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. 23
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radiation from the ground, is called the tropo- Using this, the power per area leaving z becomes
4 T 4 1zt z2
sphere. It is that part to which our daily life is con-
power
fined. We now look at radiative energy transport
c13L>c 2 1zt z22
area
from the ground into the troposphere, as out-
lined in Fig. 23.14. The ground is at a temperature L
4
T4
1zt z2
Te. We assume that the troposphere has a height 3
zt, and we are interested in energy reaching and
However, the path length zt z, divided by the
leaving some intermediate height z.
photon mean free path L, is the optical depth (as
The power per unit area reaching z from the
discussed in Chapter 6), so
4
ground is Te . The power per unit area radiated at
z is T 4, where T is the temperature at z. However, T4
power 4
some of the radiation emitted at z does not area 3
escape. It is absorbed at a higher altitude, to be re-
Equating this to the power received from the
emitted. The energy density in the layer between
4
ground Te , and solving for T, we have
z and zt, due to the energy emitted at z is 4 T 4/c.
Therefore, the energy per unit area in this layer is
a b T4
3
T4
the energy per unit volume, multiplied by the e
4
thickness, zt z, so
A more accurate treatment of the radiative
(4 T 4/c)(zt z)
energy/area transfer, originally worked out by Sir Arthur
Eddington, for the atmospheres of stars, modifies
If all of this energy left z in a time t, the power per
our result to
unit area is the energy per unit area, divided by t,
(4 T 4/ct)(zt
a b T4 a b
3 2
z)
power/area
T4 (23.14)
e
4 3
460 PART VI THE SOLAR SYSTEM



At the top of the troposphere, 0 (all photons perature, pressure and volume are T, P and V. For
escape upward). This means that T 0.84Te. If we an adiabatic process, these quantities are related by
use Te 246, then T 207 K. The infrared optical
PV P0 V 0
depth from the bottom to the top of the tropo-
sphere is about 2, giving T 1.2Te, or about 293 K. where is the ratio of the specific heats at con-
Convection also plays an important role in the stant pressure and constant volume. For a
energy transport in the lower atmosphere. As the monatomic gas, 5/3. For a diatomic gas,
air near the ground is heated, it expands. The 7/5. Since the Earth™s atmosphere is mostly N2
buoyant force on the air will make it rise until it and O2, we use 7/5. (If there is a lot of water vapor
reaches air of its own density. The more rapidly present, will be different.)
the temperature falls with altitude, the more rap- We can eliminate the volume using the ideal gas law
idly the pressure falls with altitude. A faster pres-
V NkT/P
sure fall-off means a larger pressure difference
between the top and bottom of any parcel of air. where N is the number of molecules in the parcel
The larger pressure difference provides a larger of air. This gives
buoyant force, and the air rises more quickly,
P1 1
T P0 T0
making convection more important. Therefore,
convection becomes more important when the For convenience, we will call this constant quantity
fall-off in temperature with altitude is larger. C. Solving for T gives
As the air rises, it encounters lower pressure 1
T CP
air and expands. The gas does work in the expan-
sion. It takes no energy from its surroundings, so To obtain dT/dz, we differentiate both sides with
it cools. A process in which no energy is respect to z, giving
exchanged is called adiabatic. The convection
a b 1 2P a b
dT dP
1 2
process itself will modify the temperature gradi- T C1
dz dz
ent, dT/dz, to the value appropriate for an adia-
1 2P 1 g2
2
batic process. If the temperature gradient by C1
radiation is less than the adiabatic gradient, the
g.
where we have used the hydrostatic law dP/dz
temperature gradient is not enough to drive con-
Solving for dT/dz, and substituting for C, we have
vection. However, if the temperature gradient is
a bg a b
T0 0
greater than the adiabatic gradient, convection 1
dT
will set in. Remembering that dT/dz is negative, P0
dz
we can say that the condition for significant con-
From the ideal gas law, T /P m/k, so
vection is that

a ba b
mg 1
dT
` ` 7` `
dT dT
k
dz
dz rad dz ad
9.8 K/km
As the warmer air rises from the ground, it
Example 23.2 Adiabatic temperature gradient
must be replaced by cooler air from above. This
Find the value of dT/dz when energy flow is domi-
must be replaced by cooler air from above. This
nated by convection near the ground.
leads to the general flow pattern shown in Fig.
23.15. This circulating flow is called a convection
SOLUTION
cell. We see this same pattern in a pot of boiling
We don™t have to consider the whole atmosphere to
water, or in the Sun™s atmosphere, causing the
do this. We only have to look at a small volume, or
granular appearance.
parcel, of air, and see how it behaves. Suppose we
The situation becomes more complicated
have a parcel of air with volume V0 rising from just
when there is water vapor in the air. As the air
above the ground, where the temperature is T0 and
rises and cools, it may pass through a temperature
the pressure is P0. As the parcel of air rises, its tem-
23 THE EARTH AND THE MOON 461



strengthened by the energy released when water




Cool Air Falling
Hot Air Rising
goes from a vapor to a liquid (the heat of vapor-
Troposphere
ization). In the cases of particularly strong con-
vection, thunderheads, like those in Fig. 23.16(b)




Earth
Fig 23.15. Convection cells.This motion carries hot air up
and cold air down.


at which the water condenses into liquid
droplets. When this happens, clouds form. Usually
this happens at a particular altitude where the
temperature is just right, so the cloud bottoms
form a reasonably flat layer, as in Fig. 23.16(a). If
the convection is weak, thin cloud layers will
form. However, if the convection is strong, it can
continue the cloud building upward, forming
cumulus clouds. The convection can even be




(c)




(a)




(d)
Fig 23.16. Weather phenomena resulting from convection.
(a) Normal cloud layers result from convection.The cloud
layers are the tops of the convection cells. In this photo, a
thunderstorm is forming at a higher level in the background,
with a smaller low level cloud in the foreground.
(b) Lightning under thunderclouds. (c) Flashes above
thunderstorms. (d) Very strong thunderstorms can lead to
tornadoes. [(a), (b), (d) NOAA; (c) NASA]
(b)
462 PART VI THE SOLAR SYSTEM



are formed. Thunderheads tend to form on sum- chemicals can eventually make their way from
mer afternoons, when the solar heating creates a the ground to the ozone layer. Recently, evidence
has been found for a variable and large ozone hole
very large temperature gradient, driving very
strong convection. over the Antarctic. This is indicated in Fig. 23.17.
One place where solar energy is directly
23.3.3 Retention of an atmosphere
absorbed is the ozone layer. The ozone in the upper
atmosphere is very efficient in absorbing the Even if a planet is formed with an atmosphere,
ultraviolet radiation from the Sun. This has two that atmosphere will not necessarily be retained.
important effects: (1) it shields the lower atmos- Any molecules moving faster than the escape
phere (especially us) from the harmful effects of speed can escape from the top of the atmosphere.
that ultraviolet radiation, and (2) it provides a If escaping molecules are not replaced from
source of direct heating for the upper atmos- below, by gases escaping from the planet™s inte-
phere. The direct heating makes the ozone layer rior, the atmosphere will eventually be lost.
hotter than the parts of the atmosphere immedi- At any temperature the molecules with lower
ately above or below it. molecular mass will be moving faster, so these
The shielding of the ultraviolet radiation is are the most likely to escape. We can calculate the
value of A for which the molecular speed
important, because if it is not blocked in the
(3kT/m)1/2 is equal to the escape speed for the
ozone layer, a significant amount will reach the
planet (2GM/R)1/2. Equating these, writing m as
ground. This excess solar ultraviolet radiation can
Amp, and solving for A gives
be harmful to many living things, including peo-
ple. There are concerns about the by-products of
3kTR
technology causing changes in the ozone layer, A
2GMmp
and possibly threatening the shielding effect. At
For the Earth, A 0.06. This tells us that there
first, concerns were focused on things that we do
is no molecule for which the average speed is suf-
in the stratosphere, since that is just below the
ficient for escape. However, we have seen that
ozone layer, and chemicals that destroy the ozone
because there is a distribution of speeds, many
could get from the stratosphere to the ozone layer
molecules move faster than the average.
quite easily. Normal commercial aircraft stay in
The distribution of speeds, called a Maxwell“
the troposphere, but many high performance and
Boltzmann distribution is shown for hydrogen and
supersonic aircraft (such as the Concorde) do fly
oxygen in Fig. 23.18. The distribution is such that
in the stratosphere. More recently, theoretical and
the number of molecules with speeds between v
observational studies have shown that harmful
and v dv is
N1v 2dv 1constant2v2 e mv2>2kT
dv (23.15)

Note that we encountered this expression when
talking about the fusion barrier in stars.
Note that if we go to higher and higher
speeds, the probability becomes smaller and
smaller but doesn™t reach zero. If we go out to the
escape speed, we find there are still some mole-
cules moving at or faster than that speed. This
means that a small fraction of the molecules can
escape, even though the average molecule is
securely bound to the Earth. The fraction of
hydrogen molecules capable of escaping is
greater than the fraction of oxygen molecules
Fig 23.17. This false color image shows the location of capable of escaping. This means that significantly
the ozone hole over the Antarctic. [NASA]
more hydrogen will escape than oxygen.
23 THE EARTH AND THE MOON 463



11 This escape can only take place from the
highest levels of the atmosphere. A molecule
.9
lower down in the atmosphere might start out
.8
going faster than the escape speed, but it will col-
lide with other molecules, losing energy before it
.7
can escape. The layer of the atmosphere from
.6
which the molecules can escape is called the exos-
Relative Probability
O2 phere. The thickness of the exosphere is taken to
.5
H2
be equal to the average distance between colli-
.4
sions in the gas at high altitudes.
.3
23.3.4 General circulation
.2
Just as local air flows are in response to tempera-
ture differences which cause pressure differences,
.1

global air flows are subject to the same processes.
0
0 .5 1 1.5 2 2.5 3 3.5 4 The general tendency is for air to be heated at the
Speed (1000 cm/s)
equator, rise, and then flow toward the poles,
where it cools, falls, and returns to the equator.
Fig 23.18. Maxwell“Boltzmann velocity distribution for
This simple pattern is disturbed by the effects
hydrogen and oxygen molecules at T 300 K. For each
of the Earth™s rotation. Since the rotating Earth is
curve, the vertical axis is a relative probability of ¬nding a
molecule at a given speed. In each case, notice the large an accelerating reference frame, we observe
number of molecules with speeds greater than the average. pseudo-forces. One of these is the familiar cen-
trifugal force. It doesn™t play an important role;
however, the coriolis force does.
The origin of the pseudo-force is shown in
At first you might think that once the rela-
Fig. 23.19. We are looking down from above the
tively small number of molecules moving faster
than the escape velocity leave, the process is fin-
ished. However, the remaining molecules collide A A
A
with each other, and re-establish the Maxwell“ B
B
Boltzmann distribution. Since the escaping mole-
B
cules took away more than the average energy per
molecule, if there is no replacement of the energy,
the new distribution will be at a lower tempera-
ture. The gas will be cooled by the escape of the
faster molecules (just as a liquid is cooled by the
evaporation of the faster molecules.) However, if A
there is a source of energy, such as sunlight or heat
from the ground, the equilibrium can be estab- B
A
B
lished at the same temperature as before, and the
same fraction of molecules will be moving faster
than the escape speed. It is by this process that,
over time, the atmosphere can escape. For the Fig 23.19. The coriolis pseudo-force. In the upper series,
Earth, this escape has been much more rapid for we look at an object thrown from a high latitude to the
the hydrogen than for the heavier molecules, equator. A is the point aimed at on the equator, and B is the
such as oxygen. The Earth™s atmosphere is indeed point where the object is launched. However, the horizontal
quite deficient in hydrogen. That is, the fraction motion at B is less than that at A, so the object reaches the
equator behind A.We look at an object thrown from A to B,
of hydrogen in our atmosphere is much less than
in the lower series.The object has a greater horizontal
the fraction of hydrogen in the Solar System as a
motion than does B, and therefore gets ahead of B.
whole.
464 PART VI THE SOLAR SYSTEM



lows and clockwise circulation around the highs
in the northern hemisphere. The opposite situa-
tion prevails in the southern hemisphere. This
pattern is very evident in the circulation around
hurricanes, which have very low pressure centers.
A typical hurricane is shown in Fig. 23.21.
Recently, major improvements have been made
in our understanding of the large-scale atmos-
pheric circulation. Computers are used to model
the Earth™s atmosphere in considerable detail. The
equations governing the fluid flow and energy
balance are solved, taking such details as terrain
into account. For such a model to be successful in
predicting the movement of weather systems over
periods of a few days, or even possibly weeks, it is
important to have detailed information on condi-
tions all over the Earth. Data are continuously col-
lected from a variety of ground stations, from
Fig 23.20. Circulation patterns on the Earth.The closed ships and weather buoys at sea, and from balloons
loops represent the major cells within the atmosphere, and
sent up from various places on a regular schedule.
the open arrows represent the general wind directions at
Satellite observations are also important. The
the surface.
observations over the ocean are particularly
important, since much of the energy that passes
north pole. We are standing at point B, which is into the atmosphere is stored in the water. As the
far from the equator. We want to aim at a target models are refined and the data gathering is
on the equator, at point A. We point directly at A improved, the predictions become better and bet-
and shoot. The bullet travels towards the equator ter. We have now even reached the point where we
with the speed that it left the gun. Its velocity has can use the same approach to try to understand
a slight horizontal component to the left due to the global atmospheric properties of other plan-
the Earth™s rotation. Object A has an even larger ets, as we will see in the next few chapters.
motion to the left, since it is at the equator, and
is moving faster than we are. The result is that by
the time the bullet reaches the equator, A has
moved to the left of its path, and the shot misses,
going behind the target. A similar thing happens
if someone at A shoots at B. The bullet leaves A
with a large horizontal speed, and misses B by
getting ahead of it. To the observer on Earth, it is
almost as if there is an additional force present.
In general, air flows from high to low pressure
areas in the atmosphere (Fig. 23.20). Air traveling
from the poles to the equator will reach the equa-
tor behind the point at which they were initially
launched. Air traveling from the equator to the
poles will arrive ahead of the point at which it
was aimed. This produces a circulating pattern.
Air flowing from a high to a low, going toward
the equator, will lag behind the low. Air flowing
in the other direction will get ahead. This results
(a)
in a counterclockwise circulation around the
23 THE EARTH AND THE MOON 465



Fig 23.21. In this photograph,
27N
the coriolis force results in the
spiraling motion around the large
26N
hurricane. (a) A global image
showing Hurricane Andrew, on
25 Aug 1992, from the GOES
25N
spacecraft. (b) Close view of
Hurricane Floyd (13 Sept 1999).
The false color image shows the
24N
rainfall levels.The arrows show the
wind direction and speed. [NASA]
23N




22N



21N

76W 75W 74W 73W 72W 71W 70W 69W 68W 67W


0 2 4 6 8 10 12 14 16 18
Surface Rain (mm/hr)

(b)


ticles were present. However, James van Allen cor-
23.4 The magnetosphere rectly interpreted the strange result as indicating
the presence of large numbers of charged parti-
One of the early surprises of our unmanned space cles. We call these belts of charged particles the
program was the discovery of belts of charged van Allen radiation belts (Fig. 23.22)
particles high above the Earth™ surface. The parti-
cles are a source of low frequency synchrotron
radiation. The radiation overwhelmed detectors
on early spacecraft, making it appear as if no par-

Magnetic
Rotation
axis
axis




(b)

Fig 23.22. Van Allen radiation belts and the Earth™s magne-
tosphere. (a) The radiation belts are groups of trapped parti-
cles, concentrating into two bands, each with the shape of
a doughnut that has been hollowed along the inner rim.
(b) The Earth™s magnetic ¬eld de¬‚ects the charged particles
of the solar wind.The protected region is the magnetos-
phere, and the boundary is called the magnetopause. Just
outside the magnetopause is a shock wave that looks like the
bow wave when a ship plows through the water [NASA].
(a)
466 PART VI THE SOLAR SYSTEM



BPAR, parallel to the z-axis, along which the parti-
These particles are trapped by the Earth™s
cles are drifting, and BPERP, perpendicular to the
magnetic field, and stay in spiral paths around
z-axis. If the field were constant, BPERP would be
the field lines. The region where there are large
number of charged particles trapped by the zero.
Earth™s magnetic field is called the magnetosphere. We look at the force on a proton moving in
The region dominated by charged particles is also the indicated spiral path. (The argument works
called the ionosphere. When we discussed solar just as well for negatively charged electrons. See
activity (Chapter 6), we said that charged parti- Problem 23.14.) Note that the sense of the spiral is
determined by the charge of the particle. BPAR
cles will follow helical paths around magnetic
doesn™t change the z-component of the proton™s
field lines. This is because the force on the parti-
speed. However, v x BPERP points to the left. This is
cles is perpendicular to both the field lines and
the velocity of the particle. This means that there opposite to the direction of drift of the particle.
can be no force along the field lines. The compo- The motion along the field lines is slowed, and
nent of the velocity along the field, the drift eventually reversed. It is as if the particle struck a
mirror. In fact, this phenomenon is called a mag-
speed, stays fixed, as the particles execute circu-
netic mirror. The charged particles therefore spiral
lar motion perpendicular to the field lines.
The situation is different if the particles are back and forth, trapped in the Earth™s magnetic
moving from a region of a weaker magnetic field field.
to one of a stronger magnetic field, as illustrated The particles come from interplanetary
in Fig. 23.23. The stronger field is represented by space, mostly from the solar wind. In fact the
the field lines becoming closer together. We Earth™s magnetic field shields us from most of
divide the magnetic field into two components, the solar wind by trapping the particles. Because
of the dipole nature of the Earth™s magnetic
field, these charged particles get closer to the
surface near the Earth™s magnetic poles. In these
B
v regions, there is more of the normal atmosphere
B
v
for the charged particles to strike. The fast mov-
ing charged particles lose some energy to the air
vxB
as they pass through, and the air glows. We see
vxB this glowing air as an aurora (plural aurorae). An
P
P'




Fig 23.23. Magnetic mirror. Particles spiral around mag-
netic ¬eld lines. For simplicity, we look at positively charged
particles, though the same argument can be carried out for
negative particles. In the region where the magnetic ¬eld
lines are parallel, on the left, the magnetic force on particle P
is downward.This just keeps the particle orbiting. However,
when the ¬eld lines become closer together, meaning that
the ¬eld is increasing, the magnetic force on the particle at
Fig 23.24. When the solar wind particles penetrate the
P has a slightly rearward component.This is the force that
slows and reverses the component of the motion parallel to Earth™s atmosphere, they cause the glowing aurorae.These
the ¬eld lines. are seen from above in an ultraviolet image. [NASA]
23 THE EARTH AND THE MOON 467



example is shown in Fig. 23.24. Since the mag- SOLUTION
netic mirrors are near the magnetic poles, the We can express the result as a ratio, using equation
aurorae are most prominent close to the north (23.16):
and south poles. In addition, aurorae are strongest
rME 3
a ba b
M
¢aSE
when there are a lot of charged particles arriving
MM rSE
¢aME
from the Sun.
1033 g 103 km 3
a ba b
2 385
1025 g 106 km
7.3 150
23.5 Tides 0.46

Even though the Sun exerts a greater gravitational
A number of phenomena on Earth depend on the force on the Earth than the Moon, the closeness of
fact that the gravitational force exerted by the the Moon makes its tidal effects greater.
Moon (or the Sun) on the Earth is slightly differ-
ent at different parts of the Earth. As we have The tidal effects of the Sun and Moon are
already discussed (Chapter 12), any effect which responsible for the ocean tides on Earth (Fig.
depends on the difference between the gravita- 23.25). We first look at how the Moon affects the
tional forces on opposite sides on an object is Earth. We look at three points: (1) the point clos-
called a tidal effect. est to the Moon, (2) the center of the Earth, and
If we have an object of mass m, a distance r (3) the point farthest from the Moon. We see that
from an object of mass M, the force on the object a1 a2 a3, as viewed from the rest frame of the
of mass m is Moon. However, as viewed from the Earth, all
accelerations must be relative to a2. In this frame
GMm
F the acceleration of (1) is a1 a2 towards the
r2
Moon; that of (2) is zero; that of (3) is a3 a2, and
If we move the object, the force changes. The rate is therefore directed away from the Moon. This
of change of the force is means that there will be a high tide on the side
nearest the Moon and also on the side farthest
dF 2GMm
from the Moon. We can think of the tide on the
r3
dr
near side as the water on that side being pulled
away from the Earth, and we can think of the tide
The change in force, F, in going from r to r r
on the far side as the Earth being pulled away
is
from the water.
a b ¢r
dF The Sun produces a similar effect, but only
¢F
dr half as great in size. When the Moon and Sun pull
along the same line, the difference between high
2GMm
¢r and low tides is the greatest. When they pull at
r3
right angles to each other, the difference between
The change in acceleration, a, is F/m, or high and low tides is the least. The tides on the
oceans, where the water flows without obstruc-
2GM
(23.16)
¢a ¢r tion, are well approximated by Fig. 23.25. However,
r3
in narrow or blocked waterways that run into the
Note that the tidal effects fall as 1/r 3, faster ocean, the flow of the water is quite complicated,
than the 1/r 2 fall-off of the gravitational force and the time of high tide may be delayed by a few
itself. hours behind the time of the ocean high tide.
The height of the water tide is actually not
found to be as large as one would calculate. This
Example 23.3 Tidal effects on the Earth
is because the Earth is not solid. It is therefore
Compare the strength of the tidal effects exerted
also distorted by these tidal forces. This constant
on the Earth by the Sun and the Moon.
468 PART VI THE SOLAR SYSTEM



and is illustrated in Fig. 23.26. To see how this
Water
comes about, we just consider the effect of the
Earth
Sun on the Earth. The Earth is not a perfect
1
2
3 sphere, but has a larger diameter across the equa-
Moon
tor than between the poles. This oblate, or flat-
a1 a2
a3
tened, appearance is due to the Earth™s rotation.
We can think of the non-rigid Earth as deforming
under the effects of the centrifugal force. Since
(a) the Earth is not spherical, the Sun can exert a
torque on the Earth.
To see the direction of this torque, we idealize
a3 “ a2 a 1 “ a2 the Earth as a sphere with an extra band around
(b) the equator. The side of the band closest to the


L
L
„t
F1 L'
Earth
To
Sun

F2
(c)
Fig 23.25. Tides on Earth. (a) Different accelerations (a)
caused by the gravitational attraction of the Moon. (b) We
express these accelerations relative to that of the center of L
the Earth.The result is that the water is pulled away from
L
the Earth on the side facing the Moon (c), and the Earth is
„=0 L
pulled away from the water on the far side.
2

L
reshaping of the Earth helps to heat its interior, Sun 1
3
and also dissipates the Moon™s orbital energy.
The Earth doesn™t respond instantaneously to
4
these tidal effects. Its rotation causes the bulge
„=0
in the side near the Moon to get ahead of the
(b)
Earth“Moon line. This means that the force can
act to increase the Moon™s orbital speed. This Fig 23.26. Precession caused by the Sun. (a) We treat the
Earth as a sphere with an extra bulge around the equator,
makes it move farther from the Earth. The corre-
and look at the torque exerted by the Sun on the bulge.We
sponding force on the Earth causes its rotation to
see that F1 is greater in magnitude than F2, so there is a net
slow, keeping the total angular momentum of
torque on the Earth, whose direction, given by the right-
the Earth“Moon system fixed. Similarly, the
hand rule, is out of the page.The torque would rotate the
Earth has exerted forces that distort the Moon. Earth, but the Earth already has some angular momentum, so
These forces, acting on the distorted Moon, act to the change in angular momentum must be added to the
slow its rotation, producing a rotation period existing angular momentum, as shown on the right.The
equal to the Moon™s orbital period. result is a new angular momentum whose magnitude is the
The tidal forces that the Sun and Moon exert same as before, but whose direction is different. (b) We
show the torques at different points in the Earth™s orbit.The
on the distorted Earth produce a torque that
torque can sometimes be zero, but when it is not zero it
results in a continuous change in the direction of
always points in the same direction.
the Earth™s rotation. This change is called precession,
23 THE EARTH AND THE MOON 469



Sun feels a slightly larger force than the side of positions of all stars appear to change. It may
the band farther from the Sun. When the Earth seem like the change should be small, since it
in is position 1 (first day of winter in the north- takes 26 000 years to go through a full cycle, but
ern hemisphere), the greater force on the near this amounts to 50 arc sec, or almost one-quarter
side of the band produces a torque that points minute of time each year.
out of the page. Three months later, at position 2,
the force on the near and far sides of the band
23.6 The Moon
both pass through the center of the Earth, so the
torque is zero. Three months later, at position 3,
the torque is again outward. Three months later, The Moon provides us with the most spectacular
at position 4, it is again zero. The torque is some- example of a body that has been taken from the
times zero, but is never in the opposite direction. realm of remote sensing by traditional astronom-
Therefore, averaged over the year, there is a non- ical techniques to that of up-close study, includ-
zero torque. The torque goes through two full ing the luxury of bring samples back for studies
cycles in its value over the course of the year. The in the laboratory. Various Earth- and space-based
effect of the Moon is essentially the same, with photographs are shown in Fig. 23.27.
the torque going through two full cycles in its Even from the Earth, we can see a variety of
lunar features. The highlands are lighter colored.
value each lunar orbit.
We now see the effects of this average torque They contain mountains and valleys as well as
long canyons, or rilles. The maria, once thought to
. If the Earth™s angular momentum is origi-
nally L, then after a short time dt the new torque be oceans because of their smooth, dark appear-
L is given by ance, are more level. We can also see many dif-
ferent types of craters. Some have bright rays of
L L dL
ejected material. In some areas, we see several
where layers of catering, with younger craters appear-
ing to cross the walls of older craters. One way of
dL dt
determining the relative age of surface features
Note that dL is much smaller in magnitude than on the Moon is to look at the relative numbers of
L, and is perpendicular to L, so it can only change craters.
the direction of L, not the magnitude. To change As we have said, lunar exploration has taken
the direction of L the direction of the Earth™s axis the Moon to the realm of the geologist. (In fact,
must change. The axis sweeps out a circle in the sky, the last crewed lunar landing included a geolo-
returning to its current position in 26 000 years. gist.) There were six manned lunar landings from
Since the torque on the Earth is not constant in 1969 through 1972, Apollos 11 through 17, with
time, the precession doesn™t occur smoothly. The the exception of Apollo 13, which aborted its mis-
small variations in the rate of precession are sion after an explosion en route. A variety of
called nutation. regions were visited, as shown in Fig. 23.28. These
Precession has important observational conse- missions returned 382 kg of lunar rock. In the
quences. Since the Earth™s axis is changing orien- later flights, a rover vehicle allowed studies of
tation, the Earth™s equatorial plane is doing the extended regions around the landing sites. In
same. Therefore, the point at which the celestial addition to bringing back lunar samples, the
equator and the ecliptic cross, the vernal equinox, astronauts left a variety of monitoring equip-
changes its position in the sky. It has recently ment on the Moon. This equipment included seis-
moved into the constellation Aquarius (the “Age mometers to detect moonquakes and meteor
of Aquarius”). Remember, for this precession the impacts and X-ray and radioactivity detectors.
23.5o tilt doesn™t change, just the direction in There were also unmanned Russian Luna flights
which the axis points. Since the vernal equinox is (16, 20 and 24), which, from 1970 through 1976,
the starting point of the astronomical coordinate returned 310 kg of rock from different sites. Some
system of right ascension and declination, the views are shown in Fig. 23.29.
470 PART VI THE SOLAR SYSTEM




(c)

(a)




(d)
Fig 23.27. Views of the Moon. (a) The near side, from the
Galileo spacecraft on its way to Jupiter. (b) Apollo 12
photograph of the highland area crater Ptolemaeus
(right center).The terraced crater with central peaks, in the
center of the photo, is Herschel. (c) An Apollo 11 view of the
crater Daedalus, on the lunar far side.This crater has a diam-
eter of about 80 km. (d) Lunar Orbiter 5 image of a side
view of the impact crater Copernicus.This crater is 93 km
wide, and is located within the Mare Imbrium basin. [NASA]
(b)


23.6.1 The lunar surface region. The soil contains no water. It does have a
high proportion of refractory, or high melting
The lunar soil is in a layer ranging from 1 to 20 m
in depth. It is called regolith, a combination of point, materials, such as calcium, aluminum and
titanium.
powder and broken rocky rubble. It is the result
The rocks are all basalts, meaning that they
of meteoritic bombardment. We are talking
resulted from the cooling of lava. (Remember, the
about a large number of small meteorites that
Earth has igneous rocks, but also has sedimen-
would have burned up in the Earth™s atmosphere,
tary rocks, such as limestone and shale.) The
so we see no corresponding material on the
rocks contain some silicates (compounds contain-
Earth. Larger meteor impacts have spread mate-
ing silicon and oxygen, such as olivine) as well as
rial around and mixed the material in a given
23 THE EARTH AND THE MOON 471



Fig 23.28. The locations of the
various lunar landings (red
Luna; green Apollo; yellow
Surveyor) are shown on this
photograph of the near side of
the Moon. [NASA]




oxides of iron and titanium. It also contains three Based on the properties of the rocks, and their
new materials “ named tranquilityite, armalcolite distribution, it has been possible to deduce a sce-
(after the three Apollo 11 astronauts, Armstrong, nario for the evolution of the lunar surface. Some
Aldren and Collins) and pyrroxferroite. of the parts of this scenario had been speculated
There are some differences between the maria on before the lunar explorations, but the dating of
and the highlands. The maria are basalts, like the the lunar rocks gives exact time since certain
lavas from the Hawaiian volcanoes. The highland important events. We think that the Moon formed
rocks contain anorthosites, gabbro and norrite, 4.5 billion years ago. (Some small green rock frag-
which were probably cooled slowly deep within ments in the Apollo 17 samples are that old.) The
the Moon. There are also variations in element surface underwent extensive melting and chemi-
composition. The maria rocks have more tita- cal separation for the next 200 million years. This
nium, magnesium and iron, while the highland melting resulted from the heat generated in exten-
rocks are rich in calcium and aluminum. The sive meteoritic bombardment, as well as some
highland rocks are older. The maria rocks have radioactivity. This bombardment created the large
ages in the range 3.1 to 3.8 billion years. These basins, hundreds of kilometers across. The bom-
youngest rocks on the Moon are as old as the old- bardment eased approximately 4 billion years ago.
est rocks on the Earth. The highlands have rocks Residual store heat and some radioactivity resulted
with ages ranging from 3.9 to 4.48 billion years. in melting down to 200 km below the surface.
This latter number occurs in several places. There From 3.1 to 3.8 billion years ago (the ages of
are also differences between the near and far side the rocks in the maria), lava rose and filled the
of the Moon, which is shown in Fig. 23.30. The far basins, creating the maria. Since then the Moon
side has more craters and fewer maria. It also has has been quiet, except for the occasional meteor
a higher average altitude above the Moon™s aver- impact. The footprints left on the Moon will
age radius. remain sharp for millions of years.
472 PART VI THE SOLAR SYSTEM




(a)




(b)




(c)
(d)




(e) (f)
23 THE EARTH AND THE MOON 473




(h)




(g)




( j)
Fig 23.29. Photographs from Apollo missions. (a) The rising
Earth viewed from Apollo 8. (b) The second person to step
on the Moon, Edwin E.Aldren, Jr, steps off the Apollo 11
(i) Lunar Module ladder as he prepares to walk on the surface
(20 July 1969).This photo was taken by the ¬rst person on
the Moon, Neil A. Armstrong. (c) This footprint left by the
23.6.2 The lunar interior Apollo 11 crew will last for a long time as there is no water
Much of what we know about the lunar interior is or air for erosion. (d) Wide angle view of the Apollo 17
deduced from lunar seismology. There are still landing site,Taurus-Littrow region (10 Dec 1972), including
the Lunar Rover vehicle. (e) Orange soil found by geologist/
small quakes, with about one-billionth the energy
astronaut Harrison Schmitt (12 Dec 1972) during one of the
of a typical quake on Earth. Some of the quakes
Apollo 17 surface excursions. (f) Harrison Schmitt near a
are the results of meteor impacts. By watching
large boulder (13 Dec 1972). (g) A lunar soil sample from
the propagation of seismic disturbances around
Apollo 12, back in the laboratory on Earth.This was from a
and through the Moon, we can draw some con- core sample that collected dirt as far as 225 m below the
clusions about the interior. We can also learn surface. (h) Thin section of rock brought back by Apollo 12.
about the interior from the amount of heat flow- Under polarized light, the lavender minerals are pyrexene; the
ing through the surface. This heat flow rate is black mineral is ilmenite; the white and brown, feldspar; and
about one-third of the rate on the Earth. the remainder, olivine. (i) Photomicrograph of spheres and
fragments in the “orange” soil brought back with Apollo 17.
The basic lunar structure is depicted in
The magni¬ed image shows particles in the 150“250 m
Fig. 23.31. The core has a radius of some 700 km.
range. (j) Another Apollo 17 sample, a 32 g breccia. [NASA]
At this point, we are unsure about its composition
474 PART VI THE SOLAR SYSTEM



Crust
Mantle
ly Mo
art




P


lte
n
Core




1000 km


Fig 23.31. Model for the lunar interior.
Fig 23.30. Photograph of the far side of the Moon, taken
from the Galileo spacecraft on its way to Jupiter. [NASA]
Pacific Ocean has been in its current arrange-
ment for a relatively short time, and that its cur-
or detailed structure. The absence of a magnetic rent appearance is unrelated to its appearance
field suggests that it is probably not molten. some 4 billion years ago. The fission theory is sup-
(However, some rocks show evidence for a mag- ported by the fact that, when we allow for the
netic field in the past.) Above the core is a partly effects of erosion on Earth, the surfaces of the
molten zone 400 km thick. The deep quakes are Earth and Moon have some similarities. However,
thought to originate in this zone. Above this is a there are also some major composition differ-
mantle, and above the mantle is a thin crust. The ences. Also, the Moon™s orbit is not in the plane of
crust is variable in thickness, ranging from 0 to the Earth™s orbit.
65 km thick.
Capture. In this theory, the Moon passed close to
Anomalies in the orbits of early lunar space-
the Earth, and was captured into its current
craft indicated that the mass of the Moon is not
orbit. One problem with this theory is that cap-
distributed uniformly in a sphere. Mass concen-
ture would more likely result in a much more
trations, or mascons, have been detected beneath
eccentric orbit. It is also less likely that the orbit
the circular maria, such as Imbrium, Serenitatis,
would lie so close to the plane of the ecliptic.
Crisium and Nectaris. We think that these
Theoreticians have not been able to work out a
resulted from volcanic lava filling the basins. The
reasonable dynamical scenario.
fact that these concentrations have survived the
3 billion years since the basin filling suggests Condensation. In this theory, the Moon formed as
that the lunar interior is quite rigid. If it were not, a by-product of the formation of the Earth.
the mascons would have sunk a long time ago. Perhaps it gathered after the Earth was well under
way in formation. Possibly, the Earth simply
23.6.3 Lunar origin formed as a double planet. One problem with this
Four basic theories of the origin of the Moon have idea is that we would expect the Moon to have an
been proposed. iron core, as does the Earth, and it does not.
Fission. In this theory, the Moon broke away from Impact trigger. It is possible that a large object
the Earth. This theory was particularly popular struck the Earth, and drove off enough material
when it was realized that the Moon is the right from the mantle and crust to form the Moon.
size to have left behind the Pacific Ocean basin This would explain why the Moon and Earth have
when it broke away. However, we now know that similar crusts. This is currently the best accepted
the plate tectonics on the Earth mean that the model.
23 THE EARTH AND THE MOON 475




Chapter summary
The Earth is the planet that we can study in the the upper atmosphere for most of the hydrogen to
most detail. In this chapter, we looked at the have escaped. The general flow in the atmosphere
properties of the Earth and the Moon. We also is strongly influenced by the Earth™s rotation.
looked at the physical processes responsible for The molten iron“nickel core provides the
their current properties. Earth with a magnetic field, similar to that of a
So much heat was trapped when the Earth dipole magnet. The field is strongest at the north
formed that it still has a molten core of iron and and south magnetic poles (which don™t coincide
nickel. Radioactive materials provide heating with the rotation poles). The magnetic field has
near the surface, keeping the mantle soft, like the important role of trapping charged particles
plastic. The radioactive materials also provide us that hit the Earth from outer space. These
with a way of dating various rocks. The oldest charged particles give off a lot of long wavelength
rocks on the Earth are 3.7 billion years old. The radio emission. They are also responsible for the
plastic nature of the material below the surface glowing aurorae.
allows the continental plates to drift slowly. The Moon produces tidal effects on the Earth.
These plate tectonics are responsible for most of These depend on the fact that the Moon™s gravita-
the geological activity that we see “ volcanoes, tional force is weaker on the side of the Earth
earthquakes, and mountain building. away from the Moon than on the side closer to the
The temperature of the Earth (or any planet) is Moon. This produces the water tides. The Sun has
determined by the balance between the radiation a similar effect, but it is not as large as the Moon™s.
absorbed from the Sun and that given off by the The tidal effects also cause dissipation of some of
Earth. The atmosphere plays an important role in the Moon™s orbital energy, and have resulted in
modifying that balance. The atmosphere is a very the Moon keeping the same face towards the
thin layer (relative to the radius of the Earth itself). Earth. The Sun and Moon also cause the direction
The pressure is determined by hydrostatic equilib- of the Earth™s rotation axis to precess, like a top,
rium, and drops very quickly with altitude, falling with a period of 26 000 years.
to half of its sea level value at an altitude of about The lunar surface is particularly interesting
6 km. The temperature distribution depends on the since it has been preserved over most of the his-
sources of heating for different parts of the atmos- tory of the Moon. This is because the Moon has no
phere. In the ozone layer, solar ultraviolet radiation atmospheric erosion. The lunar rocks show some
is absorbed, and directly heats that part of the differences between the maria and the high-
atmosphere. Near the ground, the source of heat- lands. An important difference is that the high-
ing is the ground, either by infrared radiation from land rocks are somewhat older (with the oldest
the ground, or convection of the air heated just being 4.48 billion years). The lunar interior is dif-
above the ground. The trapping of infrared radia- ferent from that of the Earth. In particular, the
tion near the ground produces the greenhouse Moon does not have a molten core. The question
effect, and raises the temperature near the ground of the lunar origin is still up in the air.
by about 25 K. The temperature is high enough in


Questions
23.4. If the Earth and Moon formed together, why
23.1. Why do we think that the Earth is more
are the oldest rocks on the Moon older than
than a billion years old?
the oldest rocks on Earth?
23.2. What does the fact that the Earth is differ-
23.5. What does continental drift tell us about
entiated tell us about its history?
the Earth™s interior?
23.3. Do you think that radioactive uranium would
be useful for dating the bones of a cave person?
476 PART VI THE SOLAR SYSTEM



23.22. Why has more hydrogen escaped from our
23.6. What is the relation between continental
atmosphere than oxygen?
drift and areas of geological activity (like
23.23. Why does the temperature of an atmosphere
earthquakes).
affect how much of it escapes?
23.7. Suppose we have a space probe that starts at
23.24. Draw a diagram showing how the coriolis
the Earth and goes outward. If there is no
force acts in the southern hemisphere.
temperature control, describe what happens
23.25. How do we know that the Earth has a mag-
to the surface temperature of the spacecraft
netic field?
as it moves farther from the Sun.
23.26. If you have a toy magnet that will stick to a
23.8. If sunlight passes through the atmosphere,
refrigerator, do you think that the toy mag-
why does radiation from the ground become
net or the Earth™s magnetic field exerts a
trapped?
greater force on the refrigerator?
23.9. How much of a difference does the Earth™s
23.27. Describe the motion of a charged particle
atmosphere make in its temperature?
traveling in a uniform magnetic field.
23.10. What keeps a balloon afloat? Draw a dia-
23.28. What do we mean by a magnetic mirror?
gram showing the forces on a balloon.
23.29. Why are aurorae most prominent near the
23.11. If one atmosphere is such a large pressure,
poles?
why doesn™t the air outside a building drive
23.30. In general, what do we mean by a tidal
the walls inward?
effect?
*23.12. The water in the oceans is also in hydro-
23.31. Explain, in your own words, how there can
static equilibrium. This means that the
be a high tide on the side of the Earth near
water pressure increases as you go farther
the Moon, and on the opposite side of the
below the surface. A cubic meter of water
Earth, at the same time.
weighs much more than a cubic meter of
23.32. What effect does precession have on the
air. What does this tell you about how fast
amount of tilt of the Earth™s axis?
the pressure changes with depth in the
23.33. What effect does precession have on the
water, relative to the changes in air?
apparent positions of stars in the sky?
23.13. What factors determine the temperature at
23.34. What is unique about our ability to study
a given altitude?
the Moon?
23.14. Why does the temperature decrease as we go
23.35. How does the composition of the lunar soil
up from the ground?
differ from that of the Earth?
23.15. Why is the temperature in the ozone layer
23.36. Why do we think that the Moon is more
higher than it is just above or below that
likely than the Earth to have rocks left over
layer?
from the young Solar System?
23.16. What is the importance of the ozone layer?
23.37. What are the differences between the near
23.17. How does the greenhouse effect work?
and far sides of the Moon? Why do you
23.18. Why is there concern about the possibility
think these differences occur?
of global warming?
23.38. How do we know about the lunar
23.19. What is the difference between energy
interior?
transport by convection and energy trans-
23.39. Some astronomers have suggested putting
port by radiation?
an observatory on the far side of the Moon.
23.20. If the average speed of molecules in an
Why might this be a good location?
atmosphere is less than the escape speed,
how can the atmosphere escape?
*23.21. When water evaporates from your skin, you
feel cool. Why is this? (Hint: Think about
which molecules evaporate, and which are
left behind.)
23 THE EARTH AND THE MOON 477



Problems
23.1. Show that if the number of nuclei in a sam- find an expression for the column density of
air N(z) as a function of z. The column den-
ple is given by equation (23.1b), then, follow-
ing any arbitrary starting time, the half-life sity is the number of molecules in a cylinder
of unit area from the height z to the top of
is the time for the number to be reduced to
half its value at that arbitrary time. the atmosphere.
23.2. If the number of nuclei in a sample is given 23.9. At what altitude is the pressure (1/2) atmos-
by equation (23.1a), show that the rate of phere?
23.10. If after N steps of length L in a random walk
radioactive decay (number of decays per sec-
ond) is proportional to N(t). in one dimension, the distance from the ori-
gin is N1/2L, show that this distance is
23.3. Show that 1/2 0.693 e.
(N/2)1/2L and (N/3)1/2L for two- and three-
23.4. (a) How would you measure the half-life of
an isotope whose half-life is long compared dimensional walks, respectively.
to your lifetime? (b) How would you deter- 23.11. Find the pressure distribution in an atmos-
mine the number of nuclei of that isotope phere where the temperature distribution is
in a particular sample (assuming you know
T(z) T0 bz
the half-life)?
23.5. How would the equilibrium temperature of 23.12. Using the fact that precession goes through
a planet be modified if the planet always a full cycle in 26 000 yr, calculate the
kept the same side towards the Sun? average torque on the Earth. (Hint: It is nec-
23.6. Express the equilibrium temperature of a essary to calculate the angular momentum
planet as a function of that of the Earth and of the Earth.)
the distance of the planet, in astronomical 23.13. When we applied hydrostatic equilibrium to
units. Use your equation to construct a table the Earth™s atmosphere, we found that the
of the equilibrium temperatures for the pressure falls off exponentially. However,
nine planets. when we apply it to the ocean, the pressure
23.7. If we are interested in measuring distances varies only linearly with depth. How can you
through the atmosphere to within 1%, down account for the difference?
to what angle from the zenith can we 23.14. Show that positively and negatively charged
approximate the Earth™s atmosphere as a particles are reflected the same way by a
plane parallel layer? magnetic mirror. (Hint: Remember that
23.8. Assuming the pressure in the Earth™s atmos- oppositely charged particles spiral magnetic
phere varies according to equation (23.13), field lines in opposite directions.)


Computer problems

23.1. Suppose we have two radioactive nuclei, one 23.3. Make a graph showing the variation of partial
with a half-life of 1000 yr and the other with a pressures with altitude for O2, CO and CO2,
half-life of 2000 yr. Suppose we start with equal assuming a temperature of 300 K.
amounts of both nuclei. Make a graph showing 23.4. Make a graph showing the variation of partial
the abundance of each element from the start pressure with altitude for O2, for temperatures of
point to until 10 000 yr after. Also make a graph 275 K, 300 K and 325 K.
of the ratio of the abundances (shorter/longer 23.5. Calculate the normalization for the Maxwell“
Boltzmann distribution by integral N.
half-life).
23.2. Calculate the equilibrium temperatures for all 23.6. Reproduce Fig. 23.18 for temperatures of 275 K
nine planets. and 350 K.
Chapter 24




The inner planets

The Solar System naturally divides into two planet is rotating. Astronomers were surprised to
groups of planets, separated by the asteroid find that Mercury™s rotation period is 59 days,
belt. The four inner planets have many things in two-thirds of the 88 day orbital period. This rela-
common with the Earth, whereas the next four tionship means that when Mercury is favorably
planets present worlds of an entirely different placed for observations, it often has the same side
type. (Pluto is an additional enigma.) In this towards Sun, making it seem that it always had
chapter, we look at Mercury, Venus and Mars, the same side towards the Sun. This simple ratio
comparing their properties with each other, is probably no accident. It may result from the
and with the Earth. varying tidal effects, as Mercury has a rather
eccentric orbit.
Our only close-ups of Mercury have come from
24.1 Basic features the Mariner 10 spacecraft, which made three fly-
bys of Mercury. The orbit of Mariner 10 was
24.1.1 Mercury arranged to bring it back close to Mercury peri-
Mercury is the closest planet to the Sun, and is odically. The closest flyby was within 300 km, and
not much larger than our Moon. There is an inter- allowed a very detailed study of surface features.
esting story concerning its rotation period. Since Mariner 10 photos of the whole planet are shown
Mercury is so close to the Sun, we never have a in Fig. 24.1.
really good view of it, and surface features are
24.1.2 Venus
hard to recognize. By noting the positions of
large surface features, it appeared that the rota- Venus has been referred to as our “sister planet”.
tion period was 88 days, the same as the planet™s Its size is very close to that of Earth. It should also
orbital period. This would have meant that encounter somewhat similar solar heating condi-
Mercury always keeps the same face towards the tions. It receives twice as much solar energy as
Sun ( just as the Moon keeps the same face does the Earth. For some time it was thought that
towards the Earth). Since Mercury is so close to Venus might be a good candidate for finding life.
the Sun, it seemed plausible that some tidal At the very least, it is a good candidate for testing
effect could keep its rotation period synchro- theories which explain various aspects of the
nized with its orbital period. Earth. Study of the surface is hindered by thick
However, the situation was corrected follow- clouds.
ing radar observations. Radio waves were The planet™s rotation period is 243 days. The
bounced off Mercury and then detected back on sense of the rotation is opposite to that of the
orbital motion. This is called retrograde rotation. If
Earth. The planet™s rotation causes a spread in
the Doppler shifts of the reflected waves. From the planets simply condensed out of a rotating
the amount of spread, we can tell how fast the nebula, conservation of angular momentum tells
480 PART VI THE SOLAR SYSTEM




(b)
Fig 24.1. (a) Mercury from Mariner 10, at a distance of
5 million km. (b) Mercury from Mariner 10, at a distance of
60 000 km, showing the other hemisphere. [NASA]



More recently the Pioneer Venus Orbiter produced
radar maps of most of the surface. Earth- and
space-based images of Venus are shown in Fig. 24.2.
(a)
24.1.3 Mars
Mars has long held our fascination. Except during
us that all of the planets should be rotating on dust storms, there is no thick cloud cover, so we
their axes in the same sense as they are orbiting have had a comparatively good view of the sur-
about the Sun. It has been suggested that the face. Particularly intriguing to Earth-based
drag of a heavy atmosphere and tidal effects of observers were color changes with season that
the Sun could be responsible for the unusual suggested some vegetation. The axis of Mars is
rotation of Venus. tilted by 25 , so we would expect the seasons to be
Exploration of Venus by spacecraft has been like those on Earth. The white polar caps also
quite extensive. Especially notable are the Soviet change size with the season, raising the possibil-
Venera landers, which also sampled the atmos- ity that they hold water ice. In addition, the rota-
phere on the way down to the surface. In addition, tion period is very close to that of the Earth.
the Pioneer Venus probes (in 1978) provided a Mars has also been the subject of extensive
wealth of information. One spacecraft went into exploration. The Mariner spacecraft flew by, send-
orbit and used radar to map the surface features. ing back photos of a barren, crater-marked sur-
The other sent probes through the atmosphere. face. Earth- and space-based images of Mars are
24 THE INNER PLANETS 481




(b)
Fig 24.2. (a) HST image of Venus. (b) Galileo image of
Venus from a distance of 1 million km.This has been
enhanced to show small-scale cloud structure. [(a)
STScI/NASA; (b) NASA]

We first look at the time delay. Since the sur-
face is round, different parts of the surface are
different distances from our radio telescope.
These different distances mean that the light (or
radio wave) travel times will be different for
(a)
waves bouncing off different parts of the sur-
face. We can express time delays relative to that
shown in Fig. 24.3. Two Viking spacecraft left of a wave bouncing off the closest point.
orbiters around Mars and also sent landers to the According to the figure, the extra distance that
surface. These landers sampled the soil, and even the signal has to travel is 2x, so the time delay is
carried out a preliminary search for life. Some
2x c
¢t
chemical effects were found that mimic some
simple aspects of certain living things, but no We can see that
evidence for life was found.
x R y
24.1.4 Radar mapping of planets R R cos

cos 2
Since we obtain so much information from radar
R11
mapping of planetary surfaces, it is useful to go
over the basic ideas behind it. A raised surface fea- This means that the time delay is
ture may show up as a stronger than average
a b 11 cos 2
2R
radar echo. However, the problem is to determine (24.1)
¢t
c
where on the surface the feature is. There are two
effects that help us locate the feature. One is the If we measure the time delay, and we know
the planet™s radius, R, then we can solve equation
time delay for the signal returning to Earth, and
the other is the Doppler shift. These are illus- (24.1) for . This does not give a point uniquely.
trated in Fig. 24.4. There is a whole ring of points that all have the
482 PART VI THE SOLAR SYSTEM




Incoming R
Signals
θ
y
x




(a)
φ
cos
v0
θ
v0 v cos φ v0 cos φ sin θ
0 θ


θ




φ
de
it u
C ir
cle at lat
(a)

(b)
Fig 24.4. Radar mapping of a planet: (a) time delay;
(b) Doppler shift.



Now we view this point from the above the
pole, assuming the line from the point to the pole
makes an angle with the line from the pole to
the telescope. The Doppler shift depends on the
radial velocity, vr( , ), which is given by
vr 1 , 2 v1 2 sin

v0 cos sin (24.2)
As seen from the telescope, lines of constant
Doppler shift form concentric rings about the
point on the equator that is just appearing from
(b) the back side, and the point on the equator that
is just about to disappear.
Fig 24.3. (a) HST image of Mars at closest approach to
By combining time delay and Doppler shift
Earth. (b) Space-based image of Mars. [(a) STScI/NASA;
(b) NASA] data, we limit the source of the echo to two pos-
sible points. These are the two points where the
same . As viewed from the radio telescope, lines
time delay circle intersects the Doppler shift cir-
of constant time delay appear as concentric rings
cle. The remaining ambiguity in the location of
about the closest point.
the feature can be removed by observing at a dif-
We now look at the Doppler shift. If a point on
ferent time where the feature™s location and
the equator moves with a speed v0, then a point at
Doppler shift have changed. More recently, orbit-
latitude moves with speed (see Problem 24.2)
ing spacecraft have also been used for higher res-
v1 2 v0 cos olution radar mapping on Venus and Mars.
24 THE INNER PLANETS 483




(c)
Fig 24.5. Surface features of Mercury. (a) Mariner 10
image of rayed crater. (b) Mariner 10 image of large crater.
(c) Mariner 10 image of scarp. [NASA]
(a)



There are a number of unusual features. For
example, there are long fractures, called scarps.
These indicate some large-scale compression of the
surface. There are also a series of irregular features
called weird terrain. These were probably caused by
the shocks from impacts by large objects.
Infrared observations (also from Mariner 10)
indicate that the surface has a layer of fine dust,
which is several centimeters thick. These obser-
vations also indicate surface temperatures rang-
ing from about 700 K on the day side to 100 K on
the night side. This large temperature difference
tells us that there is very little heat flow, via soil

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