Error Analysis For Circular Isophotal Magnitudes

T. Chester, T. Jarrett

Table of Contents

Introduction
Theoretical Error In Radius Determination
Theoretical Photometric Error Due To Radius Determination
Simulation Results
Observed Error Due To Radius Determination
Conclusions

Introduction

To determine an isophotal circular aperture magnitude, the radius of that aperture must first be determined. Errors in the determination of that radius are a significant component of the total photometric error. Thus we must first analyze the error in determining the radius.

This memo analyzes theoretically the errors in radius and photometry, presents simulation results, and then compares predictions to actual 2MASS results.

Theoretical Error In Radius Determination

The radius corresponding to a circular magnitude using a given isophote is determined by finding the annulus whose mean flux equals that isophote. Thus one must step through progressively larger annuli centered on a source and determine the mean flux in each annulus.

Typical values for radii for the circular isophotal magnitude at 20 mag per square arcsecond are shown in the lower plot in Fig. 1. For sources within one mag of the completeness limit, the radii are generally less than 9".

The following table gives the photometric error corresponding to annuli defined as the set of pixels whose radius from the source position is greater than r_min and less than or equal to r_max, for radii less than 9":

The "circular" theoretical values are just

pi * (r²_max - r²_min).

The "actual" number of pixels is higher than the "circular" theoretical values due to including pixels with radius equal to r_max in the annulus for the actual 2MASS algorithm. This is a minor detail which does not affect any of the conclusions of this memo.

A K mag of 20 in a 1 square arcsecond pixel corresponds to a DN value of 1. It is clear that 2MASS is woefully lacking in SNR in order to determine that isophote very well.

Theoretical Photometric Error Due To Radius Determination

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 = dr * r / { e^r - (1+r) }

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 sources within one magnitude of the completeness limit.

Note how rapidly this term decreases with increasing r. For galaxies with r = 5, df/f has dropped to 0.04 dr.

Simulation Results

We have simulated sources corresponding to K_s = 12.75 mag, using an exponential profile with a scale length of 2" and a coadd noise of 1 DN per pixel, appropriate to the best K conditions. The realized annuli surface brightnesses are shown for 10 of these sources. The 2MASS algorithm finds the radius corresponding to the 20 mag per square arcsecond isophote where the profile first crosses 1 DN.

Note especially the source profile that first dips below the 1 DN value at a radius of 5.96". Noise brings the profile back up and it next crosses the 1 DN value at a radius of 8.4"!

The simulation input has a true isophotal radius of 6.84". The simulation results, using the actual 2MASS algorithm, give a mean radius of 6.85", with a population sigma of 0.72" (10.4% in dr/r). The simulated observed integrated flux to that radius has a population sigma of 8.4%, essentially the same as the 9.4% predicted by the above theoretical formula, using the observed population sigma of 0.72".

The photon noise in an aperture of radius 6.84" corresponds to a sigma of 4.2%. Hence the photometric error for the isophotal magnitude is doubled compared to the fixed aperture noise.

Observed Error Due To Radius Determination

The upper plot in Fig. 1 shows the error in the radius determination for the 20^th mag per square arcsecond isophotal K magnitude. The values were determined for the five times repeated observations of the Hercules galaxy cluster. Note that above a radius of 8", the uncertainty ranges from 0.6 - 1.8".

In particular, the uncertainty for a radius of 6.84", the value used in the simulation, is consistent with the value of 0.7" found in the simulation.

Note that below a radius of ~6.5", the uncertainty is artificially smaller because values smaller than about 5.5" were not allowed by the calculation in the 2MASS galaxy code. The reasons for this date back to the prototype camera era when the PSFs were poor and variable. In any case, radii smaller than this value are heavily influenced by the PSF and the observed seeing rather than the actual isophote for each source.

Simulation shows that the fainter sources with smaller true radii should in fact have slightly larger uncertainties. A simulation of a 13.2 mag source with a true isophotal radius of 6" has an uncertainty of 0.86", not the much lower value of around 0.4" shown in the figure.

The following series of plots shows how this uncertainty in the radius dominates the photometric error. All of the plots show the observed scatter from the Hercules data (white dots), and the theoretical photon errors (pink crosses).

• Mean Radius less than 6"
• Mean Radius greater than 6"
• Mean Radius greater than 7"
• Mean Radius greater than 8"

Note the following:

• Galaxies with mean radii less than 6" track the theoretical photon errors quite well for the most part. In fact, if anything, the measured errors fall slightly below the theoretical curve.
• Virtually no galaxies with mean radii greater than 8" fall below the theoretical photon errors. The average photometric uncertainty is about a factor of 2 higher, as predicted by the simulations.
• The galaxies with mean radii greater than 8" reach a 10% photometric uncertainty about a half magnitude brighter than the spec of 13.8 mag.

The reason that the plots of isophotal magnitude looked like they met the Level 1 Specification for galaxy photometry is due almost entirely to artificial truncation of the radii below 6.5".

Conclusions

The major conclusions of this memo are:

• 2MASS is woefully lacking in SNR in order to determine the 20 mag per square arcsecond isophote very well.

• The photometric error for the correctly-determined 20 mag per square arcsecond isophotal magnitude is doubled compared to the fixed aperture noise.

• The above two results are observed in the actual 2MASS data, and have been verified in simulated data.

• Due to a fortuitious enforced minimum radii in the 2MASS galaxy processing code, the reported 2MASS isophotal magnitude are actually closer to fixed aperture magnitudes for galaxies less than a half magnitude brighter than the completeness limit. This created the appearance of meeting the Level 1 Specification for photometry for isophotal magnitudes.

These results call into question the usefulness of isophotal magnitudes for over 75% of all the galaxies in 2MASS, the ones in the last magnitude bin. These magnitudes do not correspond to true isophotes on the sky for smaller galaxies, and are poorly determined for larger galaxies.

Fortunately, fixed circular aperture magnitudes easily meet the observed Level 1 Specification, since they contain only the photon noise. We currently report fixed circular aperture magnitudes using apertures of 5, 7, 10" and larger, and apertures of 5-10" meet the spec.

Fixed circular apertures have several other advantages. It is much easier to understand contamination of these magnitudes by stars and neighboring galaxies. They are also affected much less by the increase in the noise caused by confusion in regions of high source density.

Because these isophotal magnitudes are still useful for larger galaxies, we do not advocate any change to the 2MASS Galaxy processor. However, we probably want to replace the magnitude in the galaxy short form database report with the 7" fixed aperture magnitude, and to use that aperture as the index field in the galaxy database.