. 25
( 28)


before Uranus was supposed to block the
The first evidence for rings around Jupiter,
starlight, a number of brief dips in the starlight
shown in Fig. 25.16, came from Pioneer 11 in
were noticed. These dips were repeated after the
1974. The discovery was confirmed by Voyager 2.
occultation. They were the result of the rings
The rings are different from those of Saturn. They
passing in front of the star.
fall within a narrow range of radii, 1.72 to 1.81
This observation, and many follow-up obser-
planetary radii. Their width is 6000 km. Their
vations, have revealed a system of nine narrow
thickness is less than 30 km. The rings are very
rings. They extend from 1.60 to 1.95 planetary
faint. The rings appear smooth, except for a 600
radii. Each ring is about 10 km wide, except for
km wide enhancement at 1.79 planetary radii.
one which is about 100 km wide. The ring parti-
The particles are much smaller than in
cles orbit with a range of eccentricities. There are
Saturn™s ring, being a few micrometers across.
also some changes in width. For example, one
They have low albedos (reflectivities), suggesting
ring varies in distance from the planet by about
that they are silicates, rather than ice. Such small
800 km, and in width from 20 to 100 km. The gaps
particles are not expected to stay in the ring for
between the rings are quite clear. The rings them-
very long, so it seems that they must be replaced
selves have a low albedo, making them difficult
continuously. It is possible that some ring mate-
to see.
rial is being thrown off the closest large moon, Io,
Voyager 2 provided close-up information on
and from two embedded smaller moons.
the rings. New rings were discovered as well. Radio
The March 1974 discovery of a ring system
observations of one ring showed that the particles
around Uranus (Fig. 25.17) was a great surprise. An
in it are probably larger than 30 cm across.
occultation of a star by Uranus was being observed.
Backlighted views show the regions away from the
Some observations were being carried out from

F(r + x)

R to Planet x

F(r) “ F(r + x)
Fig 25.18. Neptune™s rings (Voyager 2 image). [NASA]
Fig 25.19. Roche limit.We compare the attractive forces
between the two spheres with the difference between the
forces that the planet exerts on each sphere.
known rings to be filled with very thin rings. The
particles in these rings may be only a micrometer
across. Voyager 2 also provided images of Neptune™s
rings; one is shown in Fig. 25.18. other object is a distance R x from the planet.
We assume that x R. The attractive force
25.4.2 Ring dynamics between the two particles is
These ring systems pose a number of interesting
questions One obvious question is that of their Fatt G
origin. Other questions center around their struc-
ture. Why are they so thin? Why are there multi- The tidal force Ftid is the difference between the
ple rings? Why is the appearance so different forces exerted by the planet on the two objects.
from planet to planet? All of these questions We can write this as
relate to the dynamics of the particles in the
a b ¢R
rings. We must understand the forces that per- Ftid
turb the motions of the ring particles, and the
response of those particles to those forces. Since the force on the particle of mass m a dis-
When we study rings, our first dynamical tance R from the planet is
consideration is the role of tidal effects. The
result is to cause different parts of an object to F G
accelerate differently. If some particles are very
close to a massive object, the tidal effects due to we have
the massive object can prevent the particles from
staying together under the influence of their 2G
mutual gravitational attraction. We define the
Roche limit of a planet to be the minimum dis- This makes the tidal force
tance that an object can be from that planet and
still be held together by gravitational forces only. Ftid 2G
We are within the Roche limit of the Earth, but
we are not torn apart. That is because we are held where we have taken ¢R x.
together by electrical forces, not gravitational We find the limiting value of R, the Roche
forces. limit, by equating the magnitudes of the tidal
We can get a feeling for the Roche limit by force and the attractive force between the parti-
looking at the simple situation in Fig. 25.19. We cles. This gives
have an object represented by two small particles
of equal mass m, a distance x apart. One particle
2G 3
is a distance R from a planet of mass M and the R x

Simplifying, we have We find that all of the ring systems lie within
the Roche limits for their respective planets. This
M m
leads to the idea that rings are made up of parti-
R3 x3
cles that would have formed moons. However, the
We further note that the mass of the two par- particles were too close to the planet for the grav-
ticles is 2m, and the volume they occupy is itational attraction to allow the moons to form.
approximately x3, so m/x3 is approximately the The arrangements of various rings relative to the
density . Making this substitution and solving Roche limits are shown in Fig. 25.20.
for R gives the Roche limit as This begins to give us some picture of how
rings may have evolved. The scenario is illustrated
a b
in Fig. 25.21. As each planet started to form, the
RRoche (25.1)
material around it formed into a disk. For the
material far enough from the forming planet,
In calculating the Roche limit for a planet, we
the collection into a moon was possible. However,
have to enter the value for the density of the mate-
for the material inside the Roche limit, a moon
rial that is trying to hold itself together. Obviously,
could not form. Therefore, the material contin-
the greater the density, the closer it can venture to
ued to orbit in a disk. Not all the particles were
the planet without disruption. As a conservative
originally in the thin disk. However, those that
limit, we often take the density of the planet itself.
weren™t had to pass through the disk twice per
That is, we find the Roche limit for an object with
orbit. During these passages, collisions with par-
the same density as the planet. In general, an
ticles in the disk changed their orbits. Eventually,
object just forming around a planet will have a
the orbits were changed sufficiently that the
lower density than that of the planet, so it must be
stragglers joined the disk. The flattening was not
farther away than objects as dense as the planet.

Jupiter Saturn

(a) (b)



(c) (d)
Fig 25.21. Ring evolution. (a) Particles start in a rotating
cloud. Some sample orbits are shown. Particles in such orbits
Fig 25.20. Rings and Roche limits for Jupiter, Saturn and collide, and the collisions tend to reduce the motions per-
Uranus. Again, each diagram is given in units of that planet™s pendicular to the equatorial plane. (b) The cloud ¬‚attens as a
radius.The shaded areas are actual rings. (For Saturn, the result of these collisions. (c) Eventually most of the cloud is
rings are indicated by their letter designations.) The dashed in a disk, with a few particles having large motions perpendi-
line labeled R is the Roche limit for material with a density cular to the equatorial plane. However, the particles pass
of 1 g/cm3.The dashed line labeled S is the radius for syn- through the plane twice per orbit, and collisions eventually
chronous orbit about the planet. Certain electromagnetic bring them into the plane, producing the thin distribution
effects change sign at this radius. shown in (d).

total, since collisions in three dimensions always
leave some residual random motion in the direc-
tion perpendicular to the plane.
4' 2'
Within the plane of the disk, the ring could
spread out over a range of radii. One mechanism
for this spreading involved collisions between 2
particles in which one was closer to the planet
than the other. The closer one would overtake the
farther one in their orbits. A gravitational P
5' 73 1 59 1' 9'
encounter would then slow the inner one and
speed up the outer one. This would cause the
inner one to fall farther in and the outer one to 4
move farther out, meaning a spreading of the
ring. Another effect, Poynting Robertson drag,
discussed in Chapter 27, causes smaller particles 6' 8'
to move inward after losing energy under the
impact of photons. 7'
The final appearance of the rings around a
Fig 25.22. Resonant orbits. In this case, the inner satellite
planet depends on the position of the ring rela-
has half the period of the outer satellite. Positions of the
tive to the orbits of the planet™s satellites. The inner satellite are marked 1 through 9, while the positions of
gravitational effects of the satellites sculpt the the outer satellite are marked 1 through 9 .The important
rings by perturbing the orbits of the ring parti- point is that at every orbit of the outer satellite, the two are
cles. This may have the effect of producing the close together, so any perturbation can be ampli¬ed, much
detailed structure in Saturn™s rings, or of confin- the way pushing a swing at its resonant frequency can
amplify its motion.
ing the rings to some range of radii, as happened
in the case of Uranus. Satellites embedded within
the rings can serve as sources of particles in the
the appearance of Jupiter™s ring. (It also has
ring, replacing particles removed by other effects.
important effects on Io, as we will see in the next
The moons can also remove particles from the
Some other effects of satellites on rings are
The effects of multiple moons sculpting the
shown in Fig. 25.23. In Saturn™s rings the Cassini
rings are particularly important when the
division is at a radius where the orbital period
moons have resonant orbits. By this we mean that
would be half that of Mimas. It has been sug-
the ratios of the orbital periods are equal to the
gested that various resonances are important in
ratios of small integers, as illustrated in Fig. 25.22.
producing the many gaps seen in the A ring as
This means that certain configurations of the
well as establishing the boundaries of the A and
satellites repeat with regularity, on a relatively
B rings. Shepherding satellites may confine rings,
short time scale. Thus, the perturbing effects of
such as the F rings (shown in Fig. 25.24) but other
those repeating arrangements are strongly rein-
rings are confined without any such satellites. It
forced. (Similarly, if we push a swing in reso-
has also been suggested that satellites called
nance with its motion, the effects of pushing are
guardian satellites produce a spiral pattern that
reinforced.) An example is found in the asteroid
clear the regions of resonant orbits (Fig. 25.23b).
belt between Mars and Jupiter. There are gaps,
In addition, a satellite near a ring can push it
called Kirkwood gaps, in which few asteroids are
away. This accounts for the narrowing of the ring
found. They correspond to orbits whose period
of Uranus.
would be related to Jupiter™s by ratios of small
Another interesting effect results when two
integers. In Jupiter™s moon system, the periods of
small moons have orbits very close to each other, as
the orbits of Io, Europa and Ganymede have the
shown in Fig. 25.23(a). The inner moon overtakes
ratio 1:2:4. These are important in influencing

Orbital Source

Fig 25.24. Rings and satellites. Saturn™s F ring with shep-
herding satellite (Voyager 2 image). [NASA]

the more massive moon, the less massive moon
executes a horseshoe orbit.
A moon within a ring has a similar effect on
particles. As seen from the moon, particles
approach from the outer leading side of the inner
trailing side. Their paths are altered by tidal
effects of the planet, since the rings are inside the
Roche limit. This causes some of the particles to
follow looped paths, similar to a horseshoe orbit.
Since the tidal effects are directed towards the
planet, particles have a hard time sticking to the
moon on the side of the moon closest and far-
thest from the planet, but can stick to the leading
and trailing edges.

Guardian 25.5 Moons

The planets that we have discussed in this chapter
provide us with an interesting variety of moons.
In this section, we discuss the most important of
those moons, planet by planet.
The four largest moons of Jupiter were discov-
Fig 25.23. Satellites and rings. (a) Source satellites.The
upper ¬gure shows the motions of particles relative to the ered by Galileo, and are therefore called Galilean
satellite.The lower ¬gure shows the full orbit of the particles satellites. They are, in order of distance from
that are turned around near the satellites. (b) Guardian satel-
Jupiter: Io, Europa, Ganymede and Callisto (Fig.
lites. Satellites just inside or outside the rings con¬ne the rings.
25.25). The smallest, Europa, has a radius of 1561
km, and the largest, Ganymede, has a radius of
the outer moon. In a gravitational encounter, the 2631 km. The ones that are closest to Jupiter are
inner moon is pulled out, and the outer moon is denser. The densities of Io and Europa are com-
pulled in. The moons actually exchange orbits. parable to those of Mars and the Moon. These
The process repeats when they overtake again. As moons are close enough to Jupiter to be shielded
viewed from the rotating system orbiting with from the solar wind by Jupiter™s magnetic field.

Fig 25.25. Galileo orbiter images of the Galilean satellites:
One obvious feature of Io, shown in Fig. 25.27,
Ganymede, Callisto, Io and Europa. [NASA]
is its yellow color. We think that this results from
They are massive enough to affect each other™s the sulfur in the surface material. There are no
orbits. We have already mentioned the resonance impact features. The surface is volcanic. We have
among the inner three (with periods being in the even obtained pictures of volcanic eruptions in
ratio 1:2:4). This resonance keeps Io™s orbit elliptical. progress. The volcanic flows might also contain
It also produces large tidal disruptions of Io, which sulfur. They may be basalts colored with sulfur.
dissipate energy in Io™s interior, keeping it hot. There are also many calderas, over 200 of which
The higher densities of Io and Europa suggest are more than 20 km across. A few large shield
that they contain a significant amount of silicate volcanic mountains (like Olympus Mons on Mars)
rock. The lower densities of Ganymede and Callisto are evident. There are also non-volcanic moun-
suggest that they contain much less rock. Jupiter tains. One consequence of this volcanic activity is
condensed far enough from the Sun so that the that the surface is altered on short time scales.
temperature was low enough for water ice to form.
We therefore expect the non-rock part to be water Io
ice. The composition differences among the four
ρ = 3.0 g/cm3
ρ = 3.5 g/cm3
satellites could have occurred because Jupiter
itself was an important heat source as it collapsed. Sulfur and
Thus, the nearer satellites were heated by Jupiter H2 O
frozen SO2
more than the farther ones. This would explain
why the nearer ones do not have any water ice. Molten
In Fig. 25.26, we compare the basic internal Silicates Silicates
structure of these four moons. Of the four,
Callisto™s density is the least well known. Models
suggest that it is 40% to 60% silicate (by mass). Silicates
This silicate is concentrated in a core. Over the H2 O
H2 O
core is a 850 km thick section of liquid water,
with a 250 km thick crust of water ice. Ganymede Ice
is 60% to 80% silicates, but has a basically similar
structure. Its high albedo (0.4) also points to an
ice surface. Europa is thought to have a silicate ρ = 1.8 g/cm3 ρ = 1.9 g/cm3
core, with a water layer and a thin (150 km thick)
crust. Its high albedo (0.6) is also indicative of an Ganymede
ice surface. Io has a molten silicate interior and a
Fig 25.26. Structures of the Galilean satellites, from theo-
frozen sulfur dioxide (SO2) and sulfur surface,
retical models. Again, each is plotted relative to the radius of
with active volcanoes. We now look at these
the particular satellite.
moons in some more detail.

Fig 25.27. Galileo orbiter images of Io. (a) True color
global image. (b) Ongoing volcanic eruption at Tvarshtar
Catina. (c) This volcanic eruption stands out as Io is eclipsed
by Jupiter, and is dark. [NASA]

most geologically active areas on Earth have heat
flows as high as 1.7 W/m2.
Io is also strongly affected by Jupiter™s mag-
netic field. As Io orbits the planet, Jupiter™s mag-
netic field sweeps by at 57 km/s. This has the


3 1



The extensive volcanic activity is unexpected to
f Io
for such a small object. We would not expect such
a small object to have a molten interior. However,
we have already seen that the gravitational
effects of the other moons, enhances by orbital Fig 25.28. Orbital resonances and tidal heating of Io. An
resonances, distort Io. As shown in Fig. 25.28, this orbital resonance with Europa keeps Io™s orbit elliptical.This
distortion changes its orientation as the moon means that, even though Io rotates at a constant rate, equal
moves around in its orbit. The effect is to cause to its orbital period, it doesn™t go around the orbit at a con-
stant rate. If we think of Io as an ellipsoid, with an inner and
internal friction, heating the interior. This results
outer section, the effect of the torques caused by Jupiter is
in a very large heat flow through the surface,
for the inner layer to be out of phase with the outer layer.
about 2 W/m2. For comparison, the average heat
The two layers move against each other and generate heat.
flow on Earth is 0.06 W/m2, although some of the

same effect as the moving magnetic field in an
electric generator. It produces a large potential
difference. In the case of Io, about 600 000 V (see
Problem 25.9)! The ionized gas between Io and
Jupiter is a good conductor, so currents flow par-
allel to the magnetic field. These currents are as
high as one million amperes! This also results in
bursts of radio emission, explaining why Jupiter is
a strong radio source. The bursts are more frequent
(a) when Io is in certain positions.
Io also has an atmosphere. It is quite irregu-
larly distributed, with more gas being over the
warmer areas. This suggests that the atmosphere
is being replaced by the volcanic activity. A major
constituent is sulfur dioxide. Io also has a dense
ionosphere, with a density of about 104 to 105 par-
ticles/cm3. A large cloud has also been detected
around Io. This cloud was first detected from
Earth by observations of sodium, and is referred
to as the sodium cloud. However, sodium was just
one the easiest elements to observe, and other
constituents are present. The cloud extends tens
of thousands of kilometers along the orbit.
Our knowledge of Europa, shown in Fig. 25.29,
is less detailed. The closest flybys have produced
pictures with only 4 km resolution. However, we
can tell that the surface is relatively flat. There is
a complicated pattern of lines. There is also a
lack of large craters, suggesting a young surface.

Fig 25.29. Images of Europa. (a) True (left) and enhanced
(right) color views of the whole moon (Galileo orbiter
image). (b) Voyager 2 closest approach. (c) Large impact Fig 25.30. Galileo orbiter images of Ganymede. (a) Trailing
structures (Galileo orbiter image). [NASA] hemisphere.

Callisto, shown in Fig. 25.31, has a heavily
cratered surface. These craters are also flat, typi-
cal of an icy surface. An unusual feature is the
large ring structures. These are probably the
result of violent impacts in the past.
The moon of Saturn that we know the best is
its largest, Titan, shown in Fig. 25.32. It has the



Fig 25.30. (Continued) (b) Fresh impact craters.The image
covers an area 142 132 km. (c) Calderas. [NASA]

The surface is probably made of ice. We think
that the lines are tension patterns in the ice. This
tension may have resulted from an expansion
of the surface, probably by about 5%. There are
some dark patches; these are probably composed
of silicates.
Ganymede, shown in Fig. 25.30, also has an ice
surface. It is covered with irregular light and dark
regions. The dark regions are heavily cratered,
indicating that they are the older part of the sur-
face. The craters in these regions are relatively
Fig 25.31. Images of Callisto. (a) Combined Voyager and
flat. The lighter regions are grooved. The grooves
Galileo mosaic. (b) Possibly an oblique impact (Galileo
appear to be alternating ridges and troughs. The
orbiter image). [NASA]
pattern suggests tension, as on Europa.


Fig 25.32. Titan™s haze layer (Voyager 1 image). [NASA]

most significant atmosphere of all the moons in
the Solar System. Voyager 1 passed within
5000 km of Titan, giving us a good opportunity
to study it. Some of the other moons are shown
in Fig. 25.33. Titan™s density is quite low, only
twice that of liquid water, suggesting a mixture
of rock and ice. Its composition and structure
have been affected by heat given off by Saturn
during its formation. The model of its interior,
shown in Fig. 25.34, has 55% of the mass in a
rock core. It is possible that internal heating has
taken place. Tidal distortion should not be as
important for Titan and Saturn as for Io and
Jupiter. There may be some radioactivity, but it
seems more likely that the energy has been
stored from the time that the moon formed.
Beyond the core, there are probably layers of dif-
ferent structures of ice.
Titan™s atmosphere contains methane (CH4).
This was originally determined from the Earth.
However, closer observations have shown that
nitrogen (N2) makes up 80 to 95% of the atmos- (c)
phere. It has also been suggested that there is
Fig 25.33. Saturn™s moons. (a) Dione (Voyager 1 image).
some argon. Ultraviolet radiation triggers a chem-
(b) Iapetus (Voyager 2 image). (c) Enceladus (Voyager 2 image).
istry that produces traces of ethane, acetylene and

The surface pressure is 1.6 atmospheres. The
surface temperature is 93 K. The highest tempera-
ture in the atmosphere is about 150 K, in the
upper atmosphere. The lowest temperature is
about 70 K, about 40 km above the ground. An
interesting feature is that it contains the temper-
ature (90.7 K) at which the three phases “ gas, liq-
uid and solid “ of methane can coexist. We might
therefore expect to find a frozen methane sur-
face, with methane oceans and methane clouds
in the atmosphere. Others have speculated on
oceans of ethane rather than methane.
Voyager 2 provided our first good look at the
five known moons of Uranus, and discovered a
number of smaller moons. Close-ups of some of
the moons are shown in Fig. 25.35. As with

Fig 25.33. (Continued) (d) Tethys (Voyager 2 image). [NASA]

ethylene. These are found in the lower atmos-
phere. Higher up, hydrogen cyanide (HCN) is
formed. There is no oxygen, since it is tied up in
the frozen water. The chemistry produces a smog,
possibly with some seasonal variation.


Water and Ammonia (a)



Fig 25.34. Internal structure of Titan, as deduced from Fig 25.35. Voyager 2 images of the Moons of Uranus.
theoretical models. (a) Ariel; (b) Titania;

Fig 25.36. Global color mosaic of Triton (Voyager
2 image). [NASA]

Jupiter and Saturn, the moons of Uranus show a
variety of features. Oberon has large albedo vari-
ations over its surface. It is cratered, and a small
mountain was photographed on the limb. Titania
has craters, ridges, rills and scarps. Its cratering is
sparse compared to that of other moons. Ariel has
many bright reflected regions as well as deep
scarps and valleys. There are many filled in fea-
tures, indicating an active surface. Some of the
most intriguing structures are found on Miranda.
It has three distinct types of surface: an old sur-
face, a scored region, and a region with very com-
plex structures.
Neptune™s largest moon, Triton, is shown in
Fig 25.35. (Continued) (c) Miranda, showing a variety of Fig. 25.36.
terrains; (d) Oberon. [NASA]

Chapter summary
In this chapter, we looked at the outer planets in Both Jupiter and Saturn give off more energy
the Solar System. They are much larger than the than they receive from the Sun. It is speculated
inner planets, and have structures that are very that this comes from the gravitational potential
different from those of the inner planets. energy liberated during the collapse of the
Because of their large masses, these planets, planets.
especially Jupiter and Saturn, have been able to The general circulation on Jupiter and Saturn
retain much of their initial supply of hydrogen, is quite complicated, producing bands and varying
giving them compositions that are dominated by cloud and wind flow. Large, hurricanelike sys-
hydrogen. The minor constituents play important tems, such as the Great Red Spot, are found. The
roles in the atmospheric structure, energy trans- atmospheres of Uranus and Neptune are not as
fer and appearance. dense or as active.

The interiors of Jupiter and Saturn are mostly effects may be important in ring formation. We
liquid hydrogen, with the possibility of metallic also looked at the dynamical effects that provide
hydrogen in the center. Uranus and Neptune are the intricate structure of the rings and gaps.
thought to have rocky cores. Finally, we looked at the major moons of these
We looked at the properties of the rings planets. These moons provide a diversity of sur-
around these planets. We showed how tidal face and atmospheric phenomena.

25.1. What are the major differences between the 25.13. What are the differences in the ring systems
four inner planets and the four outer planets of Jupiter, Saturn, Uranus and Neptune?
(i.e. not including Pluto)? 25.14. What is the relationship between planetary
25.2. What are the similarities and differences rings and the Roche limit?
between Jupiter and Saturn? 25.15. Standing on the Earth, we are within its
25.3. What are the similarities and differences Roche limit. Why aren™t we torn apart by
between Uranus and Neptune? tidal effects?
25.4. Why is it hard to measure the rotation 25.16. If Saturn™s rings were known for centuries,
period of Uranus? why did it take so long to find the rings
*25.5. The outer planets are much more massive around Jupiter, Uranus and Neptune?
than the Earth, yet the accelerations of 25.17. Contrast the ways in which the rings of
gravity on their surfaces are not much Jupiter and Uranus were discovered.
different than that on Earth. How can 25.18. What is the evidence that Saturn™s rings are
that be? made of many small particles, rather than
25.6. The atmosphere of Jupiter contains a high being solid?
proportion of hydrogen. This is not true of 25.19. How do moons affect the appearance of
the Earth™s atmosphere. Why is this? planetary rings? What is the evidence that
25.7. Why would you expect Jupiter™s atmosphere this is happening?
to have a composition similar to that in the 25.20. What is the evidence that Io has a molten
early Solar System? interior?
25.8. What accounts for the different colors in 25.21. What is the cause of Io™s molten interior?
the bands on Jupiter (or Saturn)? 25.22. Contrast the properties of the largest moon
25.9. What is the explantion for the Great Red Spot? around each of the giant outer planets.
25.10. Compare the compositions of the atmos- 25.23. List the 15 most massive objects in the Solar
pheres of Jupiter and Saturn. System (not including the Sun), in order of
25.11. Compare the compositions of the atmos- decreasing mass.
pheres of Uranus and Neptune. 25.24. List the moons of the Solar System in order
25.12. What is the source of the excess energy that of decreasing size. Where do the smallest
Jupiter gives off? planets fit in?

25.1. Find the gravitational potential energy of 25.2. Confirm Kepler™s third law for the moons of
Jupiter. Assume that this amount of energy Jupiter. Use the information to derive
has been released over a period of 4 billion Jupiter™s mass.
years. How does the average rate of energy 25.3. Calculate the rate of temperature fall-off for
release compare with that in the sunlight an adiabatic process near the Jupiter cloud
received by Jupiter?

tops. Compare it with the similar number planet. This introduces pseudo-forces in the
for Saturn. rest frame of the orbiting material. How do
25.4. From the data given in the chapter, show these pseudo-forces affect the Roche limit
that Saturn gives off approximately twice as calculation?
much power as it receives from the Sun. 25.9. Use Faraday™s law to derive an expression for
25.5. What is the ratio of solar energy per second the potential difference across a planet by a
magnetic field B sweeping across the
per unit surface area reaching Uranus to
planet™s surface at a speed v. Take the planet
that reaching Neptune?
radius to be R.
25.6. Voyager 2 was five planetary radii from
Uranus. At that time, what angle was 25.10. Compare the magnitude of the tidal effects
subtended by the planet as viewed from the that Jupiter exerts on Io with those that
spacecraft? Saturn exerts on Titan.
25.7. Show that all the rings of Jupiter, Saturn 25.11. Estimate the adiabatic temperature gradi-
and Uranus lie within the Roche limits for ents at the altitude of minimum tempera-
these planets. ture for both Jupiter and Saturn.
25.8. In deriving the Roche limit, we ignored the
fact that the particles are orbiting the main

Computer problem

25.1. (a) Show that the major moons of each outer mass of each outer planet from the orbital data of
planet obey Kepler™s 3third law. (b) Calculate the the most massive moon.
Chapter 26

Minor bodies in the Solar System

Pluto™s mass is now known reasonably accu-
There are a vast number of smaller objects in our
rately. This is because a moon was discovered
Solar System, not as substantial as our Moon, but
orbiting Pluto in 1978. The moon is named
which provide important clues on the history of
Charon, and is shown in Fig. 26.2. By studying
the Solar System. These are asteroids, comets and
its orbital motion we can determine Pluto™s mass.
meteoroids. We have also included the ninth planet,
Actually, using the Hubble Space Telescope, it
Pluto, in this chapter. As we will see below, recent
has been possible to look at the motions of
determinations of Pluto™s mass make it by far the
both Pluto and Charon about their common
least massive planet, and it has more properties
center of mass. From this it has been deduced
in common with the other less massive objects in
1025 g, and Charon™s
that Pluto™s mass is 1.3
the Solar System.
mass is about 1/12 of that. Pluto™s mass is only
about 1/500 of the Earth™s mass, or one-fifth
26.1 Pluto that of our Moon.
Pluto™s size has been estimated from its fail-
ure to occult certain stars. (See Section 26.4 for
Pluto was discovered in 1930, following an exten-
how this technique is used for asteroids.)
sive search, by Clyde Tombaugh. The search was ini-
However, our best measurements now come
tiated by Percival Lowell after it was thought that a
from optical interferometry techniques. Using
planet beyond Neptune might be perturbing
this size, and the measured mass, we find that
Neptune™s orbit. Calculations narrowed the range
Pluto has a very low density, about 0.5 to
of possible locations on the sky, and a search was
1.0 g/cm3. Charon™s density is even lower, only
carried out. As Fig. 26.1 shows, Pluto doesn™t
20% greater than that of liquid water. This sug-
stand out very well against the background of
gests that its composition is similar to that of
stars. It is detectable as a planet only by its very
the moons of the giant planets. It has been sug-
slow motion with respect to the stars.
gested that Pluto™s surface is frozen methane
For Pluto to have a perturbing effect on other
and that its atmosphere is also composed of
planets, its mass must be greater than that of the
Earth. For this reason, since its discovery, Pluto™s
Pluto™s size, density and orbit raise questions
mass has been overestimated. We now know that
about its status as a planet. Its orbit is the most
its mass is much less than previously thought,
eccentric of the planets. It even spends part of its
and that it has no measurable effect on other
orbit closer to the Sun than Neptune. It has been
planets. In a sense, Pluto™s discovery was acciden-
suggested that Pluto may actually be an escaped
tal. It was a result of an extensive search of a par-
moon of Neptune. This would explain its small
ticular region in the sky. For this reason, other
size, low density, and crossing of Neptune™s orbit.
searches have been carried out for a “tenth
However, when we trace back the orbits of
planet”, none with success.

Fig 26.1. (a) HST images of Pluto, showing both hemi-
spheres.The insert images are the unprocessed images.
(b) Surface map of Pluto, based on the HST images.







0° 60° 120° 180° 240° 300° 360°
East Longitude


Neptune and Pluto, we find no time when they
26.2 Comets
were actually close together. Thus, if Pluto did
escape from Neptune, its orbit must have been
Every now and then, a spectacular comet, like
perturbed since then. Thus, Pluto™s origin is still
those shown in Fig. 26.3, is visible for a few weeks,
a mystery.

Fig 26.2. Pluto, with Charon.
This is a near infrared image
(2.2 m).The separation
between the objects is 0.9 arc
sec. [STScI/NASA]



(b) (d)
Fig 26.3. (a) Comet Halley in 1910.This is a computer-reconstructed image. (b) Comet Giacobini“Zinner, on 1 Nov, 1998.
(c) Comet Ikeya“Seki, 1996. (d) HST time sequence showing the evolution of the core of Hale“Bopp (1995).
[(a)“(c) NOAO/AURA/NSF; (d) STScI/NASA]

and we are reminded of this phenomenon. reflected signal. Since the early 1950s the con-
ventional picture, advanced by Fred L. Whipple, is
However, most comets are faint, and do not attract
attention. They are still very important in our that the nucleus is a “dirty snowball”. It con-
understanding of the history of the Solar System. tains dust, plus ices of water, carbon dioxide,
Over 600 comets are known. Their masses are less ammonia and methane. When a comet is close
than one-billionth of the mass of the Earth. to the Sun, material is ejected from the nucleus.
Figure 26.4 shows the basic structure of a This material acts like the exhaust of a rocket,
comet. The smallest part is called the nucleus. It and provides thrust for the comet. This may
is only a few kilometers across. One way of deter- actually alter the orbit of the comet. It is one of
mining sizes is by bouncing radio waves off the the reasons why the exact prediction of comet
surface and measuring the strength of the orbits is difficult.

CO is responsible for the blue color of the tail.
Coma The gas tail can be up to 108 km long. (This is
almost 1 AU.) The dust tail is material that is left
104 km
behind in the orbit. We see it as a smooth curve,
tracing out the comet™s orbit. It is ejected by pres-
sure from sunlight. When it is free of the comet,
the dust continues in a Keplerian orbit, per-
x1000 turbed by radiation pressure. The dust tail can be
up to 107 km long. Sometimes, a tail appears to be
Dirty Ice
pointing toward the Sun. This is an illusion,
Ice caused by the appearance of the tail pointing
10 km
away from the Sun, as viewed from the Earth in
particular positions with respect to the comet
and the Sun.
Fig 26.4. Structure of a comet.
We can estimate the effect of radiation pres-
sure on the dust grains. The source of the radiation
Outside the nucleus is an extended region of pressure is the momentum carried by photons. For
gas and dust, called the coma. It is 105 to 106 km a photon of energy E, the momentum is
in extent, and shines by reflected sunlight. The
p Ec (26.1)
material in the coma flows outward at about
If this photon is absorbed by an object, then all of
0.5 km/s. The outflow of gas drags it away from
the momentum is transferred to the object. If the
the nucleus. The sunlight reflects off both gas and
photon is reflected back off an object, then the
dust in the coma. The coma can also emit radia-
momentum delivered to the object is twice this.
tion from excited gas. Spacecraft observations
That is, since the photon reverses direction the
have shown Lyman alpha emission in the ultravi-
magnitude of its momentum change is 2p.
olet. These indicate that the hydrogen cloud is up
Suppose we want to calculate the pressure a dis-
to a factor of ten larger than the coma itself. It is
tance r from the Sun. Imagine a spherical shell at
thought that this hydrogen comes from the break-
this radius. The force F on the shell is just the
up of water molecules and OH radicals by solar
momentum per second carried by the photons
ultraviolet radiation. Spectra of the coma have
reaching the shell. That is
indicated the presence of a number of simple mol-
ecules, such as NH3, H2O, OH and NH.
F dp dt
When a comet moves relatively close to the
The pressure P is the force, divided by the area of
Sun, it may develop a large tail. This tail can be up
the shell 4 r 2, so
to 1 AU in extent, but is of such low density that
it doesn™t contain an appreciable fraction of the
dp dt
mass of the comet. Only about 1/500 of the mass P
4 r2
of the comet is in the tail.
There are actually two tails, as shown in
Fig. 26.5. The gas tail is blown straight out by the Mot
interaction of the solar wind and the comet. The of
gas tail always points away from the Sun. et
Variations in the solar wind produce a varied
appearance along the length of the tail. A num-
ber of molecular ions have been detected in the un
To S Dus
t Ta
gas tail: CO , CO2 , CH , CN , N2 , OH and il
H2O . It also contains some more complex mole-
cules, such as formaldehyde (H2CO). Since it con-
Fig 26.5. The two comet tails.
tains ions, it is called the ion tail. Emission from

The radiation pressure is
By equation (26.1),
dE dt
1 d2 2
dp dt Frad
c 4 r 2c
But dE/dt is just the energy per second emitted by Equating these and solving for d gives
the Sun, or the solar luminosity L . Therefore
L 16 GM c
P (26.2)
4 rc
1033 erg s2
16 16.67 dyn cm2 g2 2 12 1033g2 11 g cm3 2 13 1010 cm s 2
At a distance of 1 AU from the Sun, the pressure 8
is 5 10 5 dyn/cm2, which is 5 10 11 atm.
6 10 cm
Despite this small pressure, if a grain is
small enough, the force on the grain due to radi- For grains smaller than this, the radiation pressure
ation pressure can exceed the gravitational will push out more strongly than gravity pulls in.
attraction by the Sun on the grain. Small grains
Our current picture of the origin of comets
are important, because the gravitational force
places them in a cloud, called the Oort cloud, some
depends on the mass of the grains, which is pro-
50 000 AU from the Sun. The cloud is like a spher-
portional to d3 (where d is the grain size), and the
ical shell around the Solar System. The existence
radiation pressure force is pressure multiplied
of such a cloud is suggested by the fact that
by area, so it goes as d2. Therefore, the ratio of
comets appear to be bound to the Solar System
(no hyperbolic orbits), and that they come from
Frad all directions, orbiting with equal frequency in
Fgrav d either direction relative to the orbital motions of
the planets. The cloud may contain 1012 to 1013
comets, giving it a total mass in the range of one
to ten Earth masses. From time to time, one of
the objects in the cloud has its orbit severely per-
Since the ratio of the forces is proportional to 1/d,
turbed, and starts to head for the inner Solar
the radiation pressure is more effective for
System. At this time, the comet is just the mate-
smaller grains. Note that equation (26.2) tells us
rial that will be seen as a nucleus when the
that the radiation pressure force is proportional
to 1/r 2 (where r is the distance from the Sun). This comet is in the inner Solar System. There is no
coma or tail. As the comet moves closer to the
is the same dependence as the gravitational force
Sun, it develops the coma, and then the tail, as
on r. Therefore, the ratio of the forces is inde-
discussed above. The coma begins to appear when
pendent of the distance from the Sun. It only
the comet is about 3 AU from the Sun. At this
depends on grain size.
point, the temperature is about 215 K, which is
Example 26.1 Radiation pressure
right for the sublimation of water ice to form
For grains of a given density 1 g/cm3, find the grain
water vapor.
size for which gravity and radiation pressure are
The comet is brightest at perihelion. The tail
has its longest physical extent then. However, the
apparent length of the tail depends on the view-
ing angle from the Earth. Therefore, the tail may
The gravitational force on a grain of mass m is
not appear to be longest at perihelion. Once the
comet has passed through the inner Solar
Fgrav System, it continues outward in its orbit. Most of
the orbits are ellipses, so the comets will return,
3 2d3 M
G14 unless the orbit is perturbed as the comet passes
r2 near a planet. Orbital eccentricities and periods

4.6 million km (Fig. 26.6). This comet was of inter-
est because of the simultaneous discovery by the
IRAS satellite, in the infrared, and two more tra-
ditional ground-based optical observers.
Astronomers and the general public were
treated to a rare astronomical phenomenon
when a large comet, which had actually broken
into fragments, struck Jupiter in 1994. The comet
is called Shoemaker“Levy, for its co-discoverers.
When its orbit was analyzed, it was found that it
was on a collision course with Jupiter.
Fig 26.6. IRAS image of comet IRAS“Araki“Alcock. [NASA]
The Hubble Space Telescope provided very
sharp images of the comet. One of these is shown
vary considerably. For example, Comet Encke has in Fig. 26.7. The comet had broken into several
a period of 3.3 years and Comet Kohoutek has a fragments that were seen stretched out along its
period of 80 000 years. It is thought that most orbit. It was speculated that this break-up was
short-term comets have had their orbits severely caused by the tidal effects of Jupiter. That is, the
perturbed by Jupiter. side of the comet that was closer to Jupiter was
The appearance of a particular comet near pulled with a stronger gravitational force than
perihelion is often hard to predict in advance. For the side that was farther away. This would first
example, it was predicted that Comet Kohoutek simply stretch the comet, but could eventually
(1974) would be very bright. This was based on the pull it apart into pieces of various sizes.
fact that it appeared to be bright when it was far Astronomers and the rest of the news follow-
away. The comet was not as bright as predicted. It ers were fascinated with the possibility of wit-
is now speculated that the comet was making its nessing such a catastrophic event as a comet
first pass by the Sun, and therefore behaved dif- striking another planet. Also, since the comet
ferently than a comet making a return visit. On had broken into pieces, there would be a series of
the other hand, Comet West (1975) put on a better impacts over a period of several hours. Apart from
show than expected. People who remember the the spectacular nature of such an event, it would
spectacular view of Halley™s comet in 1910 were allow astronomers to study the effects of this
disappointed by an unfavorable viewing angle in impact on the Jovian atmosphere. This would
1986. The closest approach of Halley™s comet to allow atmospheric specialists to test their theo-
Earth in 1986 was much farther than in 1910. In ries about the structure and composition of the
addition, the view in 1986 was marred for many Jovian atmosphere. It would also allow them to
observers by the spread of light pollution in the study the general effects of such a catastrophic
previous 76 years. event, with an eye towards understanding how
Some comets pass quite close to the Earth. For such events might affect other planets, including
example, IRAS“Araki“Alcock (1983) passed within the Earth.

Fig 26.7. Photograph (from
HST) of fragments of Comet
Shoemaker“Levy. [STScI/NASA]

This encounter was also fascinating because heating in the Solar System itself. Therefore, the
it could be predicted, and the times and loca- composition of comets should reflect the compo-
tions of impacts could be calculated. Normally, sition of the original solar nebula. That is one
by their very nature, catastrophic events (from reason why there was considerable activity in
earthquakes to supernovae) happen with little or launching spacecraft to fly near comets as they
no warning. In this case, astronomers could pre- were close to the Earth, including Halley™s comet
pare well in advance to watch the events unfold. in 1986.
In the anticipation of the collisions, there was Also far out in the Solar System is a group of
objects called Kuiper Belt objects. These icy objects
one cautionary aspect. The impact was going to
occur on the side of Jupiter facing away from the are found between the orbit of Neptune (30 AU)
Earth. So we could not watch the actual impacts out to 50 AU. So far, more than 300 have been dis-
from Earth (or even from a telescope in orbit covered, and it is estimated that there are at least
around the Earth). We would have to wait for 70 000 such objects larger than 100 km across.
Jupiter™s rotation to bring the impact around to Unlike the Oort cloud, Kuiper Belt objects are
the side facing the Earth. There was concern that more tightly confined to the ecliptic, forming a
if the impacts were not very strong, then their thick band. It is likely that these are left over
effects would quickly fade, and we would not see
too much by the time the impact sites came into
view. However, Jupiter rotates very quickly (taking
only ten hours to make one rotation). It was cal-
culated that the impacted sites would rotate into
view roughly an hour after each impact. The
other major uncertainty was about the masses of
the fragments. The greater the masses, the more
they would affect the atmosphere.
When the impacts occurred very few people
were disappointed. At the high speed of the
impact (remember, kinetic energy is (1/2)mv2), the
masses of most of the fragments were sufficient
to cause major disruptions, as shown in Fig. 26.8. (a)
As each impact site came around into view, a
plume of material could be seen that had been
ejected from the atmosphere. Also, infrared
observations confirmed that the impact sites
were hotter than their surroundings.
In the aftermath of these spectacular events,
many were wondering about how such an impact
would affect the Earth, and how often such
impacts on the Earth might occur. There is
already growing speculation that the impact of
an object from space caused the extinction of the
dinosaurs, some 70 million years ago.
Comets are important in our understanding
of the Solar System. We think that the Oort cloud
is left over from the material that condensed to
form the Solar System. The current idea is that Fig 26.8. HST images of Jupiter during and after the
Comet Shoemaker“Levy impacts. (a) Jupiter after the frag-
the comets formed near Jupiter and Saturn, and
ment G impact. (b) Images (at various wavelengths) of
were ejected out to the Oort cloud location. The
fragment G impact spot.
cloud is far enough out to be unaffected by


Fig 26.8. (Continued) (c) Blemish caused by fragment G
impact. (d) Plume from fragment G impact. (e) Fragment G
impact plume growing. [(a)“(c) NASA; (d), (e) NASA IRTF
Comet Collision Team]

brightly as it streaks across the sky, as shown in
Fig. 26.10. At this point, we refer to it as a meteor.
Most meteors burn up as they pass through the
atmosphere. However, some do reach the ground.
The ones that reach the ground are called mete-
orites. The largest meteorites produce craters,
(e) including the large one shown in Fig. 26.11.
Some very small meteoroids, much less than
1 mm across, may settle into the upper atmos-
from the formation of the Solar System, provid-
phere, to be collected by balloons. These are of
ing information on the conditions under which
interest because there are many more small mete-
the outer part of the Solar System was formed.
oroids than large ones.
We also think that the Kuiper Belt is the source of
Most meteoroids that produce meteors are
short period comets (as opposed to the long
probably the debris of comet tails. They are there-
period comets that come from the Oort Cloud).
fore left behind in the orbit of the comet, as
shown in Fig. 26.12. When the Earth crosses the
26.3 Meteoroids comet™s orbit, we see a large number of meteors “
a meteor shower. These showers occur at the same
time each year, since they represent the passage
Meteoroids are small chunks of matter left in
of the Earth through the orbit of the comet. This
space. They are up to tens of meters in diameter.
scenario explains why we see most meteors after
When the Earth encounters a meteoroid
midnight. After midnight, an observer is on the
(Fig. 26.9), the meteoroid may fall through the
side of the Earth facing in the direction of the
Earth™s atmosphere. It is then heated by the fric-
orbital motion of the Earth.
tion between the air and the meteoroid. It glows

group, called irons, are mostly iron and nickel. The
other type, called stones, have an appearance simi-
lar to ordinary rocks and are hard to find on the
ground unless the fall has been witnessed. There is
another type, stony irons, which is a combination of
two types of material. Most meteorites are stones.
30 km/s Most of the stones contain small rounded
glasslike particles, called chondrules. Meteorites
containing chondrules are called chondrites. The
others are called achondrites. Chondrites with large
amounts of carbon are called carbonaceous chon-
drites. It is believed that these are the oldest mete-
orites. An example of a famous carbonaceous
chondrite is the Murchison meteorite, which fell in
To Australia in 1969. This meteorite contains amino
Sun acids of a type (left-handed vs. right-handed) not
found on Earth. The largest carbonaceous chon-
Fig 26.9. Earth moving through meteoroids. Note that the
activity is greatest when the observer is on the leading edge drite is the Allende meteorite, which fell on
of the Earth, which occurs after midnight.The maximum Mexico in 1969. We think that the carbonaceous
effect is actually at dawn, so the best time to see meteors is
chondrites were never strongly heated after for-
just before it begins to get light in the morning.
mation. They therefore preserve the original mate-
rial out of which the Solar System formed. The
Allende meteorite has centimeter-sized inclusions
Since we think that comets are left over from
of minerals rich in calcium and aluminum.
the formation of the Solar System, meteorites give
It has been suggested that there is some rela-
us a chance to examine that material directly. Two
tionship between asteroid types and meteorite
different compositions have been identified. One

Fig 26.10. Views of the
Leonid meteor shower. [NOAO/

Radioactive dating tells us that meteorites are
4.5 billion years old. This confirms our idea that
they are part of the debris left over from the for-
mation of the Solar System. Studies are now
being done of the relative abundances of various
elements to understand the composition of the
material out of which the Solar System formed.
So far, some unusual abundances have been
found. For example, xenon-129 is formed by the
beta decay of iodine-129, with a half-life of 17 mil-
lion years. A lot of xenon-129 has been found, sug-
gesting a large production of iodine-129 just prior
Fig 26.11. Meteorite crater on Earth.This is the Barringer
Crater, just east of Flagstaff, Arizona.The bowl has a 1 km to the formation of the Solar System. However,
diameter. Notice the elevated rim. It is the ¬rst terrestrial this is still quite speculative, and some claim that
crater recognized as coming from a meteor impact. [USGS] they can explain the abundances without invok-
ing any unusual events.
types. In this picture, the C type asteroids are
like the carbonaceous chondrite meteorites. The S
26.4 Asteroids
type asteroids are like the ordinary chondrites or
the stones.
The distribution of asteroids is shown in Fig.
26.13. Most of the asteroids lie in a band between
the orbits of Mars and Jupiter. This band is called
the asteroid belt. Over 3000 asteroids have been cat-
aloged to date, but there are many more. The ones
that have been cataloged are the brightest, and
presumably the largest. We would expect there to
be many more small ones. The combined mass of
the asteroids is less than that of the Moon.
The sizes of the asteroids are determined from
Comet Orbit

stellar occultations, as shown in Fig. 26.14. The
orbits of many asteroids are well known. When
we talk about the orbit of an asteroid (or any
other object), we are talking about the path of its
center. When the center comes close to passing in
front of a star, the asteroid can only occult the
star if the asteroid is large enough. When an
occultation is expected, astronomers from vari-
ous parts of the Earth watch. If they all see the
occultation, then the asteroid is larger than some
size. If none sees the occultation, the asteroid is
smaller than some size. If some see it and others
don™t, the size of the asteroid can be determined
quite accurately.

r t h's O r Of the asteroids that have been studied in this
way, only six are larger than 300 km, 200 are
Fig 26.12. Meteoroids and comets.The meteoroids are larger than 100 km, and there are many smaller
debris left by the comet along its orbit.We see a meteor than 1 km. If we know the size and how bright
shower as the Earth passes through the debris.
they appear, we can estimate the albedo of the

6 6 Star
(not occulted)

5 5

Path of
d1 d2
1:1 asteroid
4 4
Relative Number


. 25
( 28)