4 Results from the mean image analysis

When the mean image in each band is fitted, yielding at the same time the double system features (see Table 3 and Fig. 4) and the PSF's global structure (Fig. 5, bottom), the results are more reliable than the averages obtained in Sect. 2. This because the fit now uses improved values of the pixel intensities, requires two parameters less than the right-hand side of Eq. (2) (namely x₁ and y₁), and starting with the above averages as initial parameters requires lower stabilizing constraints, so that its results are less biased. But the way these are obtained does not allow an easy estimate of their errors, which are conservatively approximated with upper limits equal to the standard deviations in Table 3, although realistic simulations (see Sect. 4) indicate values about half of these.

  

Table 3: $\tau$ CMa features and PSF parameters from fitting the mean set images (first row in each band) and previous results by Christou & Bonaccini 1996b (last row)

The smooth radial approximation of the mean PSF (see Fig. 5, bottom) is simply obtained by inserting in the right hand side of Eq. (1) the suitable fitting parameters of the mean image. The accurate bidimensional light distribution of the mean PSF (Fig. 5, top) is instead derived by deconvolution of the mean image from the pair of impulse functions, with appropriate amplitude and position parameters, which represents the "true" photometric structure of a distant binary system, i.e., is given by

$${\rm PSF}_{\rm m}(x,y) = {\cal F}^{-1} \left[ \frac{ I_{\r... ...x_2-x_1 \right)+ v \left(x_2-x_1 \right) \right)} } \right]$$ (4)

where $I_{\rm m}(u,v)$ and the denominator in the right-hand side are respectively the $\cal F$s of the mean image and of the impulse sum. It must be stressed that the $\cal F$ of the latter does not have zeroes nor values approaching zero, thus ${\cal F}^{-1}$ in Eq. (4) can be performed without the need of any numerical stabilization. The global similarity between the detailed mean PSF and its radially symmetric approximation is clearly shown in Fig. 5, while the local disagreements can be seen in Fig. 6, which displays the actual intensity distribution along the axes in the mean PSF and in its circular approximation. Also, it is worth mentioning that the mean images in Fig. 4 are very similar to good images in Fig. 2, which implies a small occurrence of bad images. Finally, the mean performances of ADONIS during the $\tau$ CMa observations may be easily inferred by comparison (see Fig. 7) of the intensity profile and the encircled energies in the circular PSF approximation with the corresponding values in the diffraction pattern of the ESO 3.6 m telescope.

  

Figure 5: Mean PSF in the J, H and K bands (first row) and their circular approximation (second row). Note, as in Fig. 4, the accurate centering

  

Figure 6: Comparison of N-S (first row) and E-W scans (second row) of the detailed mean PSF in the J, H and K bands (solid line) with its radially symmetric approximation (dashed line)

  

Figure 7: Global performances of ADONIS during the $\tau$ CMa observations. Top: mean K PSF intensity profile (dashed line) and the diffraction pattern(solid line). Bottom: the related encircled energy

  

Figure 8: Appearance of a binary star observed with the partially compensated PSF of ADONIS in the following cases: i) the components magnitude difference and separation is the same as for $\tau$CMa, but the position angle of the secondary varies (top); ii) the secondary is placed as in $\tau$CMa, but its magnitude difference is, from left to right, 1.5, 2.0 and 2.5 (bottom images)