Hubble Space Telescope
A Possible Non-linearity in WFPC2 Observations

Abstract:

The analysis of a set of repeated observations of the same stellar field with different exposure times indicates that the photometry of faint stars can be affected by an apparent charge loss similar to a CTE effect. This effect takes the form of a photometric discrepancy between long and short exposures for the same star, in the sense of the magnitude being fainter in shorter exposures. This effect has been reported previously.

More precisely, the discrepancy appears to be a function of the total counts for a star. The photometric error is very small above 1000 total counts and rises above 0.05 mag - possibly up to 0.15 mag - below 100 total counts. The discrepancy does not appear to be a function of row number for the crowded field studied here.

A quantitative characterization of the photometric discrepancy is offered, as well as a simple-minded fitting formula which might help correct for the discrepancy.

Introduction:

This is a preliminary report on the results of detailed aperture photometry in a field about 5 arcmin from the core of omega Centauri. This field was observed in F814W as a pure parallel for the MDS project on May 7, 1994. The observations consist of 16 exposures with exposure times ranging between 140s and 1200s, taken over 6 orbits between 3:07 and 10:37 UT. All exposures were taken in equal-length pairs, the two equal exposures always being in the same orbit. Orbits 4 and 5 contained 2 pairs, the others only 1 pair. Orbits 1 and 6 were much shorter than the others because of the slew time.

This field was chosen to investigate the reported photometric difference between long and short exposures (Hill et al., preprint) because of the large number of stars available and of the broad range in exposure times. Because of the need for accurate photometry and of the extreme crowding in the WF cameras, only the PC was used. The coordinates of the center of the PC are RA 13:27:14.2, dec -47:29:29. Similar results have been obtained for the other cameras by Massimo Stiavelli, using observations of the globular cluster NGC 2419.

Photometry:

Fluxes were determined using aperture photometry, via the IRAF task noao.apphot. Stars were measured on individual frames, without combination for the purpose of cosmic ray rejection. Combination of frames was seen to introduce a small, exposure-dependent systematic effect on the photometry that could interfere with the effects we wish to measure. This problem is due to slight offsets between images in each series, and it can be minimized by allowing a non-zero noise sensitivity parameter in the combination task; however, for the purpose of maintaining the results as clean as possible, we decided to perform photometric measurements on individual images, and to reject stars for which the exposures in each series give substantially discrepant results.

The series of commands used to carry out the photometric measurements is as follows:

Target Selection:

Stars selected using noao.apphot.find on a master combined frame; stars within 20 pixels of about 20 saturated stars rejected. Total number of accepted targets: 1853

Raw Photometry:

Aperture photometry for radii of 1,2,3,4,5 pixels using noao.apphot.phot. Each star individually recentered in each image; sky computed with median in the annulus (10,15).
Actual command for file name:
phot name output=name.mag coords=starlist.dat readnoi=7 epadu=14 aperture=1,1.5,2,3,4,5,6 salgori=median zmag=28.14 annulus=10 dannulus=5 calgori=centroid interac=no verify=no > name.log

Table Manipulation:

Convert output of aperture photometry into ascii file.
Actual command for file name:
txdump name.mag xcenter,ycenter,msky,flux,mag,merr mode=h > name.dat

Combination and Rejection:

For each aperture and exposure time, average the magnitude obtained in the two executions, rejecting stars for which photometry differs by more than three combined sigma in the two executions (probable cosmic ray)

Photometric Uncertainty:

The main area of concern is the determination of the sky level. Since we will see that the photometric discrepancy, whatever the cause, can be interpreted as an error in the sky, special care was given to this determination. A number of different methods were tried---median, average, optimal filter---and it was found that all yield essentially the same results for these data. Also, the measured sky value does not appear to correlate with the photometric discrepancy (see below), which would be expected if an underestimate of the sky level---presumably different from star to star---were responsible for the discrepancy. Note that any contribution of the star PSF to the sky in the chosen annulus would be proportional to the star signal and thus result only in an overall zeropoint offset - independent of count level or exposure time.

Basic Photometry Comparison:

The immediate result of the comparison between photometry measurements for the same object at different exposure times is that count rates are smaller (stars appear fainter) in short exposures. The discrepancy is small for bright stars, and increases for progressively fainter stars, up to 0.10 mag or more.

The magnitude discrepancy is clearly illustrated in the following series of figures, which show the difference between magnitude in short and long exposures as a function of magnitude. At a fixed magnitude, the discrepancy seems larger the larger the aperture.

The points shown in the following figures are the result of averaging the two available exposures for each exposure time, and rejecting any objects that differ by more than 3 times the combined sigma in the two equal-length exposures. Most points lie before the horizontal line of zero difference, indicating a fainter magnitude in the shorter exposure.

Comparison of 140 and 600s exposures within 3 pixels, uncorrected

Comparison of 300 and 600s exposures within 3 pixels, uncorrected

Comparison of 140 and 600s exposures within multiple apertures, uncorrected

Comparison of 140 and 1200s exposures within multiple apertures, uncorrected

Comparison of 300 and 1200s exposures within multiple apertures - uncorrected

Low vs. High Row Number:

The apparent similarity with the well-known CTE problem (Holtzman et al 1995), which results in a row-dependent throughput variation of about 4% between the top and the bottom of each WFPC2 chip, suggested that this discrepancy could also have a row dependence. However, the problem exists at both low and high row numbers, and it appears to vary little with row number, especially at the faint end.

Comparison of 300 and 1200s exposures within 3 pixel for y > 600

Comparison of 300 and 1200s exposures within 3 pixel for y < 200

Comparison of 300 and 1200s exposures within 4 pixel for y > 600

Comparison of 300 and 1200s exposures within 4 pixel for y < 200

Comparison of 300 and 1200s exposures within 5 pixel for y > 600

Comparison of 300 and 1200s exposures within 5 pixel for y < 200

Binning in magnitude makes the comparison clearer - the figure below shows the median and quartiles of the magnitude discrepancy for low and high y in the same plot.

Comparison of 300 and 1200s exposures within 4 pixel for both low and high y - median and quartiles

Distance to Nearest Star in the Same Column:

It has been suggested (Burrows, private comm.) that the lack of an obvious row dependence could be due to the crowded nature of this field; a possible test of this would be to correlate the magnitude discrepncy not with row number, but rather with the distance between each star and the nearest star below. This is shown in the next two figures, which plot the magnitude discrepancy (aperture 4 pixel, 300s minus 1200s) vs. the distance to the nearest star underneath. This is defined as follows: consider the column centered on the star; check the pixels beneath the star on the same column, excluding the first 10; and mark the first pixel exceeding a given threshold. The threshold used here is 3 times the sky (any star) or 10 times the sky level (bright star).

Unfortunately, because of the density of stars in this field, the results are inconclusive - the vast majority of stars has another bright object close to it.

Magnitude discrepancy vs. distance to nearest star underneath

Magnitude discrepancy vs. distance to nearest bright star underneath

Discrepancy vs. Measured Sky Level:

As mentioned above, an error in the measured sky level can have serious consequences on the accuracy of the magnitudes obtained, especially for the short exposures. However, if this were the cause of the observed magnitude discrepancy, we would expect a strong correlation between measured sky level (in the short exposure) and magnitude discrepancy - as shown below, this is not the case. Plotted are median and quartiles of the magnitude discrepancy, binned in half-magnitude intervals, for sky level below and above the median value of 3.0 DN. No obvious correlation is seen.

Magnitude discrepancy (300 vs 1200 s) for low and high sky levels

Magnitude or Counts?:

The next question is whether the photometric discrepancy is driven mainly by exposure time or by count levels. This can be answered by comparing the discrepancy measured at the same count level for different exposure times, amintaining the same ratio between long and short exposure. For example, we compare the discrepancy between 140s and 600s and that between 300s and 1200s - in both cases the ratio of exposure times is about 4. The same discrepancy is observed at the same number of counts in the short exposure, thus indicating that total counts are the key element driving the discrepancy. The discrepancy is about 0.07 mag for stars with 100 counts (DN, gain 7) in the short exposure, slightly larger than the theoretical 1-sigma uncertainty in the magnitude measurement itself. (Of course, this is a systematic error, and it should not be treated as an additional scatter.) It seems to depend weakly on the exposure time ratio, indicating that most of the discrepancy is due to the short exposure.

Comparison of photometry within 3 pixel aperture for exposure time ratio = 2

Comparison of photometry within 3 pixel aperture for exposure time ratio = 4

Simple Correction (Fixed Charge per Pixel):

Everything seen this far is compatible with a very simple-minded interpretation: a small amount of charge is lost in each pixel used to determine the magnitude. This simple scheme explains the dependence on total counts, the increase in the discrepancy for increasing aperture size, and the weak dependence on exposure time ratios. In fact, a model based on the loss of somewhere between 2 and 2.5 electrons per pixel, independent of magnitude, counts, exposure time, or distance from the star's center, is in fairly good *quantitative* agreement with the observed discrepancy. The plots below show how the discrepancy is reduced, and essentially disappears, when a fixed number of electrons is added to every pixel in the aperture used for the magnitude measurement.

Of course, this simple model is unphysical. How can charge be lost only within the aperture, and not in the sky? (If the same amount of charge per pixel were lost in the sky, then the net result would be nil.) Thus this model is to be considered only as a rough approximation for a more complex, and more physical, process - charge might be lost as a function of the count level in each pixel, but with a threshold that would make pixels within the aperture - with their increased signal level with respect to the sky - more sensitive to such a loss. One such model, still very phenomenological but at least more realistic, is considered later.

However, regardless of how unphysical the model is, it is useful to remember that, within the range of parameters included here, a fixed charge loss of 2-2.5 e/pixel reproduces most of the features of this photometric anomaly.

Comparison of 140 and 600s exposures within multiple apertures with simplistic charge-loss correction of 2.0 e/pixel

Comparison of 140 and 600s exposures within multiple apertures with simplistic charge-loss correction of 2.5 e/pixel

Comparison of 140 and 600s exposures within multiple apertures with simplistic charge-loss correction of 3.0 e/pixel

Comparison of 140 and 1200s exposures within multiple apertures with simplistic charge-loss correction of 2.0 e/pixel

Comparison of 140 and 1200s exposures within multiple apertures with simplistic charge-loss correction of 2.5 e/pixel

Comparison of 140 and 1200s exposures within multiple apertures with simplistic charge-loss correction of 3.0 e/pixel

Comparison of 300 and 1200s exposures within 3 pixel with simplistic charge-loss correction of 2.5 e/pixel

Comparison of 300 and 1200s exposures within 4 pixel with simplistic charge-loss correction of 2.5 e/pixel

Comparison of 300 and 1200s exposures within 5 pixel with simplistic charge-loss correction of 2.5 e/pixel

Comparison of 300 and 1200s exposures within multiple apertures with simplistic charge-loss correction of 2.0 e/pixel

Comparison of 300 and 1200s exposures within multiple apertures with simplistic charge-loss correction of 2.5 e/pixel

Comparison of 300 and 1200s exposures within multiple apertures with simplistic charge-loss correction of 3.0 e/pixel

"Better" Correction Formula:

A more realistic correction is based on the assumption that the flux measured in each pixel reflects a "loss" with respect to the incident flux. In order to reproduce the behavior seen so far, this loss should increase less than linearly with the incident flux, thus resulting in a proportionally smaller count rate for lower signal levels.

A simple formula that seems to satisfy these requirements is:

Delta f = A * f / sqrt (f² + B²)

This formula does not work perfectly (see Figures below). Depending on the choice of parameters, the "corrected" magnitudes appear to be over-corrected for some apertures and under-corrected for others. We are currently experimenting with some additional parameters which might help change the shape of the correction and obtain a better fit to the observed discrepancy, however none of the formulae tried thus far are as successful as the simple "loss of 2 e/pixel" law.

Curves for Fixed Background:

The curves shown here indicate the magnitude error predicted by the correction formula for fixed sky background (in counts/s), when applied to stars of different magnitudes for various exposure times and apertures. The general trend in the magnitude correction is right - larger errors for larger apertures and shorter exposure times.

Nominal correction curves for A=2, B=3, fixed sky (0.01 DN/s)

Nominal correction curves for A=2, B=5, fixed sky (0.01 DN/s)

Nominal correction curves for A=3, B=10, fixed sky (0.01 DN/s)

Nominal correction curves for A=3, B=3, fixed sky (0.01 DN/s)

Nominal correction curves for A=3, B=5, fixed sky (0.01 DN/s)

Comparisons of Actual Magnitudes after Correction:

Here are the curves of actual corrected magnitudes, taking into account the true measured sky level for each star, for two choices of parameters. In both cases the correction is "optimized" for the largest aperture (5 pixel), but the discrepancy appears overcorrected in the small apertures. If the parameters are chosen so as to optimize the correction in the small apertures, the large apertures are undercorrected. The reason is that the correction given by this formula is related to the flux, and thus the effect fades rapidly with increasing distance from the center - contrary to the observed discrepancy, which increases significantly with aperture.

Comparison of 300 and 1200s exposures within multiple apertures after revised correction formula, A = 2, B = 10

Comparison of 300 and 1200s exposures within multiple apertures after revised correction formula, A=2.5, B=5

-Stefano Casertano (stefano@stsci.edu)