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Europa's opposition surge in the near-infrared: interpreting disk-integrated observations by Cassini VIMS
Icarus 172 (2004) 149–162 www.elsevier.com/locate/icarus
Europa’s opposition surge in the near-infrared: interpreting disk-integrated observations by Cassini VIMS Damon P. Simonelli ∗ , Bonnie J. Buratti Jet Propulsion Laboratory, MS 183-501, 4800 Oak Grove Drive, Pasadena, CA 91109, USA Received 9 August 2003; revised 10 April 2004 Available online 3 September 2004
Abstract Near-infrared observations of Europa’s disk-integrated opposition surge by Cassini VIMS, first published in Fig. 4 of Brown et al. (2003, Icarus, 164, 461), have now been modeled with the commonly used Hapke photometric function. The VIMS data set emphasizes observations at 16 solar phase angles from 0.4◦ to 0.6◦ —the first time the < 1◦ phase “heart” of Europa’s opposition surge has been observed this well in the near-IR. This data set also provides a unique opportunity to examine how the surge is affected by changes in wavelength and albedo: at VIMS wavelengths of 0.91, 1.73, and 2.25 µm, the geometric albedo of Europa is 0.81, 0.33, and 0.18, respectively. Despite this factorof-four albedo range, however, the slope of Europa’s phase curve at < 1◦ phase is similar at all three wavelengths (to within the error bars) and this common slope is similar to the phase coefficient seen in visible-light observations of Europa. The two components of the opposition surge—involving different models of the physical cause of the surge—are the Shadow Hiding Opposition Effect (SHOE) and the Coherent Backscatter Opposition Effect (CBOE). Because of sparse VIMS phase coverage, it is not possible to constrain all the surge parameters at once in a Hapke function that has both SHOE and CBOE; accordingly, we performed separate Hapke fits for SHOE-only and CBOE-only surges. At 2.25 µm, where VIMS data are somewhat noisy, both types of surges can mimic the slope of the VIMS phase curve at < 1◦ phase. At 0.91 and 1.73 µm, however—where VIMS data are “cleaner”—CBOE does a noticeably poorer job than SHOE of matching the VIMS phase coefficient at < 1◦ phase; in particular, the best CBOE fit insists on having a steeper phase-curve slope than the data. This discrepancy suggests that Europa’s near-IR opposition surge cannot be explained by CBOE alone and must have a significant SHOE component, even at wavelengths where Europa is bright. 2004 Elsevier Inc. All rights reserved. Keywords: Cassini spacecraft; Europa; Near-infrared; Photometry
1. Introduction The surge in brightness seen from most solid surfaces in the Solar System near zero solar phase angle—the opposition surge—is one of the more intriguing, but at the same time enigmatic, photometric effects in planetary science. Two main components of the opposition surge have been identified, involving different models of the physical cause of the surge; these are the shadow-hiding opposition effect (SHOE), where regolith particles hide their own shadows (e.g., Hapke, 1981, 1986), and the coherent backscatter * Corresponding author.
E-mail address: [email protected] (D.P. Simonelli). 0019-1035/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2004.06.004
opposition effect (CBOE), which involves constructive interference in the regolith between incoming and scattered light beams (e.g., Hapke, 2002). The two different surge models make different predictions about how the strength of the surge should vary with the albedo of the surface: the multiple scattering that is prevalent in a brighter regolith washes out the interparticle shadows that are at the heart of SHOE and provides more scattered light for constructive interference, meaning that a brighter surface should nominally have less SHOE and more CBOE (Hapke, 2002; Hapke et al., 1993, 1998; Helfenstein et al., 1997, 1998; Hillier et al., 1999; Mishchenko, 1992; Nelson et al., 2000; Shepard and Arvidson, 1999; Shkuratov et al., 1999). Accordingly, albedo dependence is one of several effects that have been proposed
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as a tool for discriminating between the two kinds of opposition surges. The jovian satellite Europa, with its comparatively young surface rich in water ice, has to date provided oppositionsurge information from mostly the high-albedo end of the scale; at visible wavelengths, disk-integrated observations of Europa reveal a very bright object whose opposition surge becomes prominent only when the phase angle α is 1◦ or less (geometric albedo = 0.6 to 0.9 depending on wavelength, what longitude is being observed, how the extrapolation to zero phase is performed, etc.; Buratti and Veverka, 1983; Domingue et al., 1991; Thompson and Lockwood, 1992; Buratti, 1995; Domingue and Verbiscer, 1997). Attempts to make other albedos available in Europa opposition-surge observations have so far emphasized either working in the ultraviolet, where Europa as a whole is darker (Buratti et al., 1988), or obtaining close-up, disk-resolved images that include local patches of low-albedo material (Helfenstein et al., 1998). However, these attempts have involved limited phase coverage, with just one or two phase angles covered close to α = 0◦ and the next coverage out at α = 3◦ to 5◦ — not exactly optimum conditions for delineating the shape of the prominent, α < 1◦ surge. Thanks to Cassini’s December 2000 flyby of Jupiter, however, we can now add a new source of low-albedo information to Europa’s opposition-surge record: the spacecraft’s Visual and Infrared Mapping Spectrometer (VIMS) observed this moon’s disk-integrated surge at various wavelengths across the near-infrared, where Europa experiences a wide range of albedos thanks to prominent water-ice absorption bands (e.g., see the composite Europa spectrum in Sill and Clark, 1982). Although the new observations have their own limitations where phase-angle coverage is concerned, the centerpiece of the VIMS data is an important new measurement: the slope of the phase curve at α = 0.4◦ to 0.6◦ —deep within the prominent part of the surge—has been determined accurately at each near-IR wavelength by using coverage at multiple phase angles instead of just a pair of phase angles. The VIMS Europa data set has already been published in Fig. 4 of Brown et al. (2003); Section 2 below describes these new VIMS data in more detail than was possible in that earlier publication. These data are then modeled and interpreted with the aid of the commonly used Hapke photometric function in Section 3. Finally, the resulting implications about the physical state of Europa’s surface—and about the relative contributions made to Europa’s surge by SHOE and by CBOE—are discussed in Section 4.
2. The data set Europa, as a sub-pixel point of light, was observed 16 times by VIMS as the solar phase angle α varied between approximately 0.4◦ and 0.6◦ , and was also observed at two phase angles near 2◦ and two phase angles between 6◦ and
Table 1 The VIMS Europa data set Filename
Start time, UT (year, day, time)
Phase (◦ )
Subspacecraft longitude (◦ W)
1355377995.1 1355378444.1 1355378893.1 1355379342.1 1355379791.1 1355380240.1 1355380689.1 1355381138.1 1355381587.1 1355382036.1 1355382484.1 1355382933.1 1355383382.1 1355383831.1 1355384280.1 1355384729.1 1355530771.1 1355536132.1 1355689433.1 1355696029.1
2000-348T05:41:39 2000-348T05:49:08 2000-348T05:56:37 2000-348T06:04:06 2000-348T06:11:35 2000-348T06:19:04 2000-348T06:26:33 2000-348T06:34:02 2000-348T06:41:31 2000-348T06:48:60 2000-348T06:56:28 2000-348T07:03:57 2000-348T07:11:26 2000-348T07:18:55 2000-348T07:26:24 2000-348T07:33:53 2000-350T00:08:01 2000-350T01:37:22 2000-351T20:12:21 2000-351T22:02:17
0.386 0.390 0.395 0.403 0.410 0.420 0.430 0.442 0.454 0.468 0.482 0.498 0.512 0.530 0.545 0.564 1.929 1.807 6.397 6.836
17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 194 201 16 23
Day 348 is December 13
7◦ (see Table 1). All subspacecraft longitudes were either near 20◦ W or near 200◦ W (a difference of 180◦ ), meaning that there was no need to correct for the effects of this moon’s rotational lightcurve. All the VIMS data were radiometrically and geometrically calibrated using bench calibrations obtained prior to launch and inflight calibrations of standard infrared stars (Brown et al., 2004). At each α, data in 16 VIMS wavelength channels were averaged together to produce an image with an effective central wavelength of 0.91 µm, and similar channel-averaging produced images at effective wavelengths of 1.73, 2.25, and 2.56 µm, respectively. At a particular wavelength, the easiest way to derive an accurately shaped disk-integrated VIMS phase curve is to (1) sum the brightnesses of the pixels in a 4 × 4 box surrounding Europa in each image (thanks to the point-spread function of the detectors, there was some signal in the pixels surrounding Europa) and then (2) correct for changes in the size of Europa’s disk from image to image by multiplying each sum by the square of that image’s Europa-to-Cassini range. The resulting “sum × range2” phase curve must then be given an absolute scaling—e.g., an absolute reflectance at one particular phase angle—using a source of information whose absolute brightness is trusted. At each wavelength, we elected to fit a straight line—a constant magnitudes/degree slope— to the data points at α = 0.4◦ to 0.6◦ , and apply an absolute calibration at the point where this line hits zero phase. The absolute scaling of this “y-intercept” is based on the geometric albedos derived for Europa by Buratti and Veverka (1983), where geometric albedo p is the average I /F over a satellite’s zero-phase disk, I is the intensity of light scattered from the satellite’s surface, and πF is the plane-parallel so-
Europa’s opposition surge
lar flux reaching the satellite. Specifically, the Buratti and Veverka leading- and trailing-side p’s in the 0.55-µm V filter were averaged (since the longitude observed by VIMS is in between the leading and trailing sides) and the resulting albedo was then extrapolated to the various VIMS wavelengths using the shape of the Europa spectrum in Sill and Clark (1982). Note also that the Buratti and Veverka albedos were derived from data at α 3◦ and thus do not include the prominent surge that has been seen subsequently for Europa at α < 1◦ ; thus they underpredict the albedo that is appropriate to the zero-phase intercept of our α = 0.4◦ –0.6◦ straight line. The Europa phase curves in Domingue and Verbiscer (1997) suggest that this underprediction is approximately a 0.15-mag effect, and therefore we boosted all albedos used in our calibrations by 15%. The end result is that the zero-phase intercepts at 0.91, 1.73, and 2.25 µm were set to geometric albedos of 0.81, 0.33, and 0.18, respectively. This dropoff in albedo by more than a factor of four—made possible by Europa’s nearinfrared water-ice absorption bands—will be extremely useful when analyzing the opposition surge in the next section, since different candidates for the physical effect behind the surge should, as already noted, be affected differently by changes in albedo. A further dropoff in albedo by an additional factor of ∼ 2 would be possible on going from VIMS’s 2.25-µm data to its 2.56-µm data; however, (1) by this point the dropoff in albedo in a linear sense (the “ albedo” gotten by subtracting one wavelength’s albedo from the next) is getting quite small, and (2) by 2.56 µm the noise in the VIMS α = 0.4◦ –0.6◦ data is getting especially noticeable (see the lower-left portion of Fig. 4 in Brown et al., 2003). Accordingly, we elected to focus our analysis on the data at 0.91, 1.73, and 2.25 µm, and the already wide range of albedos that these data represent. We are confident in the absolute scaling and relative calibration (phase-curve shape) of the VIMS data at α = 0.4◦ – 0.6◦ , and also trust the relative calibration of the data from 2◦ to 7◦ phase (the relative photometry of measurements collected closely in time should be good to a few percent). Unfortunately, however, we have not been able to tie the former and latter data together as well as we would have liked because of the gap in time between them (see Table 1). Preliminary results from standard star measurements and observations obtained during the 1998 lunar flyby suggest that the intrinsic absolute photometry on VIMS is good to about 15–20%, so the placement of the α = 2◦ –7◦ data relative to the α = 0.4◦ –0.6◦ data has brightness uncertainties this large, i.e., there are uncertainties of up to 15–20% in our knowledge of the shape of the VIMS phase curve between α = 0.6◦ and α = 2◦ . Accordingly, in our modeling of the data in the next section we have been forced to use the 2◦ –7◦ data only in a relative sense; that is, we leave the absolute brightness of these latter data free to “float” and the only aspect of these data used in our modeling below is the shape or slope of the phase curve from α = 2◦ to 7◦ .
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It is important to try and give our modeling procedure information on the shape of Europa’s phase curve beyond α = 7◦ ; this need becomes even more important given the limitations in the VIMS data discussed in the previous paragraph. We at first elected to get this additional information from Fig. 4 of Buratti and Veverka (1983); specifically, we wanted to add in the 14 data points that are shown in that figure for the Voyager orange filter (0.59 µm, the longest wavelength available in Voyagers’ cameras), which combine observations from a wide range of longitudes to provide information on the shape of Europa’s phase curve from 3◦ phase to 28◦ phase. There are obvious dangers to adding such data to our modeling process; for example, we have no guarantee that the shape of the phase curve out in the VIMS near-infrared bears any resemblance to its shape in the orange part of the spectrum. However, we soon found another, more subtle “danger” present in the Voyager orange data: at phase angles of ∼ 15◦ and beyond, and in particular at α = 20◦ –30◦ , the Buratti and Veverka data have a much shallower slope (drop more slowly with increasing phase) than the more complete and more recent Europa visiblelight phase curves in Domingue and Verbiscer (1997) (see Fig. 2 below). This inconsistency probably occurs because Buratti and Veverka imposed a straight-line phase behavior (a constant mag/deg slope) on their data when separating out Europa’s phase curve from its rotational lightcurve. In our modeling process, it is not feasible to simply replace the Buratti and Veverka data with data from Domingue and Verbiscer because there are too many data points in the latter’s data set; the VIMS data, with only 20 data points per wavelength, could easily be “swamped” by the Domingue and Verbiscer points in a least-squares fitting process. Instead, we elected to make an alternate version of the Voyager orange data where all the points at phase angles of ∼ 15◦ and beyond are dropped in brightness to mimic the phase-curve shape seen at these α’s by Domingue and Verbiscer (this alternate data set can be seen in Figs. 3–6 below).
3. Modeling using Hapke’s function B. Hapke developed a commonly used theoretical photometric function with SHOE as its opposition-surge model (Hapke, 1981, 1984, 1986) and then later added a CBOE surge to this function (Hapke, 2002). As already noted, SHOE and CBOE make different predictions about how the strength of the surge should vary with the albedo of the surface, a fact which might be used to help predict which type of surge dominates over the other on a specific Solar System surface. However, in the VIMS Europa data the strength of the opposition surge does not appear to vary strongly with wavelength or albedo; at α = 2◦ –7◦ the slope of the VIMS phase curve—the so-called “phase coefficient”—is constant independent of wavelength/albedo, and even at α = 0.4◦ –0.6◦ the phase-coefficient values at different wavelengths/albedos are all the same to within the size
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Fig. 1. Phase-coefficient-vs.-albedo trends seen in the VIMS Europa data, compiled from results at wavelengths of 0.91, 1.73, and 2.25 µm. The phase coefficients, and their error bars, are taken from Fig. 4 of Brown et al. (2003); the corresponding albedo values are the geometric albedos adopted at each wavelength in the current work (see text).
of their error bars (see Fig. 1, adapted from a figure in Brown et al., 2003). The lack of an obvious dependence on albedo gives us no a-priori reason to suspect that one particular type of surge is dominating over the other, and thus it is important to investigate both types of opposition surges when analyzing the VIMS observations. The parts of Hapke’s function that model the SHOE surge and the CBOE surge each have two free parameters: one modeling the opposition-surge width and one modeling the surge amplitude (for the definitions of these, and other, Hapke parameters see Hapke, 1981, 1984, 1986, 2002). Because of the limited phase coverage of our data set and in particular the fact that the data beyond α = 1◦ can only be used in a relative, shape-of-phase-curve sense, it is not possible to believably constrain the values of all four of these parameters at the same time; i.e., it is not possible to uniquely constrain a model that combines both types of surges. Accordingly, we elected to perform two sets of Hapke fits: one in which we assume a SHOE surge only and derive best-fit values of the SHOE width and amplitude parameters (as per Hapke, 1981, 1984, 1986), and one which assumes a CBOE surge only and derives best-fit values for the CBOE width and amplitude (achieved by using the Hapke (2002) model with the amplitude of SHOE pinned at zero). In both types of fits, the single-scattering albedo w of the regolith is allowed to vary to provide the freedom to match the different albedos seen at our different VIMS wavelengths. The other free parameters in the Hapke function model the macroscopic roughness of the regolith and the average phase function of regolith particles. Our data’s limited phase coverage means that we cannot independently constrain any of these Hapke parameters, with the possible exception of the backscattering portion of the phase function. The para-
meters that cannot be constrained must be pinned at values previously derived for Europa; the best source of the latter is Domingue and Verbiscer (1997). Domingue and Verbiscer published Europa Hapke fits for two different visible wavelengths and two different forms of the particle phase function; we use the parameters for the longer of the two wavelengths provided (0.55 µm), and in our particular modeling software it is most convenient to use the parameters provided for the case of the so-called “2-term, 3-parameter Henyey-Greenstein” particle phase function. Domingue and Verbiscer also published separate Hapke fits for the leading and trailing sides of Europa. We elected to use the parameters provided for the trailing side, not because of the longitude observed by VIMS (which is in between the leading and trailing sides) but because this side’s phase curve comes slightly closer to mimicking the unusually shallow slope seen at α = 20◦ –30◦ in the Voyager data and mimicking the steepness of the slope seen at α = 0.4◦ –0.6◦ in the VIMS data (see Fig. 2). In the end, we decided that each combination of VIMS wavelength and type of opposition surge (SHOE or CBOE) should be given four different Hapke modeling runs: – A standard run which uses all three basic data sets (VIMS absolute data at α = 0.4◦ –0.6◦ , VIMS relative data on the phase-curve shape at α = 2◦ –7◦ , and Voyager relative data on the phase-curve shape at α = 3◦ – 28◦ ) and where the only parameters allowed to “float” are single-scattering albedo, surge width, and surge amplitude. – Same as the first run above except the parameter in the particle phase function that describes backscattering is also allowed to float (parameter d3 in Domingue and
Europa’s opposition surge
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(a)
(b) Fig. 2. (a) Disk-integrated Europa data from the shortest VIMS wavelength investigated (0.91 µm), combined with comparable data on this same moon from the longest Voyager wavelength available (Buratti and Veverka, 1983) and compared with Europa Hapke-model phase curves from Domingue and Verbiscer (1997). The bold lines show the Domingue and Verbiscer phase curves at their original absolute brightness values; in making the lighter, non-bold lines, the Domingue and Verbiscer curves have been moved upwards or downwards in magnitude to approximately match the absolute brightness of the α < 1◦ VIMS data (the VIMS data whose absolute brightness has been carefully calibrated). The data points which are used in our modeling effort in only a relative, shape-of-the-phase-curve sense (the open and solid circles) have here been temporarily assigned an absolute brightness that allows them to track along the light, non-bold curves at phase angles from 2◦ to 10◦ . (b) Same as (a), except zooming-in on phase angles below 1◦ . In this figure and subsequent figures, the turnover in the Hapke curves at extremely low phase angles is caused by the finite angular width of the Sun, which, as seen from Jupiter, has a radius of approximately 0.05◦ .
Verbiscer’s terminology, called g1 in our software’s terminology), to better model the unusually shallow slope seen in the Voyager phase curve at α = 20◦ –30◦ . – Same as the first run above, except we use the alternate version of the Voyager relative data where the points at phase angles of ∼ 15◦ and beyond have been dropped in
brightness to better mimic the phase-curve shape seen in Domingue and Verbiscer (1997). – Same as the first run above, except the Voyager data are left out (so that no data points are constraining the fit at phase angles beyond α = 7◦ , the highest VIMS phase angle).
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Table 2 Hapke fits at 0.91 µm Type of run
Values are shown for: single-scattering albedo w; opposition-surge width h; opposition-surge amplitude B0 [and, where necessary, backscattering parameter g1 ] SHOE
CBOE
Basic run
w = 0.97 h = 0.0009 B0 = 1.00
w = 0.96 h = 0.0047 B0 = 0.62
Run where g1 is allowed to vary
w = 0.97 h = 0.0031 B0 = 0.98 [g1 = −0.33]
w = 0.97 h = 0.0033 B0 = 1.00 [g1 = −0.35]
Run where Voyager data are altered to mimic Domingue and Verbiscer shape
w = 0.96 h = 0.0015 B0 = 1.00
w = 0.95 h = 0.0041 B0 = 0.99
Run where Voyager data are left out
w = 0.95 h = 0.0018 B0 = 1.00
w = 0.95 h = 0.0046 B0 = 0.85
Note: In this and all subsequent tables, the following parameters were used unless otherwise specified, taken from the 0.55-µm trailing-side Hapke fit in Domingue and Verbiscer (1997): 2-term, 3-parameter Henyey–Greenstein particle phase function: backscattering parameter g1 (Domingue and Verbiscer’s parameter d3) = −0.386; forward-scattering parameter g2 (D+V’s parameter b3) = 0.083; partition coefficient f (this is 1−D+V’s parameter c3) = 0.216; macroscopic roughness mean slope angle θ¯ = 11◦ .
The Hapke modeling software in use, kindly provided by P. Helfenstein, uses a system of grid and gradient searches to minimize the root-mean-squared residuals between model and data and thus find the best-fitting values of the Hapke free parameters (cf., Helfenstein and Veverka, 1989; Helfenstein et al., 1994, 1996). 3.1. 0.91-µm data As shown in Fig. 2 (and especially Fig. 2b), the α < 1◦ VIMS data at this wavelength have a phase-curve slope that is generally consistent with the phase coefficient seen for Europa at 0.55 µm by Domingue and Verbiscer (1997). In other words, Europa’s opposition surge in the near-infrared—at least the α < 1◦ “heart” of the surge—is similar to its surge in the visible. Such a result is not surprising at 0.91 µm, since the albedo of Europa changes little on going from 0.55 to 0.91 µm; see, e.g., the Europa spectrum in Sill and Clark (1982). (What is surprising is the fact that this same α < 1◦ phase-curve slope is also seen at the other two VIMS wavelengths longward of 0.91 µm, where the albedo of Europa is much lower than it was at 0.55 µm or 0.91 µm; see Fig. 1.) 3.1.1. SHOE Hapke fits All four SHOE runs at this wavelength do a reasonable job of mimicking the slope of the VIMS α < 1◦ data (Fig. 3b), although the “basic” run does a slightly poorer job of this than the other three, more flexible runs. The four
different fits have opposition surges of noticeably different widths (Fig. 3a), with the surge width parameter h ranging from 0.0009 to 0.0031 (Table 2). This variation by a factor of more than three is a sign of just how uncertain the width of the surge can be when the data at α > 1◦ can only be used in a relative, shape-of-phase-curve sense (and is one of the reasons why we have elected not to bother determining rigorous error bars on the Hapke parameters in any one individual fit). This range of h values nicely encompasses the SHOE h value of 0.0016 determined for Europa in the visible by Domingue and Verbiscer (1997), emphasizing again that there are definite similarities between Europa’s near-IR and visible-light opposition surges. Note, however, that this wavelength’s near-IR data seem to require a larger SHOE amplitude B0 than Europa’s visible-light data do (see the B0 values near unity in Table 2, vs. the B0 values near 0.5 in Domingue and Verbiscer, 1997). 3.1.2. CBOE Hapke fits The CBOE runs at this wavelength do a noticeably poorer job of matching the α < 1◦ slope of the VIMS data than the corresponding SHOE fits did (Fig. 4). In particular, these CBOE fits insist on having a steeper phase-curve slope than what is seen in the data (see Fig. 4b, and compare it with Fig. 3b). In the fits that use VIMS only and leave out the Voyager data, the best-fit residual drops by a factor of more than two on switching from CBOE to SHOE; the drop in residual on going from CBOE to SHOE is smaller for the fits that include Voyager data, because in these cases the α < 1◦ VIMS data make up a smaller part of the overall data set. 3.2. 1.73-µm data 3.2.1. SHOE Hapke fits Two of the four SHOE runs “failed” in the sense that the Hapke-fitting software found best fits that make no attempt to mimic the slope seen in the α < 1◦ VIMS data; the fits match the basic absolute brightness of these VIMS data well enough, but have an almost flat phase curve at α < 1◦ thanks to a surge amplitude B0 close to zero. On running through a grid of B0 values and optimizing the other free parameters at each gridpoint, we found that the goodness-of-fit rms residual jumps back-and-forth randomly around a constant value as B0 varies; noticeably missing is the U-shaped minimum that is seen in the residual when there is a well-defined best fit. This kind of behavior occurs when we ask the Hapke function to model a phase-curve shape that it simply cannot mimic well; since the runs in question both included Voyager data (see Table 3), it may well be that the Hapke function has trouble fitting the Voyager slope and the steep, α < 1◦ VIMS slope at the same time. We will see below that this kind of fitting failure occurs again—and in fact occurs even more often—when we move on to this wavelength’s CBOE fits, and when we move on to the longest VIMS wavelength and its associated lower albedo.
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(a)
(b) Fig. 3. (a) The same 0.91-µm data set shown in Fig. 2, now compared with the phase curves of the SHOE Hapke models that best fit this data set. Note that an additional new version of the Voyager data has been added (triangles) that diverges from the original Voyager data at phase angles of ∼ 15◦ and above, to better mimic the shape of the Domingue and Verbiscer phase curves at those α. In the labels for the Hapke curves, “g1 floats” refers to a run where the backscatter parameter g1 is allowed to vary, “D+V mimic” refers to a run that uses the new alternate version of the Voyager data, and “toss Voyager” refers to a run that leaves the Voyager data out of the fitting process. The data points that are used in the fitting process in only a relative, shape-of-the-phase-curve sense (circles and triangles) have here been temporarily assigned an absolute brightness that allows the VIMS relative data to track along the topmost Hapke phase curve (i.e., allows the open circles to track along the topmost curve). (b) Same as (a), except zooming-in on phase angles below 1◦ .
The other two SHOE runs at this wavelength—the ones that produced successful, well-defined best fits—do a good job of mimicking the slope seen in the α < 1◦ VIMS data (Fig. 5b) and again have wildly different opposition-surge widths, this time covering h values from 0.0005 to 0.0033 (see Fig. 5a and Table 3).
3.2.2. CBOE Hapke fits Three of this wavelength’s four CBOE runs failed (Table 3) in a similar fashion to the failed fits mentioned above. As for the one successful CBOE fit, it again does a poorer job of mimicking the α < 1◦ slope of the VIMS data than this wavelength’s corresponding SHOE fits; the discrepancy
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(a)
(b) Fig. 4. Same as Fig. 3, except now moving on to the 0.91-µm CBOE fits. Again, all the relative data points have been temporarily assigned an absolute brightness that allows the open circles to track along the topmost curve.
is in the same direction as it was at the previous wavelength, i.e., the CBOE fit again insists on having a steeper slope than what is seen in the data (Fig. 5b). In the fits at this wavelength that use VIMS only and leave out the Voyager data, the best-fit residual drops by a factor of more than three on switching from CBOE to SHOE. 3.3. 2.25-µm data 3.3.1. SHOE and CBOE Hapke fits At this final, longest wavelength, only two of the eight runs were successful, one each for SHOE and CBOE. Not
surprisingly, these were the runs that leave out the Voyager data points entirely (Table 4), the runs which are constrained by the smallest amount of data and thus allow the Hapke function the most flexibility. Unfortunately, by the time we reach this wavelength, the noise in the VIMS instrument and in its data has grown enough that it is not clear whether SHOE or CBOE provides a better fit at α < 1◦ ; given the scatter in the data, about all that can be said is that both types of surges do a reasonable job of mimicking the slope seen at these phase angles in the VIMS data (see Fig. 6 and especially Fig. 6b).
Europa’s opposition surge
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(a)
(b) Fig. 5. Same as Figs. 3 and 4, except now focusing on the VIMS data at 1.73 µm. Since several of the Hapke fits at this wavelength “failed” (see text) and the number of successful fits that need to be displayed has dropped, the SHOE and CBOE fits can now be shown on the same plot.
4. Discussion At 2.25 µm, where VIMS data are somewhat noisy, both types of opposition surges can mimic the slope of the VIMS phase curve at < 1◦ phase. At both 0.91 and 1.73 µm, however—where VIMS data are “cleaner”—CBOE does a noticeably poorer job than SHOE of matching the VIMS phase coefficient at α < 1◦ ; in particular, the best CBOE fit insists on having a steeper phase-curve slope than the data. Given this discrepancy, it is tempting to conclude that CBOE is less likely than SHOE to be the primary cause of Europa’s
near-IR opposition surge. However, such an all-or-nothing, “SHOE instead of CBOE” result is not completely conclusive, for several reasons: (1) the sparse nature of the VIMS phase coverage, and in particular the fact that the data at phase angles beyond 1◦ could only be used in a relative, shape-of-phase-curve sense; (2) the not-necessarily-correct assumption that Europa’s photometric behavior in the nearIR is similar to its behavior in the visible (i.e., the fact that we included additional phase-curve data from—and adopted some Hapke-parameter values from—visible wavelengths); and (3) the likelihood that Europa’s opposition surge, like
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(a)
(b) Fig. 6. Same as Fig. 5, except now focusing on the VIMS data at 2.25 µm.
surges elsewhere in the Solar System, may well be a combination of both SHOE and CBOE. A more balanced conclusion to be drawn from the phasecurve slope discrepancies—one that is more conclusively supported by the data at hand—is that Europa’s strong nearIR opposition surge at phase angles α < 1◦ cannot be explained by CBOE alone and must have a significant SHOE component, even at wavelengths where Europa is bright. This result, though more measured in tone, is still telling us something important; for example, it warns researchers who deal with bright icy bodies against trying to fit α < 1◦ data
with CBOE alone (a tempting simplification of the surgemodeling process). To test the robustness of this conclusion, we performed two additional sets of Hapke-fitting runs on the data at 0.91 µm, the wavelength where the VIMS data have the highest signal-to-noise ratio and the highest albedo (and where the presence of a significant SHOE component is thus the most surprising). These runs were not part of the main fitting process in the previous section because they push the bounds of what even this wavelength’s data quality can support. First, to get more information on whether it is truly
Europa’s opposition surge
Table 3 Hapke fits at 1.73 µm Type of run
Values are shown for: single-scattering albedo w; opposition-surge width h; opposition-surge amplitude B0 [and, where necessary, backscattering parameter g1 ] SHOE
CBOE
Basic run
Failed
Failed
Run where g1 is allowed to vary
w = 0.79 h = 0.0033 B0 = 0.59 [g1 = −0.19]
Failed
Run where Voyager data are altered to mimic Domingue and Verbiscer shape
Failed
Failed
Run where Voyager data are left out
w = 0.61 h = 0.0005 B0 = 0.99
w = 0.61 h = 0.0021 B0 = 1.00
Table 4 Hapke fits at 2.25 µm Type of run
Values are shown for: single-scattering albedo w; opposition-surge width h; opposition-surge amplitude B0 SHOE
CBOE
Basic run
Failed
Failed
Run where g1 is allowed to vary
Failed
Failed
Run where Voyager data are altered to mimic Domingue and Verbiscer shape
Failed
Failed
Run where Voyager data are left out
w = 0.37 h = 0.0003 B0 = 1.00
w = 0.36 h = 0.0052 B0 = 0.37
“intrinsic” to CBOE to have a steeper phase coefficient than the VIMS α < 1◦ data, we performed CBOE fits where the VIMS α < 1◦ data were the only data constraining the fit (thus giving CBOE its best opportunity ever to match these data’s slope). If we do not carefully limit the number of free Hapke parameters—if we let the backscattering parameter g1 “float,” for example—then there are not enough data to constrain the free parameters and the fit fails in the same way as the failed fits described in the previous section. If, however, the only parameters allowed to “float” are the single-scattering albedo, surge width, and surge amplitude, then the fitting process runs to completion and produces a phase curve very similar to those seen in the 0.91-µm CBOE fits in the previous section, i.e., the best-fit phase curve still insists on having a steeper slope than the VIMS α < 1◦ data. Thus in all the fits we have performed, we have not yet found anything to counter the idea that CBOE intrinsically has a steeper phase-curve slope than the VIMS α < 1◦
159
data. This discrepancy could be a limitation specific to the Hapke mathematical formulation of CBOE, or it could be a broader limitation of the CBOE surge in general; determining which of these is the case would require a better data set (more complete near-opposition phase coverage, etc.) than the one used here. Second, given the likelihood that Europa’s opposition surge is actually a combination of SHOE and CBOE, we tried to fit our 0.91-µm data set using a Hapke function that contains both SHOE and CBOE (i.e., the “full up” function from Hapke, 2002). Given our strong suspicion, stated earlier, that our data set is not good enough to uniquely constrain the widths and amplitudes of both kinds of surges, it made little sense to perform runs that weaken the constraints by throwing out data or allowing additional Hapke parameters to float; accordingly, out of the four kinds of fitting runs described in the previous section, the only runs performed in this case were the “basic” or “standard” run and the run where the Voyager data have been altered to mimic the Domingue and Verbiscer phase-curve shape. The results of these two runs are shown in Fig. 7. Although both of the model phase curves in that figure fit the slope of the α < 1◦ VIMS data comparably well, they represent very different relative importances of the two kinds of opposition surges: in one curve the amplitude B0 for SHOE is larger than the amplitude B0 for CBOE, and in the other curve the relative amplitudes of the two kinds of surges are reversed. This supports our intuition that the data set is not good enough to uniquely constrain the widths and amplitudes of both kinds of surges. More important, though, is the fact that in both of the fits displayed, the amplitude B0 for SHOE is always 0.6 or above; this supports our main conclusion that Europa’s near-IR opposition surge at α < 1◦ cannot be explained by CBOE alone and must have a significant SHOE component. (Note that neither model curve in Fig. 7 fits the α < 1◦ data’s slope quite as well as the SHOE-only fits shown in the previous section. The fitting runs in that figure used a starting guess where the two different kinds of surges had amplitudes that were equal and significantly greater than zero; accordingly, it is not surprising that these runs did not make their way to the corner of Hapke-parameter space where the CBOE amplitude goes to zero and the surge is entirely SHOE.) Clearly, it would be extremely useful in the future to observe Europa’s opposition surge at the same range of near-IR wavelengths/same range of albedos as was done here, but this time obtain more complete phase coverage and in particular tie-down the shape of the phase curve continuously from near zero phase out to phase angles of 5◦ or 6◦ or 7◦ . That way, we could model the data with a Hapke function that has both SHOE and CBOE with some hope of being able to constrain the width and amplitude parameters of both kinds of surges—and thus be able to more conclusively measure the relative contributions made by SHOE and CBOE in Europa’s near-IR surge, and determine whether the surge is dominated by one physical cause at one wavelength/albedo
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Fig. 7. The same 0.91-µm, α < 1◦ data set shown in Figs. 2b, 3b, and 4b, this time compared with the phase curves of Hapke fits that include both SHOE and CBOE. Although both curves shown fit the slope of the data comparably well, they represent different relative importances of the two kinds of opposition surges: in the dashed curve the amplitude B0 for SHOE is larger than the amplitude B0 for CBOE, and in the other curve the relative amplitudes of the two kinds of surges are reversed.
and dominated by the other physical cause at another wavelength/albedo. Such a data set could also be fit to coherentbackscatter models other than Hapke’s, to test whether the problems in matching the observed α < 1◦ phase coefficient are specific to Hapke’s CBOE model or more broadly affect coherent backscatter in general. The Europa work performed here also reminds us of the limitations of trying to use albedo alone as a tool for discriminating between SHOE and CBOE. The core of the problem is the lack of any firm evidence that the strength of Europa’s opposition surge varies with wavelength/albedo: i.e., the fact that the slope of the VIMS phase curve at α = 2◦ –7◦ , and more importantly, at α = 0.4◦ –0.6◦ , is constant independent of wavelength/albedo to within the associated error bars (Fig. 1), and the fact that the α = 0.4◦ –0.6◦ slope seen by VIMS is similar to the slope seen previously at these phase angles in the visible (Fig. 2b). If Europa’s opposition surge is dominated at all wavelengths by one physical cause over the other—if, for example, SHOE dominates at all wavelengths, something that would be consistent with the Hapke fitting performed above—then one must explain why the dependence of surge strength on albedo predicted for that physical cause is not seen clearly in the VIMS data. If Europa’s surge is, instead, a combination of both SHOE and CBOE, then there may have to be an intricate trade-off between the two physical causes that would seemingly disobey Occam’s Razor: as the near-IR wavelength rises and the albedo drops, is it possible for SHOE to increase in importance, and CBOE decrease in importance, in a cooperation so perfect that there is no noticeable change in the strength of the surge? (And isn’t such a trade-off inconsistent with
the fact that CBOE did its best job of matching the slope of the VIMS phase curve at 2.25 µm, the longest wavelength/lowest albedo investigated?) In any case, it seems clear that the best approach is not to rely on albedo dependence as something that can discriminate between the two types of surges on its own, but to instead augment albedo information with other “tools” that can potentially differentiate between SHOE and CBOE; other potential discriminators that have been proposed include the polarization state of opposition-surge light, and the suggestion that the CBOE surge generally has a narrower angular width than SHOE (e.g., Hapke, 2002; Hapke et al., 1993, 1998; Helfenstein et al., 1997, 1998; Hillier et al., 1999; Mishchenko, 1992; Nelson et al., 2000; Shepard and Arvidson, 1999; Shkuratov et al., 1999). If an opposition-surge data set has the phase coverage, albedo leverage, detector type, etc., that can make use of the largest possible number of these tools, this will increase the likelihood that modeling the data will successfully discern the relative contributions of SHOE and CBOE. Finally, a note about the porosity of the surface of Europa: The value of the Hapke SHOE model’s angular-width parameter h is a function of both the porosity of the regolith and its grain size distribution (Hapke, 1986). The range of SHOE h values determined from the VIMS data—h = 0.0003 to 0.0033, depending on the wavelength, and type of fitting run, involved—encompasses a factor of ten and thus encompasses a range of possible porosities. However, since this range of values is centered on the SHOE h value of 0.0016 determined for Europa by Domingue and Verbiscer (1997), our results are certainly consistent with the porosity results
Europa’s opposition surge
of those earlier authors, who derived a very high porosity for Europa commensurate with a frost surface. (Note that the early shadow-hiding surge model by Irvine (1966), when fit to the phase-curve slope seen by VIMS at α < 1◦ , also predicts a very high porosity, specifically a porosity of ∼ 99% (Brown et al., 2003). However, the simple Irvine model, which does not include multiple scattering, does not have the correct phase-curve shape for Europa in the sense that it cannot reproduce the radically different phase coefficients seen at α < 1◦ and at α = 2◦ –7◦ ; if the Irvine model is fit to the shallower VIMS phase-curve slope seen at α = 2◦ –7◦ , it predicts a much different, lower porosity, on the order of 10%.)
5. Conclusions Near-infrared observations of Europa’s disk-integrated opposition surge by Cassini VIMS, first published in Brown et al. (2003), have now been modeled with the commonly used Hapke photometric function. The VIMS data set emphasizes observations at 16 solar phase angles from 0.4◦ to 0.6◦ —the first time the α < 1◦ “heart” of Europa’s opposition surge has been observed this well in the near-IR. This data set also provides a unique opportunity to examine how the surge is affected by changes in wavelength and albedo: at VIMS wavelengths of 0.91, 1.73, and 2.25 µm, the geometric albedo of Europa is 0.81, 0.33, and 0.18, respectively. Despite this factor-of-four albedo range, however, the slope of Europa’s phase curve at α < 1◦ is similar at all three wavelengths (to within the error bars) and this common slope is similar to the phase coefficient seen at this α in visible-light observations of Europa. Because of sparse VIMS phase coverage, it was not possible to constrain all the surge parameters at once in a Hapke function that has both the Shadow Hiding Opposition Effect (SHOE) and the Coherent Backscatter Opposition Effect (CBOE); accordingly, we performed separate Hapke fits for SHOE-only and CBOE-only surges. At 2.25 µm, where VIMS data are somewhat noisy, both types of surges can mimic the slope of the VIMS phase curve at α < 1◦ . At 0.91 and 1.73 µm, however—where VIMS data are “cleaner”— CBOE does a noticeably poorer job than SHOE of matching the VIMS phase coefficient at α < 1◦ ; in particular, the best CBOE fit insists on having a steeper phase-curve slope than the data. This discrepancy suggests that Europa’s near-IR opposition surge cannot be explained by CBOE alone and must have a significant SHOE component, even at wavelengths where Europa is bright. Future observations of Europa’s opposition surge at these same near-IR wavelengths, but with phase coverage that better delineates the shape of the phase curve over the entire range of α in the surge, could go a long way toward more conclusively separating out the relative importance of SHOE and CBOE on this moon.
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Acknowledgments This research was carried out at JPL with funding from the National Academy of Sciences/National Research Council resident research associateship program. We profusely thank P. Helfenstein for the use of his Hapke-fitting programs, and also appreciate helpful reviews from P. Geissler and an anonymous referee.
References Brown, R.H., 22 colleagues, 2004. The Cassini Visual and Infrared Mapping Spectrometer investigation. Space Sci. Rev. In press. Brown, R.H., 26 colleagues, 2003. Observations with the Visual and Infrared Mapping Spectrometer (VIMS) during Cassini’s flyby of Jupiter. Icarus 164, 461–470. Buratti, B.J., 1995. Photometry and surface structure of the icy galilean satellites. J. Geophys. Res. 100, 19061–19066. Buratti, B., Veverka, J., 1983. Voyager photometry of Europa. Icarus 55, 93–110. Buratti, B.J., Nelson, R.M., Lane, A.L., 1988. Surficial textures of the galilean satellites. Nature 333, 148–151. Domingue, D., Verbiscer, A., 1997. Re-analysis of the solar phase curves of the icy galilean satellites. Icarus 128, 49–74. Domingue, D.L., Hapke, B.W., Lockwood, G.W., Thompson, D.T., 1991. Europa’s phase curve: implications for surface structure. Icarus 90, 30– 42. Hapke, B., 1981. Bidirectional reflectance spectroscopy. 1. Theory. J. Geophys. Res. 86, 3039–3054. Hapke, B., 1984. Bidirectional reflectance spectroscopy. 3. Correction for macroscopic roughness. Icarus 59, 41–59. Hapke, B., 1986. Bidirectional reflectance spectroscopy. 4. The extinction coefficient and the opposition effect. Icarus 67, 264–280. Hapke, B., 2002. Bidirectional reflectance spectroscopy. 5. The coherent backscatter opposition effect and anisotropic scattering. Icarus 157, 523–534. Hapke, B.W., Nelson, R.M., Smythe, W.D., 1993. The opposition effect of the Moon: the contribution of coherent backscatter. Science 260, 509– 511. Hapke, B., Nelson, R., Smythe, W., 1998. The opposition effect of the Moon: coherent backscatter and shadow hiding. Icarus 133, 89–97. Helfenstein, P., Veverka, J., 1989. Physical characterization of asteroid surfaces from photometric analysis. In: Binzel, R.P., Gehrels, T., Matthews, M.S. (Eds.), Asteroids II. Univ. of Arizona Press, Tucson, pp. 557–593. Helfenstein, P., 23 colleagues, 1994. Galileo photometry of Asteroid 951 Gaspra. Icarus 107, 37–60. Helfenstein, P., Veverka, J., Thomas, P.C., Simonelli, D.P., Klaasen, K., Johnson, T.V., Fanale, F., Granahan, J., McEwen, A.S., Belton, M., Chapman, C., 1996. Galileo photometry of Asteroid 243 Ida. Icarus 120, 48–65. Helfenstein, P., Veverka, J., Hillier, J., 1997. The lunar opposition effect: a test of alternative models. Icarus 128, 2–14. Helfenstein, P., Currier, N., Clark, B.E., Veverka, J., Bell, M., Sullivan, R., Klemaszewski, J., Greeley, R., Pappalardo, R.T., Head III, J.W., Jones, T., Klaasen, K., Magee, K., Geissler, P., Greenberg, R., McEwen, A., Phillips, C., Colvin, T., Davies, M., Denk, T., Neukum, G., Belton, M.J.S., 1998. Galileo observations of Europa’s opposition effect. Icarus 135, 41–63. Hillier, J.K., Buratti, B.J., Hill, K., 1999. Multispectral photometry of the Moon and absolute calibration of the Clementine UV/Vis camera. Icarus 141, 205–225. Irvine, W.M., 1966. The shadow effect in diffuse reflection. J. Geophys. Res. 71, 2931–2937.
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Mishchenko, M.I., 1992. The angular width of the coherent backscatter opposition effect: an application to icy outer planet satellites. Astrophys. Space Sci. 194, 327–333. Nelson, R.M., Hapke, B.W., Smythe, W.D., Spilker, L.J., 2000. The opposition effect in simulated planetary regoliths. Reflectance and circular polarization ratio change at small phase angle. Icarus 147, 545–558. Shepard, M.K., Arvidson, R.E., 1999. The opposition surge and photopolarimetry of fresh and coated basalts. Icarus 141, 172–178.
Shkuratov, Y.G., Kresavsky, M.A., Ovcharenko, A.A., Stankovich, D.G., Zubko, E.S., Pieters, C., Arnold, G., 1999. Opposition effect from Clementine data and mechanisms of backscatter. Icarus 141, 132–155. Sill, G.T., Clark, R.N., 1982. Composition of the surfaces of the galilean satellites. In: Morrison, D. (Ed.), Satellites of Jupiter. Univ. of Arizona Press, Tucson, pp. 174–212. Thompson, D.T., Lockwood, G.W., 1992. Photoelectric photometry of Europa and Callisto 1976–1991. J. Geophys. Res. 97, 14761–14772.
Europa’s opposition surge in the near-infrared: interpreting disk-integrated observations by Cassini VIMS Damon P. Simonelli ∗ , Bonnie J. Buratti Jet Propulsion Laboratory, MS 183-501, 4800 Oak Grove Drive, Pasadena, CA 91109, USA Received 9 August 2003; revised 10 April 2004 Available online 3 September 2004
Abstract Near-infrared observations of Europa’s disk-integrated opposition surge by Cassini VIMS, first published in Fig. 4 of Brown et al. (2003, Icarus, 164, 461), have now been modeled with the commonly used Hapke photometric function. The VIMS data set emphasizes observations at 16 solar phase angles from 0.4◦ to 0.6◦ —the first time the < 1◦ phase “heart” of Europa’s opposition surge has been observed this well in the near-IR. This data set also provides a unique opportunity to examine how the surge is affected by changes in wavelength and albedo: at VIMS wavelengths of 0.91, 1.73, and 2.25 µm, the geometric albedo of Europa is 0.81, 0.33, and 0.18, respectively. Despite this factorof-four albedo range, however, the slope of Europa’s phase curve at < 1◦ phase is similar at all three wavelengths (to within the error bars) and this common slope is similar to the phase coefficient seen in visible-light observations of Europa. The two components of the opposition surge—involving different models of the physical cause of the surge—are the Shadow Hiding Opposition Effect (SHOE) and the Coherent Backscatter Opposition Effect (CBOE). Because of sparse VIMS phase coverage, it is not possible to constrain all the surge parameters at once in a Hapke function that has both SHOE and CBOE; accordingly, we performed separate Hapke fits for SHOE-only and CBOE-only surges. At 2.25 µm, where VIMS data are somewhat noisy, both types of surges can mimic the slope of the VIMS phase curve at < 1◦ phase. At 0.91 and 1.73 µm, however—where VIMS data are “cleaner”—CBOE does a noticeably poorer job than SHOE of matching the VIMS phase coefficient at < 1◦ phase; in particular, the best CBOE fit insists on having a steeper phase-curve slope than the data. This discrepancy suggests that Europa’s near-IR opposition surge cannot be explained by CBOE alone and must have a significant SHOE component, even at wavelengths where Europa is bright. 2004 Elsevier Inc. All rights reserved. Keywords: Cassini spacecraft; Europa; Near-infrared; Photometry
1. Introduction The surge in brightness seen from most solid surfaces in the Solar System near zero solar phase angle—the opposition surge—is one of the more intriguing, but at the same time enigmatic, photometric effects in planetary science. Two main components of the opposition surge have been identified, involving different models of the physical cause of the surge; these are the shadow-hiding opposition effect (SHOE), where regolith particles hide their own shadows (e.g., Hapke, 1981, 1986), and the coherent backscatter * Corresponding author.
E-mail address: [email protected] (D.P. Simonelli). 0019-1035/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2004.06.004
opposition effect (CBOE), which involves constructive interference in the regolith between incoming and scattered light beams (e.g., Hapke, 2002). The two different surge models make different predictions about how the strength of the surge should vary with the albedo of the surface: the multiple scattering that is prevalent in a brighter regolith washes out the interparticle shadows that are at the heart of SHOE and provides more scattered light for constructive interference, meaning that a brighter surface should nominally have less SHOE and more CBOE (Hapke, 2002; Hapke et al., 1993, 1998; Helfenstein et al., 1997, 1998; Hillier et al., 1999; Mishchenko, 1992; Nelson et al., 2000; Shepard and Arvidson, 1999; Shkuratov et al., 1999). Accordingly, albedo dependence is one of several effects that have been proposed
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as a tool for discriminating between the two kinds of opposition surges. The jovian satellite Europa, with its comparatively young surface rich in water ice, has to date provided oppositionsurge information from mostly the high-albedo end of the scale; at visible wavelengths, disk-integrated observations of Europa reveal a very bright object whose opposition surge becomes prominent only when the phase angle α is 1◦ or less (geometric albedo = 0.6 to 0.9 depending on wavelength, what longitude is being observed, how the extrapolation to zero phase is performed, etc.; Buratti and Veverka, 1983; Domingue et al., 1991; Thompson and Lockwood, 1992; Buratti, 1995; Domingue and Verbiscer, 1997). Attempts to make other albedos available in Europa opposition-surge observations have so far emphasized either working in the ultraviolet, where Europa as a whole is darker (Buratti et al., 1988), or obtaining close-up, disk-resolved images that include local patches of low-albedo material (Helfenstein et al., 1998). However, these attempts have involved limited phase coverage, with just one or two phase angles covered close to α = 0◦ and the next coverage out at α = 3◦ to 5◦ — not exactly optimum conditions for delineating the shape of the prominent, α < 1◦ surge. Thanks to Cassini’s December 2000 flyby of Jupiter, however, we can now add a new source of low-albedo information to Europa’s opposition-surge record: the spacecraft’s Visual and Infrared Mapping Spectrometer (VIMS) observed this moon’s disk-integrated surge at various wavelengths across the near-infrared, where Europa experiences a wide range of albedos thanks to prominent water-ice absorption bands (e.g., see the composite Europa spectrum in Sill and Clark, 1982). Although the new observations have their own limitations where phase-angle coverage is concerned, the centerpiece of the VIMS data is an important new measurement: the slope of the phase curve at α = 0.4◦ to 0.6◦ —deep within the prominent part of the surge—has been determined accurately at each near-IR wavelength by using coverage at multiple phase angles instead of just a pair of phase angles. The VIMS Europa data set has already been published in Fig. 4 of Brown et al. (2003); Section 2 below describes these new VIMS data in more detail than was possible in that earlier publication. These data are then modeled and interpreted with the aid of the commonly used Hapke photometric function in Section 3. Finally, the resulting implications about the physical state of Europa’s surface—and about the relative contributions made to Europa’s surge by SHOE and by CBOE—are discussed in Section 4.
2. The data set Europa, as a sub-pixel point of light, was observed 16 times by VIMS as the solar phase angle α varied between approximately 0.4◦ and 0.6◦ , and was also observed at two phase angles near 2◦ and two phase angles between 6◦ and
Table 1 The VIMS Europa data set Filename
Start time, UT (year, day, time)
Phase (◦ )
Subspacecraft longitude (◦ W)
1355377995.1 1355378444.1 1355378893.1 1355379342.1 1355379791.1 1355380240.1 1355380689.1 1355381138.1 1355381587.1 1355382036.1 1355382484.1 1355382933.1 1355383382.1 1355383831.1 1355384280.1 1355384729.1 1355530771.1 1355536132.1 1355689433.1 1355696029.1
2000-348T05:41:39 2000-348T05:49:08 2000-348T05:56:37 2000-348T06:04:06 2000-348T06:11:35 2000-348T06:19:04 2000-348T06:26:33 2000-348T06:34:02 2000-348T06:41:31 2000-348T06:48:60 2000-348T06:56:28 2000-348T07:03:57 2000-348T07:11:26 2000-348T07:18:55 2000-348T07:26:24 2000-348T07:33:53 2000-350T00:08:01 2000-350T01:37:22 2000-351T20:12:21 2000-351T22:02:17
0.386 0.390 0.395 0.403 0.410 0.420 0.430 0.442 0.454 0.468 0.482 0.498 0.512 0.530 0.545 0.564 1.929 1.807 6.397 6.836
17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 194 201 16 23
Day 348 is December 13
7◦ (see Table 1). All subspacecraft longitudes were either near 20◦ W or near 200◦ W (a difference of 180◦ ), meaning that there was no need to correct for the effects of this moon’s rotational lightcurve. All the VIMS data were radiometrically and geometrically calibrated using bench calibrations obtained prior to launch and inflight calibrations of standard infrared stars (Brown et al., 2004). At each α, data in 16 VIMS wavelength channels were averaged together to produce an image with an effective central wavelength of 0.91 µm, and similar channel-averaging produced images at effective wavelengths of 1.73, 2.25, and 2.56 µm, respectively. At a particular wavelength, the easiest way to derive an accurately shaped disk-integrated VIMS phase curve is to (1) sum the brightnesses of the pixels in a 4 × 4 box surrounding Europa in each image (thanks to the point-spread function of the detectors, there was some signal in the pixels surrounding Europa) and then (2) correct for changes in the size of Europa’s disk from image to image by multiplying each sum by the square of that image’s Europa-to-Cassini range. The resulting “sum × range2” phase curve must then be given an absolute scaling—e.g., an absolute reflectance at one particular phase angle—using a source of information whose absolute brightness is trusted. At each wavelength, we elected to fit a straight line—a constant magnitudes/degree slope— to the data points at α = 0.4◦ to 0.6◦ , and apply an absolute calibration at the point where this line hits zero phase. The absolute scaling of this “y-intercept” is based on the geometric albedos derived for Europa by Buratti and Veverka (1983), where geometric albedo p is the average I /F over a satellite’s zero-phase disk, I is the intensity of light scattered from the satellite’s surface, and πF is the plane-parallel so-
Europa’s opposition surge
lar flux reaching the satellite. Specifically, the Buratti and Veverka leading- and trailing-side p’s in the 0.55-µm V filter were averaged (since the longitude observed by VIMS is in between the leading and trailing sides) and the resulting albedo was then extrapolated to the various VIMS wavelengths using the shape of the Europa spectrum in Sill and Clark (1982). Note also that the Buratti and Veverka albedos were derived from data at α 3◦ and thus do not include the prominent surge that has been seen subsequently for Europa at α < 1◦ ; thus they underpredict the albedo that is appropriate to the zero-phase intercept of our α = 0.4◦ –0.6◦ straight line. The Europa phase curves in Domingue and Verbiscer (1997) suggest that this underprediction is approximately a 0.15-mag effect, and therefore we boosted all albedos used in our calibrations by 15%. The end result is that the zero-phase intercepts at 0.91, 1.73, and 2.25 µm were set to geometric albedos of 0.81, 0.33, and 0.18, respectively. This dropoff in albedo by more than a factor of four—made possible by Europa’s nearinfrared water-ice absorption bands—will be extremely useful when analyzing the opposition surge in the next section, since different candidates for the physical effect behind the surge should, as already noted, be affected differently by changes in albedo. A further dropoff in albedo by an additional factor of ∼ 2 would be possible on going from VIMS’s 2.25-µm data to its 2.56-µm data; however, (1) by this point the dropoff in albedo in a linear sense (the “ albedo” gotten by subtracting one wavelength’s albedo from the next) is getting quite small, and (2) by 2.56 µm the noise in the VIMS α = 0.4◦ –0.6◦ data is getting especially noticeable (see the lower-left portion of Fig. 4 in Brown et al., 2003). Accordingly, we elected to focus our analysis on the data at 0.91, 1.73, and 2.25 µm, and the already wide range of albedos that these data represent. We are confident in the absolute scaling and relative calibration (phase-curve shape) of the VIMS data at α = 0.4◦ – 0.6◦ , and also trust the relative calibration of the data from 2◦ to 7◦ phase (the relative photometry of measurements collected closely in time should be good to a few percent). Unfortunately, however, we have not been able to tie the former and latter data together as well as we would have liked because of the gap in time between them (see Table 1). Preliminary results from standard star measurements and observations obtained during the 1998 lunar flyby suggest that the intrinsic absolute photometry on VIMS is good to about 15–20%, so the placement of the α = 2◦ –7◦ data relative to the α = 0.4◦ –0.6◦ data has brightness uncertainties this large, i.e., there are uncertainties of up to 15–20% in our knowledge of the shape of the VIMS phase curve between α = 0.6◦ and α = 2◦ . Accordingly, in our modeling of the data in the next section we have been forced to use the 2◦ –7◦ data only in a relative sense; that is, we leave the absolute brightness of these latter data free to “float” and the only aspect of these data used in our modeling below is the shape or slope of the phase curve from α = 2◦ to 7◦ .
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It is important to try and give our modeling procedure information on the shape of Europa’s phase curve beyond α = 7◦ ; this need becomes even more important given the limitations in the VIMS data discussed in the previous paragraph. We at first elected to get this additional information from Fig. 4 of Buratti and Veverka (1983); specifically, we wanted to add in the 14 data points that are shown in that figure for the Voyager orange filter (0.59 µm, the longest wavelength available in Voyagers’ cameras), which combine observations from a wide range of longitudes to provide information on the shape of Europa’s phase curve from 3◦ phase to 28◦ phase. There are obvious dangers to adding such data to our modeling process; for example, we have no guarantee that the shape of the phase curve out in the VIMS near-infrared bears any resemblance to its shape in the orange part of the spectrum. However, we soon found another, more subtle “danger” present in the Voyager orange data: at phase angles of ∼ 15◦ and beyond, and in particular at α = 20◦ –30◦ , the Buratti and Veverka data have a much shallower slope (drop more slowly with increasing phase) than the more complete and more recent Europa visiblelight phase curves in Domingue and Verbiscer (1997) (see Fig. 2 below). This inconsistency probably occurs because Buratti and Veverka imposed a straight-line phase behavior (a constant mag/deg slope) on their data when separating out Europa’s phase curve from its rotational lightcurve. In our modeling process, it is not feasible to simply replace the Buratti and Veverka data with data from Domingue and Verbiscer because there are too many data points in the latter’s data set; the VIMS data, with only 20 data points per wavelength, could easily be “swamped” by the Domingue and Verbiscer points in a least-squares fitting process. Instead, we elected to make an alternate version of the Voyager orange data where all the points at phase angles of ∼ 15◦ and beyond are dropped in brightness to mimic the phase-curve shape seen at these α’s by Domingue and Verbiscer (this alternate data set can be seen in Figs. 3–6 below).
3. Modeling using Hapke’s function B. Hapke developed a commonly used theoretical photometric function with SHOE as its opposition-surge model (Hapke, 1981, 1984, 1986) and then later added a CBOE surge to this function (Hapke, 2002). As already noted, SHOE and CBOE make different predictions about how the strength of the surge should vary with the albedo of the surface, a fact which might be used to help predict which type of surge dominates over the other on a specific Solar System surface. However, in the VIMS Europa data the strength of the opposition surge does not appear to vary strongly with wavelength or albedo; at α = 2◦ –7◦ the slope of the VIMS phase curve—the so-called “phase coefficient”—is constant independent of wavelength/albedo, and even at α = 0.4◦ –0.6◦ the phase-coefficient values at different wavelengths/albedos are all the same to within the size
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Fig. 1. Phase-coefficient-vs.-albedo trends seen in the VIMS Europa data, compiled from results at wavelengths of 0.91, 1.73, and 2.25 µm. The phase coefficients, and their error bars, are taken from Fig. 4 of Brown et al. (2003); the corresponding albedo values are the geometric albedos adopted at each wavelength in the current work (see text).
of their error bars (see Fig. 1, adapted from a figure in Brown et al., 2003). The lack of an obvious dependence on albedo gives us no a-priori reason to suspect that one particular type of surge is dominating over the other, and thus it is important to investigate both types of opposition surges when analyzing the VIMS observations. The parts of Hapke’s function that model the SHOE surge and the CBOE surge each have two free parameters: one modeling the opposition-surge width and one modeling the surge amplitude (for the definitions of these, and other, Hapke parameters see Hapke, 1981, 1984, 1986, 2002). Because of the limited phase coverage of our data set and in particular the fact that the data beyond α = 1◦ can only be used in a relative, shape-of-phase-curve sense, it is not possible to believably constrain the values of all four of these parameters at the same time; i.e., it is not possible to uniquely constrain a model that combines both types of surges. Accordingly, we elected to perform two sets of Hapke fits: one in which we assume a SHOE surge only and derive best-fit values of the SHOE width and amplitude parameters (as per Hapke, 1981, 1984, 1986), and one which assumes a CBOE surge only and derives best-fit values for the CBOE width and amplitude (achieved by using the Hapke (2002) model with the amplitude of SHOE pinned at zero). In both types of fits, the single-scattering albedo w of the regolith is allowed to vary to provide the freedom to match the different albedos seen at our different VIMS wavelengths. The other free parameters in the Hapke function model the macroscopic roughness of the regolith and the average phase function of regolith particles. Our data’s limited phase coverage means that we cannot independently constrain any of these Hapke parameters, with the possible exception of the backscattering portion of the phase function. The para-
meters that cannot be constrained must be pinned at values previously derived for Europa; the best source of the latter is Domingue and Verbiscer (1997). Domingue and Verbiscer published Europa Hapke fits for two different visible wavelengths and two different forms of the particle phase function; we use the parameters for the longer of the two wavelengths provided (0.55 µm), and in our particular modeling software it is most convenient to use the parameters provided for the case of the so-called “2-term, 3-parameter Henyey-Greenstein” particle phase function. Domingue and Verbiscer also published separate Hapke fits for the leading and trailing sides of Europa. We elected to use the parameters provided for the trailing side, not because of the longitude observed by VIMS (which is in between the leading and trailing sides) but because this side’s phase curve comes slightly closer to mimicking the unusually shallow slope seen at α = 20◦ –30◦ in the Voyager data and mimicking the steepness of the slope seen at α = 0.4◦ –0.6◦ in the VIMS data (see Fig. 2). In the end, we decided that each combination of VIMS wavelength and type of opposition surge (SHOE or CBOE) should be given four different Hapke modeling runs: – A standard run which uses all three basic data sets (VIMS absolute data at α = 0.4◦ –0.6◦ , VIMS relative data on the phase-curve shape at α = 2◦ –7◦ , and Voyager relative data on the phase-curve shape at α = 3◦ – 28◦ ) and where the only parameters allowed to “float” are single-scattering albedo, surge width, and surge amplitude. – Same as the first run above except the parameter in the particle phase function that describes backscattering is also allowed to float (parameter d3 in Domingue and
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(a)
(b) Fig. 2. (a) Disk-integrated Europa data from the shortest VIMS wavelength investigated (0.91 µm), combined with comparable data on this same moon from the longest Voyager wavelength available (Buratti and Veverka, 1983) and compared with Europa Hapke-model phase curves from Domingue and Verbiscer (1997). The bold lines show the Domingue and Verbiscer phase curves at their original absolute brightness values; in making the lighter, non-bold lines, the Domingue and Verbiscer curves have been moved upwards or downwards in magnitude to approximately match the absolute brightness of the α < 1◦ VIMS data (the VIMS data whose absolute brightness has been carefully calibrated). The data points which are used in our modeling effort in only a relative, shape-of-the-phase-curve sense (the open and solid circles) have here been temporarily assigned an absolute brightness that allows them to track along the light, non-bold curves at phase angles from 2◦ to 10◦ . (b) Same as (a), except zooming-in on phase angles below 1◦ . In this figure and subsequent figures, the turnover in the Hapke curves at extremely low phase angles is caused by the finite angular width of the Sun, which, as seen from Jupiter, has a radius of approximately 0.05◦ .
Verbiscer’s terminology, called g1 in our software’s terminology), to better model the unusually shallow slope seen in the Voyager phase curve at α = 20◦ –30◦ . – Same as the first run above, except we use the alternate version of the Voyager relative data where the points at phase angles of ∼ 15◦ and beyond have been dropped in
brightness to better mimic the phase-curve shape seen in Domingue and Verbiscer (1997). – Same as the first run above, except the Voyager data are left out (so that no data points are constraining the fit at phase angles beyond α = 7◦ , the highest VIMS phase angle).
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Table 2 Hapke fits at 0.91 µm Type of run
Values are shown for: single-scattering albedo w; opposition-surge width h; opposition-surge amplitude B0 [and, where necessary, backscattering parameter g1 ] SHOE
CBOE
Basic run
w = 0.97 h = 0.0009 B0 = 1.00
w = 0.96 h = 0.0047 B0 = 0.62
Run where g1 is allowed to vary
w = 0.97 h = 0.0031 B0 = 0.98 [g1 = −0.33]
w = 0.97 h = 0.0033 B0 = 1.00 [g1 = −0.35]
Run where Voyager data are altered to mimic Domingue and Verbiscer shape
w = 0.96 h = 0.0015 B0 = 1.00
w = 0.95 h = 0.0041 B0 = 0.99
Run where Voyager data are left out
w = 0.95 h = 0.0018 B0 = 1.00
w = 0.95 h = 0.0046 B0 = 0.85
Note: In this and all subsequent tables, the following parameters were used unless otherwise specified, taken from the 0.55-µm trailing-side Hapke fit in Domingue and Verbiscer (1997): 2-term, 3-parameter Henyey–Greenstein particle phase function: backscattering parameter g1 (Domingue and Verbiscer’s parameter d3) = −0.386; forward-scattering parameter g2 (D+V’s parameter b3) = 0.083; partition coefficient f (this is 1−D+V’s parameter c3) = 0.216; macroscopic roughness mean slope angle θ¯ = 11◦ .
The Hapke modeling software in use, kindly provided by P. Helfenstein, uses a system of grid and gradient searches to minimize the root-mean-squared residuals between model and data and thus find the best-fitting values of the Hapke free parameters (cf., Helfenstein and Veverka, 1989; Helfenstein et al., 1994, 1996). 3.1. 0.91-µm data As shown in Fig. 2 (and especially Fig. 2b), the α < 1◦ VIMS data at this wavelength have a phase-curve slope that is generally consistent with the phase coefficient seen for Europa at 0.55 µm by Domingue and Verbiscer (1997). In other words, Europa’s opposition surge in the near-infrared—at least the α < 1◦ “heart” of the surge—is similar to its surge in the visible. Such a result is not surprising at 0.91 µm, since the albedo of Europa changes little on going from 0.55 to 0.91 µm; see, e.g., the Europa spectrum in Sill and Clark (1982). (What is surprising is the fact that this same α < 1◦ phase-curve slope is also seen at the other two VIMS wavelengths longward of 0.91 µm, where the albedo of Europa is much lower than it was at 0.55 µm or 0.91 µm; see Fig. 1.) 3.1.1. SHOE Hapke fits All four SHOE runs at this wavelength do a reasonable job of mimicking the slope of the VIMS α < 1◦ data (Fig. 3b), although the “basic” run does a slightly poorer job of this than the other three, more flexible runs. The four
different fits have opposition surges of noticeably different widths (Fig. 3a), with the surge width parameter h ranging from 0.0009 to 0.0031 (Table 2). This variation by a factor of more than three is a sign of just how uncertain the width of the surge can be when the data at α > 1◦ can only be used in a relative, shape-of-phase-curve sense (and is one of the reasons why we have elected not to bother determining rigorous error bars on the Hapke parameters in any one individual fit). This range of h values nicely encompasses the SHOE h value of 0.0016 determined for Europa in the visible by Domingue and Verbiscer (1997), emphasizing again that there are definite similarities between Europa’s near-IR and visible-light opposition surges. Note, however, that this wavelength’s near-IR data seem to require a larger SHOE amplitude B0 than Europa’s visible-light data do (see the B0 values near unity in Table 2, vs. the B0 values near 0.5 in Domingue and Verbiscer, 1997). 3.1.2. CBOE Hapke fits The CBOE runs at this wavelength do a noticeably poorer job of matching the α < 1◦ slope of the VIMS data than the corresponding SHOE fits did (Fig. 4). In particular, these CBOE fits insist on having a steeper phase-curve slope than what is seen in the data (see Fig. 4b, and compare it with Fig. 3b). In the fits that use VIMS only and leave out the Voyager data, the best-fit residual drops by a factor of more than two on switching from CBOE to SHOE; the drop in residual on going from CBOE to SHOE is smaller for the fits that include Voyager data, because in these cases the α < 1◦ VIMS data make up a smaller part of the overall data set. 3.2. 1.73-µm data 3.2.1. SHOE Hapke fits Two of the four SHOE runs “failed” in the sense that the Hapke-fitting software found best fits that make no attempt to mimic the slope seen in the α < 1◦ VIMS data; the fits match the basic absolute brightness of these VIMS data well enough, but have an almost flat phase curve at α < 1◦ thanks to a surge amplitude B0 close to zero. On running through a grid of B0 values and optimizing the other free parameters at each gridpoint, we found that the goodness-of-fit rms residual jumps back-and-forth randomly around a constant value as B0 varies; noticeably missing is the U-shaped minimum that is seen in the residual when there is a well-defined best fit. This kind of behavior occurs when we ask the Hapke function to model a phase-curve shape that it simply cannot mimic well; since the runs in question both included Voyager data (see Table 3), it may well be that the Hapke function has trouble fitting the Voyager slope and the steep, α < 1◦ VIMS slope at the same time. We will see below that this kind of fitting failure occurs again—and in fact occurs even more often—when we move on to this wavelength’s CBOE fits, and when we move on to the longest VIMS wavelength and its associated lower albedo.
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(a)
(b) Fig. 3. (a) The same 0.91-µm data set shown in Fig. 2, now compared with the phase curves of the SHOE Hapke models that best fit this data set. Note that an additional new version of the Voyager data has been added (triangles) that diverges from the original Voyager data at phase angles of ∼ 15◦ and above, to better mimic the shape of the Domingue and Verbiscer phase curves at those α. In the labels for the Hapke curves, “g1 floats” refers to a run where the backscatter parameter g1 is allowed to vary, “D+V mimic” refers to a run that uses the new alternate version of the Voyager data, and “toss Voyager” refers to a run that leaves the Voyager data out of the fitting process. The data points that are used in the fitting process in only a relative, shape-of-the-phase-curve sense (circles and triangles) have here been temporarily assigned an absolute brightness that allows the VIMS relative data to track along the topmost Hapke phase curve (i.e., allows the open circles to track along the topmost curve). (b) Same as (a), except zooming-in on phase angles below 1◦ .
The other two SHOE runs at this wavelength—the ones that produced successful, well-defined best fits—do a good job of mimicking the slope seen in the α < 1◦ VIMS data (Fig. 5b) and again have wildly different opposition-surge widths, this time covering h values from 0.0005 to 0.0033 (see Fig. 5a and Table 3).
3.2.2. CBOE Hapke fits Three of this wavelength’s four CBOE runs failed (Table 3) in a similar fashion to the failed fits mentioned above. As for the one successful CBOE fit, it again does a poorer job of mimicking the α < 1◦ slope of the VIMS data than this wavelength’s corresponding SHOE fits; the discrepancy
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(a)
(b) Fig. 4. Same as Fig. 3, except now moving on to the 0.91-µm CBOE fits. Again, all the relative data points have been temporarily assigned an absolute brightness that allows the open circles to track along the topmost curve.
is in the same direction as it was at the previous wavelength, i.e., the CBOE fit again insists on having a steeper slope than what is seen in the data (Fig. 5b). In the fits at this wavelength that use VIMS only and leave out the Voyager data, the best-fit residual drops by a factor of more than three on switching from CBOE to SHOE. 3.3. 2.25-µm data 3.3.1. SHOE and CBOE Hapke fits At this final, longest wavelength, only two of the eight runs were successful, one each for SHOE and CBOE. Not
surprisingly, these were the runs that leave out the Voyager data points entirely (Table 4), the runs which are constrained by the smallest amount of data and thus allow the Hapke function the most flexibility. Unfortunately, by the time we reach this wavelength, the noise in the VIMS instrument and in its data has grown enough that it is not clear whether SHOE or CBOE provides a better fit at α < 1◦ ; given the scatter in the data, about all that can be said is that both types of surges do a reasonable job of mimicking the slope seen at these phase angles in the VIMS data (see Fig. 6 and especially Fig. 6b).
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(a)
(b) Fig. 5. Same as Figs. 3 and 4, except now focusing on the VIMS data at 1.73 µm. Since several of the Hapke fits at this wavelength “failed” (see text) and the number of successful fits that need to be displayed has dropped, the SHOE and CBOE fits can now be shown on the same plot.
4. Discussion At 2.25 µm, where VIMS data are somewhat noisy, both types of opposition surges can mimic the slope of the VIMS phase curve at < 1◦ phase. At both 0.91 and 1.73 µm, however—where VIMS data are “cleaner”—CBOE does a noticeably poorer job than SHOE of matching the VIMS phase coefficient at α < 1◦ ; in particular, the best CBOE fit insists on having a steeper phase-curve slope than the data. Given this discrepancy, it is tempting to conclude that CBOE is less likely than SHOE to be the primary cause of Europa’s
near-IR opposition surge. However, such an all-or-nothing, “SHOE instead of CBOE” result is not completely conclusive, for several reasons: (1) the sparse nature of the VIMS phase coverage, and in particular the fact that the data at phase angles beyond 1◦ could only be used in a relative, shape-of-phase-curve sense; (2) the not-necessarily-correct assumption that Europa’s photometric behavior in the nearIR is similar to its behavior in the visible (i.e., the fact that we included additional phase-curve data from—and adopted some Hapke-parameter values from—visible wavelengths); and (3) the likelihood that Europa’s opposition surge, like
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(a)
(b) Fig. 6. Same as Fig. 5, except now focusing on the VIMS data at 2.25 µm.
surges elsewhere in the Solar System, may well be a combination of both SHOE and CBOE. A more balanced conclusion to be drawn from the phasecurve slope discrepancies—one that is more conclusively supported by the data at hand—is that Europa’s strong nearIR opposition surge at phase angles α < 1◦ cannot be explained by CBOE alone and must have a significant SHOE component, even at wavelengths where Europa is bright. This result, though more measured in tone, is still telling us something important; for example, it warns researchers who deal with bright icy bodies against trying to fit α < 1◦ data
with CBOE alone (a tempting simplification of the surgemodeling process). To test the robustness of this conclusion, we performed two additional sets of Hapke-fitting runs on the data at 0.91 µm, the wavelength where the VIMS data have the highest signal-to-noise ratio and the highest albedo (and where the presence of a significant SHOE component is thus the most surprising). These runs were not part of the main fitting process in the previous section because they push the bounds of what even this wavelength’s data quality can support. First, to get more information on whether it is truly
Europa’s opposition surge
Table 3 Hapke fits at 1.73 µm Type of run
Values are shown for: single-scattering albedo w; opposition-surge width h; opposition-surge amplitude B0 [and, where necessary, backscattering parameter g1 ] SHOE
CBOE
Basic run
Failed
Failed
Run where g1 is allowed to vary
w = 0.79 h = 0.0033 B0 = 0.59 [g1 = −0.19]
Failed
Run where Voyager data are altered to mimic Domingue and Verbiscer shape
Failed
Failed
Run where Voyager data are left out
w = 0.61 h = 0.0005 B0 = 0.99
w = 0.61 h = 0.0021 B0 = 1.00
Table 4 Hapke fits at 2.25 µm Type of run
Values are shown for: single-scattering albedo w; opposition-surge width h; opposition-surge amplitude B0 SHOE
CBOE
Basic run
Failed
Failed
Run where g1 is allowed to vary
Failed
Failed
Run where Voyager data are altered to mimic Domingue and Verbiscer shape
Failed
Failed
Run where Voyager data are left out
w = 0.37 h = 0.0003 B0 = 1.00
w = 0.36 h = 0.0052 B0 = 0.37
“intrinsic” to CBOE to have a steeper phase coefficient than the VIMS α < 1◦ data, we performed CBOE fits where the VIMS α < 1◦ data were the only data constraining the fit (thus giving CBOE its best opportunity ever to match these data’s slope). If we do not carefully limit the number of free Hapke parameters—if we let the backscattering parameter g1 “float,” for example—then there are not enough data to constrain the free parameters and the fit fails in the same way as the failed fits described in the previous section. If, however, the only parameters allowed to “float” are the single-scattering albedo, surge width, and surge amplitude, then the fitting process runs to completion and produces a phase curve very similar to those seen in the 0.91-µm CBOE fits in the previous section, i.e., the best-fit phase curve still insists on having a steeper slope than the VIMS α < 1◦ data. Thus in all the fits we have performed, we have not yet found anything to counter the idea that CBOE intrinsically has a steeper phase-curve slope than the VIMS α < 1◦
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data. This discrepancy could be a limitation specific to the Hapke mathematical formulation of CBOE, or it could be a broader limitation of the CBOE surge in general; determining which of these is the case would require a better data set (more complete near-opposition phase coverage, etc.) than the one used here. Second, given the likelihood that Europa’s opposition surge is actually a combination of SHOE and CBOE, we tried to fit our 0.91-µm data set using a Hapke function that contains both SHOE and CBOE (i.e., the “full up” function from Hapke, 2002). Given our strong suspicion, stated earlier, that our data set is not good enough to uniquely constrain the widths and amplitudes of both kinds of surges, it made little sense to perform runs that weaken the constraints by throwing out data or allowing additional Hapke parameters to float; accordingly, out of the four kinds of fitting runs described in the previous section, the only runs performed in this case were the “basic” or “standard” run and the run where the Voyager data have been altered to mimic the Domingue and Verbiscer phase-curve shape. The results of these two runs are shown in Fig. 7. Although both of the model phase curves in that figure fit the slope of the α < 1◦ VIMS data comparably well, they represent very different relative importances of the two kinds of opposition surges: in one curve the amplitude B0 for SHOE is larger than the amplitude B0 for CBOE, and in the other curve the relative amplitudes of the two kinds of surges are reversed. This supports our intuition that the data set is not good enough to uniquely constrain the widths and amplitudes of both kinds of surges. More important, though, is the fact that in both of the fits displayed, the amplitude B0 for SHOE is always 0.6 or above; this supports our main conclusion that Europa’s near-IR opposition surge at α < 1◦ cannot be explained by CBOE alone and must have a significant SHOE component. (Note that neither model curve in Fig. 7 fits the α < 1◦ data’s slope quite as well as the SHOE-only fits shown in the previous section. The fitting runs in that figure used a starting guess where the two different kinds of surges had amplitudes that were equal and significantly greater than zero; accordingly, it is not surprising that these runs did not make their way to the corner of Hapke-parameter space where the CBOE amplitude goes to zero and the surge is entirely SHOE.) Clearly, it would be extremely useful in the future to observe Europa’s opposition surge at the same range of near-IR wavelengths/same range of albedos as was done here, but this time obtain more complete phase coverage and in particular tie-down the shape of the phase curve continuously from near zero phase out to phase angles of 5◦ or 6◦ or 7◦ . That way, we could model the data with a Hapke function that has both SHOE and CBOE with some hope of being able to constrain the width and amplitude parameters of both kinds of surges—and thus be able to more conclusively measure the relative contributions made by SHOE and CBOE in Europa’s near-IR surge, and determine whether the surge is dominated by one physical cause at one wavelength/albedo
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Fig. 7. The same 0.91-µm, α < 1◦ data set shown in Figs. 2b, 3b, and 4b, this time compared with the phase curves of Hapke fits that include both SHOE and CBOE. Although both curves shown fit the slope of the data comparably well, they represent different relative importances of the two kinds of opposition surges: in the dashed curve the amplitude B0 for SHOE is larger than the amplitude B0 for CBOE, and in the other curve the relative amplitudes of the two kinds of surges are reversed.
and dominated by the other physical cause at another wavelength/albedo. Such a data set could also be fit to coherentbackscatter models other than Hapke’s, to test whether the problems in matching the observed α < 1◦ phase coefficient are specific to Hapke’s CBOE model or more broadly affect coherent backscatter in general. The Europa work performed here also reminds us of the limitations of trying to use albedo alone as a tool for discriminating between SHOE and CBOE. The core of the problem is the lack of any firm evidence that the strength of Europa’s opposition surge varies with wavelength/albedo: i.e., the fact that the slope of the VIMS phase curve at α = 2◦ –7◦ , and more importantly, at α = 0.4◦ –0.6◦ , is constant independent of wavelength/albedo to within the associated error bars (Fig. 1), and the fact that the α = 0.4◦ –0.6◦ slope seen by VIMS is similar to the slope seen previously at these phase angles in the visible (Fig. 2b). If Europa’s opposition surge is dominated at all wavelengths by one physical cause over the other—if, for example, SHOE dominates at all wavelengths, something that would be consistent with the Hapke fitting performed above—then one must explain why the dependence of surge strength on albedo predicted for that physical cause is not seen clearly in the VIMS data. If Europa’s surge is, instead, a combination of both SHOE and CBOE, then there may have to be an intricate trade-off between the two physical causes that would seemingly disobey Occam’s Razor: as the near-IR wavelength rises and the albedo drops, is it possible for SHOE to increase in importance, and CBOE decrease in importance, in a cooperation so perfect that there is no noticeable change in the strength of the surge? (And isn’t such a trade-off inconsistent with
the fact that CBOE did its best job of matching the slope of the VIMS phase curve at 2.25 µm, the longest wavelength/lowest albedo investigated?) In any case, it seems clear that the best approach is not to rely on albedo dependence as something that can discriminate between the two types of surges on its own, but to instead augment albedo information with other “tools” that can potentially differentiate between SHOE and CBOE; other potential discriminators that have been proposed include the polarization state of opposition-surge light, and the suggestion that the CBOE surge generally has a narrower angular width than SHOE (e.g., Hapke, 2002; Hapke et al., 1993, 1998; Helfenstein et al., 1997, 1998; Hillier et al., 1999; Mishchenko, 1992; Nelson et al., 2000; Shepard and Arvidson, 1999; Shkuratov et al., 1999). If an opposition-surge data set has the phase coverage, albedo leverage, detector type, etc., that can make use of the largest possible number of these tools, this will increase the likelihood that modeling the data will successfully discern the relative contributions of SHOE and CBOE. Finally, a note about the porosity of the surface of Europa: The value of the Hapke SHOE model’s angular-width parameter h is a function of both the porosity of the regolith and its grain size distribution (Hapke, 1986). The range of SHOE h values determined from the VIMS data—h = 0.0003 to 0.0033, depending on the wavelength, and type of fitting run, involved—encompasses a factor of ten and thus encompasses a range of possible porosities. However, since this range of values is centered on the SHOE h value of 0.0016 determined for Europa by Domingue and Verbiscer (1997), our results are certainly consistent with the porosity results
Europa’s opposition surge
of those earlier authors, who derived a very high porosity for Europa commensurate with a frost surface. (Note that the early shadow-hiding surge model by Irvine (1966), when fit to the phase-curve slope seen by VIMS at α < 1◦ , also predicts a very high porosity, specifically a porosity of ∼ 99% (Brown et al., 2003). However, the simple Irvine model, which does not include multiple scattering, does not have the correct phase-curve shape for Europa in the sense that it cannot reproduce the radically different phase coefficients seen at α < 1◦ and at α = 2◦ –7◦ ; if the Irvine model is fit to the shallower VIMS phase-curve slope seen at α = 2◦ –7◦ , it predicts a much different, lower porosity, on the order of 10%.)
5. Conclusions Near-infrared observations of Europa’s disk-integrated opposition surge by Cassini VIMS, first published in Brown et al. (2003), have now been modeled with the commonly used Hapke photometric function. The VIMS data set emphasizes observations at 16 solar phase angles from 0.4◦ to 0.6◦ —the first time the α < 1◦ “heart” of Europa’s opposition surge has been observed this well in the near-IR. This data set also provides a unique opportunity to examine how the surge is affected by changes in wavelength and albedo: at VIMS wavelengths of 0.91, 1.73, and 2.25 µm, the geometric albedo of Europa is 0.81, 0.33, and 0.18, respectively. Despite this factor-of-four albedo range, however, the slope of Europa’s phase curve at α < 1◦ is similar at all three wavelengths (to within the error bars) and this common slope is similar to the phase coefficient seen at this α in visible-light observations of Europa. Because of sparse VIMS phase coverage, it was not possible to constrain all the surge parameters at once in a Hapke function that has both the Shadow Hiding Opposition Effect (SHOE) and the Coherent Backscatter Opposition Effect (CBOE); accordingly, we performed separate Hapke fits for SHOE-only and CBOE-only surges. At 2.25 µm, where VIMS data are somewhat noisy, both types of surges can mimic the slope of the VIMS phase curve at α < 1◦ . At 0.91 and 1.73 µm, however—where VIMS data are “cleaner”— CBOE does a noticeably poorer job than SHOE of matching the VIMS phase coefficient at α < 1◦ ; in particular, the best CBOE fit insists on having a steeper phase-curve slope than the data. This discrepancy suggests that Europa’s near-IR opposition surge cannot be explained by CBOE alone and must have a significant SHOE component, even at wavelengths where Europa is bright. Future observations of Europa’s opposition surge at these same near-IR wavelengths, but with phase coverage that better delineates the shape of the phase curve over the entire range of α in the surge, could go a long way toward more conclusively separating out the relative importance of SHOE and CBOE on this moon.
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Acknowledgments This research was carried out at JPL with funding from the National Academy of Sciences/National Research Council resident research associateship program. We profusely thank P. Helfenstein for the use of his Hapke-fitting programs, and also appreciate helpful reviews from P. Geissler and an anonymous referee.
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