Appendix C. Stellar Number Density and Confusion Noise

As the surface density of stars exponentially increases near the disk of the Galaxy, the probability of source contamination increases accordingly. Likewise, near the plane, the ‘sky’ background is elevated in surface brightness (relative to pole, e.g.) due to faint undetected stars pumping up the overall sky brightness (and consequently the mean noise amplitude). Both surface density effects limit the source detection or completeness, and overall reliability. A convenient gauge for the severity of contamination or simply confusion level, is the "noise", or change in surface brightness relative to the pole, expressed in magnitude units. As the confusion noise becomes appreciable, it is one of the primary deterrents toward galaxy detection and separation from stars and grouping of stars. It is therefore important to understand the confusion noise in terms of the ability to detect isolated sources and in terms of identification of real extended sources, both of which require threshold adjustment with confusion noise level.

C.1. Estimation of Confusion Noise

At high stellar number density the contribution of stars to the total background noise becomes appreciable, effectively raising the surface brightness level of the sky. To estimate the additional component of "confusion" noise to the total, we adopt the method of hacking et al. (1987). The idea is to integrate the expected number of sources (with some flux distribution) within a ‘beam’ or area of the sky comparable to the pixel resolution of the 2MASS survey. We may approximate the flux distribution with a power law of index a,

where N is the integrated number density at flux f. The aggregate confusion noise, derived from df and tied to the integrated stellar number density, is then

where is the beam size and q is the n-s cutoff at which the distribution outliers are excluded from the noise calculation. The change in surface brightness due to the confusion noise is given by s_conf per beam W , expressed in mag units, or Dmag, where ‘minimum’ confusion noise corresponds to the galactic pole. The index a is approximately equal to unity as derived from the slope of the N vs. Kmag cumulative

starcount curve, ~0.35 for the NGP and slightly steeper at higher densities (cf. Jarrett 1992). The K-band confusion noise as a function of the total integrated stellar number density (K < 14 mag) is plotted in Figure C.1, described below, assuming a = 1 and q = 5 (i.e., 5s cutoff of outliers) and a beam size of 5".

C.2. Confusion Noise, Stellar Density and Galactic Coordinates

For relatively moderate flux ranges (e.g., V < 18; K < 14) basic three-component models of the Galactic stellar distribution adequately describe the number density of dwarf and giants stars comprising "disk" and "spheroid" populations (cf. Elias 1978; Bahcall & Soneira 1980;Garwood & Jones 1987). Here we employ a near-infrared-modified variation on the Bahcall & Soneira model described in Jarrett (1992), which predicts the stellar number density with ~90% reliability for most of the sky (glat > 30) and K < 14 mag, and performs adequately (~80%) for the galactic plane where patchy extinction ultimately limits the utility of these simple exponential models (e.g., the extinction is model as an exponential with a disk scale length of about 3 - 4 kpc and a disk scale height of about 100 pc, neither of which address molecular clouds). The starcount model predicts the stellar number density as a function of the galactic coordinates (glong, glat), which can then be used to calculated the approximate confusion noise (also model dependent, see C.1).

A plot of the stellar number density as a function of the galactic latitude (glat) along two separate longitudes (50° and 130° ) is shown in Figure C.1. The dotted lines represent the thresholds for what is considered low stellar number density (<10^3.1 stars per deg²), moderate density (<10^3.6 stars per deg²), and high density (>10^3.6 stars per deg²). The latter is partly driven by the relative number density of triple stars versus double or single stars. As triple+ stars become appreciable (high density), the ability to distinguish real galaxies from close groupings of stars is greatly diminished. Finally, the confusion noise (Dmag) corresponding to the stellar number density is plotted in Figure C.1, with the confusion noise axis located at the top of the plot.

Fig. C.1.-- The predicted stellar number density, log [stars per deg2] brighter than 14^th mag at K, as a function of the galactic latitude. Two sets of solutions are shown, that of 50° galactic longitude (solid line) and 130° galactic longitude (dashed line). The dashed lines define regions of low (< 3.1), moderate (3.1 to 3.6) and high (>3.6) stellar number density. The estimated confusion noise (Dmag) as a function of the stellar number density is denoted by the dot-dashed line, assuming index a = 1, outlier cutoff q = 5s, and the beam size is 5".