Earth as a PlanetEarth as a Planet

Mass M = 6 x 10Mass M = 6 x 102727 g g.. Radius R = 6371 kmRadius R = 6371 km.. Mean density = M/(4/3 Mean density = M/(4/3 RR33) = 5.5 g/cm) = 5.5 g/cm33

Moment of inertia I of Moment of inertia I of the Earth: the Earth: • I = I = r r22 dm dm

• I/(MRI/(MR22) = 0.331. ) = 0.331.

• for a uniform sphere for a uniform sphere I/(MRI/(MR22) = 0.4.) = 0.4.

Differentiation early in Earth’s history

Interior of the EarthInterior of the Earth

CrustCrust: : variable thickness with an average value of 35 variable thickness with an average value of 35 km in the continents and 7-8 km in the oceanic km in the continents and 7-8 km in the oceanic regions. Volume ~10regions. Volume ~101919 m m33 Mass 2.8 x 10 Mass 2.8 x 1022 22 kg.kg.

MantleMantle: between the Moho discontinuity (crust-: between the Moho discontinuity (crust-mantle) and the core-mantle boundary (R = 3480 km). mantle) and the core-mantle boundary (R = 3480 km). Volume 9 x 10Volume 9 x 102020 m m33 Mass 4 x 10 Mass 4 x 1024 24 kg.kg.

CoreCore: from the center of the Earth to the core-mantle : from the center of the Earth to the core-mantle boundary. Volume 1.77 x 10boundary. Volume 1.77 x 102020 m m33 Mass 1.94 x 10 Mass 1.94 x 1024 24 kg.kg.

More Details about Layering…More Details about Layering…

What is Earth made of?What is Earth made of?

Why do we need to look outside the Earth Why do we need to look outside the Earth to learn about what is inside the Earth?to learn about what is inside the Earth?• We know the Earth is layered and that what we We know the Earth is layered and that what we

can sample on the outside is not typical.can sample on the outside is not typical.• By studying members of the Solar System, it is By studying members of the Solar System, it is

possible to estimate its original composition possible to estimate its original composition and the physical and chemical processes that and the physical and chemical processes that have led to its present state.have led to its present state.

The Solar System: A highly diverse zoo!

Overview of the Solar System

SunSun MercuryMercury VenusVenus EarthEarth

• MoonMoon MarsMars

AsteroidsAsteroids JupiterJupiter SaturnSaturn UranusUranus NeptuneNeptune PlutoPluto

The Origin of the Solar System The Origin of the Solar System Frank Crary, CU BoulderFrank Crary, CU Boulder

A cloud of interstellar gas and/or dust (the "solar nebula") is disturbed and 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 collapses under its own gravity. The disturbance could be, for example, the shock wave from a nearby supernova. shock wave from a nearby supernova.

As the cloud collapses, it heats up and compresses in the center. It heats 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 enough for the dust to vaporize. The initial collapse is supposed to take less than 100,000 years. than 100,000 years.

The center compresses enough to become a protostar and the rest of the gas 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 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 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 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 an "accretion disk" around the star. The disk radiates away its energy and cools off. cools off.

Here is a brief outline of the current theory of the Here is a brief outline of the current theory of the events in the early history of the solar system: events in the early history of the solar system:

The Origin of the Solar System The Origin of the Solar System Frank Crary, CU BoulderFrank Crary, CU Boulder

First break point. Depending on the details, the gas orbiting First break point. Depending on the details, the gas orbiting star/protostar may be unstable and start to compress under its star/protostar may be unstable and start to compress under its own gravity. That produces a double star. If it doesn't ... own gravity. That produces a double star. If it doesn't ...

The gas cools off enough for the metal, rock and (far enough The gas cools off enough for the metal, rock and (far enough from the forming star) ice to condense out into tiny particles. from the forming star) ice to condense out into tiny particles. (i.e. some of the gas turns back into dust). The metals (i.e. some of the gas turns back into dust). The metals condense almost as soon as the accretion disk forms (4.55-condense almost as soon as the accretion disk forms (4.55-4.56 billion years ago according to isotope measurements of 4.56 billion years ago according to isotope measurements of certain meteors); the rock condenses a bit later (between 4.4 certain meteors); the rock condenses a bit later (between 4.4 and 4.55 billion years ago). and 4.55 billion years ago).

The dust particles collide with each other and form into larger The dust particles collide with each other and form into larger particles. This goes on until the particles get to the size of particles. This goes on until the particles get to the size of boulders or small boulders or small asteroidsasteroids. .

The Origin of the Solar System The Origin of the Solar System Frank Crary, CU BoulderFrank Crary, CU Boulder

Run away growth. Once the larger of these particles get big enough to have 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 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 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 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 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 distance from the star and the density and composition of the protoplanetary nebula. In the solar system, the theories say that this is large protoplanetary nebula. In the solar system, the theories say that this is large asteroid to asteroid to lunarlunar size in the inner solar system, and one to fifteen times the size in the inner solar system, and one to fifteen times the EarthEarth's size in the outer solar system. There would have been a big jump in 's size in the outer solar system. There would have been a big jump in size somewhere between the current orbits of size somewhere between the current orbits of MarsMars and and JupiterJupiter: the energy : the energy from the from the SunSun would have kept ice a vapor at closer distances, so the solid, would have kept ice a vapor at closer distances, so the solid, accretable matter would become much more common beyond a critical accretable matter would become much more common beyond a critical distance from the Sun. The accretion of these "planetesimals" is believed to distance from the Sun. The accretion of these "planetesimals" is believed to take a few hundred thousand to about twenty million years, with the take a few hundred thousand to about twenty million years, with the outermost taking the longest to form. outermost taking the longest to form.

The Origin of the Solar System The Origin of the Solar System Frank Crary, CU BoulderFrank Crary, CU Boulder

Two things and the second break point. How big were those Two things and the second break point. How big were those protoplanets and how quickly did they form? At about this protoplanets and how quickly did they form? At about this time, about 1 million years after the nebula cooled, the star time, about 1 million years after the nebula cooled, the star would generate a very strong solar wind, which would sweep would generate a very strong solar wind, which would sweep away all of the gas left in the protoplanetary nebula. If a away all of the gas left in the protoplanetary nebula. If a protoplanet was large enough, soon enough, its gravity would protoplanet was large enough, soon enough, its gravity would pull in the nebular gas, and it would become a gas giant. If not, pull in the nebular gas, and it would become a gas giant. If not, it would remain a rocky or icy body. it would remain a rocky or icy body.

At this point, the solar system is composed only of solid, At this point, the solar system is composed only of solid, protoplanetary bodies and gas giants. The "planetesimals" protoplanetary bodies and gas giants. The "planetesimals" would slowly collide with each other and become more would slowly collide with each other and become more massive. massive.

The Origin of the Solar System The Origin of the Solar System Frank Crary, CU BoulderFrank Crary, CU Boulder

Eventually, after ten to a hundred million years, you end up Eventually, after ten to a hundred million years, you end up with ten or so planets, in stable orbits, and that's a solar with ten or so planets, in stable orbits, and that's a solar system. These planets and their surfaces may be heavily system. These planets and their surfaces may be heavily modified by the last, big collision they experience (e.g. the modified by the last, big collision they experience (e.g. the largely metal composition of Mercury or the largely metal composition of Mercury or the MoonMoon). ).

Note:Note: this was the theory of planetary formation as it stood this was the theory of planetary formation as it stood before the discovery of before the discovery of extrasolarextrasolar planets planets. The discoveries . The discoveries don't match what the theory predicted. That could be an don't match what the theory predicted. That could be an observational bias (odd solar systems may be easier to detect observational bias (odd solar systems may be easier to detect from Earth) or problems with the theory (probably with subtle from Earth) or problems with the theory (probably with subtle points, not the basic outline.) points, not the basic outline.)

The Big QuestionsThe Big Questions What is the What is the originorigin of the solar system? It is generally agreed that of the solar system? It is generally agreed that

it condensed from a nebula of dust and gas. But the details are far it condensed from a nebula of dust and gas. But the details are far from clear. from clear.

How common are planetary systems around How common are planetary systems around other starsother stars? There is ? There is now good evidence of Jupiter-sized objects orbiting several nearby now good evidence of Jupiter-sized objects orbiting several nearby stars. What conditions allow the formation of terrestrial planets? It stars. What conditions allow the formation of terrestrial planets? It seems unlikely that the Earth is totally unique but we still have no seems unlikely that the Earth is totally unique but we still have no direct evidence one way or the other. direct evidence one way or the other.

Is there Is there lifelife elsewhere in the solar system? If not, why is Earth elsewhere in the solar system? If not, why is Earth special? special?

Is there life beyond the solar system? Is there life beyond the solar system? IntelligentIntelligent life? life?

Is life a rare and unusual or even unique event in the evolution of Is life a rare and unusual or even unique event in the evolution of the universe or is it adaptable, widespread and common? the universe or is it adaptable, widespread and common?

Solar abundance of elementsSolar abundance of elements

Determined from spectral absorption Determined from spectral absorption lineslines• Light from visible surface of sun passing Light from visible surface of sun passing

through cooler gases above the surfacethrough cooler gases above the surface This is thought to represent total This is thought to represent total

solar abundance because nuclear solar abundance because nuclear reactions powering the star take reactions powering the star take place deep inside and there is little place deep inside and there is little convection there to mix modified convection there to mix modified material up with original material.material up with original material.

MeteoritesMeteorites

Meteorites: SummaryMeteorites: Summary The fabric of chondrites is quite unlike that of any terrestrial The fabric of chondrites is quite unlike that of any terrestrial

rock and required very different conditions in which to form. rock and required very different conditions in which to form. These are identified with early stages in the development of These are identified with early stages in the development of the Solar nebula.the Solar nebula.

The carbonaceous chondrites are a close approximation to The carbonaceous chondrites are a close approximation to the material of the Solar Nebula, having lost only the most the material of the Solar Nebula, having lost only the most volatile elements. It is, therefore, plausible to regard them volatile elements. It is, therefore, plausible to regard them as a starting point from which the composition of the Earth as a starting point from which the composition of the Earth has evolved. This leads to the Chondritic Earth Model.has evolved. This leads to the Chondritic Earth Model.

The meteorites derive from the asteroids by collision.The meteorites derive from the asteroids by collision. The differentiated meteorites were formed within minor The differentiated meteorites were formed within minor

planets, or asteroids, which heated sufficiently to segregate planets, or asteroids, which heated sufficiently to segregate into layers, forming an iron core, silicate mantle and into layers, forming an iron core, silicate mantle and transitional region between. Subsequent break-up due to transitional region between. Subsequent break-up due to collisions produced iron, achondrite, and stony-iron collisions produced iron, achondrite, and stony-iron meteorites.meteorites.

Next Question:

What is the origin of the distribution of elements in the

Solar System?

Hydrogen, the simplest element, is the basic building

block.

•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.

•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.

•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.

•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.

•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!

•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.

•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.

Differentiation of the EarthDifferentiation of the Earth Differentiation is the process by Differentiation is the process by

which random chunks of primordial which random chunks of primordial matter were transformed into a body matter were transformed into a body whose interior is divided into whose interior is divided into concentric layers that differ from one concentric layers that differ from one another both physically and another both physically and chemically.chemically.

This occurred early in Earth’s history, This occurred early in Earth’s history, when the planet got hot enough to when the planet got hot enough to melt.melt.

What was the starting point for What was the starting point for differentiation?differentiation?

Heterogeneous/Hot starting modelHeterogeneous/Hot starting model• Initial layering as Earth solidified from gasInitial layering as Earth solidified from gas

Homogeneous/Cold starting modelHomogeneous/Cold starting model• Little or no initial layering because Earth Little or no initial layering because Earth

formed from the agglutination of cold, uniform formed from the agglutination of cold, uniform particlesparticles

Neither model seems to work completelyNeither model seems to work completely

When did differentiation happen?When did differentiation happen?

About 4.5 billion years agoAbout 4.5 billion years ago After beginning of Earth’s accretion After beginning of Earth’s accretion

at 4.56 billion years agoat 4.56 billion years ago Before the formation of the Moon’s Before the formation of the Moon’s

oldest known rocks, 4.47 billion years oldest known rocks, 4.47 billion years agoago

Sources of heat to melt EarthSources of heat to melt Earth

Frequent and violent impactsFrequent and violent impacts• There was likely one particularly large There was likely one particularly large

impactimpact Moon aggregated from the ejected debrisMoon aggregated from the ejected debris Earth’s spin axis was tiltedEarth’s spin axis was tilted

Decay of radioactive elementsDecay of radioactive elements• This heat generation was greater in the This heat generation was greater in the

past than todaypast than today

Basic processes of differentiationBasic processes of differentiation

In a liquid or soft solid sphere, denser In a liquid or soft solid sphere, denser material sinks to the center and less dense material sinks to the center and less dense material floats to the top.material floats to the top.

When rock is partially melted, the melt When rock is partially melted, the melt and the remaining solid generally have and the remaining solid generally have different chemistry and density. The melt different chemistry and density. The melt is usually less dense than the “residue.” is usually less dense than the “residue.” The melt is enriched in “incompatible” The melt is enriched in “incompatible” elements. The residue is enriched in elements. The residue is enriched in “compatible” elements.“compatible” elements.

Earth’s CoreEarth’s Core Iron, nickel, and other heavy elements Iron, nickel, and other heavy elements

were the densest material and formed the were the densest material and formed the core. Core radius is 2900 km.core. Core radius is 2900 km.

They are about 1/3 of the planet’s massThey are about 1/3 of the planet’s mass Inner core is solid. Inner core radius=1200 Inner core is solid. Inner core radius=1200

km. Inner core is solid because pressure is km. Inner core is solid because pressure is too great for iron to melt at Earth’s current too great for iron to melt at Earth’s current temperature.temperature.

Outer core is liquid. Some of the iron in the Outer core is liquid. Some of the iron in the outer core is iron sulfide.outer core is iron sulfide.

The Iron, Oxygen, Sulfur, The Iron, Oxygen, Sulfur, Magnesium, and Silicon Magnesium, and Silicon

storystory

There were large amounts of these five elements in the early EarthThere 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 The fate of the iron was controlled by its affinity for bonding with oxygen

and sulfur.and sulfur. Iron bonds preferentially with sulfur. All available sulfur is consumed. Iron Iron bonds preferentially with sulfur. All available sulfur is consumed. Iron

remains.remains. Oxygen bonds preferentially with magnesium and silicon. This uses up the Oxygen bonds preferentially with magnesium and silicon. This uses up the

magnesium and silicon. Oxygen remains.magnesium and silicon. Oxygen remains. Iron then combines with oxygen. Oxygen is now used up. Iron remains as Iron then combines with oxygen. Oxygen is now used up. Iron remains as

elemental iron.elemental iron. The iron, magnesium, and silicon oxides are light and form the Earth’s The iron, magnesium, and silicon oxides are light and form the Earth’s

crust and mantle.crust and mantle. The iron sufide is dense, but less dense than iron, so it forms the outer The iron sufide is dense, but less dense than iron, so it forms the outer

part of the core of Earth.part of the core of Earth. The elemental iron is densest of all, so it forms the inner core of the 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 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 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?to the core radius? What does adding sulfur do to the core radius?

Earth’s CrustEarth’s Crust

Lighter rocks floated to the surface of the magma Lighter rocks floated to the surface of the magma ocean.ocean.

The crust is formed of light materials with low The crust is formed of light materials with low melting temperature and is up to 40 km thick.melting temperature and is up to 40 km thick.

These are generally compounds of silicon, These are generally compounds of silicon, aluminum, iron, calcium, magnesium, sodium, aluminum, iron, calcium, magnesium, sodium, and potassium, mixed with oxygen.and potassium, mixed with oxygen.

Fragments of crustal rocks (zircons) of age 4.3-Fragments of crustal rocks (zircons) of age 4.3-4.4 billion years were found recently in western 4.4 billion years were found recently in western Australia. If this is confirmed, we can conclude Australia. If this is confirmed, we can conclude that Earth cooled enough for a solid crust to form that Earth cooled enough for a solid crust to form only 100 million years after the large impact.only 100 million years after the large impact.

Earth’s MantleEarth’s Mantle

Lies between the crust and the core.Lies between the crust and the core. Depth range is 40 km to 2900 km.Depth range is 40 km to 2900 km. The mantle consists of rocks of The mantle consists of rocks of

intermediate density, mostly intermediate density, mostly compounds of oxygen with compounds of oxygen with magnesium, iron, and siliconmagnesium, iron, and silicon

New continental crust may be New continental crust may be produced during partial melting of produced during partial melting of mantle material.mantle material.

Radiometric Dating: General TheoryRadiometric Dating: General Theory

The radioactive decay of any radioactive The radioactive decay of any radioactive atom is an entirely random event, atom is an entirely random event, independent of neighboring atoms, independent of neighboring atoms, physical conditions, and the chemical physical conditions, and the chemical state of the atom.state of the atom.

It depends only on the structure of the It depends only on the structure of the nucleus.nucleus.

λλ, the decay constant, is the probability of , the decay constant, is the probability of an atom decaying in unit time. It is an atom decaying in unit time. It is different for each isotope.different for each isotope.

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

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)

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.

http://www.chemguide.co.uk/analysis/masspec/howitworks.html

Major radioactive elements used in radiometric datingMajor radioactive elements used in radiometric dating

Parent Parent IsotopeIsotope

Daughter Daughter IsotopeIsotope

Half Life of Half Life of Parent (years)Parent (years)

Effective dating Effective dating range (years)range (years)

Materials that Materials that can be datedcan be dated

238238UU 206206PbPb 4.5 billion4.5 billion 10 million – 10 million – 4.6 billion4.6 billion

ZirconZircon

ApatiteApatite

235235UU 207207PbPb 0.7 billion0.7 billion 10 million – 10 million – 4.6 billion4.6 billion

ZirconZircon

ApatiteApatite

4040KK 4040AA 1.3 billion1.3 billion 50,000 – 4.6 50,000 – 4.6 billionbillion

MuscoviteMuscovite

BiotiteBiotite

HornblendeHornblende

8787RbRb 8787SrSr 47 billion47 billion 10 million – 10 million – 4.6 billion4.6 billion

MuscoviteMuscovite

BiotiteBiotite

Potassium Potassium FeldsparFeldspar

1414CC 1414NN 57305730 100 - 70,000100 - 70,000

Wood, charcoal, peat, Wood, charcoal, peat, bone and tissue, shell bone and tissue, shell

and other calcium and other calcium carbonate, carbonate,

groundwater, ocean groundwater, ocean water, and glacier ice water, and glacier ice containing dissolved containing dissolved

COCO22

Radiometric dating is not always that simple!Radiometric dating is not always that simple!

There may have been an initial concentration of the daughter in the There may have been an initial concentration of the daughter in the samplesample

Not all systems are closed. There may have been exchange of parent Not all systems are closed. There may have been exchange of parent and/or daughter with surrounding material.and/or daughter with surrounding material.

If dates from different isotope systems match within analytical error, 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 we say the ages are concordant. If they are not, then we say they are discordant.discordant.

When discordant, we suspect problems like those above with one or When discordant, we suspect problems like those above with one or all of the systems.all of the systems.

The date The date tt obtained is not always the date of formation of the rock. It 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 may be the date the rock crystallized, or the date of a metamorphic event which heated the rock to the degree that chemical changes event which heated the rock to the degree that chemical changes took place.took place.

Radioactive decay schemes are not all as simple as a parent and Radioactive decay schemes are not all as simple as a parent and exactly one daughter. exactly one daughter. 8787Rb to Rb to 8787Sr is a simple one step decay. The two Sr is a simple one step decay. The two U to Pb series have a number of intermediate daughter products.U to Pb series have a number of intermediate daughter products.

Fission Track DatingFission Track Dating As well as decaying to As well as decaying to 206206Pb as described before, Pb as described before,

238238U is also subject to spontaneous fission.U is also subject to spontaneous fission.

It disintegrates into two large pieces and several It disintegrates into two large pieces and several neutrons. This is a very rare event, occurring just neutrons. This is a very rare event, occurring just once per 2 million once per 2 million αα decays. decays.

Each event is recorded as a trail of destruction Each event is recorded as a trail of destruction about 10 about 10 m long through the mineral structure.m long through the mineral structure.

These “fission tracks” can be observed by etching These “fission tracks” can be observed by etching the polished surface of certain minerals. The the polished surface of certain minerals. The tracks become visible under a microscope.tracks become visible under a microscope.

Spontaneous

Fission Tracks

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.

)1(][238 tnowr eUD

)1(][238 tnow

ss eUD

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.

nUD nowI ][235

I

s

I

st

now

nows

N

N

D

D

n

e

U

U

)1(

][

][235

238

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.

88.1371log

1 n

N

Nt

sI

se

Spontaneous

Fission Tracks

Induced

Fission Tracks

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.

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.

http://www.geotrack.com.au/ttinterp.htm

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).

Heat in the EarthHeat in the Earth

Volcanoes, magmatic intrusions, Volcanoes, magmatic intrusions, earthquakes, mountain building and earthquakes, mountain building and metamorphism are all controlled by metamorphism are all controlled by the generation and transfer of heat the generation and transfer of heat in the Earth.in the Earth.

The Earth’s thermal budget controls The Earth’s thermal budget controls the activity of the lithosphere and the activity of the lithosphere and asthenosphere and the development asthenosphere and the development of the basic structure of the Earth.of the basic structure of the Earth.

Heat arrives at the surface of the Earth from its interior and Heat arrives at the surface of the Earth from its interior and from the Sun.from the Sun.

The heat arriving from the Sun is by far the greater of the twoThe heat arriving from the Sun is by far the greater of the two

Heat from the Sun arriving at the Earth is 2x10Heat from the Sun arriving at the Earth is 2x101717 W W

Averaged over the surface this is 4x10Averaged over the surface this is 4x1022 W/m W/m22

The heat from the interior is 4x10The heat from the interior is 4x101313 W and 8x10 W and 8x10-2-2 W/m W/m22

However, most of the heat from the Sun is radiated back into However, most of the heat from the Sun is radiated back into space. It is important because it drives the surface water space. It is important because it drives the surface water cycle, rainfall, and hence erosion. The Sun and the biosphere cycle, rainfall, and hence erosion. The Sun and the biosphere keep the average surface temperature in the range of keep the average surface temperature in the range of stability of liquid water.stability of liquid water.

The heat from the interior of the Earth has governed the The heat from the interior of the Earth has governed the geological evolution of the Earth, controlling plate tectonics, geological evolution of the Earth, controlling plate tectonics, igneous activity, metamorphism, the evolution of the core, igneous activity, metamorphism, the evolution of the core, and hence the Earth’s magnetic field.and hence the Earth’s magnetic field.

Heat Transfer MechanismsHeat Transfer Mechanisms ConductionConduction

Transfer of heat through a material by atomic or Transfer of heat through a material by atomic or molecular interaction within the materialmolecular interaction within the material

RadiationRadiation Direct transfer of heat as electromagnetic radiationDirect transfer of heat as electromagnetic radiation

ConvectionConvection Transfer of heat by the movement of the molecules Transfer of heat by the movement of the molecules

themselvesthemselves

Advection is a special case of convectionAdvection is a special case of convection

Conductive Heat FlowConductive Heat Flow Heat flows from hot things to cold Heat flows from hot things to cold

things.things. The rate at which heat flows is The rate at which heat flows is

proportional to the temperature proportional to the temperature gradient in a materialgradient in a material Large temperature gradient – higher Large temperature gradient – higher

heat flowheat flow Small temperature gradient – lower heat Small temperature gradient – lower heat

flowflow

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:

z

TTT

)(

z

TkzQ

z

TTTkQ

)(

z

TkzQ

)(

In the limit as δz goes to zero:

Heat flow (or flux) Q is rate of flow of heat per unit Heat flow (or flux) Q is rate of flow of heat per unit area.area. The units are watts per meter squared, W mThe units are watts per meter squared, W m-2-2

Watt is a unit of power (amount of work done per unit Watt is a unit of power (amount of work done per unit time)time)

A watt is a joule per secondA watt is a joule per second Old heat flow units, 1 hfu = 10Old heat flow units, 1 hfu = 10-6-6 cal cm cal cm-2-2 s s-1-1

1 hfu = 4.2 x 101 hfu = 4.2 x 10-2-2 W m W m-2-2

Typical continental surface heat flow is 40-80 mW mTypical continental surface heat flow is 40-80 mW m-2-2

Thermal conductivity k Thermal conductivity k The units are watts per meter per degree centigrade, W mThe units are watts per meter per degree centigrade, W m--

1 °1 °CC-1-1

Old thermal conductivity units, cal cmOld thermal conductivity units, cal cm-1 -1 ss-1 °-1 °CC-1-1

0.006 cal cm0.006 cal cm-1 -1 ss-1°-1°CC-1 -1 = 2.52= 2.52 W mW m-1 °-1 °CC-1-1

Typical conductivity values in W mTypical conductivity values in W m-1 °-1 °CC-1-1 : : SilverSilver 420420 MagnesiumMagnesium 160160 Glass Glass 1.21.2 RockRock 1.7-3.31.7-3.3 WoodWood 0.10.1

Consider a small volume element of height δz and area a.

Any change in the temperature of this volume in time δt depends on:

1. Net flow of heat across the element’s surface (can be in or out or both)

2. Heat generated in the element

3. Thermal capacity (specific heat) of the material

Let’s derive a differential equation describing the conductive flow of heat

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

...

!3!2)()(

3

33

2

22

z

Qz

z

Qz

z

QzzQzzQ

The net change in heat in the element is (heat entering across z) minus (heat leaving across Z+δz):

z

Qza

zzaQzaQ

)()(

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.

z

QzazAa

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 1C.

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:

t

Tzacp

z

QzazAa

t

Tzacp

z

QzazAa

t

Tzacp

z

QA

t

Tcp

z

QA

t

Tcp

In the limit as δt goes to zero:

z

TkzQ

)(

Several slides back we defined Q as:

2

2

z

TkA

t

Tcp

pp c

A

z

T

c

k

t

T

2

2

This is the one-dimensional heat conduction equation.

Simplifies to:

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:

pp c

A

z

T

y

T

x

T

c

k

t

T

2

2

2

2

2

2

pp c

AT

c

k

t

T

2

The symbol in the center is the gradient operator squared, aka the Laplacian operator. It is the dot product of the gradient with itself.

z

T

y

T

x

TT

zyx

,,

,,

2

2

2

2

2

22

2

2

2

2

2

22

z

T

y

T

x

TT

zyx

pp c

AT

c

k

t

T

2

This simplifies in many special situations.

For a steady-state situation, there is no change in temperature with time. Therefore:

k

AT 2

In the absence of heat generation, A=0:

Tc

k

t

T

p

2

Scientists in many fields recognize this as the classic “diffusion” equation.

Talk at board about the Talk at board about the qualitative behavior of the qualitative behavior of the Heat Conduction equationHeat Conduction equation

Equilibrium GeothermsEquilibrium Geotherms

The temperature vs. depth profile in The temperature vs. depth profile in the Earth is called the geotherm.the Earth is called the geotherm.

An equilibrium geotherm is a steady An equilibrium geotherm is a steady state geotherm.state geotherm.

Therefore:Therefore: 2

20,

T T Aand

t z k

Boundary conditionsBoundary conditions

Since this is a second order Since this is a second order differential equation, we should differential equation, we should expect to need 2 boundary expect to need 2 boundary conditions to obtain a solution.conditions to obtain a solution.

A possible pair of bc’s is:A possible pair of bc’s is: T=0 at z=0T=0 at z=0 Q=QQ=Q00 at z=0 at z=0 Note: Q is being treated as positive upward and z is positive downward in this Note: Q is being treated as positive upward and z is positive downward in this

derivation.derivation.

SolutionSolution

Integrate the Integrate the differential differential equation once:equation once:

Use the second bc Use the second bc to constrain cto constrain c11 Note: Q is being treated as positive Note: Q is being treated as positive

upward and z is positive downward in upward and z is positive downward in this derivation.this derivation.

Substitute for cSubstitute for c11::

1

T Azc

z k

01

Qc

k

0QT Az

z k k

SolutionSolution

Integrate the Integrate the differential equation differential equation again:again:

Use the first bc to Use the first bc to constrain cconstrain c22

Substitute for cSubstitute for c22::

Link to spreadsheetLink to spreadsheet

20

22

Q zAzT c

k k

2 0c

20

2

Q zAzT

k k

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.

These data have been empirically modeled in two ways:

d=2.5 + 0.35t2 (0-70 my)

and

d=6.4 – 3.2e-t/62.8 (35-200 my)

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.

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.

Range of Continental and Oceanic Geotherms in the crust

and upper mantle

Convection

Conductive Geotherm

~10-20 C per km

Adiabatic Geotherm

~0.5-1.0 C per km

Convective Geotherm

Adiabatic “middle”

Thermal boundary layer

at top and bottom

Illustration of mantle melting

during decompression

Solid and liquid in the Earth

Rayleigh-Benard ConvectionRayleigh-Benard Convection Newtonian viscous fluid – stress is proportional to strain Newtonian viscous fluid – stress is proportional to strain

raterate A tank of fluid is heated from below and cooled from aboveA tank of fluid is heated from below and cooled from above

Initially heat is transported by conduction and there is no Initially heat is transported by conduction and there is no lateral variationlateral variation

Fluid on the bottom warms and becomes less denseFluid on the bottom warms and becomes less dense When density difference becomes large enough, lateral When density difference becomes large enough, lateral

variations appear and convection beginsvariations appear and convection begins The cells are 2-D cylinders that rotate about their horizontal The cells are 2-D cylinders that rotate about their horizontal

axesaxes With more heating, these cells become unstable by themselves With more heating, these cells become unstable by themselves

and a second, perpendicular set formsand a second, perpendicular set forms With more heating this planform changes to a vertical With more heating this planform changes to a vertical

hexagonal pattern with hot material rising in the center and hexagonal pattern with hot material rising in the center and cool material descending around the edgescool material descending around the edges

Finally, with extreme heating, the pattern becomes irregular Finally, with extreme heating, the pattern becomes irregular with hot material rising randomly and vigorously.with hot material rising randomly and vigorously.

Rayleigh-Benard ConvectionRayleigh-Benard Convection The stages of convection The stages of convection

have been modeled have been modeled mathematically and are mathematically and are characterized by a “non-characterized by a “non-dimensional” number dimensional” number called the Rayleigh numbercalled the Rayleigh number

isis the volume coefficient the volume coefficient of thermal expansionof thermal expansion

g gravityg gravity d the thickness of the layerd the thickness of the layer Q heat flow through lower Q heat flow through lower

boundaryboundary A, A, κκ, k you know, k you know is kinematic viscosityis kinematic viscosity

4gd Q AdRa

k

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

Ra for both the upper and lower mantle seems to Ra for both the upper and lower mantle seems to be consistent with vigorous convectionbe consistent with vigorous convection

While R-B convection models are very useful, they While R-B convection models are very useful, they do not approximate the Earth very well. The do not approximate the Earth very well. The biggest problem is that they model “uniform biggest problem is that they model “uniform viscosity” materials. The mantle is not uniform viscosity” materials. The mantle is not uniform viscosity!viscosity!

Reynold’s number – indicates whether flow is Reynold’s number – indicates whether flow is laminar or turbulentlaminar or turbulent All mantle convection in the Earth is predicted to be All mantle convection in the Earth is predicted to be

laminarlaminar

Mantle convection movies from CaltechMantle convection movies from Caltech More mantle convection moviesMore mantle convection movies MoreMore MoreMore

Modern tomographic images give a very different picture!

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

Illustration of slab pull and ridge push

Plate Driving Forces

Plate boundaries are marked in several ways:

Names of the plates:

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

Types of plate boundaries:

Assumptions of Plate TectonicsAssumptions of Plate Tectonics The generation of new plate material occurs by The generation of new plate material occurs by

seafloor spreading; that is, new oceanic lithosphere seafloor spreading; that is, new oceanic lithosphere is generated along the active mid-ocean ridges.is generated along the active mid-ocean ridges.

The new oceanic lithosphere, once created, forms The new oceanic lithosphere, once created, forms part of a rigid plate; this plate may or may not part of a rigid plate; this plate may or may not include continental material.include continental material.

The Earth’s surface area remains constant; The Earth’s surface area remains constant; therefore, seafloor spreading must be balanced by therefore, seafloor spreading must be balanced by consumption of plate elsewhere.consumption of plate elsewhere.

The lithospheric plates are capable of transmitting The lithospheric plates are capable of transmitting stresses over great horizontal distances without stresses over great horizontal distances without buckling; in other words, the relative motion buckling; in other words, the relative motion between plates is taken up only along plate between plates is taken up only along plate boundaries.boundaries.

Plate motions can be determined in several ways.

The traditional way is using marine magnetic anomalies:

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

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.

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.

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.