The galaxy flux is integrated using a suite of apertures, including large ones
to capture the total flux of the source:
elliptical, isophotal, elliptical, Kron, and elliptical, extrapolated "total".
The isophotal aperture, *r _{20}*, corresponds to the 2MASS XSC
standard
aperture, derived from the K

The standard 2MASS aperture is the ellipse fit to the K_{s} = 20 mag
arcsec^{-2} isophote, corresponding to roughly
1- of the typical background noise in the
K_{s} images. This aperture is determined using the axis ratio
(*b/a*) and position angle () of the fit to
the 3- isophote, allowing the semi-major axis
(*r _{20}*) to vary, so that the mean surface brightness along the
ellipse is 20 mag arcsec

Large apertures are used to capture the lower surface brightness galaxy flux. We employ two techniques: (1) Kron apertures, and, (2) curve of growth, or 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 near-infrared, galaxies are, for the most part, smooth and axisymmetric (cf. Jarrett 2000). Deducing the "total" flux, with robust repeatability, is therefore possible using large apertures (e.g., Kron) and curve-of-growth techniques.

The Kron (1980) aperture corresponds to a scaling of the intensity-weighted
first moment radius. It was designed to robustly measure the integrated flux
of a galaxy. In an attempt to recover most of the underlying flux of the
galaxy, 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 about four disk scale lengths, detailed
below.

We employ what is effectively a Sersic (1968) modified exponential function to trace the elliptical radial light distribution,

*f* = *f*_{0} * [exp (-*r* / )^{1 / }],

where *r* is the radius (semi-major axis), *f*_{0} is the
central surface brightness, and and
are the scale length parameters. In practice,
the 2MASS PSF completely dominates the radial surface profile for small radii
(*r* < 5´´), so the exponential function is only fit to those
points beyond the PSF and nuclear influence. The fit extends from
*r* >> 5´´ to the point at which the SNR > 2. The best fit is
weighted by the SNR, 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
^{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 about four
scale lengths. For a pure exponential disk, = 1,
thus fixing

*r _{tot}* =

where *r'* is the starting point radius (typically > 5-10´´).
For robustness, the total aperture radius is not allowed to exceed five times
the isophotal radius, *r _{20}*. The intensity-weighted first
moment radius,

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 about four 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 SNR extents, recovers the underlying flux of the galaxy, which
in combination with the isophotal photometry, leads to the "total" flux of the
galaxy. Hence, we will refer to this photometry as the "total" aperture
photometry (not to be confused with the Kron aperture photometry). For
consistency across bands, we adopt the J-band integration limit,

To summarize, for the curve-of-growth technique, we quote one integration
radius, *r _{tot}*, common to all three bands,
and
radial surface brightness solutions for each band, reduced

The de Vaucouleurs "effective" aperture measures the galaxy half-light, as
computed from the "total" flux. For the total flux, we adopt the surface
brightness profile extrapolation method. Using the elliptical shape of the
galaxy, we integrate in small annular steps starting from the center
(*r* > 5´´), until we reach the integrated half-light point.
We then interpolate across the surface brightness profile to arrive at a more
precise half-light radius. We report the half-light radius for each band, and
the corresponding half-light mean surface brightness (in units of
mag arcsec^{-2}). We do not correct for PSF effects: For small
galaxies, the half-light radius is susceptible to circularizing effects from
the PSF, although this is generally not a concern for the Large Galaxy Atlas.

The concentration index characterizes the nuclear-to-bulge concentration of the galaxy. The index corresponds to the ratio of the three-quarter light-radius to the one-quarter light-radius (the de Vaucouleurs convention). These radial points are derived in a similar fashion as to the half-light radius.

Photometric repeatibility tests were carried out to access the performance of the large aperture algorithms. See IV.5a5.

Examples of radial profiles and their fits are given in
IV.5e1. The question arises,
what to do when *n* < 5 ? In order to avoid introducing discontinuous
jumps in the extrapolation (see below), we force a fit to the
profile. We assume a pure exponential
( = 1) and fit to at least two points
in the profile. The errors may be large, but in practice
the techique works adequately. Some examples of a forced fit are given
for "object 6" in IV.5e1.

[Last Updated: 2002 Jul 15; by T. Jarrett]

Return to Section IV.5.