LECTURE 14: EARTH: AGE, COMPOSITION AND STRUCTURE

• How do we estimate the age of the Earth? What is it?
• What does the density of the Earth tell us about its composition?
• What is the structure of the Earth? How do we infer it from seismology and from the Earth's magnetic field?
• What keeps the center of the Earth hot?

EARTH: BASIC NUMBERS

Orbital Properties:

• Semi-major axis: 1.000 Astronomical Units (AU) = 150 million km
• Period: 1 year
• Eccentricity: 0.017

• Diameter: 12,800 km
• Mass: 6 x 10²⁴ kg
• Rotation period: 1 day

ISOTOPES AND RADIOACTIVITY

Isotopes:

• Consider two atomic nuclei with same number of protons, different number of neutrons.
• Same chemical properties, but different mass.
• Called isotopes.
• Example: carbon-12 (6 protons + 6 neutrons), carbon-13 (6 protons + 7 neutrons), carbon-14 (6 protons + 8 neutrons).

• Isotopes with too many neutrons are unstable, and "decay," emitting particles and changing into other kinds of atoms.
• Two common processes:
  • Neutron emits an electron and converts to a proton. Proton number goes up by one.
  • Nucleus emits an "alpha-particle", 2 neutrons + 2 protons. Proton number goes down by two.
• Decaying nuclei often emit gamma-rays and other particles called neutrinos, in addition to electrons and alpha-particles.
• These particles carry energy and can heat their surroundings.
• Gamma-rays are responsible for the most damaging physical effects of radioactivity.

RADIOACTIVE HALF-LIFE

• Radioactive decay is a random process.
• Characterize speed by half-life, t_1/2, average time for half the atoms to decay.
• After t_1/2, 1/2 of original atoms are left, other 1/2 have decayed into new kind of atom.
• After 2 t_1/2, 1/4 of original atoms are left. After 3 t_1/2, 1/8 are left.
• The more unstable the nucleus, the shorter the half-life.
• Examples:
  • hydrogen-3 to helium-3: half-life 12.26 years
  • carbon-14 to nitrogen-14: half-life 5730 years
• For dating rocks on Earth, usually want isotopes with half-lives of billions of years.

RADIOACTIVE AGE DATING

Uranium-Lead method:

• Uranium-238 has 92 protons, 146 neutrons.
• Unstable, decays to Lead-206 (via several intermediate products).
• Half-life of 4.5 billion years, measured in laboratory.
• Suppose we find a rock that we know had no Lead-206 when it formed.
• Today we find equal numbers of Uranium-238 and Lead-206 atoms in the rock.
• When did the rock form?

• Assumption that rock started with no Lead-206 is wrong.
• Can correct for this using other isotopes of uranium and lead.
• Details complicated.

Potassium-Argon method:

• Potassium-40 decays into Argon-40, half-life of 1.3 billion years.
• Argon is inert (chemically inactive) gas, doesn't form molecules.
• When "free" Potassium-40 decays, Argon-40 just leaves.
• Measure Argon-40 trapped in rocks.
• It must have come from decay of Potassium-40.
• Compare amounts of Argon-40 and Potassium-40, infer age of rock.
• With sophisticated analysis, can get accuracy of 0.1%.

Several other radioactivity methods also used to date old rocks.
Consistent answers from different methods.

AGE OF THE EARTH AND THE SOLAR SYSTEM

• Rocks have wide range of ages.
• Oldest are 4.2 Gyr (4.2 billion years).
• Conclusion: Earth older than 4.2 Gyr.
• Moon rocks as old as 4.3 to 4.5 Gyr.
• Meteorites measured to be 4.5 to 4.6 Gyr.
• Accepted as age of solar system, age of Earth.
• Took 0.3 Gyr for Earth to cool enough to form first surviving solid rocks.
• Moon cooled faster, slightly older rocks.

Can estimate age of Sun from its structure and luminosity.
Completely independent method, different assumptions.
Get consistent answer, 4.5 Gyr, uncertainty of 0.1 Gyr.

DENSITY AND ITS IMPLICATIONS

• Density = mass / volume.
• Measure radius R_earth from improved version of Eratosthenes experiment.
• Volume = (4 \pi / 3) x R_earth³
• Measure mass M_earth from gravitational acceleration, g = G M_earth / R_earth²
• Divide mass by volume to get average density: 5500 kg/m³ = 5.5 g/cm³.
• Density of water: 1000 kg/m³.
• Typical density of rock: 3000 kg/m³.

• Much of Earth's interior is something denser than water or rock.
• A good candidate: iron.
• Relatively abundant in universe, and fairly dense.
• At atmospheric pressure, 7000-8000 kg/m³.

STRUCTURE OF THE EARTH

Established view of Earth's structure, from outside in:

• Crust: light rock, 0 - 100 km
• Mantle: denser rock, 100 - 2900 km
• Outer core: liquid iron (and some nickel), 2900 - 5100 km
• Inner core: solid iron (and some nickel), 5100 km - center (6400 km)
• How do we know?

• Makes reasonable sense if early Earth was very hot, molten.
• Heaviest elements sink to bottom.
• Lightest elements float to top.
• Core hotter than surface, keeps iron liquid.
• Very high pressure makes central iron solid.
• What keeps the core of the Earth hot?

SEISMOLOGY AND THE EARTH'S STRUCTURE

Earthquakes and pressure waves:

• Earthquakes produced by slippage of plates on Earth's crust.
• Create pressure waves ("sound" waves) that travel through Earth.
• Two kinds:
  • Compression waves (P waves), like sound waves in air.
  • Transverse (bending) waves (S waves), only possible in solids.
• Wave speed depends mainly on density of medium, also pressure and composition.
• Waves bent (refracted) by Earth's different sound speeds.
• Same phenomenon as bending light by glass.
• Study of earthquakes and their transmission through Earth called seismology.

Decoding the Earth's structure:

• Wait for earthquakes (small ones will do).
• Measure quake arrival times at points on Earth's surface.
• Reconstruct sound speed at every point inside the Earth.
• Infer density profile, changes in material.
• Transverse waves do not make it to other side of the Earth.
• Implies central part of Earth is liquid.
• Gives good determination of mantle-core boundary.
• Solid inner core inferred from deflection of compression waves.

EARTH'S MAGNETIC FIELD

• Earth has magnetic field, sort of like giant bar magnet.
• Magnetic fields are produced by circulating electric currents.
• Studies of magnetized rock show Earth's magnetic field flips, changing direction every million years or so.
• Understandable if field arises from currents flowing in liquid iron.
• Not understandable if entire core is solid.
• Flipping magnetic field is independent evidence for liquid core.

WHAT KEEPS THE CENTER OF THE EARTH HOT?

• Volcanos show mantle is hot.
• Liquid iron must be hotter still.
• Earth would have been hot when formed, from collisions of material, sinking of heavy stuff to center.
• But 4.5 Gyr is lots of time to cool.
• Calculations show: need an energy source to keep core as hot as it is.
• The energy source is radioactive decay, of uranium, thorium, potassium, and other elements.