The Density of Faint Galaxies and the Optimal Size of Spectrograph Pixels

The number of spectra which can be obtained in a survey increases when more targets are available within the field of view of the detector. However, once the number of targets is high enough so that the spectra fill a large fraction of the detector, the benefit of additional available targets is small. Monte-Carlo simulations were used to estimate the number of observable spectra for a given object density (objects/detector). A 2048x2048 pixel detector was assumed. Spectra with a length of 200 pixels and a width of 10 pixels were randomly placed on the detector. This is meant to correspond to spectra with a resolution of 100. For each simulation, the "observable" spectra were selected. The criterion was that the spectra do not overlap with each other or the edge of the detector. The selected spectra are marked in red in the two examples.

figure 1: 400 spectra on detector figure 2: 2000 spectra on detector

Simulations were carried out varying the number of objects within the detector area. The number of "observable" spectra was then counted in each simulation. The plots shows the the mean number of observable spectra as a function the initial number of objects.

figure 3: number of observable spectra vs. number of objects within the detector area.

The number of target objects within the detector area depends on the object density (objects per square arcmin) and the size of the detector (in square arcmin). Assuming a survey wants to obtain spectra for all galaxies up to a given magnitude limit, the density of target objects can be estimated from deep galaxy counts. The HDF-N (William, 1996) was used as a model of galaxy counts as a function of V magnitude limit.

figure 4: galaxy counts model

Arribas' model gives the relative s/n ratio as a function of pixel size for isolated objects. This model is shown in the upper panel of the figure. Increased pixel sizes lead to a lower s/n for a given spectrum. However, larger pixel sizes imply large fields of view which again increases the number of objects which can be observed.

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.

figure 5: merit.

A more typical survey might aim to obtain spectra only of pre-selected targets. From the previous discussion it is clear that such a survey will favor bigger pixels because the lower number density of objects will benefit from a larger field of view.

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.

figure 6: counts

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.

figure 7: merit for survey of galaxies with 5 < z < 10.