When trying to detect a faint structure around a star, either a companion or an extended object, the easiest solution is to subtract a point spread function, properly scaled and placed, from the image of the main star. This procedure should theoretically remove the light from the star and leave only the surrounding structures. But here again some problems appear. The adaptive optics PSF is the sum of two components: a diffraction-limited core surrounded by a large halo due to variations of uncorrected high order modes and imperfectly corrected low order modes. This halo is therefore affected by strong variations with time which are independent of the fluctuations of the central core. These variations are even higher when two different objects are compared because of the halo's dependence on parameters like the brightness of the guide star.
To assess the limits for detection of a faint companion next to a star, we used observed images of the single stars HR 2019 and HR 2076, and subtracted one from the other. For each image of HR2019, the integration time was 75 seconds, the signal to noise ratio 7500 and the Strehl ratio 0.26. For HR2076, these parameters were respectively 200 seconds, 10000 and 0.22. The delay between the two sets of images was 10 minutes. The subtraction method was applied to four different couples of images in order to obtain different residuals. Finally, we computed the standard deviation of these residuals. To work out the average detection limit for a point like object at a given distance of a bright star, we computed an azimuthally averaged radial profile of the standard deviation of the different residuals.
Figure 5 (click here) shows the radial dependence of the detection limit. The full line corresponds to the profile a star. The dotted line shows the radial profile of the standard deviation of the four different residuals. For example, we could detect on this basis a companion fainter by about 7 magnitude at 1 arcsec from the main star and fainter by more than 9 magnitudes at 2 arcsecs. This result is in excellent agreement with the observations by Tessier (1995). Note that far from the star (more than about 2 arcsec), the uncertainties are dominated by readout and sky noises. An important point is that Fig. 5 shows azimuthally averaged profiles. But the shape of the PSF is far from having a circular symmetry. Lots of non-axisymmetric small features appear, especially on the first diffraction rings. Figure 5 should therefore be considered as an over-optimistic estimate.
As an illustration of this method, we used adaptive optics data taken on January 21st 1996 to look for a possible companion to the star Betelgeuse. We subtracted the images of different calibration stars from images of Betelgeuse. No companion was visible down to the approximate limits indicated above. Near to the star, the main problem was the presence of several residual features whose position and intensity changed depending on the calibration star used. One of these features stood out by being always present at a position approximately constant. If this feature were real, it would lie at about 0.5 arcsec to the south-west from Betelgeuse and be 4.5 magnitude fainter. These data would be consistent with a reported detection by Karovska et al. (1986) using speckle observations. More details are given in Esslinger (1997).
Figure 5: Three radial profiles (in magnitude
and relative to the main star). The full line is the
profile of the star before subtraction. The dotted line
shows the radial profile of the standard deviation
of four different residuals.
This gives us the limiting magnitude for detection of a companion
at a given distance from the main star. The dashed line
(the lowest) corresponds to the standard deviation of a set of four images
taken on the same star
The previous test gives us an estimate of the global noise affecting the subtraction procedure. It would be interesting to work out the part due to the each main cause of noise. With this aim in mind, we compute the standard deviation of our set of 75-second exposures of HR 2019. This gives us the detection limit if we only had to deal with variations of high order modes and photon noise, and did not have the difficulty of mismatch between PSFs from different stars. The radial dependence of this limit corresponds to the dashed line in Fig. 5 (click here). This shows that the main source of error in our procedure is indeed the variation of the PSF between the observations of the science object and the reference star, confirming the results by Tessier (1995).
We can also compare the relative contribution of photon noise compared to the noise introduced by uncorrected or imperfectly corrected modes. We can work out the photon noise from the number of photons per pixel in the original image at a given distance from the peak. We find for instance that, at the centre of the PSF, the level of photon noise is about 7 magnitudes fainter than the peak intensity, thus 3 to 4 magnitudes fainter than the dashed line in Fig. 4. At a distance of 1 arcsec, photon noise is 10 magnitudes fainter than the peak intensity and thus 1 magnitude below the same line. From this, we see that photon noise is usually not important compared to the noise introduced by uncorrected or imperfectly corrected modes.
These results are obtained in a direct mode. Obviously the search for faint structures around a bright object can be much improved by using a coronagraph. For example, Beuzit et al. (1997) used a coronagraph with the Adonis system to study the circumstellar disk around Beta Pictoris. In their observations, they would have been able to detect a companion fainter by 11 magnitude at 1 arcsec and by 13 magnitudes at 2 arcsecs. This is an improvement of 4 magnitudes compared to our results.
The subtraction procedure can be used to carry out photometric measurements. Another possibility is to use a photometric method designed for such cases: point spread function fitting, as for example in DAOPHOT (Stetson 1987). To assess the accuracy of photometry in this case, we took images of the star HD5980 and its calibration star SAO255763 observed in December 1995 in the K band. For HD5980, the integration time was 200 seconds, the signal to noise ratio was 7000 and the Strehl ratio 0.32. For SAO255763, these parameters were respectively 120 seconds, 45000 and 0.35. The delay between the images of the two objects was 20 minutes and the pixel scale 0.050 arcsec. With the image of HD5980, we artificially created an image of a main star and its faint companion situated at a given distance and with a given difference in magnitude. The position of the companion was chosen to avoid the diffraction spikes of the main star and to limit the influence of the other residual features. We also performed the same procedure on images of Betelgeuse and its calibration star HR2076. The integration time for the images of Betelgeuse and HR2076 were respectively 6 and 200 seconds, the signal to noise ratios 25000 and 10000, and the Strehl ratio 0.22 for both objects. The delay between the two observations was 10 minutes and the pixel scale 0.035 arcsec.
We then applied the point spread function fitting algorithm DAOPHOT to these images, providing it with an image of the right calibration star. Comparing the result of DAOPHOT and the known magnitude of the companion then enabled us to work out the error in the magnitude estimation. As a comparison, we also performed the same procedure with other stars but no significant differences appeared in the results.
Figures 6 (click here) and 7 (click here) present results of such investigations as contour plots. They show, for given separations and differences in magnitude, the error which is made in the magnitude estimated by DAOPHOT. Figure 6 was created using an image of Betelgeuse and Fig. 7 using an image of HD5980. For Betelgeuse, we varied the separation from 0.25 to 1.98 arcsec with a step of 0.25 arcsec. In the case of HD5980, this was done for a separation between 0.21 and 2.12 arcsec with a step of 0.21 (the difference is mainly due to the different pixel size). In both cases, we considered differences in magnitude between 0 and 10 with a step of 1.25 magnitude. The contour plots were smoothed to minimise the small irregularities induced by residual features in the PSF.
Figure 6: Error in the photometry of a faint companion as a function
of the distance and the difference in magnitude. The original image was
created with an image of Betelgeuse and DAOPHOT used an image of HR2076.
The contours are 0.001 (1), 0.003 (2),
0.01 (3), 0.03 (4), 0.1 (5), 0.3 (6), 1 (7) and 3 (8) magnitudes
Figure 7: Error in the photometry of a faint companion as a function
of the distance and the difference in magnitude. The original image was
created with an image of HD5980 and DAOPHOT used an image of SAO255763.
The contours are 0.001 (1), 0.003 (2),
0.01 (3), 0.03 (4), 0.1 (5), 0.3 (6), 1 (7) and 3 (8) magnitudes.
Note that the range in separation is not exactly the same as in the
previous figure
Figure 7 (click here) can be considered representative of the performances of DAOPHOT in good conditions. It gives an estimate of the best performance an adaptive optics system can achieve when measuring the light of a faint companion next to a bright star. The figure shows that a very good photometric accuracy (an error of about 0.01) can only be achieved further than about 1 arcsec and for a difference in magnitude not larger than 3 or 4. Good photometric accuracy (an error between 0.01 and 0.1) can only be achieved for a difference in magnitude less than 2 near the star, and for a difference of 6 or 7 magnitudes further than 1 arcsec. Only a poor accuracy (more than 0.1) is available near the main star in most cases and, beyond 1 arcsec, for differences in magnitude larger than 6 or 7.
When comparing the two figures, it can be noted that the results are slightly better in the case of HD5980. This is due to a longer exposure time (200 seconds instead of 6), therefore smaller fluctuations in the halo, and also to a more careful choice of calibration star. But note that the two figures, even though they were obtained in very different conditions and during different runs, are nevertheless very similar, which indicates that they can be considered typical and unlikely to be much improved in similar atmospheric conditions. We carried out a similar procedure using images of HD5980 and SAO255763 obtained in the H-band. The integration times were respectively 200 and 100 seconds, the Strehl ratios 0.13 and 0.15, the signal to noise ratios 5500 and 35000, and the delay between the images 15 minutes. Again the results were very similar, only slightly worse. For a difference in magnitude of 2.5 between the main star and its companion, the errors in the magnitude estimation were respectively 0.15, 0.02 and 0.002 at 0.5, 1 and 2 arcsec, compared to 0.15, 0.015 and 0.001 in the K band for the same couple of objects. For a difference in magnitude of 5, the errors were 0.6, 0.2 and 0.02 in H, compared to 0.6, 0.15 and 0.01 in K. This result confirms that only slight improvement is obtained when the Strehl ratio is higher, at least in the range 0.10 to 0.35. Better results could obviously be obtained for higher Strehl ratios, say above 0.4.