Thursday, September 21
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Today we studied the effects of the tidal force on planetary satellites.
If the tidal force can tear apart a strengthless fluid object (as in the
derivation of the classical Roche limit), then it still applies some stretching
force to solid, intact bodies, such as moons. First consider the
static situation where neither the moon nor the planet are moving or rotating:
the tidal force stretches each object resulting in bulges along the line
connecting the two objects. These are called tidal bulges.
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Now allow the moon to rotate. Because the bulge is produced by the
differential force of gravity across the moon, it wants to stay aligned
with the line connecting the two bodies. But if the moon is rotating,
different parts of the physical body must go through this tidal bulge.
The result is that as the moon rotates it is constantly being stretched
and deformed. This of course takes energy, and that energy comes
out of the rotation energy of the moon and results in heat energy (frictional
dissipation of energy as the solid moon is deformed). As energy is
removed from the moon's rotational energy its rotation slows to the point
where it always keeps the same real estate pointed at the planet so it
doesn't have to do any stretching or deforming. For it to keep the
same real estate pointed at the planet as it goes around the planet, its
rotation period must equal its orbit period. This is called synchronous
rotation, and all major satellites in the solar system exhibit synchronous
rotation.
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The planet also spins. Because there is a finite amount of time for
the planet to deform in response to the tidal force from the satellite,
the tidal bulge on the planet will not be exactly lined up with the line
connecting the centers of the objects. If the planet rotates faster
than the orbit speed of the moon, then the planet's tidal bulge will be
in front of the moon. The extra mass in the tidal bulge in front
of the moon results in a gravitational acceleration in the direction of
motion of the moon which causes the moon to move into a slightly higher
orbit. Thus, if the planet's rotation period is less than the moon's
orbit period, the orbit of the moon expands. This is the case with
the Earth's moon which orbits in about 28 days compared to the Earth's
rotation period of 1 day. The Earth's tidal bulge is primarily manifested
in the oceans. As the Earth rotates through this tendency of the
Earth's oceans to want to stretch out toward the Moon, beaches experience
high and low tides.
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If, on the other hand, the moon orbits in less time than a planetary rotation
period, the planet's tidal bulge will lag behind the moon and cause a gravitational
retardation on the moon's orbit and the orbit will shrink. The same
situation holds for moons orbiting in the opposite direction of the planet's
rotation. These moons will eventually hit the surface of the planet.
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The size of the lag angle of the tidal bulge is related to the amount of
energy dissipated in the interior of the planet. We introduced a
specific dissipation function Q-1=1/(2p)DE/E.
Observations of large moons near the giant planets shows that tidal evolution
is slower for them than for the Earth's moon, for example, which is relatively
far from Earth. The giant planets have large values of Q which results
in a small lag angle q: sinq=Q-1.
Q for the giant planets is estimated to be on the order of 104
to 107, while for the terrestrial planets it is on the order
of 10-100. E is the peak elastic energy stored in the system over
a cycle, and DE is the energy dissipated per
cycle.
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We looked at the radial dependence of the tidal force. The mass of
the tidal bulge goes as the inverse cube of the separation of the two bodies.
The tidal force responsible for changing a moon's orbit goes as the inverse
sixth power of the distance between the bodies. See also your reader
for mathematical notes on tidal forces and the Roche limit.
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Some examples: Pluto and Charon are mutually synched. Each object
rotates at a rate equal to their mutual orbit rate. We considered
the mass of the tidal bulge raisedon the Earth by the Moon and the Sun
and saw that because of the strong radial dependence the Moon is more important
than the Sun.