A Librational Model for the Propeller Blériot in the Saturnian Ring System
AA Librational Model for the Propeller Bl´eriot in theSaturnian Ring System
M. Seiler *, M. Seiß, H. Hoffmann and F. Spahn
Theoretical Physics Group, Institute of Physics and Astronomy, University of Potsdam* [email protected]
October 11, 2018
Abstract
The reconstruction of the orbital evolution of the propeller structure Bl´eriot orbiting inSaturn’s A ring from recurrent observations in Cassini ISS images yielded a considerable offsetmotion from the expected Keplerian orbit (Tiscareno et al., 2010). This offset motion can becomposed by three sinusoidal harmonics with amplitudes and periods of 1845, 152, 58 km and11.1, 3.7 and 2.2 years, respectively (Sremˇcevi´c et al., 2014). In this paper we present resultsfrom N-Body simulations, where we integrated the orbital evolution of a moonlet, which isplaced at the radial position of Bl´eriot under the gravitational action of the Saturnian satellites.Our simulations yield, that especially the gravitational interactions with Prometheus, Pandoraand Mimas is forcing the moonlet to librate with the right frequencies, but the libration-amplitudes are far too small to explain the observations. Thus, further mechanisms are neededto explain the amplitudes of the forced librations – e.g. moonlet-ring interactions. Here, wedevelop a model, where the moonlet is allowed to be slightly displaced with respect to itscreated gaps, resulting in a breaking point-symmetry and in a repulsive force. As a result, theevolution of the moonlet’s longitude can be described by a harmonic oscillator. In the presenceof external forcings by the outer moons, the libration amplitude gets the more amplified, themore the forcing frequency gets close to the eigenfrequency of the disturbed propeller oscillator.Applying our model to Bl´eriot, it is possible to reproduce a libration period of 13 years withan amplitude of about 2000 km.
One of the most puzzling discoveries of the spacecraft Cassini – orbiting around Saturn since itsarrival in June 2004 – has been the observation of disk-embedded moons orbiting within Saturn’smain rings (Tiscareno et al., 2006; Spahn and Schmidt, 2006; Sremˇcevi´c et al., 2007; Tiscarenoet al., 2008). The typical density variations downstream the embedded moonlet’s orbit are createdby the gravitational interaction of a small sub-kilometer-sized object (called moonlet ) with thesurrounding ring material and reminds of a two-bladed propeller giving the structure its name(Spahn and Sremˇcevi´c, 2000; Sremˇcevi´c et al., 2002). Meanwhile more than 150 propeller structureshave been detected within the A and B ring. The largest propeller structure which is a fewthousands km in azimuth is called Bl´eriot and is caused by a moonlet with a diameter of around800 meters. However, the moonlet is still too small to allow its direct observation by the Camerasaboard the spacecraft Cassini.Nevertheless, the propeller structure permits the observation and orbital tracking of the largestmoonlets. The reconstruction of the orbital evolution of Bl´eriot revealed an offset motion withrespect to a Keplerian motion of considerable amplitude (Tiscareno et al., 2010). Tiscareno et al.(2010) found, that one possibility to describe the longitudinal excess motion of Bl´eriot is a harmonicfunction of 300 km and a period of 3.6 years.It is still an ongoing debate, whether Bl´erioti) is librating due to gravitational interactions with the other moons in the Saturnian system(resonances), 1 a r X i v : . [ a s t r o - ph . E P ] J a n i) is suffering from stochastical interactions with the surrounding ring material (Rein and Pa-paloizou, 2010; Tiscareno, 2013),iii) is in a ’frog resonance’ (Pan and Chiang, 2010, 2012),iv) or if it is even perturbed by the combined effects of all the above.Considering hypothesis (ii), several attempts, have been started in order to explain this excessmotion (Pan and Chiang, 2010, 2012; Tiscareno, 2013), where Rein and Papaloizou (2010) deliv-ered an explanation considering a stochastic migration and Pan et al. (2012) showed in N-Bodysimulations, that such a mechanism could generate an maximal excess motion of 300 km over atime of 4 years.Another mechanism (iii) has been introduced with a so-called ”frog resonance” model, wherePan and Chiang (2010) consider a resonance between the moonlet and its created gap edges, beingmodeled as two co-orbital point masses. This interaction is causing the moonlet to librate within itsgap. However, the approximation of the gaps as coorbital point masses of almost half of the massof the moonlet is a strong simplification, which does not reflect the true structure of a propeller.Meanwhile newer orbital fits of the orbital evolution of Bl´eriot have been performed, trackingand reconstructing the orbit from a larger set of ISS images over a larger time span. The mostrecent investigation by Sremˇcevi´c et al. (2014) yielded, that Bl´eriot’s orbital excess motion canbe fitted astonishingly well by three harmonic functions with amplitudes and periods of 1845, 152and 58 km and 11.1, 3.7 and 2.2 years, respectively, where the standard deviation of the remainingresidual is about 17 km (see also Spahn et al., 2017).The harmonic behavior of the excess motion might suggest that resonant interactions with theother Saturnian moons serve as a reason for the excess motion.Such harmonic systematic deviations from the expected Keplerian orbit are a known phe-nomenon in the Saturnian system. Some of the outer moons are librating systematically, like themoon Enceladus which is in a 2:1 ILR with Dione and the moon Atlas, which is perturbed by the54:53 CER and 54:53 ILR by Prometheus (Goldreich, 1965; Goldreich and Rappaport, 2003a,b;Spitale et al., 2006; Cooper et al., 2015).Following these examples, we perform simulations where the moonlet is perturbed by the moonsof Saturn to characterize the orbital motion of Bl´eriot. It will turn out, that this approach canexplain the observed frequencies, but not the large libration amplitudes. Thus, we propose amodel of ring-moonlet interactions which is capable to explain the amplification of the perturbedexcess motion to observable excursions. Favoring hypothesises (i) and (iv) outer gravitational nearresonant excitations and ring-moonlet interactions are needed to explain the observations.The paper is organized as follows: In section 2 we present the N-Body integrations and theirresults. In section 3 we introduce our model of the moonlet-gap interaction and give a connectionto the azimuthal motion in section 3.1, and then apply our model to the moonlet Bl´eriot. Finally,we will conclude and discuss our results in section 4. For the numerical integrations we consider the gravity of 15 Saturnian moons of masses m i andthe oblate (up to J ) Saturn of mass m c and radius R c determining the dynamics of the moonlet¨ (cid:126)r = ∇ Gm c r − ∞ (cid:88) j =2 J j (cid:18) R c r (cid:19) j P j (cos ϑ ) − N (cid:88) i Gm i (cid:18) | (cid:126)r i − (cid:126)r | − (cid:126)r i · (cid:126)rr i (cid:19) . (1)The considered moons are: Atlas, Daphnis, Dione, Enceladus, Epimetheus, Hyperion, Iapetus,Janus, Mimas, Pan, Pandora, Prometheus, Rhea, Tethys and Titan, where the initial values forthe 15 moons were taken from the SPICE kernels sat375.bsp and sat378.bsp at the initial time2000-001T12:00:00.000.Next, we apply our N-Body integration routine to the propeller structure Bl´eriot, which wemodel as a test particle, placed in the Saturnian equatorial plane ( i = 0 deg) on a circular orbit2 . e + 01 − . e + 01 − . e + 01 − . e + 01 − . e + 01 − . e + 01 − . e + 000 . e + 002000 2005 2010 2015 2020 2025 2030 ∆ λ [ k m ] Time [years] − . − . − . − . . . . . . ∆ a [ k m ] Time [years] . . . . . . .
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50 1 .
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50 2 . ∆ ˆ λ [ k m ] Frequency [1/years] . . . . . . . .
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Figure 1:
Evolution of the orbital elements of a test particle, placed at the orbital position of Bl´eriot ( a =134912 .
125 km, λ = 259 .
45 deg, e = 0 and i = 0 deg). Top Left: Mean longitude residual for the time spanfrom 2000 to 2030, where a linear trend with mean motion of n av = 616 . − has been subtracted.Top Right: Residual in semi-major axis for the same time span, with a av = 134912 .
24 km. Bottom Left: Fourierspectrum of the mean longitude residual of Bl´eriot from a 300 Earth years simulation run. The simulation data hasbeen divided into 4 frames of 148 Earth years length and an overlay of 70%. It is clearly visible, that the Fourierspectrum is not stationary and thus its periods and amplitudes are changing with time. Bottom Right: Detailedview of Fourier spectrum in the frequency range 0 to 0.6 years − of Bl´eriot from the left panel. As already visiblefrom the top left panel, the half year libration is the most dominant oscillation, being a result of the 14:13 resonancewith Pandora. More libration periods can be found and are listed in Table 1. ( e = 0) at the expected orbital position (initial semi-major axis a = 134912 .
125 km and meanlongitude λ = 259 .
45 deg, private communication Sremˇcevi´c, 2013, extrapolation of results fromISS images for the initial time given above ).The numerical deviation of the mean longitude and the semi-major axis of Bl´eriot over a timespan of 30 years are shown in the upper panels of Figure 1.Analyzing the frequency spectrum (see in the lower panels) a clear librational behavior withdifferent frequencies and amplitudes is visible in the residuals of the mean longitude and semi-major axis. The gravity of the acting 15 moons induce a mean eccentricity of e av = 2 . · − andinclination of i av = 7 . · − deg on the test moonlet. f [years − ] T [years] ∆ˆ λ [km] ∆ˆ n [m s − ] ∆ˆ a [m] Resonance1.7 0.6 4.6 1.5 · − · − · − · − Libration periods and their amplitudes in the mean longitude estimated from the Fourier spectral analysisof the simulation data. Additionally, the amplitudes in mean motion, and semi-major axis and the correspondingresonance are given. Additionally to the 14:13 CER of Pandora also three-body-resonances between Mimas, Pandoraand Bl´eriot seem to play an important role. . . . . . . . .
00 0 .
50 1 .
00 1 .
50 2 . ∆ ˆ λ [ k m ] Frequency [1/years] . . . . . . . . . .
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00 1 .
50 2 . ∆ ˆ λ [ k m ] Frequency [1/years]
Figure 2:
Fourier spectral analyses to find the dominating moons perturbing the orbit of Bl´eriot. Left: Amplitudespectrum of the mean longitude considering only Pandora and Mimas. Mimas and Pandora are in a 3:2 resonance andadditionally form a 3:16:13 three body resonance with Bl´eriot. Right: Spectrum, considering Prometheus, Pandora,Mimas, Tethys and Titan. The gravity of Prometheus, Titan and Tethys is influencing the orbital evolution of thecomplete system. Additionally, the gravity of Prometheus results in a non-stationary signal in the Fourier spectrum.Thus, the libration frequencies are changing due to the interaction between Pandora and Prometheus.
The amount of moons has been decreased systematically in order to identify the ones, which causeconsiderable resonant perturbations on Bl´eriot. It turns out, that the satellites Pandora and Mimasare dominating the resonant behavior, resulting in four characteristic peaks in the Fourier spectrumof the mean longitude residual (see Figure 2, left panel).The most dominating influence is clearly found in the 14:13 corotation-eccentricity resonance(short CER) of Pandora, which is forcing Bl´eriot to librate with a period of 0.6 years with anamplitude of 5 km. A complete list of the important periods, amplitudes and resonances, causingthe librations can be found in Table 1.The presence of Mimas is resulting in a three-body resonance between Mimas, Pandora andBl´eriot. Mimas and Pandora are known to be in a 3:2 resonance, resulting in a libration periodof 1.8 years. Additionally, Pandora is perturbing the orbital evolution of Bl´eriot with its 14:13resonance. Considering the resonant arguments (Murray and Dermott, 1999) of both resonances(3:2 and 14:13) one can construct related three-body resonances with libration periods of 0.6, 0.8,2.5 and 14.3 years (compare with Figure 2 and Table 1) by subtracting the corresponding resonantarguments: ϕ , = 3 λ Mim − λ P nd + 13 λ Ble (2) ϕ , = 3 λ Mim − λ P nd + 13 λ Ble − (cid:36) Mim + (cid:36) P nd (3) ϕ , = 3 λ Mim − λ P nd + 13 λ Ble − (cid:36) P nd + (cid:36) Ble (4) ϕ , = 3 λ Mim − λ P nd + 13 λ Ble − (cid:36) Mim + (cid:36) Ble . (5)Adding Prometheus, Titan and Tethys, all having influence on the orbital dynamics of Pandoraand Mimas, leads to a non-stationary signal in the Fourier spectrum. This could be caused by thechaotic and strong interactions between Prometheus and Pandora resulting in wandering librationfrequencies and changing amplitudes in the orbital dynamics of Bl´eriot. Performing long-timesimulations for the test moonlet considering a time span of up to 100 Saturnian years, the 42:40IVR of Prometheus seems to become more and more important forcing the moonlet to librate witha period of about 4 years with a radial amplitude of 20 meters and an amplitude around 1 km inmean longitude. This signal is clearly visible in the Fourier spectrum and stationary in contrastto the other signals.Although the libration periods of the test moonlet from our simulations agree fairly well withthe observational data, the resulting amplitudes are far too small.4 𝑦𝑦 𝑚 𝑥 𝑚 −𝑥 𝑔 𝜎 𝑔 𝐿 𝑥 𝑔 ⏞Δ−𝐿 Figure 3: Sketch of our propeller toy model, illustrating the symmetry breaking of the propellerlobes. The central moonlet is located at position ( x m , y m ), defining the beginning of the gapstructures in this way. The partial gaps, created by the moonlet-disc interactions are located at x = x g and x = − x g with lengths L ± y m and width ∆. Next, we will consider the gravitational interaction between the embedded moonlet with its createdgap region. Imagine a non-symmetric propeller structure so that the moonlet gets acceleratedby the ring gravity. We will show, that the evolution of the related longitudinal residual can bedescribed by a harmonic oscillator. This harmonic oscillator feels a periodic external forcing causedby the gravity of the outer moons. The amplitudes of the external forced frequencies get the moremagnified, the closer the forcing frequency matches the eigenfrequency of the harmonic oscillator(propeller-moonlet system).Consider a moonlet located at ( x m , y m ) as illustrated in Figure 3. Interacting with the sur-rounding viscous ring material, the propeller moonlet gravitationally scatters ring particles to largerand smaller orbits creating two gaps in its vicinity in the ring material (regions of reduced surfacemass density) and which are decorated by two density enhanced regions pairwise downstream ofits orbit (Spahn and Wiebicke, 1989). Viscous diffusion of the ring material counteracts this gap-creation, smoothing out the structure with growing azimuthal distance downstream the moonlet(Spahn and Sremˇcevi´c, 2000; Sremˇcevi´c et al., 2002). For simplicity, we assume the gap shape tobe a rectangular area with reduced density σ g , illustrated by the shaded regions at | x | = x g andwidth ∆ in Figure 3. The diffusion process defines the length L ± y m of the gaps, while the widthof the propeller-blades is set by the moonlet’s mass m or its Hill radius h ∆2 ∝ h ( m ) ∼ a (cid:18) m m c (cid:19) , (6)and Saturn’s mass is labeled by m c . We assume that both gaps are anchored at the position y m following the moonlet’s motion. Further, the ends at L and − L are fixed, because the imprintof the moonlet motion will be averaged out along the azimuth due to the diffusion process anddragged along with the Kepler shear.In the symmetric case ( x m = 0 , y m = 0) the torque on the moonlet due to the gravitationalinteraction with the gap is zero because of point symmetry. When the moonlet is leaving its meanposition, the gravitational force (cid:126)F g of the ring material on the moonlet is (cid:126)F g ( (cid:126)r m ) = − G (cid:90) ∞−∞ dx (cid:90) ∞−∞ dy σ ( (cid:126)r ) (cid:126)r m − (cid:126)r | (cid:126)r m − (cid:126)r | , (7)where σ ( (cid:126)r ) , (cid:126)r = ( x, y ) T and (cid:126)r m = ( x m , y m ) T are the surface mass density and the positionvectors of the ring material and of the moonlet. 5plitting the integrals, subtracting the unperturbed background density σ and evaluating theintegrals up to first order in x m and y m , the azimuthal force acting on the moonlet reads F g,y ( x m , y m ) ≈ G | δσ | ∆ (cid:18) x m x g + y m L (cid:19) , (8)where δσ = σ g − σ . In the following we assume that the azimuthal excursions of the moonletare negligibly small compared to the azimuthal extent of the propeller structure ( y m (cid:28) L ). We alsoassume that the radial displacement of the moonlet is very small compared to the radial distanceof the gaps to the moonlet ( x m (cid:28) x g ). The mean motion is given by the change in the mean longitude n = dλdt . Thus, the change in azimuthal direction of the moonlet is given by the Gaussian perturbationequation dndt = d λdt ≈ − a F y , (9)neglecting higher orders in eccentricity, where we used the relation λ = y m a − . Inserting F g,y from Eq. (8) results in d λdt = 6 a Gδσ ∆ y m L = 6 Gδσ ∆ L λ = − ω λ , (10)with ω = 6 G | δσ | ∆ L the libration frequency and T = 2 πω − the libration period. In this lowestorder of the perturbation expansion we arrive at a harmonic oscillator. Here, the eigenfrequencycontains the properties of the propeller feature as the lengths ∆ and L , but also the drop in thesurface mass density δσ . In order to estimate the eigenlibration period for the moonlet, we use values, which have beeninspired by theoretical and observational data (Tiscareno et al., 2007; Spahn and Sremˇcevi´c, 2000).We can estimate the length of the gaps from the analytical model derived by Sremˇcevi´c et al.(2002). The relative mass moved out of the gaps along the azimuth is (cid:90) L σ g − σ σ dy = − . aK , which can be obtained from a direct numerical integration. Assuming δσ/σ = − . L = 4 . aK and thus, for Bl´eriot L ≈ km ,where the scaling length aK can be calculated by aK = 160 km (cid:34) cm s ν (cid:35) (cid:20) h m (cid:21) (cid:20) Ω10 − s (cid:21) with ν, h and Ω denoting the viscosity, the Hill radius and the Kepler frequency. The value of ν fits to extrapolated values from observations of Tiscareno et al. (2007) and simulations of Daisakaet al. (2001).Further, the width of the gap ∆ = 2 h ≈ km .As a result, we estimate a eigenlibration period of that ring-moonlet harmonic oscillator: T = 2 πL (cid:112) G | δσ | ∆ ≈ yr (cid:34) L km (cid:18) | δσ | gcm ∆1 km (cid:19) − (cid:35) . h of the propeller moonlet has a strong influence on the gap length L ∝ h ∼ m and thus on the libration period. With h = 480 m ± m the libration period is T = (13 ± .
03 years. The longestperiod from Table 1 falls well into the uncertainty interval of the estimated libration period T , sothat our model can indeed induce much larger libration amplitudes.One should note, that our simple model is generally not fully consistent, because the librationamplitude y m is not much smaller than the used gap length L . Thus, the condition to approximatethe force for small amplitudes is easily violated. However, in reality the gap extents for more thanseveral thousand km and L is rather a decaying length of the gap if one for example assumes aexponential relaxation of the gap. A more complex model, would remove this inconsistency, butthe oscillatory behavior due to the repulsive azimuthal force should persist. We will address sucha comprehensive non-linear model in the future. In this paper we have presented results of N-Body simulations, which characterize the orbitalevolution of the moonlet of the propeller Bl´eriot, being perturbed by the gravitational interactionswith 15 outer and inner moons of Saturn. We found, that the 14:13 CER of Pandora and the 3:16:13three-body resonances of Mimas and Pandora are the dominating perturbations of Bl´eriot’s orbit.Additionally, the chaotic interaction between Pandora and Prometheus has an effect on Bl´eriot’sorbital evolution as well, resulting in changing libration frequencies over time.Our simulations yielded, that the gravitational interactions with the other moons cause similarlibration frequencies of Bl´eriot as concluded from the Cassini ISS images, but the correspondingamplitudes are too small. Considering propeller-moonlet interactions with a new model, we havebeen able to find a mechanism to amplify certain modes in form of a harmonic oscillator which isperiodically driven by an external forcing.The Eigenfrequency of this oscillating system contains key-properties of the propeller structure.In the presence of an external forcing, this oscillating system amplifies the induced forced oscilla-tions by the outer satellites up to several orders of magnitude. The closer the forcing frequency isto the eigenfrequency the stronger the amplification can be. Applying our moonlet-propeller-gapinteraction model to Bl´eriot, reasonable results have been obtained for the libration period, whichfit the observations fairly well. Combining our model with the simulation results, we are able toreproduce the largest observed mode for the libration of Bl´eriot. The smaller observed modes arenot reproduced, but may evolve if one regards processes of non-linear mode coupling in our model,which we did not consider yet.Even this first simplifying theoretical investigation demonstrates, that outer gravitational per-turbations in combination with ring-moonlet interactions are necessary to address the propeller-moonlet migration problem.In order to verify our results, we plan to incorporate our model into hydrodynamical simulationsin the future. Additionally, we want to add higher order terms in the evaluation, i.e. nonlinearterms, in order to study mode coupling effects.Besides the improvements on the propeller model, we also plan to apply our N-Body integrationsto the other propeller structures located in the outer A ring.