5. Error estimation
The standard deviation of the mean off-center distance within the adopted surface brightness range, of the order of 0.1", is of course only a lower limit to the true uncertainty of any measured value. The adopted procedure, when applied to a noisy galaxy image, is likely producing large systematic and random errors in . To get a handle on these errors, we have run Monte Carlo simulations of the measuring procedure with artificial galaxies. We have chosen a typical dE with = 16 mag and ellipticity 0.25 (between E2 and E3) and formed a purely exponential model galaxy from its measured exponential scale-length and central surface brightness as given in BC93. Having added a suitable sky background and Gaussian nucleus at a certain off-center distance (to be varied), a Poissonian pixel-to-pixel noise was mimicked by a Gaussian sigma weighted with the square root of the local total intensity, and then again added to the artificial image.
Using that Gaussian sigma as a probability distribution, a single artificial galaxy was produced, pixel by pixel, with a random number generator. Then, we determined its nuclear off-center distance exactly in the same way as with a real galaxy, i.e. according to the procedures described in Sect. 3 (including a convolution of the model galaxy with a Gaussian seeing function and 55 smoothing). For a given (true, input) displacement of the nucleus from the center of the underlying (exponential) galaxy, this application was repeated for 500 representations of the galaxy, and a mean (output) and its associated standard deviation were calculated. The results of such a Monte Carlo run are shown in Fig. 3, where we have varied the input nuclear offset - along the major axis in this case - in steps of 0.2".
The main characteristics of this simulation are as follows. Exactly central and nearly central (up to about 0.5") true nuclear positions, when measured with our procedure, appear systematically larger, i.e. displaced from the center. The more central a nucleus truly is, the more displaced it appears. There is a minimal apparent displacement of 0.5". This result of image noise had to be expected. In the process truly displaced nuclei can get more or less displaced, but truly central nuclei can get only more displaced. On the other hand, large true nuclear displacements ( 1") get systematically, but insignificantly too small. The random errors for a given single case, shown as error bars in Fig. 3, are quite large - again as expected -: from 0.22" at = 0" up to 0.4" at larger displacements (note that the errors of the means would be times smaller).
We have of course also varied the total magnitude, and hence surface brightness and exponential scale length (see BC91) of the artificial galaxy, but found rather little variance in the Monte Carlo results, except that for bright dwarfs with = 14 mag (which are in the minority, however), both random and systematic errors are significantly smaller. Also, displacing the nucleus along the minor, rather than the major axis, as well as altering the ellipticity of the galaxy made little difference. As there is a degeneracy of for small true nuclear displacements, and in view of the generally large random errors, any correction of the measured for an individual galaxy is infeasible. We therefore, in the following, take resort to a very rough and global, statistical accounting for the uncertainties in , drawing on the results for an artificial typical dE,N ( = 16 mag, ellipticity E2.5) shown in Fig. 3.
© European Southern Observatory (ESO) 2000
Online publication: July 7, 2000