## 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/α)^{1/β}],

where r is the radius (semi-major axis), f_{0} the central
surface brightness, and α (alpha) and β (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 ≥ 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 Χ^{2} (chi-square) 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 + [α * ln (148)^{β}]

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.

- M87 radial surface brightness profile: aperture radii and fit solution indicated

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 Χ^{2} (chi-square)
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).