A) The Earth
 

1. The body we know the most about is the Earth.  If the Earth is typical of celestial bodies in general, we will be able to better understand other bodies by first understanding the Earth.
Some data concerning the Earth:  Radius = 6378 km = 6.378x108 cm Mass = 5.974x1027g. Tsurf = 290 K. Density = 5.52 g cm-3.

2. Density tells us something about composition, but we need to know the effects of pressure.  This can be avoided for the Earth using seismic data.  2.1 Seismology allows us to measure the speed of seismic waves through different parts of the Earth.  This speed depends on how the material responds to applied stresses and strains.  This is determined by two parameters: Bulk modulus, K = r dP/dr Shear modulus,  μ 2.2  There are two kinds of waves, pressure (compression) waves which have a speed, νp, and shear waves which have a speed, &nus. Shear waves also have two polarizations.  The speeds of the two types of waves depend on the bulk and shear moduli, and are given by  2.3 We define a seismic parameter, φ=K/ρ.  This can be related to the speeds of seismic waves by so that if we measure the speed of seismic waves as a function of depth in the Earth, we can automatically find φ(r).  Now take the equation of hydrostatic equilibrium This gives the density as a function of radius throughout the planet without the need for an equation of state, or even a knowledge of the composition.  We can thus get a profile of the structure of the Earth.

3. Inversion of seismic data gives the PREM (Preliminary Reference Earth Model), the overall structure of the Earth. 3.1 The model (and the Earth) is divided up into a number of sections:
a.  Crust (0.005 MEarth)  Oceanic crust: 5-15 km thick, 60% by area, 20% by volume. Basaltic composition.  Continental crust: 30-50 km thick.  Granitic composition.  Transitional crust: 15-30 km thick, Islands, island arcs, continental margins.

b. Mantle (0.67 MEarth). There is a definite discontinuity between the crust and the mantle called the Mohorovicic Discontinuity (Moho). The boundary appears to be chemical.  Upper mantle: between 10-400 km.  Composed mostly of olivine(Fe,Mg)2SiO4 (0.103 MEarth).  Transition region: 400-650 km.  Marked by discontinuities at the boundaries.  At 400 km it seems to be the change from olivine to spinel.  The 650 km discontinuity doesn't seem to be simply a phase change, but may involve some chemical change as well (0.075MEarth). Lower mantle: 650-2890 km possibly pyroxene (Fe,Mg) SiO3 (0.49 MEarth).    c. Core (0.33 MEarth) - definitely a composition change probably to Fe-Ni-S  Outer core: 2890-5150 km (0.31 MEarth) liquid (no shear waves). Inner core: 5150-6370 km (0.017 MEarth) solid because of effect of pressure on melting temperature. We guess the composition from equation of state considerations and from comparison between crust and solar composition.

4. Solar composition. 4.1 We can infer the composition of the Sun and other stars from  Spectroscopy: we can measure the composition in the outer layers of stars directly. Direct measurement of elemental abundances in meteorites. Interstellar gas clouds. Cosmology: the primordial H/He ratio can be inferred from the big bang temperatures. Solar cosmic rays. Theory based on stellar evolution. 4.2  If we take a gas of this composition and look for equilibrium at around 400 K, we can ask what materials are solid under those conditions.  The Earth should be made up of that kind of material.  Meteorites should be similar but there are substantial differences.

5. The Earth's core 5.1 One of the major differences is that the Earth's crust is poor in Fe compared to meteorites.  We explain the lack of Fe in the Earth's crust by saying that the overall value of Fe in the Earth is similar to that in meteorites, but most of the Fe resides in the core. 5.2 There is a seismic indication of the presence of a core - the shadow zone, which is formed by the refraction of seismic rays through the core in such a way that there will be a region that will be free of waves.  5.3 The shear modulus in the outer part of the core is zero, indicating that this region is a liquid. 5.4    Why is the center of the Earth liquid?    a. Radioactives release heat, about 2.4x1020erg/s today, and there were more radioactive materials in the past.     b. If N0 is the original amount, then today there is N(t) = N0 e-αt , where α is the reciprocal of some typical decay time, roughly 109 years for the important nuclei.  Integrating N(t) from 0 to τ(the present) gives a total energy production proportional to N0/α(1-e-αt).  For the present we have at » 4.5, so the exponential is small compared to 1 and the total heat produced since the Earth was formed is proportional to N0/α= N(τ) eατ/α.  If current heat production is as given above, then the total heat produced over the Earth's history is about 7x1038ergs.  If the heat capacity of rock is about 9x106erg/g/K, this is enough to raise the temperature of the Earth to nearly 13,000 K.  5.5  Where did we go wrong?    a. The Earth radiates heat into space, and if the temperature of the Earth is 300 K then over its history it radiated 3.3x1041erg.  What is wrong now?    b. The Sun also heats the Earth, and the temperature of 300 K is in equilibrium with sunlight.  Most of the heat that is reradiated is sunlight; only a small fraction of the Earth's internal heat is radiated.    c. The rate of re-radiation is very strongly dependent on T, and could have been much higher in the past.    d. The rate-limiting step is how fast heat can come out of the interior.  The flux is related to the temperature gradient by

6. Convection:        If the temperature at the Earth's surface is 300 K, then the flux radiated is 4.6x105 erg/cm²/s.  The flux coming from the interior of the Earth is only some 80 erg/cm²/s.  If K = 2x105 erg/cm/K, then the temperature gradient inside the Earth must be 40 K/km.  If this continues to the center of the Earth, then at a depth of 6000 km the temperature should be 240,000 K.  This is much too high! The gradient must be much smaller inside. There are two reasons:  Not all the sources are at the center, so the flux decreases towards the center. The mantle is convecting. The convection is slow, and a turnover time is of the order of 108years.

7.   Plate tectonics 7.1 We see evidence of convection in continental drift. In 1912 Alfred Wegner noted that South America seemed to fit well with Africa.  Fossils on both sides of the Atlantic were very similar. He suggested they were once attached, but he did not have a good mechanism for moving such large masses. In the late 1950's magnetic studies of the mid-oceanic ridge showed that there was seafloor spreading.  2.2x108 years ago, the Atlantic Ocean was very small.

7.2 The mechanism for this motion is the convection of the Earth's mantle. The Earth's crust consists of a series of plates that float on the mantle, and as the mantle convects, the plates move around.
7.3 There are two kinds of plates, oceanic and continental, corresponding to the two basic kinds of crust.  The Table, taken from Anderson, gives typical compositions in weight percent. Because the continental crust is richer in less dense minerals, it tends to stay above the more dense oceanic crust.	Oxide Oceanic Crust Continental Crust Continental Crust SiO2 TiO2 Al2O3 Fe2O3 FeO MgO CaO Na2O K2O H2O 47.8 0.59 12.1 - 9.0 17.8 11.2 1.31 0.03 1.0 63.3 0.6 16.0 1.5 3.5 2.2 4.1 3.7 2.9 0.9 58.0 0.8 18.0 - 7.5 3.5 7.5 3.5 1.5 -

7.4 Plate boundaries: Where two continental plates collide there is mountain building (Himalayas).   Where they "rub" against each other you get a transform fault like the San Andreas.   If an oceanic plate collides with a continental plate the oceanic plate is forced under the continental plate and into the mantle.  This region is called a subduction zone.  These are found at the boundaries of the Pacific Ocean.  7.5 As the oceanic crust is subducted and melts, some of the lighter material comes to the surface through volcanoes. This is why volcanoes are found near plate margins.  An example of this is Mount Fujiama.  Some volcanoes are formed when there is a hot spot that puts out lava.  As the plate moves over the hot spot a chain of volcanoes are formed like the Hawaiian Islands. What other forces shape the Earth?

8. Impacts of space debris. 8.1 Space is full of bodies of all sizes.  In 1991 a 30m rock traveling 30 km/s passed within 100,000 km of Earth. On November 30, 1954 a 4.5 kg meteorite crashed through a house in Sylacauga, Alabama and bruised Mrs. Hewlett Hodges. A human built structure is hit roughly once every 3 years.  But two houses in the town of Wethersfield, Connecticut separated by less than a mile were hit in a span of 12 years. No human has been killed by a meteorite, but a dog was killed in Egypt by the fall of the Nahkla meteorite in the 19th century. 8.2 What would have happened if the 1991 meteor hit?  If we assume it was roughly a sphere, then its volume was 4πr3/3, or 1.13x1011 cm3.  If we take a typical density of meteoric material to be about 3 g cm-3, then the mass of the meteorite was about 3.4x 1011 g and its kinetic energy about 1.5x 1024 ergs.  A 1-kiloton atomic bomb produces about 4.2x 1019 erg so the meteor had the energy of a 36-megaton bomb.  This energy increases as the cube of the meteor radius, so a 10 times bigger meteor has 1000 times more energy. 8.3 There are theoretical relations relating meteor energy and crate size.  In our case, we might expect a crater of roughly 450 m diameter.  This is just a bit smaller than the Meteor Crater in Arizona.  A small asteroid some 10 km in diameter would make a crater with a diameter of 36 km. This is roughly the size of Gush Dan. 8.4 What happens to the debris? The energy heats the impactor and the target to high temperatures causing shocking of quartz (which was first seen in nuclear explosions and later in meteor craters, and shows intersecting lines not found in ordinary quartz), and production of drops of melted glass.  This material gets blown into the stratosphere, above the region of the atmosphere that is well mixed by convection.  It might stay there for years causing shading of the Sun. This could lower temperatures.  A case in point is the series of very cold winters and summers following the explosion of the Tambora volcano.  It even snowed in New England on July 4, 1816.  This could also occur after a nuclear war, and is called nuclear winter. Mount St. Helens in Oregon, 1980 had an energy of roughly 10 megatons.  This is about ¼ the energy released by our 30 m meteor.  Krakatoa in 1883 and Tambora were roughly 100 megatons. 8.5 What other consequences are there? The impact with the atmosphere doesn't lead to significant slowing, but does produce a shock wave.  The energy would be enough to drive chemical reactions, breaking up the N2 and O2 and forming products that would combine with atmospheric water to form H2NO3 (nitric acid).  This would cause acid rain and turn lakes and oceans acidic.  Any abundance differences in the meteor material would be deposited on the Earth, and lead to abundance anomalies in the layer where that impact occurred. 8.6 What important clues do we have to past large impacts? There are tektites, little blobs of melted glass that have a shape that appears to have been imposed under      conditions of aerodynamic stress.There are several fields around the Earth (North America, Pacific Ocean around Southeast Asia) where there are tektites 34 million years old.  Tektites found in Czechoslovakia are 14.7 million years old.  Tektites from the Ivory Coast are 1,000,000 years old, and those from Australia, Southeast Asia, and the Indian Ocean are about 700,000 years old. Tektites of different ages show distinct differences in composition.  The compositions are similar to other rocks in the area of the field.  Apparently they are Earth rock that have been melted and scattered by the impact of some large body.  Some of the impacts seem to have occurred at the same time as a reversal of the Earth's magnetic field, but not all, so it is not clear there is a connection. There is some indication that the temperature of the Earth dropped around the time of the impact, but there have been many temperature drops without a preceding impact, so again, there is no clear connection. 8.7 Mass extinctions:   There have been several instances where large numbers of species became extinct in a period short compared to geologic time.     One important such event included the destruction of the dinosaurs around 65 million years ago at the boundary between the Cretaceous and Tertiary periods (K/T boundary).  In addition the ocean temperature dropped by several degrees and many other life forms became extinct. Evidence includes: Layers of clay dating from this period from different places in the world (around 40 of them) that show abundance anomalies of iridium.  Iridium is known to be abundant in some types of meteorites, so it lends support to the argument that the impact of a large meteorite triggered the extinction. Shocked quartz, micortektites. An apparent crater off the coast of the Yucatan Peninsula in Mexico with a diameter of at least 180 km.  It is called Chicxulub after the town near which it was found.  It is dated at 65 million years, and there are thick layers of tektites of the same age in nearby areas in Mexico and Haiti.  Such an impact would release between 1 and 3x1031ergs.  The resultant rock vapor would go into the atmosphere and cool slowly by radiating into space.  Some of the radiation would heat the water, and vaporize it as well.  Around a meter of ocean would be vaporized and another half a meter would be boiled. Nearly all of the surface ocean life would have been destroyed. There is still some controversy surrounding this interpretation. Other mass extinctions may be explainable by the same mechanism. 8.8 Indeed it is possible to see a sort of periodicity in the dating of the extinctions.  What could cause such a periodicity? Perhaps the Sun has a small stellar companion (sometimes called Nemesis), or there is a 10th planet (Planet X) as yet undiscovered.  There is some evidence that Neptune is not following precisely the orbit we would expect, but this is still unclear.  Such a companion could perturb the Kuiper belt or the Oort cloud and cause comet showers, leading to the periodic impacts and extinctions.  The Nemesis idea is problematic because the orbit would be unstable to perturbations caused by galactic tides.  The periodicity is also open to question.

9. Climate change 9.1 In 1842 Louis Agassiz was studying the grooves glaciers made in the Alps, and found grooves where it is too hot for there to be any ice.  He speculated that the climate on the Earth was much colder in the past. We now know that there were ice ages in the past when large parts of Europe and North America were covered with ice.  The last ice age was about 12,000 years ago. 9.2 The dating is done by studying foraminifera, small marine organisms that have shells made of CaCO3.  When they die, they fall to the bottom of the oceans, and we find them in the sediments.  During an ice age water evaporates from the ocean, but much of it freezes out as land ice and snow.  The rate of evaporation from the ocean is higher than the rate of re-supply from rain, and the ocean level drops.  Since 16O is lighter than 18O, vapor containing 16O evaporates more easily, and the 16O/18O ratio in the ocean drops.  This is reflected in the ratio found in the foraminifera shells, so they act a thermometer for the surface temperature. 9.3 The data show that there are three important cycles with periods of 105,000, 41,000, and 23,000 years.  The first corresponds to the time it takes to change the Earth's orbit from nearly circular to an ellipse with maximum eccentricity and back again.  The second to the period of nutation from 21 to 24° and back again.  The third is the period of precession (measured relative to the perihelion of the Earth's orbit which is itself prיcising, giving a period relative to the fixed stars of 26,000 years).  All of these effects are due to the action of the Sun and planets.  The connection between orbit and climate was first suggested by Milutin Milankovitch in the 1920's. 9.4 There are still some unanswered questions.  One is why the 105,000-year period is the most important when that should have the least effect?

B) The Moon
 

1.  We can do the basic measurements.  Radius = 1738 km. Mass = 7.35x1025g = 0.0123 Mearth

The mass of the Moon is tricky since it doesn't have any natural satellites. Until the age of spacecraft, this was done by measuring the Earth's motion around the common center of mass. This is 4600 km from the center of the Earth.  As a result of this motion, the stars appear to make circles of diameter 6'' once a month.  Accurate measurements with a transit telescope can give the value of the distance to the center of mass and hence the Moon's mass. In 1972 the orbits of Explorers 35 and 49 gave a more accurate mass of GM = 4.90284 x 1018 cm3 s-2. Albedo = 0.07 - 0.24 Distance from Earth = 380,000 km Surface Temperature = 380 K (day), 120 K (night) Density = 3.36 g/cm3

2. Other surface data    a. Parts of the surface are covered by a white powdery sand and a rock called a breccia, a conglomeration of unrelated bits of stone packed into boulder-sized rocks.     b. Surface is divided into two major classes: Maria are flat plains hundreds of kilometers across with occasional craters and long cracks called rilles.  A rill can be from a few hundred meters to hundreds of kilometers in length. Highlands are mountainous regions with brighter rock and covered with craters.    c. Craters are caused by meteorite impacts. They range in size from microscopic to 100 km in diameter.  The meteorites also cause erosion.  With no atmosphere to stop them, there are micrometeorites grinding away the lunar surface, and turning it into a fine soil tens of meters deep called a regolith.    d. The continual gardening by meteorites overturns the soil in a period of a few million years.  In this way rocks from different parts of the Moon may be brought together.    e. Early impacts were much more numerous and the highlands were cratered at this time.  Large impacts formed basins (a typical energy of a basin forming impact is some 1031ergs.  Recall that a 1-megaton bomb is 4x1022ergs).  Later these basins were filled with molten rock that erupted from beneath.  When this rock froze, it made the smooth maria.  Because they are younger they have fewer craters.    f. The maria are nearly all on the side of the Moon that faces the Earth and the highlands on the farside.  The center of mass of the Moon is also displaced some 3 km from the center of figure towards the Earth.  Apparently this is somehow connected with the gravitational field of the Earth.     g. A similar asymmetry can be found in the thickness of the Moon's crust.  It is only about 60 km thick on the nearside, and up to 120 km thick on the farside.  The rocks in the maria are rich in silicates (olivine).    h. The nearside of the Moon contains mass concentrations (called mascons) whose effect on the gravitational field can be detected in the orbits of satellites.  They show that the Moon is not completely in hydrostatic equilibrium.  The amount of mass in these mascons can be used to estimate their thickness.  The values range from about 0.5 km over Oceanus Procellarum to about 5 km over Mare Imbrium.     i. The lunar rocks can be dated in one of two ways.  The first is based on the age of the surrounding region, which comes from crater counts.  The more exact method is by radioactive dating.

3. Dating: 3.1 If you have a radioactive nucleus, it has some probability of undergoing decay.  If I put a large number of such nuclei together, the rate of decay will be proportional to the intrinsic decay rate, as well as the number of nuclei available.  If N is the number of nuclei and l is proportional to the decay probability, which has the solution We can now see the meaning of l.  It is the inverse of the time for the number of remaining nuclei to decrease to 1/e.  If we can measure the amount of some isotope in a rock today, and can say something about the amount that was there originally, we can determine the age of the rock.  Consider the case of 87Rb which decays to 87Sr.  Usually such rocks also contain 86Sr which is stable.  Let N = 87Rb/86Sr, and let D = 87Sr/86Sr.  Originally, D = D0, but as the Rb decays D increases.  At any time, t, we have Now consider two minerals in the same rock that were formed at the same time.  Even though the amount of Sr will be different in the two minerals, we expect the value of D0 to be the same.  We can therefore write: Since the values of D and N are different for both minerals, we can eliminate D0 and solve for the age of the rock, Note that this method of dating assumes that D0was indeed the same for the two minerals, and it only gives the time since the minerals cooled from the melt.  The ages of the highland rocks are sometimes> 4x109 years.  The ages of the maria rocks are between 3.1x109 and 3.95x109 years.

4. The mare rocks have a residual magnetic field even though the Moon presently does not.  Apparently there was a lunar magnetic field in the past.  What does this mean? 4.1 Magnetic fields are usually formed in some sort of dynamo action. The basic ingredients are: Convection Conducting material Rotation Molten region    When the conducting material flows in response to thermal gradients, the resulting motion through a magnetic field produces an induced current.  If the conditions are right, this current can in turn produce an additional magnetic field.    The equations governing this phenomenon are complicated because the motion of the fluid depends on the temperature gradient, the rate of rotation, and the strength of the magnetic field.  The motion, however, affects the temperature gradient and the magnetic field, (and to a small extent, the rate of rotation) so that the system is highly non-linear.  Detailed solutions require Monte Carlo simulations on a supercomputer.

5. Craters:  5.1 The most obvious surface feature of the Moon is its craters.  At one time they were thought to be volcanic, but there are several indications that they are not: You need a huge heat source to produce as many volcanoes as there are craters. Some craters are several hundred kilometers in diameter.  This is too big for any known volcanic mechanism. The cross section of the crater is similar to an impact crater, not to a volcanic crater.  The profile of a crater could be determined from measurements of shadows. 5.2 It is now accepted that the lunar craters are caused by impacts of meteorites. The problem with this is that you needed a huge cratering rate in the past. On Earth we have very many fewer craters for three reasons: 70% of the Earth's surface is covered with water, so that 70% of the impacts do not leave craters.  The Earth's surface is constantly being reworked by tectonics. The Earth's surface is constantly being eroded by wind and rain.  None of these arguments apply to the Moon. 5.3 The shape of a crater is determined by the physics of hypervelocity impacts. The impactor is traveling at a speed of around 10 km/s.  The speed of sound in rock is less than this.  As a result, the rock cannot deform to get "out of the way" of the impactor, and the impactor stops suddenly.   Nearly all the kinetic energy is transformed into heat.  The heat capacity of rock is roughly 107 erg/g/K.  In order to raise a gram of meteoritic rock from 200K to 1800 K requires about 1.6x109 erg.  An additional 6x1010 erg/g is required for the heat of vaporization (which is the dominant term).  This amount of energy can be supplied by kinetic energy when the meteorite is moving faster than about 3-4 km/s.  In other words, an impact of the type we are discussing will vaporize itself and a good chunk of surrounding area. 5.4 The shock waves from the impact compress the surrounding rock, and when the explosion is over, and the shock subsides, the walls of the crater expand back inward.  This results in very steep walls that eventually slump back into the crater.  5.5 Not all of the meteorite is destroyed in the impact, and fragments may be thrown off in all directions.  Parts of the excavated crater are also included.  These ejecta cause many secondary craters nearby.  They also tend to form rays of ejecta extending from the crater.  Since impacts by energetic cosmic rays tend to darken the lunar soil, these rays of newly excavated material are brighter than the surroundings. 5.6 The shock waves move in all directions and give the crater its round appearance.  In addition, the waves move downward through the crust and into the denser mantle, where they are reflected back.  This tends to throw material back up into a central peak, which is very prominent in lunar craters.  On planets where water or other liquids may be present, the central region can sink causing a central pit.

6. The Apollo landers left 6 seismographs on the lunar surface, which allowed a more detailed mapping of the interior. They measured 12,000 events including 1700 meteor impacts. Only one weak event on the far side passed through the center. They were turned off in 1977. They saw: In the outermost km. the velocity increases from about 200 m s-1 to about 4 km s-1 probably due to increasing compaction of the regolith. Between 1 and 21 km the velocity increases smoothly to 6.7 km s-1.  This is typical of speeds in the Earth's crust.  Between 21 and 55 km the velocity stays fairly constant.  Between 55 and 60 km it increases to 8.9 km s-1.  This is typical of speeds in the Earth's mantle.  These regions are therefore called the lunar crust and mantle.  There was some seismic indication of an Fe-rich core, but it is small, less than 350 km in radius.  This is consistent with the Moon's low density.

7. There is evidence that the Moon was once hot: Seismic studies show that there is a discontinuity at a depth of about 400 km.  This may be due to the upper 400 km being melted.  The highland crust is about 25% Al2O3.  Primitive mantle material on Earth is only about 4% Al2O3.  If we assume the crust was formed through a differentiation of the mantle, and that the Earth and Moon are similar in this respect, then 7 times more material must have been processed on the Moon.  This requires a great deal of energy. This evidence for high temperatures in the past led people to talk about a magma ocean on the surface of the early Moon.     The magma on the Moon, is thought to have consisted of two phases, feldspar and olivine.  Since feldspar is lighter than the mixed magma, it floated to the surface of the magma ocean.  The heavier olivine sank to the bottom.      The melting was not complete, and the layer cooled in about 2x108 yrs.  In this way the lunar crust differentiated.  Similar processes lead to the differentiation of a mantle and core.

8. Any theory describing the origin of the Moon has to deal with two very important issues: Why is the Moon's composition so different from that of the Earth? Where did you get the energy to heat the Moon? The theory favored today is that a Mars-sized object crashed into the Earth, and the Moon formed from the debris.