## 5. Error estimationThe 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") 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
© European Southern Observatory (ESO) 2000 Online publication: July 7, 2000 |