To see how well LUMINOUS works and which accuracies can be expected, several simulations were made. The simulations were chosen to resemble an exposure of a crowded stellar field with the Hubble Space Telescope (HST) on a single CCD of the Wide Field Camera (WFC) of the Wide Field Planetary Camera 2 (WFPC2).
Figure 3: 256
256 pixel sub-image of the 800800 pixel
simulated image. The LF for this image is given in Fig. 4 (click here)
Figure 4: Input, detected, derived and smoothed derived LF for the
simulation
An input LF with arbitrary magnitude offset (Fig. 4 (click here)) was used to generate a simulated test image, shown in Fig. 3 (click here). This image was then analysed with DAOPHOT (Stetson 1987) and the completeness of detection was calculated as a function of magnitude. At magnitude 20 the completeness was still near unity, and dropped off to reach zero at magnitude 23, passing 0.5 at 21.5. The LF of the detected stars (detected LF hereafter) is shown in Fig. 4 (click here).
The LF and diffuse background level were derived with LUMINOUS (derived LF hereafter), using the histogram and true PSF of the image. All counts for magnitudes of 20 and higher (up to 24, in 0.5 magnitude increments) could be adjusted.
A first test was done to see if LUMINOUS would work with no other errors than those due to noise. All parameters describing the LF were left to vary freely, but values for the bias level, read-out noise, shape of the PSF, flat field, etc., were unchanged, using the same values as used to create the test image.
Figure 4 (click here) shows that the derived LF was not identical to the input one.
All deviations result from noise
effects and limitations of LUMINOUS. The large fluctuations
in the faint part of the LF were the result of noise combined with the fact
that few histogram bins could be used to derive the number of stars in
these faint magnitude bins. The derived LF fluctuated around the true
LF curve, the residuals being alternately positive and negative. When the
derived LF was smoothed, the result looked
much more acceptable. The goodness of the fit was not significantly affected
(an increase in 
from 
to 
),
since the numbers of stars in adjacent bins were strongly
correlated with negative correlation factors.
The smoothing applied was an interpolation. Each point was replaced by the average of the magnitude of two neighbouring points, with the ordinate the intensity weighted counts of the neighbouring points. The smoothing can be omitted, but the counts in adjacent bins have large correlation coefficients, and the results have to be interpreted accordingly.
The uncertainty in the number of stars grew roughly exponentially with
magnitude. For bright stars the slope was lower, resulting in uncertainty
of the same order of magnitude as the value of the LF, thus defining
a lower limit on the magnitude range for this method. For faint stars,
the uncertainty in the number of stars grew approximately with
, whereas the LF grew with 
.
To investigate the effects of some of the fixed parameters these were changed by a small amount where possible, and the effect on the LF was studied.
The effect of the bias image is to add an "intensity" offset to the individual pixels of the image, without affecting the noise of the intensities of these pixels. A true intensity (like diffuse sky background) would affect the noise. If there are errors in the bias image, the effect on the histogram of the image can be approximated as having two components: a shift in the entire bias level of the image (resulting in an identical shift of the histogram along the intensity axis), and an increase in the read-out noise. The errors in the bias image will effectively cause a widening of the distribution of pixel intensities around the true bias level. Both effects, a change in bias level and a change in read-out noise, are covered below.
Small errors in the bias level are to be expected. The effect is compensated for by the diffuse background level parameter. The additional intensity alters the noise properties of the simulated image (through Poisson noise), which in turn affects the derived number of stars for the faintest magnitudes. An increase of 1 ADU in the bias level caused the derived LF to contain twice the number of stars for magnitude 24.0, and about 70 percent of the number of stars for magnitude 22.5.
An increase in the read-out noise from 5.0 to 5.5 electrons
caused a change in the width of the peak
in the histogram around the bias level. There was no obvious way to take this
into account changing the background level or number
of stars. For the very faintest pixels of the image, the slope in the
histogram was not correct, and could not be corrected. On the bright side
of the peak, corrections could be made changing appropriate parameters.
was not as low as without the read-out noise error.
The effects on the derived LF were similar as for the error in the bias level.
The code to generate the simulated image uses a constant radius for the PSF. In realistic cases, the PSF is extracted from the image, and may contain undetected faint stars. A large PSF radius has the advantage of taking into account the wings of the PSF, including relatively bright diffraction spikes, but has the disadvantage of including undetectable stars, which can affect the fainter part of the derived LF. Figure 5 (click here) shows the derived LFs, with PSFs determined from the image, defined over a radius of 15 pixels (approximately 10 times the full width at half maximum, FWHM, of the PSF) and 4.5 pixels. For the PSF with large radius, there is a significant dip around magnitude 22, and the total number of stars is underestimated over a large range of magnitudes.
Figure 5: Input and smoothed derived LFs for PSFs with radii 4.5 and 15 pixels,
derived from the image. The FWHM of the PSF is approximately 1.5 pixels
With a smaller PSF radius, the number of undetected stars in the PSF decreased, but so did the influence of the wings. Two effects could be expected: a decrease of the derived number of faint stars due to extra stars in the PSF, and an increase of faint stars to account for missing intensity in the truncated wings of brighter stars. For a PSF radius of 4.5 pixels, the derived LF shows a slight increase in slope compared to the input LF, but the derived number of stars is more realistic than for the PSF with the large radius.
Analog to digital converter effects are normally not particularly important for photometry. The effects are very small deviations from linearity between electrons in the CCD and measured ADU. For the histogram, the effects become more important. The number of pixels in one bin may be over-represented, while under-represented in an adjacent bin. The ADC effects have a similar influence on the histogram as flat field effects have on a CCD image. An error in the ADC effects can show up in the derived LF as a large gradient. The error can partially be eliminated by a change in the number of stars for two adjacent magnitude bins, increasing the one, while decreasing the other. The fit to the histogram of the observed image will be improved at the expense of a systematic error in the derived LF. This can also happen for relatively bright stars, depending on where exactly in the histogram the largest parts of the ADC effects are located.
An error in the ADC effects was introduced by shifting the
entire observed histogram along the intensity axis, and adding the same
intensity to the bias level in the model.
Figure 6 (click here) shows two smoothed LFs, derived with the ADC effects error in
the model. The best values for were significantly higher than
for the previous simulations, indicating a deteriorated
fit to the histogram. There were at least two local minima in ,
with about equal values, represented by the two curves in the figure.
The uncertainties in the derived numbers of stars are large, even for stars of
magnitude 20.
Figure 6: Input and derived LFs for a simulation with errors in the ADC
effects. The two curves for the derived LF are two solutions with very similar
The LF in the simulation above increased linearly in a logarithmic plot. Simulations with a turned-down and a turned-up LF were made to see if LUMINOUS could derive the LF also in these cases. The results for the simulations are given in Fig. 7 (click here).
For the turned-down LF, the general appearance of the smoothed derived LF seems correct, with the exception of some small fluctuations for the highest magnitudes. The other simulation, with the turned-up LF and higher crowding than the one with the straight LF, showed some larger fluctuations around magnitude 21. The smoothed derived LF followed the general features of the input LF reasonably well for the faintest magnitudes.
Figure 7: LFs for simulated images with a steeper or lower slope in the
faint part of the LF
Both spatial (pixel) and intensity (gain) sampling were poor for the simulation discussed above. To see how the sampling affects the results, simulations were made with changed spatial and intensity resolution, again using the LF from Fig. 4 (click here).
Figure 8: Input, detected and derived LF for an image with
a PSF with large FWHM, resulting in extreme crowding. Estimates for the
uncertainties in the derived LF are indicated by the error bars, for solutions
with 
For ground based images of crowded fields the main limitation for detection of fainter stars is the confusion problem, not the signal to noise ratio with respect to the background noise. Figure 8 (click here) shows the LFs derived from a simulated image with a well sampled PSF (FWHM 11 pixels), together with the LF of the stars detected with DAOPHOT. Changes of crowding were made through increment of the FWHM of the PSF, while leaving the positions and relative intensities of the stars unchanged. The change in the PSF did not conserve flux. The stars in the simulation with the wider PSF had higher intensity, and thus, with all other parameters unchanged, a higher signal to noise ratio. The incompleteness of detection was strictly due to crowding effects. The uncertainty in the number of stars in each magnitude bin could, just as in the previous simulations, be approximated with an exponential function of the magnitude.
As seen from Fig. 8 (click here), the uncertainty in the derived LF became larger than the derived value for magnitude 22.0 and higher. A change in the number of stars for these magnitudes could not be discerned from a change in the diffuse background level.
The intensity sampling of the simulated image was changed from 7 electrons per ADU to 1.7. When expressed in electrons, the intensities of the stars and amplitude of the read-out noise were unchanged. The derived LF was smoother in appearance than for the simulation with gain 7. When smoothed, the derived LF was very close to the input LF. The estimated uncertainties in the numbers of stars in the derived LF were similar to the uncertainties for the simulation with gain 7.