2 Binary star AO image analysis

If within the isoplanatic field ($n\times n$ pixels wide) there is only a close binary star, the observed short image is the superposition of two shifted and unnormalized nearly instantaneous PSFs or, in other words, the convolution of the PSF with two unnormalized impulse distributions. When the PSF is not known and not radially symmetric, the successful analysis of the image data would give positions and luminosities of the two stars and the PSF intensity in each pixel (i.e.,a total of 6+n² values); thus, whatever the approach followed, an approximate estimate of these values can be obtained only by using prior knowledge or suitable hypotheses (in definitive solution constraints) to overcome the initial lack of information. The availability of a set of short images does not conceptually improve the situation, owing to the PSF time-space variability and the matching effects of residual image motion (see Christou & Bonaccini 1997) and pixel size, which in the actual case is about half of the Airy disk. Since the pixel intensity is an integrated value, really small structures such as the cores of the two stars or the bumps due to the triangular coma can appear somewhat different from image to image, even with very similar PSFs, if their centers do not coincide with the center of the corresponding pixel of relative maximum intensity, and the derived photometry can be misleading. But in practice these drawbacks are compensated for sufficiently by the use of statistical tools, because the set of N short images can be regarded as a large random sample whose variability is due to many unknown combining effects (the instantaneous parameters of the atmospheric turbulence, the fine reponse of the AO system, etc.). The first step of our procedure is the fit of each short image as a superposition of two weighted radially symmetric approximations of the PSF. This yields an estimate of the position and intensity of the two stars, as well as of the PSF's global shape and of the partition of its light between the 3 characteristic zones: the central core, the defaced first diffraction ring and the extended halo. At this stage it is possible to obtain either the features of the binary system by averaging the intensity and position parameters, or the set of the detailed nearly instantaneous PSFs by deconvolution of each short image from its joint two-impulse distribution. But the short PSF set thus obtained (and used below in Sect. 4) shows the expected large variability, and hence is scarcely useful to understand the performances of the AO system during the observations, while the scatter of the parameters, which will be discussed below, enables an accurate determination only of the binary separation, not of the components' intensity ratio. Thus, in the second step we obtain the set's mean image by shift-and-add of the short images, using for each of these, as shifting parameters, the coordinates of the center of the primary star. Then the mean image is fitted as the short images in the first step, yielding the adopted features of the binary system (relative position and luminosity of the components) and the azimuthally averaged behaviour of the set's mean PSF, while its detailed bidimensional structure is derived by deconvolution of the mean image from the two-impulse distribution with the suitable parameter values.

2.1 Short image parametrization

To fit the N short images of the double star we use as a smooth approximation of the instantaneous PSFs (denoted as p_k(x,y), with k=1,2,...,N) the sum of a centered Gaussian and of a Moffat's torus with a core parameter $R\rm _{c}$ and the same radius $R\rm _{t}$ of the first diffraction ring of the ESO 3.47 meter telescope, which has a 0.479 obstruction factor, as given by the Lyot stop inside the IR camera. It must be stressed that the "ad hoc'' circular approximation of the PSF, inappropriate e.g. to represent the PUEO star images (Rigaut et al. 1991), was chosen after some trial on the more suitable fitting function. In particular, we have first used the sum of a central Gaussian, a Gaussian torus with free radius and an exponential. But the behaviour of the wings of a short PSF in mean better belongs to a power law, (see Fig. 6), and the torus radius was found to be very close to that of the diffraction ring in almost all cases, thus its estimate was left out. The adopted expression of the PSF (containing the four parameters $a_1, \sigma, R_{\rm c}$ and $\beta$) explicitly reads as

$$\begin{eqnarray} p_{k}(x,y) & = & a_{1} {\rm e}^{-\left(x^{2}+y^{2}\right)/2 \si... ...(x^{2}+y^{2}\right)} -R_{\rm t}\right)/R_{\rm c}\right]^{- \beta}\end{eqnarray}$$ (1)

where a₂=1-a₁ and the normalizing factor K of the torus must be evaluated by a numerical integration, because this cannot be done analytically. It is readily apparent that the behaviour of Eq. (1) mimics a deformed diffraction pattern showing the remainders of the central maximum and of the first ring, but with a large fraction of the total light (>50%) falling in an extended, roughly circular halo. When the short PSF is characterized by the bumps due to the uncompensated triangular coma, as in the present case, the extension of Eq. (1) to represent also these features would contain many more non linear parameters, whose estimate, though possible in principle, is very difficult in practice. Above all, this extension should be of interest if the primary goal of the analysis were the search of an accurate analytical approximation of the PSF, not the differential astrometry and photometry of a binary star. In our opinion, also, the effects and limitations on the AO image analysis caused by the said bumps are less important than generally believed because they contain only a few percent of the total light, are located close to the first Airy ring, and are more irregular and extended in shape than the round and spiky core of a star image produced by an AO system. Also, it must be pointed out that owing to the amount of light in the PSF halo the image of a close binary is elongated along the centers'joining line, if the secondary luminosity is not too low, with quite a uniform distribution of the outer light; the strongest bumps of the primary are well visible, while those of the secondary progressively merge into the primary light as they become fainter. In conclusion, the discrimination of the core of the secondary star from one of the bumps of the primary seems possible by means of a visual inspection (see e.g. Figs. 1 and 2) or by an expert program, with the exception of the critical case in which the two features are very close or overlap. When this occurs the image must be still elongated, but the features separation is not very reliable. The only way to study such a binary star is a previous rotation of the AO system, if technically possible, so that these features do not overlap. The dependence of the AO image of a binary system on the values of position and intensity of its secondary star will be again considered in the section on tests and numerical simulations, as well as the influence of the choice of the secondary core among the other bright features surrounding the primary center. Using the PSF approximation of Eq. (1), the intensity distribution i_k(x,y) in a short image (after the usual pre-processing of CCD data, background subtraction and neglecting the pixel integration) reads as

$$\begin{eqnarray} i_{k}(x,y)& = & L_{\rm t} [l_{1}p_{k}\left(x-x_{1},y-y_{1}\right)+ \nonumber \\ & & ~~~~~~~~l_{2}p_{k}\left(x-x_{2},y-y_{2}\right)]\end{eqnarray}$$ (2)

where $L_{\rm t}$ is the known total image intensity, l₁ and l₂ (with l₁+l₂=1) are the components' relative intensities, and x₁, y₁, x₂, y₂ the centers. To estimate the nine non-linear parameters in Eq. (2) (i.e.: $a_{1}, e_{1}, \sigma, R_{\rm c}, \beta, x_{1}, y_{1}, x_{2}$ and y₂), we use the Newton-Gauss regularized method (Bendinelli et al. 1987, hereinafter NGR), but nearly the same result could be obtained by any current non-linear fitting routine. The implemented procedure improves the classical linearization by using the Moore-Penrose pseudo inverse of ill-conditioned matrices (see Penrose 1956) and Tikhonov first-order regularization (see Tikhonov & Arsenin 1977), so that all parameters usually converge at 0.5% level in less than 10 iterations, starting from a suitable set of initial parameters, chosen as follows. For all the short images we adopted the same starting values $l_{1}=0.7 L_{\rm t}$, $a_{1}=0.3 L_{\rm t}$, $\beta=1$, on the grounds of some preliminary fits of a few good and bad images, i.e., with the relevant details more or less stable both in position and luminosity as clearly shown in Figs. 1 and 2 (top). The pairs (x₁,y₁) and (x₂,y₂) were instead taken as the coordinates of the maximum intensity pixels in the unmistakable primary core and in the assumed secondary one. Finally, the band-dependent value of $\sigma$ was obtained by a bilinear interpolation of the intensity of the pixels surrounding the center of the primary star, and $R_{\rm c}$ was always taken equal to $\sigma$.

  

Figure 1: Good J, H and K short exposure images of $\tau$ CMa (first row), and their fit by Eq. (1) (second row). Parameter values in Table 1

  

Figure 2: Bad J, H and K short exposure images of $\tau$ CMa (first row), and their fit by Eq. (1) (second row). Parameter values in Table 1

  

Table 1: Binary star features by fit of images in Figs. 1 and 2

  

Figure 3: Dependence of derived features of the binary system in the J, H, K bands along the set's image sequence

The usefulness of Eq. (2) to obtain a good smooth approximation of the intensity distribution in short images is shown in Figs. 1 and 2 (bottom). The correlation coefficients c between observed and parametrized images, which are reported in Table 1 with the relevant fitted features of the model, strongly prove the procedure's reliability. But the behaviour of the fitting parameters and of the derived binary system features along the image sequence must be pointed out, recalling that all the obtained values have bias components which are very hard to quantify, generated either by the numerical stabilization of the non-linear fit, or by pixel integration and residual image motion effects. The scatter of the parameters related to the PSF shape (see for instance the sets of a₁ values in Fig. 3) is mainly due to its well known short-time scale variability and wavelength dependence. The apparent deplacement of the centers'coordinates, instead, follows the residual image motion, which is the same in any point of the isoplanatic field; thus the binary components' separation is really the quantity best estimated by the fit (see Fig. 3 and Tables 1 and 2) in good agreement with the previous result of $\rho$ = 0$.\!\!^{\prime\prime}$151 obtained by Hipparcos (see The Hipparcos and Tycho Catalogues 1997). The estimate of the components' luminosity ratio in a close binary system worsens as the secondary becomes fainter, because its halo and the diffraction features merge into the primary halo, while the secondary core, the only detail still visible, becomes more contaminated and thus the fitting parameters more uncertain and dependent on the fine structure of the PSF halo. The lovering of the compensation degree acts in a similar way since it decreases the central core's intensity and yields a more irregular light distribution in the intermediate PSF zone, thus making the analytical approximation used less adequate. Finally, it must be stressed that the actual position of the secondary star can strongly twist the fitting parameters, particularly when it falls near the defaced first diffraction ring or, even worse, if it overlaps one of its bumps. In these cases the secondary luminosity derived by the fit of an image may result either underestimated or overestimated, as seems to be confirmed by the conflicting magnitude difference values reported in Table 1, recalling that the binary separation is very close to $R_{\rm t}$ in the H band. In conclusion, the fit of short images gives a good global radially symmetric approximation of the instantaneous PSFs and quite a satisfactory estimate of the stars' positions, but a poor differential photometry, as occurs in the interferometry of double stars (see for instance Ten Brummelaar et al. 1996). Averaging the fit values, whose large scatter is due either to PSF variability or to non-linear numerical effects, there is a slightly improvement in the photometry (see Table 2), while a larger one seems attainable by neglecting the bad images with parameters far from the set mode (see Fig. 3); but the correctness of this procedure should be secured only by the normality of the parameter set.

  

Table 2: Binary star features and PSF paramters for J, H and K sets of images averaging the individual fits

• Note: first row values, in each band, were derived by averaging all individual fit parameters, second row values by using only good images; quoted errors are standard deviations, recalling that the pixel size is 0$.\!\!^{\prime\prime}$035.