Analysing the New Saturnian Rings, R/2004 S1

and R/2004 S2

S. M. GIULIATTI WINTER, R. SFAIR, D. C. MOURAO andT. A. BASTOS

UNESP – Campus de Guaratingueta, Av. Ariberto P. da Cunha, 333, Guaratingueta, Sao Paulo,

12500000, Brazil(E-mail: silvia@feg.unesp.br)

(Accepted 4 May 2006)

Abstract. The Cassini-Huygens arrival into the Saturnian system brought a large amount of data about

the satellites and rings. Two diffuse rings were found in the region between the A ring and Prometheus.

R/2004 S1 is coorbital to Atlas and R/2004 S2 is close to Prometheus. In this work we analysed the

closest approach between Prometheus and both rings. As a result we found that the satellite removes

particles from R/2004 S2 ring. Long-term numerical simulations showed that some particles can cross

the F ring region . The well known region of the F ring, where small satellites are present and particles

are being taking from the ring, gains a new insight with the presence of particles from R/2004 S2 ring.

The computation of the Lyapunov Characteristic Exponent reveled that the R/2004 S2 ring lies in a

chaotic region while R/2004 S1 ring and Atlas are in a stable region. Atlas is responsible for the

formation of three regimes in the R/2004 S1 ring, as expected for a satellite embedded in a ring.

Keywords: Lyapunov Characteristic Exponent, numerical simulations, planetary rings

1. Introduction

The gravitational interaction between ring particles and satellites is respon-sible for a variety of structures found in the planetary rings and also for themaintenance of narrow rings against dissipative forces (Goldreich and Tre-maine, 1979). Cassini images have revealed the existence of two diffuse rings,R/2004 S1 and R/2004 S2, in the region between the A ring and the satellitePrometheus. The satellite Atlas is embedded in the R/2004 S1 ring andPrometheus is very close to R/2004 S2 ring (Porco et al., 2005). Therefore,these two satellites can significantly affect the ring particles behaviour. Theserings are formed by particles of very small size implying that dissipativeforces, such as the Poynting–Robertson drag, can also be important for theirevolution.

Many works have analysed the effects on the ring particles due to thesatellite perturbations. We will describe some of them related to the F ring ofSaturn, since the dynamics between this ring and the close satellite Prome-theus is widely explored due to the unusual features found in the F ring. The

Earth, Moon, and Planets (2005) 97: 189–201 � Springer 2006

DOI 10.1007/s11038-006-9082-x

first work on a numerical study this singular ring was made by Showalter andBurns (1982). They studied the short-term effects of the satellite on thisnarrow ring, both in eccentric orbits. They suggested the formation ofclumps, one of the structures found in the F ring, produced by this mecha-nism. Lissauer and Peale (1986) presented two models to explain the braidedappearance of the F ring. One model consisted of a small satellite embeddedin the ring and the other model consisted of a single shepherd satellite dis-turbing the ring with the appropriate initial conditions. They also explainedthe waves and kinks by radial displacement of the ring particles generated bythe passage of shepherd moons.

Borderies et al. (1983) analysed the long-period variations of the eccen-tricity and apse precession rate imposed on a ring by a nearby satellite andapplied this result to the F ring and its ‘‘shepherd satellites’’, Prometheus andPandora. They verified that the F ring and Prometheus will experienceperiodic closest encounters. Murray and Giuliatti Winter (1996) carried out amore detailed analysis of the secular interactions of this system. They con-cluded that the satellite can enter the ring every about 19 years and only amassive ring can avoid this closest approach configuration.

Using the orbits of the multiple structure of the F ring derived by Murrayet al. (1997), Giuliatti Winter et al. (2000) analysed the behaviour of the ringparticles strands after being disturbed by Prometheus at closest approach.Their results suggested that a gap can be formed in the inner three strandsdue to changes in the eccentricity and semi major axis of the ring particlecaused by the encounter with Prometheus. The primary contribution to theformation of the gap is caused by particles which are scattered to higher orlower orbits. This scattering of ring particles due to Prometheus perturbationhas been recently detected by Cassini spacecraft (Murray et al., 2005).Murray et al. (2005) also analysed the strands configuration disturbed byPrometheus taking into account the new data on the F ring derived byCassini images.

In this paper we analyse the gravitational perturbation caused by thesatellites, Prometheus and Atlas, on these two new discovered rings. Thispaper is organized as follows. In Section 2 we analyse the perturbations dueto both satellites, Prometheus and Atlas, at closest approach with these tworings. Section 3 deals with long-period perturbations on the ring particles andin the last section the results will be discussed.

2. Perturbations at the Close Approach

Two satellites are close to the newly discovered rings. Prometheus is close toR/2005 S2 and Atlas is embedded in the R/2005 S1 ring. Figure 1 illustratesthe orbital paths of the satellites, Prometheus and Atlas, the new rings and

190 S. M. GIULIATTI WINTER ET AL.

also the multiple structure of the F ring. The multiple structure, formed bythree strands, is located outside the orbit of Prometheus. The orbital elementsof the satellites, Prometheus and Atlas, and the locations and width (w) of thenew rings are displayed in Table I, where a is the semi major axis, e theeccentricity, r the radius (midpoint) and w the width of the ring. These valuesare derived from Porco et al. (2005). The values for the F ring were takenfrom Murray et al. (2005).

As can be seen in Figure 1 the satellite Prometheus, at this particularconfiguration, approaches the R/2004 S2 ring. Since the radius of Prome-theus is about 50 km, it is likely that the satellite enters the ring. Similarconfiguration can be seen between Prometheus and the strands of the F ringat the closest approach configuration, when the satellite is at the apocentre ofits orbit and the ring particles are at the pericentre of their orbits (GiuliattiWinter et al., 2000).

137000

137500

138000

138500

139000

139500

140000

140500

141000

0 50 100 150 200 250 300 350

r (k

m)

True Longitude (deg)

Figure 1. The orbital paths of the satellites Prometheus and Atlas (solid lines) and the rings(dot dashed lines). The multiple structure (three ringlets) of the F ring is located outside the

orbit of Prometheus. The R/2004 S1 ring is coorbital to Atlas and R/2004 S2 ring is close tothe satellite Prometheus. The radii are in km and the longitude is in degrees. The outer andinner edges of R/2004 S1 and R/2004 S2 are also illustrated in this figure (dashed lines).

TABLE IThe values of the semi major axis (in km) and eccentricity of the satellites and the radius(midpoint) (in km) and width (in km) of the new rings (Porco et al., 2005)

Elements Prometheus Atlas R/2004 S1 R/2004 S2

a (km) 139,380 137,665 – –

e 0.0023 0.0012 – –

r (km) – – 137,630 138,900

w (km) – – 300 300

191ANALYSING THE NEW SATURNIAN RINGS

In this section we analyse, by numerical simulation, the effects ofPrometheus during the close encounter with the rings. We analyse thisencounter by using the ‘‘box’’ model, where the ring particles are randomlydisplaced in a small region (box), radially and azimuthly limited. The box isradially limited in the range aring±(w/2), where aring is the semi major axis ofeach ring, and azimuthly limited in a small angle on either side of the lon-gitude of the close approach of the perturber. Since the ring particle spendsmost of its time on an unperturbed orbit, with only an occasional change inits orbital elements at the approximation with the satellite, an analysis of thisperturbation in the neighbourhood of the satellite is a good approximation.The particle is removed when the distance between the satellite and theparticle is about the radius of the satellite. At this time a new particle starts atthe edge of the box. The time of the integration is three orbital periods of thesatellite therefore collisions between the particles and the precession of theorbits, caused by the oblateness of the planet, can be neglected.

Each numerical simulation has been performed using the model of thecircular restricted three body problem: Saturn–Prometheus-ring particle. Theequations that describe the motion of the particle in a frame rotating withthe same angular velocity as the satellite are given by Murray and Dermott(1999):

€x� 2 _y ¼ @U@x

€yþ 2 _x ¼ @U@y

where

U ¼ 1

2ðx2 þ y2Þ þ 1� l

rsþ lr1

and rs ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðxþ lÞ2 þ y2q

is the distance between the particle and the planet,

r1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðx� 1þ lÞ2 þ y2q

is the distance between the particle and the satelliteand l is the mass of the satellite divided by the mass of the planet.

Since the eccentricity of the satellite is small, the guiding centre approxi-mation has been used to describe its motion. In the guiding centre approxi-mation, the motion of the satellite in an elliptical orbit around a planet isviewed in a reference frame that is centred on a point, the guiding centre, thatrotates about the planet in a circle of radius equal to the satellite’s semi majoraxis, with the angular speed equal to the satellite’s mean motion. The coor-dinates of the satellite centred on the planet are (Murray and Dermott, 1999):

xsat ¼ asatð1� esat cosMsatÞ;ysat ¼ 2asat esat sinMsat;

192 S. M. GIULIATTI WINTER ET AL.

where asat, esat, and Msat are the semi major axis, eccentricity and meananomaly of the satellite.

The density and diameter of the satellites are: for Atlas the mean density isassumed to be 0.5 g/cm3 and the diameter to be 20 km (Porco et al., 2005),for the satellite Prometheus the mean density is 0.5 g/cm3 (Porco et al., 2005)and the diameter is 100 km (Murray et al., 2005).

Figure 2 shows the position of the particles of the two discovered ringsdisturbed by Prometheus after three orbital periods of the satellite. Thesample of 14,000 particles were randomly divided between the two rings andtheir orbits were numerically simulated. We have used the Burlish Stoernumerical integrator (Press et al., 1990). Simple boundary conditions havebeen used to return the particle which leaves the box in the azimuthaldirection and no constrain was adopted for that particle which leaves thebox in the radial direction. When the distance between the particle and thesatellite is about the radius of the satellite, the particle is removed and a newparticle starts at the edge of the box.

We can verify that the perturbation caused by Prometheus on R/2004 S1ring is negligible since this ring is further from the satellite. However, eachtime Prometheus encounters R/2004 S2 ring it scatters particles in directionof the F ring region and few particles are scattered in the direction of Saturn.As has been discussed in Giuliatti Winter et al. (2000) the geometry of theencounter is responsible for scattering particles closer to the satellite whileothers are scattered for orbits closer to the planet. The radial perturbation ofthe satellite on the ring, at the closest approach, is equivalent to its width.

As has been described before, Giuliatti Winter et al. (2000) analysed theeffects of Prometheus on the multiple structure of the F ring at the closestapproach configuration. As a result they found that the particles are scatteredin direction of Prometheus. Taking into account the new data of the strands(Murray et al., 2005) we presented, in Figure 3, the positions the F ringstrands and the two new rings after being disturbed by Prometheus. A sampleof 40,000 particles was randomly divided among the three strands and thedust envelope. Each time Prometheus disturbs the rings, particles are scat-tered in its direction. Therefore, Prometheus’ perturbation is responsible forpopulated the F ring region with scattered particles from both rings, the Fring inner strand and R/2004 S2 ring.

We have used the circular restricted three body problem to numericalsimulate a sample of 600 particles, initially located at aring±150 km, dis-turbed by Atlas. We have also used the box model, as described before, sincethe particle will suffer the perturbation in the neighbourhood of the satellite.

Since Atlas is embedded in R/2004 S1 three well-known regions areformed: (a) horseshoe orbits, (b) empty region, where particles will be scat-tered after a short period of time and (c) further from the satellite, waves will

193ANALYSING THE NEW SATURNIAN RINGS

0.98

0.985

0.99

0.995

1

1.005

-0.1 -0.05 0 0.05 0.1

X

Y

0.98

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0.985

0.99

0.995

1

1.005

-0.1 -0.05 0 0.05 0.1

X

Y

(a)

(c)

(b)

Figure 2. The position of the ring particles after: (a) one, (b) two and (c) three orbital periodsof the satellite.

194 S. M. GIULIATTI WINTER ET AL.

0.98

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1

1.005

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X

Y

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1

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0.99

0.995

1

1.005

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-0.1 -0.05 0 0.05 0.1

X

Y

(a)

(b)

(c)

Figure 3. The positions of the particles of the F ring strands and the two new rings after (a)

one, (b) two and (c) three orbital periods of the satellite. It can be seen that particles from R/2004 S2 ring are scattered in direction of the F ring region due to the perturbation of thesatellite Prometheus.

195ANALYSING THE NEW SATURNIAN RINGS

be generated due to gravitational interaction between distant particles andthe satellite. Figure 4 shows these three different regimes.

An embedded satellite can keep a particle in a horseshoe orbit. Dermottand Murray (1981) derived the width (Whs) for the horseshoe region as

Whs � 0:5l1=3asat

The chaotic region, formed by perturbations in the vicinity of the satellite,has a width (Hanninen, 1993) given by Wc � 2:1l1=3asat.

The width of the horseshoe and chaotic regions derived from theseequations are ~10 km and ~45 km, respectively. These values are comparablewith the values obtained from the numerical simulation. The values of Whs

and Wc obtained from Figure 4 correspond to ~15 km and ~50 km,respectively.

As has been verified Atlas can retain a sample of coorbital particles inhorseshoe orbits. However, since the particles are very small particles (orderof 1 lm) the effects of the Solar Radiation Pressure will change this config-uration. The particles will stay in this horseshoe region although in confinedchaotic orbits, as has been discussed in Giuliatti Winter et al., (2004). A gapof width ~40 km separate these horseshoe orbits from the wave region. Theamplitude of these waves will decrease from the satellite’s orbit (Murray andDermott, 1999) and they can not be seen in the edges of the ring.

Our results showed that the perturbation of Atlas on R/2005 S2 isnegligible.

0.9994

0.9996

0.9998

1

1.0002

1.0004

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

X

Y

Figure 4. Plot of position of the ring particles during the numerical simulation. The formation

of three different regimes caused by the presence of an embedded satellite (Atlas) in a ring (R/2004 S1) can be seen: particles in horseshoe (coorbital to the satellite) and chaotic orbits and,waves after the chaotic region.

196 S. M. GIULIATTI WINTER ET AL.

3. Long-term Perturbations

The long-term perturbations due to the satellites have been analysing bycomputing the variation of the semi major axis and eccentricity for a sampleof three particles located in the R/2004 S2 ring. In these numerical simula-tions the particle is in orbit around an oblate planet and it suffers the per-turbation of the satellites Prometheus and Atlas. At the initial time theparticles were located at 138,800, 138,900 and 139,000 km. The initialeccentricity for each particle was taken to be zero.

Figure 5 shows the variation of the eccentricity and the semi major axisfor a time span of 160 years. The variation of the semi major axis of theparticles, showed in Figure 5b and f, permit them to cross the F ring regionwhile the particle in Figure 5d approaches the outer edge of the R/2004 S1ring.

As we can see there is no pattern in the variation of these two orbitalelements (semi major axis and eccentricity). Although this is a characteristicof a chaotic trajectory, these plots alone do not necessary mean that theparticles are in chaotic orbit.

In order to verify if these orbits are chaotic we calculated the maximumLyapunov Characteristic Exponent. The system is formed by the particle-anoblate Saturn–Atlas–Prometheus. We have included the gravity coefficientsJ2, J4 and J6 (Null et al., 1981).

The divergence of two nearby trajectories was analysed by plotting log tversus log c, where c is given by Murray and Dermott (1999)

c ¼ limt!1

lnðd=d0Þt� t0

where d and d0 are the initial and the final displacements between the twonearby orbits and t and t0 are the initial and the final times.

Figure 6 shows the plots of log c versus log time, where the time is given inorbital periods of the satellite, for a sample of three particles located at semimajor axis 138,800, 138,900 and 139,000 km (interior to the R/2004 S2 ring).It can be seen that these particles present chaotic behaviour. The LyapunovCharacteristic Exponent obtained for a range of semi major axis from137,000 to 139,000 km indicates that the region where the R/2004 S2 ring liesis chaotic. The Lyapunov time is ~100 orbital periods. This value wasobtained of c=10)2/orbital period for the maximum Lyapunov Character-istic Exponent of the chaotic orbit.

The long period behaviour of Atlas due to the perturbation of Prometheuswas analysed. In this simulation (Figure 7) the effects of the oblateness ofSaturn was also included. The slope of the curve is ~ )1, indicating that Atlashas a regular orbit for this period of integration.

197ANALYSING THE NEW SATURNIAN RINGS

4. Discussion

Two diffuse rings were found in the region between Prometheus and the Aring of Saturn. In this paper we analysed the effects on them due to thepresence of two close satellites, Atlas and Prometheus.

Our numerical simulations show that Prometheus can scatter particles inthe direction of the F ring region at every closest approach with R/2004 S2ring. Since Prometheus is also removing particles from the F ring in directionof Saturn, the region between Prometheus and the F ring can be populated byparticles from both rings. Long period variations of the semi major axis of a

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a (k

m)

t (years)

(a)

(c)

(e) (f)

(d)

(b)

Figure 5. Variation of the eccentricity and semi major axis for a time span of 160 years for asample of three particles located in the R/2004 S2 ring.

198 S. M. GIULIATTI WINTER ET AL.

-2.5

-2

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0

1 1.5 2 2.5 3 3.5 4 4.5 5log t

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log

γlo

g γ

log

γ

log t

(a)

(b)

(c)

Figure 6. Plots of log c versus log time, where the time is given in orbital periods of thesatellite, for a sample of three particles located at (a) 138,800 km, (b) 138,900 km and (c)139,000 km. The initial eccentricity of each particle was adopted to be zero.

199ANALYSING THE NEW SATURNIAN RINGS

sample of particles show that they can cross the F ring region. If R/2004 S2ring is only populated by small particles this effect will not pose as a damagefor the F ring. However, if small moonlets share the orbit of the R/2004 S2ring, these moonlets, after being scattered to higher orbits, can provokefeatures in the F ring.

Atlas, which does not cause any significant change in the outer ring, isresponsible for the formation of three regimes in the R/2004 S1 ring.Horseshoe orbits, chaotic zone due to scattered particles and waves.

However, from the numerical simulations it is clear that is the satellitePrometheus which play an important rule in the dynamics of this region, byremoving particles in the opposite direction of Saturn. The well known regionof the F ring, where small satellites are present and particles are being takingfrom the ring, gains a new insight with the presence of particles from R/2004S2 ring.

Due to Prometheus’ perturbations and dissipatives effects, such as thePoynting Robertson drag which can scatter particles in the direction of the Aring, a confinement mechanism is necessary to maintain these rings. Newdata from the Cassini spacecraft can help to identify the source of these tworings and enlighten the dynamics present on them.

Acknowledgements

SMGW thanks FUNDUNESP, RS and DCM thank CNPq and TAB thanksPAE/UNESP for the financial support.

-4

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0

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γ

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Figure 7. Plot of log c versus log time, where the time is given in orbital periods ofPrometheus, for the satellite Atlas disturbed by Prometheus.

200 S. M. GIULIATTI WINTER ET AL.

References

Borderies, N. et al.: 1983, Icarus 53, 84–89.Dermott, S. F. and Murray, C. D.: 1981, Icarus 48, 1–11.Giuliatti Winter, S. M. et al.: 2000, Plan. Space Sci. 48, 817–827.Giuliatti Winter, S. M. et al.: 2004, Astron. Astrophys 418, 759–764.

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Murray, C. D. and Dermott, S.: 1999. Solar System Dynamics, Cambridge Univ Press,London.

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Murray, C. D. et al.: 1997, Icarus 129, 304–316.Murray, C. D. et al.: 2005, Nature 437, 1326–1329.Null, G. W. et al.: 1981, Astron. J. 86, 456–468.Press, W. H., et al.: 1990, Numerical Recipes, Cambridge University Press, Cambridge.

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