions tex2html_wrap_inline2347 and tex2html_wrap_inline2357 are represented as well with their radial integrations.

Figure 5: Left: linear gray-level representation of tex2html_wrap_inline2347 and tex2html_wrap_inline2357 for tex2html_wrap_inline2429, together with the plots of tex2html_wrap_inline2327 and tex2html_wrap_inline2433. Right: linear gray-level representation of tex2html_wrap_inline2201 and tex2html_wrap_inline2357 for tex2html_wrap_inline2439, together with the plots of tex2html_wrap_inline2133, tex2html_wrap_inline2327, tex2html_wrap_inline2445 and tex2html_wrap_inline2433


input tex2html_wrap_inline1947 input tex2html_wrap_inline2157 Q tex2html_wrap_inline2347 tex2html_wrap_inline2201 tex2html_wrap_inline2357
1.5 56.31 tex2html_wrap_inline2465 tex2html_wrap_inline2467 tex2html_wrap_inline2469 tex2html_wrap_inline2471
10 84.29 tex2html_wrap_inline2473 tex2html_wrap_inline2475 tex2html_wrap_inline2473 tex2html_wrap_inline2479
1.01 45.28 tex2html_wrap_inline2481 >45 tex2html_wrap_inline2485 >45
Table 1: Values of tex2html_wrap_inline2157 (and corresponding intrinsic errors tex2html_wrap_inline2451) found for the numerical simulations

The procedure to quantitatively find the extrema of these radial integrations and to estimate the respective errors makes use of a polynomial fit of these quantities around the extrema. We chose for this purpose to fit the tex2html_wrap_inline2489 region surrounding the extrema by a polynomial of the second degree, since close to its maximum a convex function is supposed to have a quadratic-like behavior.

The values of tex2html_wrap_inline2165 found from the four quantities tex2html_wrap_inline2133, tex2html_wrap_inline2445, tex2html_wrap_inline2327 and tex2html_wrap_inline2433 are reported in the first row of Table1. The values found for Q and tex2html_wrap_inline2201 are rather less than the input value of tex2html_wrap_inline2157. This means that there is a systematic error in detecting the right value of tex2html_wrap_inline1947 from the maximum of IQ and tex2html_wrap_inline2511. However this systematic error is not present, or at least in a very small way, in the case, not presented in this paper, where the separation d is actually large with respect to s . Nevertheless, this systematic error is avoided by considering the values of tex2html_wrap_inline2157 found for tex2html_wrap_inline2347 and tex2html_wrap_inline2357. So while Q or tex2html_wrap_inline2201 gives us a first (but biased) estimate of tex2html_wrap_inline2157, the computation of tex2html_wrap_inline2347 or tex2html_wrap_inline2357 then gives us a good value of it. The general procedure will be to consider directly tex2html_wrap_inline2347 or tex2html_wrap_inline2357 to estimate the intensity ratio of a binary star. Finally, this method gives equivalent results by using it in its reference-less version or in its standard version.

We made several numerical simulations in order to test the validity and limits of the method. This showed us that two kind of limiting cases exist depending on tex2html_wrap_inline1947.

5.3. Limiting cases

In order to better test the limiting cases of the method, we chose to make the simulation with a larger number of frames per set: 1000. The first limiting case is when tex2html_wrap_inline1947 is large (or small) with respect to 1. This corresponds to a large magnitude difference between the components of the binary star. This is already a well-known limit of speckle observations but in our present case, this corresponds to a ridge of the Q function close to the axis tex2html_wrap_inline1973 (or tex2html_wrap_inline1969), implying then a difficult determination of the right value of tex2html_wrap_inline1947. We found, with the typical parameters taken here, that the useful limit of the method is for tex2html_wrap_inline2553 (and for tex2html_wrap_inline2555), i.e. for a magnitude difference of tex2html_wrap_inline2557. As shown in Table1, second row, the computations Q and tex2html_wrap_inline2201 could only give an idea of tex2html_wrap_inline2157, and we deduce from IQ and tex2html_wrap_inline2511 that tex2html_wrap_inline2157 is greater than or of the order of tex2html_wrap_inline2571. The estimate of tex2html_wrap_inline2157 is still available from tex2html_wrap_inline2319 and tex2html_wrap_inline2577 but gives a slight under-estimate.

The second limiting case is when tex2html_wrap_inline1947 is close to 1. This problem occurs when the two components of a binary system are of close magnitudes, implying then an ambiguity in the determination of the PA. In that case, the quantities tex2html_wrap_inline2347 and tex2html_wrap_inline2357 become very small but still contain the information about the orientation of the binary, even if the precise determination of tex2html_wrap_inline1947 is no longer possible. Nevertheless, Q and tex2html_wrap_inline2201 can in this case give a good estimate of it, as shown in Table1, third row, where we report the result of a simulation made for tex2html_wrap_inline2439 (i.e. for a magnitude difference of tex2html_wrap_inline2593 0.01).

Figure 5 (click here) illustrates the procedure used to analyze these two limiting cases. In the first case ( tex2html_wrap_inline2429), tex2html_wrap_inline2157 is directly determined from the extrema of tex2html_wrap_inline2319 or tex2html_wrap_inline2577. In the second ( tex2html_wrap_inline2439), tex2html_wrap_inline2157 is determined from IQ and tex2html_wrap_inline2511 and the orientation is checked from tex2html_wrap_inline2319 or tex2html_wrap_inline2577. An interesting case is presented by the reference-less version of the method. In fact, while tex2html_wrap_inline2511 shows a maximum for tex2html_wrap_inline2157 slightly smaller than 45tex2html_wrap2657 (but with an error large enough to include the value 45tex2html_wrap2657), the shape of tex2html_wrap_inline2577 clearly denotes a value of tex2html_wrap_inline2157 greater than 45tex2html_wrap2657. In conclusion, a good estimate of tex2html_wrap_inline1947 can be found by using our method if the following procedure is performed:

  1. Compute Q and tex2html_wrap_inline2201.
  2. If tex2html_wrap_inline1947 is not close to 1: estimate it with tex2html_wrap_inline2319 or tex2html_wrap_inline2577.
  3. If tex2html_wrap_inline1947 is close to 1: estimate it with IQ or tex2html_wrap_inline2511 and check the orientation (i.e. check if tex2html_wrap_inline2157 is greater or not than tex2html_wrap_inline2649) by using tex2html_wrap_inline2319 or tex2html_wrap_inline2577.

An application of this procedure to real data of close visual binary stars is performed in the next section, together with a comparison with the results found elsewhere.

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