121
Recap from previous lecture
8th October 2008• Pressure at altitude h in planetary atmosphere in hydrostatic equilibrium
– Assumes constant scale height H
p– Pressure at planet surface:
– For an ideal gas:
• A particular component of a planet’s atmosphere will be lost if, for that component,
– For an ideal gas, translates into an escape temperature
• Jovian planets have low average density, abundant H and He in their atmosphere, are gaseous / liquid with a rocky core deep inside…
−
=
P
S H
P h h
P( ) exp
) 0 ( =
=Ph PS
g m
T k g m
T H k
H
P = =µ
escape 6 2 1
rms υ
υ = v >
P P
kR m M T G
54 1
escape =
122
Rotation of the Jovian Planets
Jupiter, Saturn, Uranus and Neptune rotate very rapidly, given their large radii, compared with the terrestrial planets.
Also, the Jovian planets rotate differentially – not like a solid body (e.g. a billiard ball) but as
a fluid (e.g. grains of rice).
We see this clearly on Jupiter:
cloud bands and belts rotate at different speeds
Planet Rotation Period *
Mercury 58.6 days
Venus 243 days
Earth 24 hours
Mars 24 h 37 m
Jupiter * 9 h 50 m Saturn * 10 h 14 m
Uranus 17 h 14 m
Neptune 16 h 7 m
Pluto 6.4 days
* At Equator
124
On Jupiter we also see that the cloud belts contain oval structures.
These are storms; the most famous being the Great Red Spot.
This is a hurricane which has been raging for at least 300 years. It measures about 40000km by 14000km
Winds to the north and south of the Great Red Spot blow in opposite directions; winds within the Spot blow counterclockwise,
completing one revolution in about 6 days.
126
The Jovian planets are also significantly flattened, or oblate, due to their rapid rotation and fluid interior.
The effect is most pronounced for Jupiter and Saturn, which have relatively smaller cores e.g. Jupiter’s polar diameter = 133708km
(6.5% less than equatorial diameter)
Smaller core: larger oblateness Larger core: smaller oblateness
132
Section 9: Ring Systems of the Jovian Planets
All four Jovian planets have RING SYSTEMS. e.g. Saturn’s rings are easily visible from Earth with a small telescope, and appear solid.
Its rings consist of countless lumps of ice and rock, ranging from
~1cm to 5m in diameter, all independently orbiting Saturn in an incredibly thin plane – less than 1 kilometre in thickness.
( Diameter of the outermost ring – 274000 km. If Saturn’s rings were the thickness of a CD, they would still be more than 200m in diameter! )
In 1859 James Clerk Maxwell proved that Saturn’s rings couldn’t be solid; if they were then tidal forces would tear them apart. He concluded that the rings were made of ‘an indefinite number of unconnected particles’
137
Saturn’s rings are bright; they reflect ~80% of the sunlight that falls on them. Their ice/rock composition was confirmed in the 1970s when absorption lines of water were observed in the spectrum of light from the rings.
(See A1 Stellar astrophysics for more on spectra and absorption lines) Ground-based observations show only the
A, Band C rings.
In the 1980s the Voyager spacecraft flew past Saturn, and observed thousands of
‘ringlets’– even in the Cassini Division (previously believed to be a gap).
They also discovered a D ring, (inside the C ring), and very tenuous E, F and G rings outside the A ring, out to ~5 planetary radii.
A ring Cassini division
B ring
C ring
139
The structure of the F ring is controlled by the two
‘shepherd moons’ – Pandora and Prometheus – which orbit just inside and outside it.
The gravitational influence of these moons confine the F ring to a band about 100km wide
The F ring shows braided structure, is very narrow, and contains large numbers of micron-sized particles.
144
Jupiter’s ring system is much more tenuous than Saturn’s. It was only detected by the Voyager space probes. The ring material is primarily dust, and extends to about 3 Jupiter radii.
Uranus’ rings were discovered in 1977, during the occultation of a star, and first studied in detail by Voyager 2 in 1986 There are 11 rings, ranging in width from 10km to 100km. The ring particles are very dark and ~1m across. Some rings are
‘braided’, and the thickest ring has shepherd moons. There is a thin layer of dust between the rings, due to collisions.
Neptune’s rings were first photographed by Voyager 2 in 1989.
There are 4 rings: two narrow and two diffuse sheets of dust.
The outermost ring has 5 ‘arcs’ of concentrated material.
Ring Systems of the other Jovian Planets
146
Why are the ring systems so thin?
Collisions of ring particles are partially inelastic.
Consider two particles orbiting e.g. Saturn in orbits which are slightly tilted with respect to each other.
Collision reduces difference of ycomponents, but has little effect on xcomponents
⇒ this thins out the disk of ring particles
x y
147
The ring systems of the Jovian planets result from tidal forces.
During planetary formation, these prevented any material that was too close to the planet clumping together to form moons. Also, any moons which later strayed too close to the planet would be disrupted.
Consider a moon of mass and radius , orbiting at a distance (centre to centre) from a planet of mass and radius .
Section 10: Formation of ring systems
r A
MP
MS RS
r MP RP
( Assume that the planet and moon are spherical ) MS
151
We assume, as an order-of-magnitude estimate that the moon is tidally disrupted if
In other words, if
This rearranges further to
2 3
2
S S S
P
R M G r
R M
G >
G
T
F
F >
(10.3)(10.4)
S S
P
R
M r M
3 / 1 3
/
2
1
<
(10.5)153
We can re-cast eq. (10.5) in terms of the planet’s radius, by writing mass = density x volume.
Substituting
So the moon is tidally disrupted if
More careful analysis gives the Roche Stability Limit
3 3
4
P P
P
R
M =
πρ
3 3
4
S S
S
R
M =
πρ
P S
P
R
r
3 / 1 3
/
2
1
<
ρ
ρ
(10.6)P m
P
R
r
3 / 1
456 .
2
<
ρ
ρ
(10.7)154
e.g. for Saturn, from the Table of planetary data Take a mean density typical of the other moons
This implies
Mostof Saturn’s ring system does lie within this Roche stability limit. Conversely all of its moons lie further out!
m
-3kg 700
P
≈ ρ
m
-3kg 1200
m
≈ ρ
P P
RL
R R
r 2 . 05
1200 456 700
. 2
3 / 1
=
×
×
=
(10.8)157
Tidal forces also have an effect (albeit less destructive) outside the Roche stability limit.
Consider the Moon’s tide on the Earth (and vice versa).
The tidal force produces an oval bulge in the shape of the Earth (and the Moon)
There are, therefore, two high and low tides every ~25 hours.
(Note: not every 24 hours, as the Moon has moved a little way along its orbit by the time the Earth has completed one rotation)
Section 11: More on Tidal Forces
158
The Sun also exerts a tide on the Earth.
Now, so
and
so that
r
3F
T∝ M
P3
Sun Moon Moon
Sun Moon
, Sun
,
= r
r M
M F
F
T
T (11.1)
kg 10 989 .
1 30
Sun = ×
M MMoon =7.35×1022kg m
10 496 .
1 11
Sun= ×
r rMoon=3.844×108m
5 . 0
3
Sun Moon Moon
Sun Moon
, Sun
,
≈
= r
r M
M F
F
T
T (11.2)
159
Spring tides occur when the Sun, Moon and Earth are aligned (at Full Moon and New Moon). High tides are much higher at these times.
Neap tides occur when the Sun, Moon and Earth are at right angles (at First Quarter and Third Quarter). Low tides are much lower at these times.
The Sun and Moon exert a tidal force similar in magnitude.
The size of their combined tide on the Earth depends on their alignment.
160 Earth
Moon Even if there were no tidal force on the Earth from the Sun, the Earth’s tidal bulge would not lie along the Earth-Moon axis. This is because of the Earth’s rotation.
161 Earth’s rotation
Earth
Moon The Earth’s rotation carries the tidal bulge ahead of the Earth- Moon axis. (The Earth’s crust and oceans cannot instantaneously redistribute themselves along the axis due to friction)
Moon’s orbital motion
162 Earth’s rotation
Earth
Moon
A
‘Drag’ forceThe Moon exerts a drag force on the tidal bulge at A, which slows down the Earth’s rotation.
The length of the Earth’s day is increasing by 0.0016 sec per century.
Moon’s orbital motion
163
At the same time, bulge A is pulling the Moon forward, speeding it up and causing the Moon to spiral outwards. This follows from the conservation of angular momentum.
The Moon’s semi-major axis is increasing by about 3cm per year.
Earth’s rotation
Earth
Moon
A
‘Drag’ forceMoon’s orbital motion
164 Earth’s rotation
Earth
Moon
A
‘Drag’ forceGiven sufficient time, the Earth’s rotation period would slow down until it equals the Moon’s orbital period – so that the same face of the Earth would face the Moon at all times.
(This will happen when the Earth’s “day” is 47 days long) In the case of the Moon, this has already happened !!!
Moon’s orbital motion
166
Given sufficient time, the Earth’s rotation period would slow down until it equals the Moon’s orbital period – so that the same face of the Earth would face the Moon at all times.
(This will happen when the Earth’s “day” is 47 days long) In the case of the Moon, this has already happened !!!
Tidal locking has occurred much more rapidly for the Moon than for the Earth because the Moon is much smaller, and the Earth produces larger tidal deformations on the Moon than vice versa.
The Moon isn’t exactly tidally locked. It ‘wobbles’ due to the perturbing effect of the Sun and other planets, and because its orbit is elliptical. Over about 30 years, we see 59% of the Moon’s surface.
168
Asteroid belt and Kirkwood gaps
• Kirkwood gaps
– regions in the main asteroid belt cleared of asteroids by the perturbing effects of Jupiter – due to resonances with Jupiter's orbital period
186
Many of the satellites in Solar System are in synchronous rotation, e.g.
Mars: Phobos and Deimos
Jupiter: Galilean moons + Amalthea
Saturn: All major moons, except Phoebe + Hyperion Neptune: Triton
Pluto: Charon
Pluto and Charon are in mutual synchronous rotation: i.e. the same face of Charon is always turned towards the same face of Pluto, and vice versa.
Triton orbits Neptune in a retrograde orbit (i.e. opposite direction to Neptune’s rotation).
Satellites in the Solar System
187 Neptune’s rotation
Neptune
Triton
A
Triton’s orbital motion
In this case Neptune’s tidal bulge acts to slow down Triton. The moon is spiralling toward Neptune (although it will take billions of years before it reaches the Roche stability limit)
