The Error Tree For Perfect Conditions
The fundamental limits to 2MASS galaxy photometry are set by the accuracy to which we can determine the background level and by the total signal to noise in the aperture we use to report the flux of a galaxy.
Error in determining the background level. The 2MASS Galaxy processor tracks background variations on spatial scales greater than about 4' in space, which is about 5-10 seconds of time. If the background varies more slowly, our accuracy in determining the background will be set simply by noise, and is
0.025 * npixel,
where npixel is the measured individual pixel noise in the coadded images. (See Analysis of Photometric Noises for 2MASS Galaxies. The coefficient has been updated from that memo to reflect the change to track higher frequency changes in the background.)
Thus in an aperture of N pixels in the coadd, the photometric error due to the error in determining the background is
0.025 * N * npixel.
Noise in the galaxy aperture. The measurement error in summing up the flux over N pixels in the coadd is
{ 1.72*4*N }1/2 * npixel,
in the case where the galaxy flux is negligible compared to the background flux, where:
Total Noise. The total photometric noise is the sum of these two terms
npixel * { 0.025 * N + 3.4 * N1/2 }
The plot of these two terms versus N shows that the noise in determining the background never becomes greater than 31% of the noise in the galaxy aperture. This value is only reached for N=1800, which corresponds to aperture diameters of 0.8', the largest value at which the 2MASS galaxy survey is complete (the size of the overlap region between scans). Even for these large galaxies, this increases the total noise by less than 5% over the noise in the aperture alone.
To determine an isophotal circular aperture magnitude, the radius of that aperture must first be determined. Errors in the determination of that radius dominate the photometric error tree.
Consider the best case of a galaxy with an exponential profile. The integral flux over a circular aperture is
C * { 1 - (1+r)e-r },
where r is the aperture radius divided by the radial scale length of the profile. The flux error caused by the error in radius determination is
df/f = r / { er - (1+r) } dr
For a scale length of 2", and typical isophotal 20 mag per square arcsecond aperture radii of 6-8,
df/f ~ 0.19 - 0.08 dr.
Since dr ~ 1", the radius determination gives a flux error of 8-19%. The photon noise in this aperture gives typical flux errors of 5-7%, and hence this error dominates the photometric error for this particular magnitude. See Error Analysis For Isophotal Magnitudes.
Of course, circular fixed aperture magnitudes do not have this extra flux error.
