figure 1: 400 spectra on detector figure 2: 2000 spectra on detector
figure 3: number of observable spectra vs. number of objects within the detector area.
The number of spectra which can be observed for a given pixel size and magnitude limit is shown in the middle panel of the figure. For each magnitude limit, the number of target galaxies was computed for each pixel size using the model shown in figure 4. In this, a detector size of 2048 x 2048 pixels was assumed. The number of observable spectra was then computed from the curve shown in figure 3. The middle panel of the figure uses different colors for the used magnitude limits. For fainter magnitudes, where the galaxy density is higher, the maximum number of observable spectra is reached even with relatively small field of view, i.e. small pixel size. By contrast, for the less dense brighter galaxy, larger pixel sizes increase the number of observable spectra.
If s/n is e.g. decreased by a factor of 2 due to the large pixel size, one would need to increase the exposure time by 2^2=4. This is only acceptable if the number of galaxies which can be observed is increased by the same factor of 4. The merit of varying the pixel size can therefore described by the number (s/n)*sqrt(n_obs), where n_obs is the number of observable galaxies. This figure of merit is plotted for the different magnitude limits in the lower panel of the figure. For each magnitude limit, the curve is normalized so that the peak of the curve is unity. Very faint magnitude limits imply large number of targets, and the s/n reachable on a single spectra is the decisive criterion to determine the optimal pixel size. The optimal pixel size is around 0.1 arcsec per pixel. For brighter magnitude limits, where the objects are less densely distributed, observations benefit from a larger pixel size. The optimal pixel size is around 0.15 arcsec per pixel at a magnitude limit of V=22.
As an example, J.MacKenty et al. 2000, (in "Next Generation Space Telescope Science and Technology", ASP Conference Series, Vol. 207, pg. 251) estimated the number counts for high-redshift galaxies. Their plot is reproduced here.
The same figure of merit as shown in figure 5 was computed for the number density of galaxies with 5 < z < 10. The upper dashed curve of figure 6 was used for the object density. The result is shown in figure 7. Even at fairly deep surveys would benefit from pixels significantly larger than 0.1 arcsec.