A more detailed understanding of the issue requires a more complete simulation
taking into account the surface brightness radial profiles for E and D
galaxies as they were modelled in Chapter 4 and the different
possible positions of the galaxy center with respect to the aperture center.
Numerically integrating the profiles of E and D galaxies of different
magnitudes over square areas of different sizes and whose centers are
randomly displaced from the galaxy center, one can obtain a clearer picture
of which galaxies will be detected and with which probability.
The percentages of detected galaxies obtained from these simulations are given
as function of total magnitude and aperture size in Tables 5.4
and 5.5 for E and D galaxies, respectively, where a of 4
has been assumed to indicate a detection.
As expected, the number of detected galaxies increases steadily with the decrease in the aperture size both for Es and Ds. Below a certain aperture size, however, this estimation becomes rather uncertain due to the present poor knowledge of brightness profiles in the galaxy innermost regions. Besides, since some kind of median filtering will be required in order to discriminate between bright stars and galaxies, the measured signal will in fact be smaller than estimated, and this effect will become significant as the aperture size decreases. In any case, a lower limit to the aperture size must be set depending on the number of false detections that are deemed acceptable.
The thorough understanding of the problems connected with false detections requires the design, implementation and testing on real fields of a dedicated algorithm for galaxy detection, but this was beyond the scope of the present study. It is however believed that an aperture size of arcsec is large enough to be safely used in the following considerations. With such a choice, E and D galaxies of, e.g., are detected with a probability of about and respectively, whereas for the detection probability quickly falls to zero. With an average number of scans of 75 per astrometric instrument (see Figure 2.4), should galaxies be observed for the whole mission in one Astro, an average of 60 and 45 scans would be obtained for Es and Ds, respectively, which, as we shall see in Chapter 7, are largely sufficient to reconstruct a high-resolution two-dimensional image. It can therefore be concluded that galaxies brighter than would be detected during the 60% of the scans with a in the ASM1 using an area of arcsec for detection. According to Table 4.4, there are about 4 million galaxies brighter than this limit on the whole sky. At low Galactic latitudes, though, galaxy detection becomes increasingly tricky due to the presence of Galactic nebulae and lots of stars. While it would be desirable to observe galaxies as well as Galactic nebulae down to very low Galactic latitudes, it is suspected that this could yield a large amount of false detections and thus loss of telemetry. Very conservatively, the galaxy detection could be carried out only when , i.e. over 75% of the sky, thus leaving a total of 3 million observable galaxies. It must also be noted that the readnoise appears to dominate the total error budget, and that therefore the adoption of a larger sample size in the ASM1 could result in a much higher number of detected galaxies.
A typical, intrinsically bright, galaxy from the ``Third Reference Catalogue of Bright Galaxies'' ([de Vaucouleurs et al. 1991], RC3 in the following) has an absolute magnitude of , according to Figure 2 in [Impey and Bothun 1997], which using the average color index obtained for bright galaxies by [Prugniel and Héraudeau 1998] gives . Under this conservative assumption, for mag we obtain a distance modulus of mag and therefore a distance of Mpc or a redshift of , using for the Hubble constant the value of Km/sMpc recently obtained by [Mould et al. 2000]. Clearly, since the RC3 only contains galaxies brighter than , i.e. on average , most detected galaxies will be intrinsically fainter, and therefore lie correspondingly farther, than assumed above, thus increasing the horizon of galaxy observations.
These numbers were derived under a number of assumptions, both optimistic and pessimistic. On the whole this gives a rather large uncertainty on the final numbers, but probably not more than a factor of two in each direction, i.e. from 1.5 to 6 million galaxies. Note, however, that following a different line of reasoning based on the analysis of several HST MDS fields, [Lindegren 2000] obtained a total number of 6 million detected galaxies, with an estimated uncertainty of about three in each direction. This remarkable agreement between estimations obtained with different methods confirms the reliability of the combination of statistical modelling and numerical siulation in mission planning.