Draft version November 5, 2018 Typeset using LATEX preprint2 style in AASTeX62
Hydrodynamic Simulations of Asymmetric Propeller Structures in Saturn’s Rings M. Seiler,1 M. Seiß,1 H. Hoffmann,1 and F. Spahn1 1Theoretical Physics Group, Institute of Physics and Astronomy, University of Potsdam, Germany
Submitted to ApJS
ABSTRACT The observation of the non-Keplerian behavior of propeller structures in Saturn’s outer A ring (Tiscareno et al. 2010; Seiler et al. 2017; Spahn et al. 2018) raises the question, how the propeller responds to the wandering of the central embedded moonlet. Here, we study numerically how the induced propeller is changing for a librating moonlet. It turns out, that the libration of the moonlet induces an asymmetry in the structural imprint of the propeller, where the asymmetry is depending on the moonlet’s libration period and amplitude. Further, we study the dependence of the asymmetry on the libration period and amplitude for a moonlet with 400 m Hill radius, which is located in the outer A ring. In this way, we are able to apply our findings to the largest found propeller structures – such as Bl´eriot– which are expected to be of similar size. For Bl´eriot,we can conclude that, supposed the moonlet is librating with the largest observed period of 11.1 years and an azimuthal amplitude of about 1845 km (Seiler et al. 2017; Spahn et al. 2018), a small assymetry should be measurable but depends on the moonlet’s libration phase at the observation time. The excess motions of the other giant propellers – such as Santos Dumont and Earhart – have similar amplitudes as Bl´eriotand thus might allow the observation of larger asymmetries due to their smaller azimuthal extent. This would permit to scan the whole gap structure for asymmetries. Although the librational model of the moonlet is a simplification, our results are a first step towards the development of a consistent model for the description of the formation of asymmetric propellers caused by a freely moving moonlet.
Keywords: hydrodynamics — methods: numerical — methods: data analysis — planets
arXiv:1811.00682v1 [astro-ph.EP] 1 Nov 2018 and satellites: dynamical evolution and stability — planets and satellites: rings — planets and satellites: individual(Saturn)
1. INTRODUCTION composed of differently sized icy particles. The Planetary rings are one of the most beautiful dense main rings of Saturn are composed of par- structures in our solar system, which are mainly ticles ranging from centimeters to a few meters (Cuzzi et al. 2018) and beside this ensemble of particles, even larger boulders (called moonlets) Corresponding author: M. Seiler [email protected] are orbiting within the ring environment. By their strong gravitational interaction with the 2 surrounding ring material, these objects try to larger propellers being created by moonlets with sweep free a gap around their orbits. Due to the radii between 0.15 km and 1.0 km have been ∼ . different angular speeds of the ring material for found (Tiscareno et al. 2010). Due to their rel- varying semi-major axis the gap appears inside atively large size, 11 of these giant-propellers and outside the moonlet’s mean radial position were able to be identified in different subsequent with a slight radial shift. Therefore, the inner images, allowing the reconstruction of their or- gap – due to the higher angular speed of the bital motion. The analysis of their orbital evolu- ring material – overtakes the embedded moon- tion revealed a longitudinal deviation from their let while the outer one falls behind. Viscous expected Keplerian location – called excess mo- diffusion of the ring material counteracts this tion (Tiscareno et al. 2010; Spahn et al. 2018). gravitational scattering process and thus results The largest propeller structure is called in a closing of the opened gap with growing Bl´eriotand is created by a moonlet of about azimuthal distance to the embedded moonlet 400 m in Hill radius (Hoffmann et al. 2016; Seiß (Spahn and Sremˇcevi´c 2000; Sremˇcevi´cet al. et al. 2017). Its excess motion can be recon- 2002; Seiß et al. 2005; Hoffmann et al. 2015). structed by a superposition of three sinusoidal For moonlets larger than a critical radius, its harmonics with periods of 11.1, 3.7 and 2.2 years gravity compensates the viscous diffusion along and amplitudes of 1845, 152 and 58 km with the whole circumference of the orbit and thus a standard deviation of about 17 km of the re- the gap is kept open. For smaller moonlet radii, maining residual (Seiler et al. 2017; Spahn et al. the gravity of the moonlet does not suffice to 2018). Bl´eriot’sexcess motion resembles the compensate the viscous diffusion and therefore typical librational motion of a resonantly per- the gap closes. As a result, two partial gaps turbed moon (Goldreich 1965; Goldreich and inside and outside the orbit of the moonlet dec- Rappaport 2003a,b; Spitale et al. 2006; Cooper orated with wakes remain and resemble an S- et al. 2015) and thus supports the hypothesis, shaped, two-bladed propeller. Those propellers that this excess motion might result from a res- act as structural insignia to detect the embed- onant or near-resonant driving by one or several ded moonlet in the dense rings. of the outer satellites. In numeric simulations After the postulation of their existence in 1 it has been shown, that the collective per- Saturn’s ring to explain optical depth varia- turbation by the outer satellites Prometheus, tions in the Voyager data (Henon 1981; Lissauer Pandora and Mimas is able to induce correct et al. 1981), the cameras onboard the spacecraft libration frequencies to explain the three mode Cassini finally revealed their presence within the fit, but the induced libration amplitudes are by dense rings of Saturn (Tiscareno et al. 2006; far to small to explain the observations (Seiler Spahn and Schmidt 2006; Sremˇcevi´cet al. 2007; et al. 2017). Tiscareno et al. 2008). Meanwhile, different Alternatively, a stochastic migration of the populations of propeller structures inside Sat- embedded moonlet due to collisions or den- urn’s A ring have been identified in the images. sity fluctuations in the rings has been pro- Between 127 000 and 132 000 km from Saturn’s posed as well (Crida et al. 2010; Rein and Pa- center three propeller belts have been observed, paloizou 2010; Pan and Chiang 2010, 2012; Pan which contain several thousand propellers, all being generated by moonlets of radii . 0.15 km 1 In these simulations a test moonlet, which has been (Tiscareno et al. 2008). Further outwards, be- placed on the expected orbital position of Bl´eriot,has tween the Encke and Keeler gap, about 37 been driven gravitationally by Saturn and 15 of its larger moons. 3 et al. 2012; Bromley and Kenyon 2013; Tis- shear. Thus, the perturbation by the moonlet’s careno 2013). The most promising results have motion only arrives at the gap ends after a cer- been obtained by Rein and Papaloizou(2010), tain delay time (hereafter called Tgap), while the who considered a random walk in the semi- beginning of the gap more or less follows the major axis of the moonlet, and in N-Body simu- moonlet motion instantaneously. For Bl´eriot lations Pan et al.(2012) have been able to gen- this delay time would be about T 0.5 years gap ≈ erate an amplitude in the moonlet’s excess mo- considering a gap length of 6500 km, respec- tion of about 300 km over a time span of 4 years. tively (Seiler et al. 2017). As a consequence of Nevertheless, their resulting amplitude is also the moonlet’s wandering and the retardation in too small. the reaction of the gap ends, the gap structures In order to solve this amplitude problem Seiler are expected to become asymmetric. Here, this et al.(2017) suggested a propeller-moonlet in- expectation is tested for a given moonlet libra- teraction model, where the usually assumed tion to answer the question, whether such an point-symmetry of the propeller structure is asymmetry might be detectable in the Cassini broken by a slight displacement of the central ISS images. Therefore, we search the simulated moonlet. This breaking of the point-symmetry gap structures for properties which allow a di- causes a restoring force to the moonlet, resulting rect measurement of the asymmetry. In addi- in a harmonic oscillation in the moonlet’s mean tion, we will study the dependence of the asym- longitude. The libration depends on the phys- metry on the libration amplitude and period. ical parameters of the propeller and the sur- The plan of this paper is as follows: First, rounding ring material. This oscillating system we give an overview of our hydrodynamic sim- has its own eigenfrequency so that for an ex- ulation code and how we model the moonlet li- ternal resonant driving with a frequency, which bration (see Section2). Afterwards, we present is sufficiently close to the eigenfrequency of the the results of our simulations and show, how propeller-moonlet oscillator, an originally small the appearance of the propeller and the differ- libration mode can be amplified by several or- ent properties of the gap structures change with ders of magnitude. the libration of the moonlet. For this purpose, However, the suggested propeller-moonlet in- we directly compare our results to simulations teraction model is oversimplifying the physi- of a symmetric propeller structure (see Section cal problem in the aspect, that it reduces the 3). The propeller-moonlet interaction model gap structures to two radially separated rect- predicts, that the restoring force by the gap- angular shaped boxes with fixed azimuthal end asymmetry follows the moonlet motion. This points, where the beginning of the gap follows fact will be checked against our simulation data the moonlet motion. Further, the surface mass (see Section4). As a next step, we study the de- density inside these boxes is assumed to be con- pendence of the asymmetry on the libration pe- stant. In reality, the gap structure has an az- riod and amplitude and test, under which con- imuthal extent of up to several thousand kilo- ditions the asymmetry becomes detectable (see meters and the surface mass density within the Section5). On the basis of our analysis, we gap structure is rather a decay than constant. will apply our findings to the propeller Bl´eriot Due to the radial separation of the gaps from the in Section 5.3. In the end of this paper, we con- moonlet, the purturbation caused by the moon- clude and discuss our findings (see Section6). let’s motion first needs time to be transported along the gap structure, given by the Kepler 2. HYDRODYNAMIC SIMULATION 4
Our hydrodynamic simulation routine bases be reduced to a two-dimensional problem us- on the code, developed by Seiß et al.(2017). ing vertically integrated quantities (Spahn et al. This integration routine will be modified so that 2018). Additionally, the moonlets in the outer the moonlet can librate around its mean orbital A ring are expected to evolve on nearly circular position within the simulation grid. orbits in the equatorial plane of Saturn. We consider a moonlet embedded in the gran- In comparison to the spatial extent of the rings ular environment of Saturn’s rings with sur- the propellers are tiny. As a consequence, only face mass density Σ. The evolution of the sur- a small ring area, centered around the moon- face mass density is described by the continuity let’s orbital position needs to be considered for equation our simulations. Therefore, the acting forces in the vicinity of the moonlet can be linearized and described by the Hill problem (Hill 1878). ∂tΣ + Σv = 0 (1) Here, we choose x to represent the radial dis- ∇ · tance from the moonlet and y stands for the and the momentum balance is quantified by azimuthal direction. As a result, the gravita- the Navier Stokes equation tional potential of the central moonlet is given by ˆ ∂tΣv + (Σv v) = Σ (Φp + Φm) + fi P . GMm ∇ · ◦ − ∇ · − ∇ · Φm = , (3) (2) − x2 + y2 + 2
Both equations above are presented in the flux with the smoothingp radius which limits the conserved form, where the tensor product is de- gravitational potential in the close vicinity of noted by the symbol. The symbol ∂ = ∂/∂t the moonlet’s center. Here, we choose = 0.2 h. ◦ t denotes the partial time derivative. The gravi- The mass of the moonlet Mm is defined by its tational potentials of the central planet and the Hill radius moonlet are described by Φp and Φm. Further, Mm we choose our simulation’s coordinate system to h = a0 , (4) co-rotate with the moonlet’s mean orbital fre- s3Mp 3 quency Ω = GMp/r0, where G and Mp denote where a0 denotes the semi-major axis of the the gravitational constant and the mass of the moonlet’s mean orbit. p central planet, and to be centered at the moon- We treat the granular ensemble forming dense let’s mean radial position r0. Here, we neglect rings as a usual (linear) fluid. Thus, the pres- the oblateness of Saturn 1. sure tensor Pˆ can be described with the Newto- In comparison to their radial and longitudi- nian ansatz as nal extent (several hundred thousand kilome- ters) the vertical dimension (. 100 m) of Sat- ∂vi ∂vj 2 urn’s rings is tiny. Thus, equations1 and2 can Pij = p δij Σν + +Σ ν ξ v δij − ∂xj ∂xi 3 − ∇· (5) 1 Including the oblateness of Saturn would result in where p, ν and ξ denote the scalar pressure, slightly different orbital frequencies and in a precession of the ascending node and the pericenter, which will not the kinematic shear and the bulk viscosities, re- have a significant impact on the results, since we are spectively. The dependence of the pressure and only considering a small ring area in the close vicinity the viscosities on the local density can be de- of the moonlet. scribed by the power laws (Spahn et al. 2000) 5 Parameter Value
α h 400 m Σ −3 −1 p=p0 (6) Ω 1.3 10 s Σ × 0 ν 300 cm2 s−1 β 0 Σ −1 ν =ν . (7) c0 0.5 cm s 0 Σ 0 Table 1. Simulation parameters for the hydrodynamic Additionally, we set the ratio between shear simulation. Those parameters are used for the scaling and bulk viscosity constant within the integration routine.
ξ ξ = 0 ν . (8) numerical methods is given in Section 3 in Seiß ν · 0 et al.(2017). In the simplest case, the unperturbed pressure The simulation scheme uses following scaled 2 is given by the ideal gas relation p0 = Σ0 c0 values: x x/h, y y/h, t Ωt t/T and → → → ∼ at equilibrium, where c0 denotes the dispersion Σ Σ/Σ . → 0 velocity. In our simulation scheme, we use an Here, we will focus on the asymmetry forma- isothermal model and therefore the dispersion tion for the giant trans-Encke propellers. For velocity c0 and the related granular temperature this reason, we will consider a Bl´eriot-sized 2 T = c0/3 are constant. moonlet with 400 m Hill radius (Hoffmann et al. 2.1. Methods 2016; Seiß et al. 2017), allowing us to predict the asymmetry of its propeller. Therefore, we The simulation area is a rectangular cut out of will simulate the moonlet at a radial location the ring environment of dimensions (x , x ) min max close to its observed orbital position and we and (y , y ), which is centered around the min max will adopt the viscosity accordingly (Seiß et al. mean orbital position of the moonlet. This 2017). The set of used parameters and their val- box is divided into N N equal-sized cells. x × y ues – if nowhere else mentioned – are given in The complete set of equations is integrated un- Table1. til a steady state is established. The advection The scaled viscosity and sound speed are cal- term is solved with the second-order scheme culated according to the moonlet size with ’MinMod’ flux limiter (LeVeque 2002) in order to better conserve the wake crests. The ν influence of pressure and viscous transport is ν˜ = 0 and (9) h2 Ω solved with an explicit scheme. At the borders c c˜ = 0 . (10) of the calculation regions, the boundary condi- h Ω tions are chosen such, that the perturbations Their value and further parameters used for can flow out of the box freely, while the inflow the simulation are given in Table2. is unperturbed. This is especially important at the azimuthal boundaries, where due to Kepler 2.2. Modeling the Moonlet Motion shear the material is flowing into the box at The observed harmonic excess motion of x < 0, y = ymin and x > 0, y = ymax and flowing Bl´eriot requires a periodic azimuthal motion out at x < 0, y = ymax and x > 0, y = ymin. The of the propeller structure. In the easiest way, influence of the moonlet is established mainly this can be achieved by a libration of the central by its gravity. moonlet. Setting the moonlet on a librational A more detailed overview of the implemen- motion around its mean orbital position fur- tation of the integration routine and the used ther allows to directly test the assumptions of 6 Parameter Value period Tm – will also set the drift rate of the ν˜ 1.441 10−3 × moonlet. c˜ 4.811 10−2 × α 1 3. VISUALIZATION OF THE ASYMMETRY β 2 For the presentation of our simulation results, ξ˜ 7ν ˜ we will define and analyze different characteris- x 10 h tics of the propeller shape and directly compare min − xmax 10 h the results for a symmetric and asymmetric pro- ymin 800 h peller. In this way, we show, how the asymme- − try of the propeller can be characterized. Our ymax 800 h results have been obtained for a moonlet with Nx 400 400 m Hill radius. For the asymmetric propeller Ny 6400 – if nowhere else mentioned –the moonlet has Table 2. Further simulation parameters used for the been forced to around its mean orbital position hydrodynamic simulation. with a radial amplitude of xm,0 = 0.5 h and pe- riod of Tm = 80 orbital periods. the propeller-moonlet interaction model (Seiler et al. 2017). 3.1. Density Profile The moonlet will be set on a librational mo- The typical imprint of an unperturbed pro- tion with amplitude xm,0 and period Tm given peller structure, where the moonlet (represented by by the star symbol) is not librating, is shown in the top panel of Figure1. A clear symmetry can be seen. The grey scale color code illustrates the 2π x = x cos t and (11) surface mass density, where darker color cor- m m,0 T m responds to regions of depleted density, while 3 2π brighter color represents denser regions. The y = x T sin t . (12) m −2 m,0 m T x and y directions in the presented plots de- m note the radial and azimuthal distances to the Note, that the above presented equations for mean orbital position of the moonlet. For a bet- the moonlet motion are already given in the ter visualization, the presented area around the scaled form, where the time t and the libration propeller is reduced radially and azimuthally to period T are given in number of orbits and the ∆x = 6 h and ∆y = 600 h. Note, that the m ± ± directions xm and ym are given in Hill radii. The moonlet’s mean orbital motion is from left to moonlet’s azimuthal amplitude is motivated by right. the proportionality of dn/dt da/dt and thus Fresh inflowing material (moving from left to ∝ it is set by the Kepler shear. right) passing by the by the moonlet from the The advantage of the librational moonlet mo- inside at ∆x < 0, are the more deflected to the tion lies in the fact, that it covers both, libra- moonlet the closer they pass it radially. The tion and migration. The resulting asymmetry deflection for the particles passing by outside for a linearly migrating moonlet can be studied at ∆x > 0 the moonlet moving from right to considering only part of the moonlet libration. left occur vice versa. For larger radial distances, In this context, the relative velocity of the per- this deflection decreases as ∆x −2. This induces | | turbed moonlet motion to its unperturbed or- a coherent motion of the ring material and thus bit – which depends on the amplitude xm,0 and results in the formation of wakes. The location 7
Moonlet Phase Space: h = 400 m , xm = 0.5 h Symmetric Propeller: h = 400 m 0.8 Propeller Structure at t = 160.00 orbits 6 Tm = 80 orbits Tm = 240 orbits 0.6 t0 = 0 4
2 0.4 ] h [ 0 0.2 m x ] ∆
h 1 3 2 [ 0.0 t1 = Tm t3 = Tm
m 4 4 x 4 0.2 − 6 600 400 200 0 200 400 600 0.4 ∆ym[h] − 0.6 t = 1T − 2 2 m 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.8 − 300 250 200 150 100 50 0 50 100 150 200 250 300 − − − − − − ym[h] Asymmetric Propeller: Asymmetric Propeller: h = 400 m, T = 80 orbits, x = 0.5 h at time t h = 400 m, T = 240 orbits, x = 0.5 h at time t Propellerm Structure at tm = 240.00 orbits 0 Propellerm Structure at t m= 480.00 orbits 0 6 6
4 4
2 2 ] ] h h [ [ m m
x 0 x 0 ∆ ∆
2 2
4 4
600 400 200 0 200 400 600 600 400 200 0 200 400 600 ∆ym[h] ∆ym[h]
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Figure 1. Comparison of a symmetric propeller(top left) and two examples for an asymmetric propeller (bottom panels). The two asymmetric examples show the propeller for a librating moonlet at the same libration phase, but for different libration periods (see top right panel). In the bottom left panel the moonlet is librating with Tm = Tgap, while in the right one the period is T = 3 T . In all panels, the origin of the coordinate system is centered to m × gap the mean orbital position of the moonlet. The moonlet’s actual position is given by the star symbol. In all cases the moonlet’s Hill radius is h = 400 m and its libration amplitude was set to xm,0 = 0.5 h. The black lines denote the density profile at 80% gap relaxation. of the moonlet within the simulation grid, which This symmetry is broken in the bottom pan- is centered to its mean orbital location, is illus- els of Figure1, where snapshots of the induced trated by the star symbol. Further, the black propeller structures for a librating moonlet are line marks the propeller surface mass density given. The azimuthal and radial scales are cho- level at 80% of gap closing (Σ/Σ0 = 0.8). At sen exactly in the same way as for the unper- this level of surface mass density, a clear sym- turbed case. The level of Σ/Σ0 = 0.8 is em- metry can be seen when comparing the inner phasized by the black line, where a clear dif- and outer gap region, where the gap lengths are ference in the gap lengths and even in the gap about 540 h and the maximum gap widths are widths is obvious. The two bottom panels in about 2 h, respectively. Figure1 show the propeller structure at the same libration phase of the moonlet but for dif- 8 ferent libration periods. In the left panel the propeller shape for the other half of the libra- moonlet’s libration period has the value Tm = tion period would look similar but mirrored 80 orbits, while its value is Tm = 240 orbits in due to the Kepler shear. The snapshots at the right one. The iso-density level Σ/Σ0 = 0.8 220 orbits and 240 orbits show the moonlet at level is suitable to illustrate the asymmetry: (xm(t = 220 orbits) = 0 h, ym(t = 220 orbits) = The bottom left panel (shorter Tm) shows a 60 h) and (xm(t = 240 orbits) = 0.5 h, ym(t = length ratio between the outer and inner gap 240 orbits) = 0 h). The strongest asymme- of 450 h/650 h 0.7 and in the other case (bot- try of the propeller can be observed for the ≈ tom right panel, T = 3 T this ratio takes moonlet at its radial amplitude (see snapshot m × gap 600 h/550 h 1.09, which makes the propeller at t = 220 orbits in Figure2). ≈ gaps look more symmetric. For the gap width at A more detailed analysis of the gap structures ∆ = 150 h the width ratio between the outer will be presented in the following sections. ± and inner gap has a value of 3 h/2 h = 1.5 for 3.2. Radial Gap Profile the bottom left panel, while for the bottom right panel this ratio is 3 h/2.5 h = 1.2. Figure2 illustrates the evolution of the asym- For the used dimension of the simulation box metry of a propeller caused by a librating moon- (see Table2) a steady state is already reached let. The grey-level density representations cor- after 80 orbits, respectively. This equals the respond to certain times t = (210, 220, 230, 240) time the fresh ring material at ∆x = 1 h needs orbits, which represent four different phases ± to drift to the moonlet. For the moonlet used (phase angles) of the moonlet libration. with Hill radius of h = 400 m the azimuthal ex- We will analyze and compare the radial gap tent is of similar size (about 600 h at Σ/Σ0 = 0.8 profiles of a symmetric and asymmetric pro- as seen in the top panel of Figure1). In this peller at different azimuthal distances to the case, the perturbation by the moonlet motion moonlet which demonstrate the changes in the would take about 63 orbits to reach the end of gap structures due to the moonlet motion. The the gaps. Therefore, the two bottom panels of minimum of the surface mass density defines Figure1 show the induced propeller structure the gap depth and its radial position gives the for T T (left) and for T 3 T gap position. At the fixed surface density level m ≈ gap m ≈ × gap (right). It has to be noted, that the timescales Σ/Σ0 = 0.8 we estimate the gap width, permit- involved depend on the mass (size) of the moon- ting the comparison of the inner and outer gap let. in this way. Due to the libration of the embedded moon- First, we reduce the noise of the simulation let, the shape of the propeller is changing all data – especially in the wake region close to the the time. Thus, at different phases of the moon- moonlet – by performing moving box averag- let’s libration the propeller structure looks dif- ing for the radial gap profiles, where the radial ferent resulting in different gap lengths and box length is set close to the wake wavelength gap depths, as presented in Figure2. Here, of about ∆x = 0.5 h. The resulting smoothed the four center and bottom panels show the radial density profiles of the inner and outer propeller shape for half a libration period at gap at azimuthal distances to the moonlet of ∆y = y y = 10, 200 and 600 h are shown times t = 210, 220, 230 and 240 orbits, while | − m| the top panel illustrates the actual position of in Figure3. Note, that for the bottom panels in the moonlet at those times. Note, that the Figure3 we use a moonlet-centered coordinate system (∆x = x x (t) and ∆y = y y (t)) moonlet libration is counter clockwise. The − m − m where we subtract of the moonlet’s actual posi- 9
Moonlet Phase Space: h = 400 m , xm = 0.5 h 0.8
0.6 t4 = 240 orbits t = 230 orbits 0.4 3
0.2 ] h [ 0.0 t2 = 220 orbits m x 0.2 − 0.4 − t1 = 210 orbits 0.6 − 0.8 − 80 60 40 20 0 20 40 60 80 100 120 − − − − ym[h] Asymmetric Propeller: Asymmetric Propeller: h = 400 m, T = 80 orbits, x = 0.5 h at t = 210 orbits h = 400 m, T = 80 orbits, x = 0.5 h at t = 220 orbits Propellerm Structure mat t = 210.001 orbits Propellerm Structure mat t = 220.002 orbits 6 4 4 2 2 ] ]
h 0 h [ [ m m
x x 0
∆ 2 ∆ 2 4 4 6 400 200 0 200 400 600 400 200 0 200 400 600 ∆ym[h] ∆ym[h]
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Asymmetric Propeller: Asymmetric Propeller: h = 400 m, T = 80 orbits, x = 0.5 h at t = 230 orbits h = 400 m, T = 80 orbits, x = 0.5 h at timet = 240t orbits Propellerm Structure mat t = 230.003 orbits Propellerm Structure at tm = 240.00 orbits4 0 6 6
4 4
2 2 ] ] h h [ [ m m x x 0 0 ∆ ∆
2 2
4 4
400 200 0 200 400 600 600 400 200 0 200 400 600 ∆ym[h] ∆ym[h]
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Figure 2. Evolution of the propeller asymmetry for half a moonlet period. Top panel: Phase space representation of the moonlet libration highlighting the four different libration phase angles shown in the center and bottom panels.
The moonlet of size h = 400 m librates with a period of Tm = 80 orbits and amplitude of xm,0 = 0.5 h. The snapshots (center and bottom panels) show the moonlet (star symbol) at different libration phases at times t1 = 210 orbits (middle left), t2 = 220 orbits (middle right), t3 = 230 orbits (bottom left), and t4 = 240 orbits (bottom right). 10 tion for better comparison to the unperturbed cations of the gap minima clearly follow the moonlet profiles. The solid and dashed curves moonlet libration (black dotted line). Further, denote the inner and outer gap profiles. Note, the radial amplitude of the gap location slightly that we mirrored the inner gap’s profile at the increases for larger azimuthal distance to the origin to compare both gap profiles in a better moonlet. For the symmetric case, the radial way. gap locations stay constant over time. The top panel of Figure3 shows the radial While the ring material in the close vicinity profile for the symmetric propeller. Here, the to the moonlet almost immediately feels the gap locations of the inner and outer gap match change in the moonlet’s motion (compare the at all azimuthal distances. Further, the gap the black solid and dashed lines in Figure4), relaxes azimuthally as theoretically predicted, the perturbation by the motion of the moonlet i.e. the gap-depth decreases for increasing az- first needs to be transported through the ring imuthal distance to the moonlet. environment to larger azimuthal distances In the bottom panels the asymmetric gap 3 profiles are illustrated at the two snapshots ∆y = Ωm t ∆x (13) T = 220 orbits and T = 240 orbits from Fig- −2 ure2. Here, the moonlet is presented at its set by the Kepler shear. Thus, when the azimuthal (left) and radial (right) libration am- moonlet reaches its radial libration amplitude plitude. While in the left panel the gap profiles of 0.5 h at T = 80 orbits the gap minima at look much alike, but are shifted radially with in- ∆y = 200 h and ∆y = 600 h are following creasing azimuth, their depths and widths dif- the moonlet motion after about 26 orbits and fer significantly in the right panel. Note, how 80 orbits, respectively. This agrees with the de- the beginning of the gaps (compare ∆y = 10 h) lay found in Figure4. stays unaffected by the moonlet libration and 3.2.2. Radial Gap Separation follows the moonlet motion instantaneously in both panels. Although the radial locations of the gap min- ima change over time and can differ by more 3.2.1. Gap Positions than 1 h (compare the solid and dashed red The changes in the gap minima locations lines in Figure4), their separation stays al- caused by the moonlet libration can be better il- most constant over time (see Figure5). The lustrated by considering their time-evolution for evolution of the radial gap separation is pre- a fixed azimuth, as shown in Figure4, where sented at azimuthal distances to the moonlet the evolution of the gap minima locations at of ∆y = y y = 10, 200 and 600 h, where | − m| ∆y = y y = 10, 200 and 600 h is presented. the dashed and solid lines denote the symmet- | − m| For the symmetric propeller the radial gap ric and asymmetric gap separation, respectively. positions for different azimuths lay on top of With increasing azimuth the scattering of the each other, where the mean radial gap position radial gap separation becomes larger (about 1 h < x > 2.83 h (at ∆y = 10 h) decreases at ∆y = 600 h and 0.5 h at ∆y = 200 h). gap ≈ ± ± ± ± to < x > 1.6 h (for ∆y = 200 and 600 h) This is illustrated in the left panel of Figure gap ≈ ± ± for larger azimuthal distances. 5, where a comparison of the gap separation Figure4 shows the time-evolution of the gap for the symmetric propeller structure (red solid minima location for the symmetric (left panel) line) along the azimuth is compared against the and the asymmetric (right panel) propeller. For asymmetric one (black lines) for three different the asymmetric case, it is visible, how the lo- snapshots. Comparing all the curves, one rec- 11
Symmetric radialT gap= 240 profiles:Orbitsh = 400 m
at ∆y = 10h 1.4 ± at ∆y = 200h ± at ∆y = 600h 1.2 ±
1.0
0 0.8 Σ / Σ 0.6
0.4
0.2
0.0 4 3 2 1 0 1 2 3 4 5 6 − − − − x x [h] − m Asymmetric radial gap profiles: Asymmetric radial gap profiles: xm,0 = 0.5 h , Tm = 80 Orbits , T = 220 Orbits xm,0 = 0.5 h , Tm = 80 Orbits , T = 240 Orbits h = 400 m, xm = 0.5 h, Tm = 80 orbits at t = 220 orbits h = 400 m, xm = 0.5 h, Tm = 80 orbits at t = 240 orbits
at ∆y = 10h at ∆y = 10h 1.4 ± 1.4 ± at ∆y = 200h at ∆y = 200h ± ± at ∆y = 600h at ∆y = 600h 1.2 ± 1.2 ±
1.0 1.0
0 0.8 0 0.8 Σ Σ / / Σ Σ 0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0 4 3 2 1 0 1 2 3 4 5 6 4 3 2 1 0 1 2 3 4 5 6 − − − − x x [h] − − − − x x [h] − m − m Figure 3. Radial gap profiles for a symmetric (Top) and asymmetric propeller structure (Bottom Left and Right) at different azimuthal distances to the moonlet of ∆y = 10 h (black), 200 h (blue), 600 h (red). The dashed ± ± ± and solid lines denote the outer and inner gap profile. The inner gap’s profile has been mirrored at the moonlet position for better comparison. For the asymmetric propeller (Bottom panels) the radial gap profile is given at two different libration phases of the moonlet. Here, the moonlet size has been set to h = 400 m and the libration period and amplitude have been set to 80 orbits and 0.5 h. For better comparison, the moonlet’s current position has been subtracted of in the case of the asymmetric propeller. ognizes, that up to a critical azimuth of about ferent snapshots at times T = 200 orbits and 250 h the variations of the radial gap separation T = 220 orbits the propagation of the pertur- for the asymmetric gaps are negligible and thus bation along the azmiuth can be observed. On it almost matches the symmetric gap structure. average, the gap separation still follows the az- For larger azimuthal distances to the moon- imuthal behavior of the unperturbed gap sepa- let, the retardation effect gets more dominant ration. resulting in larger variations of the radial gap 3.2.3. Gap Width separation along the azimuth with time. This allows to detect the perturbation by the moon- The grey-level density plots in Figure1 and let motion which gets visible as a wavy structure the radial profiles presented in Figure3 have along the azimuth. Further, comparing the dif- demonstrated, that the gap width is influenced by the motion of the moonlet. We study the 12
Aymmetric radial gap positions: x = 0.5 h , T = 80 Orbits Symmetric radial gap positions: h = 400 m h = 400 m,0xm = 0.5 h, Tmm = 80 orbits 5.0 5.0 at ∆y = 10h at ∆y = 10h ± ± at ∆y = 200h at ∆y = 200h ± ± at ∆y = 600h at ∆y = 600h 4.0 ± 4.0 ± xm
3.0 3.0 [h] [h] m m
x 2.0
x 2.0 − − x x 1.0 1.0
0.0 0.0
1.0 1.0 − 80 100 120 140 160 180 200 220 240 − 80 100 120 140 160 180 200 220 240 time [orbits]T [orbits] time [orbits]T [orbits]
Figure 4. Evolution of the radial gap minima location for the symmetric (left) and asymmetric (right) propeller at azimuthal distances to the moonlet of ∆y = 10 h (black), 200 h (blue) and 600 h (red). The coordinate system ± ± ± is moonlet-centered, meaning, that the current moonlet posotion ∆x = x x (t) and ∆y = y y (t) has been − m − m subtracted of. The dashed and solid lines denote the outer and inner gap profile. The moonlet’s radial position in the right panel is shown by the black dotted line. An increasing radial amplitude of the changing gap minima locations can be seen for larger azimuths. For the simulation the moonlet has the Hill radius h = 400 m and was librating with a period and amplitude of 80 orbits and 0.5 h.
Radial gap separation: Radial gap separation: x = 0.5 T = 80 xm,0 = 0.5 h , Tm = 80 orbits h = 400m, 0 x = 0.h5 ,h, Tm = 80 orbits h = 400 m, xm = 0.5 h, Tm = 80 orbits 5.0 m m 7.0 4.5 6.0 4.0 3.5 5.0
3.0 4.0 2.5 3.0 2.0
Gap Separation [h] 1.5 2.0 Radial Gap Separation [h] 1.0 symmetric at ∆y = 10h asymmetric at T = 200 orbits 1.0 ± at ∆y = 200h 0.5 asymmetric at T = 220 orbits ± at ∆y = 600h asymmetric at T = 240 orbits ± 0.0 0.0 0 100 200 300 400 500 600 700 800 80 100 120 140 160 180 200 220 240 y y [h] time T[orbits] [orbits] − m Figure 5. Gap Separation for a symmetric and asymmetric propeller structure for a moonlet with h = 400 h Hill radius. For the asymmetric propeller structure the moonlet was librating with a period of 80 orbits and an amplitude of 0.5 h. For all realizations, a moonlet-centered reference frame has been used. Left: Radial gap separation along the azimuth. Red and black solid lines denote the symmetric and asymmetric propeller structure. The asymmetric propeller structure is presented at different snapshots at T = 200, 220 and 240 orbits. Right: Radial gap separation for a symmetric (dashed) and asymmetric (solid) propeller structure at azimuthal distances to the moonlet of ∆y = 10 h ± (black), ∆y = 200 h (blue) and ∆y = 600 h (red). ± ± 13 effect of the moonlet motion on the gap width (about < W (∆y = 200 h) >= 2.76 h against a ± by estimating the gap width at 80 % gap closing < W (∆y = 400 h) >= 1.8 h for the inner gap a ± (Σ/Σ = 0.8). and < W (∆y = 200 h) >= 2.63 h against 0 a ± The resulting gap widths for the inner and < W (∆y = 400 h) >= 1.56 h for the outer a ± outer gap are shown in Figure6. A concave de- gap). crease for all curves can be seen, resulting from 3.2.4. Gap Depth the azimuthal gap relaxation. A similar trend holds for the gap depths. Another measurable quantity is the gap depth, given by d(t) = Σmin(r)−Σ0 . As in the In the panels of Figure6 the dashed lines refer Σ0 to the symmetric propeller structure, whereas asymmetric case the gap widths W (t) and - the solid lines represent the widths of the asym- lenghts L(t) are changing with time, an analo- metric gap structures. The red and black colors gous behavior is to be expected for d(t). denote the outer and inner gap structure. In Figure8 shows the time-evolution of d(t) for all representations the gap widths of the sym- the symmetric case (left panel) and the asym- metric and asymmetric gaps differ significantly. metric case (right panel) for different values of The moonlet libration can be identified by the ∆y. intersection points of the inner and outer gap While for the symmetric case (left panel) the width profiles (right panel, Figure6), where the values of ds(t) for the outer and inner pro- gap widths exchange their roles. peller wing coincide and slightly relax to the Figure7 shows a comparison of the time- steady state, the asymmetric value of da(t) evolution of the asymmetric Wa and symmet- (right panel) oscillates with the moonlet libra- ric Ws gap widths at two fixed azimuthal dis- tion, similarly to the width Wa(t). The ampli- tances (∆y = 200 h and ∆y = 400 h) and tude of this depth-variation is about ∆d(t) = ± ± dmax−dmin for the density level Σ/Σ = 0.8. The left panel 10%. 0
Gap widths: Gap widths: x = 0.5 T = 80 T = 220 x = 0.5 T = 80 T = 240 h = 400m, m,0 xm =h 0.5 , h,mTm = 80orbits orbits ,at t = 220 orbitsorbits h = 400m, m,0 xm =h 0.5 , h,mTm = 80orbits orbits ,at t = 240 orbitsorbits 5.0 5.0 inner: Wa(Σ/Σ0 = 0.8) inner: Wa(Σ/Σ0 = 0.8) outer: Wa(Σ/Σ0 = 0.8) outer: Wa(Σ/Σ0 = 0.8) inner: Ws(Σ/Σ0 = 0.8) inner: Ws(Σ/Σ0 = 0.8) 4.0 outer: Ws(Σ/Σ0 = 0.8) 4.0 outer: Ws(Σ/Σ0 = 0.8)
3.0 3.0
2.0 2.0 Gap Width W [h] Gap Width W [h]
1.0 1.0
0.0 0.0 0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 y y [h] y y [h] − m − m
Figure 6. Comparison of the gap width at Σ/Σ0 = 0.8 for the symmetric (dashed) and asymmetric propeller (solid) at T = 220 orbits (left) and T = 240 orbits (right). The widths of the inner and outer gap are given by the black and red colored lines. The results have been obtained from simulations of a h = 400 m sized moonlet, wich was librating with a radial amplitude and period of 0.5 h and 80 orbits for the asymmetric case.
Asymmetric gap widths: x = 0.5 T = 80 SymmetricSymmetric gap widths: gap widths:h = 400 m h = 400m, m,0 xm = 0h.5 , h, mTm = 80orbits orbits 4.0 4.0
3.5 3.5 [h] [h] 3.0 3.0 8) 8) . . = 0 2.5 = 0 2.5 0 0 Σ Σ / 2.0 / (Σ 2.0 (Σ W W 1.5 1.5
1.0 1.0 Gap Width Gap Width 0.5 0.5 at ∆y = 200h at ∆y = 200h ± ± at ∆y = 400h at ∆y = 400h 0.0 ± 0.0 ± 80.0 100.0 120.0 140.0 160.0 180.0 200.0 220.0 240.0 80.0 100.0 120.0 140.0 160.0 180.0 200.0 220.0 240.0 timeT[orbits] [orbits] timeT[orbits] [orbits]
Figure 7. Comparison of the gap width evolution of the inner (solid) and outer (dashed) propeller gap for a non- librating (left) and librating (right) central moonlet at Σ/Σ = 0.8 for ∆y = 200 h (black) and ∆y 400 h (red). 0 ± ± For the non-librating moonlet the values of W (t) fall on top of each other, giving mean values of < W > 2.7 h for s s ≈ ∆y = 200 h and < W > 1.9 h for ∆y = 400 h, respectively. For the librating moonlet, the gap widths oscillate ± s ≈ ± with the moonlet and show an anti-phase relation. Their mean values slightly differ: < W (∆y = 200 h) >= 2.76 h a ± against < W (∆y = 400 h) >= 1.8 h for the inner gap and < W (∆y = 200 h) >= 2.63 h against < W (∆y = a ± a ± a 400 h) >= 1.56 h for the outer gap. Interestingly, the amplitude of the oscillation is increasing for larger azimuth. ± 240 of the integration time, respectively. The in the same way with growing azimuth (except red and blue colors denote the outer and inner small variations due to the wakes and noise), gaps. Comparing the non-librating and librat- the gap depths change along the azimuth for ing moonlet a clear difference in the gap relax- the asymmetric propeller. ation can be observed. While for the symmetric Due to the retardation, the influence of propeller the inner and outer gaps are closing changes in the moonlet’s radial libration are 15
Asymmetric gap depths: Symmetric gap depths: h = 400 m h = 400 m, x = 0.5 h, T = 80 orbits 1.2 1.2 m m at ∆y = 10h at ∆y = 10h ± ± 1.1 at ∆y = 200h 1.1 at ∆y = 200h ± ± at ∆y = 600h at ∆y = 600h 1.0 ± 1.0 ± 0.9 0.9 0.8 0.8 0.7 0.7 0 0
Σ 0.6 Σ 0.6 / / Σ 0.5 Σ 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 80 100 120 140 160 180 200 220 240 80 100 120 140 160 180 200 220 240 time T[orbits] [orbits] time T[orbits] [orbits]
Figure 8. Evolution of the propeller gap depth for the symmetric (left) and asymmetric (right) propeller at different azimuthal positions to the moonlet ∆y = 10 h (black), ∆y = 200 h (blue) and ∆y = 600 h (red). The solid and ± ± ± dashed lines represent the inner and outer propeller gap.
Azimuthal gap profiles: Azimuthal gap profiles:
h = 400xm, m,0 =xm 0=.5 0h.5 , h,TmTm== 80 80Orbits orbits at , tT== 220 220 orbitsObits h = 400xm, m,0 =xm 0=.5 0h.5 , h,TmTm== 80 80Orbits orbits at , tT== 240 240 orbitsObits 1.00 1.00 0.90 0.90 0.80 0.80 0.70 0.70 0.60 0.60 0 0
Σ 0.50 Σ 0.50 / / Σ Σ 0.40 0.40 0.30 0.30
0.20 inner asymmetric Gap 0.20 inner asymmetric Gap outer asymmetric Gap outer asymmetric Gap 0.10 inner symmetric Gap 0.10 inner symmetric Gap outer symmetric Gap outer symmetric Gap 0.00 0.00 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800 y y [h] y y [h] − m − m Figure 9. Azimuthal gap relaxation profile for the symmetric (dashed) and asymmetric (solid) gap structure at T = 220 orbits (left) and T = 240 orbits. Comparing the inner (blue) and outer (red) gap profiles in the symmetric case, we clearly see how both profiles match perfectly. In the asymmetric case the symmetry of the propeller is broken, which can be seen from the different lengths and the different depths of the gaps at different azimuthal distances to the moonlet. found by the intersection points of the minima ima of the inner gap. This intersection point density curves (see right panel, Figure9) for the is caused at T 200 orbits, where the moonlet ≈ inner and outer gap. As an example, seen in is at its lowest radial elongation x = 0.5 h m − the right panel, the inner gap is closing faster (compare with Figure4). At this turning point, than the outer one until 400 h azimuthal dis- the moonlet starts to migrate outwards again. tance to the moonlet, respectively. There, an intersection point of both gap profile curves can 3.3.1. Gap Length be found. From this distance on, the outer gap From the azimuthal gap profile we define the is closing faster and stays over the density min- gap length L80 as the azimuthal distance to the 16 moonlet at Σ/Σ0 = 0.8 for the inner and outer sents the gap contrast of the symmetric pro- gap structure. peller structure along increasing azimuthal dis- The evolution of the gap length L80(t) = tance to the moonlet. Its maximum corre- L(Σ/Σ0 = 0.8, t) is presented in Figure 10, sponds to deviations caused by the wake region where the symmetric (red color) and asymmet- (Σ/Σ 2 10−2) and thus defines the level, 0 ≈ × ric (black color) propeller is presented. The where the symmetric and asymmetric propeller dashed and solid lines denote the outer and in- will not be distinguishable anymore. The black ner gaps. The dotted line represents the az- lines in Figure 11 represent the gap contrast for imuthal evolution of the azimuthal moonlet po- the asymmetric gap structure at different inte- sition as a reference. The asymmetric propeller gration times, where the solid, dashed and dot- gap length varies in phase with the moonlet ted lines show the gap contrast at T =200, 220 motion. For the non-librating central moon- and 240 orbits. The different snapshots for the let, the mean gap length < L80 > is about asymmetric propeller demonstrate, how the per- 539.5 h, respectively, while the for asymmetric turbation by the moonlet libration propagates propeller, the mean gap lengths slightly differ through the gaps. As an example, the minimum (< L80 >= 511 h for the inner gap against at T = 200 orbits shifts from ∆y = 200 h and < L >= 523 h for the outer gap) and the Σ/Σ = 0.1 to ∆y = 400 h and Σ/Σ = 0.12 80 0 − 0 − maximum difference of the gap lengths is about at orbit 220 and further to ∆y = 650 h and 300 h. Σ/Σ = 0.1 at orbit 240. The transport of 0 − Although the depths of the gaps only differ by perturbations along the azimuth resembles a about 10% (compare Section 3.2.4) these vari- propagating wave package. Like a dissipating ations result in large azimuthal changes in the wave package, the perturbation gets smeared gap lengths. Thus, studying the azimuthal gap along the azimuth due to diffusive effects. profiles is the favorable method to search for the imprint of the asymmetry. 3.4. The Gap Contrast 4. RING - MOONLET INTERACTIONS As already carried out, the azimuthal gap pro- The symmetry breaking of the propeller struc- files of the inner and outer gap structures dif- ture disturbs the force balance of the gravity by fer for a librating moonlet. In order to extract the ring ensemble reacting on the moonlet Seiler the perturbation by the moonlet motion and to et al.(2017). Here, we calculate the resulting study its propagation through the gaps, we de- force of the surrounding ring material on the fine the gap contrast as the difference of the in- moonlet. Therefore, we calculate the gravity of ner and outer gap profiles. every grid cell onto the moonlet and sum over all The retardation of the symmetry-breaking the contributing cells of the grid. We exclude a 1/2 perturbation by the moonlet libration along circular area r = (x x )2 + (y y )2 = − m − m the gap length sets a memory timescale T . 1.2 h around the moonlet from the force calcu- gap Changes in the radial moonlet motion, hap- lation in order to account for the physical di- pening within this timescale, become visible as mension of the moonlet, which is treated as a zero value crossings and help to differ a periodic point mass in our simulations. The resulting to- motion from a migration in this way. tal gravitional interaction between the ring and Figure 11 shows the gap contrast estimated the moonlet Fx and Fy in radial and azimuthal from the azimuthal gap relaxation profiles pre- direction is given by the sum of the contributing sented in Figure9. The red solid line repre- cells 17
Gap length:
h = 400 m, xm = 0x.5m, h,0 =Tm 0.=5 h 80 , orbitsTm = at 80tOrbits= 240 orbits 1000.00 symmetric 900.00 asymmetric y 800.00 m 700.00 ]
h 600.00 [ 500.00 400.00 300.00 Gap Length 200.00 100.00 0.00 100.00 − 80 100 120 140 160 180 200 220 240 time [orbits]
Figure 10. Comparison of the gap length at 80% relaxation for the symmetric (red) and asymmetric propeller (black). The length of the inner and outer gap are given by the solid and dashed lines. The results have been obtained from the simulation of a h = 400 m sized moonlet. For the asymmetric propeller the moonlet was librating with a radial amplitude of 0.5 h and period of 80 orbits. For comparison the azimuthal evolution of the librating moonlet is plotted as the black dotted line as well. For both profile plots a moonlet-centered reference frame has been chosen for better comparison.
Gap Contrast: x = 0.5 T = 80 h = 400 m,m,x0 m = 0.5h h, , Tm = 80 orbitsorbits 0.20 0.15 0.10 0.05 0.00 0
Σ 0.05 / − Σ 0.10 − 0.15 − 0.20 symmetric − asymmetric at T = 200 orbits 0.25 asymmetric at T = 220 orbits − asymmetric at T = 240 orbits 0.30 − 0 100 200 300 400 500 600 700 800 y y [h] − m Figure 11. Difference of the azimuthal gap relaxation of the inner and outer gap (gap contrast) for a non-librating
(red) and librating (black) central moonlet of size rh = 400 m at orbits 200, 220 and 240 (solid, dashed and dotted). let. The quantities ∆Σ = Σ Σ and ∆A = i i − 0 xm xi ∆xi ∆yi denote the difference in the surface Fx (x, y, xm, ym) = G∆A ∆Σi − × − ~r ~r 3 mass density with respect to the intial one and Grid | m − i| X (14) the surface of the equal-sized grid cells. The left panel in Figure 12 illustrates the evo- ym yi Fy (x, y, xm, ym) = G∆A ∆Σi − , lution of the normalized ring gravity, where the − ~r ~r 3 Grid | m − i| x (red) and y (blue) component of the ring grav- X (15) ity are following the moonlet motion (xm and ym with ~ri = (xi, yi) and ~rm = (xm, ym) the po- are represented by the dashed and solid black sition vectors of the i-th cell and the moon- 18
Ring-moonletxm,0 = 0.5 h interactions , Tm = 80 orbits xm,Gap0 = 0 Contrast.5 h , Tm = 80 orbits 1.50 0.15
1.00 0.10
0.50 0.05
0.00 0.00 0 Σ /
0.50 Σ 0.05 − −
Normalized Values 1.00 0.10 − − Fg,x/max(Fg,x) dy = 10 1.50 Fg,y/max(Fg,y) 0.15 dy = 200 − ym/max(ym) − dy = 600 xm/max(xm) ym/max(ym) 0.8 2.00 0.20 ∗ − 0 50 100 150 200 250 − 80 100 120 140 160 180 200 220 240 t time [orbits]t [orbits] time [orbits][orbits]
Figure 12. Left: Time evolution of the normalized moonlet motion in x (black dashed line) and y (black solid line) direction and the x (blue) and y (red) component of the ring gravity acting on the moonlet. Right: Evolution of the gap contrast at different azimuthal positions, illustrating, how the variation of the gap contrast is following the moonlet motion (black dashed line). lines). The right panel shows a comparison Note, that our analysis does not consider any of the gap contrast evolution and the moon- resonances in the vicinity of the propeller struc- let evolution, where the gap contrast is pre- ture, which might have an additional effect of sented at three different azimuthal distances to the final asymmetry for larger radial amplitudes the moonlet. The evolution for ∆y = 200 h and libration periods. ± and δy = 600 h displays the retardation set by For each simulation parameter, at each time ± the Kepler shear and the decreasing amplitude the above quantities will be determined. Fur- of the gap contrast with increasing azimuths. ther, the mean value, the maximum and mini- The force and the maximum contrast are fol- mum at each time will be estimated along the lowing the moonlet motion with a slight delay azimuth. Finally, for the complete simulation (phase shift). time frame the global maximum, minimum and mean values will be calculated. The resulting 5. SYSTEMATIC ASYMMETRY values will be presented in the following sec- PREDICTIONS tions. In this section we test the dependence of the 5.1. Dependence of Asymmetry on the propeller’s asymmetry on the a) the libration Libration Amplitude b) amplitude and the libration period. This al- We vary the libration amplitude xm,0 from lows us to make predictions for the observability 0.125 h through 4 h. of the asymmetry of propeller gap structures in In Figure 13 the dependence of the maximum Saturn’s rings. Therefore, at first, we keep the gap contrast from the radial amplitude of the libration period constant at Tm = 80 orbits and moonlet libration is shown. Here, we can fit the vary the initial radial amplitude xm,0. In a sec- data by the exponential relation ond analysis, we perform simulations, where the initial radial amplitude xm,0 = 1 h will be kept C (x ) = C C exp [ λx ] , (16) constant, while the libration period is varied. m 0 − max − m For both scenarios we will study the gap con- with Ci,0 = 0.925, Ci,max = 0.971 and −1 trast, the gap length and width at Σ/Σ0 = 0.8. λi = 0.458 h for the maximum Co,0 = 0.923, 19
Moonlet Size: h = 400 m, Tlib = 80 orbits 0.8 max(C) min( C ) 0.7 | | 0.925 0.971 exp 0.458 1 x − × − h m 0.923 0.974 exp 0.458 1 x − × − h m 0.6
0.5
0.4
0.3 Gap Contrast
0.2
0.1
0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 xm [h]
Figure 13. Dependence of the maximal asymmetry from the moonlet’s libration amplitude for constant libration period Tm = 80 orbits. −1 Co,max = 0.974 and λo = 0.458 h for the min- To summarize, with larger radial amplitudes imum gap contrast. the perturbation by the moonlet motion in- Starting from small amplitudes the contrast creases, which results in shorter gap lengths, is increasing for growing initial amplitude until broader gaps and larger gap contrast until the the saturation of about C = 0.925 is reached. perturbation exceeds a critical level, where the From this amplitude on the gap contrast is creation of the propeller structure is prevented. not increasing anymore. For amplitudes larger xm,0 = 1.6 h the moonlet starts to migrate 5.2. Dependence of Asymmetry on the through its created gaps, and thus, destroys its Libration Period own propeller structure, explaining the maxi- We vary the libration period of the moonlet mum saturation level. from Tm = 15 orbits to Tm = 320 orbits, while Figure 14 illustrates the dependence of the gap we keep the radial amplitude fixed at xm,0 = 1 h. length L80 (left panel) and the gap width Wa Figure 15 shows the dependence of the gap (right panel) on the libration amplitude. For the contrast on the libration period, where an ex- gap length, a clear drop in the mean gap length ponential relation given by can be noticed for larger amplitudes, while the maximum difference in the gap lengths of the inner and outer gap first increases from 200 h to C (T ) = C + C exp [ λT ] , (17) lib 0 max − lib 500 h for xm,0 = 0.125 h to xm,0 = 1 h and then reduces to about 200 h. With the gap length with C0 = 0.125,Cmax = 0.587 and λ = the gap width changes as well (see right panel 0.013 orbit−1 can be found. of Figure 14), but here, the mean gap width is For fixed radial amplitude the changes for increasing from 2.25 h to 2.6 h for larger ampli- larger libration periods result in a growing az- tudes, until at about xm,0 = 1.6 h a saturation imuthal excursion of the moonlet (see Eq. 11). level is reached. A similar result can be seen for As the libration period grows, the relative ve- the maximum difference of the inner and outer locity to the unperturbed orbit decreases and gap width, which increases from 0.5 h to 3.5 h, thus the perturbation by the changing moonlet respectively. position can be easily transported to larger az- imuthal distances, resulting in a more symmet- 20
Moonlet Size: h = 400 m, Tlib = 80 orbits 5.0 900.0 mean(Wi) mean(Li) min(Wi) min(Li) 4.5 800.0 max(Wi) max(Li)
8 mean(Wo) mean(Lo) .
8 700.0 4.0 min(Wo) . min(Lo)
= 0 max(W ) max(Lo) o 0 = 0
600.0 Σ 0 3.5 / Σ Σ /
Σ 500.0 3.0 400.0 2.5 300.0 2.0
Gap Length at 200.0 Gap Width[h] at 1.5 100.0
0.0 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 xm [h] xm [h]
Figure 14. Dependency of the gap lenght (left) and gap width (right) at Σ/Σ0 = 0.8 from the moonlet’s libration amplitude for the inner and outer gap.
Moonlet Size: h = 400 m, xm = 1 h 0.8 max(C) min( C ) 0.7 | | 0.125 + 0.587 exp 0.015 1 T × − orbits lib 0.112 + 0.480 exp 0.012 1 T × − orbits lib 0.6
0.5
0.4
0.3 Gap Contrast
0.2
0.1
0.0 0 50 100 150 200 250 300 350 Tlib [orbits]
Figure 15. Dependence of the maximal asymmetry on the moonlet’s libration period for constant radial amplitude xm,0 = 1 h. ric propeller structure and therefore in a lower fect transport of the perturbation through the gap contrast (see Figure 15). gaps. At this optimal parameter set the gap In Figure 16 the dependence of the gap length width reaches its maximum of about 2.5 h, re- (left panel) and the gap width (right panel) on spectively, while the differences in the gap width the libration period is presented. between inner and outer gap are at their max- The gap length (left panel) first increases from imum of about 2 h (compare right panel). For 350 h to about 500 h with respect to the libra- increasing libration periods, the relative veloc- tion period interval from T0 = 15 orbits through ity of the moonlet to its unperturbed orbit de- T0 = 80 orbits. For this interval, the maximum creases, resulting in smaller perturbations and difference in the gap lengths between outer and therefore in a more symmetric propeller struc- inner gap increase from 400 h to about 600 h. ture. For this reason, the mean gap length drops At Tm = 80 h the gap passage time equals the to about 500 h – close to the value of the un- libration period of the moonlet allowing the per- perturbed propeller gap length. Further, the 21
Moonlet Size: h = 400 m, xm = 1 h 4.0 900.0 mean(Wi) mean(Li) min(Wi) min(Li) max(Wi) 800.0 max(Li) 3.5
8 mean(Wo) mean(Lo) .
8 min(Wo) . 700.0 min(Lo) = 0 max(W ) max(Lo) o 0 3.0 = 0 Σ 0 600.0 / Σ Σ / Σ 500.0 2.5
400.0 2.0 300.0 Gap Length at Gap Width[h] at 1.5 200.0
100.0 1.0 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 Tlib [orbits] Tlib [orbits]
Figure 16. Dependence of the gap length (left) and gap width (right) at 80% gap relaxation from the moonlet’s libration amplitude for fixed radial amplitude xm,0 = 1 h. maximum difference in the gap length drops to 11.1 years = 6941 orbits and thus, the radial an almost constant level of 200 h as well. Ex- amplitude for the moonlet is given by ceeding a libration period of Tm = 80 orbits the mean gap width stays almost constant at 2.5 h, 2 ∆y 2 1845 km δx = = = 0.177 km (18) which is very close to the unperturbed value of | | 3 Ωt 3 6941 orbits 2.6 h. Nevertheless, the moonlet motion still in- This agrees with the radial amplitude esti- duces a difference of the gap widths of about mated from orbital fits by Spahn et al.(2018), 1 h. who estimated the radial deviation to be about To summarize, for small libration periods 200 m. The Hill radius of Bl´eriot is about (Tm Tgap) the asymmetry is large because r = 450 m (Hoffmann et al. 2016; Seiler et al. ≤ H of the strong perturbation by the moonlet mo- 2017) and thus, the estimated radial amplitude tion and the retardation, while for large periods corresponds to x 0.4 h. m,0 ≈ (Tm > Tgap) the asymmetry gets smaller due For Bl´eriot Seiler et al.(2017) estimated to quasi-static changes in the moonlet motion. T 0.5 years which corresponds to 313 orbits, gap ≈ However, the asymmetry is still measurable but respectively. Thus, for Bl´eriotwe need to con- depends on the actual libration phase angle of sider the case T = 11.1 years = 6941 orbits m the moonlet at the observation time. 313 orbits = 0.5 years = Tgap. At this ratio the gap contrast is already at its lowest level of about 0.125 in comparison to the noise level of the unperturbed propeller gap contrast of 0.02. The maximum gap contrast illustrated Figure 5.3. Application to the Propeller Bl´eriot 13 can be linearly approximated in the range The excess motion of Bl´eriot has been re- xm,0 = [0, 1.5 h]. Therefore, the expected gap constructed by three overlaying harmonic func- contrast level of about C0 = 0.125 estimated for tions, where we assume the moonlet to librate xm,0 = 1 h (see Figure 15) can be interpolated to with the largest amplitude and period of about xm,0 = 0.5 h, yielding 0.063, respectively. Fol- 1845 km and 11.1 years (see e.g. Seiler et al. lowing this interpolation, the maximum varia- 2017; Spahn et al. 2018). Translating the fit- tion in gap length and gap width are expected ted period to number of orbits results Tm = to be about 125 h and 0.5 h, respectively. 22
6. CONCLUSION AND DISCUSSION iii) Tm > Tgap : The quasi-static changes in In this work, we have studied the formation the moonlet motion result in a less asym- of asymmetric propeller structures in Saturn’s metric shape of the propeller. Thus, the A ring, assuming a libration of the moonlet. gap contrast is minimal and the variations Note, that in this work, we did not consider the in the gap width and gap length are rather reason for the moonlet libration, which we sim- small. ply considered as given. In pratice, librational 6.2. Libration vs. Migration behavior sensitively depends on the perturba- The retardation limits the observability of the tion forces and sources (e.g. resonances, density asymmetry and with it the chance to distinguish fluctuations, particle collisions). It turned out, a migration of the moonlet from a libration. that the additional moonlet motion is perturb- Changes in the direction of the moonlet motion ing the induced propeller structure, causing an (inward and outward movement) result in zero asymmetry in this way. The perturbation by value crossings in the gap contrast along the az- the moonlet motion gets transported through imuth. Depending on the libration period, three the gaps with the Kepler speed and thus arrives limiting cases can be distinguished: at the gap ends after a delay time Tgap. This retardation depends on the gap length and sets i) Tm < Tgap : The frequent changes in the a memory time, which finally causes the asym- moonlet motion can be seen due to the metric appearance of the propeller structure in retardation, resulting in at minimum two the images. The main implications of our sim- intersection points in the gap profile (or ulations are: zero values crossings for the gap contrast, respectively). Thus, the periodicity of the 6.1. Timescales vs Asymmetry moonlet motion is well refelcted by the We studied the dependence of the asymmetry propeller. on the libration period and amplitude. Three ii) T = T : At maximum two (at min- general cases have been investigated: m gap imum one) intersection points in the az- imuthal gap profile can be identified, de- i) Tm < Tgap : For small libration periods the asymmetry is large. This results from pending on the libration phase of the the strong perturbation by the moonlet moonlet at the observation time.
motion, which causes large variations in iii) Tm > Tgap : The slow outward and in- all gap properties, which even can prevent ward migration of the moonlet results in the propeller formation. Thus, small li- at maximum one (minimum zero) observ- bration periods are similar to the scenario able intersection points. of large libration amplitudes. In order to identify a librational motion of ii) Tm = Tgap : The strongest asymmetry the moonlet, the libration period needs to be has been identified, if the libration period smaller than the memory time Tm < Tgap. For matches the gap timescale. In this case, larger libration periods the gap contrast will the perturbation by the moonlet motion only change its sign at the turning points of the gets transported in the most effective way radial moonlet libration. In between no change ( in a kind synchronized way), resulting in sign will be visible and therefore the moon- in the largest variations in the gap length let libration might be misinterpreted as a radial and width. drift. 23
6.3. Observability of the Asymmetry than Bl´eriotbut show similar residual ampli- In order to measure the asymmetry of the pro- tudes (Tiscareno et al. 2010; Spahn et al. 2018), peller structure, our analysis has shown, that resulting in larger radial amplitudes for an as- the azimuthal profiles are the most favorable sumed moonlet libration. Further, due to the to consider for processing Cassini ISS images. smaller size of these moonlets, their induced Especially for larger azimuthal distances to the propeller wings’ azimuthal extent is smaller by 3 moonlet (∆y > 100 h), already small changes Mm/ν0 h /ν0 L/ν0 (Spahn and Sremˇcevi´c ± ∝ ∝ in the gap minima can result in large differences 2000), allowing to record both propeller wings in the gap lengths. at the same time with one observation. However, in case of high-resolution images, 6.6. Propeller-Moonlet Interactions such as UVIS scans, small variations in the radial gap profile can be resolved, allowing to The formation of an asymmetric propeller study the asymmetry even in those profiles. structure due to the motion of the moonlet Nevertheless, in all cases both, inner and outer causes a non-balancing force of the ring material gap, need to be recorded. on the moonlet. Calculating the force of the ring material on the moonlet we find, that it is dom- 6.4. Asymmetry of the Propeller Bl´eriot inated by the azimuthal amplitude of the moon- let (Seiler et al. 2017) and follows the moonlet Our predictions yield, that an asymmetry in motion with a phase shift according to the Ke- the gap contrast should be visible by about pler shear. Thus, our simulations agree with 6.3 %, which still is larger than the noise level the simple propeller-moonlet interaction model for the unperturbed propeller. The gap width suggested by Seiler et al.(2017). differences comparing inner and outer gap struc- ture should be about 0.5 h and the gap lengths 6.7. Outlook are expected to differ by 125 h. These variations Although the implementation of a librating should be still detectable, but depend on the li- moonlet is a simplification, our simulation how- bration phase of the central moonlet at the time ever gives an insight of how the propeller struc- of observation. ture reacts to perturbations and how the re- Due to the large libration period of Bl´eriot sulting asymmetry will look like. Further, our at maximum one zero value crossing for the gap simulations have shown, that the retardation contrast can be expected, which makes it impos- along the azimuth is the main mechanism to sible to judge whether the moonlet is librating make the asymmetry measurable in the images. or migrating considering one single observation. Therefore, even for an underlaying stochastic An analysis of several images of Bl´eriotat dif- migration of the central moonlet, an asymmetry ferent times would be necessary to search for might form. The observability of the resulting moving patterns (changes in the sign and am- asymmetry then will depend on the collision fre- plitudes) in the gap contrast along the azimuth. quency and the memory time of the gap. However, in order to build a fully consitent 6.5. Asymmetry of other Propellers model, as a next step, the moonlet needs to be Although the asymmetry for Bl´eriotmight be simulated in a box where it is allowed to move rather small, the asymmetry for the other gi- freely. In this way, a stochastic moonlet mo- ant trans-Encke propellers might be easier to tion can be simulated as well, which can be di- detect. As an example, the propeller struc- rectly compared to the unperturbed and librat- tures Earhart and Santos Dumont are smaller ing moonlet. 24
Further, allowing the moonlet to move freely estimate the viscosity of the surrounding ring within the simulation box will show, whether material from the images. the back-reaction of the ring material is able Our simulation code uses an isothermal model to conserve an initial libration of the moonlet, for the ring environment. This might result in which will be the final consistency test for the larger viscosity values in the close vicinity of the suggested propeller-moonlet interaction model. moonlet. For a consistent model, the tempera- Our introduction of the gap contrast allows to ture needs to be considered in the simualtions directly observe the propagation of the pertur- as well by including the energy equation. This bation by the moonlet motion along increasing might have an additional effect on the observ- azimuth. It turned out that the perturbation ability of the asymmetry. by the moonlet’s motion appears and behaves like a moving wave package which gets smeared This work has been supported by the Deutsche along the azimuth due to the damping by the Forschungsgemeinschaft ( Sp 384/28-1,2, Ho ring material. This permits to study the disper- 5720/1-1) and the Deutsches Zentrum f¨urLuft- sion relation of the perturbation and further to und Raumfahrt (OH 1401).
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