To establish the accuracy that can be attained by the fits for all the relevant parameters, we have tested our code on a large set of simulated galaxies. We have chosen model distributions spanning the same range in effective radius and total flux - in terms of instrumental units - as our sample galaxies, and a range of values (from 1 to 5) for the shape index. Since most of the objects considered have been imaged with the wide field camera of WFPC2, we have convolved the test distributions with a typical wide-field PSF. In this case, the plate-scale is the one which mostly undersamples the PSF itself, yielding the worst conditions for the retrieval of the correct parameters' value. We expect therefore the uncertainties derived from our simulations to be basically correct for the WFPC2, and conservative estimates for the other types of data: a few simulations carried out with the characteristic sampling of the Planetary Camera and of NICMOS show that this is indeed the case.
Noise has been added to the simulations at a level typical for our data, both on the pixel scale and at smaller spatial frequencies. The resulting images have been finally fitted using our code, allowing for some error in the estimate of the PSF and using different trial values for n, as in the case of the real galaxies.
In Fig. 4 we show the region covered by the synthetic objects in the - plane, represented in instrumental units (pixels and counts per pixel respectively), one panel for each value of n, from 1 to 4. The dotted contours map the , evaluated theoretically, considering the total flux inside the isophotal radius at on the single pixel. These values do not account for the effect of the PSF, and are therefore less reliable at low surface brightness levels and for sizes of the order of the PSF width (say, ). For this reason, the detection limits for our data have been deduced a posteriori from the simulations, and are represented by the dashed lines in the lower right corners of the plots. Such limits appear to depend on n; this is partly explained by the fact that, at fixed effective radius and surface brightness, there is a slight increase of with increasing n. However, the greatest contribution (about 80%) is due to the steeper central peak that characterizes the large-n distributions, making them more visible in the background noise.
5.1. Disentangling the and distributions
A first check can be carried out assuming that all elliptical galaxies are characterized by a de Vaucouleurs brightness distribution (n = 4); we can consider this as a first order classification, since nearby ellipticals show in fact a variety of shapes, that can be quantified by different values of the exponential index n ranging from about 2 to 10 and higher (see for example Caon et al. 1993, or Khosroshahi et al. 2000). If we restrict our test to the simulations with and only, we find that the correct value of n can be retrieved, on the basis of a estimator, for the whole parameter space explored. Such a simplified classification, therefore, is possible for all of our objects.
5.2. Systematic errors
In the following step we assigned to every galaxy its best-fit n value, choosing from 1 to 6, and compare the derived parameters with the original ones. We can start our analysis of the results by looking for systematic trends. Starting with our estimates for the index n, if we plot the average measured values vs. the true ones (Fig. 5), we actually observe a tendence for the higher n's to be underestimated; in particular we find
This effect depends only slightly on the choice of the PSF; it is probably due to the fact that, increasing n at fixed total flux, the peak of the distribution gets sharper, and its wings fainter and wider, but these differences tend to be concealed by the effect of the PSF on one hand, and by the backgroud noise on the other.
Similar trends can be investigated also for and : we find that, for both quantities, small values (i.e. those approaching respectively the pixel scale and the background noise level) tend to be slightly overestimated, and large values tend to be underestimated. The behaviour is very similar for all the n values, so that we can adopt average corrections:
Since these latter corrections are significant at a 3 level, whereas Eq. 1 is significant only at 1 , and since applying Eq. 1 to the estimated n's would lead to non-integer values for this parameter, we choose to correct only and and leave n unchanged, keeping in mind that n values greater than two might be somewhat underestimated.
5.3. Mapping the parameter space
We turn now to examine how accurately the relevant parameters are retrieved in the various regions of the parameters' space, starting with the shape index n. Fig. 6 shows again the - plane; in this plot the dots in each panel represent the estimated location of our simulated galaxies for the different values of n. The accuracy with which n can be retrieved - without applying Eq. 1 - is quantified by the size of each dot, as explained in the caption.
We find that the correct value of n is retrieved in most cases for the and models; the error for the and ones is more typically 1 in large portions of the plane, partly due to the systematic effect described previously. As a consequence, exponentials are almost always recognized as such, so that if the best fit is for , the distribution is certainly non-exponential. As expected, for all values of n, low flux and low surface brightness objects tend to be affected by larger errors. The main conclusion, however, is that relying on these results we can define a region (the one above the dotted line) where exponential distributions can be reliably distinguished from the others: this is the locus where both exponentials and distributions are recognized as such, and larger n distributions are affected at most by an error of 1. A comparison with Fig. 4 shows that the limit roughly spans values between 10 and 80. We have checked this result using the theoretical approach described in Avni (1976): when one or more parameters are evaluated via a minimization, the method allows to assign a confidence level to each parameter relying on the variations of the around the minimum in the parameter space. Although the computations are exact only in the case of linear fits, the method provides anyway a useful check on our findings; indeed, we find that our estimates for the uncertainty of n are broadly consistent with the ones evaluated theoretically for a 90% confidence level. In particular, the Avni method confirms that in the area above the dotted line, exponential and non-exponential distributions can be reliably distinguished.
A mapping of the parameter space, analogous to the one plotted in Fig. 6, has been produced also to estimate the uncertainties on and , corrected according to Eqs. 2 and 3. An intersting result is that the derived errors are relatively independent of the estimate of n, in the sense that a wrong estimate of the shape index does not necessarily mean larger errors for and . Most likely, whereas the choice of n is influenced mainly by the accuracy of the PSF, the estimates of and are more strictly related to the quality of the fit a whole; in other words, if a wrong n may compensate for the effect of a wrong PSF, a good estimate for and can be achieved anyway, as long as the quality of the fit is good.
For what concerns the values of the ellipticity and position angle (that are fixed a priori), we find that the typical errors associated to their estimates do not affect significantly the accuracy of the output parameters (center coordinates, effective radius and surface brightness), nor the choice of the best n value. We estimate the typical errors on the ellipticity to be around 0.1, and from 5 to 10 degrees for the position angles. The center coordinates are usually determined with great accuracy ( pixels): due to the asymmetries in the psf, this is better than what can be achieved by fitting an ellipse-averaged profile to the galaxies.
To summarize, we have tested our fitting code on a large set of simulated galaxies, and assessed the accuracy that can be attained for the various relevant parameters in the region of the parameters' space covered by the real data. In the next section, the results presented so far will be used to estimate the errors on the parameters derived for each galaxy.
© European Southern Observatory (ESO) 2000
Online publication: December 15, 2000