Slit-losses as function of Magnitude

In HDF, even galaxies with the faintest magnitudes have been resolved. It is not clear how the size of galaxies evolves for very high (z > 5) redshifts. With most cosmology, the angular size of galaxies will only slowly change with redshifts if the physical size does not change. Therefore, the slit losses due to the angular sizes of galaxies relative to the slit width might be significant. To investigate this, the drizzled HDF-S F606 image with a drizzled pixel size of 40 mas was used to compute flux contained in slits of different width for all detected galaxies. For each galaxy, the flux contained in the slit was divided by the total flux (flux_auto computed by Sextractor). The results were binned by magnitude and are shown in figure 1.

Figure 1: fraction of flux contained in slit.

Since galaxies are smaller for fainter magnitudes, the relative flux contained in the slit increases with magnitude. It can be seen in figure 1 that for a given slit with, the behavior of the flux as a function of magnitude is similar. In other works, by multiplying each curve in figure 1 by an appropriate factor, the curves can be shifted on top of each other. This has been done in figure 2.

Figure 2: normalized curves of figure 1.

The conclusion from figure 2 is that relative gain in flux obtained by changing the width of the slit does not depend on the magnitude. Therefore, the merit of changing the slit with can be modeled as the average of each curve in figure 1. Figure 3 shows the average of each curve in figure 1 arbitrarily normalized to one for a slit with of 0.1". The smooth curve is a second order polynomial fit with will be used in computing the relative s/n as a function of slit width.

figure 3: change in signal as a function of slit width.

From the signal as a function of slit width, the relative s/n can be computed. Since spectra of faint galaxies are completely background limited (see Arribas' analysis), the Poisson noise can be neglected. Therefore, the relative s/n as function of slit width is independent of the magnitude of the galaxy. The results are shown in figure 4. Also shown is the same curve assuming that the galaxies are point sources. That curve is taken from Arribas. Both curves have been normalized so that their maximum is at one. It can be seen that the relative s/n peaks at larger beam widths for the HDF galaxies. This is because the HDF galaxies are larger than the PSF.

Figure 4: relative s/n

Finally, the figure of merit for the different slit sizes can be recomputed as has been done for the analysis of crowding. This is shown in figure 5 for the case of a survey of high redshift galaxies. Compared to Figure 7 of the crowding analysis, this figure favours larger slit widths.

For a spectroscopic survey of randomly placed objects, the design of a spectrograph will allow to position objects onto the slit only with an accuracy of 0.5 pixels. Such a misplacement leads to the loss of signal as well as to a shift in the wavelength zero point. The loss of signal was investigated again by using galaxies imaged in the the HDF-S F606. For this investigation, it was assumed that the width of the slit is 2 pixels. For each slit with, the flux was computed twice. First, when the galaxy was positioned in the center of the slit. Subsequently, the flux was recomputed with the slit displaced by 0.5 pixels. Since the effect could depend on the size of the galaxy, it was again investigated as a function of magnitude. The results are shown in figure 4. The color coding of the curves is the same as in figure 3. Note that for the smallest slit widths, the pixilation of the original image might start to impact the results. It can be seen that the effect of the displacement is small than 10% for all magnitudes bins and slit widths. Since the typical displacement will be 0.25 pixels, this does not seem to be a major concern.