New Polarimetric Data for the Galilean Satellites: Io and Ganymede Observations and Modeling

New high-precision disk-integrated measurements of the polarization of Io and Ganymede in the UBVRI bands are presented. The observations were obtained using polarimeters mounted on the Crimean Astrophysical Observatory and the Peak Terskol Observatory in 2019–2023. For Io, the negative polarization branch (NPB) reaches a minimum of P min ≈ −0.25 ± 0.02% in the V band at a phase angle of α min = 2.°1 ± 0.°5. The inversion angle is α inv = 26° ± 6° in the V and R bands. The NPB for Ganymede is an asymmetric curve, with P min = −0.34 ± 0.01% at α min = 0.°52 ± 0.°06 and α inv = 8.°5 ± 0.°2 in the V band. Although Io and Europa have similar geometric albedos (0.63 and 0.67, respectively), their NPB shapes differ. The NPB of Ganymede (albedo of 0.43) is morphologically similar to that of Europa, although it is described by different parameter values (P min, α min, and α inv). This discrepancy is likely due to the compositions of their surfaces: Europa’s with H2O ice, Io’s with sulfuric/silicate composition, and Ganymede’s with H2O ice and silicates. Numerical computations using the radiative transfer coherent backscattering method demonstrated a match to the polarimetric observations and to the geometric albedos for Ganymede with the single-scattering albedo ≈ 0.943 and mean free path length kl = 2πl/λ eff ≈ 150, where λ eff is the wavelength. For Io’s regolith, the single-scattering albedo was found to be ≈ 0.979 and kl ≈ 40.


Introduction
In the paper by Kiselev et al. (2022), the authors specified the shape of the phase-angle dependence of polarization for Europa that allowed them to characterize the composition, size of particles, and surface structure (porosity) of the satellite regolith.In this work, we continue our study of the Galilean satellites of Jupiter, focusing on the polarimetry of Io and Ganymede.
Io is located very close to Jupiter.The influence of the powerful gravity of the giant planet, significantly exceeding its influence on other Galilean satellites, leads to volcanic activity that is the strongest in the solar system.As a result, Io's surface is covered by volcanic features and is constantly renewing due to the eruption of hundreds of volcanoes that throw out lava consisting of largely molten sulfur and its compounds, providing a variety of surface colors, dominated by silicates.The Voyager and Galileo observations have shown that due to the dynamic nature of the volcanism on Io, its surface could change over time on a scale of years and even months.The high geometric albedo, ρ v = 0.63, and low temperatures due to rocky material pointed to frost deposits on the surface of Io.
Ganymede is the largest moon of Jupiter and in the solar system.The surface of Ganymede is covered mostly with water ice and silicate rocks (Anderson et al. 1996), providing a geometric albedo of ρ v = 0.43.There are two clearly distinct types of terrain: ancient dark regions with a lot of craters and large bright areas of ridges and grooves.This may suggest that Ganymede's crust has been under tension from global tectonic processes due to Jupiter's giant magnetosphere and intrinsic magnetic field.Ultraviolet observations at NASA's Hubble Space Telescope have provided strong evidence for a vast saltwater ocean under the icy mantle that forms the outer layer of Ganymede (Saur et al. 2015).The discovery of the underground liquid ocean on Ganymede has led to considerable interest in the study of this satellite.
Remote measurements of the brightness and polarization of light, scattered by the surfaces of atmosphereless bodies in the solar system at phase angles smaller than about 20°, reveal two optical phenomena: strong photometric backscattering and negative linear polarization.These phenomena are highly sensitive to the micro-and macrophysical properties of the surface regolith.Extensive studies have focused on these effects in the context of asteroids, planetary moons, and laboratory samples.A notable finding is the significant polarization dependence on the phase angle: negative polarization increases as the phase angle (α) grows, with a minimum polarization (P min ) occurring at approximately α min of 10°-12°.Beyond this point, the negative polarization decreases until an Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
inversion point is reached at α inv of about 15°-25°.At this inversion point, the polarization changes sign and becomes positive.
At very small phase angles near opposition, 0°-3°, highly reflective objects, such as icy satellites, Saturn's rings, and E-type asteroids, exhibit a very sharp and narrow brightness peak, called the brightness opposition effect (Franklin & Cook 1965;Harris et al. 1989;Thompson & Lockwood 1992;Helfenstein et al. 1998), and an extremely narrow minimum of negative polarization, called the polarization opposition effect (POE; Lyot 1929;Johnson et al. 1980;Rosenbush et al. 1997).Both of these phenomena exhibit similar regularities that were attributed to their formation by the same mechanism-a coherent backscattering effect caused by the interference of the conjugate scattered electromagnetic waves propagating in inverse directions, yielding an enhanced scattered intensity and negative polarization near the direction of backscattering (e.g., Muinonen 1990;Mishchenko 1993;Shkuratov et al. 1994).The angular width of these peaks does not depend on particle shape, but strongly depends on particle size and porosity.Muinonen (2004) developed a full radiative transfer coherent backscattering (RT-CB) algorithm for discrete random media of absorbing and nonabsorbing spherical scatterers.Using the RT-CB algorithm, Muinonen et al. (2012) showed that the coherent backscattering mechanism was indeed responsible for the backscattering peak and negative polarization for the case of spherical discrete random media of spherical constituent particles.The RT-CB method was extended to dense discrete random media of scatterers by Muinonen et al. (2018) and Markkanen et al. (2018).For more details on opposition phenomena, the mechanisms of their formation, and the modeling of these physical phenomena, see Kiselev et al. (2022).
Until the 2000s, the available polarimetric data for the Galilean satellites of Jupiter were limited and even mutually contradictory (see Veverka 1971Veverka , 1977;;Gradie & Zellner 1973;Dollfus 1975;Botvinova & Kucherov 1980;Chigladze 1989).The typical accuracy of the polarization measurements of the satellites was no better than ±(0.05-0.1)%.As a result, these polarimetric data were very noisy (Veverka 1977) and had systematic errors Rosenbush (2002).Despite this, these data have provided useful information about the negative branch of the phase-angle dependence of polarization and its parameters P min , α min , α inv , and polarimetric slope h, equal to dP/dα at the inversion angle.According to the past observations, the negative polarization branch (NPB) for Io, Europa, and Ganymede had a nearly parabolic shape with different parameters P min , α min , and α inv , depending on the albedo and surface composition.For Io (Zellner & Gradie in Veverka 1977) and Ganymede (Dollfus 1975), orbital polarization variations were found.Analyzing these results, the authors concluded that Io's surface was covered by a powder of rather transparent crystal grains with a certain amount of absorbing material, whereas the shape of the polarization curve of Ganymede implied that its surface could be interpreted as water frost deposits on the surface rock.Established small inversion angles of about 10°for Io and Ganymede suggested low-opacity surface materials.Gradie & Zellner (1973) were the first to notice that the polarization of Ganymede did not vanish at zero phase angle, as might be expected, but had a distinctly nonzero value, possibly due to icy polar caps.Much later, polarization measurements of the bright Galilean satellites at phase angles smaller than 1°by Rosenbush et al. (1997) showed that there was a sharp peak of negative polarization centered at a very small phase angle, about 0.5°, the so-called POE.This effect was later confirmed by more precise observations (Rosenbush & Kiselev 2005;Kiselev et al. 2009;Zaitsev 2016).Based on all available polarization measurements collected in the Database of Planetary Satellite Polarimetry by Zaitsev et al. (2012), the whole NPB for Io, Europa, and Ganymede could be presented by the bimodal polarization curve (Rosenbush et al. 2015).This means that a narrow polarization minimum located near the backscattering direction is superimposed on a broad parabolic curve, with the second minimum extending to the inversion angle.Detailed reviews of previous polarimetric observations of the Galilean satellites are given in Veverka (1977), Rosenbush et al. (2015), and Kiselev et al. (2022).
Polarimetric observations of the Galilean satellites were also carried out from space probes.The Pioneer 10 and 11 Imaging Photopolarimeters and the Galileo Photopolarimeter Radiometer (PPR) measured the polarization of the satellites at phase angles of 80°and 130°, respectively (Martin et al. 2000;Travis et al. 2002).The observations aimed to determine the optical and physical properties of satellite surfaces using the dependence of the polarization on the phase angle.According to these measurements, the Galilean satellites' surfaces are very heterogeneous in albedo and terrain types.It was established that the phase curve of polarization for Ganymede was close to the lunar curve, which is characteristic of dark surfaces: the maximum polarization was about 4%-5% near the phase angle 90°in the 678 nm band (see Figure 7 in Rosenbush et al. 2015).For Io, a more complex polarization distribution was observed: positive polarization <1% at a phase angle of about 20°, negative polarization ∼1% near 45°, and a linear increase of positive polarization beyond 60°, with a maximum of 3% at 130°.For both satellites, there seems to be a weak dependence of polarization on wavelength.The PPR data varied in spatial resolution from disk-integrated observations, used to establish the phase-angle dependence of polarization, to 100 km and better resolution over the disk, with the aim of evaluating the variations in the polarization degree for different terrains of the surface at specific phase angles.Because of this, a direct comparison of the disk-integrated data from ground-based observations with space-based ones is not entirely correct.
In 2018, a polarimetric survey of the Galilean satellites with new polarimeters was started, with the main goal of learning more about polarization phase curves and their exact parameters in the UBVRI passbands, which are crucial for the characterization of the regolith on the surfaces of satellites and similar objects.For this, we used the identical two-channel photoelectric polarimeters mounted on the 2.6 m Shajn reflector of the Crimean Astrophysical Observatory and the 2 m telescope of the Peak Terskol Observatory.In the study by Kiselev et al. (2022), it was found that the NPB for Europa is a highly asymmetric curve with a sharp polarization minimum P min ≈ −0.3% at phase angle α min 0°.4,after which the polarization degree gradually increases to positive values, passing the inversion angle at α inv ≈ 6°-7°.
This paper presents the results of polarimetric observations of Io and Ganymede and their computer simulation, which provides a good opportunity to further understand these moons and the properties of their surfaces.The paper is structured as follows: in Section 2, we briefly describe our observations, equipment, and data reduction; in Section 3, we present the results and summarize the findings; and in Section 4, we report the results of the computer simulation.We summarize our work in Section 5.

New Observations, Instrumentation, and Data Reduction
We have acquired polarization measurements of the Galilean satellites with the UBVRI filters with the identical two-channel photoelectric polarimeters "POLSHAKH" (Shakhovskoy et al. 2022), mounted on the 2.6 m Shajn reflector of the Crimean Astrophysical Observatory and the 2 m Ritchey-Chrétien-Coudé telescope of the International Center for Astronomical, Medical and Ecological Research (Peak Terskol Observatory, North Caucasus) during the years 2019-2023.These polarimeters are based on the principle of synchronous detection, which makes the polarization measurements practically independent of the weather conditions and ensures their high accuracy.The errors in the polarization degree did not usually exceed 0.05% in the UB bands and 0.02% in the VRI bands.The range of phase angles at which the observations were obtained extends from 0°.1 to 11°.2.
To determine the instrumental polarization, we measured standard stars with strictly zero polarization (P < 0.01%) from Serkowsky (1974) for each season of observations.The values of the instrumental Stokes parameters turned out to be <0.05%.A small temporal variation of the instrumental polarization parameters at a level of ∼0.02% was found for different sets of observations.The instrumental polarization and its temporal variability were taken into account in the calculations of the Stokes parameters q and u for all observed objects.The standards with a high degree of polarization taken from Bailey & Hough (1982), Hsu & Breger (1982), Turnshek et al. (1990), Schmidt et al. (1992), and Wolff et al. (1996) were also observed in order to determine the correction for the zeropoints of the instrumental positional angle in each observation set, which was stable within the uncertainties of <2°for different sets of observations.
We also investigated the influence of the proximity of Jupiter, which makes a noticeable contribution to the measured sky background.Therefore, the specific features of measuring the sky background were that the background was measured before and after observing the satellite from both sides and at equal distances in the direction perpendicular to the "planetsatellite" direction, following Lockwood (1983).It was corrected by interpolating the background counts to the middle time of the object observation.The sky background correction allows one to remove light scattered by the planet.In addition, we carried out special measurements of the polarization of the satellite Io at different distances from Jupiter during one night and did not detect a change in the polarization of the satellite with distance from the planet.Moreover, in the work by Rosenbush (2006), it was shown that the contribution of the Jupiter radiation reflected from the surface of Io did not exceed P ≈ 0.003%, which is smaller than the measurement errors, and thus did not noticeably affect the polarization value.Additionally, Morozhenko (2001) carried out special observations of Io at the phase angle of 0°. 5 during 3 hr (25 sets) and did not obtain any evidence of the dependence of polarization on the planetary phase angle.Observations showed only random changes in the degree of polarization in the range of 0.30%-0.35%,so these observations were used to more accurately estimate the value of P = 0.320% ± 0.004%.This allows us to conclude that the contribution of scattered Jovian light to the observed polarizing properties of the satellite is smaller than the typical error bars of our observations and can be neglected.In our opinion, the difference between the data points at a similar phase angle for Io, as well as Ganymede, is the result of local heterogeneities of the surfaces of satellites, which are superimposed on global differences in the polarimetric properties of the leading and trailing hemispheres.
A detailed description of the instrumentation, the method of polarization measurements of satellites and sky background, and the calculation of polarization parameters can be found in the paper by Kiselev et al. (2022).The results of our current measurements of the polarization of Io and Ganymede in the UBVRI bands are summarized in the Appendix (Tables A1  and A2).

The Phase-angle Dependences of Polarization for Io and Ganymede
In Figures 1 and 2, we show the NPBs for Io and Ganymede, respectively, where the open circles represent observations for the leading hemisphere (0° L 180°) and the filled circles indicate data for the trailing one (180° L 360°).The data derived earlier with comparable accuracy by Rosenbush & Kiselev (2005) and Kiselev et al. (2009) are also shown.
Io. Figure 1 shows that the negative polarization of Io sharply decreases within the range of phase angles from 0°to approximately 2°, and then it nearly linearly increases up to ∼12°(the maximum phase angle of Jupiter as viewed from Earth), reaching the value of about −0.15%.Since the relative contributions of different mechanisms of light scattering at different ranges of phase angles can be different, it is practically impossible to describe the phase dependence of the NPB in the entire range of phase angles by a single expression.Additional limitations are the possible longitude dependence of polarization, which can vary with the phase angle, insufficient data coverage, and the inadequate accuracy of the measured degree of polarization.Therefore, to estimate the parameters P min , α min , and α inv , we approximated the NPB in the UBVRI bands by a curve consisting of two parts: within the range of phase angles from 0°to ∼2°, the data were fitted by the inverse cubic polynomial, while all data in the phaseangle range of about 2°-11°.8 were fitted by a linear polynomial.The limited number of measurements in the U band allows one to make only a very approximate linear fit within this range of phase angles.It is obvious that it was not possible to reach an inversion angle that lies far beyond 12°.Note that these formal values of α inv are obtained by averaging the data without taking into account the longitude dependence of the polarization, which causes the scatter of data points from ∼0.1% to ∼−0.3%.The obtained values of α inv contradict the previously found values α inv = 9°.7 ± 0°.1 (Rosenbush et al. 2015) and 10°.5 (Veverka 1977;see Figure 10.6).Fitting curves allowed us to estimate the NBP parameters P min , α min , and the slope h', which was determined from the linear fit within the range of phase angles from ∼2°to ∼12°.We have also roughly estimated extrapolated values of the inversion angles α inv .They are presented for each band in Table 1.Rough estimates of some parameters, found directly from the figure, are in brackets.Thus, it can be concluded that the minimum degree of polarization for Io varies within −0.25% to 0.31% and that the minimum takes places at phase angles within 1°.4-2°.4.
Ganymede.Figure 2 displays the phase-angle dependence of polarization for Ganymede in the UBVRI photometric bands that combine our observations with data sourced For comparison, the available data of similar accuracy derived by Rosenbush & Kiselev (2005) and Kiselev et al. (2009) are also shown by the red and green symbols, respectively.The open circles denote data related to the leading hemisphere of the satellite (longitude L < 180°), and the filled circles, respectively, to the trailing one (longitude L > 180°).The fit shown by the solid curve consists of two parts derived within the range of phase angles from 0°to ∼2°and in the range from ∼2°to 11°.8, using polynomials with different exponents.
from Rosenbush & Kiselev (2005) and Kiselev et al. (2009).This figure clearly indicates that the NPB for Ganymede does not look like a bimodal curve, as described above.It is a sharply asymmetric curve with one polarization minimum resembling the NPB for Europa.Near the opposition, in the range of phase angles of about 0°-2°, there is a large scatter of data in the U, B, and V bands, while in the R and I bands the minimum of polarization is very clearly outlined.Similar to Io, the parameters of the NPB for Ganymede, P min , α min , and α inv , were obtained by fitting data using an inverse cubic polynomial within the range of phase angles from 0°to ∼2°.The inversion angle α inv was determined from an approximation of the data by a quadratic polynomial within the range of phase angles of about 2°-11°.8.Despite the specified fit to the data points, we do not have enough coverage of the NPB by homogeneous data to determine the exact parameters of the phase curve of polarization in the spectral bands used.The obtained NPB parameters together with their errors are given in Table 2.

The Longitude Dependences of Polarization for Io and Ganymede
As it follows from Figures 1 and 2, there is no strong difference in polarization between the data for the leading and the trailing hemispheres, especially in the case of Io.Nevertheless, we tried to determine the longitude dependence of polarization for both satellites.Due to the limited number of observations, we do not analyze the longitudinal dependences of the polarization of satellites in the U and B spectral bands.Both figures demonstrate that the scatter of individual measurements for Io, and for Ganymede, noticeably exceeds the observational errors.However, the scatter in our data for both satellites is much smaller than the polarization variations found in the previous measurements by Veverka (1971), Zellner & Gradie (1975), andDollfus (1975), especially at phase angles larger than 8°(see the details in the review by Veverka 1977).This scatter can be explained by the fact that the observed polarization depends not only on the solar phase angle (the phase-angle dependence of polarization), but also on the planetocentric longitude L of the satellite (the longitude dependence of polarization or longitude effect).Separating variations in the polarization degree related to orbital longitude and variations associated with changes in the solar phase angle is quite difficult, given a limited number of observations (e.g., Rosenbush 2002).It has not yet been solved, because there are no observations for a wide interval of longitudes within the same range of phase angles.Therefore, averaged measurements for the leading and trailing hemispheres of a satellite are usually used for the investigation of the phase-angle dependence of polarization, despite a significant scatter of the NPB data points, which exceeds the measurement errors.According to our measurements (Figure 1), on average, the difference in the polarization of the leading (open circles) and trailing (filled circles) hemispheres of Io is very small at similar phase angles, if exists at all, although the degree of polarization seems to be somewhat higher for the leading side.On the contrary, as can be seen in Figure 2, the trailing hemisphere of Ganymede, on average, is more strongly polarized than the leading hemisphere.
Io.We tried to reveal the dependence of Io's polarization on longitude using our observations within the phase-angle range of about 4°-11°.8. Assuming that the polarization varies linearly with solar phase angle in this range, as Figure 1 displays, we applied a two-parameter least-squares fit to the data with the equation: P(L) = P(α, L)-(A + Bα), where P(L) is the dependence of the polarization on the orbital longitude at a specified phase angle, P(α, L) is the observed degree of polarization at phase angle α and longitude L, and A and B are the coefficients of the linear fit to the data within the range from 4°to 11°.8.In addition, we assumed that the polarization changes sinusoidally within orbital longitude and made a threeparameter least-squares fit to the data according to the expression: P(L) = P 1 + P 2 sin(L-L 0 ), where P 1 , P 2 , and L 0 are the sought-for parameters.To find the best fit, we used a standard way of choosing values of parameters such that the sum of squares of the data-point deviations from the theoretical curve was minimal (χ 2 minimization).Solving simultaneously for sinusoidal orbital polarization variations and the linear phase-angle dependence of polarization at phase angles 4°-11°.8, we obtained the P 1 , P 2 , and L 0 parameters of the longitude curve of polarization for Io.The longitude of maximum orbital variations is equal to L max = L 0 + 90°.The obtained parameters and their uncertainties are given in Table 3.The longitude dependence of polarization for Io obtained in this way in the VRI bands is shown in Figure 3.As  (1975), the variations of the polarization degree of Io due to changes in orbital longitude are (0.4-0.5)% at the phase angles α > 10°at the wavelength of 520 nm (close to the V band).These authors found that the NPB for Io is deepest near the orbital longitude of the satellite L = 160°and shallowest near L = 300°.Veverka (1977) proposed that the orbital variations of the polarization can be caused by the spotted surface of Io, i.e., the local inhomogeneities (composition and structure) over the moon's surface, superimposed on the global distinctions in polarimetric properties of the leading and trailing hemispheres.As it is now known, the volcanoes on Io could form such a spotty surface.About 400 volcanoes have been discovered on the surface of Io.The large difference between the Zellner & Gradie data (see Figure 5 in Veverka 1977) and ours most likely indicates variability in the surface properties of Io over time due to volcanic activity on its surface.Ganymede.Figure 2 shows that the scatter of polarimetric measurements for Ganymede, as well as for Io, noticeably exceeds the observational errors.Therefore, we applied the same method for determining the longitude dependence of polarization for Ganymede as for Io. Figure 4 presents the longitude dependence of the polarization of Ganymede, after taking into account the polynomial phase dependence of polarization in the range of phase angles ∼(2-11.8)°,for all    4 and Table 4 make it possible to conclude that for Ganymede, in contrast to Io, a more pronounced longitude dependence of polarization is observed, the amplitude of which is about 0.2%.The leading hemisphere of Ganymede has systematically larger (in absolute terms) polarization than the trailing side.The orbital variations in the polarization of Ganymede increase with wavelength, reaching a maximum amplitude of about 0.15% at L ≈ 100°in the I band.We compared our results with previous studies, such as Dollfus (1975), who also found the orbital polarization variations of Ganymede measured at a phase angle of 11°, with a maximum of about 0.2% near L = 20°and a minimum of 0.0% near L = 200°.The author suggested that the scatter of the individual measurements most likely reflected local variations in surface properties with orbital longitude, specifically between water frost deposits and surface rocks.

Modeling and Comparison with Observations
As presented in the paper Kiselev et al. (2022), as with other Galilean satellites, our understanding of the surfaces of Io and Ganymede is limited, due to the lack of high-precision polarization measurements and fundamental issues with lightscattering models concerning densely packed discrete random media of particles.
Electromagnetic wave scattering by high-albedo discrete media of small particles, both natural and artificial in origin, results in a distinctive coherent backscattering phenomenon, characterized by a narrow intensity peak centered in the backscattering direction and a corresponding polarization peak near opposition.This phenomenon arises due to the interference of reciprocal rays, which travel along the same optical path, but in opposite directions.As such, the coherent backscattering makes the opposition phenomena depend on the medium properties, specifically on the size, refractive index, shape, and packing density of the scatterers in the medium.
The radiation scattered by the discrete random medium can be represented as a diffuse component that can be described with radiative transfer theory complemented by coherent backscattering.For light scattering by particles beyond the Rayleigh regime, a computational approach to accounting for coherent backscattering in discrete random media was developed by Muinonen (2004).That was extended to parameterized, phenomenological scatterers by Muinonen & Videen (2012).For different input parameters of the phenomenological model, listed below, the model can produce a variety of shapes of the NPB, from polarization curves similar to those measured in the laboratory to those observed for Ganymede and Io.

Ganymede
Ganymede was modeled similarly to Europa in the paper by Kiselev et al. (2022).We employed the method developed by Muinonen & Videen (2012).This approach involves constraining the properties of particles phenomenologically through the single-scattering albedo (ω) and the maximum degree of linear polarization, specifically for the case of single scattering by the regolith surface (P max ).A single-scattering phase function is represented by a double Henyey-Greenstein function (see, e.g., Muinonen & Videen 2012) defined by the asymmetry parameters g, g 1 , and g 2 , which describe the scattering characteristics of the particles in the medium.
The RT-CB computations were performed using the Monte Carlo method.Ray tracing within the medium considers exponential extinction using the mean free path length of rays in the homogeneous medium, described by the mean free path length specified in the size parameter range kl, where k is the wavenumber 2π/λ and l is the extinction mean free path length in the medium.The forward path of interactions corresponds to radiative transfer, and the forward and reciprocal paths together, via interference, give the coherent backscattering contribution.All small grains, populating a scattering volume mimicking the entire Ganymede, have the same scattering characteristics.The numerical values of the parameters of our Ganymede model, ω, P max , g, g 1 , g 2 , and kl are given in Table 5.For the scattering volume, the parameters result in the same geometric albedo ρ v = 0.43 measured for Ganymede.The number of rays used in the Monte Carlo modeling was 10 7 and a computation for a single set of parameter values required 1320 hr of sequential computing time on a modern tabletop or laptop computer.
The best modeling results are shown in Figure 5, with a comparison to observational data in the V and R bands.There are model phase functions of polarization for three cases: kl = 100, 150, and 200.It should be noted that for larger kl, the inversion angle shifts toward smaller values.The observations show that, within the observational uncertainties, the inversion angle does not depend strongly on the wavelength in the visible range (see Table 2).The weak dependence on the wavelength can be explained by the self-similarity (fractal characteristics) of the Ganymede surface structure across the distance scales set by the present wavelengths, resulting in a change of extinction mean free path, so that kl remains unchanged as a function of wavelength.Figure 5 illustrates the comparison of the observational data in the V and R filters with the best modeling result to these data.Within the errors, the theoretical curve describes quite well the negative branch of polarization up to 4°, whereas after that it goes slightly higher than the positive branch of polarization.

Io
Io was modeled by single-scatterer polarization that was modified from the one used for Europa in the paper by Kiselev et al. (2022) and what was used for Ganymede.The numerical parameters for the single-scattering polarization are given in Table 6.For the scattering volume, the modified parameters result in the same geometric albedo ρ v = 0.63 being measured for Io.The number of rays used in the Monte Carlo modeling The best modeling results are shown in Figure 6 and compared to the observational data in the V and R bands.There are model phase functions of polarization for two cases: kl = 40 and kl = 50.It should be noted that, like in case of Ganymede, for larger kl the inversion angle shifts toward smaller values.The observations show that, within the observational uncertainties, the inversion angle does not depend strongly on the wavelength in the visible range (see Table 3).Within the errors, the theoretical curve describes quite well the negative branch of polarization in both bands V and R.
As depicted in Figure 7, all three moons were modeled using the same P 11 element of the scattering matrix.However, when it comes to modeling Europa and Ganymede, it is possible to model both with a single-scatterer element −P 12 /P 11 by only varying the single-scattering albedo.In contrast, modeling Io requires more extensive adjustments, including negative polarization and a lower maximum polarization (P max ).The differing single-scatterer polarization is likely to be due to differences in surface composition and not due to a different  particle size distribution, a conclusion that is justified as follows.Polarimetric observations show only slight or no dependence on wavelength (see Figures 1 and 2).The real part of the refractive index (Re (m)) remains relatively constant across the present wavelengths for water ice, silicates, and sulfuric compounds.High geometric albedos result from a small imaginary part (Im (m)), and as such it has almost no effect on the effective size of the particles.Additionally, narrow particle size distributions would lead to polarimetric dependence on wavelength, which is not observed.Consequently, light at different wavelengths (λ) encounters a similar distribution in the size parameter defined as x = 2πr/ λx = 2πr/λ, where r is the radius of the particle.The higher (lower) single-scatterer polarization for Europa and Ganymede (Io) is also in agreement with the lower Re (m) for water ice as compared to sulfuric/silicate materials on Io, although differences for particle size distributions can also play a role.As for Europa in Kiselev et al. (2022), the RT-CB reproduces the NPB of Ganymede without negatively polarizing single scatterers.We justify the merely positively polarizing single-scattering model by the less pronounced negative polarization of single scatterers with lower real parts of refractive indices.This is shown, for example, by the numerical Discrete Dipole Approximation results for roughened Gaussian random particles by Zubko et al. (2007), assuming both icy and silicate refractive indices.
We emphasize that the RT-CB approach can be valid for densely packed media of scatterers by taking into account the corrections introduced by Markkanen & Penttilä (2023) and by Väisänen et al. (2020).It turns out that the fundaments of the RT-CB method do not change due to the corrections.Thus, the corrections become crucial when the phenomenological scattering matrices are interpreted in terms of physical scatterers.This will be a topic of forthcoming research.
In conclusion, we can assume wide power-law-type particle size distributions with different refractive indices (m), particularly concerning the real part, for the Europa and Ganymede icy surfaces (Re (m) ∼1.3) and for Io's surface containing sulfuric compounds due to its volcanic activity (Re (m) > ∼ 1.4).
The present modeling results for Io resemble the scattering matrix measurements found in the Granada-Amsterdam Light Scattering Database (Muñoz et al. 2012).First, our retrieved single-scattering characteristics for Io in Figure 7 resemble the measurement results for small olivine particles by Muñoz et al, (2000) and Frattin et al. (2019), although the measurements indicate a higher positive polarization.Second, the RT-CB computations for Io (as well as the Io observations) can be compared to the measurements for a white porous 5 mm diameter ball composed of cotton fibers (Muñoz et al. 2020).It is intriguing how the overall polarization characteristics for the cotton ball resemble the Io space-based observations at large phase angles, also suggesting a large inversion angle and shallow NPB, as indicated by the present work

Conclusions
New high-precision disk-integrated measurements of the polarization of Jupiter's moons Io and Ganymede in the UBVRI bands within the range of phase angles from 0°.12 to 11°. 8 are presented.The observations were obtained using the identical two-channel photoelectric polarimeters "POLSHAKH," which were mounted on the 2.6 m Shajn reflector of the Crimean Astrophysical Observatory and the 2 m Ritchey-Chrétien-Coudé telescope at the Peak Terskol Observatory between 2019 and 2023.As a result, we have established the most accurate shapes of the NPBs for the satellites.
1.The negative polarization of Io, starting from zero phase angle, sharply decreases, reaching a minimum of P min ≈ −0.25 ± 0.02% at a phase angle of α min = 2°.1 ± 0°.5 in the V band.The negative polarization then increased linearly to −0.15% at a phase angle of ∼12°.
The extrapolated value of the inversion angle in the V and R bands is approximately α inv = 26°± 6°, which is the largest among the satellites.2. The NPB of Ganymede has the shape of an asymmetric curve with a polarization minimum of P min = −0.34± 0.01% at α min = 0°.52 ± 0°.06 and inversion angle α inv = 8°.5 ± 0°.2 in the V band.3. Small amplitude changes in orbital polarization are found for Io, 0.05%, and Ganymede, 0.2%, which most likely reflect local changes in surface properties with longitude.4.Although Io and Europa have very similar geometric albedos, 0.63 and 0.67, respectively, the shapes of their NPBs are significantly different.The polarization curve for Ganymede, which has an albedo of 0.43, shows morphological similarities to Europa's, although the curves are described by different values of P min , α min , and α inv .This discrepancy is likely due to the different compositions of their surface regolith, as Europa is covered by H 2 O ice, Io by sulfuric/silicate material, and Ganymede by H 2 O ice and silicates.5. Computer modeling using the RT-CB approach demonstrated that the best match to the polarimetric observations and to the geometric albedos is provided for the Ganymede regolith layer with a single-scattering albedo ∼0.943 and a mean free path length kl = 2πl/λ eff ≈ 150, representing the effective wavelength in the UBVRI spectral bands.For Io's regolith, the single-scattering albedo was found to be ∼0.979 and kl ≈ 40.It follows that individual particles in Ganymede's regolith are, on average, more absorbing than those in Io's regolith.
We have presented preliminary RT-CB modeling for the polarimetry of Io and Ganymede following the approach utilized earlier for Europa.The modeling results suggest that differences between Io's polarimetric phase curve and those of Europa and Ganymede result from variations in the refractive index, rather than differences in the particle size distributions in the regolith.Future work will involve a simultaneous modeling of photometry and polarimetry, together with a strictly physical interpretation of the single-scattering characteristics retrieved phenomenologically for the three satellites.

Figure 1 .
Figure1.The phase-angle dependence of polarization for Io in the UBVRI bands obtained from observations carried out during 2018-2023 (black symbols).The data obtained byRosenbush & Kiselev (2005;  red symbols) andKiselev et al. (2009;  green symbols) are also displayed.The open circles denote data related to the leading hemisphere of the satellite (longitude L < 180°), and the filled circles, respectively, to the trailing one (longitude L > 180°).The fit shown by the solid curve consists of two parts derived within the range of phase angles from 0°to ∼2°using the inverse polynomial of the third degree and within the range of approximately 2°-11°.8 using the linear polynomial.

Figure 2 .
Figure 2. The phase-angle dependence of polarization for Ganymede in the UBVRI bands obtained from observations carried out during 2019-2023 (black symbols).For comparison, the available data of similar accuracy derived byRosenbush & Kiselev (2005) andKiselev et al. (2009) are also shown by the red and green symbols, respectively.The open circles denote data related to the leading hemisphere of the satellite (longitude L < 180°), and the filled circles, respectively, to the trailing one (longitude L > 180°).The fit shown by the solid curve consists of two parts derived within the range of phase angles from 0°to ∼2°and in the range from ∼2°to 11°.8, using polynomials with different exponents.

Figure 3 .
Figure 3.The longitude dependence of polarization for Io in the VRI bands after taking into account the correction for the linear phase-angle dependence of the polarization in the range of phase angles of ∼4°-11°.8.The fits to the data, according to the expression P(L) = P 1 + P 2 sin(L-L o ), are shown by a solid curve.

Figure 4 .
Figure 4.The longitude dependence of polarization for Ganymede in the VRI bands after taking into account the correction for the phase-angle dependence of polarization in the range of phase angles from ∼2°to 11°.8.The fits to the data performed according to the expression P(L) = P 1 + P 2 sin(L-L o ) are shown by the solid curves.

Figure 5 .
Figure 5. Shifting the inversion point with the increase of kl, in comparison to observations obtained with the V (on the left) and R (on the right) filters.

Figure 6 .
Figure 6.Modeled degree of polarization for Io in the V (left) and R (right) bands.

Figure 7 .
Figure 7. Single-scatterer P 11 element modeled identically for all the moons (on the left).The single-scatterer −P 12 /P 11 element (on the right) depicts the one used for the modeling of Io (red line) and the one used for both Europa and Ganymede (blue line).

Table 1
The Parameters of the Phase-angle Dependence of Polarization for Io in the UBVRI Bands

Table 2
The Parameters of the Phase-angle Dependence of Polarization for Ganymede in the UBVRI Bands be seen in the table and figure, the data do not show significant variations in the polarization between the leading and trailing hemispheres at phase angles 4°-11°.8.The maximum amplitude of the longitude dependence (parameter P 2 * 2) is about 0.04%, which is approximately twice larger than the measurement error and is located near a longitude of about 147°.To eliminate errors in the longitude dependences of polarization for Io, arising from the approximation of polarization curves for each hemisphere in the wide range of phase angles from ∼4°to 11°.8, we determined the difference in the polarization for the leading and trailing hemispheres in a narrow range of phase angles ∼(10-11.8)°indifferent spectral bands.Despite the limited number of observations, the results confirm the small amplitude of the orbital variations in Io's polarization, <0.05%, which contradicts previous observations of Io.According to Zellner & Gradie can

Table 4
The Fitting Parameters of the Longitude Dependence of Polarization for Ganymede in the VRI Bands Ganymede in the VRI bands.The parameters of the approximation function are given in Table4.Both Figure

Table 5
Parameters for the Ganymede Model

Table A1
Results of Polarimetric Observations of Io in the UBVRI Filters