The USNO observations cover about half a square degree to r ~ 19.5. This is similar to the limiting magnitude we expect in the cap region of the SDSS. The data was taken on clear, but not photometric nights, and the data from separate fields were shifted so that the stellar loci lined up (this is thought to reduce the internal systematic errors). The maximum shift was about 0.1 magnitudes. I used only 828 objects that have u, g, r, and i measurements in this data set; this is about half of the 1630 objects detected to r ~ 19.5. A locus was fit to this data set, using the algorithm of Newberg and Yanny (1997). Near the blue end of the locus, the ``one sigma" widths of the locus are 0.04 magnitudes in the thick direction, and 0.02 magnitudes in the thin direction. The distribution is of course not Gaussian, especially in the tails.
Figure 1 shows that there are half a dozen to a dozen objects that are truly inconsistent with being in the stellar locus (12 objects are circled). Since the data only covers half a square degree and we are only considering half of the objects to the limiting magnitude, it is reasonable to assume we cannot dig any further into the stellar locus without going over the ~24 fibers per square degree that are allocated in the cap region. The twelve objects were chosen using an eight ``sigma" stellar locus width and a blue endpoint of (u-g, g-r, r-i) = (0.80, 0.25, 0.10).
Not all of the point sources in the diagram have similar errors. I selected only those points which had photometric errors smaller than 0.03 in all three colors. This yielded 465 stars - again cutting the sample in half. All of these stars are within a tube that is 0.20 magnitudes in the wide direction, and 0.10 magnitudes in the thin direction. The relevant direction vectors on the blue end are:
Direction along locus: 0.88 (u-g) + 0.44 (g-r) + 0.17 (r-i) Thick direction: -0.44 (u-g) + 0.64 (g-r) + 0.63 (r-i) Thin direction: 0.17 (u-g) - 0.63 (g-r) + 0.76 (r-i)This is reasonably consistent with the results of Simulation #1, which used simulated stars. In that case we excluded the area inside a tube which was 0.22 magnitudes in the wide direction and 0.14 magnitudes in the thin direction.
We now apply this minimum volume of exclusion around the stellar locus to the simulated QSOs, and select 1955/2034 = 96% of the QSOs in the 2.5 < z < 3.0 range. It is notable that the locus of simulated QSOs comes exactly up to the actual blue endpoint of the locus of stars of similar magnitude. This begs the question, in my mind, of whether the simulations are based on a luminosity function for QSOs that was generated from observations of QSOs that are the easiest to find in color space. Nevertheless, I will assume that the simulations are a fair sample of QSOs.
We now have everything we need to estimate the effects of systematic photometry errors. If the locus moves about due to systematics of the photometry, then we will have to exclude a larger volume in color space. That volume will be larger than the minimal volume by about the largest expected systematic error in photometry. We cannot significantly increase the number of fibers allocated to QSO targets, so we will take the hit in completeness. So, I calculate the completeness in the 2.5 <= z < 3.0 range for various maximal systematic errors. Note that the errors not only increase the width of the tube, but also move the blue endpoint.
Err_max thick width thin width blue endpoint completeness 0.00 0.18 0.09 0.8, 0.25, 0.1 96 % 0.02 0.20 0.11 0.78, 0.24, 0.10 94 % 0.05 0.23 0.14 0.76, 0.23, 0.09 86 % 0.10 0.28 0.19 0.71, 0.21, 0.08 70 %
Caption: The top figure is a u-g vs g-r diagram for 828 stars observed with the USNO 1 meter. Outliers (at the 8 ``sigma" level) are circled. The locus points are shown as squares. The bottom figure shows simulated QSOs in the 2.5 <= z < 3.0 range. QSOs not detected with the minimal tube are indicated as + marks.