Formation of a Propeller Structure by a Moonlet in a Dense PlanetaryRing
MICHIKOSHI, Shugo, KOKUBO, Eiichiro,(NAOJ)
A moonlet embedded in a planetary ring tends to open
a gap, such as Keeler or Encke gap, due to gravitational
scattering [1]. Conversely, viscous diﬀusion of ring particles tends to close a gap. If a moonlet is suﬃciently
large, it will form a fully circular gap, whereas a small
moonlet will form only a partial gap that consists of two
azimuthally aligned lobes shaped like a propeller. Usingthe viscous fluid model, the formation of propellers bysmall moonlets was predicted. The Cassini spacecraft
discovered propellers in Saturn’s A ring [3].
The optical depth of the A ring is as high as 0.3–0.5,
while that of the B ring is larger than unity. Distinct
and large gravitational wakes form in rings with large
optical depths [4]. Such large gravitational wakes canalter the structures around an embedded moonlet. We
investigated the condition for propeller formation in adense ring in which gravitational wakes are prevalent.
Figure 1 shows snapshots of the low and high surface
density models [5]. In the low surface density model,
a propeller-shaped feature is clearly visible in the weakgravitational wakes. The surface density decreases considerably in the two lobes downstream of the moonlet.
They are aligned in the orbital direction and are symmetric about the moonlet. In the high surface density
model, strong wake structures due to gravitational instability formed but no propeller structures are observed.

moonlet size that depends on the surface density aroundthe moonlet.
We performed 25 simulations with diﬀerent ring surface densities and moonlet radii so as to confirm thecondition. As shown in Figure 2, we confirmed that apropeller-shaped structure is clearly observed when thiscondition is satisfied. If this condition is not satisfied, nodistinct steady propeller structure is observed although
the ring particles are aﬀected by the moonlet to someextent.
700
600
Σ (g/cm2)

500
400

0.98

0.61

0.63

0.45

0.32

0.94

0.85

0.43

0.29

0.27

0.97

0.61

0.42

0.15

0.10

0.57

0.21

0.10

0.070.05

0.25

0.120.08

0.08

0.09

50

100

150
R (m)

200

250300200
10000300

Figure 2: Condition for propeller formation in the R-Σ plane.
Filled squares denote models in which clear propellers formand crosses denote models in which no propellers form. We
show the ratio of the time-averaged surface density in thepropeller region to the initial surface density Σ̄/Σ0 at eachpoint. The solid line indicates the condition for propeller
formation.

References[1] Lissauer, J. J., Shu, F. H., & Cuzzi, J. N. 1981, Natue,292, 707

Figure 1: Snapshots of numerical simulations of low and highsurface density models.

[2] Spahn, F., & Sremčević, M. 2000, Astronomy andAstrophysics , 358, 368

The numerical simulation results indicate that propeller formation depends on the ring surface density. The
clumps in gravitational wakes typically have a mass of∼ Σλ2 , where Σ is the surface density and λ is the critical
wavelength of the gravitational instability. If the clump
mass is greater than the moonlet mass, the gravitational
wakes may not be aﬀected by gravitational scattering
due to the moonlet. Accordingly we derived the critical[3] Tiscareno, M. S., Burns, J. A., Hedman, M. M., etal. 2006, Nature, 440, 648[4] Salo, H. 1995, Icarus, 117, 287[5] Michikoshi, S., & Kokubo, E. 2011, ApJ , 732, L231