Changes in a Dusty Ringlet in the Cassini Division after 2010
DDraft version August 24, 2020
Typeset using L A TEX preprint2 style in AASTeX62
Changes in a Dusty Ringlet in the Cassini Division after 2010
Mathew Hedman and Bill Bridges Department of Physics, University of Idaho, Moscow Idaho 83843
ABSTRACTA dusty ringlet designated R/2006 S3, also known as the “Charming Ringlet”, islocated around 119,940 km from the center of Saturn within the Laplace Gap in theCassini Division. Prior to 2010, the ringlet had a simple radial profile and a predictableeccentric shape with two components, one forced by solar radiation pressure and theother freely precessing around the planet. However, observations made by the Cassinispacecraft since late 2012 revealed a shelf of material extending inwards from the ringletthat was not present in the earlier observations. Closer inspection of images obtainedafter 2012 shows that sometime between 2010 and 2012 the freely-precessing componentof the ringlet’s eccentricity increased by over 50%, and that for at least 3 years after2012 the ringlet had longitudinal brightness variations that rotated around the planetat a range of rates corresponding to ∼
60 km in orbital semi-major axis. Some eventtherefore disturbed this ringlet between 2010 and late 2012. INTRODUCTIONSaturn’s complex ring system includes mul-tiple broad rings and narrow ringlets that areprimarily composed of particles less than 100microns across. Unlike the millimeter-to-metersized particles that dominate the main ring sys-tem, these dusty rings are sensitive to a vari-ety of non-gravitational forces that can influ-ence their structure and dynamics. Further-more, several of these dusty rings have been ob-served to change significantly over timescales ofyears to decades. Most often these changes in-clude the formation of bright clumps of material(French et al. 2012; Hedman et al. 2013; Hed-man 2019), but in other cases they involve largerscale structural changes that can be attributedto variations in periodic perturbing forces or dis-crete disturbances spanning broad ring regions(Chancia et al. 2019; Hedman and Showalter2016). This paper describes a new type of time-variable phenomena in dusty rings, where the radial profile and orbit shape of a narrow ringletappears to have suddenly changed.The particular ringlet we will focus on here islocated within the Laplace Gap of the CassiniDivision in Saturn’s rings, and is officially des-ignated R/2006 S3 (Porco 2006), but informallyreferred to as the “Charming Ringlet”. Theringlet has a peak optical depth of 10 − andis strongly forward-scattering, indicating thatit is composed primarily of dust-sized parti-cles (Hor´anyi et al. 2009; Hedman et al. 2011).Observations from early in the Cassini missionshowed that this ringlet exhibited heliotropicbehavior (i.e. the ringlet’s peak brightness isfound further from the planet at longitudesmore closely aligned to the Sun), which is mostlikely due to solar radiation pressure perturbingthe orbits of the small particles that form thisringlet (Hedman et al. 2010).Images obtained later in the Cassini missionreveal a noticeable change in this ringlet’s radialprofile. Prior to 2010, the ringlet’s brightness a r X i v : . [ a s t r o - ph . E P ] A ug declined roughy symmetrically around its peak,but after 2012 a “shelf” of material can be seenon its inner flank. Figure 1 shows the best imageof this feature, while Figure 2 shows two imagesof the Charming Ringlet taken at similar phaseangles eight years apart, where the shelf is vis-ible in the later image and is absent from theearlier image. Similarly, Figure 4 shows radialbrightness profiles of the ringlet obtained beforeand after 2010 at nearly identical phase angles,demonstrating that the shelf is seen repeatedlyin images taken after 2012, and is always absentfrom earlier observations. These data there-fore indicate that something happened to thisringlet over the course of the Cassini missionthat changed its overall structure.In this paper we use the full span of theCassini data to quantify and characterize thesechanges to the ringlet, and to explore what pro-cesses might be responsible for disturbing thisringlet. Section 2 describes the Cassini imagingdata used in this study and how these data wereprocessed to generate radial brightness profilesof this ringlet. Section 3 then examines boththe total brightness of the ringlet and its ra-dial position to show that while the particlecontent of the ringlet did not change dramat-ically over the course of the Cassini mission, itsshape and orbital properties underwent a clearshift between 2010 and 2012. Section 4 takesa closer look at the brightness profiles obtainedafter 2010, which reveal that the formation ofthe shelf was associated with significant longitu-dinal variations in the ringlet’s brightness thatrotated around the planet at speeds consistentwith the expected mean motion of orbiting ma-terial. Finally, Section 5 discusses the processesthat could have been responsible for inducingthis change, which include collisions with in-terplanetary debris and sudden changes in thering’s electromagnetic environment. OBSERVATIONS AND PRELIMINARYDATA REDUCTION
Figure 1.
The highest resolution and signal-to-noise image of the ringlet obtained in 2017 at aphase angle of 152 ◦ . Radius increases from upperleft to lower right, and the ringlet’s location is in-dicated by the arrow. This image shows a clear“shelf” of material extending from the ringlet’s in-ner flank that was not present in earlier images (seeFigure 2). This study uses images obtained by the Nar-row Angle Camera of the Image Science Sub-system (ISS) onboard the Cassini Spacecraft(Porco et al. 2004). We conducted a com-prehensive search of images containing theLaplace Gap with resolutions better than 10km and ring opening angles above 5 ◦ using theOPUS search tool available on the Ring-MoonSystems Node of the Planetary Data System( https://pds-rings.seti.org/search ). Thissearch yielded 1228 images. A relatively smallnumber of the above images were found not to Later searches found a few additional image se-quences that contained the ringlet, but these werejudged to have too low signal-to-noise and/or resolutionfor this particular study.
Figure 2.
Changes in the morphology of the dusty ringlet within the Laplace gap. The two images shownhere were both obtained at phase angles of 143 ◦ and have been independently rotated, cropped to facilitatecomparisons. The ringlet is marked by the arrows. The left image was obtained in 2009 while the rightimage was obtained in 2017 (see Figure 4 for radial brightness profiles derived from these images). Notethat in the earlier image the ringlet appears as a relatively symmetric bright band, while in the later imagethere is again a faint “shelf” of material extending from the ringlet’s inner flank. be suitable for this analysis. First of all, weexcluded 93 images that were made when theSun was within 2 ◦ of the ringplane because dur-ing this timeframe shadows from nearby ringmaterial could fall across the ringlet, compli-cating the analysis. A further 17 images wereremoved from consideration because the ringletwas either not completely captured in the imageor the image could not be properly navigated.These issues caused the total radially-integratedbrightness of the ringlet to be extremely high orlow (above 100 m or below 0.1 m) or the appar-ent ringlet position to be over 80 km from itsexpected location. This left a total of 1118 im-ages deemed suitable for this particular study.Figure 3 shows the distribution of these ob-servations as functions of phase angle and time.Note that there is a large gap in the data be-tween late 2009 and late 2012. This corresponds Figure 3.
Distribution of the ringlet observationsused in this analysis. Each point corresponds tothe time and phase angle of an individual image.Note the large data gap between late 2009 and late2012. Also note that the data prior to 2010 containa relatively large fraction of isolated observations,while the data obtained after 2012 is mostly in theform of a few sequences of many images.
Table 1.
Notable movie sequences of the Charming Ringlet
Observation ID Files Date Duration (hours) Phase Angle (deg) Emission Angle (deg)ISS 103RI SHRTMOV001 PRIME N1613254649-N1613295800 2009-044 11 161.897-150.715 96.8-116.18ISS 104RI SHRTMOV002 PRIME N1614278585-N1614302748 2009-056 6 161.54-159.198 93.13-104.57ISS 172RI MOONLETCD001 PIE N1727207218-N1727222674 2012-268 4 90.207-61.55 56.36-58.03ISS 189RI BMOVIE001 PRIME N1746791718-N1746835578 2013-129 12 103.14-120.27 148.22-152.91ISS 206RI BMOVIE001 PRIME N1784298322-N1784343322 2014-198 12 94.96-117.23 134.37-136.76ISS 213RI BMOVIE001 PRIME N1804922788-N1804944201 2015-072 5 72.71-67.14 82.86-83.51ISS 231RI CDMOVIE001 PRIME N1832677373-N1832690125 2016-028 3 78.22-76.485 86.15-86.07ISS 276RI HPMONITOR001 PRIME N1874437615-N187471965 2017-145 9 144.19-141.58 81.68-79.25 to an extended period of time when the Cassinispacecraft remained close to the planet’s equa-tor plane, and so could not easily image themain rings. Also, note that while prior to 2010a fair fraction of the data points are isolatedobservations, after 2012 virtually all the im-ages of the ringlet came from movie sequenceswhere the camera stared at the Cassini Divi-sion for a significant fraction of an orbital pe-riod. These sequences are particularly usefulfor investigating longitudinal variations in theringlet’s brightness and so Table 1 summarizesthe properties of several of these observations.All these images were calibrated using theCISSCAL routines to remove instrumentalbackgrounds, apply flat-fields, and convert themeasured data to
I/F , a standard unit of re-flectance that is unity for a perfect Lambertiansurface viewed and illuminated at normal in-cidence (Porco et al. 2004; West et al. 2010).These images are also geometrically navigatedusing the appropriate SPICE kernels (Acton1996). The image pointing was refined basedboth on the positions of stars within the field ofview and the location of the outer edge of theJeffries Gap. This geometric information wasalso used to determine the phase, incidence andemission angles at the ring.Since the radial structure of the ringlet didnot obviously change across the limited range oflongitudes visible in a single image, we deriveda radial brightness profile from each image byaveraging over a range of longitudes for each im-age. These profiles of observed
I/F were then converted to “normal
I/F ” = µI/F , where µ is the cosine of the emission angle. For a lowoptical depth ringlet like the Charming Ringlet,this quantity should be independent of emissionangle and so is a more useful quantity for com-paring observations.Figure 4 shows some example radial profilesderived from images obtained at similar phaseangles before and after the shelf appeared. Theshelf can be clearly seen on the inner flank ofthe three images obtained after 2012, but is notpresent on any of the profiles taken before 2010.These data also show that the shelf is a rathersubtle feature, and that the ringlet itself canhave an asymmetric shape. These aspects ofthe ring’s structure informed how we analyzedthese profiles. SURVEY OF RINGLET PROPERTIESOVER THE COURSE OF THE CASSINIMISSIONWhile the shelf can be seen in all the post-2010 observations in Figure 4, quantifying itsabsolute brightness for all the observations ischallenging because the shelf is a rather subtlefeature that needs to be detangled from back-ground trends from both the ringlet and straylight from both edges of the gap. For this rea-son, our initial investigation of these data in-stead focused on the total integrated brightnessof the ringlet plus shelf, as well as the locationsof both the ringlet and the shelf. These param-eters all provide important information aboutwhat happened to the ringlet between 2010 and2012.
Figure 4.
Examples of ringlet profiles obtained before and after the formation of the shelf. Each of thethree pairs of images were obtained at nearly the same phase angle, and so should be directly comparableto each other. Statistical uncertainties in the brightness profiles are typically between 1 × − and 2 × − and so are comparable to the line width. In the data taken in 2006-2007, the location of the ringlet varies,and shows some asymmetry in its overall shape, but still appears as a simple peak. By contrast, all theprofiles obtained in 2013-2017 show a clear shelf on their inner flank. Note also that the radial extent of thisshelf does not seem to change much as the position of the ringlet changes. The steps we used to determine these param-eters for each profile are illustrated in Figure 5.First, we isolate the signal from the ringlet andthe shelf from background trends associatedwith the surrounding ring material, so thatwe could determine the ringlet’s total bright-ness. Second, we fit the background-subtractedringlet signal to an asymmetric Lorentzianmodel in order to quantify the location andshape of the ringlet, and compute the residu-als to this model. The background-subtractedresidual signal largely isolates the signal fromthe shelf, enabling the location of this featureto be quantified. Details of these proceduresare provided in the following subsections. Sec-tion 3.1 focuses on the resulting estimates ofthe ringlet’s overall brightness, which showsno dramatic changes over the course of theCassini mission, while Section 3.2 focuses onthe ringlet’s orbital properties, which do shiftbetween 2010 and 2012.3.1.
No evidence for large-scale changes in theringlet’s particle content
In order to properly quantify the brightnessand shape of the ringlet, we first need to isolatethe ringlet signal from the backgrounds associ-ated with nearby ring material. To accomplishthis we take each profile and find the bright-ness peak between radii 119,900 km and 119,930km. We then find the radii with the minimumbrightness on either side of this peak and de-fine these as r min and r max . We then take thedata outside the region between r min and r max and interpolate the observed brightness trendsin those regions into the region between r min and r max using a cubic spline interpolation onthe log-transformed data. Examples of the esti-mated background trends are shown as the bluedotted lines in Figure 5.In order to verify that this method properlyisolates the signal from the ringlet, as well ascheck for possible changes in the overall bright-ness of the ringlet over time, we compute the radially integrated brightness of the ringlet plusshelf from each background-subtracted profileto obtain a quantity known as the normal equiv-alent width:NEW = (cid:90) µ ( I/F − I/F back ) dr. (1)This quantity should be independent of resolu-tion, emission and incidence angle, but shouldvary with phase angle (since the particles aremore efficient at scattering light in the forwarddirection) and potentially time (if the change inthe ringlet’s structure also changed the amountof material in the ringlet). Note that the uncer-tainties in these parameters are dominated bysystematic errors in the background levels, andso cannot be reliably determined a priori . In-stead, the uncertainties in these parameters areestimated based on the scatter in measurementsobtained at similar viewing geometries.Figure 6 shows the resulting estimates of theringlet’s integrated brightness as a function ofphase angle and time, and Table 2 provides theaverage NEW values within different phase an-gle bins, with uncertainties based on the ob-served scatter in the measurements. Whilethere is a dispersion of the brightness estimatesaround the mean trend, this dispersion is rea-sonably small ( < ◦ . At lower phase angles the dispersion islarger (closer to 50%), probably because theringlet is much fainter in those viewing geome-tries making the NEW estimates more sensi-tive to small errors in the estimated backgroundtrends. Even so, it is important to note that themean brightness level at phase angles below 90 ◦ is not a strong function of phase angle, eventhough the brightness of the nearby ring mate-rial decreases significantly between 20 ◦ and 90 ◦ phase. These results indicate that the above al-gorithms are isolating the desired ringlet signalrelatively well. As mentioned above, there were13 images that yielded integrated brightness es-timates well off this trend, with NEW values Figure 5.
Illustrations of how the signal from the shelf and ringlet are isolated. The six panels show thesame radial brightness profiles of the ringlet shown in Figure 4 as black solid lines. The blue dotted linesshow the background signal interpolated from the region surrounding the ringlet, while the green dashedline shows the best asymmetric Lorentzian model to the ringlet’s signal above this background. The orangeline shows the residuals to this model relative to the background model. Note that for all the post-2010profiles these residuals show a positive peak due to the shelf, while this feature is absent from the earlierprofiles without this shelf.
Figure 6.
The radially-integrated brightness of theringlet. The top panel shows the measured Nor-mal Equivalent Width versus phase angle for theavailable images of the Charming ringlet. TheseNEW values are also provided in Table 4. Coloredpoints are individual measurements, with colors in-dicating phase angles. The points with error barsare average brightness values for each phase anglerange (see Table 2), and the solid curve is a two-component Henyey-Greenstein function (see text).The bottom panel shows the NEW values, normal-ized by the best-fit phase function as a function oftime (still color coded by phase angle). This plotshows that there is no obvious shift in the ringlet’stypical brightness around 2011. above 100 meters or below 0.1 meters. Theseaberrant observations were excluded from therest of this analysis.To check whether there are subtle temporalvariations in the integrated brightness measure-
Table 2.
Average integrated brightness measure-ments of the Charming Ringlet.
Phase Range Mean Phase NEW(degrees) (degrees) (meters)15-25 19.7 3 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . ments, we fit the integrated brightness data toa two-component Henyey-Greenstein function: N EW = C π (cid:18) w (1 − g )(1 − g − g cos θ ) / +(1 − w )(1 − g )(1 − g − g cos θ ) / (cid:19) . (2)with fit parameters C , w , g and g . The best-fit values for these quantities were C = 0 . w = 0 . g = 0 .
643 and g = − . ◦ and 160 ◦ quite well. The mostnotable issue is that the data at 20 ◦ phasefalls below the model trend, but even in thiscase the model is only about 20% above thedata. This discrepancy most likely representsa limitation of the photometric model. Forexample, other dusty rings required a three-component Henyey-Greenstein function to re-produce the observed phase curve (Hedman andStark 2015). Alternatively, it could indicatethat our background-subtraction algorithm isstarting the remove some of the real ringlet sig-nal in cases where the surrounding ring mate-rial is sufficiently bright. In principle, we couldfit these data to a more complex model and/orfurther refine our background-subtraction pro-cedures to address this issue. However, in prac-tice the only observations obtained at such lowphase angles happened before the formationof the shelf, so we decided it was not worthfurther complicating the model/analysis to fitthese data for this particular study.Dividing the observed brightness data by theabove phase curve yields a phase-correctedNEW that should be around one for all theobservations. The bottom panel in Figure 6plots these phase-corrected brightness estimatesas function of time. These data do not showstrong evidence for any obvious changes in theringlet’s overall brightness over the course ofthe Cassini mission. More specifically, if wecompute the mean phase corrected brightnessvalues for all phase angles above 30 ◦ from theobservations made before and after the shelf’sappearance between 2010 and 2012, we findvalues of 1 . ± .
005 and 0 . ± .
008 (un-certainties are based on the observed scatterin the measurements). The brightness of theringlet therefore changed by less than 10% be-tween these two time periods. This not onlyconfirms that these background-removal proce-dures are robust, but also indicates that theformation of the shelf was not associated witha major change in either the number densityor typical size of the visible particles in thisringlet.3.2.
Evidence for changes in the ringlet’sorbital properties
In addition to changes in the ringlet’s parti-cle content, we also wanted to search for vari-ations in the ringlet’s orbital properties. Wetherefore fit the background-subtracted bright-ness profile to an appropriate functional formto estimate the ringlet’s location. After someexperimentation, we found that fitting the peakof the ringlet’s brightness profile to an asym-metric Lorentzian function both fit the vari-able shape of the main ringlet reasonably welland allowed the signal from the shelf to be iso-lated from the fit residuals. We therefore fit thebackground-subtracted brightness data to thefollowing functional form (Stancik and Brauns2008): µI/F = 2
A/πγ r − r ) /γ + B (3)where A, B and r are constants, but the width γ is a function of radius: γ = 2 γ e α ( r − r ) (4)where γ and α are both constants, with α be-ing the parameter that quantifies the asymme-try of the curve. The fit is performed usingthe mpfitfun IDL program (Markwardt 2012),and considers the background-subtracted pro-file wherever the brightness is over 1/3 the peakbrightness, which helped to ensure that the shelfdid not bias the fit.The green dashed lines in Figure 5 show ex-amples of the best-fit functions for the selectedprofiles, with the background signal added backin to facilitate comparisons with the data. Thisfunction clearly fits the shape of the curve verywell close to the peak, and so the fit parameter r should provide a robust and reliable estimateof the peak location. However, it is also im-portant to note that the asymmetric Lorentzianform does not perfectly reproduce the trendsseen further from the peak. Alternative fittingfunctions, including an asymmetric Gaussian,did not significantly improve these aspects of0the fit. While such issues should not systemati-cally affect the estimates of the peak locations,they do imply that estimates of the parameteruncertainties from the fit are not reliable, andso we will instead estimate the uncertainties inthese parameters based on the scatter in thedata.Furthermore, the signal from the shelf is stillvisible as a bump in the residuals from thebackground-subtracted brightness data (shownas a solid orange curves in Figure 5, again withthe background brightness added back in to fa-cilitate comparisons). These residuals thereforecapture the signal from the shelf. For the dataobtained prior to 2010, the residuals interior tothe peak are either flat or monotonic, consis-tent with the lack of an observable shelf. Bycontrast, the residuals for the profiles obtainedfrom 2012 on all show a broad peak interior tothe ringlet’s peak brightness whose shape andamplitude are consistent with the shelf. Forthis particular analysis, we are most interestedin the location of the shelf, which we estimateas the location of the peak in the residual profileinterior to r − γ .Hedman et al. (2010) showed that this ringlethad both an eccentricity and an inclination, andthat the eccentricity at least had two compo-nents, a “forced” component due to solar ra-diation pressure and a “free” component thatprecessed around the planet at roughly the ex-pected rate due to Saturn’s non-spherical shape.Together, these aspects of the ringlet can causethe apparent position of the ringlet to shift backand forth by over 20 km. Initial investigationsof all the observations indicated that the Hed-man et al. (2010) models for the ringlet’s or-bital properties were not accurately predictingthe position of the ringlet in the more recentdata.In order to investigate these apparent discrep-ancies and to ensure that they were not dueto small pointing errors, we estimated the po- sition of the Laplace Gap’s inner edge as thepoint of maximum slope interior to the ringlet,and compared this to a model of its expectedposition that included the mean radius and ec-centricity given by French et al. (2016). Thedifferences between the observed and expectedpositions had a standard deviation of around 4km, with some outliers as large as 40 km. Wetherefore applied a constant offset to each radialprofile to bring the observed edge position intoagreement with these predictions. These cor-rected values for the peak and shelf positionsare provided in Table 4.Even after these corrections, the ringlet’s peaklocation still failed to match predictions in ob-servations made after 2010. In order to quan-tify these changes in the ringlet’s location, weconsider the following simplified model of theringlet’s radial location r as a function of longi-tude λ and time t : r = a − ae ( t ) cos( λ − λ (cid:12) − (cid:36) (cid:48) ( t )) , (5)where λ (cid:12) is the sub-solar longitude, a is the (as-sumed constant) ringlet semi-major axis, whilethe eccentricty e and pericenter location relativeto the sub-siolar longitude (cid:36) (cid:48) are both implicitfunctions of time given by the following rela-tionships: ae cos (cid:36) (cid:48) = ae f cos (cid:36) (cid:48) f + ae (cid:96) cos( (cid:36) (cid:48) (cid:96) + ˙ (cid:36) (cid:48) (cid:96) t ) , (6) ae sin (cid:36) (cid:48) = ae f sin (cid:36) (cid:48) f + ae (cid:96) sin( (cid:36) (cid:48) (cid:96) + ˙ (cid:36) (cid:48) (cid:96) t ) , (7)where ae f , ae (cid:96) , (cid:36) (cid:48) f , (cid:36) (cid:48) (cid:96) and ˙ (cid:36) (cid:48) (cid:96) are all constants,with ae f representing the eccentricity forced bysolar radiation pressure and ae (cid:96) representing thefree component of the eccentricity that precesses The inclination and structure forced by the 2:1 Reso-nance with Mimas are not included in these calculationsbecause they would only produce sub-kilometer varia-tions in the apparent edge position. ae (cid:96) or ae f . Fitsto these data indicated that (cid:36) (cid:48) f (cid:39) ◦ and˙ (cid:36) (cid:48) (cid:96) (cid:39) . ◦ /day, values that are consistent withtheoretical expectations for a forced eccentric-ity induced by solar radiation pressure and thefree precession rate for particles orbiting in thevicinity of the ringlet (after correcting for theapparent motion of the Sun). Also, the twocomponents of the eccentricity were found tohave the amplitudes ae f = 17 . ± . ae (cid:96) = 7 . ± . rms difference between the observed and pre-dicted ringlet positions for a range of orbital pa-rameters. In order to keep the parameter spacemanageable, we assume that (cid:36) (cid:48) f = 180 ◦ and˙ (cid:36) (cid:48) (cid:96) = 4 . ◦ /day, so that we only have to con-sider the parameters ae f , ae (cid:96) and (cid:36) (cid:48) (cid:96) (the meanradius a being a constant offset that falls outof the rms calculation). We evaluated the rms misfit for the data taken before and after 2010for an array of ae f , ae (cid:96) and (cid:36) (cid:48) (cid:96) values sampledevery 1 km, 1 km and 10 ◦ , respectively.Figure 7 shows the minimum rms misfit asfunctions of the three parameters. For the ob- Figure 7.
Changes in the best-fit orbit parame-ters for the ringlet over the course of the Cassinimission. Each panel shows the rms misfit be-tween the observed ringlet peak locations and thepredicted values for a three-parameter model ofthe ringlet’s shape assuming a free precession rate˙ (cid:36) (cid:48) (cid:96) = 4 . ◦ /day and forced pericenter location (cid:36) (cid:48) f = 180 ◦ . The curves for the free pericenter loca-tion at each give the minimum rms misfit over allpossible values of the free and forced eccentricity,both of which show a clear minimum at 230 ◦ . Thesolid curves for the forced and free eccentricitiesshow the minimum rms misfit assuming a pericen-ter at epoch of 230 ◦ , while the dashed lines showthe minimum rms misfit allowing the pericenterposition to float. These plots show clear evidencethat the ringlet’s free eccentricity increased sub-stantially sometime between 2010 and 2012. Figure 8.
Orbital changes in the ringlet. Thetop two panels show the differences between theobserved and predicted ringlet positions assumingthe best-fit pre-2010 model as functions of the pre-dicted radial offset from the best-fit semi-majoraxis. The top panel shows the data obtained before2010, which are scattered around the zero line. Bycontrast, the data obtained after 2010 show largesystematic differences indicating that this model forthe ringlet’s position in no longer adequate. Thelower four panels show the differences between thepredicted and observed positions of the ringlet andthe shelf for the two different models of the ringlet’slocation given in Table 3. Fit 1 is the best-fit over-all model, while Fit 2 assumes ae f = 17 km. Bothmodels show a much tighter dispersion around zero,and in both cases the shelf location is consistentlyabout 37 km interior to the peak ringlet signal. servations obtained before 2010, we find thebest-fit (i.e. minimum rms misfit) solution has ae f (cid:39)
17 km, ae (cid:96) (cid:39) (cid:36) (cid:48) (cid:96) (cid:39) ◦ , whichare consistent with prior results. However, forthe observations made after 2010 this solution isnot the one that gives the minimum rms misfit.These later observations still prefer (cid:36) (cid:48) (cid:96) (cid:39) ◦ ,but now the minimum rms misfit occurs where ae f (cid:39) ae (cid:96) (cid:39)
14 km While the twominima in ae f are fairly broad, the minima in ae (cid:96) are narrower and suggest that the free ec-centricity increased sometime between 2010 and2012.Assuming the minimum rms misfit providesa reasonable estimate of the statistical uncer-tainties in the position measurements, we cantranslate the profiles shown in Figure 7 into χ statistics and then compute relative probabili-ties that can be fit to gaussians in order to es-timate both the best-fit parameters and theiruncertainties. These numbers are provided inTable 3, along with estimates of the ring radiusderived from the mean value of the residuals af-ter removing the variations due to the ringlet’seccentricity. The parameters for the pre-2010data are perfectly consistent with those derivedby Hedman et al. (2010), albeit with slightlylarger error bars. The Post-2010 best-fit so-lution (designated “Fit 1” here) is also signif-icantly different from the Pre-2010 solution.In addition to providing the absolute best-fit solution for the post-2010 data as “Fit 1”,Table 3 also provides parameters for a “Fit 2”where the forced eccentricity ae f is required tohave its Pre-2010 value. This is done becausethe forced eccentricity should only depend onthe average ringlet particle size, and there isno evidence that this changed substantially overthe course of the Cassini mission (see the pre-vious subsection). Note that even in this case ae (cid:96) is significantly higher after 2010. Also notethat the best-fit mean radius of the ringlet for3 Table 3.
Orbital parameters for the ringletCase ae f (km) (cid:36) (cid:48) f (deg) ae (cid:96) (km) (cid:36) (cid:48) (cid:96) (deg) ˙ (cid:36) (cid:48) (cid:96) (deg/day) a (km)Hedman et al. 2010 Model 1 17.0 ± a ± ± ± ± a ± ± a ± ± a ± ± a ± a a ± ± a ± a parameter held fixed in the fit. this model is about 8 km interior to its value forthe best-fit solution.Figure 8 further illustrates these changes inthe ringlet’s orbit properties. The top panelsshow the differences between the observed andexpected positions of the ringlet for the Pre-2010 fit model as functions of the predictedringlet position. The pre-2010 data show afairly random scatter around zero difference,while the post-2010 data show systematic dif-ferences of up to 10 km. The differences forthe two Post-2010 fits are both much tighter,although there are clear trends in subsets of thedata for Fit 2, as is to be expected given thisis not the fit with the minimal rms misfit. Inaddition, we can note that the two models pre-dict very different values for the ringlet’s radialoffsets, with Fit 2 generally predicting the datafall closer to apocenter, which is consistent withFit 2 having the lower value of a .Figure 8 also shows the locations of the shelffor these two fits. These positions have a consid-erably larger scatter because the shelf is a moresubtle feature, but there is not a strong trendin the shelf position relative to the ringlet, con-sistent with the appearance of the profiles inFigure 4. This implies that the shelf consistsof particles with roughly the same eccentricityas the ringlet, but a smaller semi-major axis.At the same time, there are hints of a trendin the shelf position within the data around 0km in Fit 1 and 10 km in Fit 2. This suggeststhat there might be longitudinal variations in the shelf’s structure. Both of these findings aresupported by a more detailed investigation ofselected movie sequences made after 2012. LONGITUDINAL VARIATIONS IN THESTRUCTURE OF THE RINGLET ANDSHELF BETWEEN 2013 AND 2015As mentioned above, most of the data onthis ringlet obtained after 2010 were movie se-quences where the camera took multiple imagesof the ringlet over the course of a timespan com-parable to the orbital period of the ring mate-rial. Detailed examinations of the different im-ages within several of these sequences revealedsubtle variations in the brightness profiles thatappear to reflect longitudinal variations in thestructure of both the ringlet and the shelf.Documenting these brightness variations ischallenging because they are subtle, amount-ing to only 5% of the ringlet’s peak brightness,and so can easily be obscured by slight radialshifts in the ringlet’s position and variationsin the ringlet’s overall brightness due to smallchanges in the observed phase angle. Hence,in order to document the longitudinal varia-tions in the structure of this ringlet, we pro-cess the relevant brightness profiles to createaligned, background-subtracted and normalizedprofiles that can easily be co-added and com-pared. We first align the brightness profiles byinterpolating the brightness data for each pro-file onto a common regular grid of radii rela-tive to the peak location estimated from theasymmetric Lorentzian fit discussed in the pre-4vious section. To improve signal-to-noise, weaverage together all profiles obtained withinpre-determined ranges of co-rotating longitudes,which are computed for each profile assumingthe material had a mean motion consistent witha particle orbiting at a specified semi-majoraxis. A linear background is then removedfrom these combined profiles based on the sig-nal levels at the local minima on either side ofthe ringlet. Finally, the brightness values arenormalized so that the peak brightness of theringlet was unity.These procedures were applied to data fromfive of the movies in Table 1. Specifically,we consider Rev
103 SHRTMOVIE, Rev189 BMOVIE, Rev 206 BMOVIE, Rev 213BMOVIE and Rev 276 HPMONITOR. Thefirst of these is the longest movie obtained priorto 2010 at high phase angles, and so provides auseful baseline for the later observations. Rev189 BMOVIE, Rev 206 BMOVIE and Rev 276HPMONITOR each cover roughly one orbitalperiod of the ringlet, and so provide the clear-est picture of the longitudinal variations in theringlet after 2011. The Rev 213 BMOVIE obser-vation is also included because it covers abouthalf an orbit period and does preserve infor-mation about the longitudinal structure of theringlet/shelf. The other movies in Table 1 werefound to be too short to provide clear infor-mation about the longitudinal structure of theringlet.Figure 9 shows the resulting normalized,background-subtracted profiles derived fromthese five observations, along with the differ-ences between each profile and the average ofall the profiles to better show the variations.Staring with the Rev 103 SHRTMOVIE, we seethat all the normalized profiles have very sim-ilar shapes, with residual differences less than0.02 the peak brightness. This demonstrates “Rev” designates each orbit of Cassini around Saturn Figure 9.
Normalized, background-subtractedprofiles derived from five movie sequences. For eachmovie, the profiles shown in different colors corre-spond to different co-rotating longitudes assumingthe mean motion of 736.388 ◦ /day (correspondingto semi-major axis of 119,930 km) and an epochtime of 2011-341T00:00:00 UTC. For each movie,the top panel shows the actual profiles, while thelower panel shows the differences between each pro-file and the average of all profiles. The data plottedhere is provided in Table 5. Figure 10.
Comparisons of the residual bright-ness variations in the ringlet as functions of co-rotating longitude for the Rev 189, 206 and 213BMOVIE sequences, shown as diamonds, stars andtriangles, respectively. Each panel shows the resid-ual brightness variations at a particular radiusversus co-rotating longitude assuming a rotationrate of 736.388 ◦ /day and an epoch time of 2011-341T00:00:00, so that the color code is the sameas for Figure 9. Note that the data in each panelare repeated twice for clarity. The brightness vari-ations in the panels within ±
10 km of the peak arewell aligned, but those further away from the peakare not well aligned for this particular rotation rate.The data plotted here is provided in Table 5.
Figure 11.
Comparisons of the residual brightnessvariations in the ringlet as functions of co-rotatinglongitude for the Rev 189, 206 and 213 BMOVIEsequences, shown as diamonds, stars and trian-gles, respectively. Each panel shows the residualbrightness variations at a particular radius versusco-rotating longitude. Note that the data are re-peated twice for clarity. For each movie, the profilesshown in different colors correspond to differentco-rotating longitudes assuming the mean motionprovided and an epoch time of 2011-341T00:00:00UTC. Note that the brightness variations in theshelf are well aligned assuming a rotation rate ofaround 736.388 ◦ /day, while the variations aroundthe ringlet peak are better fit by a rate around736.758 ◦ /day, and the variations exterior to thepeak are aligned with a rate of 736.203 ◦ /day. ◦ versus less than 120 ◦ ), so one couldargue that the ringlet is just more homogeneousin higher-phase observation geometry. However,we regard this explanation as unlikely, and willinstead argue that whatever event disturbed theringlet between 2010 and 2012 produced a seriesof localized disturbances in the ring that rotatedaround the planet at different rates and there-fore gradually smeared out around the ring,causing the longitudinal variations to dissipatebetween 2013 and 2017.A closer look at the BMOVIE sequences re-veals three distinct radial zones in the bright-ness variations. The innermost zone falls be-tween -50 km and -20 km from the peak lo-cation, and corresponds to the shelf. Indeed,it appears that the profiles with the minimumbrightness in this region basically do not have ashelf, while the highest brightness profiles havea shelf comparable to that seen at all longitudesin 2017. Next, there are the brightness varia-tions within about 15 km of the peak location.These brightness variations go to zero at thepeak because of how the profiles are normalized,and indicate that the width of the ringlet varieslongitudinally. Finally, around +20 km out-side the peak there are variations in the ringletbrightness that are out of phase with the varia-tions closer to the peak, which cause an inflec-tion in the normalized brightness profiles at cer-tain longitudes. This last feature is clear in boththe Rev 189 and Rev 206 BMOVIE data, but is not clearly visible as a distinct feature in theRev 213 BMOVIE data, most likely because inthis particular observation the variations withinand outside the peak happen to be aligned witheach other.It is important to note that these three bright-ness variations do not appear to be rotatingaround the planet at the same rate. All thedata shown in Figure 9 uses the same color-code, and so the same colored curves in the dif-ferent panels are at the same co-rotating longi-tudes assuming a mean motion of 736.388 ◦ /day(this rate is the expected mean motion for ma-terial at a semi-major axis of 119930 km). Inthis frame, the brightness variations around thepeak are well aligned between the various ob-servations, with the green/cyan-colored profilesbeing high and the red/pink-colored profiles be-ing low. However, if we look at the variationsin the shelf and outside the main peak, we seethey are not aligned in the same way, indicatingthat the brightness maxima in these regions arenot traveling around the planet at exactly thisrate.These variations in the pattern alignment canalso be seen in Figure 10, which shows the samedata in a different form. Here we show thebrightness variations at particular radii as func-tions of co-rotating longitude (using the samecolor code as in Figure 9), but data from thedifferent observations are shown with differentsymbols. In general, the longitudinal bright-ness variations are quasi-sinusoidal. Also, forradii within ±
15 km of the peak the varia-tions from the different observations are prettywell aligned. However, this is not the case forradii further from the peak. At +20 km, thereare clear offsets between the three observations,and for radii between −
20 and −
50 km in theshelf, the 189 and 206 BMOVIE data show peakbrightnesses at points roughly 180 ◦ apart.After some experimentation, we found thatthe brightness variations in the shelf are bet-7ter aligned if we assume a rate of 736.758 ◦ /day,while the variations outside the peak are wellaligned assuming a rate of 736.203 ◦ /day. Theserates correspond to semi-major axes of 119,890km and 119,950 km, or about -40 km and +20km from the semi-major axis that aligns thevariations within the peak. These numbers aretherefore consistent with the observed radial po-sitions of these features. Figure 11 shows thebrightness variations aligned using these differ-ent rotation rates. The variations at +20 km areclearly much better aligned using the slower ro-tation rate, and the variations around −
30 and −
40 km are also reasonably well aligned withthe faster rate. Note the situation at -20 km isless clear, most likely because the longitudinalvariations in the ringlet and the shelf are inter-fering with each other. Still, the data indicatethe ringlet’s brightness variations involve mate-rial with a range of semi-major axes and meanmotions, which is consistent with the idea thatthese variations sheared out between 2015 and2017. DISCUSSIONThe above analyses provide clear evidencethat something happened to this dusty ringletsometime between 2010 and 2012 that bothproduced a shelf of material interior the ringlet,and altered the average orbital properties ofthe particles in the ringlet. Determining whatcould have caused these changes is challeng-ing because we do not know exactly whenthese changes were initiated. Since the Rev172 MOONLETCD observation in 2012 alreadyshows the ringlet’s position systematically de-viates from the pre-2010 model, we know thatwhatever disturbed the ring started before this.Unfortunately, Cassini was orbiting close toSaturn’s ring-plane for most of this time pe-riod, and so there are virtually no images thatcould directly document what happened to thering between 2010 and 2012. Instead, we have to use the available data to indirectly constrainwhat might have happened to the ringlet.When evaluating possible mechanisms forchanging the shape and orbit of this ringlet,it is important to remember that the particlesin the ringlet and the shelf are very small. Themagnitude of the heliotropic eccentricity before2010 implies that the average effective particleradius in the ringlet is around 20 µ m (Hed-man et al. 2010), which is consistent with theobserved strength of the Chirstainsen featurein occultations of this ringlet (Hedman et al.2011). The lack of an obvious change in theringlet’s phase function over the course of theCassini mission suggests that the changes inthe ringlet’s orbital properties did not alter theringlet’s particle size distribution much. Fur-thermore, the shelf is about 10% the ringlet’speak brightness at multiple phase angles (seeFigure 4), so there does not also appear to be amajor difference in the particle size distributionbetween the shelf and the ringlet after 2012.Hence we may assume here that the typicalparticle in the ringlet and the shelf has a radiusof roughly 20 µ m.In principle, there are a wide variety of pro-cesses that could perturb the orbits of thesmall particles found in this ringlet, includingsolar radiation pressure, Poynting-Robertsondrag, changes in the local electromagnetic fieldsand/or plasma environment, and collisions withinterplanetary debris. However, in practice, col-lisions with interplanetary debris is the optionthat seems most consistent with the longitudi-nal brightness variations seen between 2012 and2016 and the orbital properties of the particlesin the shelf.The longitudinal brightness variations indi-cate that this process did not affect all parts ofthe ringlet equally. This argues against the ideathat these changes were produced by a global,uniform process, and instead suggests that theperturbation was a discrete event that was suf-8ficiently localized in time and space to affectcertain longitudes more than others. This dis-favors phenomena like solar radiation pressureand Poynting-Robertson drag because they de-pend on solar flux, which is unlikely to vary onsufficiently short timescales to only affect partsof the ringlet. Also, while changes in the globalconfiguration of the electric and magnetic fieldshave been observed (Andriopoulou et al. 2014;Provan et al. 2014), such large-scale variationsmight have trouble producing sufficiently localperturbations within this ringlet.Meanwhile, even tough both the mean ec-centricity and semi-major axis of the ringletchanged during this event, the largest orbitalchanges were experienced by the material thatgenerated the shelf. Both the radial locationsof the shelf and the rotation rates of its lon-gitudinal asymmetries indicate that the parti-cles in the shelf are on orbits with semi-majoraxes around 40 km closer to Saturn than theringlet. If we make the reasonable assumptionthat the shelf material originally came from theringlet, then this finding has clear implicationsregarding the direction of the forces that actedon these particles.In order for the particles’ semi-major axes toshift inwards, the particles need to lose orbitalenergy, so they need to experience a force thatopposed their orbital motion. This providesfurther evidence against the perturbation beingdue to sudden localized shifts in the planet’smagnetic field since magnetic fields do no work.This also suggests that the disturbance was notdue to a sudden change in the plasma environ-ment. While one could imagine some event sud-denly releasing a large amount of plasma intoa small part of the ringlet, that plasma wouldnaturally be picked up by the magnetic fieldand so move around the planet at Saturn’s rota-tion rate, which is faster than the ring-particles’Keplerian orbital rate. Momentum exchangebetween such a plasma and the ring particles would therefore cause the ringlets to move out-wards rather than inwards.Given the above challenges associated with at-tributing the changes in the Charming Ringlet’sstructure and orbit to interactions with theplanet’s electromagnetic field, the local plasmaenvironment and solar radiation pressure, weare left with the possibility that the ringlet wasdisturbed by collisions with interplanetary de-bris. So long as the debris only passed through asmall part of the ringlet and approached the ringfrom an appropriate direction, such collisionscould easily produce the localized disturbancesand the orbital energy reduction required by theobservations.In principle, these collisions could have in-volved a small number larger objects thatcrashed into source bodies within the ringletand released new material, or a larger numberof tiny particles that perturbed the orbits of thedust-sized particles within the ringlet. However,given the overall brightness of the ringlet didnot change much between 2010 and 2012, thelatter option appears to be more likely. Fur-thermore, there are precedents for attributingstructural changes in dusty rings to collisionswith debris clouds. Collisions with debris fromShoemaker-Levy 9 appears to have created ver-tical spiral patterns in Jupiter’s dusty rings(Showalter et al. 2011), and similar collisionswith cometary debris is a plausible explanationfor spiral patterns in Saturn’s D ring (Hed-man et al. 2011, 2015; Hedman and Showalter2016). Furthermore, one of these D-ring pat-terns was likely formed in late 2011, so thereis even evidence that a debris cloud could havebeen passing through the rings at roughly theright time to initiate the observed changes inthe Charming Ringlet.However, one potential concern with this sce-nario emerges when we estimate the magnitudeof the impulses that needed to be imparted tothe particles in the Charming Ringlet in order9to produce the shelf. The standard orbital per-turbation equations for near-circular orbits saythat the semi-major axis of a particle evolves ata rate (Burns 1976; Hedman 2018): ∂a∂t = 2 na F p F G (8)where F p is the azimuthal component of the per-turbing force, n = (cid:112) GM/a (cid:39) . ◦ /dayis the particle’s mean motion, and F G = GM m p /a = n am p is the planet’s central grav-itational force on the particle, where G is thegravitational constant, M is the planet’s massand m p is the particle’s mass. Thus the particlewill undergo a semi-major axis change δa whenit receives an azimuthal impulse given by thefollowing formula: F p δt = F G na δa = 12 nm p δa (9)Assuming the particles in the shelf are com-posed primarily of water ice with negligibleporosity and have an average radius of 20 µ m,we may conclude that the typical mass of theseparticles is around 3 × − kg, and so therequired impulse to produce a semi-major axisshift of 40 km is 10 − kg m/s.This impulse can be delivered by a collisionwith a small piece interplanetary debris. So longas the impactor is much smaller than the ringparticle, the impulse delivered to the ring parti-cle will be comparable to the momentum of theimpactor m i v i , where m i is the mass of the im-pactor and v i is the impact speed. Furthermore,for any collision between a ring particle and in-terplanetary debris, the impact speed will becomparable to the ring-particle’s orbit speed, sowe can estimate that v i ∼
20 km/s. In this case,the required momentum could be delivered bya debris particle with a mass of m i (cid:39) × − kg, which corresponds to a solid ice grain witha radius of around 1 micron. Note that this in-ferred impactor mass is consistent with the priorassumption that m i << m p . The problem with this particular scenario isthat the collision may not just change the par-ticle’s orbit, but could also cause the particle tobreak apart. The standard metric for whether acollision disrupts a body is the ratio of the col-lision energy to the mass of the target particle Q , which is this case can be well approximatedas: Q (cid:39) m i m p v i (cid:39) δaa v i (10)Again assuming impact speeds of around 20km/s, we find that the collisions required toshift the particle’s semi-major axes by 40 kmhave Q (cid:39) × J/kg or 3 × erg/g. This isclose to the expected threshold energy for catas-trophic disruption for solid ice grains.Extrapolating the disruption energy thresh-olds computed by Benz and Asphaug (1999)down to 20 µ m yields a critical disruptionthreshold Q ∗ D ∼ × erg/g for icy objects col-liding at speeds of 3 km/s. Scaling this thresh-old by v . i , as recommended by Krivov et al.(2018) based on work by Stewart and Lein-hardt (2009) increases this threshold slightlyto around 4 × erg/g. This number is alsoroughly consistent with the expected fragmen-tation threshold for ice-rich dust derived byBorkowski and Dwek (1995), which is estimatedto be roughly the ratio of particle’s dynamic ten-sile strength to its mass density. Assuming massdensities of around 1 g/cm and dynamic tensilestrengths of around 20 MPa (Lange and Ahrens1983), one also obtains disruption thresholdsaround 2 × erg/g. However, it is importantto note that laboratory experiments collidingcentimeter-scale ice blocks generally yield or-ders of magnitude lower disruption thresholdsthan these numbers (see Stewart and Leinhardt2009). While this may in part be because theseexperiments are at much lower speeds and of-ten involve polycrystalline ice, we certainly donot have sufficient information to determinewhether the collisions needed to move material0into the shelf would actually destroy the parti-cles or not. Future constraints on the strengthof tiny ice grains therefore could either supportor refute this particular scenario.Despite the challenges associated with any ofthe above mechanisms for producing the ob-served shifts in the structure of the ringlet, fur-ther examination of the total required momen-tum input into the ring yields evidence for aconnection between the changes in the ringletand the nearly contemporaneous appearance ofspiral patterns in D ring mentioned above.Consider a small part of the shelf that contains N particles per unit area, the total impulse perunit area needed to move all those particles intothe shelf is J = N F p δt = 12 N m p nδa (11)For a tenuous ring consisting of particles ofsize r p , the optical depth is τ = N πr p , whilethe mass of such particles is m p = (4 π/ ρ p r p ,where ρ p is the the particle’s mass density. Wecan therefore re-write this impulse density as: J = N F p δt = 23 τ ρ p r p nδa (12)This expression can be generalized to a ringletwith a range of particle sizes by simply re-interpreting r p as the effective average size of therelevant particles. For this particular ringlet, wemay assume r p (cid:39) µ m, and since all the par-ticles are composed primarily of water ice, wemay also say ρ p (cid:39) . Furthermore, sincethe shelf is about 10% as bright as the ringlet,which has a peak optical depth of around 0.001,we may also assume τ (cid:39) − . This yields aimpulse density of around 8 × − kg/m/s.This number is interesting because it is com-parable to the impulse density required to gen-erate the spiral pattern in the D ring in late2011, which Hedman and Showalter (2016) es-timated was around 40 × − kg/m/s. Further-more, if interplanetary material was responsible for both structures, the impulse density into theD ring would naturally be higher due to its lo-cation deeper in Saturn’s gravitational poten-tial well. These numbers therefore provide ad-ditional support for the idea that interplanetarymaterial might have perturbed both these dustyrings.Finally, we can further explore this potentialconnection between the two rings by consider-ing what the longitudinal brightness variationsin the ringlet would have looked like when theD ring was being disturbed. The date of theD-ring disturbance can be accurately estimatedbased on the predictable evolution of the spiralpattern that it generated, and turns out to bewithin a week of December 7, 2011 (Hedmanand Showalter 2016). Figure 11 uses this as theepoch time to define the co-rotating longitudes,so this plot shows the locations of the bright-ness variations in the various regions at thatparticular time. While the absolute longitudesof these features change rapidly as the parti-cles orbit the planet, the relative motion of thebrightness variations is much slower, and so thisgraph should provide a good sense of what theradial distribution of the brightness of the fea-tures would be at this particular time. Interest-ingly, the brightness maxima in the shelf interiorto the ringlet are roughly aligned with bright-ness minima in the ringlet itself. This could beconsistent with roughly 10% of the material inthe ringlet being thrown inwards by about 40km. However, we caution that this could alsobe a chance alignment. For example, near thestart of 2011 the brightness asymmetries at allradii are nearly aligned with each other.At this point, we cannot unambiguously provethat the disturbances to the Charming Ringletwere caused by the same event that produceda spiral pattern in the D ring. However, if thesame event was responsible for disturbing boththese rings, then that event may have effectedthe structures and orbits of other dusty rings1in the Saturn system. Detailed studies of thosedusty systems should therefore be a productiveavenue for future work. ACKNOWLEDGEMENTSWe thank the Cassini Project and ImagingTeam for acquiring the data used in this analy-sis, as well as the Planetary Data System Ring-Moon System Node for making these data easilyavailable. This work was supported in part bya NASA Cassini Data Analysis Program GrantNNX15AQ67G.REFERENCES Acton, C. H., 1996. Ancillary data services ofNASA’s Navigation and Ancillary InformationFacility. Planet. Space Sci. 44, 65–70.Andriopoulou, M., Roussos, E., Krupp, N.,Paranicas, C., Thomsen, M., Krimigis, S.,Dougherty, M. K., Glassmeier, K.-H., 2014.Spatial and temporal dependence of theconvective electric field in Saturn’s innermagnetosphere. Icarus 229, 57–70.Benz, W., Asphaug, E., 1999. CatastrophicDisruptions Revisited. Icarus 142 (1), 5–20.Borkowski, K. J., Dwek, E., 1995. TheFragmentation and Vaporization of Dust inGrain-Grain Collisions. ApJ 454, 254.Burns, J. A., 1976. Elementary derivation of theperturbation equations of celestial mechanics.American Journal of Physics 44, 944–949.Chancia, R. O., Hedman, M. M., Cowley,S. W. H., Provan, G., Ye, S. Y., 2019. Seasonalstructures in Saturn’s dusty Roche Divisioncorrespond to periodicities of the planet’smagnetosphere. Icarus 330, 230–255.French, R. G., Nicholson, P. D., McGhee-French,C. A., Lonergan, K., Sepersky, T., Hedman,M. M., Marouf, E. A., Colwell, J. E., 2016.Noncircular features in Saturn’s rings III: TheCassini Division. Icarus 274, 131–162.French, R. S., Showalter, M. R., Sfair, R.,Arg¨uelles, C. A., Pajuelo, M., Becerra, P.,Hedman, M. M., Nicholson, P. D., 2012. Thebrightening of Saturn’s F ring. Icarus 219 (1),181–193.Hedman, M. M., 2018. An Introduction toPlanetary Ring Dynamics. In: Tiscareno, M. S.,Murray, C. D. (Eds.), Planetary Ring Systems.Properties, Structure, and Evolution, pp. 30–48. Hedman, M. M., 2019. Bright clumps in the D68ringlet near the end of the Cassini Mission.Icarus 323, 62–75.Hedman, M. M., Burns, J. A., Evans, M. W.,Tiscareno, M. S., Porco, C. C., 2011. Saturn’scuriously corrugated C ring. Science 332,708–712.Hedman, M. M., Burns, J. A., Hamilton, D. P.,Showalter, M. R., 2013. Of horseshoes andheliotropes: Dynamics of dust in the EnckeGap. Icarus 223 (1), 252–276.Hedman, M. M., Burns, J. A., Showalter, M. R.,2015. Corrugations and eccentric spirals inSaturn’s D ring: New insights into whathappened at Saturn in 1983. Icarus 248,137–161.Hedman, M. M., Burt, J. A., Burns, J. A.,Tiscareno, M. S., 2010. The shape anddynamics of a heliotropic dusty ringlet in theCassini Division. Icarus 210 (1), 284–297.Hedman, M. M., Nicholson, P. D., Showalter,M. R., Brown, R. H., Buratti, B. J., Clark,R. N., Baines, K., Sotin, C., 2011. TheChristiansen Effect in Saturn’s narrow dustyrings and the spectral identification of clumpsin the F ring. Icarus 215 (2), 695–711.Hedman, M. M., Showalter, M. R., 2016. A newpattern in Saturn’s D ring created in late 2011.Icarus 279, 155–165.Hedman, M. M., Stark, C. C., 2015. Saturn’s Gand D Rings Provide Nearly CompleteMeasured Scattering Phase Functions of NearbyDebris Disks. ApJ 811 (1), 67.Hor´anyi, M., Burns, J. A., Hedman, M. M., Jones,G. H., Kempf, S., 2009.
Diffuse Rings , pp. 511. Table 4.
Ringlet Parameters derived from the Cassini Images. See supplemental information for a machine-readable version of this entire table.
Image Name Image Midtime Observed Sub-Solar Emission Phase NEW Peak Radius Shelf Radius(seconds) a Longitude (deg) b Longitude (deg) b Angle (deg) Angle (deg) (km) (km) c (km) c N1477740094 152319435.4 79.9 164.6 99.0 145.9 0.02809 119946.6 -1.0N1537949486 212528441.0 194.6 -169.4 57.8 113.3 0.00874 119951.9 -1.0N1540572432 215151370.6 146.4 -168.3 105.0 133.4 0.01345 119957.5 -1.0N1540573152 215152090.6 146.4 -168.3 105.2 133.0 0.01351 119958.7 -1.0N1540598112 215177050.5 132.1 -168.3 113.4 124.9 0.01093 119955.2 -1.0N1546193738 220772641.1 221.4 -166.1 144.0 76.4 0.00273 119953.1 -1.0N1547768529 222347422.4 135.3 -165.5 87.1 41.8 0.00466 119952.7 -1.0 a Ephemeris time in seconds past J2000 epoch b Longitudes measured in an intertial frame in Saturn’s equator plane relative to the plane’s ascending nodeon the J2000 coordinate system. c Positions after correcting profiles to match the predicted location of the Laplace Gap’s inner edge. Shelfradius is set to -1 km prior to 2011.
Table 5.
Normalized, aligned ringlet profiles from selected movie sequences. See supplemental informationfor a machine-readable version of this entire table
Distance from Co-rotating Normalized brightness for ObservationPeak (km) Longitude a (deg) Rev 103 SHRTMOVIE Rev 189 BMOVIE Rev 206 BMOVIE Rev 213 BMOVIE Rev 276 HPMONITOR-75.0 -170.0 0.0006 0.0120 0.0713 -NaN 0.0473-74.0 -170.0 0.0009 0.0098 0.0633 -NaN 0.0389-73.0 -170.0 0.0013 0.0075 0.0554 -NaN 0.0306-72.0 -170.0 0.0017 0.0054 0.0476 -NaN 0.0222-71.0 -170.0 0.0023 0.0035 0.0409 -NaN 0.0139-70.0 -170.0 0.0030 0.0017 0.0349 -NaN 0.0093-69.0 -170.0 0.0038 0.0000 0.0290 -NaN 0.0070-68.0 -170.0 0.0048 -0.0015 0.0242 -NaN 0.0047-67.0 -170.0 0.0059 -0.0031 0.0200 -NaN 0.0023-66.0 -170.0 0.0070 -0.0044 0.0157 -NaN 0.0000 a Assuming a = 119 , km and an epoch time of 2011-341T00:00:00Krivov, A. V., Ide, A., L¨ohne, T., Johansen, A.,Blum, J., 2018. Debris disc constraints onplanetesimal formation. MNRAS 474 (2),2564–2575.Lange, M. A., Ahrens, T. J., 1983. The dynamictensile strength of ice and ice silicate mixtures.J. Geophys. Res. 88 (B2), 1197–1208.Markwardt, C., 2012. MPFIT: Robust non-linearleast squares curve fitting.Porco, C. C., 2006. Rings of Saturn (R/2006 S 1,R/2006 S 2, R/2006 S 3, R/2006 S 4). IAUCircular 8759, 1.Porco, C. C., West, R. A., Squyres, S., McEwen,A., Thomas, P., Murray, C. D., Del Genio, A.,Ingersoll, A. P., Johnson, T. V., Neukum, G.,Veverka, J., Dones, L., Brahic, A., Burns, J. A.,Haemmerle, V., Knowles, B., Dawson, D.,Roatsch, T., Beurle, K., Owen, W., 2004.Cassini Imaging Science: InstrumentCharacteristics And Anticipated ScientificInvestigations At Saturn. Space ScienceReviews 115 (1-4), 363–497. Provan, G., Lamy, L., Cowley, S. W. H.,Dougherty, M. K., 2014. Planetary periodoscillations in Saturn’s magnetosphere:Comparison of magnetic oscillations and SKRmodulations in the postequinox interval.Journal of Geophysical Research (SpacePhysics) 119, 7380–7401.Showalter, M. R., Hedman, M. M., Burns, J. A.,2011. The impact of Comet Shoemaker-Levy 9sends ripples through the rings of Jupiter.Science 332, 711–.Stancik, A., Brauns, E., 2008. A simpleasymmetric lineshape for fitting infraredabsorption spectra. VibrationalSpectroscopy 47, 66–69.Stewart, S. T., Leinhardt, Z. M., 2009.Velocity-Dependent Catastrophic DisruptionCriteria for Planetesimals. ApJL 691 (2),L133–L137.3