version 1.0: 2016 Mar 15 — Supplementary Material

Noncircular features in Saturn’s rings III: The Cassini Division –

Supplementary Material

Richard G. French∗,1

Philip D. Nicholson2

Colleen A. McGhee-French1

Katherine Lonergan1

Talia Sepersky1

Mathew M. Hedman3

Essam A. Marouf4

Joshua E. Colwell5

∗ Corresponding author, rfrench@wellesley.edu

1 Department of Astronomy, Wellesley College, Wellesley MA 02481

2 Department of Astronomy, Cornell University, Ithaca NY 14853

3 Department of Physics, University of Idaho, Moscow, ID 83844

4 Department of Electrical Engineering, San Jose State University, San Jose CA 95192

5 Department of Physics, University of Central Florida, Orlando FL 32816

– 2 –

REFERENCES

Spitale, J. N., Hahn, J. M. 2016. The shape of Saturn’s Huygens ringlet viewed by Cassini

ISS. Icarus, in press.

This preprint was prepared with the AAS LATEX macros v5.2.

– 3 –

Huygens IER

-10 -5 0 5 10r - 117820 (km)

-0.2

0.0

0.2

0.4

0.6

0.8

1.01−

exp(

−τ)

Huygens OER

-10 -5 0 5 10r - 117842 (km)

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1−ex

p(−τ

)

Strange (R6) COR

-10 -5 0 5 10r - 117908 (km)

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

1−ex

p(−τ

)

Huygens OEG

-10 -5 0 5 10r - 117927 (km)

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

1−ex

p(−τ

)

Herschel IEG

-10 -5 0 5 10r - 118191 (km)

-0.1

0.0

0.1

0.2

0.3

1−ex

p(−τ

)

Herschel IER

-10 -5 0 5 10r - 118232 (km)

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

1−ex

p(−τ

)

Herschel OER

-10 -5 0 5 10r - 118260 (km)

-0.05

0.00

0.05

0.10

0.15

1−ex

p(−τ

)

Herschel OEG

-10 -5 0 5 10r - 118283 (km)

-0.05

0.00

0.05

0.10

0.15

0.20

1−ex

p(−τ

)

Russell IEG

-10 -5 0 5 10r - 118587 (km)

-0.02

0.00

0.02

0.04

0.06

0.08

1−ex

p(−τ

)

Russell OEG

-10 -5 0 5 10r - 118627 (km)

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

1−ex

p(−τ

)

Jeffreys IEG

-10 -5 0 5 10r - 118931 (km)

-0.01

0.00

0.01

0.02

0.03

0.04

1−ex

p(−τ

)

Jeffreys OEG

-10 -5 0 5 10r - 118965 (km)

-0.05

0.00

0.05

0.10

0.15

1−ex

p(−τ

)

Kuiper IEG

-10 -5 0 5 10r - 119400 (km)

-0.02

0.00

0.02

0.04

0.06

0.08

1−ex

p(−τ

)

Kuiper OEG

-10 -5 0 5 10r - 119405 (km)

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

1−ex

p(−τ

)

Laplace IEG

-10 -5 0 5 10r - 119840 (km)

-0.01

0.00

0.01

0.02

0.03

0.04

1−ex

p(−τ

)

Laplace IER

-10 -5 0 5 10r - 120038 (km)

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

1−ex

p(−τ

)

Laplace OER

-10 -5 0 5 10r - 120076 (km)

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

1−ex

p(−τ

)

Laplace OEG

-10 -5 0 5 10r - 120083 (km)

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

1−ex

p(−τ

)

Bessel IEG

-10 -5 0 5 10r - 120229 (km)

-0.02

0.00

0.02

0.04

0.06

1−ex

p(−τ

)

Bessel OEG

-10 -5 0 5 10r - 120242 (km)

-0.05

0.00

0.05

0.10

0.15

0.20

1−ex

p(−τ

)

Barnard IEG

-10 -5 0 5 10r - 120299 (km)

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

1−ex

p(−τ

)

Barnard OEG

-10 -5 0 5 10r - 120315 (km)

-0.05

0.00

0.05

0.10

0.15

1−ex

p(−τ

)

Fig. 1.— A gallery of profiles showing each of the 22 ringlets and gap edges in our study,

as observed in the RSS X-band occultation egress profile from rev 7. The vertical dashed

lines mark the approximate locations of the measured features.

– 4 –

Huygens ringlet

1.0 0.5 0.0 -0.5 -1.0Cos(f)

-40

-20

0

20

40

∆r (

km)

Envelope of modes

OER

IER

-40 -20 0 20 40<∆r> (km)

0

10

20

30

40

Wid

th (

km)

Fig. 2.— Huygens ringlet width-radius relation. In the upper panel, we plot the individual

observations of the radii of the inner and outer edges as a function of cos f , which to

first order in e should fall on the straight lines shown (thin for the IER and thick for the

OER). There is considerable scatter about these lines, but nearly every point falls within

the boundaries marked by dashed lines, which represent the maximum excursions expected

when all of the normal modes for each edge are in phase. Statistically, this means that the

amplitudes of the radial excursions do not exceed those expected from normal modes that

we have shown to be present on each edge. In the lower panel, we show that there is a

weak trend of increasing ringlet width with orbital radius, again with quite a bit of scatter,

presumably associated with the contribution of normal modes to the ring shape.

– 5 –

Huygens ringlet outer edge

0 60 120 180 240 300 360Mean anomaly (deg)

-10

-5

0

5

10

dr (

km)

Huygens ringlet inner edge

0 60 120 180 240 300 360Mean anomaly (deg)

-10

-5

0

5

10

dr (

km)

A B

Fig. 3.— Post-fit residuals for the edges of the Huygens ringlet, plotted as a function of mean

anomaly. We mark the approximate predicted locations of features A and B of Spitale and

Hahn (2016), which they propose are long-lived radial disturbances on the ringlet’s inner

edge, rotating at the local Keplerian rate, resulting from two embedded small satellites. Our

occultation results do not show clusters of larger residuals near these or other locations,

as would be expected if there were significant localized radial features in either ring edge

rotating at the mean orbital rate.)

– 6 –

Huygens ringlet

0 60 120 180 240 300 360Longitude (deg)

-100

-50

0

50

100

dr (

km)

↑

↑

↑

Fig. 4.— Three simulations of the shape of the Huygens ringlet based on the coaddition

of normal modes observed in our occultation data, with random relative phases (see text

for details). These are to be compared to Cassini ISS observations of the ringlet shape

shown in Fig. 11 of Spitale and Hahn (2016). The inner edge features labeled A and B in

their figure qualitatively resemble signatures in these simulations (marked by arrows) that

result from the coaddition of normal modes that have been convincingly identified in the

occultation data.

– 7 –

Strange ringlet (R6)

-15 -10 -5 0 5Node rate (deg/day)

0

2

4

6

8R

MS

(km

)

-15 -10 -5 0 5Node rate (deg/day)

0

2

4

6

8

a si

n(i)

(km

)

Fig. 5.— Strange ringlet node rate scan, after fitted models for normal modes m =

0, 1, 2, 2M , and 3 have been subtracted from the data. The inclination and the nodal re-

gression rate are tightly constrained by the observations.

– 8 –

Strange ringlet (R6)

1.0 0.5 0.0 -0.5 -1.0Cos(f)

-20

-10

0

10

20

∆r (

km)

Envelope of modes

OER

IER

-15 -10 -5 0 5 10 15<∆r> (km)

0

1

2

3

4

5

Wid

th (

km)

Fig. 6.— Width-radius relation for Strange ringlet. In the upper panel, the measured

orbital radius relative to the ring’s mean orbital radius is plotted as a function of cos f ;

bold lines and filled circles correspond to the OER; normal lines and open circles to the

IER. Solid lines mark the best-fitting elliptical model of each edge, and the dashed lines are

offset from these by the sum of the mode amplitudes for each edge. Although this ringlet

has a substantial eccentricity, this does not dominate the observed width variations, as is

evident in the lower panel.

– 9 –

Herschel OEG

0 2 4 6 8 10Pattern speed (deg/day)

0.00

0.05

0.10

0.15

0.20R

MS

(km

)

m=1

0 2 4 6 8 10Pattern speed (deg/day)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Am

plitu

de (

km)

Fig. 7.— Herschel outer gap edge normal mode scan for secondary m = 1 mode. Note the

zero amplitude at the nominal pattern speed for the appropriate radius of the gap edge,

confirming that the best-fitting eccentric model has already been removed.

– 10 –

Herschel IER

-15 -10 -5 0 5Node rate (deg/day)

0.0

0.5

1.0

1.5

2.0

RM

S (

km)

-15 -10 -5 0 5Node rate (deg/day)

0.0

0.5

1.0

1.5

a si

n(i)

(km

)

Herschel OER

-15 -10 -5 0 5Node rate (deg/day)

0.0

0.5

1.0

1.5

2.0

2.5

RM

S (

km)

-15 -10 -5 0 5Node rate (deg/day)

0.0

0.5

1.0

1.5

2.0

a si

n(i)

(km

)

Fig. 8.— Herschel ringlet IER and OER node rate scans. Both inner and outer edges show

similar inclinations, with fitted a sin i ≃ 1.5 and 2.1 km, respectively, and nearly identical

nodal regression rates that are close to the predicted values at the IER. The secondary

maxima at Ω ∼ −15 d−1 and +5 d−1 are due to aliasing.

– 11 –

Herschel ringlet

-3 -2 -1 0 1 2 3<∆r> (km)

20

25

30

35

40

Wid

th (

km)

Fig. 9.— Herschel ringlet width-radius relation, showing very little correlation of width

with the ringlet’s mean radius, owing primarily to the misalignment of the pericenters of

the two edges.

– 12 –

Laplace ringlet

1.0 0.5 0.0 -0.5 -1.0Cos(f)

-40

-20

0

20

40∆r

(km

)

Envelope of modes

OER

IER

-3 -2 -1 0 1 2 3<∆r> (km)

20

30

40

50

60

Wid

th (

km)

Fig. 10.— Width radius relation for the Laplace ringlet. In the upper panel, the measured

orbital radius relative to the ring’s mean orbital radius is plotted as a function of cos f ;

bold lines and filled circles correspond to the OER; normal lines and open circles to the

IER. Solid lines mark the nominal elliptical edges, and the dashed lines are offset by the

sum of the mode amplitudes for each edge. The observed width variations have a relatively

weak correlation with the orbital radius of the ring’s midline, owing to the nearly perfect

anti-alignment of the apses of the two edges, as shown in the lower panel.

– 13 –

Barnard IEG

586.0 586.5 587.0 587.5 588.0 588.5Pattern speed (deg/day)

0.0

0.2

0.4

0.6

0.8

1.0

RM

S (

km)

m=5

586.0 586.5 587.0 587.5 588.0 588.5Pattern speed (deg/day)

0.0

0.2

0.4

0.6

0.8

1.0

Am

plitu

de (

km)

Barnard OEG

586.0 586.5 587.0 587.5 588.0 588.5Pattern speed (deg/day)

0.00

0.05

0.10

0.15

0.20

RM

S (

km)

m=5

586.0 586.5 587.0 587.5 588.0 588.5Pattern speed (deg/day)

0.00

0.05

0.10

0.15

0.20

Am

plitu

de (

km)

Fig. 11.— Normal mode scans for the m = 5 edge modes on the inner (left) and outer

(right) edges of the Barnard Gap, forced by the Prometheus 5:4 ILR. The best-fitting

pattern speeds are both very close to the mean motion of Prometheus. The amplitude of

the ILR mode on the inner edge is substantially larger than for the outer edge, owing to

the proximity of the resonance just exterior to the IEG.