5 Tests and simulations

The agreement shown in Table 3 between our findings and the previous ones by IBD is even too good in view of the fact that neither method is rigorous. A way to settle which of them gives the more correct results in this particular application should be the comparison with HST results on the $\tau$ CMa system, but so far these are lacking. Our procedure requires much less iterations and CPU occupation than IBD, but also the preliminary identification of the secondary core, thus it is worth checking how it works when the binary star components are characterized by different values of intensity and position. To address the matter we have first used (in the J, H and K bands) a set of 48 simulated images of binary stars, each with a random magnitude difference of the components in the range 1.0 - 2.5, random position of the secondary star within a square box of size 3 $R_{\rm t}$ centered on the primary and a random PSF drawn out of the PSF set obtained above by deconvolution of short images from the related two-impulse distribution. For instance, Fig. 8 shows what would be the instantaneous appearance of a binary star, observed with the partially compensated PSF of ADONIS, if its components where characterized by:

1.

the same separation and magnitude difference $\Delta$$_{\rm m}$ as in $\tau$ CMa but with different secondary positions (top),

2.

the same positions as in $\tau$ CMa but $\Delta$$_{\rm m}$ respectively of 1.5, 2.0 and 2.5 (bottom, from left to right).

In both cases the mean PSF in the H band (Fig. 3, top) was used. These simulated images, in our opinion, sufficiently prove that an expert eye is able to discriminate between an artifact of the AO system and a secondary core reasonably differing (in position and luminosity) from the primary one. But a less subjective way to do this, in dubious cases, is to fit within a small box centered on all the secondary maxima of the image, as was already done in Sect. 2.1 to derive the initial $\sigma$ value. The fit of Eq. (2) to the simulated image set always converges, enabling us to derive the magnitude differences of the components with a standard deviation of about 0.1 mag (for each band, with a systematic decrease from J to K), and the largest deviations occurring when the secondary is faint and is close to one of the bumps. A second check of the method was done by fitting the good and bad images of Figs. 1 and 2, by taking successively as initial secondary core the three main aberration bumps. Using the brightest bump the fit always converged, restoring the true $\Delta \rm _m$ value within 0.1 mag and the true secondary position within a fifth of a pixel (i.e., 0$.\!\!^{\prime\prime}$007), while for the lowest bump the fit diverges or converges to unreliable values, yielding a parametrized image which is clearly different from the observed one and overall with a much smaller correlation (0.2-0.3). To understand the magnitude of the errors (in relative photometry and astrometry) related to the present method, the successive 48 images in each band (see Sect. 1) were analyzed, obtaining results differing from those of Table 2 only by $3-4\%$ and practically the same standard deviations. This may be considered as an indication that the errors in the procedure are really about half or less of the relative standard deviations, but also as evidence that the atmospheric turbulence was constant, in general, during the image set acquisition. Thus other simulated sets were constructed assuming a binary star model similar to $\tau$ CMa (i.e., with the same magnitude difference 0.9 and the same relative position and separation 0$.\!\!^{\prime\prime}$151 of the components in each band), while the primary centers were randomly placed within the same pixel, and the same PSF sets of the first simulation in this section were used.

  

Table 4: Simulated image results: i) mean and standard deviations of the individual fits (top) and ii) final values derived from mean images (bottom)

The results from the simulated images, summarized in Table 4 after the rounding off, show that the deviations from the true values of the estimates of the components' separation and magnitude difference obtained from the mean images are about half of the corresponding ones obtained by averaging the individual fits, and even less of their standard deviations. But the results in Table 4 and those in Tables 2 and 3 have a similar behaviour, allowing us to assume errors of the same order of magnitude also when dealing with real images, i.e., for the final estimates of $\tau$ CMa reported in Table 3. Nevertheless the comparison of results from real and simulated images deserve two further comments. The first is that the standard deviations in Table 2 are calculated by using the mean set values, hence contain a bias component and are larger than those in Table 4, calculated with respect to the true values, i.e., those assumed in doing the simulations. The second refers to the magnitude difference of the components of $\tau$ CMa, which in H seems larger than in other bands (color effects apart, here not considered), but the simulations also give the same, thus it must be a procedure artifact occurring when the secondary center overlaps the first deformed diffraction ring, as in the present case.