-Erich Karkoschka, Lunar and Planetary Lab, November 1995
For more information, send E-mail to: erich@pirl.lpl.arizona.edu.
For each WFPC2 filter, calibration constants for the albedo of Solar System objects are given. The transmission properties of filter FQCH4N-D, the filter most often used by Solar System observers, is analyzed.
Calibration Constants:
Solar System observers constitute a small minority among the users of the WFPC2. It is not surprising that the WFPC2 Handbook (Version 3.0, June 1995) does not have the planetary observer in mind. For example, the estimation of exposure times is given in detail for galaxies with known brightness in magnitude/arc-second2, but Solar-system objects are not mentioned.
Photometric calibration of a WFPC2 exposure yields the average flux of the object over a certain wavelength range. This wavelength range is given by the mean wavelength and filter width listed in Table 6.2 of the WFPC2 Handbook, or more accurately by the filter+system response curves shown in section 8.2. For Solar System objects, most published spectra are calibrated in terms of albedos or I/F, not in flux. The wavelength range of the albedo spectrum probed by each filter is not the same as for the flux spectrum unless the Solar flux can be considered constant with wavelength. For many filters in the visible and near infrared, this may not be a bad approximation. However, at ultraviolet wavelengths, the filter+system response has to be multiplied by the Solar flux spectrum first before a useful interpretation of a calibrated albedo is possible. In Fig. 1, the product of filter+system response and Solar flux is displayed for six ultraviolet filters. It is evident that these curves are significantly different and shifted towards longer wavelengths compared to the filter+system response curves shown in the WFPC2 Handbook.
The ordinate in Fig. 1 has been converted from Solar flux to count rates per unit wavelength for a large object of I/F=1 at 1 AU from the Sun, imaged in PC1 with gain=14. Areas below the curves correspond to count rates. Note that the horizontal and vertical scales change at wavelength 500 nm by a factor of five in the opposite way so that equal areas remain the same. Considering the inverse square law, these count rates can be easily adapted to objects at different distances from the Sun. Exposures with gain=7 give twice as many counts, exposures in Wide Field Cameras give 4.8 times as many counts as with PC1. If the object is smaller than the extent of the point spread function, the count rate in the center of the object is smaller according to the enclosed energy given in Fig. 5.2 of the WFPC2 Handbook. The throughput varies also with time since the last decontamination. Its spectral dependence was estimated on the basis of data in Whitmore (1995). The Figure assumes an average date, 15 days after decontamination. Table I gives a summary of the count rates for all filters, also for 15 days after decontamination. Where the decontamination cycle is important, data is given for 0 and 30 days after decontamination. Since the throughput of Wide Field Cameras decreases faster than that of the Planetary Camera, the factor of 4.8 in count rate between both cameras decreases somewhat after decontamination.
Three special wavelengths are marked in Fig. 1 by vertical dashed lines for each filter. The central one is the average wavelength of all deteceted photons. The left one is the average wavelength of the 50% of photons of shorter wavelength, and the right one is the average wavelength of the 50% of photons of longer wavelength. The dotted rectangle shown is the one rectangle with the same three averages and the same area (= count rate) as the filter. Average wavelengths and width of these fitted rectangles are listed in Table I.
Filters F300W and F336W have red leaks of 1.3 and 1.8 percent for an object of constant albedo. These leaks can be quite important for objects bright in the red but dark in the ultraviolet.
Methane Absorption:
The last column of Table I lists the mean methane absorption coefficient for each filter. Data was taken from Karkoschka (1994) below 1000 nm wavelength and from Fink et al. (1977) above 1000 nm. For narrow filters, this is the effective methane absorption coefficient. If the albedo of the object varies significantly inside the width of the filter, the listed number is an upper limit of the effective methane absorption coefficient since the albedo as a function of absorption coefficient is not linear, it has positive curvature. The effective methane absorption coefficient is the coefficient yielding the observed average I/F over the filter bandpass.
Three of the four methane filters and many other narrow filters are narrow enough, so that mean and effective coefficients are essentially the same. However, one methane filter, FQCH4N-D, probes wavelengths from deep methane absorptions to continuum wavelengths which has to be considered when interpreting images taken with this filter. The long wings of the filter transmission curve are not shown nor mentioned in the WFPC2 Handbook. Fig. 2 displays the distribution of detected photons with this filter for several objects. At 885 and 900 nm, the horizontal and vertical scales change by a factor of five, so that areas (=count rates) are preserved. The contribution from the wings is very dependent on the position on a planetary disk. For the three positions on Saturn's disk shown in Fig. 2, the contribution from the wings varies between 42% and 160% of the contribution from the main band. This causes an error of up to 160% in the measured I/F if the contribution from the wings is neglected. The relative contribution from the wings is 72% for Saturn and 21% for Titan (not shown).
The contribution of the wings can be subtracted if another image probing the region near the methane band is available. Filters F850LP and F953N are best suited. This requires the accurate knowledge of the filter transmission in the wings. This may not be possible currently as discussed in the next section.
The importance of the wings of the other three methane filters has not been studied, though they seem to be less important than the wing of FQCH4N-D.
Photometric Accuracy:
The accuracy of the listed count rates depend on the accuracy of both, the filter+system response functions and the Solar flux spectrum. The filter+system response function were taken from files at STScI dated 21 August 1995. Above 330 nm wavelength, the Solar flux was taken from Neckel and Labs (1983). It is better known (1%) than the filter+system response (estimated at 3%). At shorter wavelengths, the accuracy of the filter+system response may be similar to that of the Solar flux (6%) taken from Mount and Rottman (1982).
Count rates for filters used in the July/August 1994 imaging sequence of Jupiter were independently calculated using Jupiter's albedo derived from ground-based spectrophotometry by Karkoschka (1994) of 4% accuracy. With an estimated accuracy of 3% of the filter+system response, both calibrations should agree to about 5%. The differences are: F336W: +2%, F410M: +2%, F547W: -1%, F555W: -1%, F953N: +7%, FQCH4N-D: -36% (positive numbers mean that Jupiter's observations favor a higher count rate than that in the table). The ultraviolet and visible filters agree all within 2%, much better than expected. The value for filter F547W was derived by averaging many exposures, while total counts between exposures varied by 6% due to shutter timing of the short, 0.1s exposures. Calibration numbers for filter F953N changed a lot from 1994 to 1995. Last year, the discrepancy was +26%, this year only +7%, showing the progress in calibration.
The methane filter with a discrepancy of -36% clearly needs some more consideration. It is definitely the filter most often used by Solar System observers. Nevertheless, Whitmore (1995) includes calibration data for almost all WFPC2 filters, but he does not mention the methane filters. The analysis of the Jupiter data yielded the following. There seems to be a spatial variation in the sensitivity of about 30%. However, this is only seen for Jupiter while Galilean satellites yieled more consistent counts across the field. One explanation for this discrepancy is, that the transmission of the wings of the filter is spatially variable. Also, counts on Galilean satellites showed a much smaller than 36% discrepancy in absolute calibration. This may be due that most of the error in the filter+system transmission is in the wings of the filter. The observations indicate that across the location where Jupiter's disk was placed, the wings are shallower than current data files suggest. Further analysis is required before filter FQCH4N-D can be used photometrically.
Below 300 nm wavelength, the same technique (using Jupiter images) was applied to albedos of Jupiter by Wallace et al. (1972) of 10% accuracy, adjusted to the Solar flux by Mount and Rottman (1983). With the uncertainty in the solar flux and in the filter+system response, one would expect agreement to about 13%. The observed differences are: F218W: +15%, F255W: -5%. While it is hard to prove anything with statistics of two data points, it is clear that an improvement of the calibation below 300 nm wavelength is desireable. The listed calibration requires a jump of almost 30% in Jupiter's albedo near 255 nm wavelength which is not reasonable. The jump needs to be stronger than the 20% relative discrepancy because both filters significantly overlap. Spectra of Jupiter do not display a discontinuity at that wavelength. The listed count rate for the F218W filter is almost certainly too low. This discrepancy may be solved by WFPC2 observations of a solar analog star such as 16 Cyg B, or of Uranus or Neptune. Both planets are known to have albedo variations of not more than a few percent in the 200-350 nm wavelength range (Wagener et al. 1986). Such calibration exposures could be done within one orbit and would improve the interpretation of more than 30 orbits of existing images.
Appendix: Definitions of Average Wavelength and Width:
In section 8.1 and 8.2 of the WFPC2 Handbook, filter transmission curves are approximated by rectangular functions. This is done in a slightly inconsistent way. Since the mean wavelength is calculated via logarithms of wavelength, the calculation of the left and right edge of the approximated rectangular function should be done also via logarithms. Otherwise, the approximation of symmetric functions is always shifted towards shorter wavelengths. For example, the F702W transmission curve (section 8.1.6) is close to symmetric, but the approximated rectangle is apparently slightly shifted to the left. If the average wavelength is used (as it is done in this text) instead of the unconventional definition for the mean wavelength, this bias disappears.
The definition of the width in the WFPC2 Handbook yields only 82% of the width of a rectangular function. The values listed in the Table 6.2 and used in the filter transmission curves is only 83% of that value (probably a misprint in the definition of width). The result is that many filters with close to rectangular curves get approximated by rectangles seemingly too narrow. The definition of filter width used here yields the correct width for a rectangular curve.
Average Wavelength = Integral{Q(l)T(l)F(l) l dl} / Integral{Q(l)T(l)F(l) dl}
Width = 4 (Average Deviation)
= 4 Integral{Q(l)T(l)F(l) |l-m| dl} / Ingetral{Q(l)T(l)F(l) dl}
(m=median, F(l)=solar photon flux, l=wavelength).
References:
Fink, U., D.C. Benner, and K.A. Dick 1977. Band model analysis of laboratory mathane absorption spectra from 4500 to 10500 A. J. Quant. Spectrosc. Radiat. Transfer 18, 447-457. Karkoschka, E. 1994. Spectrophotometry of the Jovian planets and Titan at 300- to 1000-nm wavelength: The methane spectrum. Icarus 111, 174-192. Mount, G.H. and G.J. Rottman 1983. The solar absolute spectral irradiance 1150-3173 A: May 17, 1982. J. Geophys. Res. 88, 5403-5410. Neckel, H. and D. Labs 1983. The solar radiation between 3300 and 12500 A. Solar Phys. 90, 205-258. Wagener, R., J. Caldwell, and K.-H. Fricke 1986. The geometric albedos of Uranus and Neptune between 2100 and 3350 A. Icarus 67, 281-288. Wallace, L., J.J. Caldwell, and B.D. Savage 1972. Ultraviolet photometry from the Orbiting Astronomical Observatory. III. Observations of Venus, Mars, Jupiter, and Saturn longward of 2000 A. Astrophys. J. 172, 755-769. Whitmore, B. 1995. Photometry with the WFPC2. Space Telescope Science Institute.
Table I:
WFPC2-Filter Data for Solar-System Objects (August 1995)
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Filter Mean Fitted Count Rate Mean Methane Comment
Name Air- Rectang. I/F=1 1AU Absorption
Wavelength Width PC1 Gain14 Coefficient
(nm) (nm) (DN/s) (1/km-am)
F122M 657 614 46.5 1.1
F130LP 641 429 1200000 .61
F160BW d0 186 57 0.15 .000 decontamination
F160BW d1 189 48 0.11 .000 decon + 1 month
F165LP 641 428 1210000 .60
F170W 573 715 49.3 .80
F185W d0 423 754 10.9 .48 decontamination
F185W d1 436 772 10.3 .51 decon + 1 month
F218W d0 241 85 33.0 .000 decontamination
F218W d1 243 88 29.6 .000 decon + 1 month
F255W d0 275 61 236 .000 decontamination
F255W d1 276 62 224 .000 decon + 1 month
F300W 329 120 7160 .015
F300W uv 323 99 7070 .000 UV part only
F300W rl 764 211 90 1.2 red leak only
F336W 344 76 11900 .015
F336W uv 338 51 11700 .000 UV part only
F336W rl 720 55 200 .89 red leak only
F343N 342.7 3.2 183 .000
F375N 373.2 3.5 241 .000
F380W 407 88 43600 .000
F390N 389.0 6.5 1320 .000
F410M 409.1 20.3 12600 .000
F437N 436.9 3.5 1880 .001
F439W 434 69 46400 .000
F450W 465 120 159000 .003
F467M 467.1 23.8 27200 .001
F469N 469.4 3.6 2870 .000
F487N 486.5 3.8 3520 .022
F502N 501.2 3.9 4850 .002
F547M 549 69 188000 .021
F555W 549 172 412000 .034
F569W 567 137 339000 .042
F588N 589.3 7.1 22500 .004
F606W 604 209 703000 .074
F622W 620 132 468000 .070
F631N 630.6 4.5 14100 .007
F656N 656.4 3.2 7490 .077
F658N 659.1 4.1 11500 .089
F673N 673.2 6.8 19600 .047
F675W 672 125 405000 .33
F702W 691 200 599000 .32
F785LP 865 190 158000 3.4
F791W 785 175 309000 1.6
F814W 796 215 351000 1.7
F850LP 910 139 81500 6
F953N 954.5 7.6 2420 .32
F1042M 1022 56 2840 6
FQUVN-A 376.6 10.3 1200 .000
FQUVN-B 382.8 8.8 1100 .000
FQUVN-C 391.2 8.2 1670 .000
FQUVN-D 399.7 8.6 2840 .000
FQCH4N-A 543.6 4.9 11400 .11
FQCH4N-B 619.9 4.9 14600 .59
FQCH4N-C 727.9 5.7 11900 3.5
FQCH4N-D 889.4 23.0 5410 23
FQCH4N-D bw 848 91 464 2.8 blue wing only
FQCH4N-D mb 893.1 7.4 4920 25 methane band only
FQCH4N-D rw 935 106 27.0 3.5 red wing only
POLQ_par 650 434 866000 .68
POLQ_per 786 361 225000 2.0