Analysis of Photometric Noises for 2MASS Galaxies

T. Chester, T. Jarrett

Table of Contents

Conclusions

As part of our investigation into whether photometric or other errors could cause problems with the 2MASS Galaxy J-H / H-K color-color plot, T. Jarrett and T. Chester have been exploring and characterizing the various sources of noises in 2MASS galaxy photometry. This memo reports the results.

The main conclusions are as follows:

• The noises related to galaxy fluxes are relatively well-understood.

• It was known before that the method used by GALWORKS to compute the background is very much more robust than using an annulus around each source to get the background, for two reasons: extra noise is introduced into galaxy photometry due to the noise in the computation of the annular flux, and the annular flux is more biased by point sources in regions of high source density.

  We have found that the annulus around a galaxy is more awful place than expected even at high galactic latitudes, full of extra flux due to faint galaxies, faint stars, and flux from the outer parts of the galaxies themselves.

  In particular:

  • Due to the enhanced probability of finding a galaxy near another galaxy, the density of galaxies in the annulus with radius of 10-20" from the center of a galaxy is ~3-5 times higher than Poisson statistics would predict.
  • Elliptical galaxies have 7% of their total flux in that annulus, with values of 0.1 DN or below for each pixel for a 13^th total mag galaxy. The exponential disk of a spiral galaxy will contribute no flux to that annulus, but 7% of its bulge component will be present.

• The error in computing the background is sufficiently small that it may be worthwhile to reduce the area over which the background is determined in order to track background changes better. There is no evidence of any error in the Hercules data caused by untracked background variations, but we have observed that problem in other data.

Introduction

In the following, assume that the individual pixel noise in the coadded images is n_pixel, which ranges from 1.0 - 1.3 DN at H and K to around 0.6 DN at J. Also, remember that:

• the error of the median is 1.25 times the error of the mean for a Gaussian distribution;
• the pixels in the coadd have correlated noise. One must divide the number of coadd pixels by 4 to get independent pixels; and
• the noise variance caused by Poisson fluctuations in the background and by read noise in an aperture with N coadd pixels containing no other flux is

  1.7²*4*N*(n_pixel)^2,

  in the case where the galaxy flux is negligible compared to the background flux, where:

  • The factor of 1.7 corrects for the smoothing introduced in the coadd by the Weinberg kernel (see Analysis of Coadd Noise for a derivation.); and
  • The factor of 4 results from the normalization of the flux of the coadd pixels, being the flux of the camera pixels divided by 4, and the correlation of the flux in the coadd caused by dividing the area of a camera pixel into 4 coadd pixels.

Technical note: In this memo, whenever the noise is calculated for an aperture with N coadd pixels, we compute the noise from first principles using the Poisson noise of the total flux from sources in that aperture, the read noise, and the Poisson noise in the background, using a gain of 8 and a read noise of 55 electrons. The formula above approximates that calculation in the case where the total flux from sources in the aperture is small compared to the background flux, which is the usual case. We use a background of 100 DN for J, and 400-800 DN for H and K, which corresponds to a coadd noise of 0.6 DN for J, and 1.0 - 1.3 DN for H and K.

Photometric Error due to Noise in Determining the Background

Error in Determining the Background

Currently, which will change with the next software update, GALWORKS determines the background as follows:

• A single coadd (512 cross-scan x 1024 in-scan) is blocked into 8 x 8 groups after source blanking.
• The median is determined for each 8 x 8 block.
• A cubic polynomial is fit to the 128 8 x 8 blocks in the in-scan direction for each of the 32 columns.
• A cubic polynomial is fit to the 64 values of the 32 in-scan polynomials along each cross-scan row.

The software change which will occur is to fit the cubics to half of the in-scan extent of the coadd, separately for the top half, the middle half, and the bottom half, and interpolate between those solutions to get the background.

The noise of the background determination depends on the structure of the background. For example, if there is no variation in background across the entire coadd, then all cubic polynomial coefficients will be zero except for the mean level, which will be determined from the mean of all the 8 x 8 medians. The error in the determination of that mean is:

1.25 * n_pixel / { (64/[4*1.7²]) * 128 * 64 }^1/2 = 0.0059 n_pixel

If the background variation requires a cubic, and is well-fit by a cubic, which is normally the case except for H-band observations with high airglow, then intuitively the coadd is effectively divided into 4 x 4 regions with the resulting errors then being 4 times higher. ("intuitively" means that we don't really know if this is the answer, but it seems logical! if our reasoning is faulty, please let us know.) The resulting noise in the determination of the background is then:

0.024 n_pixel

If the background variation requires a higher-power polynomial than a cubic, then the error in the background will no longer be determined by the pixel noise. Instead, the background error will result from the residual error after a cubic polynomial is fit. This will be the case for airglow variation, which can vary on high spatial frequency scales.

Resulting Photometric Error

The following table gives the photometric error caused by several levels of noise in the determination of the background error, along with ancillary information, for galaxies at the "10% flux accuracy" Level-1 Specification at H, and scaled for the other bands. The isophotal mag at the 20^th mag per square arcsecond contour is used, and all bands have a radius of 6-10 pixels to that contour, containing 110 to 310 pixels.

The range for the Noise Due to Background (DN) is for 110-310 pixels. We used the following zero points from the prototype camera, which need to be updated to the actual processing values: j=20.84, h=20.63, k=19.94.

In order to keep the noise in determining the background level from increasing the total photometric noise by more than 10%, that noise must be less than half the Poisson noise. The noise in determining the background level must then be less than about 0.05 DN, using the largest aperture, the highest background and the usual case where a cubic fit is needed. Fortunately, the above results show that the typical noise in determining the background is ~0.01-0.02 DN, safely below that number. This will be true even if the change is made to fit half the coadd at a time, increasing the noise by sqrt(2), and will probably lead to an overall improvement since it will lower the "untracked" background amount.

Measured and Expected Noise From Annuli Around Galaxies

In order to investigate the actual accuracy of determining the background, T. Jarrett changed GALWORKS to report the median flux in an annulus centered on each galaxy candidate. For most sources, the annulus spans a radius of 10 to 20 pixels from the center of each source.

The 5 duplicate scans over the Hercules Supercluster on 970521n were used for the analysis, since these scans have been properly calibrated, are photometric, and are free from hardware, operational or airglow problems.

Three problems prevent these annuli from being a true measure of the background:

• Galaxies with an r^1/4 profile will have a significant amount of their own flux in that aperture. In particular, 7% of the total flux of an elliptical galaxy will be present, all with values of 0.1 DN or below for a 13^th total mag galaxy. The exponential disk of a spiral galaxy will contribute no flux to that annulus, but 7% of its bulge component will be present.

• Neighboring galaxies will have some or all of their flux in that aperture, at a significantly higher rate than would be found at random positions, due to the non-zero 2-point positional correlation coefficient of galaxies.

• Faint stars will create increased flux in the aperture and give the distribution a tail toward positive fluxes. (Reminder: stars brighter than {~2 mags fainter than the galaxy flux requirement} are blanked in the aperture and for the coadd background. Stars fainter than this contribute negligibly to the coadd background, but significantly to the annular flux when they are present.)

The duplicate scans allow us to examine both the repeatability of the annuli fluxes using fixed positions on the sky as well as to examine the distribution in area as well as repeat. Using the fixed positions on the sky eliminates the dispersion caused by the above effects, but the mean flux will remain biased by them.

The statistics for those scans are presented in the following table. The sigma due to sources is the scatter required in addition to the sigma using fixed sources in order to equal the sigma using all sources. The sigma due to sources presumably results from the combination of the three factors mentioned above.

The sigma using fixed sources, 5 repeats was calculated by first computing the unbiased estimate of the sigma for every source and then averaging those sigmas. |Means for individual sources| above 0.4 DN were tossed prior to that computation.

The sigma using all sources was calculated by computing the scatter of all measurements using all sources, after tossing the data points with |mean backgrounds| above 0.4 DN.

The bias values for each case would be identical if no points were tossed. Since outliers are tossed somewhat differently for each computation, they differ slightly. Note that points do exist off the boundaries of the plots.

The expected scatter in total flux for an annulus between a radius of 10" and 20", containing 942 coadd pixels, is:

1.7 * 2 * 31 * n_pixel = 105 n_pixel,

which is a scatter of 0.111 n_pixel in the mean flux of the annulus, which is 0.066 at J and 0.105 - 0.141 at H and K (from detailed computation). These are actually upper limits to the value of the noise, since blanking can remove some of the pixels from the aperture, resulting in lower noise. With this is mind, as can be seen from the above table, the sigma using fixed sources, 5 repeats is in agreement with those values.

The sigma due to sources is essentially the same in all bands.

Photometric Noise From Sources in the Annulus

For all the computations below, we calculate the contribution of various sources of noise in the annulus from a radius of 10" to a radius of 20" centered on each galaxy. The goal is to understand the source of the mean annuluar bias and the sigma due to sources found in the previous table.

The conclusion is that all three of the sources considered below contribute to the flux mean and scatter of the annulus. The table below gives a summary of the effect of each source alone at K-band:

The mean bias is easily achievable due to a combination of all of these factors. The sigma due to sources is not so easily understood, but it is certainly possible that the contribution of neighboring galaxies is larger than calculated below, which might close the gap between the sum of these sources and the observed effect. At face value, another source of noise with sigma of 0.050 DN is needed to reach 0.070 DN.

Flux From Unblanked Point Sources

Point Sources are blanked in all computations if they are brighter than ~2 mags (a namelist parameter) below the galaxy flux requirement. Thus point sources below ~17.0, 16.3, and 15.5 mag, for J, H and K, respectively, contribute flux to the annulus if they are present. Their effect scales directly with stellar density.

Consider first K band at high galactic latitudes, where the density of sources is 330 sources per square degree brighter than 14^th mag. Assume that the density scales as 10^0.4*m_K for fainter magnitudes. The density of sources above 15.5 mag is then ~1300 per sq. deg., and between 15.5 and 16.0 mag is 770 per sq. deg.

The maximum effect will occur when a star with 15.5 mag appears in the annulus, which makes the mean flux of the annulus brighter by 58 DN / 942 = 0.06 DN. This occurs for 0.06 of all annuli.

Of course, stars fainter than this flux will also be found in the annulus. The following table gives some rough numbers of the contribution of stars with various magnitudes:

Given these probabilities and fluxes, we can then randomly assign those sources among a set of 100 annuli and get the expected distribution below:

Summing up the contributions from the table gives a mean bias of 0.015 DN and a sigma due to sources of 0.022 DN.

Extra credit discussion: It is true that the total flux from faint stars diverges if the number of sources scales as 10^0.4*m_K. However, when the expected number of sources grows beyond one per annulus, the flux from those sources becomes part of the background and thus does not affect the aperture fluxes.

Flux From the Galaxy Itself

As mentioned above, galaxies with an r^1/4 profile will have a significant amount of their own flux in that aperture. In particular, 7% of the total flux of an elliptical galaxy will be present, all with values of 0.1 DN or below for a 13^th total mag galaxy. The exponential disk of a spiral galaxy will contribute no flux to that annulus, but 7% of its bulge component will be present.

To get an upper limit to this effect, assume that half of all galaxies are ellipticals. Half of all annuli will then have an extra 7% of the elliptical galaxy flux, corresponding to 42 DN for a galaxy with m_K = 13 mag. This results in an average 21 DN extra flux in the total annulus, which is a mean bias of 0.022 DN for the mean annulus flux, about 40% of the observed mean bias. The scatter resulting from this distribution of fluxes is 0.024 DN.

Flux From Neighboring Galaxies

It is well-known that the probability of finding a galaxy near another galaxy is significantly higher than that expected due to the average density of galaxies. This is true of galaxies in "the field", whatever that means, as well as of galaxies in a "cluster".

This enhanced probability is usually stated in terms of a two-point correlation function w(theta), such that the probability of finding any galaxy at an angular distance theta from another galaxy is {1 + w(theta)} times the normal Poisson probability density. The factor {1 + w(theta)} is the ratio of the true probability of finding a galaxy to that found by chance for objects placed at random.

To get a rough order of magnitude for this effect, we blindly apply the 2-point correlation coefficient found for optical galaxies with b_j = 19.5 from the EDSGC (we forget what that stands for...) Assuming that m_K - b_j is ~4 implies that the b_j = 19.5 mag value might roughly represent the 2-point correlation function for galaxies with m_K ~ 15.5, which would not be blanked out in the annulus.

The value of w(theta) found in the EDSGC is 0.12 * theta^-0.58, where theta is expressed in units of degrees. The values are then 3.6-2.4 for a radii of 10-20", implying a factor of 4.6-3.4 times the random probability.

Assuming that the density of galaxies brighter than m_K ~ 13.5 is 25 sources per square degree, appropriate for the Hercules cluster studied here, and wildly extrapolating using a power law of 1.5, the density of galaxies with m_K ~ 15.0 - 15.5 is 200 per square degree. This implies 0.015 such galaxy per randomly-chosen annulus, and 0.051 - 0.069 such galaxies per annulus centered on a 2MASS galaxy.

Of course, galaxies fainter than this flux will also be found in the annulus. Further, even when the number of such galaxies in the annulus becomes large, they still affect the mean flux of the annulus since their density is higher in these annuli than their mean density in the coadd. The following table gives some rough numbers of the contribution of galaxies with various magnitudes:

Given these probabilities and fluxes, we can then randomly assign those sources among a set of 100 annuli and get the expected distribution below:

Summing up the contributions from the table gives a mean bias of 0.041 DN and a sigma due to sources of 0.040 DN.

Analysis of the "postage stamp" images could provide some evidence for a nonzero correlation coefficient for 2MASS galaxies. A postage stamp is 100" x 100". For the typical galaxy densities of 25 per square degree above m_K = 13.5 mag, the expected number of postage stamps with two such galaxies is 0.02 from chance. This implies that among 25 galaxies, half of all such fields should have two galaxies found in the same postage stamp. The value of the 2-D correlation coefficient is still 1.0 even at a radius of 100", and hence the expected number among 25 galaxies is greater than unity.