Aperture photometry involves measuring the amount of light contained in a given aperture for each object. The same procedure is usually applied on the science objects and on a few photometric standards. In most cases, the science objects and the standards have to be observed at different times. This introduces two problems. Because of the constantly changing PSF of an adaptive optics system, the fraction of light contained in a given aperture also varies, which introduces an error in the calibration procedure. Note that this is a drawback because in order for adaptive optics to produce a gain, the size of the aperture has to be reduced. The second problem is also related to PSF variations, but this time because of a different correction for the science objects and the photometric standards. The difference is essentially created by a different level of noise in the wavefront sensor and can be minimised by carefully selecting the brightness of the standard stars and their spectral type, even if some slight differences caused by their colour or brightness might remain. Tessier (1995) showed that when the different objects are carefully matched, the main source of variations in the PSF is seeing fluctuations. In our assessment of the accuracy of aperture photometry, we therefore assumed that the second source of error was negligible compared to the first one - but note that this does require careful matching.
In order to estimate the errors induced by PSF variations on
aperture photometry, we analysed several sets of data
of Betelgeuse and massive stars using the IRAF/NOAO APPHOT package.
We considered sets of 4 or 5 successive frames of single
stars taken in the H and K bands or in a narrow band filter at
m during these runs. The
individual observations had integration time varying from 1 to 90 seconds.
For each set of frames, we worked out the rms variation of the
magnitude measured in a given aperture.
To define the sizes of these apertures, we decided to use areas
defined by the three first dark rings of the unobstructed
diffraction limit of the
telescope (even if the images were not actually diffraction-limited).
The corresponding sizes were given by Born & Wolf (1970):
In K for example,
this corresponded to aperture diameters of 0.31, 0.56 and 0.82 arcsecs.
Figure 2: Rms fluctuations of the magnitude measured in a given aperture
as a function of the integration time of a single frame.
The size of the aperture is defined by the second dark ring
of the diffraction limit of the telescope. Crosses are data
taken in the H band and diamonds data taken in the K band.
The data were obtained during four different runs with ADONIS
Figure 2 (click here) presents the results in the case of an aperture defined by the second dark ring. The rms variation of the magnitude measured in an aperture is given as a function of the integration time for one frame. The pixel size was either 35 or 50 milliarcsec and the four observing runs gave results consistent with each other (for example, for an integration time of 1 minute, different data from the four runs all yielded fluctuations of 0.015 magnitude or so). The signal to noise ratios of the images (defined by the peak of the PSF and the background noise throughout this article) varied between 300 and 45000, with most of them above 7000. In H, the Strehl ratios were between 0.05 and 0.15, and in K, between 0.15 and 0.35. Each rms fluctuation of the flux was computed from measurements on 4 or 5 successive images. We verified in each case that the error due to photon or background noise was clearly smaller than the total error (it always amounted to less than 0.01 magnitude).
Figure 2 (click here) shows that globally the rms variations
behave like 1/
, where T is the integration time.
Such a behaviour can be expected if the fluctuations
of the magnitude are completely uncorrelated from one frame to another.
The data globally follow this trend even if the spread is quite large and
some points stand out. The spread is not surprising as parameters other
than the integration time (such as the seeing fluctuations or the
brightness of the guide star) are also important.
Figure 2 (click here) gives an estimate of the precision of aperture photometry when the science objects and the photometric standards have to be observed at different times. Note that the results apply only to near-infrared observations, as they depend on the rate of change of the Strehl upon the coherence length of the atmosphere, and therefore on the wavelength. For exposure times between 20 and 90 seconds, seeing fluctuations impose a limit to the accuracy of photometry at a level of about 0.01 or 0.02 magnitudes. For exposure times of a few seconds, this limit increases to 0.04 magnitude and more. We also tried to work out the magnitude variations for longer exposure times. These attempts showed that the variations tend to get larger than 0.02 magnitudes rather than smaller. This can be explained by the fact that for long exposure times, the amount of light in an aperture is determined by the slow variation in the atmosphere turbulence and the successive magnitude estimates are no longer uncorrelated. The fluctuations are then controlled by the amplitude of the slow variations of the seeing and increase with the exposure time. A first conclusion that can be drawn from Fig. 2 (click here) is the fact that aperture photometry measurements with adaptive optics are unlikely to yield results more precise than 0.01 or 0.02 magnitude. The theoretical accuracies obtained in the previous section for a 20-mode correction are, for instance, far too optimistic. Nevertheless, this limitation still allows much better precision than in the seeing-limited case.
An important parameter which determines the amplitude of the fluctuations is the size of the chosen aperture. The larger the aperture, the smaller the variations. In order to estimate the influence of the aperture size, we analysed a set of 8 frames from our observations of Betelgeuse. The integration time was 6 seconds and the observations were spread over a little more than 1 hour. The signal to noise ratios varied between 25000 and 30000, with one exception at 17000. The Strehl ratios were between 0.20 and 0.26, with the same exception at only 0.14, and the average value was 0.22. In each frame we measured the Strehl ratio and the flux contained in three different apertures, defined as before by the three first dark rings.
Figure 3: Variation in magnitude as a function of the variation in Strehl ratio.
All values are given relative to the averages on the whole set of data.
Three different aperture are show: one defined by the first dark ring
(diamonds), the second dark ring (crosses) and the third one (squares).
The data were taken in the K band with an exposure time of 6 seconds
and a pixel size of 0.035 arcsec. The average Strehl was 0.22
Figure 3 (click here) presents the result. The magnitude computed in
the three different apertures are given as a function of the Strehl
ratio in the frame.
Magnitudes are given relative to the average magnitude for each aperture
and Strehl ratios relative to the average Strehl ratio.
Figure 3
gives variations in flux induced by fluctuations in Strehl ratio.
Parameters like the exposure time or the brightness of the object
do not influence these variations.
Figure 3 (click here) is therefore very general and will be the
same for any other observation (at least for a similar range of
Strehl ratios and a similar PSF shape). The line denoted Strehl variations
represents a case where the variation in flux is exactly
equal to the variation in Strehl ratio. Its equation is:
Figure 3 (click here) shows that the actual variations are smaller,
which could be expected as the use of a large aperture attenuates
the fluctuations. It also shows that the variation in flux for an aperture
is approximately proportional to the variation in Strehl.
We could plot three lines which approximately fitted the data for the
three different apertures. Each line has the following equation:
where 
is a factor depending on the size of the aperture.
The values of that best fit the data are 0.61 for the first dark
ring, 0.55 for the second and 0.45 for the third, with an uncertainty
of about 0.02. Note that another frame with a poor Strehl ratio
(
about 0.64, not shown on the figure) also fitted these
lines very well.
Figure 3 (click here) gives a way of estimating the influence of
Strehl ratio or seeing fluctuations on the variations of the
magnitude measured in an aperture. It therefore makes a correction
possible from one frame to another.
But given that some data do not fit the three lines well,
an uncertainty remains in this procedure, of
about 0.02 magnitude in this particular case.
Moreover, to correct for the Strehl ratio effect, a way to measure
this parameter is needed: the Strehl ratio cannot be measured in the
usual way from the images, because this method requires an absolute
measurement of the total flux, which we do not have.
From the estimates of the factor , we can also assess
the improvement brought by using a larger aperture.
First the use of an aperture defined by the first dark ring decreases
the fluctuations of the magnitude of 40 per cent compared to pure Strehl
variations. When going from the first to the second dark ring, magnitude
variations still decrease by 10 per cent or so. When going to the
third dark ring, the variations still decrease by about 25 per cent.
Finally, we also tried to apply point spread function fitting. Of course, this method normally applies to crowded fields,
but it was nevertheless interesting
to assess its behaviour. The results showed that
the magnitude variations could also be approximately
described by Eq. (3), even though the fit was
less accurate than before. The coefficient was found to be
0.77 or so, well above the values for aperture photometry,
vitiating the use of the method in the case of a single source in an
uncrowded field.
One possible way to reduce the fluctuations between different frames would be to perform a real-time selection during the observations. For example, during each exposure, the Strehl ratio or another parameter could be checked at a very fast rate, say every hundredth of a second, and a fast shutter would close the camera when this parameter is beyond a threshold value. Different selection criteria could be applied. For example, only the Strehl ratios above a given level can be taken into account. Or, in order to favour consistency, only the ones near an average value could be chosen. Obviously, such a procedure should only be applied to bright objects, when seeing fluctuations dominate the effects of photon or read-out noise.
We assessed these two possibilities using a set of 300 20-ms exposures of Betelgeuse. We considered 5 sets of 60 frames in order to simulate 5 successive 1.2-second frames. We then applied a selection criterion to the frames. In one case, we rejected all the frames with a Strehl ratio below a certain level, in the other one, we rejected all the frames with a Strehl ratio too far from the average value (on the 300 frames). We could then compute an average image for each 1.2 s-exposure frame and work out the rms variation of the magnitude measured in these 5 frames. We considered 3 different levels of selection: rejection of 50, 75 or 150 frames, corresponding to one frame out of 6, one out of 4 or half the frames. The results are presented in Table 3.
| Selection | Best Strehl sel. | Average Strehl sel. |
| 1 out of 6 | 0.038 | 0.041 |
| 1 out of 4 | 0.033 | 0.036 |
| 1 out of 2 | 0.025 | 0.023 |
|
|
Table 3 (click here) shows that such a selection can be quite effective in reducing magnitude fluctuations. For example, keeping the shutter open during only half the exposure time reduces the magnitude fluctuations by about 60 per cent. In our simulations, we kept a constant integration time without taking into account the time during which the shutter was actually open, which leads to a variable effective integration time. For instance, when we rejected an average of half the frames, the number of frames actually used in each 1.2-second exposure varied between 12 and 47 (instead of the theoretical average value of 30). A better way to perform the selection would be to consider a variable integration time, dependent on the variations of the Strehl ratio, in order to obtain a fixed effective integration time. This would not greatly affect the level of magnitude fluctuation and yield a coherent set of images with the same effective integration times and signal to noise ratios.
The second problem as far as aperture photometry is concerned is angular anisoplanatism. Because atmospheric phase distortions depend on the direction in the sky, the PSF varies over the field of view. Basically, the further from the centre of the image, the lower the Strehl ratio and the larger the FWHM. This means that the fraction of light contained in a given aperture is going to vary and that the brightness of the objects far from the centre will be underestimated.
To assess the consequences of anisoplanatism on accurate photometry, we
used simulated point spread functions constructed using the method
described in Wilson & Jenkins (1995) and kindly provided by the authors.
We simulated PSFs corresponding to different
distance from the reference star (0, 5, 10, 15 and 20 arcsec).
For each PSF, we applied the IRAF/NOAO APPHOT package to
work out the flux contained in the three different
apertures defined as before by the first dark rings.
We performed the operations for three values of D/r0: 5, 7.5 and
10. These correspond approximately to the K, H and J bands on
a 4-meter telescope in good seeing conditions
(0.75 arcsec seeing at 
m).
The variation in magnitude due to anisoplanatism is given in
Table 4 for the three bands and the four separations.
All results are relative to the magnitude of an object placed at the
centre of the image (the guide star).
| Band | Ring | 5'' | 10'' | 15'' | 20'' | |
| 1 | 0.035 | 0.130 | 0.25 | 0.44 | ||
| J | 2 | 0.006 | 0.020 | 0.04 | 0.10 | |
| 3 | 0.006 | 0.023 | 0.04 | 0.08 | ||
| 1 | 0.022 | 0.082 | 0.16 | 0.29 | ||
| H | 2 | 0.002 | 0.015 | 0.03 | 0.06 | |
| 3 | 0.003 | 0.013 | 0.02 | 0.04 | ||
| 1 | 0.012 | 0.045 | 0.09 | 0.16 | ||
| K | 2 | 0.002 | 0.008 | 0.02 | 0.03 | |
| 3 | 0.001 | 0.005 | 0.01 | 0.02 | ||
|
|
Figure 4: The effect of anisoplanatism on aperture photometry.
The error on the estimated magnitude is given as a function of the angular
separation in the J, H and K bands and for apertures defined by the
second and the third dark ring
Table 4 (click here) shows that the variations in magnitude are much higher when the size of the aperture is defined by the first dark ring, rather than the other ones. The reason for such a difference is easy to understand: the effects of anisoplanatism consist not only in a decrease of the Strehl ratio with the distance to the centre but also a distortion and a broadening of the core of the PSF. When an object is far from the centre, a significant part of the core spreads beyond the first dark ring, thus the large variations in the magnitude. For larger apertures, this effect has no significant consequence and the variations are much smaller. Globally, this means that the use of the smallest aperture should be avoided in most cases, except maybe in the H and K band for a small field of view.
Figure 4 (click here) presents the variations of the magnitude as a function of the angular separation for apertures defined by the second and the third dark ring. The curves for an aperture limited to the first dark ring have the same shape but at a higher level. Figure 4 allows an estimate of the variations due to anisoplanatism for apertures defined by the second and third dark ring. Of course, the variations in a real case depend on several factors like the seeing, the number and strengths of turbulent layers, the central obscuration of the telescope and other parameters of the adaptive optics system. For this reason, Fig. 4 cannot be used to compensate accurately for the effects of anisoplanatism. But it can still be used to illustrate the behaviour of the error. Basically the variations in magnitude are quite small near the centre of the image and start to increase to significant levels at a distance which depends on the band and the chosen aperture. As an example in our case, for an aperture defined by the second dark ring, the variations reach a level of 0.02 magnitude at 9 arcsec in J, 12 arcsec in H and 16 arcsec in K.