- Email: [email protected]
The Phase-Angle and Longitude Dependence of Polarization for Callisto
Icarus 159, 145–155 (2002) doi:10.1006/icar.2002.6879
The Phase-Angle and Longitude Dependence of Polarization for Callisto Vera K. Rosenbush Main Astronomical Observatory, National Academy of Science of Ukraine, Golosiiv, 03680 Kiev-127, Ukraine E-mail: [email protected] Received June 28, 2001; revised March 14, 2002
A detailed study of the linear polarization of Callisto in the UBVR filters has been carried out using the results of the author’s observations and all the data available. The variations of polarization with phase angle, longitude, and wavelength have been investigated. Systematic shifts of different sets of observations have been detected, and all data have been consequently reduced to a unified system by introducing a correction to the degree of polarization. The separation of the phase-angle and orbital curves of polarization is of special interest, because the observed polarization depends on the solar phase angle as well as on the longitude of the central meridian. The amplitude of longitudinal variations of polarization depends also on the phase angle. A method of separating the solar and orbital curves of polarization has been proposed, and their analytical description is given. The major sources of uncertainty in the problem of extracting the phase-angle dependence of polarization and the longitude one from the Callisto observations have been analyzed. The results obtained show that the satellite surface is polarimetrically heterogeneous. There is a cluster of seven data points that have very small values of polarization for the leading hemisphere. In our opinion, the Valhalla ring system, which has a high-albedo palimpsest in the center (≈600 km diameter), may cause this particularly low polarization. c 2002 Elsevier Science (USA) Key Words: satellites of Jupiter; Callisto; polarimetry.
INTRODUCTION
A polarimetric study of such distant objects as the Galilean satellites is quite difficult because of small phase-angle coverage accessible for ground-based observations. Nevertheless, the polarization measurements have shown that these satellites differ by their microtexture and surface albedo. Furthermore, some differences between the polarization curves measured at eastern and western elongation were detected for each satellite. This asymmetry arises from the fact that the satellites orbit Jupiter in synchronous rotation, and areas of different reflectivity come into view. Different polarization curves observed for two hemispheres and the significant scatter of polarization data points for each hemisphere indicate a variation in satellites polarization with orbital longitude, which might be caused by intrinsic differences in the surface characteristics between the two hemispheres for each satellite.
The most considerable difference between the polarization curves for two hemispheres is observed for Callisto. Polarimetric measurements made almost simultaneously by Dollfus (1971) and Veverka (1971) showed for the first time that the leading hemisphere of Callisto (in its orbital motion phase-locked to Jupiter) differs from the trailing one. To explain this difference, Dollfus (1975) suggested that the leading hemisphere of Callisto is covered with a thick, dusty, lunar-like regolith, whereas the trailing hemisphere presents larger grains or large areas of clean, dust-free rocky surface. Distinguishing between the polarization variations related to the position of a satellite in its orbit and the variations caused by the solar phase angle is a difficult problem. So far, it has not been solved because observations for a wide interval of longitudes at a fixed phase-angle range have been lacking. To investigate the phase dependence of polarization, measurements relating to different longitudes of the central meridian (L) were used: 0◦ ≤ L ≤ 180◦ for the leading hemisphere and 180◦ ≤ L ≤ 360◦ for the trailing one. Such averaging leads to a significant scattering of the polarization data points on the phase-angle dependence, which is significantly greater than the measurement errors. This scatter might be attributed to the local inhomogeneity of the satellite surface, which is superimposed on the global distinctions of polarimetric properties of the leading and trailing hemispheres. The purpose of this study is twofold. First, to separate the phase-angle dependence of polarization from the longitude one and to obtain their empirical analytic description on the basis of observations made by the author and using the results available in the literature. Second, to study the variations of polarization with the phase angle, the orbital longitude, and the wavelength. OVERVIEW OF OBSERVATIONS
In the 1960s and 1970s, the polarimetric measurements of the Galilean satellites of Jupiter, including Callisto, were carried out by Veverka (1971) and Dollfus (1971, 1975). It was found that Callisto is polarimetrically quite different from the other Galilean satellites. The polarization curve for the leading face of Callisto (i.e., at its eastern elongation when the orbital phase angle is 90◦ ) is characterized by the polarization minimum Pmin ≈ −0.9%. Extrapolation suggests an inversion angle
145 0019-1035/02 $35.00 c 2002 Elsevier Science (USA) All rights reserved.
146
VERA K. ROSENBUSH
at αinv = 22◦ . At the western elongation, at the orbital longitude L = 270◦ corresponding to the trailing hemisphere, the polarization minimum of about −0.6% is reached at the phase angle near 5◦ , and the inversion angle is 13◦ . A significant scattering of the polarization data points exceeding the measurement errors is observed for the leading hemisphere as well as for the trailing one. In 1972–1973, a series of measurements of linear polarization was fulfilled by Gradie and Zellner (1973). Their measurements supplemented the results of the previous works and confirmed the distinctions in polarimetric characteristics of the leading and trailing hemispheres of Callisto. The multicolor polarimetry of the Galilean satellites in six spectral bands in the range of 390–685 nm was performed for the first time by Botvinova and Kucherov (1980). These authors concluded that the minimum of polarization is essentially wavelength independent for all four Galilean satellites. It was also confirmed that there is a strong dependence of polarization degree of Callisto on the position of a satellite in its orbit. In 1981–1988, Chigladze (1989) carried out an extensive multicolor polarimetric observations of the Galilean satellites in the BVRI filters at phase angles ranging from 11.5◦ to 0.2◦ . It was found that Pmin for the leading hemisphere of Callisto exceeds Pmin for the trailing one by a factor of almost 1.5 and this distinction is essentially wavelength independent. At the same time, the values of Pmin for both hemispheres of the satellite were detected to show a substantial spectral trend that does not agree with the results obtained by Botvinova and Kucherov (1980). In 1988–1991, Rosenbush et al. (1997) performed polarimetric observations of the Galilean satellites within the phase-angle interval from 0.2◦ to 11.8◦ . For the leading and trailing hemispheres of each satellite, the phase curves of polarization degree and its parameters Pmin (the minimum polarization), αmin (the phase angle at which the minimum of polarization is observed), and αinv (the inversion angle defined as the phase angle where polarization changes its sign) were obtained in the UBVR filters. The polarization phase curves of Io, Europa, and Ganymede exhibit the polarization opposition effect in the form of a sharp peak of negative polarization centered at a very small phase angle of α ≈ 0.6◦ –0.7◦ . This peak, which is likely to be produced by the coherent backscattering mechanism, is superimposed on the regular negative branch of polarization. For Callisto, the polarization of the leading hemisphere is significantly larger than that for the trailing one, but a sharp negative polarization peak at α < 1◦ is apparently absent. For both hemispheres of Callisto, the observed degree of polarization near α ≈ 1◦ is about 0.3– 0.4%. At smaller phase angles the degree of polarization tends to 0. Observations in different spectral bands at very small phase angles are still rare, and the behavior of polarization near opposition is unknown in detail. That is why extensive measurements of polarization of the Galilean satellites around opposition were made in the UBVR filters at the 70-cm reflector of the Astronomical Observatory of Kharkiv National University on September 9–25, 1998 (Rosenbush et al. 2001) and November
19–December 7, 2000. In 1998, CCD imaging polarimetry of satellites was carried out using the one-channel polarimeter with a quasi-continuous rotating polaroid. The used CCD camera has 752 × 582 pixels2 (60 × 60 arcsec2 on the sky). A 15 × 15 arcsec2 area was selected for the polarization measurements of the satellites and the sky background. The instrumental polarization produced by this apparatus was determined with the accuracy ±0.038%. During the 2000 opposition, a one-channel photoelectric photometer-polarimeter with a rapidly rotating polaroid was used to observe the satellites in the UBVR spectral bands. The instrumental polarization was determined with the maximum possible accuracy ±0.010% for the filter V, and therefore the measurement error of the polarization degree of the satellite did not exceed ±0.02%. These observations confirmed the coherent polarization opposition effect for Io, Europa, and Ganymede found earlier by Rosenbush et al. (1997). For Callisto such a peak of negative polarization was not detected. We did not find any real deviation of the polarization-plane position from 90◦ relative to the scattering plane for Callisto. Hence, for this satellite as well as for most atmosphereless Solar System bodies (ASSBs), the phase changes of polarization are described by the second Stokes parameter only. Thus, the present study is based on the whole set of measurements of the polarization degree for Callisto in the UBVR filters made by the author to the present time and on all available results obtained by the other observers (Gradie and Zellner 1973, Dollfus 1975, Botvinova and Kucherov 1980, Chigladze 1989). There are 32 observations in the U band, 142 in the B band, 213 in the V band, and 133 measurements in the R band. The averaged error in the degree of polarization for the B, V, and R bands is estimated to be about ±0.05%. Only 17 measurements in the V band have an accuracy about ±0.1%. In the U band, the average error is about ±0.1%. Figures 1a and 1b represent the degree of polarization in the V filter versus the phase angle α and the central-meridian longitude L of the satellite, respectively. The scatter of the data points in these plots is considerably larger than the accuracy of the measurements. It is possible that there is a systematic shift of different sets of observations, which should be taken into consideration in a proper way. However, the scatter of the polarization data points in the plots makes it clear that the observed polarization depends both on the solar phase angle and on the orbital longitude. First, we investigated different datasets to search for possible systematic shifts. A longitude dependence of polarization was derived for two intervals of the phase angle: α = 9.7–11.3◦ and α = 10.7–11.8◦ . The number of observations made by different authors is large enough for comparison. We have assumed that the longitude dependence of polarization does not depend on the phase angle within the selected phase-angle ranges. The systematic shifts of different sets of observations in the V band are clearly seen in Fig. 2. Since the corrections have different signs we have found the best fit for all observations according to χ 2 minimization. The curve of best fit has been adopted as
147
P o la rization (%)
POLARIZATION OF CALLISTO
0.0
0.0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1.0
-1.0
a
F ilter V
-1.2
b
F ilter V
-1.2 0
2
4
6
8
10
P h ase an gle ( d e g )
12
0
90
180
270
0
90
180
L o n g itu d e ( d e g )
FIG. 1. Measured degree of linear polarization of Callisto versus solar phase angle (a) and longitude of the central meridian of the satellite (b) for the V filter. In (a), the open circles show polarization for the trailing hemisphere and the filled circles are for the leading one. In (b), the data points at the end are repeated from the beginning.
a “standard” curve. Thus, we reduced all the data available to a unified system using the corrections for the degree of polarization. For the V filter, all data of Dollfus (1975) were shifted by +0.11%, Gradie and Zellner (1973) by −0.04%, and Rosenbush et al. (1997) by −0.07% and −0.04% for observations in Bolivia and in Uzbekistan, respectively. There are also systematic shifts in the B filter (−0.11% for observations in Uzbekistan) and in the R filter (+0.08% for Chigladze’s observations, −0.08% for those in Bolivia, and −0.05% for those in Uzbekistan).
As we can see in Fig. 1, the observed polarization depends on the solar phase angle as well as on the longitude. Besides, the -0.4
P o larization (% )
-0.5 -0.6 -0 .7 -0 .8 -0 .9 -1 .0 90
18 0 2 70
0
90
P(α, L) = P1 (α) + P2 (L , α),
(1)
where P1 (α) and P2 (L , α) are the phase and longitude functions of polarization, respectively. P2 (L , α) can be given by P2 (L , α) = A(α)P(L),
METHOD DESCRIPTION
0
amplitude of longitude variations of polarization also depends on the phase angle. Therefore, the observed degree of polarization may be represented as a superposition of the two functions:
(2)
where P(L) is the dependence of polarization on the orbital longitude at a specified phase angle, and A(α) is the amplitude of the longitude dependence of polarization (assuming this variation with longitude is of sinusoidal type). The function A(α) should reflect the fact that the polarization does not depend on the orbital longitude and is identically equal to 0 at the phase angle α = 0◦ . If the longitude dependence of polarization is taken into account, P1 (α) is the unified phase function of polarization P(α) for the two hemispheres. The separation of the phase and longitude curves of polarization was performed by a method of successive iterations. In the first approximation, we supposed that the polarization does not depend on the longitude, that is, P2 (L , α) = 0. Then for each hemisphere, the observed data P(α, L) were approximated by the trigonometrical polynomial of Lumme and Muinonen (1993),
18 0
L on gitud e (de g) FIG. 2. Longitude dependence of polarization of Callisto for the interval of phase angles α = 10.7◦ –11.8◦ , which demonstrates the systematic shifts of different sets of observations in the V band. Filled circles are the measurements of Chigladze (1989); plus signs, Dollfus (1975); open triangles, Rosenbush et al. (1997), Bolivia; filled triangles, Rosenbush et al. (1997), Uzbekistan; open circles, Gradie and Zellner (1973).
Pfit (α) = b sinc1 (α) cosc2 (α/2) sin(α − αo ),
(3)
where the required parameters b, c1 , c2 , and αo are determined by the method of nonlinear least-squares fitting. The parameter c2 reflects the asymmetry of the phase curve of polarization at large phase angles. Therefore, it is very small in the domain of negative polarization and the term cosc2 (α/2) is close to unit at
148
VERA K. ROSENBUSH
small phase angles. αo = αinv is the inversion angle. The best fit curves for the leading and trailing hemispheres for the UBVR filters are shown in Fig. 3. To find this best fit, we used a standard way of choosing such parameters values where a sum of the squares of the data-point deviations from the theoretical curve was minimal (χ 2 minimization): P(L) = P(α, L) − Pfit (α).
(4)
For each hemisphere, the deviations P(L) represent the superposition of the longitude curves of polarization at different phase angles, which cannot be separated (Figs. 4a and 4b). In order to compare the observations under the same conditions of satellite illumination, the obtained deviations were adjusted to the phase angle α = 6◦ according to the fitted curve Pfit (α) from Eq. (3). Figures 3c, 3e, 3g, and 4a show that there is a cluster of seven data points (measurements made by Chigladze (1989) and
Leading hem isphere
Trailing hem isphere
0.0
0 .0
U
P olariza tion (%)
-0.2 -0.4
-0 .4
-0 .6
-0 .6
-0 .8
-0 .8
-1 .0
-1 .0
-1.2
-1 .2
a
-1.4 0.0
B
P olariza tion (%)
-0.4
-0 .4
-0 .6
-0 .6
-0 .8
-0 .8
-1 .0
-1 .0
-1.2
B
-0 .2
-1 .2
c
-1.4 0.0
d
-1 .4 0 .0
V
-0.2
P olariza tion (%)
b
-1 .4 0 .0
-0.2
-0.4
-0 .4
-0 .6
-0.6
-0 .8
-0.8
-1 .0
-1.0
-1.2
V
-0 .2
-1 .2
e
-1.4
f
-1 .4 0 .0
0.0
R
-0.2
P o lariza tion (%)
U
-0 .2
-0.4
-0 .4
-0 .6
-0.6
-0 .8
-0.8
-1 .0
-1.0
-1.2
R
-0 .2
-1 .2
g
-1.4
h
-1 .4 0
2
4
6
8
P ha se a ng le ( d e g )
10
12
0
2
4
6
8
10
12
P h ase an gle ( d e g )
FIG. 3. Phase-angle dependence of the degree of linear polarization of Callisto in the UBVR filters separately for the leading (left) and trailing (right) hemispheres. Solid lines are the best fit by the two-parameter model of Lumme and Muinonen (1993). Filled squares are the measurements of Botvinova and Kucherov (1980), and asterisks correspond to the Morozhenko measurements from Rosenbush et al. (1997). The other denotations are the same as described in the legend to Fig. 2.
149
POLARIZATION OF CALLISTO
L ea din g he m isp he re
Trailing he m isph ere
0.8
0 .8
V
Polarization (%)
0.6 0.4
0 .4
0.2
0 .2
0.0
0 .0
-0 .2
-0.2
-0 .4
-0.4
-0 .6
-0.6
a
-0.8 0.8
0
b
-0 .8 2
4
6
8
10
12
0
2
4
6
8
10
12
8
10
12
0 .8
V
0.6
Polarization (%)
V
0 .6
V
0 .6
0.4
0 .4
0.2
0 .2
0.0
0 .0
-0 .2
-0 .2
-0 .4
-0 .4
-0 .6
-0 .6
c
-0.8
d
-0 .8 0
2
4
6
8
10
12
0
2
P h ase an gle (deg)
4
6
P h a s e a n g le (deg)
FIG. 4. Deviations of the observed polarization from the best fit curve for the leading (left) and trailing (right) hemispheres of Callisto for the V filter. (a, b) are obtained after the first iteration and (c, d) after the second iteration.
If the differences of the polarization dependence on longitude for two hemispheres are taken into account, a common longitude dependence of polarization of the satellite at α = 6◦ can be derived. The solid curve in Fig. 5 shows that this dependence may 0 .0 F ilter V
-0 .2
P olariza tion (%)
Dollfus (1975)), which have very low values of polarization for the leading hemisphere. Close inspection of the data revealed that all the measurements, which were obtained at the phase angles 8.88◦ –11.52◦ and longitudes from 14◦ to 42◦ , give the degree of polarization −0.2–0.3%, and it does not relate to the instrumental effects. In our opinion, the Valhalla ring system, which has a size of about 4000 km, with a high-albedo palimpsest in the center (≈600-km diameter), may cause this particular low polarization. In fact, the Valhalla basin is located towards the center of the leading hemisphere of Callisto at 15.1◦ N, 55.6◦ W. The large-scale irregularities of the surface can have a profound influence on the total flux of the sunlight scattered by the particular area at given phase angles. At small phase angles, the total brightness of the satellite is governed essentially by the albedo, while at larger phase angles it depends also on the contribution of shadows, which arise from the large-scale irregularities of the surface. Since the degree of linear polarization is the ratio of the linearly polarized flux to the total flux, a very low polarization value detected for the leading hemisphere at phase angles around ∼10◦ is likely to be actually connected to the impact basin Valhalla. Thus, these seven data points were excluded from the further analysis.
-0 .4
-0 .6
-0 .8
-1 .0
-1 .2
0
90
18 0
27 0
0
90
18 0
L o ng itu de (deg) FIG. 5. Longitude dependence of polarization of Callisto at phase angle α = 6◦ after taking into account the phase-angle dependence of polarization in the first iteration.
150
VERA K. ROSENBUSH
L ea din g he m isp he re
Trailing he m isph ere
0.0
0 .0
P o la rization (%)
F ilter V
Filter V
-0.2
-0 .2
-0.4
-0 .4
-0.6
-0 .6
-0.8
-0 .8
-1.0
-1 .0
a
b
-1.2
-1 .2 0
2
4
6
8
10
12
0
2
P ha se a ng le (deg)
4
6
8
10
12
P h a s e a n g l e (deg)
FIG. 6. Phase-angle dependence of polarization for the leading (left) and trailing (right) hemispheres of Callisto after taking into account the longitude dependence of polarization in the first approximation (filled circles). Open circles are the observed polarization.
pendence at α = 6◦ , and P2 is the shift of the unified longitude dependence for both hemispheres. There is no longitude dependence at L = L o . It means that the values of the phase function of polarization coincide for both hemispheres. Therefore, the
be fitted, in a first approximation, by the function P(α = 6◦ , L) = P1 sin(L − L o ) + P2,
(5)
where the parameter P1 is the amplitude of the longitude de-
0 .0
0 .0
P o la rizatio n ( % )
U
B
-0 .2
-0.2
-0 .4
-0 .4
-0 .6
-0.6
-0 .8
-0 .8
-1 .0
-1.0
a
b
-1 .2
-1 .2 0
2
4
6
8
10
12
0.0
0
2
4
V
P o la rizatio n ( % )
6
8
10
12
8
10
12
0 .0
R
-0 .2
-0.2
-0.4
-0 .4
-0 .6
-0.6
-0.8
-0 .8
-1 .0
-1.0
d
c -1.2
-1 .2 0
2
4
6
8
P h a se an g le (deg)
10
12
0
2
4
6
P h ase a n gle (deg)
FIG. 7. Phase-angle dependence of polarization for the leading (filled circles) and trailing (open circles) hemispheres of Callisto for the UBVR filters after taking into account the variations of polarization with orbital longitude.
151
POLARIZATION OF CALLISTO -0 .4
-0.4
B
P ola rizatio n ( % )
U -0.6
-0 .6
-0.8
-0 .8
-1.0
-1 .0
b
a -1 .2
-1.2 0
90
1 80
2 70
0
90
-0 .4
90
0
1 80
18 0
2 70
1 80
-0 .4
V
P o la riza tio n ( % )
90
0
R
-0 .6
-0 .6
-0 .8
-0 .8
-1 .0
-1 .0
c
d
-1 .2
-1 .2 0
90
180
2 70
0
90
18 0
0
L o n g itu d e (deg)
90
1 80
2 70
0
90
1 80
Lo ng itu d e (deg)
FIG. 8. Longitude dependence of polarization of Callisto at phase angle α = 6◦ for the UBVR filters after taking into account the phase-angle dependence of polarization.
parameter P2 corresponds to the phase function P(α), for two hemispheres, at a given phase angle. After taking into account the longitude dependence of polarization to a first approximation, we obtained the phase dependence of polarization for the leading and trailing hemispheres of Callisto; the corresponding curves are shown in Figs. 6a and 6b. One can see that the scatter of the data points decreased considerably. The essential decreasing of scattering is also seen from Figs. 4c and 4d, where the residuals between the fitted curve and P(α = 6◦ , L) are plotted. In a similar manner, we make the second iteration using the phase and longitude functions derived in the first approximation. The resulting phase-angle and longitude curves of polarization for the UBVR filters are shown in Figs. 7 and 8, respectively. The best-fit curves are also plotted in the figures. The parameters of these curves for the V filter are given in Table I as an example. Since the amplitude and the shift of the longitude curve vary with a phase angle, the procedure described above was applied to the narrow interval of the phase angles (α = 1.5◦ ) with a step of 1◦ through the phase-angle range of observations (0.5◦ – 11.5◦ ). All the data were reduced to the middle of the selected intervals: 0.5◦ , 1.5◦ , . . . , 11.5◦ . As a consequence, the set of the parameters P1(α), P2(α), and L o (α) and their variations with the satellite phase angle are shown in Fig. 9 for the UBVR filters.
P1(α) and P2(α) have clear functional relations. Parameter L o is found to have only a slight dependence on the phase angle at α > 4◦ . The fitting polynomials for the phase dependence of the amplitude and the shift of the longitude curve of polarization
TABLE I Model Parameters of the Phase-Angle and Longitude Curves of Polarization for Callisto in the V Filter Phase-angle dependence of polarization b
C1
αo
P(α)
7.37 ± 1.01
Leading hemisphere 0.59 ± 0.02
(29.0 ± 2.0)a
P(α)
12.46 ± 0.72
Trailing hemisphere 0.48 ± 0.02
14.4 ± 0.2
Longitude dependence of polarization P1 P2 P(L)
0.129 ± 0.001
−0.694 ± 0.001
Lo 186.5 ± 0.3
Note. All observations are reduced to α = 6◦ . This value is apparently overstated because of the small phase-angle coverage of the negative branch of polarization. a
152
VERA K. ROSENBUSH 4 00
0.0
b
U B V R
3 50
0 .2 0 .0 U B V R
-0 .2
L o (deg)
P1 (%)
a
3 00 2 50 2 00
0
2
4
6
8
2
4
6
8
10 12
P ha se a ng le (deg)
P ha se a ng le (deg)
-0 .4 -0 .6
-1.0 0
10 12
U B V R
-0 .8
15 0
-0 .4
c -0 .2
P 2 (%)
0 .4
0
2
4
6
8
10 12
P h a s e a n g le (deg)
FIG. 9. Variations of the parameters P1 (a), P2 (c), and L o (b) with the phase angle of Callisto for the UBVR filters. Solid lines are derived from the model calculations and symbols mark the values of parameters, which were obtained from the observations.
were derived from them. For example, for the V filter they are P1(α) = 11.985 sin0.58 (α − 18.4),
(6)
P2(α) = −0.034 + 0.0115α + 0.00205α . 2
(7)
The deviation L o from 180◦ can be apparently explained by the fact that we used the harmonic function (Eq. (5)) for approximation of the longitude dependence, while the observation data show a small asymmetry of the longitude curve for the two hemispheres. Comparison of Eqs. (1), (2), and (5) shows that P1(α) ≡ A(α) and P2(α) ≡ P1 (α). Thus, the polarimetric observations of Callisto for any phase angle and any longitude can be described by the general expression P(α, L) = A(α) sin(L − L o ) + P1 (α),
ization made here. The analytical description for the dependence of polarization on the phase angle and on the longitude is given in Table II for the UBVR filters.
(8)
where P1 (α) and A(α) are described by Eqs. (6) and (7) for each filter, respectively. If we eliminate the first term in Eq. (8), which represents the longitude dependence of polarization, we derive the phase-angle dependence of polarization P(α). It testifies to the correct consideration of the longitude dependence of polarTABLE II Model Phase-Angle and Longitude Dependence of Polarization for Callisto P(α, L) = A(α) sin(L − L o (α)) + P1 (α) Filter U P1 (α) = 5.74 sin0.39 α sin(α − 25.5) A(α) = −0.211 + 0.1038α − 0.00512α 2 L o (α) = 375.1 − 45.6α at α < 4◦ L o (α) = 225.4 − 6.4α + 0.316α 2 at α > 4◦
Filter B P1 (α) = 9.84 sin0.49 α sin(α − 18.0) A(α) = 0.007α + 0.00184α 2 L o (α) = 150.2 − 7.93α at α < 4◦ L o (α) = 181.5 at α > 4◦
Filter V P1 (α) = 11.98 sin0.58 α sin(α − 18.4) A(α) = −0.034 + 0.0115α + 0.00205α 2 L o (α) = 375.1 − 45.6α at α < 4◦ L o (α) = 225.4 − 6.4α + 0.316α 2 at α > 4◦
Filter R P1 (α) = 10.74 sin0.55 α sin(α − 19.9) A(α) = 0.074 − 0.0215α − 0.00366α 2 L o (α) = 198.1 + 2.33α at α < 4◦ L o (α) = 211.8 − 2.42α at α > 4◦
COMPARISON OF THE MODELED AND OBSERVED LONGITUDE CURVES
The validity of presentation of the observed data by Eq. (8) was tested in the following manner. We modeled the longitude dependence of polarization for the phase-angle intervals in which the phase function of polarization P(α) does not change significantly. This is correct for very narrow ranges of phase angles. We estimate such a phase-angle interval to be from 1◦ to 1.6◦ . It is impossible to choose a narrower interval because of the lack of observation data, especially for small phase angles. On the other hand, even for a narrow interval as small as α = 2◦ , the longitude dependence of polarization is masked by the influence of the phase dependence, which must be taken into account. Figure 10 shows the variations of the measured degree of polarization in the V filter as a function of the planetocentric longitude of the satellite for selected intervals of the solar phase angles; the approximating function described by Eq. (5) is also displayed. For large phase angles, the longitude dependence of polarization is well pronounced and has large amplitude, which decreases with decreasing phase angle. At small phase angles, the longitude dependence is less noticeable and poorly determined because of the small number of observations. Variations of the parameters P1, P2, and L o with the satellite phase angle (Fig. 9) are very close to those derived from the model calculations. This fact testifies in favor of the suggested method of separation of the phase and longitude curves of polarization. UNCERTAINTY IN MODELING
There are several major sources of uncertainty in the extraction of the phase-angle and longitude dependencies of polarization from the Callisto observations available. First, the phaseangle range, which is accessible to observations from the Earth, is too small—about 12◦ . Consequently, a satisfactory fit of the
POLARIZATION OF CALLISTO
153
FIG. 10. The longitude dependence of polarization for the narrow intervals of the phase angles (α = 1.6◦ ), in which P(α) is considered to be constant. The solid line represents the best fit of the observed polarization; the dashed line is the fit of all data points corrected to the phase angle in the center of the interval selected. The small numbers in square brackets give the range of the phase angle over which the data for the particular curve was selected. The data obtained in the 1998 and 2000 oppositions are marked by filled and open diamonds, respectively. The other denotations are the same as described in the legends to Figs. 2 and 3.
phase-angle curve of polarization cannot be obtained. Although the minima of polarization Pmin and phase angle αmin for the leading hemisphere are probably not much in error, the inversion angle αinv may be overstated very much. That is why the values for αinv are given in parantheses in Table III. Besides, the data are not uniformly distributed both over the accessible
phase angles and over the whole longitude range. Therefore, we could not derive a reliable value of the inversion angle for the trailing hemisphere in the U filter because of a small number of measurements. However, as it seems to us, the primary uncertainty in determining the phase-angle and longitude curves of polarization
154
VERA K. ROSENBUSH
TABLE III Parameters of the Phase-Angle Dependence of Polarization for Callisto in the UBVR Filters Filter
Pmin (%)
αmin (◦ )
αinv (◦ )
U (360 B (460 V (550 R (690
nm) nm) nm) nm)
Leading hemisphere −1.02 9.0 ± 0.4 −0.79 8.8 ± 0.5 −0.85 10.1 ± 0.3 −0.87 ≥11.2
(23.1 ± 3.0) (26.7 ± 1.3) (26.2 ± 1.2) (35.4 ± 2.7)
U (360 B (460 V (550 R (690
nm) nm) nm) nm)
Trailing hemisphere −0.66 3.7 ± 0.4 −0.62 4.5 ± 0.3 −0.63 4.9 ± 0.4 −0.69 5.8 ± 0.2
(22.7 ± 3.75) 14.0 ± 0.2 14.1 ± 0.1 14.6 ± 0.1
is due to the fact that there is no way to fit the rotational polarization curves of Callisto very accurately. To do this, it is necessary to know very accurately the solar phase curve. It is clear that we are still short of polarimetric observational data of a sufficient amount and quality. As a result, the model function (Eq. (5)) describes the dependence of polarization on longitude only approximately at small phase angles. In the future it will be useful to measure the rotational polarization curve while the solar phase angle remains relatively fixed. RESULTS
Using the model relations (Table II), the parameters of the phase-angle dependence of polarization were determined for the
FIG. 11. Spectral dependence of the polarization minimum Pmin normalized to the V filter for the Galilean satellites. For Callisto, data are given for the leading (filled circles) and trailing (open circles) hemispheres.
leading and trailing hemispheres of Callisto in the UBVR filters. Pmin , αmin , and αinv are given in Table III. Evidently, the leading and trailing hemispheres of Callisto differ not only by the values of Pmin but also by their dependence on the wavelength. The variations of negative polarization minimum, normalized to the V filter, with wavelength are shown in Fig. 11. One can see that for both hemispheres of Callisto, |Pmin | varies insignificantly in the BVR filters. However, the absolute value of polarization for the leading hemisphere increases more sharply in the U filter than that for the trailing one. Analyzing the function Pmin (λ), it is necessary to take into account that the values Pmin are less reliable in the U filter than those in the other filters because of a small number of observations, which also require further refinement. Rosenbush et al. (1997) showed that there are some systematic differences in the depth of the negative branch of polarization for Io, Europa, Ganymede, and Callisto. For comparison, the curves Pmin (λ) for Io, Europa, and Ganymede are also illustrated in Fig. 11. In comparison with Callisto, the spectral distribution of polarization minimum for both hemispheres of Io, Europa, and Ganymede are much less pronounced, and Pmin (λ) can be averaged over the disks of these satellites. For Io and Europa, the negative branch of polarization is significantly deeper in the U filter than in other filters. The values of polarization minimum Pmin are about −0.6% for Io and −0.4% for Europa. There is a noticeable increase of polarization in the R filter for these satellites. Ganymede and Callisto show a low value of Pmin even in the U filter. From Fig. 7 and Table III one can see that the leading hemisphere of Callisto is characterized in all the spectral bands by values of the phase angle of polarization minimum αmin being considerably higher than those for the trailing hemisphere (approximately by a factor of 2 in the BVR bands). Besides, there is the systematic wavelength dependence of αmin for the leading as well as for the trailing hemisphere. It is seen from Fig. 7 that the point of inversion lies at phase angles α > 12◦ for both hemispheres of Callisto, and its position can be found only by extrapolation. Using the expressions listed in Table II we have obtained the inversion angle for the trailing hemisphere in the BVR filters, which is close enough to the phase angle, at which observations of the satellite can be performed from Earth (Table III). Therefore, these values of αinv are quite reliable. However, we could not derive a reliable value of the inversion angle in the U filter because of a small number of measurements. For the leading hemisphere, the determination of the inversion angle becomes even less reliable due to a narrow phase-angle range accessible for ground-based observations. The orbital variations of polarization reflect the distribution of materials with different physical properties over the satellite surface. Up to the present, the dependence of polarization on the longitude has been derived only for large phase angles α = 11◦ (Gradie and Zellner 1973, Dollfus 1975, Mandeville et al. 1980) and α ≈ 7◦ –10◦ (Botvinova and Kucherov 1980) in a single spectral band (see Table IV). The scarcity of the
155
POLARIZATION OF CALLISTO
TABLE IV Parameters of the Longitude Dependence of Polarization for Callisto in the UBVR Filters P(L min ) (%)
P (%)
−0.58 −0.57 −0.58 −0.67
0.44 0.21 0.22 0.15
λ ≈ 550
α = 10◦ –11.3◦ (Gradie and Zellner 1973) 90–100 ∼−0.98 270 ∼−0.25
0.73
λ ≈ 500
130–140
α = 11◦ (Dollfus 1975) −0.9 260–270
0.8
λ ≈ 550
α = 7◦ –10◦ (Botvinova and Kucherov 1980) 90–100 −1.05 270 −0.25
Filter (nm)
L max (◦ )
P(L max ) (%)
U (360) B (460) V (550) R (690)
108 89 98 105
α = 6◦ (present work) −1.02 287 −0.78 270 −0.80 278 −0.82 287
V (550)
98
L min (◦ )
α = 11◦ (present work) −0.93 276
much of the surface. These features are interpreted to be the result of erosion of the surface material caused by the sublimation of volatile compounds (Klemaszewski et al. 2001). Therefore, interpretation and discussion of new results from the groundbased and spacecraft data now available should be the next step in the study of Callisto. The features of the solar and orbital variations of brightness and polarization of the satellite together with the differences in color, albedo, micro- and macroscopic structure, and composition over the surface help us to retrieve information about the nature of Callisto’s surface materials and their evolution. ACKNOWLEDGMENTS
−0.1
−0.25
0.8
Special thanks to N. Kiselev for discussion and useful advice to this paper and to E. Petrova, who critically read the manuscript and suggested many improvements. I also thank K. Muinonen and A. Dollfus for their valuable comments and suggestions as reviewers.
0.68
REFERENCES
results is explained by a small number of measurements and by complexity of the separating of the solar and orbital variations of polarization. Especially, there is not enough near opposition data where the microstructure of the surface plays a dominant role in the light scattering process. The parameters of the longitude dependence of polarization of Callisto, which were received for the phase angle α = 6◦ , are given in Table IV. The orbital longitudes L min and L max , at which the corresponding minimum P(L min ) and maximum P(L max ) of polarization are observed, are given in this table. There are also the parameters for the phase angle α = 11◦ obtained by the author and other observers. The results agree very closely. It should be noted that both L min and L max seems to change with wavelength, and their spectral dependencies correlate in contrast to the degree of polarization, P(L min ) and P(L max ). Figure 10 shows that the amplitude of orbital variations of the polarization degree near opposition is not large, while it reaches ≈0.7–0.8% at α ≈ 10◦ –11◦ . Figure 8 and Table IV demonstrate that the variations of polarization degree with longitude of the satellite P (longitude effect) take place for all filters and their amplitude increases in the U filter. FUTURE WORK
High-resolution images of Callisto obtained from the Galileo spacecraft have shown that the surface of Callisto is strongly degraded and is covered with dark, ice-poor material which mantles
Botvinova, V. V., and V. A. Kucherov 1980. Multicolour polarimetry of Galilean satellites of Jupiter. Astrom. Astrofiz. 41, 59–63. Chigladze, R. A. 1989. Investigation of the Polarimetric Properties of the Galilean Satellites of Jupiter and Planet Uranus. Ph.D. Thesis, Abastumany Astrophysics. Observatory. Dollfus, A. 1971. Physical studies of asteroids by polarization of the light. In Physical Studies of Minor Planets (T. Gehrels, Ed.), NASA SP-267, pp. 100–116. Dollfus, A. 1975. Optical polarimetry of the Galilean satellites of Jupiter. Icarus 25, 416–431. Gradie, J., and B. Zellner 1973. A polarimetric survey of the Galilean satellites. Bull. Am. Astron. Soc. 5, 404–405. Klemaszewski, J. E., R. Greeley, and Galileo SSI Team 2001. Cratering and erosion on Callisto: New insights. In Jupiter: Planet, Satellites and Magnetosphere, June 25–30, 2001. Abstracts. Lumme, K., and K. Muinonen 1993. A two-parameter system for linear polarization of some Solar System objects. In Asteroids, Comets, Meteors (A. W. Harris, E. Bowell, Eds.), LPI Contribution 810, pp. 194–199. Lunar and Planetary Institute, Houston. Mandeville, J.-C., J. G. Geake, and A. Dollfus 1980. Reflectance polarimetry of Callisto and the evolution of the Galilean satellites. Icarus 41, 343–355. Rosenbush, V. K., V. V. Avramchuk, A. E. Rosenbush, and M. I. Mishchenko 1997. Polarization properties of the Galilean satellites of Jupiter: Observations and preliminary analysis. Astrophys. J. 487, 402–414. Rosenbush, V. K., N. N. Kiselev, K. Jockers, V. V. Korokhin, N. M. Shakhovskoj, and Yu. S. Efimov 2000. Optical polarimetry of the Galilean satellites, Iapetus, and 64 Angelina near opposition. In Kinematics and Physics of Celestial Bodies, Suppl. Ser. 3, pp. 227–230. Veverka, J. 1971. Polarization measurements of the Galilean satellites of Jupiter. Icarus 14, 355–359.
The Phase-Angle and Longitude Dependence of Polarization for Callisto Vera K. Rosenbush Main Astronomical Observatory, National Academy of Science of Ukraine, Golosiiv, 03680 Kiev-127, Ukraine E-mail: [email protected] Received June 28, 2001; revised March 14, 2002
A detailed study of the linear polarization of Callisto in the UBVR filters has been carried out using the results of the author’s observations and all the data available. The variations of polarization with phase angle, longitude, and wavelength have been investigated. Systematic shifts of different sets of observations have been detected, and all data have been consequently reduced to a unified system by introducing a correction to the degree of polarization. The separation of the phase-angle and orbital curves of polarization is of special interest, because the observed polarization depends on the solar phase angle as well as on the longitude of the central meridian. The amplitude of longitudinal variations of polarization depends also on the phase angle. A method of separating the solar and orbital curves of polarization has been proposed, and their analytical description is given. The major sources of uncertainty in the problem of extracting the phase-angle dependence of polarization and the longitude one from the Callisto observations have been analyzed. The results obtained show that the satellite surface is polarimetrically heterogeneous. There is a cluster of seven data points that have very small values of polarization for the leading hemisphere. In our opinion, the Valhalla ring system, which has a high-albedo palimpsest in the center (≈600 km diameter), may cause this particularly low polarization. c 2002 Elsevier Science (USA) Key Words: satellites of Jupiter; Callisto; polarimetry.
INTRODUCTION
A polarimetric study of such distant objects as the Galilean satellites is quite difficult because of small phase-angle coverage accessible for ground-based observations. Nevertheless, the polarization measurements have shown that these satellites differ by their microtexture and surface albedo. Furthermore, some differences between the polarization curves measured at eastern and western elongation were detected for each satellite. This asymmetry arises from the fact that the satellites orbit Jupiter in synchronous rotation, and areas of different reflectivity come into view. Different polarization curves observed for two hemispheres and the significant scatter of polarization data points for each hemisphere indicate a variation in satellites polarization with orbital longitude, which might be caused by intrinsic differences in the surface characteristics between the two hemispheres for each satellite.
The most considerable difference between the polarization curves for two hemispheres is observed for Callisto. Polarimetric measurements made almost simultaneously by Dollfus (1971) and Veverka (1971) showed for the first time that the leading hemisphere of Callisto (in its orbital motion phase-locked to Jupiter) differs from the trailing one. To explain this difference, Dollfus (1975) suggested that the leading hemisphere of Callisto is covered with a thick, dusty, lunar-like regolith, whereas the trailing hemisphere presents larger grains or large areas of clean, dust-free rocky surface. Distinguishing between the polarization variations related to the position of a satellite in its orbit and the variations caused by the solar phase angle is a difficult problem. So far, it has not been solved because observations for a wide interval of longitudes at a fixed phase-angle range have been lacking. To investigate the phase dependence of polarization, measurements relating to different longitudes of the central meridian (L) were used: 0◦ ≤ L ≤ 180◦ for the leading hemisphere and 180◦ ≤ L ≤ 360◦ for the trailing one. Such averaging leads to a significant scattering of the polarization data points on the phase-angle dependence, which is significantly greater than the measurement errors. This scatter might be attributed to the local inhomogeneity of the satellite surface, which is superimposed on the global distinctions of polarimetric properties of the leading and trailing hemispheres. The purpose of this study is twofold. First, to separate the phase-angle dependence of polarization from the longitude one and to obtain their empirical analytic description on the basis of observations made by the author and using the results available in the literature. Second, to study the variations of polarization with the phase angle, the orbital longitude, and the wavelength. OVERVIEW OF OBSERVATIONS
In the 1960s and 1970s, the polarimetric measurements of the Galilean satellites of Jupiter, including Callisto, were carried out by Veverka (1971) and Dollfus (1971, 1975). It was found that Callisto is polarimetrically quite different from the other Galilean satellites. The polarization curve for the leading face of Callisto (i.e., at its eastern elongation when the orbital phase angle is 90◦ ) is characterized by the polarization minimum Pmin ≈ −0.9%. Extrapolation suggests an inversion angle
145 0019-1035/02 $35.00 c 2002 Elsevier Science (USA) All rights reserved.
146
VERA K. ROSENBUSH
at αinv = 22◦ . At the western elongation, at the orbital longitude L = 270◦ corresponding to the trailing hemisphere, the polarization minimum of about −0.6% is reached at the phase angle near 5◦ , and the inversion angle is 13◦ . A significant scattering of the polarization data points exceeding the measurement errors is observed for the leading hemisphere as well as for the trailing one. In 1972–1973, a series of measurements of linear polarization was fulfilled by Gradie and Zellner (1973). Their measurements supplemented the results of the previous works and confirmed the distinctions in polarimetric characteristics of the leading and trailing hemispheres of Callisto. The multicolor polarimetry of the Galilean satellites in six spectral bands in the range of 390–685 nm was performed for the first time by Botvinova and Kucherov (1980). These authors concluded that the minimum of polarization is essentially wavelength independent for all four Galilean satellites. It was also confirmed that there is a strong dependence of polarization degree of Callisto on the position of a satellite in its orbit. In 1981–1988, Chigladze (1989) carried out an extensive multicolor polarimetric observations of the Galilean satellites in the BVRI filters at phase angles ranging from 11.5◦ to 0.2◦ . It was found that Pmin for the leading hemisphere of Callisto exceeds Pmin for the trailing one by a factor of almost 1.5 and this distinction is essentially wavelength independent. At the same time, the values of Pmin for both hemispheres of the satellite were detected to show a substantial spectral trend that does not agree with the results obtained by Botvinova and Kucherov (1980). In 1988–1991, Rosenbush et al. (1997) performed polarimetric observations of the Galilean satellites within the phase-angle interval from 0.2◦ to 11.8◦ . For the leading and trailing hemispheres of each satellite, the phase curves of polarization degree and its parameters Pmin (the minimum polarization), αmin (the phase angle at which the minimum of polarization is observed), and αinv (the inversion angle defined as the phase angle where polarization changes its sign) were obtained in the UBVR filters. The polarization phase curves of Io, Europa, and Ganymede exhibit the polarization opposition effect in the form of a sharp peak of negative polarization centered at a very small phase angle of α ≈ 0.6◦ –0.7◦ . This peak, which is likely to be produced by the coherent backscattering mechanism, is superimposed on the regular negative branch of polarization. For Callisto, the polarization of the leading hemisphere is significantly larger than that for the trailing one, but a sharp negative polarization peak at α < 1◦ is apparently absent. For both hemispheres of Callisto, the observed degree of polarization near α ≈ 1◦ is about 0.3– 0.4%. At smaller phase angles the degree of polarization tends to 0. Observations in different spectral bands at very small phase angles are still rare, and the behavior of polarization near opposition is unknown in detail. That is why extensive measurements of polarization of the Galilean satellites around opposition were made in the UBVR filters at the 70-cm reflector of the Astronomical Observatory of Kharkiv National University on September 9–25, 1998 (Rosenbush et al. 2001) and November
19–December 7, 2000. In 1998, CCD imaging polarimetry of satellites was carried out using the one-channel polarimeter with a quasi-continuous rotating polaroid. The used CCD camera has 752 × 582 pixels2 (60 × 60 arcsec2 on the sky). A 15 × 15 arcsec2 area was selected for the polarization measurements of the satellites and the sky background. The instrumental polarization produced by this apparatus was determined with the accuracy ±0.038%. During the 2000 opposition, a one-channel photoelectric photometer-polarimeter with a rapidly rotating polaroid was used to observe the satellites in the UBVR spectral bands. The instrumental polarization was determined with the maximum possible accuracy ±0.010% for the filter V, and therefore the measurement error of the polarization degree of the satellite did not exceed ±0.02%. These observations confirmed the coherent polarization opposition effect for Io, Europa, and Ganymede found earlier by Rosenbush et al. (1997). For Callisto such a peak of negative polarization was not detected. We did not find any real deviation of the polarization-plane position from 90◦ relative to the scattering plane for Callisto. Hence, for this satellite as well as for most atmosphereless Solar System bodies (ASSBs), the phase changes of polarization are described by the second Stokes parameter only. Thus, the present study is based on the whole set of measurements of the polarization degree for Callisto in the UBVR filters made by the author to the present time and on all available results obtained by the other observers (Gradie and Zellner 1973, Dollfus 1975, Botvinova and Kucherov 1980, Chigladze 1989). There are 32 observations in the U band, 142 in the B band, 213 in the V band, and 133 measurements in the R band. The averaged error in the degree of polarization for the B, V, and R bands is estimated to be about ±0.05%. Only 17 measurements in the V band have an accuracy about ±0.1%. In the U band, the average error is about ±0.1%. Figures 1a and 1b represent the degree of polarization in the V filter versus the phase angle α and the central-meridian longitude L of the satellite, respectively. The scatter of the data points in these plots is considerably larger than the accuracy of the measurements. It is possible that there is a systematic shift of different sets of observations, which should be taken into consideration in a proper way. However, the scatter of the polarization data points in the plots makes it clear that the observed polarization depends both on the solar phase angle and on the orbital longitude. First, we investigated different datasets to search for possible systematic shifts. A longitude dependence of polarization was derived for two intervals of the phase angle: α = 9.7–11.3◦ and α = 10.7–11.8◦ . The number of observations made by different authors is large enough for comparison. We have assumed that the longitude dependence of polarization does not depend on the phase angle within the selected phase-angle ranges. The systematic shifts of different sets of observations in the V band are clearly seen in Fig. 2. Since the corrections have different signs we have found the best fit for all observations according to χ 2 minimization. The curve of best fit has been adopted as
147
P o la rization (%)
POLARIZATION OF CALLISTO
0.0
0.0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1.0
-1.0
a
F ilter V
-1.2
b
F ilter V
-1.2 0
2
4
6
8
10
P h ase an gle ( d e g )
12
0
90
180
270
0
90
180
L o n g itu d e ( d e g )
FIG. 1. Measured degree of linear polarization of Callisto versus solar phase angle (a) and longitude of the central meridian of the satellite (b) for the V filter. In (a), the open circles show polarization for the trailing hemisphere and the filled circles are for the leading one. In (b), the data points at the end are repeated from the beginning.
a “standard” curve. Thus, we reduced all the data available to a unified system using the corrections for the degree of polarization. For the V filter, all data of Dollfus (1975) were shifted by +0.11%, Gradie and Zellner (1973) by −0.04%, and Rosenbush et al. (1997) by −0.07% and −0.04% for observations in Bolivia and in Uzbekistan, respectively. There are also systematic shifts in the B filter (−0.11% for observations in Uzbekistan) and in the R filter (+0.08% for Chigladze’s observations, −0.08% for those in Bolivia, and −0.05% for those in Uzbekistan).
As we can see in Fig. 1, the observed polarization depends on the solar phase angle as well as on the longitude. Besides, the -0.4
P o larization (% )
-0.5 -0.6 -0 .7 -0 .8 -0 .9 -1 .0 90
18 0 2 70
0
90
P(α, L) = P1 (α) + P2 (L , α),
(1)
where P1 (α) and P2 (L , α) are the phase and longitude functions of polarization, respectively. P2 (L , α) can be given by P2 (L , α) = A(α)P(L),
METHOD DESCRIPTION
0
amplitude of longitude variations of polarization also depends on the phase angle. Therefore, the observed degree of polarization may be represented as a superposition of the two functions:
(2)
where P(L) is the dependence of polarization on the orbital longitude at a specified phase angle, and A(α) is the amplitude of the longitude dependence of polarization (assuming this variation with longitude is of sinusoidal type). The function A(α) should reflect the fact that the polarization does not depend on the orbital longitude and is identically equal to 0 at the phase angle α = 0◦ . If the longitude dependence of polarization is taken into account, P1 (α) is the unified phase function of polarization P(α) for the two hemispheres. The separation of the phase and longitude curves of polarization was performed by a method of successive iterations. In the first approximation, we supposed that the polarization does not depend on the longitude, that is, P2 (L , α) = 0. Then for each hemisphere, the observed data P(α, L) were approximated by the trigonometrical polynomial of Lumme and Muinonen (1993),
18 0
L on gitud e (de g) FIG. 2. Longitude dependence of polarization of Callisto for the interval of phase angles α = 10.7◦ –11.8◦ , which demonstrates the systematic shifts of different sets of observations in the V band. Filled circles are the measurements of Chigladze (1989); plus signs, Dollfus (1975); open triangles, Rosenbush et al. (1997), Bolivia; filled triangles, Rosenbush et al. (1997), Uzbekistan; open circles, Gradie and Zellner (1973).
Pfit (α) = b sinc1 (α) cosc2 (α/2) sin(α − αo ),
(3)
where the required parameters b, c1 , c2 , and αo are determined by the method of nonlinear least-squares fitting. The parameter c2 reflects the asymmetry of the phase curve of polarization at large phase angles. Therefore, it is very small in the domain of negative polarization and the term cosc2 (α/2) is close to unit at
148
VERA K. ROSENBUSH
small phase angles. αo = αinv is the inversion angle. The best fit curves for the leading and trailing hemispheres for the UBVR filters are shown in Fig. 3. To find this best fit, we used a standard way of choosing such parameters values where a sum of the squares of the data-point deviations from the theoretical curve was minimal (χ 2 minimization): P(L) = P(α, L) − Pfit (α).
(4)
For each hemisphere, the deviations P(L) represent the superposition of the longitude curves of polarization at different phase angles, which cannot be separated (Figs. 4a and 4b). In order to compare the observations under the same conditions of satellite illumination, the obtained deviations were adjusted to the phase angle α = 6◦ according to the fitted curve Pfit (α) from Eq. (3). Figures 3c, 3e, 3g, and 4a show that there is a cluster of seven data points (measurements made by Chigladze (1989) and
Leading hem isphere
Trailing hem isphere
0.0
0 .0
U
P olariza tion (%)
-0.2 -0.4
-0 .4
-0 .6
-0 .6
-0 .8
-0 .8
-1 .0
-1 .0
-1.2
-1 .2
a
-1.4 0.0
B
P olariza tion (%)
-0.4
-0 .4
-0 .6
-0 .6
-0 .8
-0 .8
-1 .0
-1 .0
-1.2
B
-0 .2
-1 .2
c
-1.4 0.0
d
-1 .4 0 .0
V
-0.2
P olariza tion (%)
b
-1 .4 0 .0
-0.2
-0.4
-0 .4
-0 .6
-0.6
-0 .8
-0.8
-1 .0
-1.0
-1.2
V
-0 .2
-1 .2
e
-1.4
f
-1 .4 0 .0
0.0
R
-0.2
P o lariza tion (%)
U
-0 .2
-0.4
-0 .4
-0 .6
-0.6
-0 .8
-0.8
-1 .0
-1.0
-1.2
R
-0 .2
-1 .2
g
-1.4
h
-1 .4 0
2
4
6
8
P ha se a ng le ( d e g )
10
12
0
2
4
6
8
10
12
P h ase an gle ( d e g )
FIG. 3. Phase-angle dependence of the degree of linear polarization of Callisto in the UBVR filters separately for the leading (left) and trailing (right) hemispheres. Solid lines are the best fit by the two-parameter model of Lumme and Muinonen (1993). Filled squares are the measurements of Botvinova and Kucherov (1980), and asterisks correspond to the Morozhenko measurements from Rosenbush et al. (1997). The other denotations are the same as described in the legend to Fig. 2.
149
POLARIZATION OF CALLISTO
L ea din g he m isp he re
Trailing he m isph ere
0.8
0 .8
V
Polarization (%)
0.6 0.4
0 .4
0.2
0 .2
0.0
0 .0
-0 .2
-0.2
-0 .4
-0.4
-0 .6
-0.6
a
-0.8 0.8
0
b
-0 .8 2
4
6
8
10
12
0
2
4
6
8
10
12
8
10
12
0 .8
V
0.6
Polarization (%)
V
0 .6
V
0 .6
0.4
0 .4
0.2
0 .2
0.0
0 .0
-0 .2
-0 .2
-0 .4
-0 .4
-0 .6
-0 .6
c
-0.8
d
-0 .8 0
2
4
6
8
10
12
0
2
P h ase an gle (deg)
4
6
P h a s e a n g le (deg)
FIG. 4. Deviations of the observed polarization from the best fit curve for the leading (left) and trailing (right) hemispheres of Callisto for the V filter. (a, b) are obtained after the first iteration and (c, d) after the second iteration.
If the differences of the polarization dependence on longitude for two hemispheres are taken into account, a common longitude dependence of polarization of the satellite at α = 6◦ can be derived. The solid curve in Fig. 5 shows that this dependence may 0 .0 F ilter V
-0 .2
P olariza tion (%)
Dollfus (1975)), which have very low values of polarization for the leading hemisphere. Close inspection of the data revealed that all the measurements, which were obtained at the phase angles 8.88◦ –11.52◦ and longitudes from 14◦ to 42◦ , give the degree of polarization −0.2–0.3%, and it does not relate to the instrumental effects. In our opinion, the Valhalla ring system, which has a size of about 4000 km, with a high-albedo palimpsest in the center (≈600-km diameter), may cause this particular low polarization. In fact, the Valhalla basin is located towards the center of the leading hemisphere of Callisto at 15.1◦ N, 55.6◦ W. The large-scale irregularities of the surface can have a profound influence on the total flux of the sunlight scattered by the particular area at given phase angles. At small phase angles, the total brightness of the satellite is governed essentially by the albedo, while at larger phase angles it depends also on the contribution of shadows, which arise from the large-scale irregularities of the surface. Since the degree of linear polarization is the ratio of the linearly polarized flux to the total flux, a very low polarization value detected for the leading hemisphere at phase angles around ∼10◦ is likely to be actually connected to the impact basin Valhalla. Thus, these seven data points were excluded from the further analysis.
-0 .4
-0 .6
-0 .8
-1 .0
-1 .2
0
90
18 0
27 0
0
90
18 0
L o ng itu de (deg) FIG. 5. Longitude dependence of polarization of Callisto at phase angle α = 6◦ after taking into account the phase-angle dependence of polarization in the first iteration.
150
VERA K. ROSENBUSH
L ea din g he m isp he re
Trailing he m isph ere
0.0
0 .0
P o la rization (%)
F ilter V
Filter V
-0.2
-0 .2
-0.4
-0 .4
-0.6
-0 .6
-0.8
-0 .8
-1.0
-1 .0
a
b
-1.2
-1 .2 0
2
4
6
8
10
12
0
2
P ha se a ng le (deg)
4
6
8
10
12
P h a s e a n g l e (deg)
FIG. 6. Phase-angle dependence of polarization for the leading (left) and trailing (right) hemispheres of Callisto after taking into account the longitude dependence of polarization in the first approximation (filled circles). Open circles are the observed polarization.
pendence at α = 6◦ , and P2 is the shift of the unified longitude dependence for both hemispheres. There is no longitude dependence at L = L o . It means that the values of the phase function of polarization coincide for both hemispheres. Therefore, the
be fitted, in a first approximation, by the function P(α = 6◦ , L) = P1 sin(L − L o ) + P2,
(5)
where the parameter P1 is the amplitude of the longitude de-
0 .0
0 .0
P o la rizatio n ( % )
U
B
-0 .2
-0.2
-0 .4
-0 .4
-0 .6
-0.6
-0 .8
-0 .8
-1 .0
-1.0
a
b
-1 .2
-1 .2 0
2
4
6
8
10
12
0.0
0
2
4
V
P o la rizatio n ( % )
6
8
10
12
8
10
12
0 .0
R
-0 .2
-0.2
-0.4
-0 .4
-0 .6
-0.6
-0.8
-0 .8
-1 .0
-1.0
d
c -1.2
-1 .2 0
2
4
6
8
P h a se an g le (deg)
10
12
0
2
4
6
P h ase a n gle (deg)
FIG. 7. Phase-angle dependence of polarization for the leading (filled circles) and trailing (open circles) hemispheres of Callisto for the UBVR filters after taking into account the variations of polarization with orbital longitude.
151
POLARIZATION OF CALLISTO -0 .4
-0.4
B
P ola rizatio n ( % )
U -0.6
-0 .6
-0.8
-0 .8
-1.0
-1 .0
b
a -1 .2
-1.2 0
90
1 80
2 70
0
90
-0 .4
90
0
1 80
18 0
2 70
1 80
-0 .4
V
P o la riza tio n ( % )
90
0
R
-0 .6
-0 .6
-0 .8
-0 .8
-1 .0
-1 .0
c
d
-1 .2
-1 .2 0
90
180
2 70
0
90
18 0
0
L o n g itu d e (deg)
90
1 80
2 70
0
90
1 80
Lo ng itu d e (deg)
FIG. 8. Longitude dependence of polarization of Callisto at phase angle α = 6◦ for the UBVR filters after taking into account the phase-angle dependence of polarization.
parameter P2 corresponds to the phase function P(α), for two hemispheres, at a given phase angle. After taking into account the longitude dependence of polarization to a first approximation, we obtained the phase dependence of polarization for the leading and trailing hemispheres of Callisto; the corresponding curves are shown in Figs. 6a and 6b. One can see that the scatter of the data points decreased considerably. The essential decreasing of scattering is also seen from Figs. 4c and 4d, where the residuals between the fitted curve and P(α = 6◦ , L) are plotted. In a similar manner, we make the second iteration using the phase and longitude functions derived in the first approximation. The resulting phase-angle and longitude curves of polarization for the UBVR filters are shown in Figs. 7 and 8, respectively. The best-fit curves are also plotted in the figures. The parameters of these curves for the V filter are given in Table I as an example. Since the amplitude and the shift of the longitude curve vary with a phase angle, the procedure described above was applied to the narrow interval of the phase angles (α = 1.5◦ ) with a step of 1◦ through the phase-angle range of observations (0.5◦ – 11.5◦ ). All the data were reduced to the middle of the selected intervals: 0.5◦ , 1.5◦ , . . . , 11.5◦ . As a consequence, the set of the parameters P1(α), P2(α), and L o (α) and their variations with the satellite phase angle are shown in Fig. 9 for the UBVR filters.
P1(α) and P2(α) have clear functional relations. Parameter L o is found to have only a slight dependence on the phase angle at α > 4◦ . The fitting polynomials for the phase dependence of the amplitude and the shift of the longitude curve of polarization
TABLE I Model Parameters of the Phase-Angle and Longitude Curves of Polarization for Callisto in the V Filter Phase-angle dependence of polarization b
C1
αo
P(α)
7.37 ± 1.01
Leading hemisphere 0.59 ± 0.02
(29.0 ± 2.0)a
P(α)
12.46 ± 0.72
Trailing hemisphere 0.48 ± 0.02
14.4 ± 0.2
Longitude dependence of polarization P1 P2 P(L)
0.129 ± 0.001
−0.694 ± 0.001
Lo 186.5 ± 0.3
Note. All observations are reduced to α = 6◦ . This value is apparently overstated because of the small phase-angle coverage of the negative branch of polarization. a
152
VERA K. ROSENBUSH 4 00
0.0
b
U B V R
3 50
0 .2 0 .0 U B V R
-0 .2
L o (deg)
P1 (%)
a
3 00 2 50 2 00
0
2
4
6
8
2
4
6
8
10 12
P ha se a ng le (deg)
P ha se a ng le (deg)
-0 .4 -0 .6
-1.0 0
10 12
U B V R
-0 .8
15 0
-0 .4
c -0 .2
P 2 (%)
0 .4
0
2
4
6
8
10 12
P h a s e a n g le (deg)
FIG. 9. Variations of the parameters P1 (a), P2 (c), and L o (b) with the phase angle of Callisto for the UBVR filters. Solid lines are derived from the model calculations and symbols mark the values of parameters, which were obtained from the observations.
were derived from them. For example, for the V filter they are P1(α) = 11.985 sin0.58 (α − 18.4),
(6)
P2(α) = −0.034 + 0.0115α + 0.00205α . 2
(7)
The deviation L o from 180◦ can be apparently explained by the fact that we used the harmonic function (Eq. (5)) for approximation of the longitude dependence, while the observation data show a small asymmetry of the longitude curve for the two hemispheres. Comparison of Eqs. (1), (2), and (5) shows that P1(α) ≡ A(α) and P2(α) ≡ P1 (α). Thus, the polarimetric observations of Callisto for any phase angle and any longitude can be described by the general expression P(α, L) = A(α) sin(L − L o ) + P1 (α),
ization made here. The analytical description for the dependence of polarization on the phase angle and on the longitude is given in Table II for the UBVR filters.
(8)
where P1 (α) and A(α) are described by Eqs. (6) and (7) for each filter, respectively. If we eliminate the first term in Eq. (8), which represents the longitude dependence of polarization, we derive the phase-angle dependence of polarization P(α). It testifies to the correct consideration of the longitude dependence of polarTABLE II Model Phase-Angle and Longitude Dependence of Polarization for Callisto P(α, L) = A(α) sin(L − L o (α)) + P1 (α) Filter U P1 (α) = 5.74 sin0.39 α sin(α − 25.5) A(α) = −0.211 + 0.1038α − 0.00512α 2 L o (α) = 375.1 − 45.6α at α < 4◦ L o (α) = 225.4 − 6.4α + 0.316α 2 at α > 4◦
Filter B P1 (α) = 9.84 sin0.49 α sin(α − 18.0) A(α) = 0.007α + 0.00184α 2 L o (α) = 150.2 − 7.93α at α < 4◦ L o (α) = 181.5 at α > 4◦
Filter V P1 (α) = 11.98 sin0.58 α sin(α − 18.4) A(α) = −0.034 + 0.0115α + 0.00205α 2 L o (α) = 375.1 − 45.6α at α < 4◦ L o (α) = 225.4 − 6.4α + 0.316α 2 at α > 4◦
Filter R P1 (α) = 10.74 sin0.55 α sin(α − 19.9) A(α) = 0.074 − 0.0215α − 0.00366α 2 L o (α) = 198.1 + 2.33α at α < 4◦ L o (α) = 211.8 − 2.42α at α > 4◦
COMPARISON OF THE MODELED AND OBSERVED LONGITUDE CURVES
The validity of presentation of the observed data by Eq. (8) was tested in the following manner. We modeled the longitude dependence of polarization for the phase-angle intervals in which the phase function of polarization P(α) does not change significantly. This is correct for very narrow ranges of phase angles. We estimate such a phase-angle interval to be from 1◦ to 1.6◦ . It is impossible to choose a narrower interval because of the lack of observation data, especially for small phase angles. On the other hand, even for a narrow interval as small as α = 2◦ , the longitude dependence of polarization is masked by the influence of the phase dependence, which must be taken into account. Figure 10 shows the variations of the measured degree of polarization in the V filter as a function of the planetocentric longitude of the satellite for selected intervals of the solar phase angles; the approximating function described by Eq. (5) is also displayed. For large phase angles, the longitude dependence of polarization is well pronounced and has large amplitude, which decreases with decreasing phase angle. At small phase angles, the longitude dependence is less noticeable and poorly determined because of the small number of observations. Variations of the parameters P1, P2, and L o with the satellite phase angle (Fig. 9) are very close to those derived from the model calculations. This fact testifies in favor of the suggested method of separation of the phase and longitude curves of polarization. UNCERTAINTY IN MODELING
There are several major sources of uncertainty in the extraction of the phase-angle and longitude dependencies of polarization from the Callisto observations available. First, the phaseangle range, which is accessible to observations from the Earth, is too small—about 12◦ . Consequently, a satisfactory fit of the
POLARIZATION OF CALLISTO
153
FIG. 10. The longitude dependence of polarization for the narrow intervals of the phase angles (α = 1.6◦ ), in which P(α) is considered to be constant. The solid line represents the best fit of the observed polarization; the dashed line is the fit of all data points corrected to the phase angle in the center of the interval selected. The small numbers in square brackets give the range of the phase angle over which the data for the particular curve was selected. The data obtained in the 1998 and 2000 oppositions are marked by filled and open diamonds, respectively. The other denotations are the same as described in the legends to Figs. 2 and 3.
phase-angle curve of polarization cannot be obtained. Although the minima of polarization Pmin and phase angle αmin for the leading hemisphere are probably not much in error, the inversion angle αinv may be overstated very much. That is why the values for αinv are given in parantheses in Table III. Besides, the data are not uniformly distributed both over the accessible
phase angles and over the whole longitude range. Therefore, we could not derive a reliable value of the inversion angle for the trailing hemisphere in the U filter because of a small number of measurements. However, as it seems to us, the primary uncertainty in determining the phase-angle and longitude curves of polarization
154
VERA K. ROSENBUSH
TABLE III Parameters of the Phase-Angle Dependence of Polarization for Callisto in the UBVR Filters Filter
Pmin (%)
αmin (◦ )
αinv (◦ )
U (360 B (460 V (550 R (690
nm) nm) nm) nm)
Leading hemisphere −1.02 9.0 ± 0.4 −0.79 8.8 ± 0.5 −0.85 10.1 ± 0.3 −0.87 ≥11.2
(23.1 ± 3.0) (26.7 ± 1.3) (26.2 ± 1.2) (35.4 ± 2.7)
U (360 B (460 V (550 R (690
nm) nm) nm) nm)
Trailing hemisphere −0.66 3.7 ± 0.4 −0.62 4.5 ± 0.3 −0.63 4.9 ± 0.4 −0.69 5.8 ± 0.2
(22.7 ± 3.75) 14.0 ± 0.2 14.1 ± 0.1 14.6 ± 0.1
is due to the fact that there is no way to fit the rotational polarization curves of Callisto very accurately. To do this, it is necessary to know very accurately the solar phase curve. It is clear that we are still short of polarimetric observational data of a sufficient amount and quality. As a result, the model function (Eq. (5)) describes the dependence of polarization on longitude only approximately at small phase angles. In the future it will be useful to measure the rotational polarization curve while the solar phase angle remains relatively fixed. RESULTS
Using the model relations (Table II), the parameters of the phase-angle dependence of polarization were determined for the
FIG. 11. Spectral dependence of the polarization minimum Pmin normalized to the V filter for the Galilean satellites. For Callisto, data are given for the leading (filled circles) and trailing (open circles) hemispheres.
leading and trailing hemispheres of Callisto in the UBVR filters. Pmin , αmin , and αinv are given in Table III. Evidently, the leading and trailing hemispheres of Callisto differ not only by the values of Pmin but also by their dependence on the wavelength. The variations of negative polarization minimum, normalized to the V filter, with wavelength are shown in Fig. 11. One can see that for both hemispheres of Callisto, |Pmin | varies insignificantly in the BVR filters. However, the absolute value of polarization for the leading hemisphere increases more sharply in the U filter than that for the trailing one. Analyzing the function Pmin (λ), it is necessary to take into account that the values Pmin are less reliable in the U filter than those in the other filters because of a small number of observations, which also require further refinement. Rosenbush et al. (1997) showed that there are some systematic differences in the depth of the negative branch of polarization for Io, Europa, Ganymede, and Callisto. For comparison, the curves Pmin (λ) for Io, Europa, and Ganymede are also illustrated in Fig. 11. In comparison with Callisto, the spectral distribution of polarization minimum for both hemispheres of Io, Europa, and Ganymede are much less pronounced, and Pmin (λ) can be averaged over the disks of these satellites. For Io and Europa, the negative branch of polarization is significantly deeper in the U filter than in other filters. The values of polarization minimum Pmin are about −0.6% for Io and −0.4% for Europa. There is a noticeable increase of polarization in the R filter for these satellites. Ganymede and Callisto show a low value of Pmin even in the U filter. From Fig. 7 and Table III one can see that the leading hemisphere of Callisto is characterized in all the spectral bands by values of the phase angle of polarization minimum αmin being considerably higher than those for the trailing hemisphere (approximately by a factor of 2 in the BVR bands). Besides, there is the systematic wavelength dependence of αmin for the leading as well as for the trailing hemisphere. It is seen from Fig. 7 that the point of inversion lies at phase angles α > 12◦ for both hemispheres of Callisto, and its position can be found only by extrapolation. Using the expressions listed in Table II we have obtained the inversion angle for the trailing hemisphere in the BVR filters, which is close enough to the phase angle, at which observations of the satellite can be performed from Earth (Table III). Therefore, these values of αinv are quite reliable. However, we could not derive a reliable value of the inversion angle in the U filter because of a small number of measurements. For the leading hemisphere, the determination of the inversion angle becomes even less reliable due to a narrow phase-angle range accessible for ground-based observations. The orbital variations of polarization reflect the distribution of materials with different physical properties over the satellite surface. Up to the present, the dependence of polarization on the longitude has been derived only for large phase angles α = 11◦ (Gradie and Zellner 1973, Dollfus 1975, Mandeville et al. 1980) and α ≈ 7◦ –10◦ (Botvinova and Kucherov 1980) in a single spectral band (see Table IV). The scarcity of the
155
POLARIZATION OF CALLISTO
TABLE IV Parameters of the Longitude Dependence of Polarization for Callisto in the UBVR Filters P(L min ) (%)
P (%)
−0.58 −0.57 −0.58 −0.67
0.44 0.21 0.22 0.15
λ ≈ 550
α = 10◦ –11.3◦ (Gradie and Zellner 1973) 90–100 ∼−0.98 270 ∼−0.25
0.73
λ ≈ 500
130–140
α = 11◦ (Dollfus 1975) −0.9 260–270
0.8
λ ≈ 550
α = 7◦ –10◦ (Botvinova and Kucherov 1980) 90–100 −1.05 270 −0.25
Filter (nm)
L max (◦ )
P(L max ) (%)
U (360) B (460) V (550) R (690)
108 89 98 105
α = 6◦ (present work) −1.02 287 −0.78 270 −0.80 278 −0.82 287
V (550)
98
L min (◦ )
α = 11◦ (present work) −0.93 276
much of the surface. These features are interpreted to be the result of erosion of the surface material caused by the sublimation of volatile compounds (Klemaszewski et al. 2001). Therefore, interpretation and discussion of new results from the groundbased and spacecraft data now available should be the next step in the study of Callisto. The features of the solar and orbital variations of brightness and polarization of the satellite together with the differences in color, albedo, micro- and macroscopic structure, and composition over the surface help us to retrieve information about the nature of Callisto’s surface materials and their evolution. ACKNOWLEDGMENTS
−0.1
−0.25
0.8
Special thanks to N. Kiselev for discussion and useful advice to this paper and to E. Petrova, who critically read the manuscript and suggested many improvements. I also thank K. Muinonen and A. Dollfus for their valuable comments and suggestions as reviewers.
0.68
REFERENCES
results is explained by a small number of measurements and by complexity of the separating of the solar and orbital variations of polarization. Especially, there is not enough near opposition data where the microstructure of the surface plays a dominant role in the light scattering process. The parameters of the longitude dependence of polarization of Callisto, which were received for the phase angle α = 6◦ , are given in Table IV. The orbital longitudes L min and L max , at which the corresponding minimum P(L min ) and maximum P(L max ) of polarization are observed, are given in this table. There are also the parameters for the phase angle α = 11◦ obtained by the author and other observers. The results agree very closely. It should be noted that both L min and L max seems to change with wavelength, and their spectral dependencies correlate in contrast to the degree of polarization, P(L min ) and P(L max ). Figure 10 shows that the amplitude of orbital variations of the polarization degree near opposition is not large, while it reaches ≈0.7–0.8% at α ≈ 10◦ –11◦ . Figure 8 and Table IV demonstrate that the variations of polarization degree with longitude of the satellite P (longitude effect) take place for all filters and their amplitude increases in the U filter. FUTURE WORK
High-resolution images of Callisto obtained from the Galileo spacecraft have shown that the surface of Callisto is strongly degraded and is covered with dark, ice-poor material which mantles
Botvinova, V. V., and V. A. Kucherov 1980. Multicolour polarimetry of Galilean satellites of Jupiter. Astrom. Astrofiz. 41, 59–63. Chigladze, R. A. 1989. Investigation of the Polarimetric Properties of the Galilean Satellites of Jupiter and Planet Uranus. Ph.D. Thesis, Abastumany Astrophysics. Observatory. Dollfus, A. 1971. Physical studies of asteroids by polarization of the light. In Physical Studies of Minor Planets (T. Gehrels, Ed.), NASA SP-267, pp. 100–116. Dollfus, A. 1975. Optical polarimetry of the Galilean satellites of Jupiter. Icarus 25, 416–431. Gradie, J., and B. Zellner 1973. A polarimetric survey of the Galilean satellites. Bull. Am. Astron. Soc. 5, 404–405. Klemaszewski, J. E., R. Greeley, and Galileo SSI Team 2001. Cratering and erosion on Callisto: New insights. In Jupiter: Planet, Satellites and Magnetosphere, June 25–30, 2001. Abstracts. Lumme, K., and K. Muinonen 1993. A two-parameter system for linear polarization of some Solar System objects. In Asteroids, Comets, Meteors (A. W. Harris, E. Bowell, Eds.), LPI Contribution 810, pp. 194–199. Lunar and Planetary Institute, Houston. Mandeville, J.-C., J. G. Geake, and A. Dollfus 1980. Reflectance polarimetry of Callisto and the evolution of the Galilean satellites. Icarus 41, 343–355. Rosenbush, V. K., V. V. Avramchuk, A. E. Rosenbush, and M. I. Mishchenko 1997. Polarization properties of the Galilean satellites of Jupiter: Observations and preliminary analysis. Astrophys. J. 487, 402–414. Rosenbush, V. K., N. N. Kiselev, K. Jockers, V. V. Korokhin, N. M. Shakhovskoj, and Yu. S. Efimov 2000. Optical polarimetry of the Galilean satellites, Iapetus, and 64 Angelina near opposition. In Kinematics and Physics of Celestial Bodies, Suppl. Ser. 3, pp. 227–230. Veverka, J. 1971. Polarization measurements of the Galilean satellites of Jupiter. Icarus 14, 355–359.