IV. 2MASS Data Processing

5. Extended Source Identification and Photometry

g. 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 Galactic plane, faint undetected stars significantly increase the mean "noise" amplitude in the "background" sky. Both surface density affects limit the source detection or completeness, and overall reliability. Confusion arises from the appearance of an interloping star within close proximity to the "beam" or point spread function pattern that induces a significant flux bias (or "deflection") from the non-contaminated flux due to the intended source (in our case, galaxy) near the beam center. A convenient gauge for the severity of contamination or level of source confusion, is the confusion "noise". In more practical form, the confusion noise represents the change in surface brightness (i.e., sensitivity limit) relative to the Galactic pole (where the confusion noise is negligible), expressed in magnitude units. The confusion noise is directly related to the stellar number density. In the 2MASS database the stellar number density is referred to as the "density", representing the base-10 log of the cumulative number of stars per deg² brighter than K_s=14 mag (see also Table 1). The starcount metric is discussed in detail in IV.5c.

As the confusion noise becomes appreciable, it is one of the primary limits on galaxy detection and reliability. Moreover, confusion decreases the accuracy of both flux and position estimation. 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.

i. Estimation of Confusion Noise

To estimate the additional component of "confusion" noise, we adopt the methodology of Hacking (1987, PhDT) and Hacking & Houck (1987, ApJS, 63, 311). The idea is to integrate the expected number of sources (with some flux distribution) within the 2MASS effective beam, , where represents the effective radius of the point spread function (typically ~2´´). We may approximate the stellar flux distribution with a power law of index ,

(Eq. IV.5.13)

where N is the integrated stellar number density (in deg²) at flux f (in mJy). The aggregate variance due to background sources in the beam, derived from f, d and the differential stellar number density, is then

(Eq. IV.5.14)

where D_c represents the outlier or "deflection" cutoff point (in n- units; i.e., the detection threshold). The source density index is approximately equal to unity (more precisely ~0.85 for high latitude fields) as derived from the log-log slope of the N vs. K_s cumulative star count curve, ~0.35 for the NGP and slightly steeper at higher densities (Jarrett 1992, PhDT). Letting , we may express the confusion noise as a function of the stellar number density, N(f_lim), at the limiting flux, f_lim, and the deflection cutoff, q,

(Eq. IV.5.15)

Appropriate values for 2MASS K_s-band data are the following: = 0.9, = 13.6 arcsec² (4´´ beam), f_lim = 1.8 mJy (corresponding to K_s=14 mag), and q between 3 and 5; the confusion noise has units of mJy.

The confusion noise adds in quadrature to the already present background noise, , raising the overall noise and surface brightness of the background light. We desire to express the change in the background surface brightness due to confusion noise as a function of the stellar number density. We can turn the confusion noise into a surface brightness by dividing by . For the background sky noise we adopt a value of 20.0 mag/pixel (typical for 2MASS 2.2 µm images). To convert the sky noise per pixel to an equivalent surface brightness within the PSF beam, we need to divide by to account for the noise limit after averaging over a 4´´ diameter. Accordingly, we arrive at a sky noise surface brightness of 21.6 mag/arcsec², representing the value at the north Galactic pole (NGP), which is negligibly affected by confusion from stars. The confusion noise (in mag units) is relative to the NGP sky surface brightness noise (in a 4´´ diameter beam). The K_s-band confusion noise as a function of the total integrated stellar number density (f_lim < 1.8 mJy, or K_s < 14 mag) is plotted in Figure 30, described below, assuming a beam size of 4´´, = 0.9 and q between 3 and 5.

ii. Confusion Noise, Stellar Density and Galactic Coordinates

For relatively moderate flux ranges (e.g., V < 18; K_s < 14), basic three-component models of the Galactic stellar distribution adequately describe the number density of dwarf and giant stars comprising "disk" and "spheroid" populations (Elias 1978, ApJ, 224, 453; Bahcall & Soneira 1980, ApJS, 44, 73; Garwood & Jones 1987, PASP, 99, 453). Here we employ a near-infrared modified variation on the Bahcall & Soneira model, which predicts the stellar number density with ~90% accuracy for most of the sky (|b| > 30°) and K_s < 14 mag, and performs adequately (~80%) for the Galactic plane where patchy extinction ultimately limits the utility of these simple models. The star-count model predicts the stellar number density as a function of the Galactic coordinates, which can then be used to calculate the approximate confusion noise.

A plot of the stellar number density (see IV.5c) as a function of the Galactic latitude along two separate longitudes (50° and 130°) is shown in Figure 30. The vertical dotted lines represent the thresholds for what is deemed low stellar number density (<10^3.1 stars deg^-2), moderate density (<10^3.6 stars deg^-2), and high density (>10^3.6 stars deg^-2). The limit on high density is partly driven by the relative density of triple stars vs. double or single stars. As triple+ stars become appreciable, the ability to distinguish real galaxies from close groupings of stars is greatly diminished. Finally, the confusion noise (mag) appropriate to the stellar number density is plotted in Figure 30 (denoted with a cross-hatching, showing the detection threshold range in q between 3 and 5), with the confusion noise axis located at the right of the plot. Here the confusion noise (in mag units) is relative to the equivalent sky background surface brightness (1- detection limit) measured at high galactic latitudes (i.e., NGP), equal to 0.0016 mJy (in a 4´´ circular diameter beam) or 21.6 mag at 2.2 µm. This relative confusion noise is called the "differential" confusion noise above.

Figure 30

[Last Updated: 2003 Mar 10; by T. Jarrett, T. Chester, S. Schneider, S. Van Dyk, & R. Cutri]

Return to Section IV.5.