Profile Fitting & "Total" Mags &
"Effective" Surface Brightness

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 symmetric (see Near-Infrared Galaxy Morphology Atlas ). Computing the "total" flux, with robust repeatability, is thus possible using curve of growth or extrapolation techniques.

For 2MASS, the approach is to extrapolate the radial surface brightness profile, with the lower boundary given by the 20 mag/arcsec^2 isophotal radius and the upper boundary by the deduced extent of the galaxy (also referred to as the integration radius, see below).

Elliptical-Shape Fit & Radial Profile
The galaxy is assumed to be symmetric and elliptically-shaped (again, reasonable assumptions for most resolved objects). We also assume that the shape of the galaxy is preserved from low to high radii -- a crude approximation that is needed for robustness and processing speed. The projected shape of a galaxy is determined at the 3-sigma isophote. The algorithm is detail here. The resultant axis ratio and position angle represent the galaxy shape;

The radial profile is constructed by computing the median value in elliptical annuli of 1-pixel width (r<15"), 2-pixel width (15 < r < 30"), and 5-pixel width (30 < r < 115"). The corresponding S/N is carried for each measurement. The profile is not well determined for small radii (r < 4").

The profile is characterized by the modified exponential function:

In practice, the 2MASS PSF completely dominates the radial surface profile for small radii (<4"), so the exponential function is only applied to those points with r>4". Hence, The fit applies to those points with a S/N > 2, where we have incorporated S/N weights into the fit. A minimum of 5 points are required for the least-squares fit. The number of free terms in the fit is n/2-3, where n is the number of points (median surface brightness measurements), the "2" comes from the correlated pixels (frame to coadd conversion) and the "3" is the number of parameters to fit. The resultant reduced CHI^2 is thresholded at a value of 5.0: A poor fit, rCHI^2 > 5, is rejected. 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 and here.

Extrapolation

If we have adequately characterized the radial surface brightness profile (which in turn adequately reduces the 2-D surface brightness to a 1-D profile), it is a simple matter to integrate the fit to the profile out to radii that encompass the galaxy. This integration (or extrapolation of the profile) recovers the lost flux of the galaxy, which in combination with the isophotal photometry leads to the "total" flux of the galaxy.

The extrapolation or integration radius, rinteg, we adopt to be roughly four times the scale length of the disk (or spheroid):

The extrapolation radius cannot exceed three times the one-sigma isophotal radius. The minimum radius corresponds to 1.2 times larger than the isophotal radius. The integration radius is also discussed in the Kron Photometry document.

The total mags are constructed by adding the isophotal flux to the extrapolated flux. Repeatibility tests demonstrate their robustness:

Note 1: the dip in the total "radius" (i.e., rinteg) at ~12" is due to fits that have failed the rCHI^2 test (see above). In this case, the total radius reverts to the isophotal radius. A solution to break this discontinuuity is in the works.
Note 2: The estimated uncertainty in the total mags (see dashed magenta line) is overestimated at the faint end. A more realistic estimate is still under construction.

The total mags and the isophotal mags are systematically different by ~10-20%. That is to say, the extrapolation is recovering about 10-20% of the flux that is lost in the background noise. Morphological differences account for a significant component of the scatter (particularly at J-band):

Half-Light "Effective" Aperture

With the total flux measured, it is then a straight forward procedure to determine the half-light radius (aka de Vaucouleurs "effective" aperture).

The procedure is to integrate the source in small incremental steps, starting from R=2". The point at which the integrated flux equals 1/2 of the total flux corresponds to the half-light radius. In practice we employ a bi-linear interpolation to rectify the differences in the adjacent integration radii. GALWORKS reports the half-light radius and the half-light "effective" surface brightness (integrated flux divided by the area). The resultant half-light apertures can be seen in the samples.

Concentration Index

The concentration index characterizes the nuclear (surface brightness) concentration of the galaxy. The index corresponds to the ratio of the 3/4 light-radius to the 1/4 light-radius. These radii are determined in a similar fashion as the half-light "effective" radius (see above).