# Large Aperture Photometry: Profile Fitting, Isophotal & Total Mags

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 Ks-band 20 mag arcsec^{-2} isophote. The Kron aperture is
derived from the intensity-weighted "first moment" (discussed below). The "total" aperture
is derived from extrapolation of the radial surface brightness from the standard isophote out
to some scale length that covers the deduced extent of the galaxy, discussed further below.
The photometric measures are given in mag units (relative to the photometric zero point mag,
given in the FITS header) and include their 1-sigma uncertainties. The absolute calibration
of 2MASS is uniformly 2-3% across the sky. The photometric uncertainty includes formal
errors due to the background and target poisson noise, as well as the estimated uncertainty due to
the background removal, large-scale background variations and the fit to the surface brightness
profile (relevant to the "total" integrated flux; see below). Although the random components are
well modeled (see
Jarrett et al. 2000), the systematic components (e.g., H-band airglow variations) are not well understood and may induce large errors in the photometry. Verification of the apparent photometry is discussed below.

### a. Isophotal Photometry

The standard 2MASS aperture is the ellipse fit to the Ks-band 20 mag arcsec^{-2} isophote, corresponding to roughly 1-sigma of the typical background noise in the Ks images. This aperture is determined using the axis ratio (b/a) and position angle (P.A.) of the fit to the 3-sigma isophote, allowing the semi-major axis (r_{20}) to vary so that the mean surface brightness along the ellipse is 20 mag arcsec^{-2}. The integrated fluxes within this ellipse in the background-subtracted J, H and Ks mosaic images are then calculated..

### b. Large Apertures: Kron & Total Mags

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 NIR, galaxies are, for the most part, smooth and axi-symmetric (c.f. Jarrett 2000). 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 corresponds to a scaling of the intensity-weighted 1st moment radius. It was designed to robustly measure the integrated flux of a galaxy. In attempt to recover most of the underlying flux of the galaxy, we define the Kron radius to be 2.5 times the 1st 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 ~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 = f0 * [exp (-r/alpha)^{1/beta}],

where r is the radius (semi-major axis), f0 the central surface brightness, and alphta & beta 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 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 CHI2 represents the goodness of fit, or alternatively, the deviation from the assumed Sersic model.

For the 1st moment calculation, we adopt an effective integration radius of the total aperture, r_{tot}, that corresponds to ~four scale lengths. For a pure exponential disk, beta = 1, thus fixing f/f0 = 55. It then follows that the total integration radius is

r_{tot} = r' + [alpha * ln (55)^{beta}]

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 1st moment radius, r1, is computed from the aperture delimited by r_{tot}. The Kron radius, r_{Kron}, is then 2.5 * r1. In this way the Kron aperture is closely tied to the measured radial light distribution and thus represents an 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 effects of the background.

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 ~4 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; and 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, r_{tot} (see above), for all three bands, since the J-band images are the most sensitive to the low surface brightness galaxy signal (1-sigma = 21.4 for the J band) and consequently lead to the most precise radial surface brightness profile. The only 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, r_{tot}, 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 for each band, 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).

### c. Half-Light "Effective" Aperture & Concentration Index

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 mag/arcsec2 units). 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 3/4 light-radius to the 1/4 light-radius (the de Vaucouleurs convention). These radial points are derived in a similar fashion as to the half-light radius.

### c. Results

Photometric repeatibility tests were carried out to access the performance of the large aperture algorithms. See Repeatibility Tests.

Examples of radial profiles and their fits are given here: Examples of Radial Profile Fitting. 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 (beta = 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 here.

[Last Updated: 2002 Jul 15; by Tom Jarrett]