Before dealing with the consequences of all the problems presented
above, we first present the benefits of using adaptive optics
in two illustrative examples.
The theoretical advantages of using an adaptive optics system
in order to perform
photometry are clear. For isolated objects, the increase in Strehl
ratio means a much better contrast against the background noise, and
the higher concentration of flux allows a reduction in the
size of the aperture used for measurement, which means a significant
reduction of noise.
For crowded fields, the use of adaptive
optics has a further advantage: by sharpening the image, it reduces
confusion and separates close sources which could otherwise have been
considered as single objects by the point spread function fitting
algorithm. In this section, we give an illustration
of the theoretical benefits of adaptive optics in two such cases,
a single star
observed against the sky background and a small star cluster,
without taking into account any of the sources of noise described before.
We use the simulated PSF constructed using the method described
in Wilson & Jenkins (1995) and kindly provided by the authors.
The PSFs are obtained on-axis for three values of D/r0: 10, 7.5 and
5, corresponding approximately to the J, H and K bands on
a 4-meter telescope in good seeing conditions
(0.75 arcsec seeing at 
m).
We first simulated a single star observed against the sky background
in the three bands J, H and K. In each band, we considered three cases:
an uncorrected image, an image obtained with only tip/tilt correction,
and an image with an adaptive optics correction of 20 modes. The value
of the noise in each band was chosen to yield a signal to noise ratio
of 2 for the peak for the uncorrected image.
The IRAF/NOAO APPHOT aperture photometry package
was then applied to each image in order to
work out the flux contained in a given aperture. The
radius of this aperture was chosen to be 1.22 times the Full Width
at Half Maximum of each image.
In the diffraction-limited case, this corresponds to 1.22
, i.e. the radius of the first dark ring.
This choice of aperture is not optimised
but simply serves as an illustrative example.
| Correction | J | H | K |
| No correction | 0.05 | 0.05 | 0.05 |
| Tip/tilt | 0.015 | 0.010 | 0.010 |
| 20 modes | 0.002 | 0.003 | 0.005 |
|
|
For each band and each level of correction, we created a set of 10 images with a different statistical realisation of the background noise. We computed the flux of the star in the chosen aperture for each image, and finally worked out the rms variations of the corresponding magnitudes, which gave us an estimate of the error on the magnitude. The results are presented in Table 1 (click here), which gives the uncertainty on the magnitude in each band and for each level of correction. In the case of no applied correction, the uncertainty is approximately the same whatever the band because we set the same signal to noise for each uncorrected image. When a tip/tilt correction is applied, the signal to noise ratio of the peak is multiplied by 2.6 in J, 3.7 in H and 5.3 in K. This leads to a decrease in the uncertainty on the magnitude by a factor 2 or so in J and about 3 in H and K. When a correction of 20 modes is applied, the signal to noise is multiplied by 40 in J, 33 in H and 40 in K (compared to the uncorrected case). The uncertainty of the magnitude is then divided by 25 in J, 17 in H and 10 in K, the relative improvement being better at shorter wavelength because the increase in signal to noise is greater. This example clearly shows that the use of adaptive optics on isolated faint objects theoretically leads to a huge improvement in the accuracy of photometry, by at least a factor 10, thanks to a better contrast and a higher concentration of the flux.
In order to illustrate the advantages of adaptive optics in the very
usual case of a crowded field, for instance a star cluster,
we created artificial images of a cluster containing 15 objects:
one bright star whose magnitude was taken as a reference, 2 stars
1.25 magnitudes fainter, 4 stars 2.5 magnitudes fainter and 8 star
3.75 magnitudes fainter. One image was created with an uncorrected
H-band PSF (Strehl 0.016),
one with a tip/tilt-corrected H-band PSF (Strehl 0.061) and one
with a corrected H-band PSF (20 modes, Strehl 0.55).
The 15 stars were scattered on a area of 3 
3 arcsec2,
which yielded a density of about 1.7 stars per arcsec2.
Some gaussian noise was added on the three images with a level
chosen to be 5 magnitudes fainter than the peak of the brightest star
in the uncorrected case. Figure 1 (click here) presents the three images,
uncorrected on the left, tip/tilt corrected in the middle and
and fully corrected on the right.
In the corrected images, all the components are clearly visible, whereas
in the uncorrected one, the faintest components can only be identified after
a subtraction of the brightest stars. This clearly has consequences for
the precision of photometry.
Figure 1: Uncorrected, tip/tilt corrected and fully
corrected images of a star cluster created with simulated
PSFs in the H-band were used. A logarithmic scale is used to show the
faintest stars. The use of adaptive optics clearly brings a huge improvement,
both for detection of objects and for photometry
To assess the improvement in photometric accuracy in this case, we applied the IRAF/NOAO DAOPHOT point spread function fitting package to the three images to work out the magnitude of each component of the cluster. The PSF was supposed to be perfectly known. Comparing the results to the actual magnitudes in the original artificial image, we were able to work out the error on each measurement. We then computed the rms error for each group of star with a given flux in order to obtain a final value: the average uncertainty on the flux of a star with a given difference in magnitude relative to the brightest object of the field. The results are presented in Table 2 (click here). In this case again, the use of adaptive optics leads to a huge increase in precision. For the three differences in magnitude considered, the improvement in precision between the uncorrected and the fully corrected images varies between 15 and 35 and very accurate results can theoretically be obtained. For instance, for a difference in magnitude of 3.75, the uncertainty in the uncorrected image makes the measured values virtually useless, whereas the corrected image yields very useful results with a precision better than 5 per cent. Here again, thanks to the increase in contrast and the better concentration of energy, adaptive optics theoretically leads to much more accurate photometric results.
| Correction |  |  |  |
| No correction | 0.07 | 0.08 | 0.65 |
| Tip/tilt | 0.005 | 0.025 | 0.07 |
| 20 modes | 0.002 | 0.003 | 0.04 |
|
| |||
The problem is that the two situations in this section have been simulated without taking into account the sources of noise introduced in Sect. 2, and these results must therefore be considered optimistic. The rest of this article is an assessment of the influence of each source of noise on photometric accuracy, and it will show that the improvements illustrated above cannot be fully achieved in reality.