Today in Astronomy 111: rings
 Tour of the Solar system’s four ring systems.
 Tides, tidal disruption, and the Roche limit.
Earth
Saturn and rings A-G, seen from the planet’s shadow. The night side is illuminated faintly
by ring-light. What’s that pale blue dot between the G and F rings, around 10 o’clock?
(Cassini /JPL/NASA)
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Saturn’s rings and orbital resonances
The first rings – Saturn’s – were first seen by Galileo in 1610,
and first realized to be rings by Huygens in about 1655.
 Seven main rings, most comprised of many narrow rings.
 Extremely flat: they’re ~1000 cm thick, and ~ 1010 cm wide.
• A scale model of a ring made of printer paper would
be ~ 1 km wide.
 The narrow rings are separated by gaps that are usually
empty save for small satellites (particularly large ring
particles) orbiting therein.
 Many of the rings and gaps clearly match the meanmotion resonances of the moons within the gaps.
 Very icy particles (water ice, mostly): large albedo.
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Saturn’s rings and orbital resonances (continued)
Image from the Cassini satellite (NASA/JPL)
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Saturn’s rings and orbital resonances (continued)
Next image
Image from the Cassini satellite (NASA/JPL)
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Saturn’s rings and orbital resonances (continued)
Ring C
Cassini division:
• 1 arcsec wide, as
seen from Earth
• 2:1 mean motion
resonance with the
moon Mimas
Ring B
Ring A
Encke gap
Keeler gap
Ring F
Next image
Image from the Cassini satellite (NASA/JPL)
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Saturn’s rings and orbital resonances (continued)
The Keeler gap, and the moon that’s responsible for it
Image from the Cassini satellite (NASA/JPL)
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Saturn’s rings and orbital resonances (continued)
Keeler
gap
Pan
Spiral density waves and bending waves in the A ring; the Encke gap; and
Pan.
Image from the Cassini satellite (NASA/JPL)
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Saturn’s rings (continued)
 Spokes: radial markings are seen
on the rings, too, especially when
the Sun is illuminating the rings
nearly edge on (right: spoke movie
from Voyager 1; below: first Cassini
detection of spokes.
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Saturn ring vital statistics
Radius
Saturn equator
D inner edge
D outer edge
C inner edge
Titan ringlet
Maxwell gap
C outer edge
B inner edge
B outer edge
Cassini division
A inner edge
Encke gap
Keeler gap
A outer edge
F ring center
G inner edge
G outer edge
E inner edge
E outer edge
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Albedo
(km)
60,268
66,900
74,510
74,658 0.12 - 0.30
77,871
87,491
92,000
0.2
92,000
0.4 - 0.6
117,580
0.2 - 0.4
122,170
0.4 - 0.6
133,589
136,530
136,775
0.4 - 0.6
140,180
0.6
170,000
175,000
181,000
483,000
Thickness
Surface density Eccentricity
(m)
(gm cm-2 )
5
5
10
1.4 - 5
17
17
7
20 - 100
20
30
18 - 20
30 - 40
30
20 - 30
0.00026
0.00034
0.0026
1.E+05
1.E+07
1.E+07
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The Phoebe ring
In 2009, an even larger ring
was detected at midinfrared wavelengths by the
Spitzer Space Telescope.
 It covers radii
6-18×106 km, and is
tilted 27° from the A-BC-D ring plane.
 It contains the small,
retrograde-orbiting
moon Phoebe, and is
probably formed from
this moon’s material.
3 November 2011
Verbiscer, Skrutskie & Hamilton 2009
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Uranus’s rings
The second ring system discovered: in 1977, by Ted Dunham
and Jim Elliot using the NASA Kuiper Airborne Observatory.
 Discovered by observing Uranus occulting (=passing in
front of; eclipsing) a star. The original intent of the
experiment was to use the starlight to probe the structure
of Uranus’s upper atmosphere; Ted and Jim were mildly
surprised by the rings.
 Ten narrow rings are observed from our distance, but
viewed close up (by Voyager 2) they appear to have
structure not unlike Saturn’s.
 The rings are not shiny, though; the albedo of ring
particles is small, more like that of asteroids.
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Uranus’s rings (continued)
Voyager 2 image of
Uranus and its rings
(processing by Calvin
J. Hamilton). From the
outside: ε, δ, γ, η, β, α,
4, 5, and 6.
Area of next image
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Uranus’s rings (continued)
Voyager 2 image of
Uranus and its rings
(processing by Calvin
J. Hamilton). From the
outside: ε, δ, γ, η, β, α,
4, 5, and 6.
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Uranus’s rings (continued)
Longer exposure on
the Uranian rings by
Voyager 2. Note the
ease with which the
background stars
shine right through
the rings.
(Contrast this with
Saturn’s rings, which
cast dark shadows on
the planet.)
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Uranus’s rings (continued)
Uranus equator
6
5
4
α
β
η
γ
δ
λ
ε
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Radius Albedo Width Eccentricity
(km)
(km)
25,559
41,837 0.015
~1.5
0.001
42,235 0.015
~2
0.0019
42,571 0.015
~2.5
0.001
44,718 0.015 4 - 10
0.0008
45,661 0.015 5 - 11
0.0004
47,176 0.015
1.6
47,626 0.015
1-4
0.0001
48,303 0.015
3-7
50,024 0.015
~2
51,149 0.018 20-96
0.0079
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Jupiter’s rings
Discovered in 1979 by Voyager 1, these rings were almost
immediately thereafter also detected from the ground.
 A good way to see them is to observe at the center
wavelength of a strong atmospheric absorption in Jupiter,
which dims the planet relative to the rings. There are
several bands of methane at near-infrared wavelengths
that are generally used in this manner.
 The main ring is shepherded by two small satellites,
called Metis and Adrastea, orbiting Jupiter just inside and
just outside the ring (respectively).
 A faint halo can be seen above and below the main ring,
about a factor of ten fainter.
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Jupiter’s main ring, seen from the planet’s shadow
(Galileo/JPL/NASA)
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Jupiter, main ring, and shepherd satellites
Adrastea
Metis
Near-infrared (λ = 2.17 µm, in a methane band) image
sequence taken in 1994 at the NASA IRTF (Mauna Kea, HI).
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Jupiter’s main ring and its halo
(Galileo/JPL/NASA)
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The Amalthea
and Thebe
gossamer rings
 Two fainter rings
extend to the orbits
of Amalthea and
Thebe, and match
the vertical extent
of these satellite’s
orbital excursions.
(Galileo/JPL/NASA)
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Jupiter’s rings (continued)
 The rings have low density – even the main one – and
have very low albedo, even lower than most asteroids:
Radius (km)
Jupiter equator
71,492
Halo
100,000 122,000
Main ring
122,000 129,000
Amalthea
gossamer
129,200 182,000
Thebe
gossamer
182,000 224,900
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Albedo
Surface density
(gm cm-2)
0.015
5 x 10-6
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Neptune’s rings
So by the mid 1980s every stellar occultation of Neptune was
scrutinized for ring evidence.
 Results inconsistent and controversial: there appeared to
be incomplete arcs rather than smooth, continuous rings
like those of the other giant planets.
• Interesting false alarm: the first “ring” occultation
turned out to be the discovery of the small moon
Larissa (Reitsema et al. 1981).
 Voyager 2 arrived in 1989 to find six narrow rings that
completely circle the planet, but indeed confirmed that
one of them (“Adams”) has some sections brighter than
others, in the sense that the albedo is larger by a factor of
about 3 than the rest of the ring. The cause is still under
investigation.
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Neptune’s rings (continued)
Galle
Lassell
Adams
Arago
LeVerrier
Voyager 2 (JPL/NASA)
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Neptune’s rings (continued)
Arcs in the
Adams ring,
by Voyager 2
(JPL/NASA):
Liberté,
Egalité,
Fraternité
(left-right).
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Neptune’s rings
Ring
Radius (km)
Albedo
Width (km)
Name
Neptune equator
24,766
Galle
41,900
0.015
2000
Discoverer of Neptune.
LeVerrier
53,200
0.015
110
He who predicted Neptune’s position for Galle.
Lassell
53,200
0.015
4000
First to claim sight of a ring around Neptune (spurious).
Arago
57,200
< 100
Great French physicist who encouraged LeVerrier’s work.
Unnamed
61,950
Adams
62,933
0.015
50
Independently predicted Neptune’s position, prediscovery.
Dubiously associated with revolutionary mottos.
Arcs in Adams Ring:
Courage
62,933
15
Liberté
62,933
15
Egalité 1
62,933
0.04
15
Egalité 2
62,933
0.04
15
Fraternité
62,933
3 November 2011
One of many mottos of the first French Revolution,
perhaps coined by Robespierre.
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Common features of the rings
 None of the ring particles are very big.
• The rings can’t be solid, as first explained by Maxwell,
and first verified by Keeler.
• Lots of rocks and dust even when mostly ice.
• Moonlets and shepherd satellites no larger than ~20
km diameter, like Metis, Adrastea, Pan, etc.
 Very complex, but regular, structure determined by
orbital resonances with shepherd satellites.
• Ringlets and gaps
• Waves: bending waves, spiral waves, braids, ripples
 Not much material, in all.
• Saturn’s ring is by far most massive, at about 10 −6 M⊕ ,
one could make a 100-200 km diameter moon out of it.
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Origin of the rings
The rings are probably debris from collisions among the
system of moons of each planet, and as such may be
considered leftovers from the planet-formation process.
 As we will discuss toward the end of the semester, giant
planets (and stars) form from disks.
 Each planet’s equator and main moon system lines up
with the ring plane.
 Each ring lies within the Roche limit for its material and
its host planet: a moon made in the same location from all
the ring material would be torn apart by tidal forces,
which is why the debris hasn’t coalesced into moons.
 The Roche limit is the closest that a moon can be brought
to a planet without being ripped apart by tidal forces.
Which, finally, brings us to tides…
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Tidal accelerations
First, consider a moon of size ∆r × ∆ in the radial and
azimuthal direction, a distance r from a planet of mass M,
that isn’t in orbit (i.e. has no angular momentum):
∆
GM
g= − 2
r
dg
2GM
Radial tide
∆r
gtr =
r
∆r=
∆
(stretch)
dr
r3
θ
GM
gtφ = −2 2 cos θ
r
r
∆φ ∆φ
GM r ∆φ
GM
=
−2 2
=
− 3 ∆ Azimuthal tide
2 2
(compression)
2r
r
r
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Tidal accelerations (continued)
Mostly, though, we are interested in objects that revolve, and
in the forces in the reference frames at rest with respect to
those objects. At the center of the moon:
v2
Fcent = m
r
Inertial reference frame
= mω 2 r
v
Fgrav = −
GMm
r
2
dv
mv 2
m
=
−
=
−mω 2 r
dt
r
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dv
v= 0=
dt
Fgrav = −
GMm
r2
Revolving reference frame
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Tidal accelerations (continued)
Consider just the radial component of accelerations, and
consider a moon in orbit around planet with mass M. The
effective acceleration in the moon’s (revolving) reference
frame includes force from the planet and centrifugal force. At
the center of the body,
GM
GM
2
2
a = orbital radius
g eff =
− 2 + ω a =⇒
0
ω =
.
a
a3
And a distance r away from the planet, along a radial line
through the body’s center,
GM GM
g eff ( r ) =
− 2 + 3 r
r
a
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Tidal accelerations (continued)
The radial tidal acceleration in this case is
g=
tr
dg eff
=
∆r 2
GM
3
∆r +
GM
3
∆r ≅
3GM
3
∆r
if
R<<r
dr
r
a
a
Suppose this is such a big difference that it equals the
acceleration of the moon’s own gravity. If the moon (mass m)
is spherical and uniform in density, with radius R, this
condition is given by
Gm
2
=
3GM
3
R
R
a
G  4π 3  3G  4π 3

R
R
=
ρ
ρ

planet planet  R
2  3
3  3
 a 

R 
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Tidal accelerations (continued)
or, cancelling common factors,
a
 3 ρplanet
Rplanet 

ρ




13
 ρplanet
≅ 1.44 Rplanet 
 ρ




13
If the moon is placed in an orbit with radius smaller than
this value of a, its gravity cannot hold it together against the
tidal forces, and it will break apart. If one accounts also for
the stretching of the moon along the direction toward the
planet, there’s a slight change; one gets (Roche 1847):
aRoche
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ρ
 planet 
≅ 2.46 Rplanet 

 ρ 
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Roche
limit
32
Distribution of rings
and satellites
Scaled to each planet’s
radius.
orbit synchronous
with planetary
rotation.
Roche radius for
objects of density
1 gm/cm3.
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