1. Earth as a Planet Mass M = 6 x 1027 g. Radius R = 6371 km.
Mean density = M/(4/3 p R3) = 5.5 g/cm3
Moment of inertia I of the Earth:
I =  r2 dm
I/(MR2) =
for a uniform sphere I/(MR2) = 0.4.
As I have suggested, you can weight it if you want to know what it is inside the earth. Can anybody in the class tell me how to weight the Earth? You probably will tell me INTERNET can do it, so let me just click it to see whether it works.
In 1798, about a century after Newton’s gravitation law, Lord Cavendish of England made the first measurement of the gravitational constant G using a torsion balance.
He then used this number in a pendulum experiment and was able to determine the mass of the Earth from the period of the swinging pendulum.
The average density of the bulk Earth was obtained to be approximately 5.45 g/cm3 (the presently
accepted bulk Earth density is 5.25 g/cm3). The implications of these calculations were far-reaching. Most of the rocks on the
Earth’s surface have densities between 2.5 and 3 g/cm3, and thus somewhere in the Earth’s interior must contain a heavy component.
These calculations have been further verified and refined by seismological studies and studies of the Earth’s moment of inertia.
In particular, in 1906, Richard Oldham of the Geological Survey of India, used seismic waves to show that the Earth was largely
partitioned into two parts: the central core, with a radius ~0.4 of the Earth’s radius, and the surrounding mantle. The
boundary between the core and the mantle was found to coincide with a distinct density and seismic velocity jump (Fig. 2.1),
confirming earlier suggestions of an Earth with a dense interior.
In 1798, Cavendish determined the gravitational constant using a torsion balance. He then used this
number in a pendulum experiment and was able to determine the mass of the Earth from the period of
the swinging pendulum. Even earlier, however, Pierre Bourguer of France had visited the Andes in
1748 in order to study the Earth’s magnetic field. Instead, Bourguer made the interesting observation
that his plumb line was deflected from vertical towards the Andes. The English astronomer Nevil
Maskelyne used this observation to determine the mass of the Earth, yielding a density of 4.5 g/cm3.
The moment of inertia is an index of how difficult a body is to rotate along an axis, an analogy of mass as an index of how difficult a body is to move.
It is the product of the square of the distance of a mass dm from the rotation axis, r.

2. Differentiation early in Earth’s history

3. Interior of the Earth
Crust: variable thickness with an average value of 35 km in the continents and 7-8 km in the oceanic regions. Volume ~1019 m3 Mass 2.8 x 1022 kg.
Mantle: between the Moho discontinuity (crust-mantle) and the core-mantle boundary (R = km). Volume 9 x 1020 m3 Mass 4 x 1024 kg.
Core: from the center of the Earth to the core-mantle boundary. Volume 1.77 x 1020 m3 Mass 1.94 x 1024 kg.
Roughly speaking, there are three main domains within the earth. In other words, the earth consists of three parts: crust, mantle and core. The crust has variable thickness with an average value of 35 km in the continents and 7-8 km in the oceanic regions.

4. More Details about Layering…
More details layering in different subjects or fields is listed here. For example, petrologists further divided the mantle into an upper mantle of peridotite and a lower mantle of perovskite. The core is made of iron. Seismologists found that besides the three major boundaries, the crust-mantle boundary Moho, the core-mantle boundary CMB and the inner core boundary, ICB. There are two additional boundaries in the upper mantle. The 410-km and 660-km discontinuities. There are many seismic studies suggested other discontinuities at various depths, but all lack a consistent global confirmation. One of there, usually called the D” discontinuity in the D region of Bullen’s naming, has been actively studied in the last decade.

5. What is Earth made of?
Why do we need to look outside the Earth to learn about what is inside the Earth?
We know the Earth is layered and that what we can sample on the outside is not typical.
By studying members of the Solar System, it is possible to estimate its original composition and the physical and chemical processes that have led to its present state.

6. The Solar System: A highly diverse zoo!

7. Sun Asteroids Mercury Jupiter Venus Saturn Earth Uranus Neptune Mars
Overview of the Solar System
Sun
Mercury
Venus
Earth
Moon
Mars
Asteroids
Jupiter
Saturn
Uranus
Neptune
Pluto

9. The Origin of the Solar System Frank Crary, CU Boulder
Here is a brief outline of the current theory of the events in the early history of the solar system:
A cloud of interstellar gas and/or dust (the "solar nebula") is disturbed and collapses under its own gravity. The disturbance could be, for example, the shock wave from a nearby supernova.
As the cloud collapses, it heats up and compresses in the center. It heats enough for the dust to vaporize. The initial collapse is supposed to take less than 100,000 years.
The center compresses enough to become a protostar and the rest of the gas orbits/flows around it. Most of that gas flows inward and adds to the mass of the forming star, but the gas is rotating. The centrifugal force from that prevents some of the gas from reaching the forming star. Instead, it forms an "accretion disk" around the star. The disk radiates away its energy and cools off.

10. The Origin of the Solar System Frank Crary, CU Boulder
First break point. Depending on the details, the gas orbiting star/protostar may be unstable and start to compress under its own gravity. That produces a double star. If it doesn't ...
The gas cools off enough for the metal, rock and (far enough from the forming star) ice to condense out into tiny particles. (i.e. some of the gas turns back into dust). The metals condense almost as soon as the accretion disk forms ( billion years ago according to isotope measurements of certain meteors); the rock condenses a bit later (between 4.4 and 4.55 billion years ago).
The dust particles collide with each other and form into larger particles. This goes on until the particles get to the size of boulders or small asteroids.

11. The Origin of the Solar System Frank Crary, CU Boulder
Run away growth. Once the larger of these particles get big enough to have a nontrivial gravity, their growth accelerates. Their gravity (even if it's very small) gives them an edge over smaller particles; it pulls in more, smaller particles, and very quickly, the large objects have accumulated all of the solid matter close to their own orbit. How big they get depends on their distance from the star and the density and composition of the protoplanetary nebula. In the solar system, the theories say that this is large asteroid to lunar size in the inner solar system, and one to fifteen times the Earth's size in the outer solar system. There would have been a big jump in size somewhere between the current orbits of Mars and Jupiter: the energy from the Sun would have kept ice a vapor at closer distances, so the solid, accretable matter would become much more common beyond a critical distance from the Sun. The accretion of these "planetesimals" is believed to take a few hundred thousand to about twenty million years, with the outermost taking the longest to form.

12. The Origin of the Solar System Frank Crary, CU Boulder
Two things and the second break point. How big were those protoplanets and how quickly did they form? At about this time, about 1 million years after the nebula cooled, the star would generate a very strong solar wind, which would sweep away all of the gas left in the protoplanetary nebula. If a protoplanet was large enough, soon enough, its gravity would pull in the nebular gas, and it would become a gas giant. If not, it would remain a rocky or icy body.
At this point, the solar system is composed only of solid, protoplanetary bodies and gas giants. The "planetesimals" would slowly collide with each other and become more massive.

13. The Origin of the Solar System Frank Crary, CU Boulder
Eventually, after ten to a hundred million years, you end up with ten or so planets, in stable orbits, and that's a solar system. These planets and their surfaces may be heavily modified by the last, big collision they experience (e.g. the largely metal composition of Mercury or the Moon).
Note: this was the theory of planetary formation as it stood before the discovery of extrasolar planets. The discoveries don't match what the theory predicted. That could be an observational bias (odd solar systems may be easier to detect from Earth) or problems with the theory (probably with subtle points, not the basic outline.)

14. The Big Questions
What is the origin of the solar system? It is generally agreed that it condensed from a nebula of dust and gas. But the details are far from clear.
How common are planetary systems around other stars? There is now good evidence of Jupiter-sized objects orbiting several nearby stars. What conditions allow the formation of terrestrial planets? It seems unlikely that the Earth is totally unique but we still have no direct evidence one way or the other.
Is there life elsewhere in the solar system? If not, why is Earth special?
Is there life beyond the solar system? Intelligent life?
Is life a rare and unusual or even unique event in the evolution of the universe or is it adaptable, widespread and common?

15. Solar abundance of elements
Determined from spectral absorption lines
Light from visible surface of sun passing through cooler gases above the surface
This is thought to represent total solar abundance because nuclear reactions powering the star take place deep inside and there is little convection there to mix modified material up with original material.

16. Meteorites

17. Meteorites: Summary
The fabric of chondrites is quite unlike that of any terrestrial rock and required very different conditions in which to form. These are identified with early stages in the development of the Solar nebula.
The carbonaceous chondrites are a close approximation to the material of the Solar Nebula, having lost only the most volatile elements. It is, therefore, plausible to regard them as a starting point from which the composition of the Earth has evolved. This leads to the Chondritic Earth Model.
The meteorites derive from the asteroids by collision.
The differentiated meteorites were formed within minor planets, or asteroids, which heated sufficiently to segregate into layers, forming an iron core, silicate mantle and transitional region between. Subsequent break-up due to collisions produced iron, achondrite, and stony-iron meteorites.

18. Hydrogen, the simplest element, is the basic building block.
Next Question:
What is the origin of the distribution of elements in the Solar System?
Hydrogen, the simplest element, is the basic building block.

19. Hydrogen burning – 4 protons become alpha particle (helium nucleus)
Helium burning - 3 alpha particles become 12C (which can absorb another to become 16O
Carbon burning and oxygen burning produce 28Si (very stable), 24Mg , 32S, and other elements
Each of these requires more heat than the fusion reaction before it.

20. Silicon burning involves breaking of pieces of other nuclei and adding them to others. This produces many stable nuclei heavier than Si.
As temperature rises, this becomes the equilibrium e-process which is like shaking and breaking up all the existing nuclei and recombining them randomly to make all possible stable nuclei up to the iron group elements.
Everything bigger than the iron group is less stable and the e-process would rearrange them into iron group elements.

21. Neutron capture becomes the method that builds larger nuclei.
Slow-neutron or the s-process. Add a neutron to a nucleus. If nucleus becomes unstable because its neutron/proton ratio is too high, it has time to “fix” itself before another neutron arrives. It “fixes” things by a beta decay. A neutron converts to a proton and an electron is emitted. The nucleus has moved one element up the periodic table.
The s-process can build elements up to 209Bi. At that point there is no neutron/proton ratio stable enough to allow the one by one conversion of neutrons to protons.

22. The rapid-neutron or r-process involves adding neutrons to a nucleus faster than things can be “fixed.” Much heavier nuclei can be built up. Once the bombardment is over, the neutron-rich nucleus will undergo repeated beta decays to produce nuclei that are relatively more stable but which, in turn, are unstable to alpha decay and so break down into lighter nuclei. These include 238U, 235U, and 232Th which have half-lives comparable to the age of the Earth, and so have not yet decayed to negligible amounts.

23. The s-process also helps fill in gaps between some of the lighter elements such as between 12C and 16O. This mechanism can only produce neutron-rich nuclei, so other processes must account for known nuclei with lower than average neutron/proton ratios.
The p-process resolves this by adding protons to nuclei.
Light elements Li, Be, and B are not produced by any of the above processes. In fact, they are destroyed at the temperatures required for hydrogen burning. They are probably formed as fragments when heavy nuclei in interstellar dust are struck by cosmic rays. This is a very slow process, but interstellar dust spends a lot of time in space!

24. The most likely place for these reactions to take place is in the interior of a large star.
The Sun is not large enough to ever get beyond hydrogen burning, and therefore will not generate the distribution of elements found in the Sun or meteorites.
One or many larger stars were needed to produce these elements that formed the nebula that became the Solar System.
The matter from these stars would have been disseminated by supernova explosions at the end of their existence as stars.
There is time between the formation of our galaxy, 15x109 years ago and the formation of the Solar System 4.6x109 years ago for many generations of stars to form, explode and slowly enrich the interstellar medium. This heavy material is the 2% of the Solar System.

25. One additional observation suggests that the last of these supernovae must have occurred just 2-3 million years before the initiation of the formation of the Solar System. It must have occurred close to the dust cloud that became the Solar System.
There is evidence that certain elements like 26Al with very short half-lives were present in the material that formed the Solar System. If these had been formed gradually by many stars, these elements would have decayed away. They could only have been formed and disseminated by a very recent supernova.
It is possible that this supernova not only contributed material to the cloud, but also initiated its collapse.

26. Differentiation of the Earth
Differentiation is the process by which random chunks of primordial matter were transformed into a body whose interior is divided into concentric layers that differ from one another both physically and chemically.
This occurred early in Earth’s history, when the planet got hot enough to melt.

27. What was the starting point for differentiation?
Heterogeneous/Hot starting model
Initial layering as Earth solidified from gas
Homogeneous/Cold starting model
Little or no initial layering because Earth formed from the agglutination of cold, uniform particles
Neither model seems to work completely

28. When did differentiation happen?
About 4.5 billion years ago
After beginning of Earth’s accretion at 4.56 billion years ago
Before the formation of the Moon’s oldest known rocks, 4.47 billion years ago

30. Sources of heat to melt Earth
Frequent and violent impacts
There was likely one particularly large impact
Moon aggregated from the ejected debris
Earth’s spin axis was tilted
Decay of radioactive elements
This heat generation was greater in the past than today

31. Basic processes of differentiation
In a liquid or soft solid sphere, denser material sinks to the center and less dense material floats to the top.
When rock is partially melted, the melt and the remaining solid generally have different chemistry and density. The melt is usually less dense than the “residue.” The melt is enriched in “incompatible” elements. The residue is enriched in “compatible” elements.

32. Earth’s Core
Iron, nickel, and other heavy elements were the densest material and formed the core. Core radius is 2900 km.
They are about 1/3 of the planet’s mass
Inner core is solid. Inner core radius=1200 km. Inner core is solid because pressure is too great for iron to melt at Earth’s current temperature.
Outer core is liquid. Some of the iron in the outer core is iron sulfide.

33. The Iron, Oxygen, Sulfur, Magnesium, and Silicon story
There were large amounts of these five elements in the early Earth
The fate of the iron was controlled by its affinity for bonding with oxygen and sulfur.
Iron bonds preferentially with sulfur. All available sulfur is consumed. Iron remains.
Oxygen bonds preferentially with magnesium and silicon. This uses up the magnesium and silicon. Oxygen remains.
Iron then combines with oxygen. Oxygen is now used up. Iron remains as elemental iron.
The iron, magnesium, and silicon oxides are light and form the Earth’s crust and mantle.
The iron sufide is dense, but less dense than iron, so it forms the outer part of the core of Earth.
The elemental iron is densest of all, so it forms the inner core of the Earth.
Note: The amount of oxygen in the starting material plays a key role in determining the size of the core of a planet. What does adding oxygen do to the core radius? What does adding sulfur do to the core radius?

34. Earth’s Crust Lighter rocks floated to the surface of the magma ocean.
The crust is formed of light materials with low melting temperature and is up to 40 km thick.
These are generally compounds of silicon, aluminum, iron, calcium, magnesium, sodium, and potassium, mixed with oxygen.
Fragments of crustal rocks (zircons) of age billion years were found recently in western Australia. If this is confirmed, we can conclude that Earth cooled enough for a solid crust to form only 100 million years after the large impact.

35. Earth’s Mantle Lies between the crust and the core.
Depth range is 40 km to 2900 km.
The mantle consists of rocks of intermediate density, mostly compounds of oxygen with magnesium, iron, and silicon
New continental crust may be produced during partial melting of mantle material.

36. Radiometric Dating: General Theory
The radioactive decay of any radioactive atom is an entirely random event, independent of neighboring atoms, physical conditions, and the chemical state of the atom.
It depends only on the structure of the nucleus.
λ, the decay constant, is the probability of an atom decaying in unit time. It is different for each isotope.

37. Suppose that at time t there are N atoms and that at time t+δt, δN of those have decayed, then δN can be expressed as
δN = -λ N δt
In the limit as δN and δt go to 0, this becomes
dN/dt = -λ N
Thus, the rate of decay is proportional to the number of atoms present. Rearrangement and integration gives:
loge N = -λ t + c
If at t=0 there are N0 atoms present, then c = loge N0
N = N0 e-λt

38. N0/2 = N0 e-λT½ or T½ = (loge 2) / λ
The half-life, T½, is the length of time required for half of the original atoms to decay.
N0/2 = N0 e-λT½ or T½ = (loge 2) / λ
Consider the case of a radioactive Parent atom decaying to an atom called the Daughter. After time t, N = N0 – D parent atoms remain and
N0 – D = N0 e-λt
Where D is the number of daughter atoms (all of which have come from decay of the parent) present at time t. Thus
D = N0 (1 – e-λt)

39. However, it is not possible to measure N0, but only N
Use the previous equation and N = N0 e –λt yields
D = N (eλt – 1)
This equation expresses the number of daughter atoms in terms of the number of parent atoms, both measured at time t, and it means that t can be calculated by taking the natural log
t = loge (1 + D/N) / λ
In practice, measurements of D/N are made using a mass spectrometer.

41. Major radioactive elements used in radiometric dating
Parent Isotope
Daughter Isotope
Half Life of Parent (years)
Effective dating range (years)
Materials that can be dated
238U
206Pb
4.5 billion
10 million – 4.6 billion
Zircon
Apatite
235U
207Pb
0.7 billion
40K
40A
1.3 billion
50,000 – 4.6 billion
Muscovite
Biotite
Hornblende
87Rb
87Sr
47 billion
Potassium Feldspar
14C
14N
5730
,000
Wood, charcoal, peat, bone and tissue, shell and other calcium carbonate, groundwater, ocean water, and glacier ice containing dissolved CO2

42. Radiometric dating is not always that simple!
There may have been an initial concentration of the daughter in the sample
Not all systems are closed. There may have been exchange of parent and/or daughter with surrounding material.
If dates from different isotope systems match within analytical error, we say the ages are concordant. If they are not, then we say they are discordant.
When discordant, we suspect problems like those above with one or all of the systems.
The date t obtained is not always the date of formation of the rock. It may be the date the rock crystallized, or the date of a metamorphic event which heated the rock to the degree that chemical changes took place.
Radioactive decay schemes are not all as simple as a parent and exactly one daughter. 87Rb to 87Sr is a simple one step decay. The two U to Pb series have a number of intermediate daughter products.

44. Fission Track Dating
As well as decaying to 206Pb as described before, 238U is also subject to spontaneous fission.
It disintegrates into two large pieces and several neutrons. This is a very rare event, occurring just once per 2 million α decays.
Each event is recorded as a trail of destruction about 10 m long through the mineral structure.
These “fission tracks” can be observed by etching the polished surface of certain minerals. The tracks become visible under a microscope.

45. Spontaneous
Fission Tracks

46. Consider a small polished sample of a mineral
Consider a small polished sample of a mineral. Assume that it has [238U]now atoms of 238U distributed throughout its volume
The number of decays of 238U, Dr, during time t is:
The number of decays of 238U by spontaneous fission, Ds, which occur in time t is:
Where s is the decay constant for spontaneous fission of 238U.
To determine an age, we must count the visible fission tracks, estimate the proportion of the tracks visible (crossing) the surface, and measure [238U]now.

47. Fortunately, we do not need to do this in an absolute manner, because another isotope of Uranium, 235U, can be made to fission artificially. This is done by putting our sample in a nuclear reactor and bombarding it with slow neutrons for a specified time (hours). This provides us with a standard against which to calibrate the number of tracks per unit area (track density). The number of induced fissions is:
Where σ is the known neutron capture cross-section and n is the neutron dose in the reactor.
We assume that if the two isotopes of U are equally distributed in the sample, then the proportion of tracks that cross the surface will be the same. We can combine equations to get:
Where Ns and NI are the numbers of spontaneous and induced fission tracks counted in an area.

48. The equation can be rearranged and the known present ratio of the two isotopes of Uranium, [238U]now/[235U]now=137.88,can be inserted to give:
In practice, after the number of spontaneous fission tracks Ns has been counted, the sample is placed in the reactor and then etched again. The spontaneous tracks are enlarged and the induced tracks are exposed. The number of induced tracks NI are counted and the age calculated.
Spontaneous
Fission Tracks
Induced
Fission Tracks

49. There is an additional (and very powerful) way to use fission tracks.
Fission tracks in a mineral crystal are stable at room temperature, but can “heal” if the temperature of the crystal is high enough.
At very high temperature, the tracks heal completely very quickly. This means that the “age” of a rock can be completely “reset” by heating.
The rate at which tracks are healed varies with temperature and mineral type. Therefore there is a “closure” temperature that is a function of mineral type and rate of cooling.

50. For example, fission track ages determined from sphene are always greater than ages determined from apatite. This is because healing tracks in sphene (~300C) requires much greater temperatures than healing tracks in apatite (~90C).
Imagine that rocks are being uplifted and eroded during the creation of a mountain range. The individual rocks are cooling as they are brought closer to the surface. A progression of fission track ages in different minerals record the uplift/cooling history of the rock.
There are newer, even more sophisticated methods, that use the rate at which tracks heal, they actually shorten before disappearing, to determine more complicated temperature history curves from each mineral.

51. Heat in the Earth
Volcanoes, magmatic intrusions, earthquakes, mountain building and metamorphism are all controlled by the generation and transfer of heat in the Earth.
The Earth’s thermal budget controls the activity of the lithosphere and asthenosphere and the development of the basic structure of the Earth.

52. The heat arriving from the Sun is by far the greater of the two
Heat arrives at the surface of the Earth from its interior and from the Sun.
The heat arriving from the Sun is by far the greater of the two
Heat from the Sun arriving at the Earth is 2x1017 W
Averaged over the surface this is 4x102 W/m2
The heat from the interior is 4x1013 W and 8x10-2 W/m2
However, most of the heat from the Sun is radiated back into space. It is important because it drives the surface water cycle, rainfall, and hence erosion. The Sun and the biosphere keep the average surface temperature in the range of stability of liquid water.
The heat from the interior of the Earth has governed the geological evolution of the Earth, controlling plate tectonics, igneous activity, metamorphism, the evolution of the core, and hence the Earth’s magnetic field.

53. Heat Transfer Mechanisms
Conduction
Transfer of heat through a material by atomic or molecular interaction within the material
Radiation
Direct transfer of heat as electromagnetic radiation
Convection
Transfer of heat by the movement of the molecules themselves
Advection is a special case of convection

54. Conductive Heat Flow Heat flows from hot things to cold things.
The rate at which heat flows is proportional to the temperature gradient in a material
Large temperature gradient – higher heat flow
Small temperature gradient – lower heat flow

55. Imagine an infinitely wide and long solid plate with thickness δz .
Temperature above is T + δT
Temperature below is T
Heat flowing down is proportional to:
The rate of flow of heat per unit area up through the plate, Q, is:
In the limit as δz goes to zero:

56. Heat flow (or flux) Q is rate of flow of heat per unit area.
The units are watts per meter squared, W m-2
Watt is a unit of power (amount of work done per unit time)
A watt is a joule per second
Old heat flow units, 1 hfu = 10-6 cal cm-2 s-1
1 hfu = 4.2 x 10-2 W m-2
Typical continental surface heat flow is mW m-2
Thermal conductivity k
The units are watts per meter per degree centigrade, W m-1 °C-1
Old thermal conductivity units, cal cm-1 s-1 °C-1
0.006 cal cm-1 s-1°C-1 = 2.52 W m-1 °C-1
Typical conductivity values in W m-1 °C-1 :
Silver 420
Magnesium 160
Glass 1.2
Rock
Wood 0.1

57. Let’s derive a differential equation describing the conductive flow of heat
Consider a small volume element of height δz and area a.
Any change in the temperature of this volume in time δt depends on:
Net flow of heat across the element’s surface (can be in or out or both)
Heat generated in the element
Thermal capacity (specific heat) of the material

58. Expand Q(z+δz) as Taylor series:
The heat per unit time entering the element across its face at z is aQ(z) .
The heat per unit time leaving the element across its face at z+δz is aQ(z+δz) .
Expand Q(z+δz) as Taylor series:
The terms in (δz)2 and above are small and can be neglected
The net change in heat in the element is (heat entering across z) minus (heat leaving across Z+δz):

59. Suppose heat is generated in the volume element at a rate A per unit volume per unit time. The total amount of heat generated per unit time is then
A a δz
Radioactivity is the prime source of heat in rocks, but other possibilities include shear heating, latent heat, and endothermic/exothermic chemical reactions.
Combining this heating with the heating due to changes in heat flow in and out of the element gives us the total gain in heat per unit time (to first order in δz as:
This tells us how the amount of heat in the element changes, but not how much the temperature of the element changes.

60. The specific heat cp of the material in the element determines the temperature increase due to a gain in heat.
Specific heat is defined as the amount of heat required to raise 1 kg of material by 1C.
Specific heat is measured in units of J kg-1 C-1 .
If material has density ρ and specific heat cp, and undergoes a temperature increase of δT in time δt, the rate at which heat is gained is:
We can equate this to the rate at which heat is gained by the element:

61. Simplifies to:
In the limit as δt goes to zero:
Several slides back we defined Q as:
This is the one-dimensional heat conduction equation.

62. The term k/ρcp is known as the thermal diffusivity κ
The term k/ρcp is known as the thermal diffusivity κ. The thermal diffusivity expresses the ability of a material to diffuse heat by conduction.
The heat conduction equation can be generalized to 3 dimensions:
The symbol in the center is the gradient operator squared, aka the Laplacian operator. It is the dot product of the gradient with itself.

63. This simplifies in many special situations.
For a steady-state situation, there is no change in temperature with time. Therefore:
In the absence of heat generation, A=0:
Scientists in many fields recognize this as the classic “diffusion” equation.

64. Talk at board about the qualitative behavior of the Heat Conduction equation

65. Equilibrium Geotherms
The temperature vs. depth profile in the Earth is called the geotherm.
An equilibrium geotherm is a steady state geotherm.
Therefore:

66. Boundary conditions
Since this is a second order differential equation, we should expect to need 2 boundary conditions to obtain a solution.
A possible pair of bc’s is:
T=0 at z=0
Q=Q0 at z=0
Note: Q is being treated as positive upward and z is positive downward in this derivation.

67. Solution Integrate the differential equation once:
Use the second bc to constrain c1
Note: Q is being treated as positive upward and z is positive downward in this derivation.
Substitute for c1:

68. Solution Integrate the differential equation again:
Use the first bc to constrain c2
Substitute for c2:
Link to spreadsheet

69. Oceanic Heat Flow
Heat flow is higher over young oceanic crust
Heat flow is more scattered over young oceanic crust
Oceanic crust is formed by intrusion of basaltic magma from below
The fresh basalt is very permeable and the heat drives water convection
Ocean crust is gradually covered by impermeable sediment and water convection ceases.
Ocean crust ages as it moves away from the spreading center. It cools and it contracts.

70. These data have been empirically modeled in two ways:
d= t (0-70 my)
and
d=6.4 – 3.2e-t/ ( my)

71. Half Space Model
Specified temperature at top boundary.
No bottom boundary condition.
Cooling and subsidence are predicted to follow square root of time.
Plate Model
Specified temperature at top and bottom boundaries.
Cooling and subsidence are predicted to follow an exponential function of time.
Roughly matches Half Space Model for first 70 my.

73. The model of plate cooling with age generally works for continental lithosphere, but is not very useful.
Variations in heat flow in continents is controlled largely by changes in the distribution of heat generating elements and recent tectonic activity.

74. Range of Continental and Oceanic Geotherms in the crust and upper mantle

76. Convection

77. Conductive Geotherm
~10-20 C per km
Adiabatic Geotherm
~ C per km
Convective Geotherm
Adiabatic “middle”
Thermal boundary layer
at top and bottom

78. Illustration of mantle melting during decompression
Solid and liquid in the Earth
Illustration of mantle melting during decompression

79. Rayleigh-Benard Convection
Newtonian viscous fluid – stress is proportional to strain rate
A tank of fluid is heated from below and cooled from above
Initially heat is transported by conduction and there is no lateral variation
Fluid on the bottom warms and becomes less dense
When density difference becomes large enough, lateral variations appear and convection begins
The cells are 2-D cylinders that rotate about their horizontal axes
With more heating, these cells become unstable by themselves and a second, perpendicular set forms
With more heating this planform changes to a vertical hexagonal pattern with hot material rising in the center and cool material descending around the edges
Finally, with extreme heating, the pattern becomes irregular with hot material rising randomly and vigorously.

80. Rayleigh-Benard Convection
The stages of convection have been modeled mathematically and are characterized by a “non-dimensional” number called the Rayleigh number
a is the volume coefficient of thermal expansion
g gravity
d the thickness of the layer
Q heat flow through lower boundary
A, κ, k you know
n is kinematic viscosity
The critical value of Ra for gentle convection is about 103.
The aspect ratio for R-B convection cells is about 2-3 to 1
Ra above 105 will produce vigorous convection
Ra above 106 will produce irregular convection

81. Reynold’s number – indicates whether flow is laminar or turbulent
Ra for both the upper and lower mantle seems to be consistent with vigorous convection
While R-B convection models are very useful, they do not approximate the Earth very well. The biggest problem is that they model “uniform viscosity” materials. The mantle is not uniform viscosity!
Reynold’s number – indicates whether flow is laminar or turbulent
All mantle convection in the Earth is predicted to be laminar
Mantle convection movies from Caltech
More mantle convection movies
More

82. Two-layer vs. Whole Mantle Convection
Studies like you did in lab, seemed to show that subduction stopped at about 670 km depth. This was interpreted to mean there was mantle convection operating in the upper mantle that was separate from convection in the lower mantle.
Two-layer vs. Whole Mantle Convection
Modern tomographic images give a very different picture!

83. Illustration of slab pull and ridge push
Plate Driving Forces
Illustration of slab pull and ridge push

84. Plate boundaries are marked in several ways:

85. Names of the plates:
(Lowrie, 1997) Arrows indicate relative velocities (mm/yr) from
NUVEL-1 model of DeMets et al., 1990

86. Types of plate boundaries:

87. Assumptions of Plate Tectonics
The generation of new plate material occurs by seafloor spreading; that is, new oceanic lithosphere is generated along the active mid-ocean ridges.
The new oceanic lithosphere, once created, forms part of a rigid plate; this plate may or may not include continental material.
The Earth’s surface area remains constant; therefore, seafloor spreading must be balanced by consumption of plate elsewhere.
The lithospheric plates are capable of transmitting stresses over great horizontal distances without buckling; in other words, the relative motion between plates is taken up only along plate boundaries.

88. The traditional way is using marine magnetic anomalies:
Plate motions can be determined in several ways.
The traditional way is using marine magnetic anomalies:

89. Magnetic anomalies allow the identification of
isochrons in the worlds oceans.

90. Very Long Baseline Interferometry (VLBI)
VLBI measures the time difference between the arrival at the Earth of a radio signals emitted by quasars. The time difference between arrivals at two satellites is proportional to the distance between the two satellites and the direction of the source. These satellites may be separated by some 10,000 km. Using large numbers of time difference measurements from many quasars observed with a global network of antennas, VLBI determines the inertial reference frame defined by the quasars and simultaneously the precise positions of the antennas. Because the time difference measurements are precise to a few picoseconds, VLBI determines the relative positions of the antennas to a few millimeters and the quasar positions to fractions of a milliarcsecond. Since the antennas are fixed to the Earth, their locations track the instantaneous orientation of the Earth in the inertial reference frame.

91. Satellite and Lunar Laser ranging (SLR & LLR)
SLR targets are satellites equipped with corner cubes or retro-reflectors. Currently, the global SLR network tracks over forty such satellites. The observable is the round-trip pulse time-of-flight to the satellite.
SLR systems are equipped with short-pulse laser transmitters that can range to orbiting satellites.
Lunar Laser Ranging (LLR) systems can range to retro-reflectors located on the moon.

92. Global positioning system (GPS)
There are  24 GPS satellites currently in circular orbits some  20,200 kilometers above the Earth.  At any one time in most places six  can be "seen" by GPS receivers that  get and process signals.  
GPS receivers  calculate current position (latitude, longitude, altitude) with varying degrees of precision.   There are many thousand permanent GPS receivers located world wide.  These provide data for modeling plate motions on yearly time scales.