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Geology 130F

Lecture Twelve

Tidal Evolution of Close Satellites

  1. How tidal locking works.

    Many satellites in the solar system always show the same face to their primary. For example, we always see the same side of the moon. Other examples include the Galilean satellites of Jupiter (Io, Europa, Ganymede and Callisto), as well as most of the regular satellites of the other giant planets. Tides raised in the satellites by the gravitational field of the planet is responsible for this, hence the name tidal locking. A simplified account of the process follows. Imaging that we start with a rapidly spinning satellite with its spin axis not aligned with its orbital axis

    1. The planet raises a tidal bulge on the moon. Because the moon is rapidly rotating, this bulge is carried slightly forward of the line joining the centers of the two bodies. The planet exerts a torque on this bulge. Because the bulge moves in the body of the moon as the moon rotates, friction in the moon removes energy from the spin and turns it into heat. Two changes in the moon's rotation occur:
    2. First, the spin axis is aligned with the orbital axis
    3. Second, the rate of spin decreases, usually until the tidal bulge no longer moves in the body of the moon. In other words, the moon always shows the same face to its planet.

    A body that is tidally locked to its primary is said to be in a one to one spin orbit resonance, since the spin period and the orbital period are in a 1:1 ratio. There are other possible end states of tidal evolution. For example, Mercury is in a 3:2 spin orbit resonance.

  2. How tides length the day ("spin down")

    Tidal interactions between a satellite and its primary also affect the spin of the primary. For example, the length of the day is increasing. In outline form, this works as follows:

    1. The Moon raises a tidal bulge on Earth.
    2. This bulge exerts a torque on the Moon, and
    3. the Moon exerts a torque on the bulge.
    4. Result: the Moon moves away from the Earth, and the Earth spins more slowly (the day gets longer).

  3. Evidence for spin down and the flight of the Moon.

    1. Laser ranging to the Moon shows that it is moving away from the Earth at the rate of 3 cm/year.
    2. Eclipse observations show that the average rate of increase in the lenght of day over the last 3000 years is consistent with the radar ranging results for the rate of recession of the moon.
    3. Paleontological observations (the number of days in the year, as recorded in fossil corals) show that the rate of recession has been roughly constant for the last 500,000,000 years.

    Note that the tidal interaction that affects the spin period of the primary also affects the orbital period of the moon. Since the moon is moving away from the Earth, the month is getting longer. In other cases, the orbital period of the moon may decrease. If the orbital period of the moon is shorter than the length of day, or of the moon is retrograde, like Neptune's moon Triton, the month will shorten over time.

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