Photometric Analyses of Saturn's Small Moons: Aegaeon, Methone, and Pallene Are Dark; Helene and Calypso Are Bright

The simplest way to account for these shape-related brightness variations is to divide the observed A_eff by the plane-projected area of the object A_phys, which, for an ellipsoidal object, is given by the following formula (Vickers 1996):

$$\begin{eqnarray}\begin{array}{rcl}{A}_{\mathrm{phys}} & = & \pi \left[{b}^{2}{c}^{2}{\cos }^{2}{\lambda }_{O}{\cos }^{2}{\phi }_{O}\right.\\ & & +\ {\left.{a}^{2}{c}^{2}{\cos }^{2}{\lambda }_{O}{\sin }^{2}{\phi }_{O}+{a}^{2}{b}^{2}{\sin }^{2}{\lambda }_{O}\right]}^{1/2}\end{array}\end{eqnarray} \tag{ 4 }$$

where a, b, and c are the dimensions of the ellipsoid, while λ_O and ϕ_O are the sub-observer latitude and longitude for the object. The resulting ratio then yields the disk-averaged reflectance of the object:

$$\begin{eqnarray}&&\langle I/F\rangle =\displaystyle \frac{{A}_{\mathrm{eff}}}{{A}_{\mathrm{phys}}}.\end{eqnarray} \tag{ 5 }$$

Tables 8–28 include estimates of A_phys computed assuming that these bodies have the shape parameters given in Table 1, and Figures 4–6 show the resulting estimates of $\langle I/F\rangle$ as functions of phase and sub-observer longitude. While the fractional brightness residuals around the mean phase curve are somewhat smaller for $\langle I/F\rangle$ than they are for A_eff, the dispersion is still rather large. For Aegaeon, Prometheus, and Pandora there are still clear maxima at 90° and 270°, while for Methone, Pallene, Telesto, and Calypso the dispersion at larger phase angles has a similar magnitude for the two parameters.

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The major problem with using $\langle I/F\rangle$ is that this parameter only accounts for the viewing geometry, but not how the moon is illuminated by the Sun, which, for very elongated bodies, can strongly affect the observed brightness (Helfenstein & Veverka 1989). Muinonen & Lumme (2015) provide an analytical formula for elongated bodies assuming the surface follows a Lommel-Seeliger scattering law, which is appropriate for dark surfaces. However, the applicability of such a model for bright objects like many of Saturn's moons is less clear. We therefore use a more generic numerical method to translate whole-disk brightness data into information about the object's global average reflectance behavior, which we call the Generic Ellipsoid Photometric Model, or GEPM.

The GEPM predicts an ellipsoidal object's A_eff in any given image assuming its surface reflectance R follows a generic scattering law that is a function of the cosines of the incidence and emission angles cos i and cos e. The predicted effective area of an object whose surface obeys these scattering laws is given by the following integral:

$$\begin{eqnarray}&&{A}_{\mathrm{pred}}={\int }_{++}(\cos e){{Rr}}_{\mathrm{eff}}^{2}\cos \lambda d\lambda d\phi \end{eqnarray} \tag{ 6 }$$

where λ and ϕ are the latitude and longitude on the object, the ++ sign indicates that the integral is performed only over the part of the object that is both illuminated and visible (i.e., where cos i and cos e are both positive), the factor of cos e in this integral accounts for variations in the projected size of the area element, and r_eff is the effective radius of the object at a given latitude and longitude, which is given by the following expression:

$$\begin{eqnarray}\begin{array}{rcl}{r}_{\mathrm{eff}}^{2} & = & \left[{b}^{2}{c}^{2}{\cos }^{2}\lambda {\cos }^{2}\phi \right.\\ & & +\ {\left.{a}^{2}{c}^{2}{\cos }^{2}\lambda {\sin }^{2}\phi +{a}^{2}{b}^{2}{\sin }^{2}\lambda \right]}^{1/2}.\end{array}\end{eqnarray} \tag{ 7 }$$

The factors of cos i and cos e in the above integral are also functions of λ and ϕ that depend upon the viewing and illumination geometry, as well as the object's shape. To obtain these functions, we first use the sub-observer latitude λ_O and longitude ϕ_O to define a unit vector pointing from the center of the body toward the spacecraft:

$$\begin{eqnarray}&&\hat{{\boldsymbol{o}}}=\cos {\lambda }_{O}\cos {\phi }_{O}\hat{{\boldsymbol{x}}}+\cos {\lambda }_{O}\sin {\phi }_{O}\hat{{\boldsymbol{y}}}+\sin {\lambda }_{O}\hat{{\boldsymbol{z}}}\end{eqnarray} \tag{ 8 }$$

where $\hat{{\boldsymbol{x}}}$ points from the moon toward Saturn, $\hat{{\boldsymbol{y}}}$ points in the direction of orbital motion, and $\hat{{\boldsymbol{z}}}$ points along the object's rotation axis. Similarly, we use the sub-solar latitude λ_S and longitude ϕ_S to define a unit vector pointing from the center of the body toward the Sun

$$\begin{eqnarray}&&\hat{{\boldsymbol{s}}}=\cos {\lambda }_{S}\cos {\phi }_{S}\hat{{\boldsymbol{x}}}+\cos {\lambda }_{S}\sin {\phi }_{S}\hat{{\boldsymbol{y}}}+\sin {\lambda }_{S}\hat{{\boldsymbol{z}}}.\end{eqnarray} \tag{ 9 }$$

Finally, for each latitude λ and longitude ϕ on the surface, we compute the surface normal for the body, which depends on the shape parameters a, b, and c.

$$\begin{eqnarray}&&\hat{{\boldsymbol{n}}}=\displaystyle \frac{\tfrac{\cos \lambda \cos \phi }{a}\hat{{\boldsymbol{x}}}+\tfrac{\cos \lambda \sin \phi }{b}\hat{{\boldsymbol{y}}}+\tfrac{\sin \lambda }{c}\hat{{\boldsymbol{z}}}}{\sqrt{\tfrac{{\cos }^{2}\lambda {\cos }^{2}\phi }{{a}^{2}}+\tfrac{{\cos }^{2}\lambda {\sin }^{2}\phi }{{b}^{2}}+\tfrac{{\sin }^{2}\lambda }{{c}^{2}}}}.\end{eqnarray} \tag{ 10 }$$

The cosines of the incidence and emission angles are then given by the standard expressions $\cos \,i=\hat{{\boldsymbol{n}}}\cdot \hat{{\boldsymbol{s}}}$ and $\cos \,e=\hat{{\boldsymbol{n}}}\cdot \hat{{\boldsymbol{o}}}$.

The above expressions allow us to evaluate A_pred for any given scattering law R, including Lunar–Lambert functions or Akimov functions (McEwen 1991; Shkuratov et al. 2011; Schröder et al. 2014). However, for the sake of concreteness, we will here assume that R is given by the Minnaert function (Minnaert 1941):

$$\begin{eqnarray}&&{R}_{M}=B{\left(\cos i\right)}^{k}{\left(\cos e\right)}^{1-k},\end{eqnarray} \tag{ 11 }$$

where the quantities k and B will be assumed to be constants for each image. Note that if we assume that k = 1, the above expression reduces to the Lambert photometric function:

$$\begin{eqnarray}&&{R}_{L}=B\,\cos \,i.\end{eqnarray} \tag{ 12 }$$

For a generic Minnaert function, Equation (6) becomes:

$$\begin{eqnarray}\begin{array}{rcl}{A}_{\mathrm{pred}} & = & B{\displaystyle \int }_{++}{\left(\cos i\right)}^{k}{\left(\cos e\right)}^{2-k}{r}_{\mathrm{eff}}^{2}\cos \lambda d\lambda d\phi \\ & = & \ {{Ba}}_{\mathrm{pred}}.\end{array}\end{eqnarray} \tag{ 13 }$$

The factor of B moves outside the integral, and the remaining factor a_pred can be numerically evaluated for any specified viewing and illumination geometry, so long as we also assume values for the object's shape parameters a, b, and c, as well as the photometric parameter k. A Python function that evaluates a_pred given these parameters is provided in Appendix A. From this, we can estimate the "brightness coefficient" parameter B to be simply the ratio A_eff/a_pred. Note that B is not equal to the mean reflectance $\langle I/F\rangle$, even for spherical bodies, because it includes corrections for the varying incidence and emission angles across the disk.

Of course, the estimated value of B depends on the assumed values of a, b, c, and k, which are typically estimated from resolved images. While only a few images are needed to obtain useful estimates of a, b, and c, the parameter k can depend on the observed phase angle and location on the object, and so k is far harder to estimate reliably for the full range of observation geometries. Previous studies of resolved Voyager images of the mid-sized moons found that Mimas, Enceladus, Tethys, Dione, and Rhea had k = 0.65, 0.78, 0.63, 0.53–0.56, and 0.52–0.54, respectively, at phase angles between 5° and 17° (Buratti 1984), while a study of resolved Cassini images of Methone found that k = 0.887–0.003α (α being the solar phase angle) based on images obtained at phase angles between 45° and 65° (Thomas et al. 2013). Thus, this parameter likely varies among these moons and with observation geometry and terrain for each moon.

Figure 7 shows the fractional differences in the predicted values of a_pred between a model where k = 0.5 and a model where k = 1 (that is, a Lambertian surface). Each data point corresponds to a different combination of sub-observer and sub-solar locations scattered over the entire object. For the spherical object, there is a trend where the k = 0.5 model becomes 15% brighter than the Lambertian model at high phase angles. Very prolate and oblate objects show a similar overall trend but with considerably more scatter depending on the exact viewing and illumination geometry. This suggests that one could constrain k by minimizing the rms scatter in the estimated brightness coefficients from unresolved images of elongated bodies at each phase angle. In practice, the rms scatter in the brightness coefficients for the various moons are very weak functions of k, most likely because real surface albedo variations and/or instrumental noise dominate the dispersion in the brightness coefficients. We therefore expect that a thorough analysis of resolved images would be needed to constrain k reliably for each of the moons, and such an analysis is well beyond the scope of this paper.

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Fortunately, it turns out that the results presented below are insensitive to the exact value of k. We bracket the range of possible k-values by considering cases where k = 0.5, 0.75, and 1.0 when estimating the brightnesses of the various moons. Estimates of a_pred derived from all three cases are provided in Tables 8–28. However, since the brightness differences among these different scenarios are subtle, we will normally only plot brightness coefficients computed assuming the surface follows a Lambertian scattering law (i.e., k = 1).

The utility of this model can be most clearly demonstrated by taking a closer look at the data for Aegaeon. Aegaeon is not only the most elongated moon, it was also observed in multiple ARCORBIT sequences where the spacecraft repeatedly imaged the moon as it moved around a significant fraction of its orbit, yielding true rotational light curves. The images from eight of these sequences are listed in Table 2, while the values of A_eff are shown in Figure 8, plotted as a function of the sub-observer longitude. All show clear brightness variations that can be attributed to the changing viewing geometry of the moon, with the lowest signals seen when the sub-observer longitude is near 0° or 180°, and the highest signals seen when the sub-observer longitude is close to 90° or 270°. However, it is also clear that the maxima and minima are not precisely aligned with these four longitudes (this is most clearly seen in the Rev 201, 210, and 236 data). This result clearly demonstrates that the projected area is not the only thing affecting the moon's projected brightness. Indeed, the green curves show the variations in the projected area, scaled to match the mean signal in the data. These variations are both too subtle to match the observations, and the observed and predicted peaks and troughs do not line up. By contrast, the GEPM predictions are a much better fit to the data in all cases. This demonstrates that accounting for the lighting geometry not only causes the predicted locations of peaks and troughs to shift into alignment with the data, but it also increases the predicted fractional brightness variations.

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More generally, Figures 9–11 show the estimated brightness coefficients for the small moons derived assuming that they have the nominal shapes given in Table 1 and surfaces that follow a Lambertian scattering law (assuming the surfaces follow Minnaert scattering laws with k = 0.5 and 0.75 yield nearly identical results). Compared with Figures 4–6, the scatter around the mean trends with phase angle are much tighter for almost all of the moons. The exceptions to this trend are Helene, Polydeuces, Janus, and Epimetheus, for which the dispersions in B and $\langle I/F\rangle$ are comparable. These exceptions are likely because these four moons are the most spherical in shape, so the corrections included in the GEPM are less important. There are some outlying data points for Aegaeon, Atlas, and Prometheus, but these can be attributed to the low signals from Aegaeon and mis-estimated background levels for Atlas and Prometheus.

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Besides the reduction in dispersion, the fractional brightness residuals no longer show double-peaked trends with co-rotating longitude. Instead, the most obvious remaining longitudinal trends are that the leading sides of Pallene, Epimetheus, and Janus are 10%–20% darker than their trailing sides. Such leading-trailing asymmetries are reminiscent of the longitudinal brightness variations seen in ground-based, voyager, and Cassini images of the mid-sized icy moons (Noland et al. 1974; Franz & Millis 1975; Buratti & Veverka 1984; Verbiscer & Veverka 1989, 1992, 1994; Buratti et al. 1990, 1998; Pitman et al. 2010; Schenk et al. 2011). Figure 12 shows the brightness coefficients for these larger moons derived assuming the same Lambertian model as was used for the smaller moons.⁷ Consistent with prior work (Buratti et al. 1998; Schenk et al. 2011; Hendrix et al. 2018; Verbiscer et al. 2018), we find that Tethys, Dione, and Rhea have brighter leading sides, while Mimas has a brighter trailing side. This overall trend is thought to arise because the particles in the E ring preferentially strike the leading sides of moons orbiting exterior to Enceladus and the trailing sides of moons orbiting interior to Enceladus (Hamilton & Burns 1994). The longitudinal brightness asymmetries observed on Pallene, Janus, and Epimetheus are consistent with this pattern, but it is worth noting that Aegaeon, Methone, and the trojan moons do not appear to have such asymmetries. The implications of these findings will be discussed further in Section 4 below.

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Finally, we may consider what happens if we relax the assumption that the shapes of these moons equal the best-fit values from resolved images. Figures 13 and 14 show the rms residuals as functions of the aspect ratios b/a and c/b for Aegaeon, Anthe, Methone, Pallene, and the trojan moons, along with the aspect ratios derived from the resolved images. For the smaller moons shown in Figure 13, the two methods for determining the shape agree fairly well. The best-fit solution for Aegaeon falls where b ≃ c and b ≃ 0.3a, consistent with the constraints from the resolved images. For Methone, the two methods yield best-fit solutions with the same value of c/b ≃ 0.95 and b/a ≃ 0.65, although the photometry favors a slightly higher value for b/a. Similarly, for Pallene, we find both methods give c/b ≃ 0.9, but the resolved images favor b/a ≃ 0.7, while the photometry prefers 0.8. This discrepancy is likely related to the variations in the surface brightness with longitude mentioned above. More sophisticated photometric models could potentially resolve this difference, but even this level of consistency gives us some confidence in this method. Also, the photometric data for Anthe clearly favor a shape with b/a ≃ 0.7 and c/b ≃ 0.95, which implies that Anthe has a shape similar to Methone.

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The shapes of the trojan moons derived from the photometry and resolved images are somewhat more discrepant. For Polydeuces, both methods yield b/a ≃ 0.8 and c/b ≃ 1 with rather large uncertainties. For Calypso, both methods agree that b/a ≃ 0.6, but the photometry favors c/b ≃ 0.9 while the resolved images prefer c/b ≃ 0.7. For Telesto, both methods agree that c/b ≃ 0.8, but the photometry favors b/a ≃ 0.8 while the resolved images prefer b/a ≃ 0.7. The biggest discrepancies are found with Helene, where the photometry prefers a much more spherical shape than was found with the resolved images. These differences most likely arise because the shapes of these objects are not simple ellipsoids like the smaller moons appear to be. Again, a more sophisticated photometric model that could accommodate more complex shapes would likely reduce these discrepancies. Even so, the relatively simple GEPM clearly reduces the dispersion in the brightness estimates for Telesto, Polydeuces, and especially Calypso, and so it provides a useful basis for comparing the brightnesses of all these satellites.

The calculations described in Section 3 demonstrate that our model is a sensible way to quantify the surface brightness of elongated moons. However, the absolute values of B derived by this method are not directly comparable to traditional parameters like geometric or Bond albedos previously reported in the literature. In principle, the derived values of B as functions of phase angle could be extrapolated to zero, and then we could translate that value of B to a geometric albedo for an equivalent spherical object. However, in practice, performing such an extrapolation would be problematic because the small moons were rarely observed by Cassini at phase angles below 10°, so we have no information about the magnitude or shape of the opposition surges for these bodies. Also, since these objects are not spherical, the amount of light scattered will depend on the orientation of the body, which complicates the classical definition of albedo. On the other hand, it is relatively straightforward to estimate B for Saturn's other moons, and so we have chosen that approach for this analysis.

In order to compare the reflectances of these objects, we only consider the data obtained between phase angles of 20° and 40°, and fit the brightness coefficients in this phase range to a linear function of phase angle α:

$$\begin{eqnarray}&&B(\alpha )={B}_{0}+(\alpha -30^\circ ){C}_{B}.\end{eqnarray} \tag{ 14 }$$

Note that B₀ is the brightness coefficient at 30° phase, which is in the middle of the range of observed phase angles and so does not depend on any questionable extrapolations. For a spherical object of radius r observed at 0° and 30° phase, a_pred = 2.12r² and 1.86r², respectively, for a Lambertian scattering law (for a Minnaert scattering law with k = 0.5, these numbers become a_pred = 2.12r² and 1.91r², respectively). Hence, the average reflectance the object would have at 30° if it were a sphere is $\langle I/F{\rangle }_{\mathrm{sphere}}=0.59{B}_{0}$ for a Lambertian scattering law (or $\langle I/F{\rangle }_{\mathrm{sphere}}=0.61{B}_{0}$ for a Minnaert k = 0.5 scattering law). The corresponding geometric albedo of the object assuming that the linear phase function could be extrapolated to an appropriately small phase angle α₀ would be⁸

$$\begin{eqnarray}&&{{ \mathcal A }}_{\mathrm{sphere}}=0.67[{B}_{0}+({\alpha }_{0}-30^\circ ){C}_{B}].\end{eqnarray} \tag{ 15 }$$

For the mid-sized moons Mimas, Enceladus, Tethys, Dione, and Rhea, this expression yields ${{ \mathcal A }}_{\mathrm{sphere}}=0.60,0.92,0.75,057$ and 0.63, respectively for α₀ = 10°, which are 0.97, 0.92, 0.90, 0.87, and 1.00 times the central values reported by Thomas et al. (2018), so at this phase angle, the calculations are reasonably consistent. However, if we extrapolate to zero degrees phase, we obtain ${{ \mathcal A }}_{\mathrm{sphere}}=0.65,0.99,0.82,0.63$ and 0.55, which are 0.68, 0.72, 0.67, 0.63, and 0.74 times the geometric albedos measured by Verbiscer et al. (2007). This discrepancy most likely reflects the fact that this formula does not account for the opposition surges associated with these moons.

For the sake of simplicity, we will just use the parameters B₀ in our analyses below. Tables 3–5 tabulate B₀ and C_B for all of Saturn's moons examined here assuming either a Lambertian scattering law or a Minnaert scattering law with k = 0.75 or k = 0.50. These tables also includes statistical errors based on the scatter of the points around the mean trend, and a systematic error that is based on the uncertainty in the mean size of the object. Additional systematic uncertainties associated with the size of the pixel response function are on the order of 5% and are not included here because these systematic uncertainties are common to all of the brightness estimates and so do not affect the moons' relative brightness. Figure 15 compares the values of B₀ derived assuming the different scattering laws for each of the moons. This plot shows that models assuming lower values of the Minnaert k parameter yield systematically lower estimates of B₀, but that these differences are rather subtle and have little effect on the relative brightnesses of the various moons. Hence, for the sake of simplicity, we will only consider the estimates of B₀ computed assuming a Lambertian scattering law from here on. Figure 16 plots these central brightness values as a function of distance from the center of Saturn, along with estimates of the relative E-ring particle density and fluxes derived from images, as well as the relative energetic proton and electron fluxes (see below).

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Figure 16 shows that the brightness coefficients for the largest moons (Janus, Epimetheus, Mimas, Tethys, Dione, and Rhea) follow the same basic trends as previously reported for these moon's geometric albedos (Verbiscer et al. 2007, 2018). The brightness of these larger moons falls off with distance from Enceladus' orbit, which strongly suggests that the E-ring plays an important role in determining these moons' surface brightness. Furthermore, the leading-trailing asymmetries in these moons' surface brightness are also consistent with how E-ring particles are expected to strike the moons (Buratti et al. 1998; Schenk et al. 2011). Most of the visible E ring particles have orbital semimajor axes close to Enceladus and a range of eccentricities (Horanyi et al. 1992; Hamilton & Burns 1994). The E-ring particles outside Enceladus' orbit are therefore closer to the apocenter of their eccentric orbits and so are moving slower around the planet than the moons are. Hence, Tethys, Dione, and Rhea mostly overtake the E-ring particles, and the corresponding flux onto these moons is larger on their leading sides. On the other hand, the E-ring particles inside Enceladus' orbit are closer to their orbital pericenters, and so tend to move faster around the planet than the moons. Hence, the E-ring particles tend to preferentially strike the trailing sides of Janus, Epimetheus, and Mimas. For all of these moons, the side that is preferentially struck by E-ring particles is brighter, as one would expect if the E-ring flux was responsible for brightening the moons.

The connection between the E-ring and the moons' brightness can be made more quantitative by estimating the flux of E-ring particles onto each of the moons. The E ring consists of particles with a wide range of orbital elements, so detailed numerical simulations will likely be needed to accurately compute the fluxes onto each moon. Such simulations are beyond the scope of this report, and so we will here simply approximate the particle flux based on simplified analytical models of the E ring motivated by the available remote-sensing and in situ observations.

Prior analyses of E-ring images obtained by the Cassini spacecraft provided maps of the local brightness density within this ring as functions of radius and height above Saturn's equatorial plane. These brightness densities are proportional to the local number density of particles times the size-dependent scattering efficiency for those particles, so these maps provide relatively direct estimates of the particle density in the vicinity of the moons. For this particular study, we will focus on the E-ring density profile shown in Figure 16, which is derived from the E130MAP observation made on day 137 of 2006, and is described in detail in Hedman et al. (2012). This observation included a set of wide-angle camera images that provided an extensive and high signal-to-noise edge-on map of the E ring at phase angles around 130° (W1526532467, W1526536067, W1526539667, W1526543267, W1526546867, W1526550467, and W1526554067). These images were assembled into a single mosaic of the edge-on ring, and then onion-peeled to transform the observed integrated brightness map into a map of the local ring brightness density in a vertical cut through the ring (Hedman et al. 2012). The profile of the brightness density near Saturn's equatorial plane was extracted from this map as the average brightness in regions between 1000 and 2000 km from Saturn's equatorial plane after removing a background level based on the average brightness 20,000–30,000 km away from that plane. Note that the region used here deliberately excluded the region within 1000 km of the equatorial plane to minimize contamination from the G ring interior to 175,000 km. The vertical scale height of the E ring is sufficiently large that including data in the 1000–2000 km range should provide a reasonable estimate of the density in the plane containing all of these moons.

Consistent with previous studies, the E-ring brightness density is strongly peaked around Enceladus' orbit. However, this profile also differs from the profiles reported elsewhere in the literature. For example, this profile is much more strongly peaked than that the profile used by Verbiscer et al. (2007). This is because the Verbiscer et al. (2007) profile was of the E-ring's vertically integrated brightness, rather than the brightness density close to the equatorial plane. The latter quantity is more sharply peaked because the E-ring's vertical thickness increases with distance from Enceladus' orbit, and because the ring is warped so that its peak brightness shifts away from Saturn's equatorial plane far from Enceladus' orbit (Kempf et al. 2008; Hedman et al. 2012; Ye et al. 2016). Since all of the moons orbit close to the planet's equatorial plane, the profile shown in Figure 16 is a better representation of the relative dust density surrounding the moons. On the other hand, the profile shown in Figure 16 is also a bit broader than in situ measurements would predict. These data have generally been fit to models where the particle density falls off with distance from Enceladus' orbit like power laws with very large indices (Kempf et al. 2008; Ye et al. 2016). Assuming that the brightness density is proportional to the particle number density, these models tend to underpredict the signals seen exterior to 5 R_S and interior to 3 R_S. These discrepancies arise in part because the in situ instruments are only sensitive to grains with radii larger than 0.9 μm, while the images are sensitive to somewhat smaller particles that probably have a broader spatial distribution (Horányi et al. 2008; Hedman et al. 2012). The profile shown in Figure 16 therefore should provide a reasonable estimate of the relative densities of the particles larger than 0.5 μm across within this ring.

However, for the purposes of understanding how the E-ring affects the moons' surface properties, we are primarily interested in the E-ring particle flux into the moons, which depends not only on the particle density, but also on the particles' velocity distribution. Since a detailed investigation of the E-ring's orbital element distribution is beyond the scope of this paper, we will here consider a simple analytical model of the E-ring particles' orbital properties that can provide useful rough estimates of the particle flux into the various moons.

Prior analyses of E-ring images indicate that most of the visible particles in this ring have semimajor axes close to Enceladus' orbit, and that the large size of this ring is because the particles have a wide range of eccentricities and inclinations. Furthermore, the images indicate that mean eccentricities, inclinations, and semimajor axes were strongly correlated with each other (Hedman et al. 2012). We therefore posit that the E-ring density distribution seen in Figure 16 primarily reflects a distribution of eccentricities ${ \mathcal F }(e)$. We also allow the semimajor axes of the particles to vary, but for the sake of simplicity, we assume that the semimajor axes of these particles a is a strict function of e: a = [2400,000 + (e/0.1) × 5000] km, which is consistent with the imaging data (Hedman et al. 2012). Since we are only concerned with particles near Saturn's equatorial plane, we will not consider the inclination distributions here.

For a given eccentricity distribution ${ \mathcal F }(e)$, the particle density as a function of radius d(r) should be given by the following integral:

$$\begin{eqnarray}&&d(r)={\int }_{{e}_{\min }}^{1}\displaystyle \frac{2}{| {v}_{r}| T}{ \mathcal F }(e){de},\end{eqnarray} \tag{ 16 }$$

where v_r is the radial velocity of the particles and T is their orbital period, and the factor of two arises because particles with a finite eccentricity pass through each radius twice. The integral's lower limit is ${e}_{\min }=| 1-r/a|$ because only particles with eccentricities greater than this can reach the observed r. Similarly, the average flux of particles onto a body on a circular orbit at a given radius is given by the integral.

$$\begin{eqnarray}&&F(r)={\int }_{{e}_{\min }}^{1}\displaystyle \frac{1}{2| {v}_{r}| T}{ \mathcal F }(e)\sqrt{{v}_{r}^{2}+\delta {v}_{\lambda }^{2}}{de},\end{eqnarray} \tag{ 17 }$$

where v_r is the particle's radial velocity as defined above, and δv_λ is the particle's azimuthal velocity measured relative to the local circular orbital velocity. Note that a factor of one-fourth arises from averaging over the moon's surface. Since the radial motions of particles on any given orbit are symmetric inwards and outwards, in this model, there are no differences in the fluxes on the sub-Saturn and anti-Saturn sides of the moon. However, because the particles at a given radius can have different average azimuthal velocities, there can be asymmetries in the fluxes on the moon's leading and trailing sides. These asymmetries can be parameterized by the following integral:

$$\begin{eqnarray}&&\delta {F}_{\lambda }(r)={\int }_{{e}_{\min }}^{1}\displaystyle \frac{1}{2| {v}_{r}| T}{ \mathcal F }(e)\delta {v}_{\lambda }{de},\end{eqnarray} \tag{ 18 }$$

which is half the difference between the fluxes on the leading and trailing points.

In order to evaluate these integrals, recall the standard expressions for v_r and δv_λ in terms of orbital elements (Murray & Dermott 1999):

$$\begin{eqnarray}&&{v}_{r}=\displaystyle \frac{{na}}{\sqrt{1-{e}^{2}}}e\,\sin \,f\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&\delta {v}_{\lambda }=\displaystyle \frac{{na}}{\sqrt{1-{e}^{2}}}(1+e\,\cos \,f)-{na}\sqrt{a/r}\end{eqnarray} \tag{ 20 }$$

where n = 2π/T is the mean motion of the particle and f is the true anomaly of the particle's orbit. At a given radius r, we will be observing particles at different mean anomalies given by the standard expression:

$$\begin{eqnarray}&&r=\displaystyle \frac{a(1-{e}^{2})}{1+e\,\cos \,f}\end{eqnarray} \tag{ 21 }$$

so we can re-express $| {v}_{r}|$ and δv_λ in terms of r

$$\begin{eqnarray}&&| {v}_{r}| =\displaystyle \frac{{{na}}^{2}}{r}\sqrt{{e}^{2}-{\left(r/a-1\right)}^{2}},\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&\delta {v}_{\lambda }=\displaystyle \frac{{{na}}^{2}}{r}\left[\sqrt{1-{e}^{2}}-\sqrt{r/a}\right].\end{eqnarray} \tag{ 23 }$$

Inserting these expressions into the above integrals, we find the density, flux, and flux asymmetries can all be expressed as the following integrals over the eccentricity distribution (remember that a is implicitly a function of e):

$$\begin{eqnarray}&&d(r)={\int }_{{e}_{\min }}^{1}\displaystyle \frac{r{ \mathcal F }(e)}{\pi {a}^{2}\sqrt{{e}^{2}-{\left(r/a-1\right)}^{2}}}{de},\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&F(r)={\int }_{{e}_{\min }}^{1}\displaystyle \frac{n{ \mathcal F }(e)}{4\pi }\sqrt{1+\displaystyle \frac{{\left(\sqrt{1-{e}^{2}}-\sqrt{r/a}\right)}^{2}}{{e}^{2}-{\left(r/a-1\right)}^{2}}}{de},\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&{F}_{\lambda }(r)={\int }_{{e}_{\min }}^{1}\displaystyle \frac{n{ \mathcal F }(e)(\sqrt{1-{e}^{2}}-\sqrt{r/a})}{4\pi \sqrt{{e}^{2}-{\left(r/a-1\right)}^{2}}}{de}.\end{eqnarray} \tag{ 26 }$$

It turns out that the observed density distribution shown in Figure 16 can be matched reasonably well by assuming the eccentricity is a Lorentzian times a regularization term to avoid singularities at e = 0:

$$\begin{eqnarray}&&{ \mathcal F }(e)=\displaystyle \frac{{{ \mathcal F }}_{0}}{1+{e}^{2}/{e}_{0}^{2}}[1-\exp (-e/{e}_{c})]\end{eqnarray} \tag{ 27 }$$

with constants ${{ \mathcal F }}_{0}$, e₀, and e_c. The specific model density distribution shown in Figure 16 has e₀ = 0.17 and e_c = 0.01. Note that e_c basically only affects the density levels close to Enceladus' orbit, while e₀ determines how quickly the signal falls off away from the E-ring core.

Using this same ansatz for ${ \mathcal F }(e)$, we obtain estimates for how both the flux and flux asymmetry vary with radius, which are shown in the third panel of Figure 16. First, note that the flux is much less sharply peaked than the density. This occurs because the particles are moving faster relative to the local circular velocity at larger distances from Enceladus' orbit thanks to their higher eccentricities. Also note that azimuthal component of the flux is larger exterior to Enceladus' orbit than it is interior to that moon. This asymmetry arises because exterior to Enceladus, the particles are all moving slower than the local circular velocity, while interior to Enceladus, the particles are all moving faster than the local circular speed, but the azimuthal components of their velocity can still fall below the local circular velocity depending on their true anomaly.

Finally, we can translate these trends in the relative flux into rough estimates of the absolute flux of E-ring particles using recent in situ measurements of the particle number density in the E-ring core. In principle, the brightness density seen in images can be translated into estimates of the particle number density, but in practice, the relevant conversion factor depends on the size-dependent light scattering efficiency of the particles. By contrast, the peak E-ring particle density has been directly measured to be between 0.02 and 0.2 m⁻³ with various in situ measurements (Kempf et al. 2008; Ye et al. 2016). Now, as mentioned above, these densities are for the particles larger than the detection thresholds of these instruments, which is slightly larger than the sizes of the particles seen in remote-sensing images. Hence, the number density of visible particles could be somewhat larger than these measurements. Nevertheless, these are still the best estimates of the absolute particle density near the core of the E ring, and they should still provide a useful order-of-magnitude estimate of the visible particles. For the sake of concreteness, we will here assume a peak number density of 0.03 particles m⁻³, which is the value measured by Ye et al. (2016) for particles larger than 1 μm in radius.

Figure 17 and Table 6 give the brightness coefficients and estimated E-ring particle fluxes for all of the moons known to be embedded in the E ring. For the mid-sized moons and the co-orbitals, there is a very good correlation between the moon's brightness coefficients and the estimated fluxes. This correlation not only applies to the average values for each moon but also to their leading and trailing sides, where the estimated flux values are taken to be $F\pm \delta {F}_{\lambda }/\sqrt{2}$ (the factor of $\sqrt{2}$ accounts for the fact that the observations span a full quadrant of each body). The one mid-sized moon that falls noticeably off the main trend is Rhea. This could be related to other unusual aspects of Rhea's surface. Rhea's visible spectrum is redder than any of the other mid-sized moons (Thomas et al. 2018), and its near-infrared spectrum exhibits stronger ice bands than Dione (Filacchione et al. 2012; Scipioni et al. 2014). Furthermore, radio-wave data indicate that Rhea has a higher albedo at centimeter wavelengths than Dione (Black et al. 2007; Le Gall et al. 2019). Various explanations have been put forward to these anomalies, including differences in the moons' regolith structure (Ostro et al. 2010) and variations in the energetic particle and/or ring particle flux (Scipioni et al. 2014; Thomas et al. 2018). The latter option is probably the more likely explanation for Rhea's excess brightness at visible wavelengths, since the moon is in a region where both the predicted fluxes of E-ring grains and energetic particles are expected to be very low (see also below). Indeed, our simple E-ring model is probably not completely accurate near Rhea's orbit. Detailed numerical simulations of the E-ring particles reveal that there could be a population of sub-micron particles orbiting in the outskirts of the E ring close to Rhea's orbit (Horányi et al. 2008). This population could potentially increase the particle flux into Rhea, moving that data point closer to the trend defined by Mimas and Dione, but it is not clear whether it could also explain that moon's spectral properties. In any case, overall, these data confirm that the E-ring particle flux is an important factor for determining the surface brightness of the mid-sized and co-orbital satellites.

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Table 6.  Particle and Radiation Fluxes

Moon	B0a			E-ring Fluxb			Radiation Fluxc
				(part. m−2 s−1)			(protons cm−2 s−1)
	Ave.	Lead	Trail	Ave.	Lead	Trail	in range 25–59 MeV
Janus	0.53	0.47	0.58	1.21	1.01	1.41	5.4
Epimetheus	0.46	0.45	0.52	1.21	1.01	1.41	7.4
Aegaeon	0.18	0.16	0.24	1.60	1.32	1.88	920.4
Mimas	0.75	0.73	0.79	2.23	1.81	2.64	5.6
Methone	0.56	0.55	0.55	2.66	2.15	3.17	140.2
Pallene	0.53	0.47	0.53	3.89	3.19	4.60	162.0
Enceladus	1.15	1.12	1.17	6.44	6.25	6.63	4.4
Tethys	0.94	0.96	0.89	3.46	4.77	2.15	3.2
Telesto	0.90	0.84	0.84	3.46	4.77	2.15	3.2
Calypso	1.10	1.13	1.21	3.46	4.77	2.15	3.2
Dione	0.68	0.76	0.56	1.53	2.33	0.72	3.1
Helene	1.02	0.84	0.95	1.53	2.33	0.72	3.1
Polydeuces	0.61	0.65	0.63	1.53	2.33	0.72	3.1
Rhead	0.71	0.79	0.63	0.46	0.76	0.15	3.1

Notes.

^aBrightness coefficient at 30° computed assuming surface follows Lambertian scattering law, see Table 3. ^bE-ring flux computed assuming e₀ = 0.17, e_c = 0.01 and a peak number density of 0.03 particles per cubic meter. Leading and trailing side values given by $F\pm \delta {F}_{\lambda }/\sqrt{2}$. ^cRadiation fluxes computed assuming π steradian viewing angle. ^dRadiation fluxes at Rhea assumed to be the same as those for Dione.

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Turning to the smaller moons, however, the situation is more complicated. Two of the co-orbital moons—Telesto and Polydeuces—fall close to the same trend as their larger companions. However, the other two co-orbitals—Calypso and Helene—fall noticeably above this trend. Meanwhile, Aegaeon, Methone, and Pallene all fall well below the trend for the mid-sized moons. This implies that something besides the E-ring flux is affecting the brightnesses of these small moons.

Aegaeon, Methone, and Pallene are all considerably darker than one would have predicted given their locations and the observed trend between Janus, Epimetheus, Mimas, and Enceladus. Aegaeon in particular is exceptionally dark, with a surface brightness coefficient around 0.2. This conclusion is consistent with the few resolved images of these bodies, which show that only the dark side of Iapetus has a comparably low surface brightness as Aegaeon (see Figure 18). In principle, the darkness of these moons could be due to a number of different factors, including their small size and their associations with other dusty rings, but the most likely explanation involves their locations within Saturn's radiation belts.

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Aegaeon, Methone, and Pallene are much smaller than the mid-sized moons, and so their small sizes could somehow be responsible for their low surface brightness. Indeed, based on preliminary investigations of Aegaeon's low surface brightness, Hedman et al. (2010) suggested that a recent collision catastrophically disrupted the body, stripping off its bright outer layers to produce the debris that is now the G ring and leaving behind a dark remnant. Deeper examination, however, does not support this model. For example, if a collision was sufficient to expose the dark core of a kilometer-scale body, then small impacts should expose dark materials on the other moons, but no such dark patches have been seen in any of the high-resolution images obtained by Cassini. More fundamentally, attributing Aegaeon's extreme darkness to some discrete event in the past is difficult because the E-ring should be able to brighten this moon relatively quickly. Assuming typical particle sizes of around 1 μm, the above estimates of the E-ring flux implies that Aegaeon would acquire a layer of E-ring particles 10 μm thick in only 100,000 yr. This calculation neglects mixing within Aegaeon's regolith and any material ejected from Aegaeon, but even this basic calculation shows that E-ring particles would cause Aegaeon to brighten on very short timescales, which strongly suggests that whatever process is making Aegaeon dark is actively ongoing.

Another possibility is that Aegaeon, Methone, and Pallene are dark because they are embedded not only in the E ring, but also in narrower rings like the G ring. However, since these rings likely consist of debris knocked off the moons' surfaces, this does not necessarily explain why this material would be so dark. Another problem is that Prometheus and Pandora are rather bright, even though these moons are close to the F ring and therefore are likely coated in F-ring material (see below). Furthermore, since the rings/arcs associated with Methone and Pallene are over an order of magnitude fainter than the G ring (Hedman et al. 2018), it seems unlikely that these tenuous rings could significantly affect the surface properties of those two moons. Thus, the G ring and other dusty rings do not appear to provide a natural explanation for the darkness of these small moons.

A more plausible explanation for the low surface brightnesses of Aegaeon, Methone, and Pallene is that these moons orbit Saturn within regions that have unique radiation environments. The bottom panel of Figure 16 shows profiles of energetic proton and electron fluxes from two of the highest-energy channels of Cassini's Low-Energy Magnetospheric Measurement System (LEMMS) instrument (Krimigis et al. 2004). Note that while the electron flux has a broad peak centered around the orbit of Aegaeon, the high-energy proton flux is confined to three belts between the main rings and the orbits of Janus, Mimas, and Enceladus (Roussos et al. 2008). This disjoint distribution of high-energy protons arises because energetic protons circle the planet under the influence of the corotation electric field, but their longitudinal motion is enhanced by gradient-curvature drifts, which act in the same direction as corotation. These particles therefore orbit the planet with a period of one to a few hours. Combined with the alignment of the spin and dipole axes, this drift allows these protons to re-encounter the inner moons frequently, causing a permanent flux decrease (macrosignature) along their orbits for sufficiently energetic protons. It is important to realize that these macrosignatures are not just depressions in the energetic proton density, but are also regions where the flux of protons is greatly decreased. Proton macrosignatures along the moon orbits are visible in the data at energies of >300 keV or so but are only expected to exist at much higher energies for electrons. Since Janus and Mimas are exposed to comparable amounts of energetic electrons as Aegaeon, Methone, and Pallene, the electron flux is not a natural explanation for the smaller moons' low surface brightness. However, Aegaeon, Methone, and Pallene are exposed to much higher fluxes of high-energy protons than any of Saturn's other moons, so this is a plausible potential explanation for the darkness of these three moons.

Table 6 provides more quantitative estimates of the proton fluxes into the different moons. Here, we use the protons observed with the P8 channel on the LEMMS instrument as a proxy for the total flux of high-energy protons (Krimigis et al. 2004). In the radiation belts, this channel's sensitivity is mainly to protons of 25.2–59 MeV, and the numbers reported in Table 6 are for an acceptance solid angle of 1 steradian to facilitate comparisons with the E-ring particle flux. Since Mimas, Janus, and Epimetheus occupy narrow gaps in the radiation belts, the fluxes provided in this table are averaged over the moons' true anomalies. Furthermore, for Janus and Epimetheus, we also average over the semimajor axis variations associated with these moons' co-orbital interactions. Also note that while estimates of the Galactic Cosmic Ray background have been removed from these data, the fluxes for the moons beyond Enceladus are probably upper limits. Nevertheless, these numbers clearly show that Methone, Pallene, and Aegaeon experience exceptionally high proton fluxes.

If radiation damage from high-energy protons is the dominant darkening process and the E-ring particles are the dominant brightening process, then we would expect the moons' equilibrium surface brightness to depend on the ratio of the radiation flux to the E-ring particle flux. To make this ratio properly unitless, we consider the mass flux ratio for high-energy protons and E-ring particles. This quantity is computed by taking the number flux of protons given in Table 6 multiplied by the proton mass and dividing that number by the average flux of E-ring particles given in Table 6 multiplied by m_eff, the effective average mass of particles with radii larger than 1 μm. Specifically, we assume m_eff = 4π ln(100) × 10⁻¹⁵ kg, which is consistent with a power-law size distribution with a differential index of −4 (Ye et al. 2014) extending between 1 and 100 μm, and a particle mass density of 1 g cm⁻³. Figure 19 shows that the moons' brightnesses do indeed systematically decrease as this ratio increases. Furthermore, Methone, Pallene, and Aegaeon fall along the same basic trend in this plot as the mid-sized and co-orbital moons. This match provides strong evidence that radiation damage is the dominant process responsible for making Methone, Pallene, and Aegaeon dark.

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At the moment, we do not know precisely which components of the radiation flux are responsible for the darkening. The macrosignatures are most prominent in the highest-energy proton channels, which suggests energetic protons are the dominant agent, but other agents (e.g., electrons with even higher energies) may also contribute at some level. If the darkening agent is protons with energies greater than 1 MeV, there are several ways that they could be altering the surface. Howett et al. (2011) suggested that energetic electrons "sinter" or fuse grains in the uppermost layer, which can affect the reflectance properties at a range of wavelengths (Schenk et al. 2011; Howett et al. 2018). Electrons are mainly slowed down by interactions with electrons in materials. Protons are initially slowed down mainly by interactions with electrons in materials, although as they slow down, they interact more with nuclei. So it is possible very energetic protons (in the first fraction of their mean range) are causing the same kinds of changes to the ice as energetic electrons. Furthermore, they have large gyroradii and so they can often affect the whole body, i.e., they do not weather the surface differentially in many cases. Radiation damage can also change the chemical properties of surface materials, such as generating complex organics (Hapke 1986; Poston et al. 2018) or producing color centers in salts (Hand & Carlson 2015; Hibbitts et al. 2019; Trumbo et al. 2019), which could darken the surface and produce distinctive spectral signatures.

During its closest flybys of Aegaeon and Methone, Cassini's NAC obtained images of both moons through multiple filters. We processed these high-resolution color images with the same methods as described above for the mid-sized moons, yielding the brightness coefficients given in Tables 27 and 28. We then interpolated these measurements to a single phase angle to obtain the visible spectra shown in Figure 20. Furthermore, during the close encounter with Methone, the Visual and Infrared Mapping Spectrometer (VIMS; Brown et al. 2004) was able to obtain near-infrared spectra of the moon. While VIMS was unable to spatially resolve Methone, VIMS acquired five "cubes" with good signal-to-noise (V1716191964, V1716192051, V1716192374, V1716192461, and V1716192872) that could yield decent disk-integrated spectra. We calibrated these cubes with the standard pipelines (Brown et al. 2004; Clark et al. 2012, flux calibration RC19), co-added the signals in the pixels containing the moon signal, removed backgrounds based on adjacent pixels, normalized the resulting spectra and averaged them together to produce a single mean spectrum shown in Figure 21. (Note that this spectrum is arbitrarily normalized.) The near-infrared spectrum of Methone is dominated by water ice, while the visible spectra of the two moons are generally flat with a spectral break around 0.5 μm. These moon's spectral properties are therefore very similar to those of the rest of Saturn's moons (Buratti et al. 2010, 2019; Filacchione et al. 2012). Furthermore, the depths of the water-ice bands at 1.5 and 2.0 μm (computed following the procedures laid out in Filacchione et al. 2012) are 0.45 ± 0.04 and 0.60 ± 0.06, respectively. These values are very similar to those previously found for Janus, Epimetheus, Telesto, Dione, and Rhea (Filacchione et al. 2012). Hence, there is no evidence that these bodies have surfaces with distinctive spectrally active components, like irradiated salts (Hibbitts et al. 2019). Instead, the radiation appears to change the surface brightness relatively uniformly at all wavelengths.

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An interesting point of comparison for these spectra is Jupiter's moon Callisto. Callisto and Aegaeon are probably exposed to similar fluxes of high-energy protons (roughly 10/(cm² s sr keV) around energies of 1 MeV, Paranicas et al. 2018), and both objects are comparably dark (Callisto has a geometric albedo around 0.2 between 0.5 and 2.5 μm (Calvin et al. 1995), and extrapolating the available data suggests that Aegaeon's equivalent spherical albedo is around 0.15). Like Aegaeon, Callisto has a visible spectrum that is flat longwards of 0.6 μm and has a downturn at shorter wavelengths (Calvin et al. 1995; Moore et al. 2007). The structure in Callisto's spectrum in the near-infrared primarily consists of comparatively weak water-ice features, along with a weak carbon-dioxide band at 4.2 μm (Moore et al. 2007), which is consistent with the compositional features seen in Methone and Saturn's other moons (Clark et al. 2008; Hendrix et al. 2018; Buratti et al. 2019). It may therefore be that both Callisto and Aegaeon represent end-members of what ice-rich surfaces look like when exposed to large doses of high-energy radiation. If correct, it could mean that even if we had near-IR spectra of Aegaeon, they would not contain diagnostic spectral features of the chemicals responsible for making its surface dark. Whether the observed amount of darkening is consistent with structural changes in the regolith (Howett et al. 2011; Schenk et al. 2011) or chemical changes in a more spectrally neutral component (such as the non sulfur-bearing ices in Poston et al. 2018), will require detailed spectral models that are beyond the scope of this work.

After Aegaeon, Methone, and Pallene, the next most obvious anomalies in Figure 17 are Helene and Calypso. Calypso is brighter than both Tethys and Telesto, while Helene is brighter than both Dione and Polydeuces. These findings are actually consistent with prior estimates of these moons' geometric albedos summarized in Verbiscer et al. (2007, 2018). At the same time, it is worth noting that despite being in a region where the E-ring flux should be extremely low, Pan, Atlas, Prometheus, and Pandora are all reasonably bright, and in particular, Prometheus and Pandora are brighter than any of the other moons in their vicinity. The excess brightness of these moons is probably related to distinctive aspects of their near-infrared spectra. For the mid-sized moons, the depths of the moon's water-ice bands are reasonably well correlated with their visible brightness (Filacchione et al. 2012). However, Calypso, Prometheus, and Pandora all have deeper water-ice bands than Enceladus, while Atlas, Telesto, and Helene have water-ice bands intermediate in strength between Tethys and Dione (Buratti et al. 2010; Filacchione et al. 2012, 2013). Note that even though Helene does not appear to have as exceptionally deep water-ice bands as Calypso, both Calypso and Helene have deeper bands than their respective large companions Tethys and Dione. The anomalous spectral and photometric properties of Prometheus, Pandora, Calypso, and Helene could be attributed to a number of different factors, including the moons' size and differences in the local radiation environment. In practice, localized enhancements in the particle flux appear to be the most likely explanation, but the excess particle flux responsible for the high observed brightnesses of Helene and Calypso is far from clear.

In principle, the surface brightness of small moons could be different from the brightness of large moons in the same environment because their reduced surface gravity affects the porosity and structure of their regolith. In fact, one aspect of these moons' surface properties probably can be attributed to their small size: their general lack of a leading-trailing brightness asymmetries. In particular, all four of the trojan moons appear to lack the strong leading-trailing brightness asymmetries seen on Tethys and Dione.⁹ For Tethys and Dione, this asymmetry is thought to arise because of asymmetries in the E-ring flux, so it is perhaps surprising that similar asymmetries are not seen in the smaller moons. However, the smaller size and much lower surface gravity of these moons means that secondary ejecta from the E-ring impacts can be more easily globally dispersed over their surfaces, providing a natural explanation for why these moons have more uniform surface properties.

While detailed modeling of impact debris transport around these small moons is beyond the scope of this paper, we can provide rough order-of-magnitude calculations that are sufficient to demonstrate the feasibility of this idea. Assuming that most of the E-ring particles striking these moons are near the apocenters of their eccentric orbits, we can assume r ≃ (1 + e) a in Equation (23) and estimate the E-ring particle's impact velocities at the orbits of Tethys and Dione to be 1.4 km s⁻¹ and 3.6 km s⁻¹, respectively. Experimental studies of impacts into ice-rich targets by 20–100 μm glass beads found impact yields at these velocities of the order of 100 (Koschny & Grün 2001), and assuming roughly comparable values for micron-sized E-ring grains would imply an effective average ejection velocity of at most 14 m s⁻¹ and 36 m s⁻¹, respectively. These speeds are much less than the escape speeds from Tethys and Dione (which are 390 and 510 m s⁻¹, respectively), but are comparable to the escape speeds of Telesto, Calypso, and Helene (which are 7, 19, and 9 m s⁻¹, respectively). Hence, it is reasonable to think that the debris from E-ring particle impacts is more uniformly distributed on the smaller trojans than they are on Tethys and Dione, making their surfaces more uniform in brightness.

However, while their size is likely responsible for their lack of leading-trailing brightness asymmetries, it is unlikely that size-related phenomena can explain the overall high brightness of Prometheus, Pandora, Calypso, and Helene. The problem with such ideas is that there are no clear trends between size and brightness excess. Telesto is intermediate in size between Helene and Calypso but is roughly the same brightness as Tethys. Also, Prometheus and Pandora are themselves intermediate in size between Epimetheus and Atlas. This lack of obvious trends with mass or size suggests that the brightness excess of these moons most likely reflects something different about their environments.

Given the previous analyses of the trends among the other moons, the easiest way to increase the brightness of these moons would be either by decreasing the radiation flux or increasing the particle flux. Decreasing the radiation flux is an unlikely explanation because the high-energy proton flux onto many of the moons is already very low (see Table 6). Furthermore, for the radiation flux to be lower at Calypso than at Telesto, or lower at Helene than at Polydeuces, there would need to be a strong asymmetry in the radiation flux on either side of Tethys and Dione. While both Tethys and Dione have been observed to produce localized reductions in plasma known as microsignatures, these do not extend all the way to the trojan moons. Also, given that the plasma is bound to the planet's magnetic field, which rotates faster than the moons orbit around the planet, it is hard to imagine any interaction with charged particles that would similarly affect Calypso (which trails Tethys) and Helene (which leads Dione).

The above considerations leave variations in the particle flux as the best remaining option. For Prometheus and Pandora, this is a perfectly reasonable explanation, since these moons fall within the outskirts of the F ring, which is a natural source of dust flux into these moons. If an excess flux of F-ring particles is indeed responsible for the high brightness of Prometheus and Pandora, this is interesting because it means that the dust impacting the moons does not have to be the relatively fresh ice found in the E ring to produce increases in brightness. In this case, the lower-opacity dusty rings surrounding Pan and Atlas could be keeping those moons somewhat brighter than they would otherwise be given their locations well interior to the E ring.

Excess dust fluxes are also an attractive explanation for the excess brightness of Calypso and Helene because these moons are nearly as bright as Enceladus, an object whose brightness is certainly due to a high particle flux. However, there is no obvious source for the particles that would strike Calypso and Helene more than their co-orbital companions. The brightness differences cannot be easily attributed to differences in the E-ring flux, since Telesto, Calypso, and Tethys are at basically the same orbital distance from Saturn, as are Helene, Polydeuces, and Dione. Hence, the E-ring flux into Telesto, Tethys, and Calypso (or Helene, Dione, and Polydeuces) should be nearly the same.

In principle, asymmetries in the particle flux could arise from either subtle interactions between the larger moons and the E-ring grains, or more local particle populations sourced from the larger moons. For example, particles on nearly circular orbits launched from Tethys that are carried outward by plasma drag would preferentially fall behind Tethys, and thus potentially be more likely to strike Calypso than Telesto. The problem with such explanations is that for Calypso and Helene to fall along the same trend as the other moons in Figure 17, the excess particle flux into these moons needs to be a substantial fraction of the E ring flux, which would imply a large excess density of particles around these moons compared to Tethys, Dione, Telesto, or Polydeuces. These particles should produce localized brightness enhancements and/or longitudinal brightness asymmetries in the regions around Tethys' and Dione's orbits, and no such features have yet been seen in the Cassini images of the E ring. In principle, there could be some subtle interactions between E ring particles and the relevant moons that would cause the requisite variations in the particle flux without producing detectable asymmetries in the density of the rings. Fully exploring such possibilities would likely require numerical simulations that are beyond the scope of this work.

Given the above challenges with identifying a suitable dust population around these moons during the Cassini mission, it is worth considering whether Calypso's and Helene's current brightnesses are not their steady-state values but are instead transient phenomena caused by some event in the relatively recent past. Specifically, we can consider the possibility that a recent impact into Calypso and/or Helene released enough material to affect the global surface properties of both these moons. A thorough quantitative evaluation of such scenarios is beyond the scope of this paper. Instead we will just provide some order-of-magnitude calculations that demonstrate that this is a viable possibility.

The total amount of material needed to affect the overall surface brightness of Calypso and Helene is simply the amount of material needed to coat these moons with a layer thick enough to determine its optical properties at optical wavelengths. Since light can only penetrate a few wavelengths through the surface regolith, we may conservatively estimate that a 10 μm thick layer of material will be sufficient for these purposes. Given that Calypso and Helene have surface areas of around 400 km² and 1400 km², respectively, the total volume of material needed to coat these moons is 4000 and 14,000 m³. Conservatively assuming this is ice-rich material of negligible porosity, these volumes correspond to total debris masses of 4 × 10⁶ and 14 × 10⁶ kg, respectively.

The flux of objects large enough to produce these sorts of debris clouds is fairly well constrained thanks to observations of comparably massive impact-generated debris clouds above Saturn's rings described by Tiscareno et al. (2013). The masses of those clouds are uncertain and depend on the assumed size distribution of the debris, but the feature designated Bx probably had a mass between 2 × 10⁵ kg and 2 × 10⁷ kg, which is comparable to that required to brighten Calypso or Helene. The estimated flux of impactors able of producing this amount of debris is 3 × 10⁻²⁰ m⁻² s⁻¹ (Tiscareno et al. 2013). Assuming cross sections consistent with the observed sizes of Calypso and Helene, this means the mean time between impacts should be between 1000 and 10,000 yr.

The other relevant timescale in this problem is how long it would take E-ring material to coat these moons and erase any transient signals from an impact. Using the fluxes provided in Table 6, and assuming a typical E-ring particle radius of around 1 μm, we estimate that E-ring material would accumulate on the surfaces at rates of between 0.2 nm yr⁻¹ for Helene and 0.5 nm yr⁻¹ for Calypso. It would therefore take 20,000 yr for Calypso and 50,000 yr for Helene to accumulate a 10 μm thick layer of E-ring particles. These numbers could potentially be reduced by a factor of ∼100 if we account for the yield of secondary debris from each E-ring particle impact (see above). The resurfacing time due to E-ring particles is therefore likely comparable to the mean time between impacts capable of covering these entire moons in fresh impact debris. It is therefore not unreasonable that Calypso and Helene both experienced a recent impact that brightened their surfaces, while Telesto and Polydeuces managed to avoid such a recent collision. In principle, some of the material from a collision into Helene or Calypso could have been transported to the other moon, allowing a single impact to brighten both moons, but to properly explore the relative likelihood of such an event will require numerical simulations of both dynamics of the impact debris similar to those done by Dobrovolskis et al. (2010), perhaps including non-gravitational forces.

Note that this explanation for Calypso's and Helene's brightness means that impacts tend to increase the moons' brightness, rather than darken them. This might at first be counterintuitive because cometary debris in the outer solar system is generally assumed to be much darker than Saturn's moons. However, it is important to realize that most of the debris produced by the impacts would come from Helene and Calypso, not the impactor. Indeed, the expected impact yields for such bodies is of the order of 10,000 (Cuzzi & Estrada 1998; Tiscareno et al. 2013), and so the impactor required to produce 4–14 × 10⁶ kg of debris would be only 4–14 × 10² kg, which corresponds to an ice-rich object with a radius between 0.5 and 0.7 m. Thus, most of the debris falling on the moons would be pure ice and should be able to brighten moon surfaces like the E and F rings are apparently able to do. This bright deposit would also be much thinner than the deposits previously identified in high-resolution images, which affect both the morphology and the color of the surface (Thomas et al. 2013; Hirata et al. 2014). While such a thin deposit would probably not produce obvious morphological structures (even the original crater would be near the resolution limit of the best images), it is not clear whether a fresh global covering of fine debris is consistent with the color variations currently observed on these moons (Thomas et al. 2013). Also, it is not obvious whether recent impacts are consistent with near-infrared spectra of these moons, which show that Prometheus, Pandora, and Calypso have deeper ice bands than Telesto and Helene (Filacchione et al. 2012). Even so, the above considerations indicate that a recent impact is a possible explanation for Calypso's and Helene's high surface brightness and so merits further investigation.