The Charge‐Transfer Efficiency and Calibration of WFPC2

The ground‐based ω Cen data were those of Walker (1994), with stars fainter than V = 21.0 eliminated because of the large scatter in the data at the faint end. The ground‐based NGC 2419 data were provided by P. Stetson, in which a similar faint‐end cut had already been made. The faint cuts in both data sets, in addition to reducing the amount of low‐quality data in the fits, avoid the photometry bias that is found just above the cutoff. As the WFPC2 data are generally deeper than the ground‐based data, they also avoid this effect.

Given the superior resolution of HST over ground‐based telescopes, it is not surprising that many of the ground‐based stars were resolved into multiple star systems when observed by HST. In order to eliminate errors from this effect, all WFPC2 stars that fell within 08 of the ground‐based standard star were combined into a single star. If the combined magnitude was more than 0.05 mag brighter than that of the brightest constituent star, the standard star was thrown out. Otherwise, the star was kept, with the combined magnitude used for the WFPC2 magnitude in the analysis. If at least 25% of all images with detections of a star contained bright companions, the star was eliminated from the analysis altogether. (In other cases, it was assumed that a cosmic ray was responsible for the second detection, as no cosmic‐ray cleaning could be made.) In this process, 29 ω Cen stars and 13 NGC 2419 stars were eliminated.

For the remaining stars, the expected WFPC2 flight‐system magnitudes were calculated using the H95 zero points and color terms with the equation

where the values are defined as in H95. For this analysis, it is assumed that the color terms from H95 are correct, a necessary assumption given that these data do not contain a range of colors sufficient to recalibrate the color terms. However, filter‐dependent corrections are determined for the zero points.

WFPC2 observations of the ω Cen and NGC 2419 standard fields were obtained from the STScI archive, using on‐the‐fly calibration to process the images using the best available calibration data at the time of retrieval. The ω Cen data consist of 795 images, mostly in the standard filters (117 F439W, 271 F555W, 88 F675W, and 235 F814W), with additional images reduced but discarded because of insufficient overlap with the ground‐based field. Additionally, data in secondary filters were also obtained (eight F380W, eight F410M, four F450W, six F467M, nine F547M, seven F569W, 13 F606W, two F622W, nine F702W, seven F785LP, two F791W, two F850LP, and four F1042M). Observations in U, although available, were omitted because of the lack of ground‐based comparison photometry in the Walker (1994) data.

The NGC 2419 data consist of 51 images (seven F555W and 44 F814W). F300W images were also available for NGC 2419 but were omitted in this analysis because of the lack of interest in calibrating F300W (as per the recommendation of H95). In the interest of producing the best possible calibrations, the chips containing NGC 2419 itself (WFC4 for five of the images, WFC2 for the remainder) were omitted from the analysis because of the higher crowding. This caused a mean magnitude offset of ∼0.02 mag, which is reasonable in crowded field photometry but undesirable in a calibration study, especially since the NGC 2419 data provide most of the high‐background points.

Photometry was made using the usual HSTphot recipe, except that cosmic‐ray cleaning was omitted because the images were reduced individually. Aperture corrections were made to correct to the 05 aperture of H95. In order to reduce the number of bad points in the analysis, all stars were required to have χ≤1.5 and - 0.3≤sharpness≤0.3 to be used in the calibration solution. Additionally, all stars near the WFPC2 saturation limit were removed, as were stars with fewer than 105 electrons, which contributed more noise than signal to the solution. In all, 843 images were used for the solution, producing a total of 58,728 cleanly fit stars that were matched to the standard stars.

As noted in § 1, the most likely explanation of the apparent package dependence of the CTE loss is that different photometry packages calculate the background differently. Specifically, any package that determines a background level close to the star will invariably measure the wings of the star as part of the background. The DoPHOT sky measurement, as made by Saha et al. (1996) and SLP00, is perhaps the most severe example of this effect, with its average sky pixel containing ∼0.2% of the star's total light. HSTphot, which also determines sky values near the star (although not as close as DoPHOT), shows this dependence at a smaller level.

In an attempt to create as generally useful a CTE determination as possible (as well as to provide independent parameters for the solution), the count dependence of the background was solved and subtracted from the background levels before running the calibration solution. Because the background contamination is proportionally the most significant at low background levels, it was calculated using the low‐background ω Cen data only.

This dependence was determined to be chip dependent, in that it was about an order of magnitude smaller in PC1 but the same in the three WFCs, wavelength dependent (smallest for B and V), and temperature dependent (slightly larger at cold temperature than at warm temperature). The correction equations and coefficients are not given here, since this result almost certainly will not apply to other software packages. For example, DoPHOT background levels have a factor of ∼20 greater dependence on the number of counts than do HSTphot background levels.

Finally, as a negative background is a nonphysical result of readout noise or bias errors on a very small background, all negative values were set to zero. The background and counts were converted to electrons by multiplying by the gain (7 or 14) and the background then softened according to S98's procedure [background = (1 + N²_electrons)^1/2] to allow logarithms to be taken at zero background. For the remainder of this paper, counts will refer to the number of electrons detected for a star, while background will refer to the background level in electrons, so that the value of the gain is unimportant. If discussing values in DN rather than electrons, it will be made explicit.

The nature of the data in this present study—WFPC2 photometry with comparison ground‐based photometry—permits a more direct study of the position‐independent error ("long‐short anomaly") than what was given by CM98. The complete sample of 58,728 stars was restricted to points with limited ranges of counts, background, and epoch, thus producing a relatively homogeneous data set that should have a common set of correction factors. To maximize the ability to determine the position‐independent correction, the counts were restricted to between 210 and 350, for which the average position‐independent correction (based on CM98) should be about 0.23 mag. The background was required to be 3.5 or less, which would include the greatest amount of data. The differences between WFPC2 and ground‐based magnitudes were then fit to the formula

(Δcounts could have been used equally well, as the next section demonstrates, with the same result produced). The term ΔZP is the zero‐point correction from § 4 and is of order 0.01–0.02 mag, depending on filter and chip. (In the initial solution, the ΔZP term was omitted, producing a result less than 0.01 mag different from the one below. However, to minimize systematic errors, the term was added and the solution redone after running the CTE solution.)

A robust fit parameter was minimized using the nonlinear minimization routine dfpmin from Numerical Recipes in C (Press et al. 1992), providing the best values of the three parameters, and uncertainties estimated via bootstrap tests. In the solution, c₁ and c₂ are the Y‐CTE and X‐CTE terms, respectively, while c₀ is the position‐independent offset.

For the NGC 2419 data that were used by CM98, 21 stars were found that met the count and background criteria, producing a c₀ value of 0.02 ± 0.07 mag. Expanding the sample to all 1997 data to improve the statistics,c₀ was determined to be 0.04 ± 0.05 mag. To further improve the statistics, the count range was changed to 350–700, producing 114 points and a correction of - 0.01 ± 0.02 mag despite the expected value of 0.12 mag from the CM98 correction.

Although the uncertainties are significant, a position‐independent offset of 0.23 mag in the faint data is ruled out at almost the 4 σ level, while the expected correction of 0.12 mag for brighter points is ruled out at more than the 5 σ level. Both samples, however, are consistent to well within 1 σ with no anomaly, and therefore the CTE analysis below does not include a position‐independent correction except for the zero‐point corrections, ΔZP. Further discussion of the long‐short anomaly is given in § 5.3.

With the result of no position‐independent correction, the functional form of the charge loss will include only X‐CTE and Y‐CTE corrections. The first issue to be determined is whether the charge loss is of the form

or

As is demonstrated in Figure 1, which shows both magnitude residuals and the ratio of counts lost as a function of Y for a subset of the data, the choice of functional form does not make a significant difference. However, the plot as a function of magnitude difference gives the slightly better fit, so the form given in equation (4) will be used for this study.

Zoom In Zoom Out Reset image size

The CTE loss is assumed to depend only on counts, background, temperature, and epoch. This assumption was tested by determining the CTE solution for subsets of the data restricted by chip, gain, and filter. These subset CTE determinations produced corrections that were not significantly different from the full solution, thus verifying the assumption.

Finally, a number of numerical and observational criteria were set out, which the CTE functional form must satisfy:

• 1.  
  There must be no combination of counts, background, and epoch that would cause the formula to become undefined.
• 2.  
  As the CTE effect is one of lost charge, the correction must never be negative.
• 3.  
  The logarithms of the mean magnitude residuals appear to scale roughly linearly with the logarithms of counts and background, implying a form involving e^{c_ctln(counts)} and e^{c_bgln(background)}.
• 4.  
  A sufficiently high background will eliminate the CTE loss. For example, stars with fewer than 1000 counts and more than 1000 background have an average offset of - 0.001 ± 0.015 mag.
• 5.  
  A sufficiently high count rate will reduce but not eliminate the CTE loss. For example, stars with over 21,000 counts but under 10 background have an average offset of 0.031 ± 0.001 mag.
• 6.  
  A sufficiently early time will again reduce but not eliminate the CTE loss. For example, the average cold camera offset in 1994 is 0.020 ± 0.001 mag.
• 7.  
  High count rates do not eliminate the time dependence, nor do early epochs eliminate the count dependence. For example, stars with 21,000 or more counts increase from an average offset of 0.018 ± 0.002 to 0.041 ± 0.004 mag from 1994 to 2000. Likewise, the 1994 correction increases from 0.018 ± 0.002 mag for stars with at least 21,000 counts to 0.083 ± 0.002 mag for those with between 500 and 1000 counts.

These observations restrict the set of functional forms that can be used. Attempts were made to adopt different forms obeying the above restrictions, with the formulae below producing the best fits to the data:

The offsets of 1996.3, 7, and 1 were roughly the averages of these values in the data and were included to improve numerical stability and to produce independent coefficients. (Failing to do so would have provided correlated coefficients and thus correlated errors.) The addition of the y₆ parameter was required to eliminate an overcorrection at moderate‐ and high‐background levels seen in the first solution. No similar x₆ parameter was needed. Additionally,x₀ and x₃ terms analogous to the y₀ and y₃ terms were initially included but were determined to be insignificant (well less than 1 σ) and subsequently removed from the equation.

Note that in the above functional form, all coefficients y_i and x_i must be positive to meet the requirements itemized above. Additionally, to avoid negative corrections at early epochs (1994.3), y₁ must be at least twice y₂ and likewise for x₁ and x₂.

Separate sets of constants were determined for the warm (−76°C) and cold (−88°C) camera observations. However, all warm camera observations were made with little or no background and at similar time (1994 January through April), so no background or time dependence could be determined. Additionally, no X‐CTE loss was measurable in the warm data.

In addition to the CTE solution, a set of zero‐point corrections was also determined. The first correction was made to search for any offset between the ground‐based ω Cen and NGC 2419 data. This correction factor was determined to be well under 0.01 mag (and less than a 1 σ effect) and was eliminated.

The remaining corrections, as functions of filter, temperature, chip, and gain, were divided into two sets. This division, though made primarily for numerical reasons, was not entirely arbitrary. The "ideal" solution would naturally involve a zero‐point correction for every combination of temperature, filter, chip, and gain, or a total of 272 free parameters. This total was reduced to 25 by adopting the following assumptions and simplifications:

• 1.  
  The principal factors affecting detection efficiency are wavelength and temperature. This makes the assumption that the four chips have similar wavelength‐dependent quantum efficiencies (an overall sensitivity difference would be corrected by item 3 in this list), which was verified by determining the CTE corrections individually for the four chips. This creates 34 filter‐ and temperature‐dependent zero‐point offsets.
• 2.  
  Although the nonstandard filter offsets are temperature dependent, the differences between their corrections and the corrections of the standard magnitudes of the same color are observed to be independent of temperature. This allows the consolidation of the 26 nonstandard filter zero‐point corrections to 13 corrections relative to the nearest standard filter and the reduction of the 34 total temperature/filter zero‐point offsets to 21.
• 3.  
  The principal factors in the readout efficiency are chip and gain setting, as each of the eight combinations involves slightly different electronics. This creates eight zero‐point offsets dependent on chip and gain. After running the solution, it was determined that the gain setting made no measurable difference, further reducing these offsets to 4.

Thus the total zero‐point correction is thus

As with the CTE loss correction, all of the zero‐point corrections were computed in the sense that they should be subtracted from the WFPC2 magnitudes to produce standardized magnitudes.

The final coefficients and offsets provide the values to subtract from the observed WFPC2 magnitudes (calibrated using H95 flight‐system zero points, gain ratios, and pixel area ratios) to generate corrected WFPC2 flight‐system magnitudes. The CTE coefficients are given (with 68% confidence limits) in Tables 1 and 2 for cold and warm data, respectively. As noted above, no X‐CTE loss was detectable in the warm camera observations, but it can be characterized well in the cold observations. The time dependence of the cold X‐CTE loss is small and is consistent with the Whitmore (1998) time dependence, while the dependence on counts is much higher than that on background. The detection of any significant background dependence in the X‐CTE loss is different from S98 and WHC99, but the effect on the corrections is minimal given the size of the X‐CTE correction. The X‐CTE loss is only a minor effect, with the worst‐case combination in the calibration data (lct = -1.15;bg = 1,lbg = -1;yr = 4) producing an X‐CTE ramp of only 0.05 mag.

The Y‐CTE loss is by far the larger of the two. The primary difference between warm and cold is the Y‐CTE base value y₀, which decreased by 0.08 mag when the instrument was cooled. In contrast to the X‐CTE equation, the Y‐CTE loss is strongly dependent on counts, background, and time. Again, the time dependence is consistent with the values from Whitmore (1998). The worst‐case combination will produce a Y‐CTE ramp of 0.50 mag, consistent with the WHC99 result of a ∼40% loss for faint stars on low background in early 1999.

A second simple check can be made by determining the corrections for the conditions used by H95 (lct = 1;bg = 3,lbg = 0;yr = -1.8) to determine their corrections of 0.10–0.15 mag in the warm data and 0.04 in the cold. These values lead to Y‐CTE ramps of 0.11 mag in the warm data and 0.03 mag in the cold data, consistent with the H95 CTE loss estimates. Detailed comparisons with the recent studies are given in § 5.2.

The zero‐point offsets by color and temperature are given in Table 3. There is a clear trend of increasing offset with increasing wavelength for the cold data, and the opposite case in the warm data. This trend was extrapolated to U (F336W) to compensate for the lack of U data in the calibration sample. Because of the extrapolation, the uncertainties are likely ∼0.01 mag. The cause of these offsets is unknown, as the H95 calibrations were based on these same set of observations. Systematic differences between HSTphot and aperture photometry are unlikely, especially given the excellent agreement between the present CTE loss determination and that by WHC99. It is also not a time‐dependent effect, as this trend is still seen if only the 1994 data are used. The easiest explanation would be that H95 used both warm and cold data, as the offsets would roughly cancel out if added, but this was not the case. Compared with the synthetic zero points of Table 28.1 of the HST data handbook, the new zero points are on average 0.02 mag fainter, with rms scatter of 0.02 mag. The difference is most likely the result of the handbook zero points, which are claimed to be accurate only to 0.02 mag.

Table 4 gives the zero‐point corrections of the nonstandard filters, which should be subtracted from the observed WFPC2 flight‐system magnitudes in order to determine magnitudes on the same system as the standard filter of the same color. (In addition, the Table 3 value should be subtracted to produce magnitudes corrected to the ground‐based data.) It is assumed that these values are nonzero because the H95 transformations for these filters were synthetic rather than empirical, and thus were not directly tied to any observed system. Given the amount of observations made in F450W and F606W, it is worth noting that while the F450W magnitudes appear to be correct relative to F439W, F606W magnitudes are off by 0.02 mag relative to F555W. A comparison between the new zero points and those in Table 28.1 of the HST data handbook indicates that the handbook zero points fare quite poorly, with rms scatter of 0.08 mag between the new zero points and those of Table 28.1 for the nonstandard filters. In comparison, the rms scatter between the new zero points and the synthetic zero points of H95 is less than half of that value.

Finally, the chip‐to‐chip zero‐point corrections are 0.037 ± 0.001, - 0.012 ± 0.001,0.007 ± 0.001, and 0.004 ± 0.001 for PC1, WFC2, WFC3, and WFC4, respectively. The presence of nonzero chip‐to‐chip differences implies either a minor error in the relative pixel areas reported by H95 or a sensitivity difference between the four chips. Although other studies have determined that chip‐to‐chip offsets are a function of filter as well as gain, such an effect is not observed here at any significant level.

Rather than calibrating using H95 and then applying two or three sets of zero‐point corrections, it is simpler to apply the corrections to the calibration process. The calibration equations, analogous to equations (7) and (8) of H95 but incorporating the pixel area correction, would be

The term ΔZ_CG is the zero‐point modification for chip and gain settings, and the values can be found in Table 5. The term Z_FG can be found in Table 6, and terms Z_FS,T₁, and T₂ in Table 7, and follow the definitions of H95. The terms T₁ and T₂ in Table 7 are reproduced from Tables 7 and 10 of H95. The CTE correction should be calculated from equations (5)–(9) and Table 1 for cold data or Table 2 for warm data. Again, it should be noted that aperture corrections were made to the 05 aperture of H95 and others, rather than the nominal infinite aperture of the zero points in Table 28.1 of the HST data handbook.

Residuals (WFPC2 magnitude minus ground‐based magnitude) from before and after applying the CTE correction and revised calibration are shown in Figures 2–5, plotted against Y, X,ln (counts), and ln (background), in order to provide a preliminary test of the correction formulae. As the figures demonstrate, the correction has been successful, at least to first order, in reducing the systematic residuals to under 0.01 mag.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

A concern in adopting any functional form for the corrections is that it will not properly account for second‐order factors. For example, one frequently mentioned concern is that the count dependence in the CTE corrections changes as a function of background level. This can be readily tested, with mean residuals for combinations of low and high counts, background, and epoch shown in Table 8. None of the residuals in the table is significantly more than 0.005 mag.

However, the most significant source of concern is caused by the fact that the average calibration data are significantly different from most science data. The calibration observations are generally bright stars with little background (due to short exposure times), while most science observations are long enough to have a significant background, and most projects require accurate photometry as faint as possible. The effect of this can be tested, and the solution appears to have succeeded despite these problems, with the average residual for points with background between 35 and 105 (typical for most science exposures) being 0.003 mag.

Another potential source of error in the calibration is at the faint end, as there are insufficient stars in the calibration sample with fewer than 350 counts (and those that are present naturally have large uncertainties) to have much effect on the solution. Therefore, the solution is largely extrapolated below 350 counts and subject to significant systematic errors. Such an effect can be checked by comparing the relative photometry of the NGC 2419 field, in a manner similar to that of CM98, but using only the F814W images with no preflash. The multiple image version of HSTphot, MULTIPHOT (Dolphin 2000), was used in order to reduce the random scatter and force a common object list. The 100 and 300 s images had sufficient background (at least 1 DN pixel⁻¹) to reduce faint sensitivity and were omitted. As with the CTE study, the more crowded chip (WFC2) including the globular cluster was omitted to improve the photometry. Figure 6 shows the short (10 and 40 s exposure) magnitudes minus the 1000 s magnitudes for these stars plotted against ln (counts) scaled from the 1000 s image, before CTE correction in the top panel and after correction in the bottom. Only well‐fit stars (χ≤2 and - 0.3≤sharpness≤0.3) are shown. No systematic error is detectable, aside from the effect of the photometry cutoff beginning around ln (counts) = 5.3, or 200 electrons. Between ln (counts) of 5.3 and 5.5, the median residual after CTE correction is −0.01 mag, while the median uncorrected residual is 0.15 mag. However, even down to ln (counts) = 4.9 (134 electrons), the mode of the distribution is within 0.01 mag of zero, giving confidence in the photometry to the faintest level.

Zoom In Zoom Out Reset image size

As a final check on the corrections, the corrected WFPC2 data were used to determine combined magnitudes of the ω Cen and NGC 2419 standard stars. A comparison of ground‐based and corrected WFPC2 data is given in Figure 7, showing good agreement.

Zoom In Zoom Out Reset image size

In order to make a comparison between the present corrections and those of S98, WHC99, and SLP00, all four sets of corrections were applied to the data set used in this study. To be perfectly fair, it should be noted that this study's corrections are at somewhat of an advantage, given that the same data used to determine the corrections are now used for the comparison. However, the data set used here is identical (although larger) than that used in the earlier studies, and the quality of the photometry should be at least as good as that in the earlier studies given the use of HSTphot. Finally, the fact that all major differences are attributable to obvious causes gives confidence that the comparisons are accurate.

To eliminate the effects of zero‐point differences, the present zero‐point corrections were applied to the WHC99 study, while the S98 and SLP00 zero points were applied respectively. In order to adequately model the SLP00 background dependence, their background parameter was modified for the observed DoPHOT count‐background correlation:

Residuals (corrected WFPC2 magnitudes minus standard magnitudes) are given in Table 9.

In terms of the overall corrections in the top row of the table, the WHC99 equations produce very similar results to the present study. The S98 and SLP00 corrections are also reasonably good on average but produce considerably more scatter. The cause of this extra scatter is clear when examining the remaining rows in Table 9: neither includes a time‐dependent term and thus has a trend of increasing residual with time, in the sense that the 2000 data are significantly undercorrected in both studies. Since the WFPC2 magnitude of the average star in this data set has increased by 0.06 mag from 1994 to 2000 (and that the magnitude of the average star with 350 electrons or fewer has increased by 0.18 mag), it would have been surprising had either S98 or SLP00 been consistent with all of the available data.

A plot comparing the present CTE corrections to those of S98 using only cold F555W and F814W data obtained before 1997 and ignoring zero‐point differences is shown in Figure 8. The corrections clearly agree extremely well for this limited data set, with an average difference of 0.004 mag and scatter of 0.015 mag. However, the time dependence limits the usefulness of the S98 solution, with the 1999–2000 data producing an average difference of 0.061 mag with significantly more scatter.

Zoom In Zoom Out Reset image size

A similar comparison made with SLP00 is shown in Figure 9, using only F555W and F814W data obtained during 1997. The effect of the different functional form used for the two colors is apparent, with the F555W correction (which has no position‐independent term) producing excellent agreement (undercorrecting on average by 0.006 ± 0.010 mag) while the F814W correction (which has a position‐independent term) is considerably off (overcorrecting on average by 0.026 ± 0.025 mag). Although the zero‐point offsets will compensate for this error in conditions identical to the calibration data used by SLP00, it is dangerous to assume that this will be the case in general.

Zoom In Zoom Out Reset image size

Finally, the agreement between the present corrections and those of WHC99, shown in Figure 10 for data taken through 1999 February, is excellent. Although the data show more scatter than Figure 8, a figure using the WHC99 corrections restricted to data before 1997 (as per Fig. 8) would show only 65% of the scatter. The differences become significant only at more recent epochs, where the WHC99 functional form (with the count dependence tied to the time dependence) begins to break down. For example, the average star with fewer than 1000 counts observed between 1998 February and 1999 February has an average residual of 0.10 ± 0.14 mag with the WHC99 corrections, while those of this study produce an average residual of 0.00 ± 0.10 mag.

Zoom In Zoom Out Reset image size

Thus the differences between the results of this study and previous recent studies can be explained fairly easily. Neither S98 nor SLP00 included a time effect, producing large deviations in corrections for stars beyond the times in which the data were taken. The remaining significant scatter between this study and SLP00 is due to the use of a single parameter (the DoPHOT background level) instead of both the dark background and counts. Minor differences remain between all four corrections, due to different choices made in the functional forms.

This comparison of the different CTE solutions appears to explain the discrepancies, leading to the conclusion that, provided any count dependence is removed from the background levels, there is no package dependence in the CTE loss. It is thus suggested that the present CTE study, which contains a time dependence and corrects some of the shortfallings in the WHC99 functional form, should be applicable to all WFPC2 stellar photometry.

The previous section shows, reassuringly, that differences between previous CTE studies are primarily the result of assumptions of time dependence and the method of background calculation. However, the different results for the long‐short anomaly (a position‐independent charge loss) need to be reconciled as well.

Positive results regarding the detection of a long‐short anomaly have come from the modified zero points of Saha et al. (1996), Kelson et al. (1996), and Hill et al. (1998), all of whom found that magnitudes from long exposures are about 0.05 mag more than those from short exposures. CM98 characterized this effect, providing a correction formula that was based on the number of counts rather than the exposure time. However, S98 attempted to determine a position‐independent loss in his CTE solution, but found no evidence for it, and § 3.1 of this paper likewise finds no evidence. Finally, § 5.1, which compares magnitudes from the 1000 s exposure of NGC 2419 with those from 10 and 40 s exposures, finds no effect at more than the 0.01 mag level.

In order to understand the nature of this effect, a more detailed comparison between the present data and that of CM98 needs to be made. Specifically, Figure 6 was taken from the same data used by CM98 and compared in the same way (short magnitude minus the 1000 s magnitude) and should thus be nearly identical to the 10 s and 40 s lines of Figure 11 of CM98 (although CM98 Fig. 11 is plotted against short counts rather than scaled long counts). However, while CM98 finds a median short minus long error of 0.27 mag for ln (counts) = 5.4, the difference in this study is only −0.01 mag. The difference between the two studies is also in the pre‐CTE correction photometry, with Figure 9 of CM98 showing a magnitude difference of about +0.38 mag and the top panel of Figure 6 showing a difference of +0.15 mag at ln (counts) = 5.5, thus ruling out the CTE solution as the source of the anomaly.

The difference, then, appears to stem from the photometry. The most plausible explanation for the apparent long‐short anomaly is an overestimate of the sky by CM98, as well as by previous authors. The expected functional form of the magnitude error caused by a sky measurement error is

where Δsky is the error per pixel in the sky value and counts is the number of counts. For the 2 pixel radius used by CM98, a sky error of +0.58 DN or +4.1 electrons would match the CM98 correction formula to within 0.01 mag for stars with detections of 33 DN or more. This potential solution to the problem would also explain the aperture dependence of the effect seen by CM98, as a larger aperture should be more susceptible to a sky error. Finally, this would explain why CM98 found only a count dependence in the long‐short anomaly. This also would explain the Hill et al. (1998) remark that the effect appeared to be a constant subtraction of 2 electrons from every star pixel but not the background pixels, as a background overestimation would do exactly that.

In summary, there appears to be no convincing evidence for a long‐short anomaly. It would be quite remarkable if a real effect of 0.27 mag measured by CM98 was reduced to −0.01 mag by an error in HSTphot; rather, it is more likely that an error in the CM98 reduction was responsible for a false detection of the anomaly. Given the similarity between the CM98 correction equation and equation (14) of this paper given a Δsky error of 0.58 DN, which agree at the 0.01 mag level above 33 DN, as well as the simple explanation of the aperture dependence of the effect, the most probable cause of the CM98 result (as well as the other reports of the long‐short anomaly) is an overestimation of the background by a few electrons.

I would like to thank Alistair Walker and Peter Stetson for providing the ground‐based ω Cen and NGC 2419 photometry, respectively. This work was supported by NASA through grants GO‐02227.06‐A and GO‐07496 from Space Telescope Science Institute.