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Geology 130F

Lecture Six


Physics

Thermodynamics

  1. The First Law of Thermodynamics: Energy is conserved. ``You can't get something for nothing''. Energy can change form---for example the energy of motion (kinetic energy) can be converted into electrical energy. When moving water spins a turbine that in turn moves a wire coil in a magnetic field, electricity is generated. Similarly, matter can be converted into heat or kinetic energy, and light (a form of electromagnetic energy) can be converted into positrons and electrons.
  2. The Second Law of Thermodynamics: the entropy in a closed system can only increase. ``You can't even break even''. Entropy is a measure of the order of a system. The conversion of kinetic energy into heat (random atomic motion) is generally accompanied by an increase in entropy. As a result, it is not possible to convert the heat back into ordered motion without some external work being done, since this would require the entropy to decrease.

Fundamental Forces

  1. The strong nuclear force is responsible for the energy generated by the sun, and is the source of the radiogenic energy that heats the interiors of large bodies.
  2. Electromagnetism is responsible for chemical interactions, such as those that lead to the rigidity of rock, and body forces, such as occur in a collision. It also makes itself felt in magnetic fields.
  3. The weak nuclear force is responsible for the radioactive decay that releases the radiogenic energy mentioned above.
  4. Gravity is responsible for the collapse of the solar nebula, for tides, and for keeping our feet on the ground.

Planetary Differentiation

During the planetary accretion phase, large amounts of gravitational and kinetic energy are converted into heat concentrated in the accreting body. Heat is random motion of atoms; large amounts of random motion mean that the chemical forces responsible for forming rocks are less effective in maintaining rigidity. As a result the material in the accreting body will be more readily deformable than it would be if it were cooler. This fluidity may allow the accreting body to differentiate, or separate into distinct layers.

Gravity vs. The Second Law

The entropy of the solar system will increase if heat leaks out of the large bodies, so that is what happens. However, if the object is large enough, this heat loss will take a geologically long time. The rate at which heat leaks out of a body increases with the temperature and surface area of the body. The amount of heat accumulated in the body increases with volume. Since planets are spherical, and the volume of a sphere increases faster than it's surface area, large bodies retain their heat for longer times.

Primary Energy Sources

  1. Gravitational
    1. Accretionary heating.
    2. Core formation
    3. Tidal
  2. Nuclear
    1. Radiogenic heating
    2. Solar

Heat Sources

A) Accretionary Heating

Newton tells us that any two masses attract each other. If, because of this mutual attraction, two bodies move toward each other, they convert their gravitational potential energy into kinetic energy. If they then collide, this kinetic energy is converted in to thermal energy. For example, if we drop a rock from the Moon onto the Earth, it will hit with a velocity of v~12km/s, representing an energy of

E=(1/2)mv2.

If all this energy is converted into heat in the rock, the temperature will be given by

E=NkT

where N is the number of atoms in the rock, and k=1.38x10-16 is Boltzman's constant. The mass of the rock is just the average mass of an atom times the number of atoms, m=\mu N, and is typically about 50-100 times the mass of a proton, or about 10-22 grams. Solving for the temperature,

T={1\over2}v2{\mu\over k}=(1.2x106)2 x10-22/10-16

or about 1,000,000K. There is more than enough energy to vaporize the rock! The total energy deposited in the planet is of order GMEarth MEarth/rEarth. The mass of the earth is MEarth=6x1027g, while it's radius is rEarth=6x108cm, so the energy is 10-8x (6x1027)2 /6x108=1039ergs. Since the Earth is about 4.5 billion years, or 1017 seconds, old, the average rate of heat loss must be about 1022erg/s. It was much higher early on.

B) Core Formation and the Heat of Differentiation

Similar calculations, taking into account the loss of energy by radiation (light) suggest that the earth was more or less completely melted early on in its history. This would allow for differentiation of the interior of the planet, since denser material would settle toward the center while less dense material would rise toward the surface. This process of differentiation would release more gravitational potential energy in the form of heat.

C) Radiogenic Heating

We can dig Uranium out of the ground, while theories of stellar evolution allow us to estimate the fractional abundance of Uranium in the solar nebula. The two measures of the abundance allow us to estimate the amount of heat released by the decay of Uranium throughout the body of the planet, about 3x1019ergs/s. This is small compared to the residual accretion heat still being lost from the Earth's interior.

D) Solar Heating

The Sun puts out 4x1033erg/s. The Earth captures a fraction (rEarth/Rorbit)2~ (6x108/1.5x1013)2, or 1024erg/s. This is about a factor of 100 higher than the heat currently coming up from the interior of the earth, but it does not penetrate very far below the surface.

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