4.2 Large Aperture Photometry
Large apertures are used to capture the lower surface brightness flux. We employ two techniques: (1) Kron apertures, and (2) curve of growth, specifically, extrapolation of the surface brightness profile. A well-behaved radial surface brightness profile provides a means for recovering the flux lost in the background noise. Fortunately in the NIR, galaxies are, for the most part, smooth and axi-symmetric (c.f. Jarrett 2000). Disk galaxies typically possess low order spiral modes (m = 1 or 2; cf. Block et al. 1994), presenting a very smooth radial profile. Deducing the "total" flux, with robust repeatability, is thus possible using large apertures (e.g., Kron) and curve of growth techniques.
The Kron (1980) aperture arises from a scaling of the intensity-weighted first moment radius. It was designed to robustly measure the integrated flux of a galaxy. In attempt to recover most of the underlying flux, we define the Kron radius to be 2.5 times the first moment radius, consistent with the scaling used by the 2MASS and DENIS projects (see also Bertin & Arnouts 1996). The first-moment itself is computed from an area that is large enough to incorporate the total flux of the galaxy. This "total" aperture is determined from the radial light distribution, which is constructed from the median surface brightness computed within elliptical annuli centered on the galaxy (see Jarrett et al 2000 for more details). We define the "total" aperture radius, r_{tot}, to be the point at which the surface brightness extends down to ~five disk scale lengths, detailed below.
We employ what is effectively a Sersic (1968) modified exponential function to trace the radial light distribution,
f = f_{0} * [exp (-r/alpha)^{1/beta}],
where r is the radius (semi-major axis), f_{0} the central surface brightness, and alpha (alpha) and beta (beta) are the scale length parameters. In practice, the 2MASS PSF completely dominates the radial surface profile for small radii (r <5 arcsec), so the exponential function is only fit to those points beyond the PSF and nuclear/core influence. The spherical bulge may, however, still influence the fit at small radii, tending to enlarge (and circularize) the radial profile. The fit extends from r>>5 arcsec to the point at which the mean surface brightness in the elliptical annulus has a S/N ³ 2. The best fit is weighted by the ~S/N, as we solve for the scale length parameters and central surface brightness. The number of degrees of freedom in the fit is n/2-3, where n is the number of points in the radial distribution, the "2" comes from the correlated pixels (frame to coadd conversion) and the "3" is the number of parameters. The final reduced CHI^{2} represents the goodness of fit, or alternatively, the deviation from the assumed Sersic model.
For the first moment calculation, we adopt an effective integration radius of the total aperture, r_{tot}, that corresponds to ~five scale lengths. For a pure exponential disk, beta is unity, thus fixing f/f0 = 148. It then follows that the total integration radius is
r_{tot} = r + [alpha * ln (148)^{beta}]
where r is the starting point radius (typically >5-10 arcmin beyond the nucleus). For robustness, the total aperture radius is not allowed to exceed five-times the isophotal radius, r_{20}. The intensity-weighted first moment radius, r_{1}, is computed from the aperture delimited by r_{tot}. The Kron radius (r_{Kron}) is then 2.5 * r_{1}. In this way the Kron aperture is closely tied to the measured radial light distribution and so represents a broad integrated flux metric. On the downside, the relatively large Kron aperture, compared to the isophotal aperture, is much more sensitive to stellar contamination and other deleterious affects associated with the background removal.
For the curve of growth technique, the approach is to integrate the radial surface brightness profile, with the lower radial boundary given by the 20 mag arcsec^{-2} isophotal radius and the upper boundary delimited by the shape of the profile. As noted above, we adopt ~5 disk scale lengths as the delimiting boundary, r_{tot}, representing the full diameter of a "normal" galaxy. This integration, or extrapolation of the profile to low S/N extents, recovers the underlying flux of the galaxy, which in combination with the isophotal photometry, leads to the "total" flux of the galaxy. We will refer to this photometry as the "total" aperture photometry (not to be confused with the Kron aperture photometry).
For consistency across bands (and color comparisons), we adopt the J-band integration limit, r_{tot}, for all three bands, since the J-band images are the most sensitive to the low surface brightness galaxy signal, leading to the most precise radial surface brightness profile. The one exception to this rule is for the heavily obscured, reddened galaxies seen behind the Milky Way (e.g., Maffei 2 & Circinus), where we instead use the Ks-band surface brightness to deduce the total extent of the galaxy.
To summarize the curve of growth technique: we quote one integration radius, r_{tot}, common to all three bands, alpha and beta radial surface brightness solutions and reduced CHI^{2} fit for each band, and the integrated mags for each band. For the estimated uncertainty in the mags, we RSS the formal errors associated with the background removal, the isophotal photometry uncertainty, the ellipse fit to the 3-sigma isophote, and the fit to the radial surface brightness distribution (details given in appendices of Jarrett et al 2000).