MIGRATION OF SMALL MOONS IN SATURN's RINGS

In the A ring, satellites with r_H >~20 km are too massive for fast migration.

“Satellites with r H 20 km are too massive for fast migration.”

In the A ring, small moons with Hill radii between a few km and 20 km can migrate in fast mode.

“Between a few km and 20 km, small moons can migrate in fast mode.”

When a moon reaches the edge of the ring, migration ceases.

“When a moon reaches the edge of the ring, migration ceases.”

• Also stated in: § 5.4. Observational Tests · plain

  “When a moon reaches the edge of the ring, migration ceases.”

The viscous migration rate in the A ring is about -3×10^-6 km/yr for r_H >~24 km.

“da visc dt ≈ -3 × 10 -6 1.3 × 10 5 km a ν rad 85 cm 2 s -1 km yr -1 r H 24 km, ν rad 0. (28”

Starting with initial conditions identical to Salmon et al. (2010), the simulations yield a gradual expansion of a narrow ring of small particles into a broad disk.

“Starting with initial conditions identical to Salmon et al. (2010), our simulations yield a gradual expansion of the narrow ring into a broad disk.”

• cites Salmon et al. (2010)Not checked yet

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The fast migration rate in the A ring is approximately ±17 km/yr for 2 km < r_H < 24 km.

“da fast dt ≈ ±17 a 1.3 × 10 5 km 3/2 Σ 40 g cm -2 km yr -1 2 km r H 24 km, (26”

Moons with Hill radii r_H ∼ 2-24 km should migrate through the A ring in 1000 yr.

“We predict that moons with Hill radii r H ∼ 2-24 km should migrate through the A ring in 1000 yr.”

Numerical simulations demonstrate that orbital resonances and stirring by distant, larger moons (e.g., Mimas) may be important factors for Daphnis.

“numerical simulations demonstrate that orbital resonances and stirring by distant, larger moons (e.g., Mimas) may be important factors.”

• AssumptionThe authors state this as an assumption: The ring is not self-gravitating in the N-body simulations.

The embedded migration rate in the A ring is at most about 10^-4 km/yr for r_H ~ 2 km.

“da emb dt -10 -4 10 m h 2 a 1.3 × 10 5 km 1/2 × Σ 40 g cm -2 km yr -1 r H 2 km, (25”

For at least 100 Myr, the time evolution of the surface density distribution in the simulations closely follows Salmon et al. (2010).

“For at least 100 Myr, the time evolution of the surface density distribution in our simulations closely follows Salmon et al. (2010), who solve the 1D radial diffusion equation for a disk of small particles.”

• cites Salmon et al. (2010)Not checked yet

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Objects with Hill radii less than roughly 0.25 km migrate on a timescale comparable to the age of the solar system.

“Objects with Hill radii less than roughly 0.25 km migrate on a timescale comparable to the age of the solar system”

• cites Goldreich & Tremaine 1982Not checked yet

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For a moon with r_H = 7 km and initial semimajor axis 130,000 km in a ring with Σ = 40 g cm^-2, total annular width 56 km, and gap halfwidth 14 km, the migration rate is nearly independent of the masses of ring particles.

“Although migration is more stochastic with larger ring particles, the migration rate is nearly independent of the masses of ring particles.”

In the size range r_H ∼ 2-24 km, moons orbiting in an empty gap or in a full ring eventually migrate at the same rate.

“In this size range, moons orbiting in an empty gap or in a full ring eventually migrate at the same rate.”

After moons reach e < 10^-4, the drop in e is more gradual and includes additional stochastic variations due to perturbations in the surface density of swarm particles.

“After moons reach e < 10 -4 , the drop in e is more gradual and includes additional stochastic variations due to perturbations in the surface density of swarm particles.”

Resonant interactions with Janus and Epimetheus stir up enough ring material to create an obstacle for smaller moons.

“Resonant interactions with Janus and Epimetheus stir up enough ring material to create an obstacle for smaller moons.”

Stirring by distant moons results in a kinetic energy input per unit mass of P stir ∼ 2 × 10 -7 dyne g -1 .

“Stirring by distant moons results in a kinetic energy input per unit mass of P stir ∼ v 2 Kep de 2 /dt ∼ 2 × 10 -7 dyne g -1 .”

In several simulations, the eccentricity of ring particles grows at a rate de 2 /dt ∼ 10 -11 yr -1 , within a factor of two of the prediction from Equation (37).

“In several simulations for Figure 10, the eccentricity of ring particles grows at a rate de 2 /dt ∼ 10 -11 yr -1 , within a factor of two of the prediction from Equation (37).”

In simulations with a moon r_H = 14 km at 130,000 km embedded in rings with fixed surface density and swarm particles of various sizes, fast migration begins nearly immediately when the corotation zone is initially free of disk particles in an annulus of 3 r_H.

“In all simulations, the corotation zone is initially free of disk particles in an annulus of 3r H . Fast migration begins nearly immediately.”

In all cases, the onset of fast migration requires a short time, smaller than the synodic period of material at the gap edge.

“In all cases, the onset of fast migration requires a short time, smaller than the synodic period of material at the gap edge.”

Gravitational jostling of orbits by distant moons is comparable to the expected radial excursions of small moonlets comparable in size to Blériot and other propellers in the rings.

“Gravitational jostling of orbits by distant moons is comparable to the expected radial excursions of small moonlets comparable in size to Blériot and other propellers in the rings.”

Within 10-20 yr, gravitational scattering removes roughly half the mass of the corotation zone of a small moon.

“Within 10-20 yr, gravitational scattering removes roughly half the mass of the corotation zone.”

Over a one-year time frame, the total migration of a small moonlet is comparable to the variation in semimajor axis from jostling by distant moons.

“Over a one-year time frame, the total migration is comparable to the variation in a from jostling by distant moons.”

Without viscosity, particles with Hill radii comparable to the propeller moonlets (r H 1 km) can maintain a fairly empty corotation zone once they start migrating.

“Without viscosity, particles with Hill radii comparable to the propeller moonlets (r H 1 km) can maintain a fairly empty corotation zone once they start migrating.”

Fast migration ensues for a moonlet with r H = 0.7 km in a ring with surface density Σ = 0.1 g cm -3 when the corotation zone is initially clear and no viscous diffusion can fill in the gap.

“Thus, fast migration ensues, and there is a subsequent change in orbital distance, Δa (solid black curve; the dashed line shows the theoretical prediction).”

Without initial clearing of the corotation zone, there is a time lag ∼T_clear in the onset of fast migration.

“Without this initial clearing of the corotation zone, there is a time lag ∼T clear in the onset of fast migration (see Figure 5; Kirsh et al. 2009).”

• cites Kirsh et al. 2009Not checked yet

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Other rises in eccentricity occur at the orbits of Atlas and Prometheus and at many other labeled resonances.

“Other rises in e occur at the orbits of Atlas and Prometheus-which lie beyond the outer edge of the ring in regions of very low surface density-and at many other labeled resonances.”

Density perturbations can change migration rates substantially.

“These simulations demonstrate that density perturbations can change migration rates substantially.”

These larger satellites migrate freely.

“These larger satellites migrate freely.”

Janus/Epimetheus stir up the eccentricities of ring particles close to the 5:4 resonance.

“In all cases, Janus/Epimetheus stir up the eccentricities of ring particles close to the resonance.”

For a simulated moon with r_H = 14 km in the A ring, after the mass in the corotation zone falls to about a third of its starting value, fast migration starts.

“After the mass in the corotation zone falls to about a third of its starting value, fast migration starts.”

When a moonlet is allowed to migrate, its semimajor axis shifts in a stepwise fashion that tracks the theoretical prediction.

“When the moonlet is allowed to migrate (as in the black and dark gray curves, which have particles that are 1% and 2% of the moonlet mass, respectively), its semimajor axis shifts in a stepwise fashion which tracks the theoretical prediction (dashed curve).”

At a distance of 130,000 km from Saturn, ring particles at resonance have radial excursions of a few kilometers.

“At a distance of 130,000 km from Saturn, these ring particles have radial excursions of a few kilometers.”

At resonance, Janus and Epimetheus can stir ring particles to fairly high eccentricity (e p ∼ 10 -5).

“At resonance, Janus and Epimetheus can stir ring particles to fairly high eccentricity (e p ∼ 10 -5 ; see Figure 10).”

Moons interacting with tracer particles tend to begin migrating earlier than moons surrounded by massive swarm particles.

“Moons interacting with tracer particles tend to begin migrating earlier than moons surrounded by massive swarm particles.”

For a small moon with r_H = 7 km and a gap with half-width = 2 r_H, fast migration initiates in every simulation in the suite of tests.

“For a small moon with r H = 7 km and a gap with halfwidth = 2 r H , fast migration initiates in every simulation in this suite of tests (Figure 3).”

Several resonances lie close to but are not obviously coincident with the orbit of Daphnis within the Keeler gap.

“Several of these resonances lie close to but are not obviously coincident with the orbit of Daphnis within the Keeler gap.”

Moons with r H = 14 km migrate right through the 5:4 resonance.

“However, moons with r H = 14 km migrate right through the resonance.”

Away from resonance, ring particle eccentricity is a factor of 10 smaller.

“Away from resonance, e p is a factor of 10 smaller.”

For moons capable of undergoing fast migration, the damping timescale of e and i is well within one to two decades.

“For moons capable of undergoing fast migration, the damping timescale is well within one to two decades.”

Our calculations yield results in rough agreement with the analytic theory.

“Our calculations yield results in rough agreement with the analytic theory.”

Moons within a random distribution of ring particles appear to start migrating earlier than moons in a uniform distribution.

“Moons within a random distribution of ring particles appear to start migrating earlier than moons in a uniform distribution.”

In Saturn's rings, damping and viscous spreading smooth out the wakes left behind by migrating moons.

“In Saturn's rings, damping and viscous spreading smooth out the wakes left behind by migrating moons.”

In a pristine ring of small particles, numerical simulations show that moonlets and small moons migrate at the rates predicted by analytic theory.

“In a pristine ring of small particles, numerical simulations show that moonlets and small moons migrate at the rates predicted by analytic theory (Figures 8 and 9).”

In a basic set of simulations, Mimas rapidly stirs the e and i of the ring particle.

“In a basic set of simulations, Mimas rapidly stirs the e and i of the ring particle (left panels of figure).”

When a small moon is embedded within a continuous ring, it eventually clears enough material from its corotation zone to begin fast migration.

“When a small moon is embedded within a continuous ring, it eventually clears enough material from its corotation zone to begin fast migration (Figure 5).”

The growth rate of e follows the Weidenschilling (1989) estimate closely.

“The growth rate of e follows the Weidenschilling (1989) estimate closely.”

Off resonances, the orbit is stable on timescales of at least 10^4 yr before Daphnis begins to settle into the ring.

“Off resonances, the orbit is stable on timescales of at least 10 4 yr before Daphnis begins to settle into the ring.”

Off-resonance pumping of eccentricity is apparent throughout the ring.

“Some of the strongest resonances are labeled; off-resonance pumping of eccentricity is also apparent throughout the ring.”

After migration starts, its rate is comparable to the theoretical prediction.

“After migration starts, its rate is comparable to the theoretical prediction (dashed curve), and it halts once the moon nears the edge of the simulated disk.”

For a moon within a gap, fast migration proceeds only when the half-width of the gap is 4 r_H.

“For a moon within a gap, fast migration proceeds only when the half-width of the gap is 4 r H .”

In numerical simulations that include damping, gravitational interactions between small moons and ring particles, and viscous spreading, the moons clear their corotation zones on timescales T_clear.

“In numerical simulations that include damping, gravitational interactions between small moons and ring particles, and viscous spreading, the moons clear their corotation zones on timescales T clear .”

The constant jostling of moonlets by distant moons results in chaotic radial variations in distance from Saturn.

“The constant jostling of moonlets by distant moons results in chaotic radial variations in distance from Saturn (Figure 9).”

For most moonlets, measurements of radial drift limit the rate to below 0.1 km yr^-1.

“For most moonlets, measurements of radial drift limit the rate to below 0.1 km yr -1 .”

Saturn's moons produce quasi-periodic fluctuations in the inclination, which prevent Daphnis from settling into the ring plane.

“Saturn's moons produce quasi-periodic fluctuations in the inclination, which prevent Daphnis from settling into the ring plane.”

Evolution within the Pandora 18:17 resonance is similar but on a longer timescale.

“Evolution within the Pandora 18:17 resonance is similar but on a longer timescale.”

In simulations of Saturn's A ring, fast migration of small moons with r_H ≈ 2-20 km is ubiquitous.

“In our simulations of Saturn's A ring, fast migration of small moons with r H ≈ 2-20 km is ubiquitous.”

From our simulations, we identify sources of stirring which might maintain Daphnis's orbit for timescales exceeding 10^4 yr.

“From our simulations, we identify sources of stirring which might maintain Daphnis's orbit for timescales exceeding 10 4 yr.”

Migration halts once the moon nears the edge of the simulated disk.

“After migration starts, its rate is comparable to the theoretical prediction (dashed curve), and it halts once the moon nears the edge of the simulated disk.”

When Saturn's oblateness is included, stirring and precession yield a rough equilibrium in e and i as expected for a ring particle.

“When Saturn's oblateness is included in the calculations, the ring particle precesses rapidly.Stirring and precession yield a rough equilibrium in e and i as expected for a ring particle.”

Propeller moonlets seem to reside exclusively in the gray bands (zones).

“The gray bands are the zones where propeller moonlets seem to reside exclusively.”

When precession is slowed by a factor of 100, the variations in e and i become periodic.

“When precession is slowed by a factor of 100, the variations in e and i become periodic due to the precession of the ring particle's angular momentum vector precesses (dark gray curves).”

The growth rate of i is several times faster than predicted.

“The growth rate of i is several times faster than predicted.”

Moonlets have a range of sizes up to about r_H ∼ 1.5 km.

“Moonlets have a range of sizes up to about r H ∼ 1.5 km (Tiscareno et al. 2008(Tiscareno et al. , 2010)).”

• cites Tiscareno et al. 2008 sourceNot checked yet

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• cites Tiscareno et al. 2010 sourceNot checked yet

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Stirring grows continuously.

“Stirring grows continuously.”

In all cases, the time for each moon to become embedded in the disk is very short.

“In all cases, the time for each moon to become embedded in the disk is very short.”

Pan's Hill radius is very close to r_fast.

“Pan's Hill radius is very close to r fast .”

When a small moon lies within a gap, it eventually pulls enough material from the gap edges to begin fast migration.

“When a small moon lies within a gap, it eventually pulls enough material from the gap edges to begin fast migration (Figures 3 and 4).”

The power per unit mass dissipated by collisions is P coll ∼ 6 × 10 -7 dyne g -1 .

“P coll ∼ Σ m p h v p πr 2 p f loss m p v 2 p (39 ) ∼ 6 × 10 -7 dyne g -1 ,(40)”

Stirring by distant moons outside resonance can also overcome damping and prevent migration.

“Stirring by distant moons outside resonance can also overcome damping and prevent migration (Section 4.5).”

Migration stops after ∼170 years when the moon hits the inner edge of the simulated disk.

“In both instances, migration stops after ∼170 years, when the moon hits the inner edge of the simulated disk.”

For small moons capable of fast migration, r_H ≈ 2-5 km, these approaches yield similar results for Δa.

“For small moons capable of fast migration, r H ≈ 2-5 km, these approaches yield similar results for Δa.”

For much larger moons, torque balance consistently yields a larger half-width.

“For much larger moons, torque balance consistently yields a larger half-width.”

Results for the half-width of the gap differ from the simpler treatments by less than a factor of two.

“Although these solutions yield much better estimates for the surface density distribution, results for the half-width of the gap differ from the simpler treatments by less than a factor of two.”

For Pan, the balance is at Δa ∼ 75 km, much smaller than the observed half-width, ∼160 km, of the Encke Gap.

“For Pan, the balance is at Δa ∼ 75 km, much smaller than the observed half-width, ∼160 km, of the Encke Gap.”

Large moons achieve a balance between viscous diffusion and gravitational scattering.

“In our calculations, large moons achieve a balance between viscous diffusion and gravitational scattering.”

This limit is comparable to r_fast.

“This limit is comparable to r fast .”

Requiring Δa > 4 r_H yields the maximum Hill radius for a satellite capable of drawing material from the gap edges, ∼23 km, when scattering balances viscous diffusion.

“Requiring Δa 4 r H yields the maximum Hill radius for a satellite capable of drawing material from the gap edges, ∼23 km, when scattering balances viscous diffusion.”

The precise value of T_damp, as long as it is within an order of magnitude of 0.01 yr, does not make any noticeable difference in the outcomes of the simulations.

“As long as it is within an order of magnitude of our choice, the precise value of T damp does not make any noticeable difference in the outcomes of our simulations,”

It is challenging for patch simulations to track the response of a moonlet or small moon to the global evolution of ring particles.

“Thus, it is challenging for these simulations to track the response of a moonlet or small moon to the global evolution of ring particles.”

For more massive satellites with r_H >> r_gap, fast migration stalls.

“For more massive satellites with r H r gap , fast migration stalls.”

Including Saturn's oblateness and interactions with distant moons do not change the typical behavior established in simpler simulations.

“Once we establish a typical behavior, we verify that including oblateness and interactions with distant moons do not change this behavior.”

For Daphnis, stirring by distant moons seems the most promising mechanism to halt fast migration.

“For Daphnis, stirring by distant moons seems the most promising mechanism to halt fast migration.”

Larger moons such as Pan or Atlas do not migrate because of their own inertia.

“Larger moons-such as Pan or Atlas-do not migrate because of their own inertia.”

Moons that are too small or too large migrate very slowly.

“Thus, moons which are too small or too large migrate very slowly.”

The presence of Daphnis (r_H ≈ 5 km) inside the Keeler gap challenges the fast migration scenario.

“Although the presence of Daphnis (r H ≈ 5 km) inside the Keeler gap challenges this scenario”

• cites Porco et al. 2005Not checked yet

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Torque exchange between small moons and ring particles leads to radial drift.

“In theory, torque exchange between these moons and ring particles leads to radial drift.”

Smaller moons or moonlets such as the propellers are trapped by diffusion of disk material into corotating orbits, creating inertial drag.

“Smaller moons or moonlets-such as the propellers -are trapped by diffusion of disk material into corotating orbits, creating inertial drag.”

Satellites with r_gap < r_H < r_fast undergo fast migration.

“Thus, satellites with r gap < r H < r fast undergo fast migration.”

Because migration time scales as r_H^3, fast migration may be responsible for the lack of propeller moonlets with r_H >~1 km.

“Because the migration time scales as r 3 H , fast migration may be responsible for the lack of propeller moonlets with r H 1 km.”

Periodic boundary conditions at the ends of a patch limit the ability of patch simulations to include perturbations from material outside the patch.

“However, periodic boundary conditions at the “ends” of a patch limit the ability of these calculations to include perturbations from material outside the patch.”

Two moons, Pan and Atlas, have Hill radii very close to the fast migration cutoff r_fast.

“Two moons, Pan and Atlas, have Hill radii very close to the fast migration cutoff r fast .”

Daphnis's radius lies right in the middle of the expected range of radii for fast migration.

“Its radius lies right in the middle of the expected range of radii for fast migration.”

As long as the satellite mass exceeds the ring particle mass by a factor of a few hundred, superparticles can accurately represent smaller ring particles and their effect on an embedded satellite.

“As long as the satellite mass exceeds the ring particle mass by a factor of a few hundred, calculations suggest that these superparticles can accurately represent smaller ring particles and their effect on an embedded satellite (Figure 3; see also Kirsh et al. 2009).”

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Fast migration may be responsible for the observed large-radius cutoff of r_H ∼ 1-2 km in the size distribution of the A ring's propeller moonlets.

“may be responsible for the observed large-radius cutoff of r H ∼ 1-2 km in the size distribution of the A ring's propeller moonlets.”

The calculations accurately follow the evolution of a small moon at the expense of a large graininess in the structure of the ring.

“Thus, these calculations accurately follow the evolution of a small moon at the expense of a large graininess in the structure of the ring.”

Fast migration of 2-24 km moons should eliminate intermediate-size bodies from the A ring.

“Fast migration of 2-24 km moons should eliminate intermediate-size bodies from the A ring”

Sustained fast migration of a small moon requires that the moon draw/scatter ring material from one edge of its corotation zone to the other and then drift a sufficiently large radial distance in the opposite direction to avoid re-encountering the scattered particles.

“The moon must be able to draw/scatter ring material from one edge of its corotation zone to the other and then drift a sufficiently large radial distance in the opposite direction to avoid encountering the scattered particles again.”

When a satellite has r_H > r_gap, it undergoes fast migration.

“When a satellite has r H > r gap , it undergoes fast migration.”

In the simulations, the number of superparticles is too small to identify small-scale or short-term structures within the ring particles.

“In our simulations, the number of superparticles is too small to identify small-scale or short-term structures within the ring particles.”

As long as the disk and satellite maintain T_clear << T_fill and r_H >> r_fast, the satellite continues fast migration.

“As long as the physical properties of the disk and the satellite maintain the conditions T clear T fill and r H r fast , the satellite continues fast migration.”

The general lack of small moons in the ring is consistent with migration theory.

“Thus, the general lack of small moons in the ring is consistent with migration theory (see also Goldreich & Tremaine 1982).”

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Using the impulse approximation, Weidenschilling (1989) showed that distant moons give a particle a small kick in e and i every synodic period.

“Using the impulse approximation, Weidenschilling (1989) showed that distant moons give a particle a small kick in e and i every synodic period.”

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The fluctuations in semimajor axis of an isolated moonlet orbiting Saturn but not migrating through the disk result from the constant gravitational jostling from distant and nearby moons.

“These fluctuations result from the constant gravitational jostling from distant and nearby moons.”

Fast migration may be a common, if not ubiquitous, process in a disk similar in structure to Saturn's A ring.

“We conclude that fast migration may be a common, if not ubiquitous, process in a disk similar in structure to Saturn's A ring.”

The onset of fast migration depends on the initial orbital parameters of a satellite.

“The onset of fast migration depends on the initial orbital parameters of a satellite.”

A 7:6 resonance with the co-orbiting moons Janus and Epimetheus is responsible for the outer edge of the A ring.

“A 7:6 resonance with the co-orbiting moons Janus and Epimetheus is responsible for the outer edge of the A ring (e.g., Lissauer et al. 1985).”

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When a small moon is scattered into an otherwise unperturbed region of a ring, it will probably have a large orbital eccentricity or inclination relative to the ring particles.

“When a small moon is scattered into an otherwise unperturbed region of a ring, it will probably have a large orbital eccentricity or inclination relative to the ring particles.”

The migration of smaller moons is expected to stall close to resonance.

“Thus, we expect the migration of smaller moons to stall close to resonance.”

The migration time is extremely short compared to the ∼0.1-1 Gyr lifetime of the ring.

“The migration time is extremely short compared to the ∼0.1-1Gyr lifetime of the ring (e.g., Esposito 2010).”

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Titan tugs more on the barycenter of the entire Saturnian system instead of stirring the rings.

“With an orbital distance of 1.2 × 10 6 km, Titan tugs more on the barycenter of the entire Saturnian system instead of stirring the rings.”

The stirring expressions are valid for objects on nearly circular orbits.

“These expressions are valid for objects on nearly circular orbits.”

In a real disk, the finite scale height breaks the symmetry, allowing the satellite to draw material from the gap edges.

“In a real disk, the finite scale height breaks the symmetry, allowing the satellite to draw material from the gap edges (Hourigan & Ward 1984).”

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For the Saturn system, the impulse approximation is valid for moons with separations from the ring much smaller than the semimajor axis of ring particles.

“For the Saturn system, this approach is valid for moons with separations from the ring, δa, much smaller than the semimajor axis of ring particles.”

In a real ring, the balance that keeps a satellite on a stable orbit in a wide clear gap is metastable.

“In a real ring, however, this balance is metastable.”

Understanding barriers to migration requires a thorough understanding of the physical processes that impact the eccentricity of ring particles.

“Thus, understanding barriers to migration requires a thorough understanding of the physical processes that impact the eccentricity of ring particles.”

Interactions with larger ring particles can hinder or reverse migration of a larger moon on short timescales.

“Interactions with larger ring particles can hinder or reverse migration of a larger moon on short timescales (e.g., Tiscareno 2012).”

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The value of r H that separates the two modes of migration through a resonance depends on several factors.

“The value of r H that separates these two modes of migration through a resonance depends on several factors.”

Fast migration of satellites with r_gap < r_H < r_fast is a robust phenomenon in the A ring.

“Results for a large suite of simulations yield the same conclusion: fast migration of satellites with r gap < r H < r fast is a robust phenomenon in the A ring.”

Atlas is too far away from ring material to have any chance at migrating.

“Although it is too far away from ring material to have any chance at migrating, Atlas may have migrated out through the ring to its present location.”

As with Daphnis, orbital resonances and stirring from distant moons probably prevent any form of migration for Pan.

“As with Daphnis, orbital resonances and stirring from distant moons probably prevent any form of migration.”

Our analytic estimates and numerical simulations demonstrate that Saturn's rings are an important laboratory for testing migration theory.

“To conclude, our analytic estimates and numerical simulations demonstrate that Saturn's rings are an important laboratory for testing migration theory.”

We identify plausible mechanisms for limiting migration in Daphnis.

“Within the A ring, we can explain aspects of the radial and size distributions of propellers and we identify plausible mechanisms for limiting migration in Daphnis.”

The good agreement between r_gap and the maximum size of propellers is a strong confirmation of migration theory.

“The good agreement between r gap and the maximum size of propellers is a strong confirmation of migration theory.”

Propeller moonlets with Hill radii less than r_gap ≈ 2 km should experience slow, embedded migration with radial drift rates below 10 m yr^-1.

“With Hill radii less than r gap ≈ 2 km (Tiscareno et al. 2008), the propeller moonlets should experience slow, embedded migration with radial drift rates below 10 m yr -1 (Figure 2).”

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The onset of fast migration relies on the presence of some ring material within ∼4 r_H of a small moon.

“The onset of fast migration relies on the presence of some ring material within ∼4 r H of a small moon.”

Interactions between small moons and very large ring particles or local enhancements in ring surface density lead to stochastic migration, often reducing migration rates significantly.

“Interactions between small moons and (1) very large ring particles or (2) local enhancements in ring surface density lead to stochastic migration, often reducing migration rates significantly (Crida et al. 2010;Rein & Papaloizou 2010;Tiscareno 2012).”

• cites Crida et al. 2010 sourceNot checked yet

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• cites Rein & Papaloizou 2010Not checked yet

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• cites Tiscareno 2012Not checked yet

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Within the A ring, we can explain aspects of the radial and size distributions of propellers.

“Within the A ring, we can explain aspects of the radial and size distributions of propellers and we identify plausible mechanisms for limiting migration in Daphnis.”

Moonlets orbit Saturn in a gravitational field which is constantly modified by much more massive moons orbiting at larger distance.

“Moonlets orbit Saturn in a gravitational field which is constantly modified by much more massive moons orbiting at larger distance.”

Multiple moons can easily migrate rapidly through the rings.

“Thus, multiple moons can easily migrate rapidly through the rings.”

The torques from ring material in this configuration are too weak to cause measurable radial drift for Pan.

“The torques from ring material in this configuration are too weak to cause measurable radial drift.”

Torque balance yields a relationship between the gap half-width and the satellite's Hill radius: Δa ∼ 2 (GM a^5 / ν^2)^{1/6} r_H^{2/3}.

“From weak-scattering theory (Lin & Papaloizou 1979), this balance yields a simple relationship between the gap half-width and the satellite's Hill radius: Δa ∼ 2 GM a 5 ν 2 1/6 r 2 H .(A1)”

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• cites Lin & Papaloizou 1979Not checked yet

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The rough equality of stirring and collisional damping rates suggests that the A ring's velocity dispersion and thickness are supported by stirring from distant moons.

“The rough equality of these rates suggests that the A ring's velocity dispersion and thickness are supported by stirring from distant moons.”

Our new results suggest that the radial drift of Blériot and other moonlets can be confined by orbital resonances.

“Our new results suggest that the radial drift of Blériot and other moonlets can be confined by orbital resonances.”

Stochastic migration in a self-gravitating ring, interactions with surface density variations in the ring, and oscillatory motion from interactions with corotating material have similar low drift rates.

“Stochastic migration in a self-gravitating ring (Rein & Papaloizou 2010;Crida et al. 2010), interactions with surface density variations in the ring (Tiscareno 2012), and oscillatory motion from interactions with corotating material (Pan & Chiang 2010, 2012) have similar low drift rates.”

• cites Rein & Papaloizou 2010Not checked yet

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• cites Crida et al. 2010 sourceNot checked yet

  Not checked in the free run.

• cites Tiscareno 2012Not checked yet

  Not checked in the free run.

This motion may complicate interpretations of non-Keplerian motion of moonlets within the ring.

“Aside from complicating the measurement of a moonlet's orbital elements within our simulations, this motion may complicate interpretations of non-Keplerian motion of moonlets within the ring (e.g., Blériot, Tiscareno et al. 2010).”

• cites Tiscareno et al. 2010 sourceNot checked yet

  Not checked in the free run.

Several physical processes within the rings limit or halt migration.

“Several physical processes within the rings limit or halt migration.”

An alternative approach equates the radial distance ring material can diffuse in a synodic period with the displacement in semimajor axis a particle receives after every close encounter, yielding Δa ∼ (GM a^5 / ν^2)^{1/18} r_H^{4/3}.

“An alternative approach equates the radial distance ring material can diffuse in a synodic period, √ νT syn , with the displacement in semimajor axis a particle receives after every close encounter with the satellite (proportional to the right-hand side of Equation (A1); see also Equation ( 5) of Bromley & Kenyon 2011b).This balanced is achieved at an orbital distance from the satellite Δa ∼ GM a”

• cites Bromley & Kenyon 2011bNot checked yet

  Not checked in the free run.

Larger objects migrate out of the ring on timescales of 100-1000 yr.

“Larger objects migrate out of the ring on timescales of 100-1000 yr.”

Migration theory predicts an upper limit of r_gap ≈ 2 km.

“Migration theory predicts an upper limit of r gap ≈ 2 km.”

It is possible to eliminate the viscosity and derive a minimum gap size, as in Equation (2) for r_gap.

“If the time to fill the gap is the viscous time (e.g., T fill in Equation ( 7) in the main text), it is possible to eliminate the viscosity in this expression and derive a minimum gap size, as in Equation (2) for r gap .”

These effects may operate to help maintain the Encke gap, whose edges are relatively far, ∼8 r_H, from Pan.

“In addition, these effects may operate to help maintain the Encke gap, whose edges are relatively far, ∼8 r H , from Pan.”

Interactions with distant moons plausibly prevent fast migration of Daphnis through the A ring.

“Thus, interactions with distant moons plausibly prevent fast migration of Daphnis through the A ring.”

If some external physical mechanism cannot maintain Daphnis in its current orbit, it must be a recent, ∼10^3-10^4 yr, addition to the ring.

“If some external physical mechanism cannot maintain Daphnis in its current orbit, it must be a recent, ∼10 3 -10 4 yr, addition to the ring.”

In these simulations, a distant perturber (Mimas) continuously forces the alignment of angular momentum vectors between Mimas and the ring particle.

“In these simulations, a distant perturber (Mimas) continuously forces the alignment of angular momentum vectors between Mimas and the ring particle.”

A precise balance between stirring and collisional damping requires modest changes in the size/density of the ring particles.

“A precise balance between the two energies requires modest changes in the size/density of the ring particles.”

Smaller moons with r_H ≈ 2-20 km may undergo fast migration.

“Smaller moons with r H ≈ 2-20 km may undergo fast migration.”

These moons migrate slowly in the type II mode.

“These moons migrate slowly in the type II mode.”

Moons with r_H > 23 km can prevent material from encroaching within 4r_H.

“In principle, moons with r H 23 km can prevent material from encroaching within 4r H .”

Daphnis is a good candidate for fast migration.

“Based on this analysis, we identify Daphnis as a good candidate for fast migration.”

Barring other effects such as resonances with distant moons, Pan may be unable to halt long-term diffusion of ring material to within 4r_H.

“However, barring other effects such as resonances with distant moons, Pan may be unable to halt longterm diffusion of ring material to within 4r H .”

Pan is a poor candidate for fast migration, at least under current conditions in the Encke Gap.

“Pan is a poor candidate for fast migration, at least under current conditions in the Encke Gap.”

Material on initially circular orbits with Δa as large as 2√3 r_H can be deflected onto crossing orbits with the satellite.

“In addition to particles on horseshoe orbits, material on initially circular orbits with Δa as large as 2 √ 3r H can be deflected onto crossing orbits with the satellite(Gladman 1993).”

• cites Gladman 1993Not checked yet

  Not checked in the free run.

Viscous diffusion constantly tries to drive material into the gap.

“Because gravitational scattering falls off so steeply with orbital distance from the satellite, viscous diffusion constantly tries to drive material into the gap.”

Moons with r_H 20-25 km can migrate in the type II mode.

“Moons with r H 20-25 km can migrate in the type II mode.”