4 Applications

4.1 Simulated field

The code was first applied to a simulated image (Fig. 4) including 1000 stars, placed randomly in the frame and with a given magnitude distribution (Fig. 6). Each star is a scaled copy of a long exposure high-Strehl PSF, obtained with the ADONIS AO system at the ESO 3.6 m telescope. The image is $368\times 368$ pixels large ( $13 \hbox{$^{\prime\prime}$ }\times 13 \hbox{$^{\prime\prime}$ }$) and has a stellar density of 6 stars arcsec^-2. Photon, readout noise and a background nebulosity, normalized to the same flux of the stellar sources, were added to the image. The faintest stars have a peak signal-to-noise ratio of 5.

We performed a standard reduction of the artificial image using the "default" parameters of the method.

The PSF was estimated by superposing the images of the four brightest stars in the field. The retrieved PSF is very similar to the true one (Fig. 5).

Figure 5: Axial plot of the true PSF (continuous line) and of the retrieved PSF (dashed line)

Figure 6: Comparison between the true (continuous line) and the estimated luminosity function (dashed line)

Figure 6 shows the good agreement between the true and the observed luminosity function. The sole statistically significant discrepancy is in the bin from magnitude 7 to 8, where $\sim$25% of the stars were lost; the other small differences are due to photometric errors which shift some objects to a neighboring magnitude interval. The lost stars are $\sim$10% of the total number of sources and are generally faint: about 90% have magnitude between 7 and 8, the rest is in the bin between magnitude 6 and 7. The only bright lost star has magnitude $\sim$4.5 and is the secondary component of a very close binary, with a separation of just 1/2 pixel. Roughly 70% of the lost stars are located at a distance $\leq 1$ PSF FWHM from the nearest object, $\sim$20% are on the first diffraction ring of a brighter source and only $\sim$10% are isolated. It should be stressed however that $\sim$15% of the lost stars can be recognized by visual inspection as faint objects in the halo of the four brightest stars in the field, independently of their separation from the nearest source. It is apparent that the blending effect and the contamination by the halo of very bright stars do account for the lost stars. Note that the number of false detections in this simulated field is negligible (1 case out of 1000).

The plots in Figs. 7 and 8 show the astrometric and photometric accuracy of StarFinder.

Figure 7: Plot of astrometric errors vs. relative magnitude of detected stars; the errors are quoted in FWHM units (1 FWHM $\sim$3.6 pixel) and represent the distance between the calculated and the true position. A tolerance of 1 FWHM has been chosen to find the coincidences between the detected stars and their true counterparts. The small errors for the brightest stars are due to blending effects

Figure 8: Plot of photometric errors vs. relative magnitude of detected stars. The brightest star in the field has mag = 0 by definition

About 80% of the detected stars have both astrometric error smaller than 0.1 PSF FWHM and photometric error smaller than 0.1 magnitudes. The stars with less accurate astrometry or photometry are generally faint: $\sim$75% belong to the magnitude interval from 7 to 8, $\sim$20% to the bin between magnitude 6 and 7 and only $\sim$5% are distributed in lower bins of the luminosity function. These stars have, in $\sim$45% of the cases, a lost source in their immediate neighborhood within the first diffraction ring of the PSF. The others are faint stars dispersed in the halo of the brightest sources or in the noisy background nebulosity.

Figure 9: PUEO image of the Galactic Center. North is to the left (at an angle of $-100.6\hbox {$^\circ $ }$ from vertical) and east is $-10.6\hbox {$^\circ $ }$ from the vertical. The display stretch is logarithmic

Figure 10: Reconstructed image, given by the sum of about 1000 detected stars and the estimated background. The display stretch is logarithmic

After discussing the performance of the code with a standard analysis, it is interesting to examine how the results are affected by the main parameters of the method. Applying the de-blending strategy we detected $\sim$+10% more binaries in the separation range between 1/2 and 1 PSF FWHM, even though the overall detection gain is less than 1% referred to the total number of sources. With a higher number of iterations of the main loop we detected $\sim$+30% more binaries in the range from 1 to 2 PSF FWHM. Decreasing the detection threshold from 3 to 1 times the noise standard deviation, we found $\sim$+25% more binaries in the range between 1/2 and 1 PSF FWHM, but with 10 faint ( $\rm mag > 8$) false detections instead of 1. Increasing the threshold on the correlation coefficient, from 0.7 to 0.8, we reported no false detection, but the number of lost stars increased by about 60%; the additional lost sources were generally fainter than magnitude 7, but not necessarily in crowded groups. Lowering the correlation threshold to 0.6 we detected more faint isolated stars and binaries, at separations between 1 and 2 PSF FWHM, but with a higher probability of false detections (2 instead of 1). Finally the astrometric and photometric accuracy approaches a stable level after a few ($\sim$2) re-fitting iterations.

4.2 Galactic center

The code was run on a 15 min exposure time K band (2.2 $\mu$m) image of the Galactic Center (Fig. 9), taken with the PUEO AO system on the 3.6 m CFH telescope (Rigaut et al. 1998). The Strehl ratio in the image is $\sim$45%. The PSF FWHM is $\sim$0.13 $^{\prime\prime}$, with a sampling of $0.034 \hbox{$^{\prime\prime}$ }$/pixel. The Adaptive Optics guide star was a $m_{\rm R}=14.5$ star (called star 2 in Biretta et al. 1982) located about (to the upper left) $20 \hbox{$^{\prime\prime}$ }$ from the center of the image, out of the field of view of the figure. There is therefore a slightly elongation of the PSF towards the direction of the guide star. However, due to the fact that the isoplanatic patch was much larger than the $13 \hbox{$^{\prime\prime}$ }\times 13 \hbox{$^{\prime\prime}$ }$ shown in the figure, a space-invariant PSF fits the data very well, as we will show.

A standard analysis was performed, analogous to the one described in Sect. 4.1 for the synthetic stellar field. About 1000 stars were detected, with a correlation coefficient of at least 0.7; the reconstructed image is shown in Fig. 10.

We evaluated the accuracy of the algorithm by means of an experiment with synthetic stars. We created 10 frames adding to the original image 10% of synthetic stars at random positions for each magnitude bin of the estimated luminosity function. The 10 frames were analyzed separately. As in the simulated case, a distance tolerance of 1 PSF FWHM was adopted to find coincidences between the detected stars and their true counterparts. The lists of detected artificial stars were merged together and the astrometric and photometric errors were computed and plotted as a function of the true magnitude (Figs. 11 and 12).

The plots show no apparent photometric bias and high astrometric and photometric accuracy: the stars brighter than magnitude 5, for instance, have a mean astrometric error <0.5 mas and a mean absolute photometric error <0.01 mag. It should be stressed however that the artificial sources are contaminated by the background noise present in the observed data and by the photon noise due to neighboring stars, but no additional noise was added. Figure 13 shows a comparison between the mean luminosity function retrieved in the 10 experiments and the truth. Assuming an expected error for each bin equal to the square root of the corresponding number of counts, according to the Poisson statistic, the only significant differences occur for magnitudes fainter than 9. It is also interesting to consider the magnitude distribution of the false detection cases (dashed-dotted line in Fig. 13), i.e. the detected stars for which we found no counterpart in the original list, within a distance of 1 PSF FWHM. The mean percentage of false detections in the 10 experiments is 2% of the total number of stars. The false detections are almost always very faint ($\rm mag > 8$); their number is comparable to the square root of the total counts only in the last magnitude bin, for magnitudes fainter than 10. The percentage of false detections reported in these experiments seems to confirm the analysis performed by visual inspection on the stars detected in the original frame.